Hierarchical Bayesian Models for Estimating the Extent of Plant Pest Invasions

A thesis submitted for the degree of Doctor of Philosophy in the Faculty of Science and Technology, Queensland University of Technology

By Mark Andrew Stanaway Bachelor of Applied Science (Life Science)(Hons)

Principal Supervisor: Kerrie Mengersen Associate Supervisors: Robert Reeves, Grant Hamilton, Peter Whittle

Queensland University of Technology Faculty of Science and Technology Mathematical Sciences January 2011

Keywords

Bayesian inference, hierarchical Bayesian models, plant biosecurity, surveillance, spiraling whiteﬂy, Aleurodicus dispersus, red banded mango caterpillar, Deanolis sublimbalis, quarantine, non-indigenous species, biological invasions, entomology, agriculture, early detection, market access, area freedom, pest free areas, eradication, Markov chain Monte Carlo, detectability, insect disperal, dispersal modelling.

iii Abstract

lant biosecurity requires statistical tools to interpret ﬁeld surveillance data Pin order to manage pest incursions that threaten crop production and trade. Ultimately, management decisions need to be based on the probability that an area is infested or free of a pest. Current informal approaches to delimiting pest extent rely upon expert ecological interpretation of presence / absence data over space and time. Hierarchical Bayesian models provide a cohesive statistical framework that can formally integrate the available information on both pest ecology and data.

The overarching method involves constructing an observation model for the surveillance data, conditional on the hidden extent of the pest and uncertain detection sensitivity. The extent of the pest is then modelled as a dynamic invasion process that includes uncertainty in ecological parameters. Modelling approaches to assimilate this information are explored through case studies on spiralling white- ﬂy, Aleurodicus dispersus and red banded mango caterpillar, Deanolis sublimbalis. Markov chain Monte Carlo simulation is used to estimate the probable extent of pests, given the observation and process model conditioned by surveillance data. Statistical methods, based on time-to-event models, are developed to apply hierarchical Bayesian models to early detection programs and to demonstrate area freedom from pests.

The value of early detection surveillance programs is demonstrated through an application to interpret surveillance data for exotic plant pests with uncertain spread

iv rates. The model suggests that typical early detection programs provide a moderate reduction in the probability of an area being infested but a dramatic reduction in the expected area of incursions at a given time. Estimates of spiralling whitefly extent are examined at local, district and state-wide scales. The local model estimates the rate of natural spread and the influence of host architecture, host suitability and inspector efficiency. These parameter estimates can support the development of robust surveillance programs.

Hierarchical Bayesian models for the human-mediated spread of spiralling whitefly are developed for the colonisation of discrete cells connected by a modified gravity model. By estimating dispersal parameters, the model can be used to predict the extent of the pest over time. An extended model predicts the climate restricted distribution of the pest in Queensland. These novel human-mediated movement models are well suited to demonstrating area freedom at coarse spatio-temporal scales. At finer scales, and in the presence of ecological complexity, exploratory models are developed to investigate the capacity for surveillance information to estimate the extent of red banded mango caterpillar. It is apparent that exces- sive uncertainty about observation and ecological parameters can impose limits on inference at the scales required for effective management of response programs.

The thesis contributes novel statistical approaches to estimating the extent of pests and develops applications to assist decision-making across a range of plant biosecurity surveillance activities. Hierarchical Bayesian modelling is demonstrated as both a useful analytical tool for estimating pest extent and a natural investigative paradigm for developing and focussing biosecurity programs.

v vi Contents

Keywords iii

Abstract iv

List of Tables xiv

List of Figures xxi

Statement of Original Authorship xxiii

Acknowledgements xxv

1 Introduction 1

1.1 Primary Research Aim ...... 1

1.2 Motivation ...... 2

1.3 Research Plan ...... 3

1.4 Scope of Thesis ...... 5

1.5 Outline of Thesis ...... 6

vii 2 Literature Review 9

2.1 Inference in Biosecurity ...... 9

2.2 Hierarchical Bayesian Models ...... 13

2.2.1 Bayesian approaches ...... 15

2.2.2 Pest observation models ...... 18

2.2.3 Invasion process models ...... 22

2.2.4 Computation ...... 28

2.3 Bayesian Plant Biosecurity Applications ...... 31

2.4 Summary ...... 33

3 Early Detection Surveillance 35

3.1 Introduction ...... 35

3.2 Model ...... 38

3.2.1 Overview and notation ...... 38

3.2.2 Incursion process model ...... 40

3.2.3 Observation model ...... 42

3.2.4 Parameters ...... 44

3.2.5 Inference and interpretation ...... 46

3.2.6 Computation ...... 48

3.3 Simulations ...... 49

3.3.1 Data and methods ...... 49

viii 3.3.2 Results ...... 50

3.4 Early Detection Program ...... 54

3.4.1 Surveillance data and methods ...... 54

3.4.2 Results ...... 56

3.5 Discussion ...... 61

3.6 Summary ...... 64

4 Whiteﬂy Detection and Natural Spread 67

4.1 Introduction ...... 67

4.2 Pest Information and Surveillance Program ...... 69

4.3 Models ...... 72

4.3.1 Observation model ...... 72

4.3.2 Growth rates ...... 74

4.3.3 Spread ...... 76

4.3.4 Analysis ...... 77

4.4 Results ...... 78

4.5 Discussion ...... 83

4.6 Summary ...... 87

5 Human-Mediated Dispersal Reliability Analysis 89

5.1 Introduction ...... 90

5.2 General Incursion Model ...... 92

ix 5.2.1 Overview ...... 92

5.2.2 Reliability framework for colonisation ...... 93

5.2.3 Gravity model ...... 95

5.3 Data and Observation Model ...... 99

5.3.1 Spiralling whiteﬂy data ...... 99

5.3.2 Observation model ...... 100

5.4 Simulations and Computation ...... 103

5.5 Results ...... 105

5.5.1 Likelihoods ...... 105

5.5.2 Comparison of algorithms ...... 106

5.5.3 Scalability of models ...... 106

5.6 Discussion ...... 108

5.7 Summary ...... 113

6 Predicting Spiralling Whiteﬂy Distribution 115

6.1 Introduction ...... 116

6.2 Distribution and Data ...... 118

6.2.1 Distribution of spiralling whiteﬂy ...... 118

6.2.2 Surveillance for spiralling whiteﬂy in Queensland ...... 121

6.2.3 Data ...... 122

6.3 Mechanistic Models ...... 124

x 6.3.1 Temperature stress on growth ...... 124

6.3.2 Movement within zones ...... 127

6.3.3 Movement between zones ...... 130

6.4 Hierarchical Model ...... 132

6.4.1 Observation model ...... 135

6.4.2 Incorporating the movement model ...... 138

6.4.3 Computation ...... 139

6.5 Results ...... 140

6.6 Discussion ...... 146

6.7 Summary ...... 151

7 Mango Caterpillar Invasion Ecology 153

7.1 Introduction ...... 154

7.2 Overview ...... 156

7.3 Data and Observation Model ...... 159

7.4 Local Ecological Process Modelling ...... 164

7.4.1 Mango fruit development ...... 164

7.4.2 RBMC life history model ...... 168

7.4.3 Local population model simulations ...... 177

7.4.4 Host availability and Allee eﬀects ...... 180

7.5 Dispersal ...... 185

xi 7.5.1 Human-mediated dispersal ...... 186

7.5.2 Long distance wind assisted ﬂight ...... 188

7.5.3 Directed short distance ﬂight ...... 189

7.5.4 Dispersal model ...... 191

7.6 Discussion ...... 193

7.7 Summary ...... 196

8 Discussion 199

8.1 Conclusions and Further Work ...... 199

8.2 Recommendations to Biosecurity ...... 205

Appendices 211

A Early Detection Model Code 211

B Spiralling Whiteﬂy Natural Spread Code 214

C Gravity Model MCMC Algorithms 217

C.1 Simulated Block Algorithm ...... 218

C.2 Individual Proposal Algorithm ...... 219

C.3 MCMC for Chapter 5 and 6 Applications ...... 222

Bibliography 222

xii List of Tables

3.1 Description of informative priors to describe uncertainty in the observation and ecological characteristics of target pests...... 47

3.2 Prior and posterior means and standard deviations for selected parameters and scenarios. Scenarios B and C are used to demonstrate spatial eﬀects only and are omitted. Scenario A) N = 20, D) N = 40,

E) N = 10, F) N = 20 and xN = 1...... 50

3.3 Summary of prior and posterior estimates of latent variables and parameters for early detection surveillance of bananas in the Cairns district...... 56

3.4 Sensitivity of probability of colonisation and estimated area of infestation if colonised. Scenario 1 is the original model. Prior values for other scenarios are shown only where they diﬀer from scenario 1. . 60

4.1 Temperature dependent survival and fecundity rates for spiralling whiteﬂy from Wen et al. (1994)...... 75

4.2 Posterior estimates of global parameters for the incursion...... 78

xiii 6.1 Details of the zones, arranged roughly north to south. Date of ﬁrst detection is the date of ﬁrst collection from the zone, followed by whether it was a public report or a structured survey. Missing values indicate the pest is not known to occur. The total number of observations on all hosts and visits is given for the study period. . . 125

6.2 Mean of potential propagules introduced from source to target zones per annum if source cells were completely covered. Note that these estimates are based on connectivity only and therefore include pressure from zones that are not predicted to contain any pests. . . . . 145

7.1 Proportion of fruit induced over the season, Aw, where month subscripts refer to the 1st or 2nd half of the month...... 167

7.2 Probability of fruit advancing to the next fruiting class, Bc, for infested and uninfested fruit and the expected number entering the class...... 168

7.3 Lifestage length (days) and fecundity reported in the literature for red banded mango caterpillar in Papua New Guinea (PNG), the Philippines (PHL), Andhra Pradesh, India (APR), West Bengal, In- dia (WBN), Bangladesh (BGD)...... 170

7.4 Probability of a pre-pupa emerging from and a larva entering a dormant state over the season...... 172

7.5 Probability of ovipositing on fruit of each size class after testing. . . 174

7.6 Local simulation model parameters and variables...... 181

7.7 Priors used for the analysis of simulated population data to estimate Allee eﬀect parameters...... 183

7.8 Mean posterior estimates (SD) for Allee eﬀects model parameters. . 185

xiv List of Figures

2.1 The role and construction of hierarchical Bayesian models for biosecurity...... 14

3.1 Prior distributions (dashed) and posterior distributions (solid) of selected parameters from scenario A. a) exposure rate, λ b) velocity of spread, υ c) time of colonisation, φ d) radius of incursion at 10 years

if colonisation had occurred, ρl...... 52

3.2 Posterior distribution of colonisation point, χ (grey), given that colonisation occurs before l, after ten years of observations with 20 plants examined at each site (black). a-e) Absence outcomes at all sites. f) Absence outcomes at ﬁrst 19 sites with a detection at l. The ﬁnal

observation site for a,d-f has sN =(13.9,32.4)...... 53

3.3 Empirical cumulative distribution function for the prior (grey) and posterior (black) distribution of area of infestation over the domain of 187 km2...... 57

xv 3.4 Prior (a) and posterior (b,c) distribution of sub-district colonisation events, χ, where t > φ, for banana pests targeted by an early detection surveillance program in Cairns, Queensland. (a,b) at times t = l = 4.9 years and (c) at t = 6.0 years with no further surveillance. Contours display the prior or posterior point probabilities of infestation, Pr(Z(, t) = 1), resulting from colonisation and subsequent spread...... 59

3.5 Posterior point probability of infestation for the central part of the domain for sensitivity analysis scenarios. For comparison, the posterior for scenario 1 is displayed (grey dashed). Note diﬀerent contour interval on 2...... 61

4.1 Spiralling whiteﬂy surveillance data for periods following the detection on 18 March 1998...... 71

4.2 Estimates of global parameters (red dashed - prior, black solid - posterior)...... 78

4.3 Detectability, ηh, as the eﬀective proportion of hosts searched (red dashed - prior, black solid - posterior)...... 79

4.4 Posterior estimates of suitability, γh, as a proportional rate of increase on host plants...... 80

4.5 Estimated probability of detection on hosts at days after colonisation given the joint posterior probability of host detectability and suitability, inspector eﬀectiveness of 0.3 and intrinsic rate of increase of 0.026 (median - solid lines, 0.05 and 0.95 quantiles - grey dashed). 81

4.6 Estimated relative eﬀectiveness of selected inspectors θi. Only estimates for inspectors who surveyed more than 20 sites are shown. . . 82

xvi 4.7 Random eﬀects error in site colonisation time s. a) Median b) Stan- dard deviation...... 84

5.1 Combined Weibull (solid) and exponential (dashed) model of the

distribution of time to colonisation from a unit cell j where λij = 0.7

and υ = (.08, 0.2, 0.7, 2). Dotted line displays Ej, the period of colonisation at which source cell j is completely covered...... 98

5.2 a.) Study area with number of residential properties used to generate the mass values for the gravity model. The central Cairns cells are boxed. b.) Year that spiralling whiteﬂy was ﬁrst detected in cells. . 100

5.3 Likelihood function for detectability yim|Qim,Vim given a Beta(2,2) distribution of observer detectability and population-based detection probabilities for potential site colonisation times across the cell. From

left to right, the periods of cell colonisation m − φi are 30, 90, 365

with yim=10 and Vim=50...... 102

5.4 a) Mean likelihoods of valid draws of 100 000 simulated realisations of the spread process given the observation data within the central cells. b) Probability of the draw being valid for a range of values of ψ. Note that the ﬁrst mean likelihood estimate is based upon only two valid draws in 100 000...... 105

5.5 Comparison of the posterior estimates from the two algorithms, blue solid - simulated block φ, red dashed - individual proposal. Months with absence data are indicated by grey dashed lines while months with presence data are given by black dashed lines. Cell 3 was set as the initialising cell with a known time of colonisation...... 107

5.6 Trace and posterior density for estimates of ψ for central cells (a, b) and the full set of district cells (c,d). Trace plots are thinned to show every 100th iteration...... 107

xvii 5.7 Mean posterior estimate of colonisation time (black circle) and estimated 90% range of colonisation times from the individual proposal algorithm (lines). Cells are arranged roughly north to south. Posi- tive observation data are displayed as red circles and absence records are displayed as green circles. The ﬁrst detection within the cell is indicated by a red triangle...... 109

5.8 Posterior density estimates of gravity model parameters for central and district models. Central 26 cells - a) joint distribution of the distance coeﬃcient, ω, and the scaling factor, ψ. b) marginal distribution of ψ, c) marginal distribution of ω. District 53 cells - d) joint distribution of the distance coeﬃcient, ω, and the scaling factor, ψ. e) marginal distribution of ψ, f) marginal distribution of ω...... 110

6.1 Countries where spiralling whiteﬂy has been detected. Adminis- trative regions within some countries are shown when documented. Source CABI (2006); Monteiro et al. (2005). Personal communications (J.H. Martin, 2008, B.M. Waterhouse, 2008)...... 119

6.2 Spiralling whiteﬂy records from Florida State Collection of Arthro- pods in relation to January average minimum monthly temperature records...... 121

6.3 a.) Presence and absence records of spiralling whiteﬂy in Queensland in relation to minimum July temperatures b.) Zones used for the climate model...... 123

6.4 Half day measure of cold temperature stress accumulation for τ=10◦C as the area below a threshold of a cos curve between the maximum and minimum temperature...... 127

6.5 Daily temperature stresses, Si, given a temperature threshold, τ = 10◦C for centroids of each zone from January 1997 to June 2010. . . 128

xviii 6.6 Overview of hierarchical model components. The deterministic gravity model is deﬁned by the covariate data for the zones, X, and the scaling factor ψ. The joint colonisation times, φ, are deﬁned by

the gravity model and the intrinsic growth rate, βi, that moderates

the rate of internal spread through each zone. βi is determined by

the stress accumulation rate, δ, and the temperature data, Si. The growth model provides the detection probabilities for the observation data, Y , collected at times T ...... 134

6.7 Likelihood function for detectability yim|Qim,Vim given a Beta(0.5,0.5) distribution of observer detectability and population-based detection probabilities for potential site colonisation times across the zone. From left to right, the periods of zone colonisation are a) 30 days, b)

365 days and c) 730 days, with yim=10 and Vim=30...... 137

6.8 Estimates of the posterior densities of the stress accumulation parameter, δ, and the scaling factor for the gravity model, ψ...... 140

6.9 MCMC chains for the scaling factor, ψ, thinned to show every 100th draw...... 141

6.10 Posterior density of estimates for the time of introduction into northern zones for three MCMC chains. Black dashed line indicates the time of ﬁrst detection where applicable. Black +, positive detection for the month. Grey ×, absence record for the month...... 142

6.11 Posterior density of estimates for the time of introduction into southern zones for three MCMC chains. Black dashed line indicates the time of ﬁrst detection where applicable. Black +, positive detection for the month. Grey ×, absence record for the month...... 143

xix 6.12 Mean posterior estimates and 90 % credible intervals for intrinsic

rates of increase, βi and zone introduction times, φi. Estimates of βi less than zero suggest that colonisation will not occur...... 144

6.13 Progressive posterior estimates of the time of introduction into zones for data collected up to a particular year. Solid lines - posterior means, grey dashed - 90 % credible intervals. Red dashed indicates the time of ﬁrst detection for the twelve known colonised zones. . . 147

6.14 Progressive posterior estimates of the stress accumulation parameter δ, and the scaling factor, ψ, for data collected up to a particular year. Solid lines - posterior means, grey dashed - 90 % credible intervals. 148

7.1 Furthest extent of red banded mango caterpillar detections and distribution of mango trees within the Northern Peninsula Area. Extent contours are hand-drawn to delimit mango trees with detections up to that year. The shape of the contours is immaterial except in relation to the host distribution...... 157

7.2 Nine independent realisations from a model of temporal variability in

fruiting status Fc,w, with smallest fruit class at the back and mature fruit at the front. Seasonal time is represented as bi-monthly classes. 169

7.3 Lifestage transitions between time steps and influencing factors. The numbers in each lifestage are for eggs and the first two larval instars (N E12), third to fifth instars (N L35), dormant pre-pupae (N DPA), and pupae and adults (N PPA). Influencing factors, given in the numbered legend, are indicated on the transition lines between stages...... 171

7.4 Realisations of the number of each RBMC stage within a cell over a season from a model initialised in year 1 with 100 dormant pre-pupae on 10 trees. E12 - Eggs and early instars, L35 - later instars, DPA - dormant prepupae, PPA - prepupae / pupae / adults...... 178

xx 7.5 Realisations of the fruit infestation status of a cell over a season. Blue - uninfested, red - infested. Green - proportion of fruit infested on right hand scale...... 179

7.6 Rate of increase over a season for simulated local populations on trees

DPA starting with Ny dormant pre-pupae (jittered). Values less than zero indicate population decrease. Simulations resulting in extinctions (i.e. -∞) not shown. Log of the mean rate of increase (black) and model ﬁtted values (green)...... 184

7.7 First detections by Australian Quarantine and Inspection Service on Torres Strait islands. Most populated islands are named (Gabba Island is uninhabited)...... 187

7.8 Distribution of downwind displacement in a one hour period, based on three hourly wind data provided by Bureau of Meteorology for Horn Island from 2000 to 2007...... 190

7.9 Distribution of downwind displacement in a one hour period, based on three hourly wind data provided by Bureau of Meteorology for Horn Island from August to November. (Note the directional resolution format changed in 2003)...... 190

7.10 Example odour plume model for the period 2001 to 2007 for wind data from August to November...... 191

xxi

Statement of Original Authorship

The work contained in this thesis has not been previously submitted to meet requirements for an award at this or any other higher education institution. To the best of my knowledge and belief, the thesis contains no material previously published or written by another person except where due reference is made.

xxiii xxiv Acknowledgements

Thanks go to: Supervisors Kerrie Mengersen, Rob Reeves, Grant Hamilton and Peter Whittle for giving direction and guidance.

The Cooperative Research Centre for National Plant Biosecurity for providing support, in particular Kirsty Bayliss for her patience.

Peter Whittle and Chris Adriaansen for advocating the beneﬁts of this research and encouraging my proposal to undertake the study while working with Biosecurity Queensland within the Department of Employment, Economic Development and Innovation. Shane Campbell, Gabrielle Vivian-Smith and Jim Thompson from Biosecurity Queensland Science for continuing with this support and Dane Panetta and Joe Scanlan for their ongoing interest in modelling biosecurity problems.

The many Biosecurity Queensland officers who collected the data and provided the context. In particular, Jane Royer, Aurea King, Rebecca Sapuppo, Ceri Pearce, Mick Berridge, Sharyn Foulis and Mark Trinca for their exchange of knowledge of pests, plants, places and people. Phil Gleeson who coordinated the early spiralling whitefly data collection and managed the data during the first year of the response. Jackson Sailor and James Bond from the Australian Quarantine and Inspection Service for their knowledge of the Northern Peninsula Area and for facilitating work during the red banded mango caterpillar surveys.

xxv Greg Hodges (Florida State Collection of Arthropods), Jon Martin (National His- tory Museum, London, U.K.), Barbara Waterhouse (Northern Australia Quarantine Strategy, AQIS), Estrella Hernandez-Suarez (Departmento De Proteccin Vegetal, Instituto Canario de Investigaciones Agrarias) and Bernarr Kumashiro (Hawaii De- partment of Agriculture) for providing information on spiralling whiteﬂy collections and distribution in Florida, eastern Africa, eastern Indonesia, Canary Islands and Hawaii respectively.

Myron Zalucki, Bob Sutherst and Gunter Maywald whose enthusiasm about bugs (and ﬂies and moths), ecology and modelling laid the foundations for this thesis many years ago.

Finally, my thanks go to Rebecca and Riley, for support that never wavered and your blind belief that carried me through those overwhelming times.

xxvi t borders, as at death and in dreams, no amount of prior A planning will necessarily avail. The law of boundaries applies. In the nature of things, control is not in the hands of the traveller.

Janette Turner Hospital - Borderline

xxvii xxviii Chapter 1

Introduction

1.1 Primary Research Aim

The primary objective of this research is to develop hierarchical Bayesian models to estimate the extent of invading plant pests for more effective biosecurity. Estima- tion of pest extent from field surveillance is critical to the management of invasive species. Hierarchical Bayesian models provide a useful analytical framework to assimilate information from field surveillance and ecological knowledge. The assimilation of information to estimate pest extent requires component models for the observation of pests and for the ecological processes that drive the evolution of an invasion over space and time. In order to meet this objective, statistical methods need to be developed to produce biosecurity applications that meet pest management requirements. Furthermore, for hierarchical Bayesian applications to be adopted by biosecurity agencies, the relationship between the quality of the surveillance data and the decision support that is required must be explored to discern their potential.

To estimate the probability of pest presence or absence, observation models must accommodate the uncertainty and variability that exists between the data and the

1 2 CHAPTER 1. INTRODUCTION pest population. Statistical models for the observation process must be developed to translate imperfect surveillance data into information on pest status. To eﬀectively estimate the hidden spatial extent of incursions over time, ecological process models need to successfully link the observational evidence for pest absence to an unfolding invasion process. These dynamic invasion process models require expert opinion to support inference on pest extent within a spatio-temporal domain. The development of mechanistic growth and dispersal models to describe the change in extent of an invading pest over time must acknowledge considerable uncertainty about a complex ecological process. Within the hierarchical Bayesian modelling framework, this thesis explores issues of system complexity and computational challenges, and relates them to the biosecurity management expectations of statistical modelling.

The contribution of this thesis is to provide a framework to estimate the predictive parameters and latent spatio-temporal invasion status that are fundamental to invasive plant pest management. As such, it draws together research elements from statistics and invasion ecology for application to plant biosecurity priorities. Recommendations are made to the biosecurity community about the strengths and limitations of hierarchical Bayesian models as tactical tools for estimating the extent of invading plant pests.

1.2 Motivation

Australia has recently been exposed to a number of emergency plant pest (EPP) incursions leading to response campaigns for eradication or control costing tens of millions of dollars (Maynard et al., 2004). Successful responses to EPPs rely upon quality information to determine the extent of the incursion. These delimiting activities are predominantly based upon the results of field surveillance. The operational guide for EPP management in Australia, PLANTPLAN, prescribes that a surveillance strategy to delimit the extent of a pest must “provide biometric methods to 1.3. RESEARCH PLAN 3 specify the different confidence limits for targeted and general surveillance” (Plant Health Australia, 2010b). Despite the investment in plant biosecurity surveillance and response, the development of statistical tools to specifically address these ques- tions is still in its infancy (Leung et al., 2010).

Plant biosecurity encompasses a range of surveillance activities designed to min- imise the impact of emergency plant pests. Firstly, early detection surveillance is conducted at sites considered to have a high probability of exotic pests establishing. Its aim is to detect incursions before they are too large to eradicate. Secondly, once an EPP is detected, delimiting surveillance is used to identify areas that require management. Finally, there is a need to demonstrate pest free areas to conclude eradication activities and/or to gain trade access to markets that may have imposed restrictions based on the presence of the pest. Each of these activities attempts to spatially delineate an infested source area from pest free target areas that are connected by a recognised movement pathway.

The status quo for EPP management is for scientists and regulators to use informal ecological process models as working hypotheses for decision making. Decisions in the early part of a response must be made in the absence of strong data as well as in the face of uncertainty surrounding the ecology of the invader (Epanchin-Niell and Hastings, 2010). Bayesian analysis provides an appealing statistical methodology to assimilate ecological and observational information into estimates of pest extent. This thesis examines the potential for hierarchical Bayesian models to formalise the complex relationship between data and prior ecological knowledge on invasive species to provide the foundation for more robust EPP surveillance systems.

1.3 Research Plan

Hierarchical Bayesian models could assist biosecurity across a range of applications. In particular, they could be used in: 4 CHAPTER 1. INTRODUCTION

• early detection surveillance to assist surveillance planning

• area freedom surveillance to estimate the probability that a pest is present

• response surveillance to determine the extent of a pest control area.

The ability of hierarchical Bayesian models to estimate the extent of an invasion depends upon the surveillance data available, the spatio-temporal scale at which inference is required and how well the complexity of the ecological system is captured within the model structure. The modelling approach adopted seeks to estimate the pest colonisation status over space and time. Models are developed and assessed for several ecological and surveillance scenarios that are applicable to EPP management.

Process models to capture the ecological complexity of invasions are developed for:

• constant spread rates through a small scale continuous landscape

• human-mediated modes of spread through locally uniform but fragmented habitats

• spread through fragmented landscapes with dynamic host relationships.

Observation models that support inference include:

• growth dependent detectability

• variation in detectability from diﬀerent observers

• complex detectability relationships with hosts.

The tasks involved in developing hierarchical Bayesian models are to identify the components of observational and ecological uncertainty for biosecurity scenarios and to specify models that can incorporate the information into estimates of extent. Three themes arise out of the modelling process: 1.4. SCOPE OF THESIS 5

• the quality of information available to construct models that provide useful inference

• the ecological complexity of the invasion systems that are to be modelled

• the computational challenges to implementing hierarchical Bayesian models that adequately estimate extent.

The research is used to critically examine the role that hierarchical Bayesian models can play in providing a coherent inferential framework for a range of biosecurity applications.

1.4 Scope of Thesis

Hierarchical Bayesian approaches to model biological invasions have been developed elsewhere to predict historical and future spread of invasions at continental and national scales (Cook et al., 2007a; Hooten and Wikle, 2008). Here, models for estimating extent are investigated at more local pest management scales. The analyses presented explore modelling approaches that focus on boundary estimation in a biosecurity setting. Considerable detail is provided in the construction of the models and the presentation of results to demonstrate the operational use of modelling in the context of management decisions. Further detail on the process of constructing these types of models can be obtained from Craigmile et al. (2009).

Risk-based management of biological invasions requires both the probability and consequences of damaging events to be estimated (Hennessey, 2004). Biosecurity consequences can be diﬃcult to quantify as they encompass economic, environmental and political concerns (Cannon, 2009; Hueston, 2003). Good reviews of bioeconomic aspects of invasion ecology and management are provided by Epanchin-Niell and Hastings (2010) and Sharov (2004), while Carrasco et al. (2010d) demonstrate some recent advances. The models developed here are discussed in the context of 6 CHAPTER 1. INTRODUCTION a risk-based decision making framework, but the consequences are not quantiﬁed and analysed in detail. Instead, the statistical models estimate the probability of extent at a particular time so that this information is available to external decision making processes.

The models presented investigate the relative importance of the information available from both survey data and ecological knowledge. Models based on case studies are developed to extract parameter and extent estimates, while progressively identifying sources of ecological complexity that challenge the modelling approaches. This complexity creates computational and inferential challenges that need to be overcome to develop fully operational applications. While novel computational approaches may oﬀer solutions to these diﬃculties, more advanced algorithms are not implemented within the scope of this thesis. Rather, the contribution of this thesis is to demonstrate the structural modelling approaches that can be used to assimilate the diverse sources of information typically available to biosecurity. Its aim is to prepare a foundation for further rigorous quantitative analysis of surveillance information.

1.5 Outline of Thesis

In Chapter 2, the hierarchical Bayesian approach to understanding biological invasions is presented through a review of the literature. The biosecurity decision making context is presented, along with the data that is typically collected by the biosecurity agencies that manage risks. Literature on the complexity of invasion processes is reviewed in terms of the observation and ecological process models that can formalise the information for hierarchical Bayesian models. The theme of information assimilation for operational biosecurity applications is developed to build the foundation for future statistical modelling.

In Chapter 3, a hierarchical Bayesian approach to an early detection surveillance 1.5. OUTLINE OF THESIS 7 program demonstrates how surveillance absence data changes the state of knowledge about the extent of potential pests. Using an ecologically simpliﬁed dispersal model, the observation and process components of Bayesian inference in a biosecurity setting are introduced. This chapter provides a base model to estimate colonisation time over space, from which the probable hidden pest extent can be estimated. The model maps areas with a high probability of infestation so that surveillance can be better targeted. This work has been accepted in Environmental and Ecological Statistics,

Stanaway, M.A., Mengersen, K.L. and Reeves, R. (In press) Hierarchical Bayesian modelling of early detection surveillance for plant pest invasions. Environmental and Ecological Statistics. DOI: 10.1007/s10651-010-0152-x.

In Chapter 4, a more complex observation model for an invading pest, spiralling whitefly, is developed. The contribution of host/pest relationships and inspector efficiency to interpreting surveillance data are estimated. The process of model construction and incorporation of prior information about pest ecology is detailed, to arrive at estimates of natural spread rates at a local scale. This chapter dis- cusses the role that hierarchical Bayesian models can play in improving surveillance efficiency by learning about observation parameters.

In Chapter 5, a novel hierarchical Bayesian framework to estimate parameters for multiple modes of spread is proposed using a reliability analysis of pathways. Moti- vated by a district scale spiralling whitefly application, the model estimates human- mediated pest movements. The reliability framework is developed in generalised form before implementing a modified gravity model to describe movements between a patchwork of towns. Two computational approaches are compared to estimate colonisation times and spread parameters. The first computational approach was proposed by Robert Reeves. Issues of scalability of inference on parameters and extent are addressed and recommendations made on extending the model for multiple dispersal modes. 8 CHAPTER 1. INTRODUCTION

In Chapter 6, spiralling whitefly’s ultimate distribution in Queensland is predicted by extending the model for human-mediated dispersal to a landscape with climatic stresses. A mechanistic temperature stress model provides the support for the hierarchical model. Spatially, the model is broken down into within zone and between zone movements using a gravity model to define the connections between zones. The model estimates the gravity model parameter, temperature stress parameter and the colonisation times of the zones. A progressive analysis of the data available at different times of the invasion demonstrates how information about critical management parameters is assimilated as more data become available.

In Chapter 7, the role of statistical modelling of incursions is examined in relation to surveillance for pests with complex ecology and a high degree of uncertainty. The ecology of an invading pest, red banded mango caterpillar, is reviewed and formalised in a stochastic simulation model to demonstrate the behaviour of the pest in relation to a seasonally changing host. Assumptions behind observation models are examined. The process of investigating ecological knowledge, such as Allee eﬀects and dispersal behaviour, for complex dynamic systems is examined in terms of the inferential goals of hierarchical Bayesian models for plant pest invasions.

In Chapter 8, the contributions of the thesis to hierarchical Bayesian modelling, invasion ecology and biosecurity are discussed. The beneﬁts of hierarchical Bayesian approaches to surveillance analysis are explored, along with some of the limitations imposed by data collection, computation and ecological complexity. Priority areas for future research are highlighted and recommendations are made to biosecurity agencies on the adoption of these modelling approaches. Chapter 2

Literature Review

This chapter introduces Bayesian modelling to the broader plant biosecurity community and reviews the literature on hierarchical Bayesian models in the context of invasive plant pest applications. The role of estimating pest extent in biosecurity is explored to provide the focus for the thesis. Hierarchical Bayesian model components for surveillance and invasion ecology are reviewed to make recommendations for developing spatio-temporal plant biosecurity applications.

2.1 Inference in Biosecurity

Plant biosecurity is concerned with managing the consequences of invading arthropod and disease pests (Waage and Mumford, 2008). The importance of maintaining agricultural systems free of exotic pests through quarantine has long been recognised by governments (Nairn et al., 1996). However, it was the liberalisation of world trade in the mid 1990s that led to a call for a more scientiﬁc approach to managing the risks of moving pests (Andersen et al., 2004; McRae and Wilson, 2002). For plant pests, the International Plant Protection Convention provides broad guidance on risk analysis to negotiate quarantine related trade restrictions

9 10 CHAPTER 2. LITERATURE REVIEW

(Baker et al., 2005). As risk mitigation is a bioeconomic management issue, the strategic decision making process has beneﬁted from modelling both the probabilities and consequences of pest invasions (Carrasco et al., 2010a,d; Cook, 2005; Epanchin-Niell and Hastings, 2010; Myers et al., 1998; Waage and Mumford, 2008). However, from a tactical point of view, there are few statistical tools for post-border management of incursions at an operational level. Critical to the management of pests at the operational level is the ability to infer the likely extent of invading pests over time.

Post-border surveillance activities that feature in plant biosecurity risk management include early detection, area freedom and response surveillance (McMaugh, 2005). Early detection surveillance aims to detect pests in an area before they become too widespread to eradicate (Hulme, 2006). These programs target surveillance at areas with a predetermined high probability of a pest being present (Hadorn and Stark, 2008; Stark et al., 2006; Wotton and Hewitt, 2004). Managers are interested in how to best deploy early detection surveillance over space and time, while balancing the cost of surveillance against the expected beneﬁt of timely eradication (Myers et al., 1998; Prattley et al., 2007).

Once an exotic pest is detected, the major threat facing producers is the suspension of access to international and domestic markets until the extent of the incursion can be demonstrated. Area freedom surveillance aims to provide suﬃcient evidence to satisfy the importing markets that the probability of moving the pest through trade from particular areas is low (Aluja and Mangan, 2008; Plant Health Australia, 2010b). Guidelines for establishing areas of low pest prevalence have recognised that area freedom is not a necessary requirement for market access negotiations (IPPC, 2008; Lloyd et al., 2010). However, they have rarely been implemented due to concerns over ecological (and operational) uncertainty within quarantine systems (Aluja and Mangan, 2008).

Delimiting extent is not only necessary for maintaining trade but is also needed 2.1. INFERENCE IN BIOSECURITY 11 for managing eradication or long term containment programs (Cacho et al., 2010; Carrasco et al., 2010b). Ongoing surveillance provides information about the extent of pests over time so that movement restrictions and control measures can be targeted most eﬀectively. Each of these applications require the spatial extent of the pest to be reliably estimated over time (Cacho et al., 2010). While inference on the probability of pest extent provides the foundation for decision making, the spatial statistics for analysing the dynamic extent of invaders are not available to the agencies that manage incursions. However, recent analytical approaches have identiﬁed this as an emerging need (Leung et al., 2010)

The spatial design of any biosecurity investigation needs to consider the units used for decision making, data collection, and ecological modelling (Graham et al., 2004). One approach to spatially deﬁning extent is to construct a convex hull, which is the smallest boundary that circumscribes every individual organism at a particular time. Convex hulls can be used to bound species distributions in continuous space, but for invasions, where non-contiguous satellite populations are common, some meaningful ecological or management resolution is needed to deﬁne boundaries (Burgman and Fox, 2003). More commonly, species distribution models seek to assign a value for presence or absence to discrete cells. These cells may be arranged on a continuous regular grid (Argaez et al., 2005; Royle et al., 2007) or may consist of an irregular patchwork (Gumpertz et al., 2000).

On a local scale, the eﬀective extent of an invading plant pest is restricted to those host plants that are capable of sustaining the pest throughout its lifecycle. Other environmental constraints, such as weather conditions, operate at broader scales. As host landscapes are all fragmented at some scale (With, 2002), it may be necessary to break the spatial domain down into discontinuous habitat patches for a particular analysis (Leung et al., 2004; Moilanen, 2004). Statistical tools to estimate the extent of an incursion need to address the choice of spatial scale on two fronts. Firstly, models of the invasion ecology must be at a resolution that can adequately represent the dynamics of spread over time. Secondly, the model 12 CHAPTER 2. LITERATURE REVIEW outputs need to be at a spatial resolution that is useful for making management decisions. The choice of a geographic model is therefore integral to the modelling process and the parameterisation of these models.

Surveillance programs gather information about the presence or absence of pests of interest over space and time. As the location of each organism comprising the invasion is not known, the process of delimiting the extent becomes one of estimating the hidden or latent extent at a particular time (Clark, 2005). Pests that may be the target of a plant biosecurity surveillance program include arthropods (insects and mites) and plant pathogens (viruses, bacteria, fungi, nematodes, etc.) (Plant Health Australia, 2010a). Typically, visual inspection of host plants provides the data which are used to demonstrate that areas are pest free. However these data are far from complete. The sites or areas to which observations of presence or absence are attributed may only be partially examined. Even at a ﬁne scale, such as on a single plant, pests may be overlooked if the symptoms are not apparent to the observer (Bulman et al., 1999; Fitzpatrick et al., 2009; Gambley et al., 2009). In order to infer the probability of pest absence for an area, the observation process must be modelled to accommodate potential false absence records within the area of interest (Kery, 2002; Meats and Clift, 2005; Tyre et al., 2003).

If a pest is present in an area, false absence records are, to a large extent, dependent on the density of the pest population in that area (Cacho et al., 2010; Kery et al., 2006; Royle and Dorazio, 2006). Observation models alone can only be used to provide evidence against presence at a particular population intensity. It is this loss of power for observations to detect pests at low levels that challenges the delimiting of pest extent (Delaney and Leung, 2010). In order to infer the pest absence in an area, additional information about a pest’s likely intensity must be introduced into the analysis. This information about intensity derives from the particular population and spread characteristics of a pest and is statistically expressed in the intrinsic spatial and temporal correlation in pest populations (Wintle and 2.2. HIERARCHICAL BAYESIAN MODELS 13

Bardos, 2006). Dynamical models for the invasion process can mathematically specify spread mechanisms (and parameter uncertainty) to allow structured ecological information to be incorporated into the statistical analysis (Gibson et al., 2006; Hooten et al., 2007).

Ultimately, the managers of invading pests seek to map the probable spatial extent of a pest at the current time (or at some time in the future) based on all of the information available. These maps can deﬁne containment lines for pest control (Plant Health Australia, 2010b), help negotiate access to markets for produce from areas considered free of pests (Jorgensen et al., 2004; Lloyd et al., 2010; Martin et al., 2007) and be used to deploy surveillance resources to maximise the information required to make decisions (Barrett et al., 2009; Davidovitch et al., 2009; Prattley et al., 2007). Hierarchical Bayesian models provide a solid statistical framework for assimilating surveillance information and knowledge about the ecology of a pest over space and time (Clark, 2005; Ellison, 2004; Fabre et al., 2010; Wikle and Berliner, 2007). A hierarchical Bayesian framework for analysing invasion data involves the construction of component models for observations and invasion processes (Fig- ure 2.1). In the following section, modelling approaches to organise and analyse surveillance and ecological information for inferring invasion extent are examined.

2.2 Hierarchical Bayesian Models

Hierarchical Bayesian modelling allows components of the invasion system to be constructed individually, before being assembled as conditional probability statements in an analytical environment. In Section 2.2.1, the Bayesian paradigm is introduced with biosecurity examples and a general framework for hierarchical modelling is explored. Section 2.2.2 reviews potential models for the observation process. In Section 2.2.3, the literature on invasion process modelling is reviewed under the topics of colonisation and dispersal. The focus of this invasion modelling section is 14 CHAPTER 2. LITERATURE REVIEW

Incursion Management

Estimate of Extent

Hierarchical Bayesian Model

Detectability Observation Incursion Model Process Ecology Model Surveillance Data

Figure 2.1: The role and construction of hierarchical Bayesian models for biosecurity. 2.2. HIERARCHICAL BAYESIAN MODELS 15 on model structures that may be applied to biosecurity applications, rather than a comprehensive review of this vast area of research. Finally, an introduction to the computation of Bayesian statistical models is provided in Section 2.2.4.

2.2.1 Bayesian approaches

Bayesian and frequentist approaches differ in their fundamental approaches to inference (Clark, 2005). Accessible introductions to these differences and the practical implementation of Bayesian approaches are available for epidemiology (Basanez et al., 2004) and ecology (Clark, 2005; Ellison, 2004). The Bayesian approach is introduced with a motivating example in a biosecurity setting. In general terms, biosecurity applications are interested in estimating some latent variable, θ, that represents pest presence or absence given some observation data, y. Frequentist approaches frame this question of pest presence at a site as a test of the probability of collecting the data given a hypothesis that the pest is present at a particular density Pr(y|θ = x). If we consider a visit to a site containing N trees of which n are examined, then a binomial model, for example, could offer a simple model for the hypothesis that the number of trees infested is, θ = x, (e.g. European and Mediterranean Plant Protection Organization, 2006; Venette et al., 2002). If the pest is detected, (y = 1), then the site is infested and requires some management intervention. If the pest is not detected then the quantity of interest is,

P r(y = 0|θ = x, n) = (1 − x/N)n (2.1)

For a given x, we can reject the hypothesis if the probability is below some threshold value (e.g. 0.05). This is akin to the design prevalence approach commonly used in veterinary epidemiology (Cannon, 2002; Martin et al., 2007). To implement this, it is common to choose some arbitrarily low value for the proportion of infested units e.g. (x = 0.01N). However, the hypothesis that we wish to reject to demonstrate 16 CHAPTER 2. LITERATURE REVIEW area freedom is the probability that the pest is present at the site, x = {1, 2,...,N} given that absence was recorded.

Bayesian statistical approaches are better formulated to estimate the latent pest status, θ, given the data. Using basic probability to reorganise the joint probability of the data and the parameter, Bayes’ rule can be used to estimate the posterior probability of the parameter,

P r(y|θ)P r(θ) P r(θ|y) = . (2.2) P r(y)

The ﬁrst term in the numerator of Equation 2.2 is the likelihood function which, for the example above, is given by Equation 2.1. The next term in the numerator is the prior probability, which is a statement of the probability that the parameter has a particular value before any observations are made, that is P r(θ = {0, 1, 2,...,N}). The denominator is a normalising constant to ensure that all possible posterior probabilities sum to 1 (Gelman et al., 2004). An alternative statement of Bayes’ rule that ignores the normalising constant is,

P r(θ|y) ∝ P r(y|θ)P r(θ). (2.3)

The Bayesian approach to estimating the probability of pest absence oﬀers an obvi- ous advantage over the untestable hypothesis of the frequentist approach by treating the parameter of interest as a random variable rather than a set value (Ellison, 2004). However, it does require additional information to be brought into the model via the prior distribution of the parameter. This prior information may be vague (e.g. the number of infested trees is equally likely to be 0, 1,...,N) or the prior distribution of the parameter may be elicited from experts in relation to what is known about the pest distribution elsewhere (Suess et al., 2002). The posterior 2.2. HIERARCHICAL BAYESIAN MODELS 17 distribution provides an updated view of our prior model based on the additional data collected.

Bayes’ rule can be extended to incorporate multiple data sets and multiple parameters in a hierarchical Bayesian modelling framework (Craigmile et al., 2009; Gelman et al., 2004). Given a valid joint probability distribution for the data and parameters, hierarchical models can estimate the posterior probabilities of parameters as a product of the conditional probabilities for model components and the prior parameters. The hierarchical construction of these models provides the ﬂexibility to incorporate ecological structure and complexity as well as uncertainty in ecological and observation parameters. For reviews of hierarchical Bayesian modelling in the ecological context, see Ellison (2004), Clark (2005) and Cressie et al. (2009).

We consider a conceptual framework for hierarchical Bayesian models from Wikle (2003),

[parameters, invasion status| data] ∝

[data| invasion status, data parameters]

[invasion status| invasion process parameters] (2.4)

[all parameters].

Bracket notation is used in Equation 2.4 where, [a, b|c] refers to the joint probability distribution of a and b given c. For each biosecurity application, we need to develop the model structure and deﬁne the priors for parameters for the modelling components. Firstly, we need to identify the appropriate observation model to describe the relationship between the data and the invasion status given prior uncertainty about the observation parameters. Secondly, we need to specify a suitable model for the invasion status conditional on the invasion process parameters. The role of the invasion process model is to generate the observable pattern from the ecological process (Pysek and Hulme, 2005). These population dynamics models may contain 18 CHAPTER 2. LITERATURE REVIEW mechanistic and stochastic components (Buckland et al., 2007; Soubeyrand et al., 2009). Finally we need to consider the parameter uncertainty for both the observation and invasion models. Uncertainty may be described using uninformed or vague probability distributions for parameters (Gelman et al., 2004), or by eliciting expert opinion to describe informed priors (Kuhnert et al., 2010; Martin et al., 2005a).

2.2.2 Pest observation models

Both frequentist and Bayesian approaches use observation models to describe the imperfect signal that an observer receives about the true pest population when visiting a site. Here, the focus is on visual inspection of hosts, but similar models are applied to other signal detection data such as background community assisted surveillance (Cacho et al., 2010), trapping (Barclay and Hargrove, 2005; Meats and Clift, 2005) and remote sensing (Wang, 2009). In most plant health surveillance systems, false positives are unlikely and so observation of the pest (and subsequent diagnostic conﬁrmation) is considered suﬃcient evidence that it is truly present. Therefore the primary goal of observation models in plant biosecurity applications is to analyse evidence for pest absence at a site.

Biosecurity surveillance data consist of observational outcomes, generally presence / absence, attributed to a geographic area that is usually referred to as a site. The spatial definition of a site is somewhat arbitrary, where the area may consist of a single plant, a field, a farm or some other functional management area. Commonly, a site is subdivided into units that are assumed to be independent and identically distributed so that standard statistical models for count data may be applied. The spatial definition of a site is integral to the construction of the observation model (MacKenzie, 2005). When constructing an observation model, the analyst must be mindful of the relationship between the outcome recorded, the parameter of interest and the effect of spatial aggregation of information within the model. 2.2. HIERARCHICAL BAYESIAN MODELS 19

The probability that a pest is observed within a site can be considered a function of the search intensity (e.g. plants inspected, time spent, area covered) and the expression of the pest within the sampling frame (Kery, 2002). Consider a pest that is present on a particular number of plants at a site and is perfectly observed. If the proportion of plants inspected from the area is relatively small, the probability of not detecting the pest may be adequately modelled by the binomial distribution. Under the assumption that the plants selected are exchangeable, this model can be used in the frequentist form to arrive at a conﬁdence level for a design prevalence (Cannon and Roe, 1982). If the proportion of plants inspected is large, the observation model may instead be based on a hypergeometric distribution (Cameron and Baldock, 1998; Hanson et al., 2003). Where the measure of search intensity is the proportion of the area surveyed, or search time, a Poisson distribution provides a further option. These basic statistical functions for modelling count data from observations can be implemented in frequentist analyses or they can be used to provide the likelihood component of a Bayesian approach (Hanson et al., 2003; Johnson et al., 2004).

Imperfect examination of those units that constitute the measure of search intensity is commonly referred to in the epidemiology literature as test sensitivity (Bohning and Greiner, 2006; Cannon, 2001; Gambley et al., 2009) and in ecological studies as detectability (Royle, 2008; Wintle et al., 2005), which we adopt here. Overesti- mation of detectability will result in underestimation of pest distribution that can severely compromise population management decisions (Myers et al., 1998; Wintle et al., 2004). The simplest approach to imperfect detection is to select a point estimate of detectability, d, for the base observation model as a modiﬁer of search intensity (Martin et al., 2007). For the binomial model with n plants inspected and a probability of each selected plant from the site being infested, θ,

[y|n, d, θ] ∼ Binomial(n, θd). (2.5) 20 CHAPTER 2. LITERATURE REVIEW

Detectability on an infested unit may be inﬂuenced by a number of factors, for in- stance observability due to tree architecture (Gambley et al., 2009), terrains (Hauser and McCarthy, 2009) or diﬀerences in observer experience (Christy et al., 2010; Gambley et al., 2009). Variation in individual pest behaviour may also result in mixtures of detectability (Christy et al., 2010; Royle, 2006), as can spatial cluster- ing on units within the site (Gschlossl and Czado, 2008). Where there is epistemic uncertainty surrounding the detectability parameter, or detectability is expected to vary between units, the data are referred to as being overdispersed (Potts and Elith, 2006).

Overdispersion of data from a binomial model is commonly modelled using a beta- binomial distribution (Thebaud et al., 2006; Venette et al., 2002), and can be expressed by the distributions,

[y|n, d, θ] ∼ Binomial(n, θd) (2.6)

d ∼ Beta(a, b), (2.7)

where a and b are parameters describing the uncertainty about the detectability.

One of the advantages of the beta-binomial distribution is that it has an analytically tractable form for estimating detectability in a Bayesian framework that has lead to its widespread use (Clark, 2003; Gelman et al., 2004; Hooten et al., 2007; Thebaud et al., 2006). Data from overdispersed Poisson processes may likewise be modelled using a negative-binomial distribution (Gschlossl and Czado, 2008; Royle, 2004).

Another class of models for dealing with overdispersion are the zero-inﬂated binomial and zero-inﬂated Poisson models (Branscum et al., 2004; Hall, 2000; Martin et al., 2007; Wintle et al., 2004). For pest count data in a binomial setting, the models consider the outcomes of the observation process to be binomial with a 2.2. HIERARCHICAL BAYESIAN MODELS 21 probability z or else 0,

 Binomial(n, θd) , with P r(z) y ∼ (2.8) 0 , with P r(1 − z).

Royle (2006) recommends caution when using zero-inﬂated models to infer population sizes at low densities. Therefore, despite their simplicity, these models may have limited value in estimating pest absence for biosecurity applications.

Most biosecurity surveillance programs are limited to the collection of presence / absence data, suggesting that logistic regression on θ, could be used to predict the status of sites given some additional covariate data, x, (Kery, 2002). The logistic regression model uses a logit link function to express the probability of an event occurring based on a linear function of covariate data, x, (Gelman et al., 2004),

logit(θ) = log(θ/(1 − θ)) = α + βx (2.9)

y ∼ Bernoulli(θ), (2.10) where α and β are the intercept and regression coeﬃcients.

In a Bayesian setting, the foundation observation models discussed so far provide the likelihood function for analysis. Uncertainty in detectability can be deﬁned by specifying a prior distribution on the hyperparameters for overdispersed models or for random eﬀects in logistic regression. Priors may be derived from plausible values provided by experts, or from existing empirical evidence (Hooten et al., 2007). In addition to this uncertainty about detectability, it is also necessary to consider the variable expression of the pest in the context of the invasion process.

A major source of variation in detectability will be the size of the population within the observation unit (Harwood et al., 2009; Royle, 2006). As an area is invaded, both the number of infested units and the probability of detection on individual 22 CHAPTER 2. LITERATURE REVIEW units will increase. At the margins of the range, pest expression is expected to be poor and therefore models will lack inferential power (Barrett et al., 2009). The capacity for observation models to process the evidence for absence that is needed for extent mapping hinges upon how well they describe the detection of the pest signal by observers at these low population levels.

Pest observation outcomes recorded at some point in space and time are generally interpreted as applying to some spatio-temporal vicinity (Yoccoz et al., 2001). Observations taken at a site at one time are expected to reﬂect the true status at times in the recent past and future. Temporal discounting of surveillance data has been examined for herd based sampling in veterinary epidemiology given continued exposure to infection (Schlosser and Ebel, 2001). In a similar way, observations in one area are expected to contain information about nearby or connected areas. This autocorrelation of the pest status in space and time is a function of the invasion ecology of the organism. It is recognised that the assumptions required of simple inferential probability models are usually violated in the face of spatial autocorrelation (Legendre, 1993; Wintle and Bardos, 2006). To delimit extent, the observation process is modelled in relation to internal processes within the observation unit, but this must also be supported by the external processes that give rise to interdependencies with other units. In the following section, invasion process models are introduced to deﬁne some ecological processes that give rise to spatial and temporal correlation.

2.2.3 Invasion process models

General reviews on invasion ecology can be found in Liebhold and Tobin (2008); Mack et al. (2000); Puth and Post (2005); Simberloﬀ (2009) and With (2002). The spatial realisation of an invasion process over time is the result of the birth, dispersal and death of many individual organisms. As the extent of an invasion evolves as a dynamic process, considerable heterogeneity and spatial dependence is displayed in 2.2. HIERARCHICAL BAYESIAN MODELS 23 the resulting distributional patterns (Hastings et al., 2005). Spatial correlation can be due to similar underlying environments as well as being intrinsic to the dispersal process itself (Wintle and Bardos, 2006). When the invasion is viewed as a latent process, the probability of a pest being present in a particular area can be modelled as a function of the dispersal mediated connections with infested sites and the time over which those connections exist (Jerde and Lewis, 2007). While the presence of spatio-temporal correlation complicates the analysis, the statistical modelling of invasions using prior ecological knowledge of the processes can allow more powerful inference to be made on pest extent.

Invasion processes can be broken down in different ways, depending on the components of interest to particular applications (Simberloff, 2009). At national scales, processes of interest can include introduction (entry), colonisation, establishment and spread (Drake and Lodge, 2006; Hennessey, 2004; Hulme, 2006; Lockwood et al., 2005; Mack et al., 2000; With, 2002). Here we treat the invasion processes under the headings of colonisation and dispersal. In this simplified framework, colonisation deals with the internal processes within a defined area while dispersal deals with the exchange of organisms between areas.

Colonisation

In simplest terms, colonisation is the process of a defined area going from uninfested to infested. As any infested area poses a biosecurity risk, estimating colonisation events is fundamental to the spatial management of invasive pests. Much work has focussed on colonisation across national borders (Drake and Lodge, 2006; Holmes et al., 2009; Simberloff, 2009). The International Plant Protection Conven- tion adopts the terminology of endangered areas to identify a region that favours the establishment of a pest of concern, where establishment is defined as the perpet- uation of a pest in an area for the forseeable future (IPPC, 2006). Here we consider the colonisation process as applying to any area of interest for which the pest status 24 CHAPTER 2. LITERATURE REVIEW is sought. The process encompasses the introduction of the pest into the area followed by successful reproduction leading to permanent establishment. Considerable work has gone into identifying the intrinsic biological characteristics of successful invaders (Johnson et al., 2006). However, the most reliable indicators of invasion success across taxa appear to be extrinsic factors such as climate/environment simi- larity and the number of pest propagules introduced (Hayes and Barry, 2008; Jarvis and Baker, 2001a; Rouget and Richardson, 2003).

Habitat suitability, in particular the availability of suitable host plants and climatic requirements, has a major impact on the probability that an area will be colonised. Climate matching has proved to be one of the most useful estimators for the ultimate distribution of an invading organism (Sutherst and Maywald, 1985) but comes with some caveats. Biogeographic predictive models based on environmental covariates in the native range of a pest can lead to erroneous estimates of final extent, either due to genetic differences in the invading population or due to different relationships between the pest and unidentified covariates in the two places (Fitzpatrick et al., 2007). It also needs to be recognised that the destination areas encompass both spatial and temporal environmental variation (Jarvis and Baker, 2001b; Simberloff, 2009).

The exposure of an area to the risk of pest introduction is commonly referred to as propagule pressure (Carrasco et al., 2010d; Leung et al., 2004; Lockwood et al., 2005). A propagule is a group of one or more organisms that enters an area at a particular time, while the propagule pressure is the total exposure of an area to these over some period of time (Simberloﬀ, 2009). Exposure assessments for environmental pollutants have used epidemiological risk characterisation techniques at a sophisticated level (Nieuwenhuijsen et al., 2006) but these are yet to be investigated rigorously with respect to colonisation in invasions (Stohlgren and Schnase, 2006).

Of interest to biosecurity are estimates of the probability that an area is free of 2.2. HIERARCHICAL BAYESIAN MODELS 25 a pest at some time. One approach is to implement discrete time models for the number of propagules arriving at a destination and surviving. Jerde and Lewis (2007) adopt a Poisson model for the survivors as the sum of movements from all pathways into the destination and use a geometric distribution to estimate the waiting time for a colonisation event in discrete time. A similar approach was used by Leung et al. (2004).

The fate of organisms between entry and establishment is one of the great un- knowns of invasion biology (Puth and Post, 2005), and is perhaps the most difficult process to parameterise. Successful establishment is based upon the fates and reproductive success of what is generally a small founding population (Kawasaki et al., 2006). Therefore, stochasticity plays a central role in understanding and modelling the colonisation process. In particular, there is potential for initially low rates of population increase and spread, known as Allee effects (Drake and Lodge, 2006; Hastings, 1996). Allee effects need consideration as a mechanism for local extinction after the introduction of a propagule (Dennis, 2002; Drake and Lodge, 2006; Foley, 2000; Keitt et al., 2001). They may also impact upon assumptions of population models from spatial draining effects early on in the colonisation process (Kean and Barlow, 2000). While Allee effects may contribute significantly to the success and expression of the colonisation process, prohibitively intensive collection of data from populations at low densities may be needed to quantify this effect (Kramer et al., 2009).

Early stages of the colonisation processes are poorly understood, most notably because they are rarely observed and there is little empirical evidence on processes that lead to establishment (Simberloff, 2009). A handful of studies have attempted to quantify the number of propagules being moved along pathways (Lee and Chown, 2009; McCullough et al., 2006; Stanaway et al., 2001), however, these are difficult to relate to the establishment of populations in new areas. In order to define the hierarchical link between observations and pest status, models of colonisation need to represent the population states within the area over time. Given that it is difficult 26 CHAPTER 2. LITERATURE REVIEW to collect empirical information on the colonisation phase, much of the burden for providing prior ecological knowledge falls upon dispersal models.

Dispersal

Invasive pests can spread by a number of natural dispersal mechanisms, including along drainage lines, wind assisted ﬂights (Reynolds and Reynolds, 2009) or active ﬂight (Guichard et al., 2010). Additional human-mediated dispersal pathways may also exist on nursery stock (Smith et al., 2007), produce (Areal et al., 2008) or simply as incidental hitchhikers (Ward et al., 2006). Spatial connectivity processes for natural dispersal and human-mediated dispersal underpin biosecurity management problems (Diggle, 2006). Within a hierarchical Bayesian framework for estimating pest extent, the role of dispersal models is to map the probabilistic relationships between the pest status of connected areas. These relationships provide the spatio- temporal covariance structure that allows the extent of the latent invasion process to be conditioned by space-time surveillance information.

Several spatial frameworks that have been used to provide the scaffolding for modelling invasions and pest dispersal. On continuous space and time, the classic deterministic reaction-diffusion models of Skellam (1951), based on random movements of individuals, formed the basis of invasion research for decades. Integro-difference equations (IDE) offer a discretised methodology for implementing invasive dispersal with the flexibility of different dispersal kernels (Neubert and Parker, 2004). Dispersal kernels model the probability of movement between two areas as a function of Euclidean distance. While Gaussian kernels are commonly used (Chapman et al., 2007; Havel et al., 2002; Wikle and Hooten, 2006), other distributions used for dispersal kernels include Laplace (Lewis and Pacala, 2000; Neubert and Caswell, 2000), Cauchy (Mayer and Atzeni, 1993), exponential (Havel et al., 2002) and negative exponential (Chapman et al., 2007). Dispersal kernels with exponentially bounded tails lead to asymptotically constant rates of spread through continuous 2.2. HIERARCHICAL BAYESIAN MODELS 27 space, while others, for example the Cauchy distribution, can lead to accelerating rates of spread (Kot et al., 2004). One of the drawbacks of IDEs is that they are deterministic and provide for a continuous distribution of organisms rather than a discrete distribution of individuals. Incorporating stochasticity on discrete individuals can slow the rate of spread (Kot et al., 2004) so that even fat-tailed kernels can lead to asymptotic rates of spread (Clark et al., 2001, 2003).

While continuous space models can have attractive mathematical properties, their application to heterogeneous environments can be problematic. As biosecurity programs frequently deal with spread through geographically fragmented host landscapes, another option is to look at the transfer rates of propagules between discrete areas. Connectivity models provide a more tractable framework for working with discrete patches that may exchange propagules (Urban and Keitt, 2001). Rates of exchange may again be modelled as a function of distances to known infested sites using the same dispersal kernels as for continuous landscapes. Gravity models to predict the colonisation rates of lakes by zebra mussels have been one successful application of this approach (Bossenbroek et al., 2001). Similar approaches on lattices that deﬁne connectivity as bond strengths between neighbouring areas are commonly used in epidemiology (Dybiec et al., 2004, 2005; Gibson et al., 2006; Ot- ten et al., 2004; Sander et al., 2002; Shirley and Rushton, 2005; Zhou et al., 2006). As these models deal with the links between individual ecological units of interest, they oﬀer readily interpretable statistics for the management of spread between areas (Shirley and Rushton, 2005; Urban and Keitt, 2001).

Incursions typically spread through adjacent areas at both ﬁne and coarse scales (Scherm et al., 2006). Most invasive species arrive in countries due to the activity of people and continue to travel on similar human mediated pathways after arrival as well as by natural dispersal. Invasion processes can be highly stochastic with the colonisation of satellite sites outside of the contiguously infested area a major driver of the overall spread (Lewis and Pacala, 2000; Neubert and Caswell, 2000). Where long distance movement is a prime contributor to invasions, uncertainty in 28 CHAPTER 2. LITERATURE REVIEW predicting rates of spread and distribution can become prohibitively large (Clark et al., 2003). Empirical evidence for the dispersal distances of individuals is diﬃcult to collect but can be used to estimate dispersal kernels (Hawkes, 2009; Kareiva, 1983).

Plant biosecurity surveillance is deployed and evaluated according to some (generally informal) underlying mechanistic model of pest ecology (Plant Health Australia, 2010b). While mechanistic models may be formulated diﬀerently, the spatial realisation of these models may be similar (Wikle, 2003). What is more important for managing EPPs is that the model can be translated into operational use for the appropriate management units, whether they are farms, blocks within a farm, trees within a block or a continuous landscape of wild hosts. EPP spread across a landscape of plant host material requires critical examination as a spatial model (With, 2002). Whether it is for early detection, incursion management or to justify pest free areas, these ecological models provide the dynamic context for interpreting surveillance data.

2.2.4 Computation

Analytical solutions for the posterior distributions of parametres can be obtained for simple Bayesian models, but for complex hierarchical models with multiple parameters, integration of the normalising constant becomes intractable (Gilks et al., 1996). The advent of Markov chain Monte Carlo (MCMC) simulation on fast com- puters has enabled Bayesian techniques to be employed to solve quite complex high dimensional problems (Zhao et al., 2006). MCMC approximates the posterior distribution of parameters by sampling values from their conditional distributions (Gelman et al., 2004). Good reviews of MCMC and other computation methods are given in Gelman et al. (2004) and in an epidemiology setting by Lawson (2009). Here, two of the most common sampling algorithms, Gibbs sampling and Metropolis-Hastings sampling, are introduced. 2.2. HIERARCHICAL BAYESIAN MODELS 29

A detailed introduction to the Gibbs sampler for estimating the posterior distribution of parameters is provided in Casella and George (1992). In general, to implement a Gibbs sampler on an ordered set of parameters, θ = θ1, . . . , θn, the parameter values are initialised with plausible values for the ﬁrst iteration, t = 1. For the second iteration, a sample is drawn from the conditional distribution

[θi|θj6=i, y], (2.11)

where, y, are some observational data and, θj6=i, is the current value of all other parameters excluding the ith that is currently being sampled. As each parameter is successively sampled in the iteration, its current value is updated to become a new sample from the joint posterior distribution of the parameter given the data. The algorithm continues to cycle through the parameters on subsequent iterations until there are sufficient samples to approximate posterior distributions of all of the parameters. That is, the Markov chain will converge to the stationary distribution of the parameters. The number of iterations must be large enough to overcome any effects due to the arbitrarily chosen starting values. In order to avoid initialisation effects, it is common to exclude samples from the initial “burn-in” iterations (Brooks and Gelman, 1998a).

Often, it is not possible to draw directly from the conditional distributions given in Equation 2.11 due to the analytical intractability of the normalising constant (Chib and Greenberg, 1995). Provided the ratio of the probability densities can be evaluated, the Metropolis-Hastings algorithm overcomes this by drawing values from a proposal distribution, q, at each iteration. The proposal distribution is commonly deﬁned as a random walk with sampled values based on the current

t+1 t t parameter value (e.g. q(θi |θi) ∼ N(θi, σ)) (Gilks et al., 1996). The acceptance ratio for the proposed value to the previous iteration is calculated, 30 CHAPTER 2. LITERATURE REVIEW

t+1 t+1 t [θi |θj6=i, y]q(θi |θi) r = t t t+1 , (2.12) [θi|θj6=i, y]q(θi|θi )

and the new parameter value accepted with probability, min(r, 1). All parameters are cycled through in the same manner as for the Gibbs sampler. The choice of the proposal distribution can require some tuning. If the jumps are too large, the sampler will reject a high proportion of samples as parameter values will be drawn from low probability areas of the posterior density. If the jumps are too small, the sampler will take a long time to explore the parameter space and converge to the distribution.

Software, such as WinBUGS and OpenBUGS, has allowed for the rapid development of Bayesian models using pseudo-code to implement these and other sampling algorithms (Lunn et al., 2009; Spiegelhalter et al., 2003; Thomas et al., 2006). While some model structures are too complex for these packages, they remain a useful starting point for developing and testing components that may be later tackled in more ﬂexible programming environments (Gelman et al., 2004). In order to capture the complex ecology of invasions in space and time, the computational challenges that need to be overcome to implement hierarchical Bayesian models can be prohibitive (Banerjee et al., 2008; Latimer et al., 2009). New computational methods have been developed to deal with these problems such as approximate Bayesian computation (Cornuet et al., 2008; Sisson et al., 2007) and Kalman ﬁlters (Wang, 2009; Wikle and Berliner, 2007). As these new computational and computing approaches are developed, real time Bayesian analyses of complex invasion problems in response to changing information on fast moving invasions is becoming feasible (Jewell et al., 2008). 2.3. BAYESIAN PLANT BIOSECURITY APPLICATIONS 31

2.3 Hierarchical Bayesian Models for Plant Biose- curity Applications

Hierarchical Bayesian models can take advantage of the variety of component models available to attack the problem of deﬁning pest extent. In this section, a selection of those tools that could be applied to biosecurity applications are examined for making inference given the complex ecology of an invading pest.

A goal of many ecological studies is to estimate the extent of a species using species distribution models (Elith and Leathwick, 2009). Statistical approaches to inferring extent commonly rely on generalized linear models to relate species distribution to environmental covariate data, however such models ignore the intrinsic spatial correlation that is a feature of where individuals are found (Beale et al., 2010; Dormann, 2007b; Hoeting, 2009; Latimer et al., 2006; Wintle and Bardos, 2006). Auto-models provide an extension to regression models to allow the spatial covariance between sites within a neighbourhood to be admitted to the analysis (Besag, 1974, 1972). This autocovariance term may be based on Gaussian, binomial or Bernoulli distributions.

In the broader class of auto-models, conditional autoregressive (CAR) models have been increasingly used for disease mapping applications (Lawson, 2009), particularly as the algorithms to implement these models within MCMC software are freely available (Thomas et al., 2004). For the analysis of presence/absence data, autologistic models have become a mainstay of species distribution problems (Augustin et al., 1996; Hoeting et al., 2000; Huﬀer and Wu, 1998) although their performance under conditions of strong spatial association has been questioned (Carl and Kuhn, 2007; Dormann, 2007a). It is uncertain over what range of scenarios a predominantly spatial model can implicitly incorporate the temporal processes of invasions. Spatio-temporal extensions to the autologistic model have been applied to pest outbreaks under the assumption that the spatial and temporal components of the 32 CHAPTER 2. LITERATURE REVIEW auto-covariance are separable (Zhu et al., 2008). As dynamic processes in space and time may not be separable, explicitly modelled space-time processes must be developed for greater power of inference and interpretation of management strategies (Wikle and Royle, 1999). One of the main limitations of spatio-temporal auto- models for biosecurity applications is their inability to accommodate the dynamic nature of a pest invasion as it unfolds.

Wikle (2003) introduced reaction-diffusion equations for pest spread and reproduction into a hierarchical Bayesian framework to estimate arrival times of invading house finches. This and related studies (Hooten et al., 2007; Hooten and Wikle, 2008; Wikle and Hooten, 2006), used integro-difference equations with spatially varying diffusion coefficients to structure the spatial transition of populations in discrete time steps. As the parameter of interest was the spatially varying rate of spread, these authors specified a log-normally distributed population process, but with the population set to zero according to some reasonable boundary conditions. While the hierarchical Bayesian modelling framework they developed marked a major advance in the analysis of invasion data, further extension of their models is required to focus the inference on the estimation of pest boundaries.

To delimit pest extent in a Bayesian model, inference on the pest status of each area of interest from the surveillance data requires some underlying invasion process. Gibson et al. (2006) describe a percolation model to estimate the disease infection times of plants on a lattice by considering the diﬀerence in colonisation times between neighbouring plants to be exponentially distributed.

A feature of biosecurity data is that often thousands of spatially referenced data points are collected but the information content of these data is not known. When detection probabilities are low, the information available to distinguish parameters can lead to poor convergence properties for highly parameterised models (Webster et al., 2008). Furthermore, incorporating the space-time information contained in these points into MCMC models becomes computationally prohibitive. Banerjee 2.4. SUMMARY 33 et al. (2008) and Latimer et al. (2009) propose predictive process models that may overcome some computational hurdles by modelling invasion processes at a manageable number of points in space and time. The dimensional complexity of space-time models makes evaluation computationally intensive and requires research to determine workable incursion management scenarios. Some level of spatial and temporal model aggregation is required to partition the system into computationally (opera- tionally) manageable components for which conditional probabilities of absence can be determined.

As biosecurity programs gather information from expert ecological opinion and from large spatio-temporal surveillance data sets, hierarchical Bayesian frameworks suggest themselves as a useful analytical paradigm. While applications have explored their use in modelling invasion spread, the approaches are not tuned to the estimation of extent which is a critical requirement of biosecurity applications. The challenge to hierarchical Bayesian modelling of pest extent is to estimate the geographic tails of the population distribution in light of uncertainty in observation and the invasion ecology of pests.

2.4 Summary

The review introduces the management requirements for inference on pest extent in a biosecurity setting. Hierarchical Bayesian modelling approaches are explained in terms of their ability to assimilate information from surveillance data and prior ecological data. The modelling tools available for both the observation process and the ecological invasion process are reviewed to identify the fundamental building blocks of analysis. A framework to incorporate these component tools within a hierarchical Bayesian modelling is proposed and related to the current literature on Bayesian statistical modelling for managing invasive pests.

Hierarchical Bayesian models can provide a formal deﬁnition of biosecurity problems 34 CHAPTER 2. LITERATURE REVIEW and the information available to solve them. Their implementation as analytical applications face some hurdles. Ecological complexity must be captured by the processes model so that it adequately portrays uncertainty over space and time. The assimilation of information from large data sets into high-dimensional models also imposes computational challenges for the estimation of parameters using MCMC. Despite this, they oﬀer an analytical structure that, through the model construction process, can enrich the understanding of uncertainty in decision making as well as provide transparent inference on the probability of pest extent. Chapter 3

Hierarchical Bayesian Modelling of Early Detection Surveillance for Plant Pest Invasions

3.1 Introduction

Eradication or containment campaigns for exotic plant pest and disease incursions pose a substantial cost to agricultural producers and government biosecurity regulators (Bogich et al., 2008; Myers et al., 2000). Interception of these pests at the border is desirable, but early detection surveillance programs oﬀer a second line of defence against pests that escape interception and establish (Maynard et al., 2004). These programs are founded upon both the probability that pests will enter and spread, as well as an expected utility in early detection. Surveillance eﬀort is commonly directed towards “risk” areas with the aim of detecting incursions before they are too extensive to eradicate (Barrett et al., 2009; Hulme, 2006; Stark et al., 2006). Early detection programs often run for many years without detecting pests of concern. One such program, targeting a range of major horticultural pests, has

35 36 CHAPTER 3. EARLY DETECTION SURVEILLANCE operated in residential areas of Queensland, Australia, since 1999. Surveillance by observers (entomologists, pathologists and regulatory inspectors from Biosecurity Queensland) has been deployed unevenly over space and time, based on perceived spatio-temporal risk. To assess the worth of this and similar programs, we need to understand the value of the information that the data provide, and how this information contributes towards the management of invading pests.

Knowledge about the extent of an incursion comes from observational data under- pinned by an understanding of pest incursion dynamics. Bayesian models provide a cohesive inferential framework that can combine prior information about the ecology of a pest with information from ﬁeld observations (Buckland et al., 2007; Cook et al., 2007a; Cressie et al., 2009; Hooten et al., 2007). By hierarchically struc- turing these models, complex systems can be broken down into simpler statistical components. A useful decomposition used in invasion ecology is to develop component models for the invasion process, the observation process and the distribution of parameters for these processes (Wikle, 2003). Such models allow us to make inference on the extent of an incursion over time given the surveillance data and prior knowledge about the incursion parameters.

Prior expert opinion about the ecology of the invasion process can be used to define the state of knowledge about incursion extent over time without the benefit of observations. When no pests are detected by a program, it is this prior information that provides all of the information to support pest presence. The worth of an early detection program therefore rests on the degree to which surveillance data opposes the prior assessment of extent. Hierarchically structured models can incorporate this prior information about invasion process parameters, such as exposure rates and spread of a target pest, while recognising uncertainty in the parameter values (Hooten and Wikle, 2008). For early detection surveillance, the choice of priors for model parameters must describe the range of invasion characteristics that could belong to a potential target pest. Target pests of concern to such a program are expected to have high probabilities of entering, have the potential to increase to 3.1. INTRODUCTION 37 destructive levels and be capable of spreading from the point of colonisation. In addition, target pests must also have characteristics that allow them to be effectively eradicated or contained if there is to be some utility in early detection (Mack et al., 2000). Priors for ecological parameters must therefore be chosen to support a baseline estimate of the latent spatio-temporal invasion status that can be challenged by the pest absence data.

Observational data collected by surveillance programs will imperfectly reﬂect the invasion status at a site (Royle, 2006). Pests present at a site may be overlooked if symptoms are poorly expressed, or if only a portion of the site is examined (Barclay and Humble, 2009). In applications where presence and absence data is available, it is possible to estimate detectability by repeated sampling of closed populations (Royle and Dorazio, 2006; Royle and Kery, 2007). However, incursion processes are not at equilibrium and, when only absence data are available, there is no opportunity to learn about detectability. Expression of pest symptoms is an ecological process that is related, at least in part, to the length of time that a site has been colonised. By drawing on prior information to model changes in detectability due to population growth over time, the completed observation model can describe the distribution of presence/absence data in relation to detectability at the site. As the surveillance data is space-time referenced, it can provide information on the extent of the pest with respect to the invasion process model.

Our interest is in understanding how surveillance changes our state of knowledge about incursion extent. The initial state of knowledge is provided through an incursion process model. The model can be parameterised by ecologists and managers to reflect the characteristics of pests that will be targeted by a specific program. In this article, we use the incursion process model to estimate the prior extent of the pest. The posterior extent is estimated by using a hierarchical Bayesian interpretation of the incursion process model when conditioned on the observation data. As the cost of eradication increases with pest extent, we consider estimates of extent to be a contingent financial liability. We are thus able to contrast predictions of the 38 CHAPTER 3. EARLY DETECTION SURVEILLANCE current liability with and without the benefit of surveillance data. In addition to estimating the total area of incursion at a time, the incursion process model allows estimates of the point probability of infestation to be mapped. Future surveillance can be planned to target sites with the highest probability of infestation. We aim to demonstrate that pest absence data collected by surveillance programs, when appropriately modelled, can be justified in terms of the gains in knowledge about pest extent.

The model is analytically complex and so the posterior distributions of invasion extent and ecological parameters are estimated with Markov chain Monte Carlo (MCMC) simulation using the BUGS software (Spiegelhalter et al., 2003; Thomas et al., 2006). After introducing the model in Section 3.2, model behaviour is demonstrated using simulated datasets in Section 3.3. The framework is then extended to include explicit spatial priors for exposure rates and applied to surveillance for banana pests in a residential area in Section 3.4. The simple framework can be modiﬁed or expanded to include more detailed ecological and surveillance processes and to meet the needs of speciﬁc early detection programs.

3.2 Model

3.2.1 Overview and notation

Invasion ecology and epidemiology both deal with the introduction and spread of damaging species into a geographically distinct area. We will use the terms pest to cover both arthropod pests and diseases and ecology to refer to both epidemiology and invasion ecology. We will use incursion to refer to an invasion that is in the early stages of spread and could be considered for eradication or control. Square bracket notation is used to denote a probability distribution, for example, [a, b|c] refers to the joint probability distribution of a and b given c. 3.2. MODEL 39

Suppose Z(s, t) ∈ {0, 1}, (0 = absence, 1 = presence) is the binary status of a latent incursion process, at a location with coordinates, s, at time, t, in a continuous space-time domain of interest, A. The spatial status of the incursion at a given time, t, can be denoted by Z(, t) and the incursion status at any location, s, over time can be given by Z(s, ). Letting Z ≡ {Z(s, t):(s, t) ∈ A}, we wish to infer

Z from binary surveillance data, X = {xi : i = 1,...,N}, collected from N visits, which are indexed in space and time by si and ti.

Of secondary interest is the estimation of some invasion process parameters, θp, that describe the pest ecology and thereby, Z. Consider the joint prior distribution of the incursion status and process parameters,

[Z, θp] = [Z|θp][θp]. (3.1)

The distribution represents our prior knowledge about how a target pest incursion could manifest in the district, based upon the expert opinion contained in the process parameters and the structure of the model.

Surveillance observations are imperfect and can also be modelled with uncertainty in the observation parameters, θd. Following Wikle (2003), a general hierarchical Bayesian framework for the joint posterior distribution of the invasion status and parameters, conditional on the data, is,

[Z, θp, θd|X] ∝ [X|Z, θd][Z|θp][θd][θp]. (3.2)

The ﬁrst term on the right hand side is the observation model that speciﬁes the distribution of the observational outcomes, conditional on both the underlying invasion status and the observation parameters. The next term models the distribution of invasion status, conditional on the distribution of the invasion process parameters. 40 CHAPTER 3. EARLY DETECTION SURVEILLANCE

Finally, the parameter models describe the distribution of both the observation and process parameters.

The diﬀerence between the posterior and prior distribution of invasion status and process parameters (Z, θp) is due to the knowledge that is gained by observations from the early detection surveillance program.

3.2.2 Incursion process model

Pest pressure into a district results in the establishment of a plant pest species at (χ, φ) ∈ A, where χ is the location and φ is the time of colonisation. We adopt a continuous colonisation time model with,

[φ|λ] ∼ Exponential(λ), (3.3) where λ is the exposure rate parameter for a potential target pest. Uncertainty in the exposure rate is given by a gamma hyperprior, λ ∼ Gamma(aλ, bλ).

In the absence of prior information about the spatial distribution of colonisation points, we let χ be uniformly and randomly distributed within the bounds of the district. Later, in the surveillance program application, we adopt a discrete χ ∈

{χm : m = 1, 2,...M}, where χm is the centroid of sub-district m. We consider the probability of colonisation in each sub-district to be proportional to the number of residential properties in the sub-district. Letting Rm be the proportion of residential properties in the district that are in sub-district m, we model the distribution of the colonisation point falling in sub-district m as,

[m] ∼ Categorical(R), (3.4)

with χ = χm. 3.2. MODEL 41

Note that here we are only considering an incursion that originates from a single colonisation event. An alternative approach would be to model colonisation events independently across the sub-districts, [φm|λm]. While such a model would reﬂect the opportunity for repeated incursions over the surveillance period, computational limits prevent this approach being used for this application. The practical implications of this are discussed further in Section 3.2.4.

Incursion models based on dispersal by diﬀusion lead to asymptotically constant rates of advance for invasion fronts (Shigesada et al., 1995; Skellam, 1951). While recognising the complexity of invasive species spread (see Hastings et al. (2005) for a review), we adopt this simple model and assume spread to occur from χ at a constant, but unknown, rate υ. The prior for velocity of spread is given by,

υ ∼ Uniform(aυ, bυ), with the interval, aυ to bυ, encompassing spread rates for target pests.

The distribution of the invasion status, Z(s, t), can be calculated over space and time from the joint distribution of [χ, φ, υ] using the indicator function,

Z(s, t) = I(t − (φ + ks − χk /υ) ≥ 0). (3.5)

The resulting model for the invasion status can be visualised as a distribution of inverted cones in space and time. The apex of the cone is at (χ, φ) and is a stochastic process on (λ, R) with epistemic uncertainty surrounding λ. The angle of the cone is represented by the uncertainty surrounding υ. As each plausible incursion process can be described completely by Equation 3.5, the joint distribution of [χ, φ, υ] models the spatio-temporal correlation in colonisation times.

In order to link the observation data at a visit to the invasion process model, we extend the notation for the colonisation time of the district, φ, to refer to the colonisation time at a particular location, φs, so that, 42 CHAPTER 3. EARLY DETECTION SURVEILLANCE

φs = φ + ks − χk /υ. (3.6)

When given a suitable model for the observation process at (s, t), the posterior distribution of [χ, φ, υ] can now be estimated by conditioning on the observation data collected from the geo-referenced sites.

3.2.3 Observation model

To obtain a positive record for a colonised site, the pest must be detectable, observed and reported. Detectability is deﬁned as the probability of seeing a pest on a single selected plant at the site as determined by the visual expression of pest symptoms. To be observed, these symptoms must be perceived by the observer on one or more plants examined. For a pest to be reported, it must be observed and subsequently collected and diagnosed for a conﬁrmed positive record. Failure at any of these three points will result in a false negative observation for the site. The completed observation model consists of a biological process component for detectability and a sampling component for observation and reporting.

Rather than modelling population growth per se, we model increase in detectability as a function of time elapsed since the site was colonised. When a site is ﬁrst colonised at φs, there may be a biological latent period, γ, before any symptoms are visible. This pest latent period may be non-existent in the case of many insects but for systemic diseases, it may be months or years. The length of time for which the pest could have been detected, C(s, t), at the observation time t, will be,

C(s, t) = t − φs − γ. (3.7)

We define D(s, t) to be the pest detectability on a plant at s, t. Hooten et al. (2007) use a Ricker model for density dependent growth in population abundance. Here 3.2. MODEL 43 we use a similarly shaped logistic growth model, implemented as an inverse logit function, to deterministically model pest detectability at some time after the site is colonised,  0 if C(s, t) ≤ 0,   D(s, t) = logit−1(βC(s, t) + h) otherwise, (3.8)   −1 η if logit (βC(s, t) + h) > η where β is the rate of increase in pest detectability over time and h is an offset to define a small threshold of detection (say logit(1/1000)). In keeping with reproduc-

2 tive growth of our target species, we assume a lognormal prior, log(β) ∼ N(µβ, σβ) to prevent negative growth rates that would represent a failure to colonise. As C(s, t) increases, the pest detectability will approach one. An additional parameter, η ∼ Uniform(aη, bη), is introduced to model uncertainty in the maximum pest detectability. The model for D(s, t) provides the structure to relate observations back to the invasion status of the site.

At any particular observation visit, i, there will be local environmental variation in pest detectability. Letting Di ≡ D(si, ti) be the pest detectability on a plant at observation visit i, we include a random eﬀect by modelling the visit detectability,

∗ Di , as,

∗ logit(Di ) = logit(Di) + i, (3.9)

2 2 where, i ∼ N(0, σ ) with ﬁxed variance, σ .

For a given detectability, observation of a pest at a visit will also be a function of the search intensity. Surveillance consists of a visit to a site where a total of ni plants are examined and the binary presence/absence outcome for the visit recorded, xi. While we expect detectability between plants at a visit to vary, for this application we are generally dealing with a small number of plants in suburban backyards. As 44 CHAPTER 3. EARLY DETECTION SURVEILLANCE

we expect the variation to be small compared to i, we assume that detectability on individual plants at a visit is homogeneous and that the eﬀect of within site variation is accounted for by i. Assuming independence, the probability of observing the pest, Qi, on any of the ni plants inspected is modelled as,

∗ ni Qi = (1 − (1 − Di ) ). (3.10)

Uncertainty in the probability of reporting a pest, given that symptoms were observed, is described by an informed prior for a reporting parameter, ω ∼ Beta(aω, bω). Assuming independence, the probability of the pest being reported is given by,

Pi = ωQi. (3.11)

The observation outcome for visit i is then modelled as,

[xi|Pi] ∼ Bernoulli(Pi). (3.12)

The observation model provides for considerable parameter uncertainty from a range of sources that may be relevant to a particular early detection surveillance program. The observation parameters are poorly identiﬁed by the data but are included individually to provide a rich model that can be informed by regulators and ecologists for speciﬁc applications.

3.2.4 Parameters

Early detection surveillance programs target species which we may wish to eradicate. The selection of prior distributions for parameters to reﬂect the characteristics of target species is therefore critical for providing the baseline belief in the invasion 3.2. MODEL 45 threat. It is this information, elicited from managers and ecologists, that provides the foil against which the surveillance data is tested. Priors for the invasion process parameters must characterise organisms that have the capacity to both establish and assume pest status. A somewhat opposing constraint is that the process parameters must only characterise pests with viable management options, most notably the potential for eradication. The complete list of informed priors and the parameters used for the analysis are given in Table 3.1. In this section, we discuss the biological and management characteristics that deﬁne the parameter space over which we evaluate our data.

Rate parameters for the pest establishment process are difficult to quantify for a particular species (Koch et al., 2009; Simberloff, 2005). However, parameter uncertainty in exposure rates can be characterised in terms of the goals of a general early detection surveillance program. If exposure rates for a particular pest species are greater than one in every five years, regulators would either design a tailored detection and eradication program, or accept that repeated eradication was not feasible. For pests with low exposure rates, targeted surveillance would be unlikely to provide any return on the investment for many years. A Gamma(10,100) distribution is proposed for the exposure rate parameter, λ, which has a mean of 1/10 years and 5% and 95% quantiles of 1/18.4 and 1/6.4 years. As mentioned in Section 3.2.2, computational limits prevented us from modelling exposure rates independently at the cell level. While multiple cells in a district may be colonised before the first detection is made, we consider that inference is still valid for the low prior exposure rates that are a feature of these programs.

For this application, we focus on short distance natural spread from a colonising propagule that would be considered for eradication. If natural spread is too fast, the chance of eradication is slim, even if detected early. At the other end of the scale, pests that spread extremely slowly are unlikely to be considered for eradication as they seldom have a substantial impact on horticulture. Uncertainty is modelled using a uniform prior for spread rates of between 500 m and 5 km per year. 46 CHAPTER 3. EARLY DETECTION SURVEILLANCE

The model of detectability requires information about the latent period, growth rate, maximum detectability and variability. Long latent periods before the pest is observable will make detection and eradication prohibitively difficult (Manjunath et al., 2008). We propose a uniform prior for the pest latent period, γ, that ranges from 0 to 6 months. Applications for specific pests would benefit from modelling the population dynamics explicitly and interpreting process detectability from population size (Hooten et al., 2007). In the absence of specific life history information, we consider prior rates of increase in pest detectability, β, that give 5%, 50% and 95% quantiles of 24, 125 and 653 days to attain a pest detectability of 0.5. Even after a long period of colonisation at a site, the expected pest detectability on a plant will not necessarily approach one. A uniform prior for η is proposed over the range of 0.75 to 0.999. For the banana surveillance model, bη was reduced to 0.9 to provide greater scope for local environmental variation. Visit specific variation in the expression of pest symptoms is incorporated using a normal distribution with a variance, σ2, of 0.1 on the logit scale. The variance in detectability is greatest when the process detectability is 0.5, where two standard deviations cover a range of detectability from 0.35 to 0.65.

Finally, observers may erroneously attribute symptoms to physiological stress or endemic species and fail to collect a sample of the target pest for diagnosis and reporting. The probability of reporting a pest that is observed, ω, is expected to be high, but is included as additional uncertainty with a Beta (18,2) prior.

3.2.5 Inference and interpretation

The value of information provided by early detection surveillance data is examined in two ways. Firstly, we consider that the estimated area of incursion extent describes a contingent liability. This liability represents a management cost that pest managers must be prepared to account for at any particular time. Extent can be estimated from both the prior and posterior distribution of Z(, t). By comparing 3.2. MODEL 47

Table 3.1: Description of informative priors to describe uncertainty in the observation and ecological characteristics of target pests.

Parameter Description Prior Distribution Prior mean SD

λ District exposure rate (/yr) Gamma(10,100) 0.1 0.032 υ Spread rate (km/yr) Unif(0.5,5) 2.75 1.30 γ Pest latent period (yr) Unif(0,0.5) 0.25 0.14 β Process detectability growth rate LogNorm(3,1) 33.12 43.41 η Maximum process detectability Unif(0.75,0.999) 0.87 0.069 ω Reporting probability Beta(18,2) 0.9 0.065

the liabilities associated with these two extents, managers can determine whether the reduced liability, given surveillance data, is warranted by the expense of the program. While the true cost of responding to an incursion will not be realised until a particular pest is detected, these liabilities form a reasonable basis for comparison given the uncertainty about the threat. The second advantage that surveillance provides is through updated estimates of the point probability of infestation at a given time.

Most commonly, interest is in estimating the current incursion status, Z(, l), where l is the time that the last observation is made. We examine three statistics for incursion extent that are derived from the joint distribution of [χ, φ, υ] in both the prior and posterior model.

Firstly, we consider the latent colonisation state of the district at time l to be Vl so that Vl = 0 if the district is pest free and Vl = 1 if it is infested. By calculating

Vl = φ < l from samples taken from the prior and posterior distribution of φ, we ¯ can estimate the mean colonisation state Vl. This estimate can be interpreted as the probability that the district is colonised and that some management liability has accrued by the current time. The diﬀerence between the prior and posterior ¯ estimates of Vl, represents the change that surveillance information makes to the belief that the pest is established in the district. 48 CHAPTER 3. EARLY DETECTION SURVEILLANCE

Secondly, if the district has been invaded, eradication costs are expected to increase with the area infested. The radius of incursion, ρl = (l − φ)υ : φ < l, provides a simple measure of the incursion extent for an unbounded domain. A better measure of eradication costs is the area infested, αl, which we deﬁne as the area of Z(, l): Z = 1, l > φ across the domain. Note that this deﬁnition excludes the case where the district is free of the pest (i.e. αl > 0) and translates into the estimated liability for managing an incursion given that the district was colonised.

The ﬁnal use of the joint distribution of [χ, φ, υ] is to map the spatial posterior distribution of incursion status after surveillance, Z(, l). We use samples from the prior and posterior distributions of Z(, l) to estimate the point probability of infestation, Z¯(, l) = P r(Z(, l) = 1). These maps help future surveillance to be targeted at geographic areas with a higher probability of being infested.

3.2.6 Computation

Analysis is conducted using MCMC simulation within the WinBUGS and Open- BUGS software. BUGS software uses Gibbs sampling to simulate the joint posterior distribution of the model parameters by drawing each of the parameters in a pre- scribed order, conditional on the value of all other parameters and the data (Gelman et al., 2004; Spiegelhalter et al., 2003). All parameters are initialised with a starting value (here generated using the BUGS built-in function) and new parameter values are proposed in turn. After a suitable number of “burn in” iterations to remove initialisation eﬀects, the parameter estimates will be sampled from the joint conditional distribution of the parameters given the data. These “burn in” samples are discarded with the remaining iterations used as samples from the posterior distribution of parameters. The number of iterations needs to be large enough for the Markov chain to converge to its stationary distribution. Convergence for models may be checked by running multiple chains, each initialised with diﬀerent values, to ensure that chains are all converging to the same joint density and not, for example, 3.3. SIMULATIONS 49 becoming trapped in local minima. Preliminary runs of the models were assessed for convergence using the Gelman-Rubin statistic (Brooks and Gelman, 1998b) to determine a suitable burn-in period for the sampler.

3.3 Simulations

Simulated data sets are used to examine the model behaviour over a range of space- time visit scenarios. Our interest is in how estimates of extent change with the data characteristics of diﬀerent scenarios, and how the data inform the individual process parameters that contribute to this change.

3.3.1 Data and methods

Six scenarios were examined for a 50 km × 50 km district with observations made at N sites and each site visited on a single occasion. In scenario A, observations consisted of absence outcomes, xi = 0 : i = 1, 2,...,N, for N = 20 sites within the district, with each site visited six months apart, ti = {0.5, 1.0, 1.5,..., 10.0}.

Observations are made on ni = 20 : i = 1, 2,...,N plants for each visit. The location of sites, si, were generated randomly from a bivariate uniform distribution over the district.

Scenarios B and C examine the eﬀect of site location on the posterior distribution of the colonisation point, χ. In scenario B, a sequential series of observations runs diagonally up across the domain. In scenario C, all observations were made at the centre of the domain.

Scenarios D and E examine the eﬀect of surveillance intensity. In scenario D, N was increased to 40 with additional random sites inserted alternately over time between the data in scenario A. In scenario E, N was reduced to 10 using every second observation from Scenario A. Finally, in scenario F, the spatial arrangement 50 CHAPTER 3. EARLY DETECTION SURVEILLANCE in scenario A was retained but with a positive detection on the ﬁnal observation.

Samples from the prior and posterior distributions of φ and υ were used to estimate the colonisation status, Vl, and the area infested, αl, at l = 10 years. Prior distributions for Vl and αl were simulated using the R software package by drawing

50 000 samples from [φ|λ][λ] and [υ]. The posterior distributions of Vl and αl were calculated from samples drawn from the φ and υ chains generated by the WinBUGS model which was run with two chains for 50 000 iterations after a 10 000 iteration burn-in and thinning every second draw.

3.3.2 Results

Surveillance absence data naturally reduces our estimates of the probability of pest presence across the domain as well as updates our beliefs about the invasion param- ¯ eters. In scenario A, the estimate of the probability that the district is colonised, Vl, was modestly reduced from a prior mean estimate of 0.62, to a posterior mean estimate of 0.44 (Table 3.2). More intensive surveillance increases the estimate of the ¯ district being free from pests, although the diﬀerences in Vl between the scenarios for N=10, 20 and 40 were not substantial.

Table 3.2: Prior and posterior means and standard deviations for selected parameters and scenarios. Scenarios B and C are used to demonstrate spatial eﬀects only and are omitted. Scenario A) N = 20,

D) N = 40, E) N = 10, F) N = 20 and xN = 1.

Prior mean Posterior mean: A D E F

Vl 0.616 (0.486) 0.444 (0.497) 0.393 (0.488) 0.472 (0.499) 1 ( 0)

ρl 16 (11.4) 8.1 (5.76) 6.56 (4.45) 8.79 (6.31) 17.4 (6.31)

αl 683 ( 690) 228 ( 269) 166 ( 203) 250 ( 294) 849 ( 487) φ 11.1 (12.4) 14.9 (13.2) 16 (13.3) 14.4 (13.3) 3.68 (1.98) υ 2.75 ( 1.3) 2.53 (1.31) 2.53 (1.32) 2.52 (1.31) 2.96 (1.13) λ 0.010 (0.032) 0.097 (0.031) 0.096 (0.030) 0.097 (0.031) 0.11 (0.032) β 33.1 (43.4) 32.4 (43.0) 32.8 (43.3) 32.8 (43.0) 33.1 (43.2) 3.3. SIMULATIONS 51

¯ In addition to the reduction in Vl, there was a marked decrease in the posterior estimate of the area infested if the district were colonised, αl. Scenarios D and E demonstrate that more intensive surveillance with additional absence outcomes result in a lower posterior estimate of αl. The posterior reduction in area can be attributed to changes in two parameters. Firstly, there is a lower posterior probability that the pest has been established for a long period (Figure 3.1c), with the mode of φ approaching l. Secondly, the rate of spread has been conditioned towards lower values (Figure 3.1b). The joint distribution of these two parameters translates into a smaller posterior distribution of the radius of incursion (Figure

3.1d) and consequently, a large reduction in αl.

For each scenario, the distribution of district colonisation points, χ, is shown for those colonisation events that occur before l (Figure 3.2). Regions of lower posterior probability of colonisation occur close to the observation points. More recent absence data provides stronger evidence against colonisation in the vicinity of the observation point. This eﬀect is due to the relatively small window of opportunity for colonisation to have occurred in the vicinity before the pest would have arrived at the site by natural dispersal and been detected.

In the event of a detection, there is naturally a strong positive spatial association between the posterior distribution of χ and the detection point (Figure 3.2f). The shape of the posterior distribution of χ is then determined by the space-time arrangement of the earlier absence observations.

Observation parameters (β, γ, ω, η) register only small changes in the posterior estimates due to identiﬁability issues mentioned earlier. As would be expected, there is little diﬀerence between the prior and posterior estimates of exposure rate, λ, which, after ten years of absence data, remains close to one in ten years (Figure 3.1a). The posterior estimate of velocity of spread is weighted towards pests with characteristics for slow spread, these being less likely to be intercepted by a given surveillance program. It should be noted that the posterior distribution of υ, shown 52 CHAPTER 3. EARLY DETECTION SURVEILLANCE 14 0.30 12 10 0.20 8 6 Density 4 0.10 2 0 0.00

0.05 0.10 0.15 0.20 0.25 1 2 3 4 5

Exposure (rate / year) Spread velocity (km/year) a) b) 0.08 0.08 0.06 0.06 0.04 Density 0.04 0.02 0.02 0.00 0.00

0 10 20 30 40 50 60 0 10 20 30 40 50

Colonisation time (years) Radius of incursion (km) c) d)

Figure 3.1: Prior distributions (dashed) and posterior distributions (solid) of selected parameters from scenario A. a) exposure rate, λ b) velocity of spread, υ c) time of colonisation, φ d) radius of incursion at 10 years if colonisation had occurred, ρl. 3.3. SIMULATIONS 53

Figure 3.2: Posterior distribution of colonisation point, χ (grey), given that colonisation occurs before l, after ten years of observations with 20 plants examined at each site (black). a-e) Absence outcomes at all sites. f) Absence outcomes at ﬁrst 19 sites with a detection at l. The ﬁnal observation site for a,d-f has sN =(13.9,32.4). 54 CHAPTER 3. EARLY DETECTION SURVEILLANCE in Figure 3.1b, includes the 56% of MCMC samples from the distribution where φ > l and which therefore provide no conditional information about υ.

3.4 Early Detection Program

3.4.1 Surveillance data and methods

We analysed surveillance data for early detection of banana pests in a district covering a residential area of Cairns, Queensland between July 2003 and June 2008 (l=4.9 years). Data were collected from 272 sites on 326 occasions, with each site being visited between one and ﬁve times through the period. An average of 19 plants were examined on each occasion. To model the point of colonisation, χ, the district was gridded into 1 km × 1 km cells (sub-districts) and the number of properties in each sub-district retrieved from a cadastral database. Sub-districts containing fewer than ten properties were removed from the domain as these generally represent natural reserves or sugarcane plantations that contain few, if any, banana plants.

The proportion of the remaining properties falling in each sub-district, Rm : m = 1, 2,..., 191, provided a simple model for pest pathways into the district. The land area of the sub-districts, taking into account those overlapping the coast, is 187 km2.

A stochastic simulation model of sub-district colonisation and spread was developed to sample the prior distribution of φ and υ using the method described in Section 3.3.1. At each of 100 000 iterations, the colonisation event, χ, was assigned to sub-district m with categorical probability Rm. For each iteration, the radius of the infested area ρl was calculated. To restrict estimates of the area infested, αl, to within the irregular domain, a 100 m grid was constructed over the valid sub- districts. Those grid points that were intercepted by ρl were used to calculate

αl and to generate the empirical cumulative distribution function for the area of 3.4. EARLY DETECTION PROGRAM 55 infestation.

The prior spatial distribution of incursion status at Z(, l) was evaluated across a 100 m grid using iterations from the joint prior distribution [χ, φ, υ]. At each grid point, g, the mean of the simulated values, Z¯(g, l), was calculated as an estimate of the point probability of infestation Pr(Z(g, l) = 1). These values were then used to construct a contour map. Note that in contrast to the estimation of αl, the map has been allowed to extend outside the domain of the sub-districts to illustrate that the model does not accommodate for landscape heterogeneity during the spread process.

Values from the posterior distributions were generated in OpenBUGS from two chains run for 50 000 iterations after a 10 000 burn-in. OpenBUGS was chosen for this simulation after it was found to run the model in less than an hour rather than several hours in WinBUGS. The pseudo-code for the model is provided in Appendix A. Samples from the joint posterior distribution of [χ, φ, υ] were treated in the same manner as the prior simulation to estimate the posterior distribution of αl and the empirical cumulative distribution function. The joint posterior distribution was also used to contour map the posterior estimate of Z¯(, l) and Z¯(, t = 6.0), over the 100 m grid.

As the results of the comparison are highly dependent on the prior speciﬁcation, a sensitivity analysis was conducted over a range of prior parameter values. While the onus is on the managers of early detection programs to deﬁne the pest threat, ¯ ¯ we examined the sensitivity of Vl, αl and Z(, l) to the choice of parameters to characterise the changes in inference. Due to the large number of parameters in the model, our approach was to alter only one parameter at a time and note the impact on the estimates. 56 CHAPTER 3. EARLY DETECTION SURVEILLANCE

3.4.2 Results

The value of early detection surveillance program information was examined by comparing prior and posterior estimates of: the probability of district colonisation, ¯ Vl, the infested area if colonisation has occurred, αl, and the spatial status of the incursion at times of interest, Z¯(, t).

¯ The value of Vl represents the estimated probability that a target pest has established and that there is some unrealised cost associated with eradication. Table 3.3 ¯ shows that the posterior probability of Vl is around a quarter of the prior estimate. As there is little change in the posterior estimate of exposure rate, λ, the lower posterior probability of colonisation can be mostly attributed to stochastic outcomes of the exposure process.

Table 3.3: Summary of prior and posterior estimates of latent variables and parameters for early detection surveillance of bananas in the Cairns district.

Prior: mean Posterior: mean SD 0.025 0.975

Vl 0.38 0.090 0.286 0 1

ρl 7.26 1.261 1.065 0.05223 3.982

αl 68.03 5.957 9.208 0.01 30.65 φ 11.11 15.35 12.75 4.032 49.71 υ 2.75 2.69 1.3 0.60 4.88 λ 0.1 0.096 0.030 0.046 0.164 β 33.12 33.35 42.91 2.68 150.7

In addition to providing greater confidence in pest freedom, surveillance markedly reduces the estimated infested area if the district was colonised, αl. The combined effect of these estimates of Vl (y intercept) and αl are presented together in Figure 3.3 as an empirical cumulative distribution function of area infested. These demonstrate a substantial difference in the prior and posterior state of knowledge about incursion extent and therefore the putative eradication costs that have accrued. 3.4. EARLY DETECTION PROGRAM 57

Figure 3.3: Empirical cumulative distribution function for the prior (grey) and posterior (black) distribution of area of infestation over the domain of 187 km2. 58 CHAPTER 3. EARLY DETECTION SURVEILLANCE

Spatial estimates of the probability of colonisation events occurring in each sub- district are shown in Figure 3.4 along with contour maps of the prior and posterior estimates of the point probability of infestation. At time l, surveillance provides an order of magnitude reduction in the probability that any particular location is colonised. The reduction is a result of a shift towards estimates of later colonisation times in the posterior distribution and, to a lesser extent, a reduction in spread velocity. Ideally, a surveillance program would aim to produce a ﬂat posterior surface for the point probability of infestation. The spatial distribution of Z¯(, l) shows that areas around sub-districts with a high prior probability of colonisation retain a relatively high posterior probability of infestation even though they were intuitively targeted more heavily by surveillance.

As the probability distribution of incursion extent is deterministically described over time by [χ, φ, υ], it is a simple task to evaluate Z¯(, t) at any time. The posterior probability of infestation after six years is shown in Figure 3.4c. As surveillance data becomes older and no new data is obtained, continued exposure to pest pressure erodes conﬁdence that the invasion extent is small and manageable.

We report briefly on the impact of different prior specifications on the posterior estimates of pest extent in Table 3.4 and map the results in Figure 3.5. Doubling the ¯ prior exposure rate, λ, led to a roughly proportional increase in Vl. As the estimate of the area infested if the district was colonised was relatively unchanged, there was a fairly uniform doubling of the estimate of Z¯(, l). The pattern of surveillance deployment to address risks across the district would be expected to remain the same under reasonable changes in the prior assumptions.

Reducing the minimum velocity of spread, aυ, from 500 m to 50 m had negligible impact on the posterior distribution of extent. As expected, the estimate of the area infested given that it was colonised is lower but this is compensated for by a higher probability that the district is colonised. Doubling the maximum velocity of spread, bυ, to 10 km had a noticeable impact on the shape of the posterior estimate with a 3.4. EARLY DETECTION PROGRAM 59

Legend 120°0'0"E 140°0'0"E 160°0'0"E Pr. Infestation

Banana Surveillance 10°0'0"S Pr. Colonisation Event < 0.00001 0. 0.000010 5- 0.0001 0. 30°0'0"S 00 0.0001 - 0.001 2 0.001 - 0.005 0.005 - 0.01

0.1 Prior 2 Posterior 0 Posterior

0 Last Observation . at 6 Years 0 4.9 years

0 . 0 2 . 0

0 0 6

. 1 5 4 0 .0 . 0 1 3 .0 0 2 .0 0 0 . 0

2 0 1 0 . 0 . 0 0

0. 05 Kilometres 05 0 1 2 3 4 5 0. (a) (b) (c)

Figure 3.4: Prior (a) and posterior (b,c) distribution of sub-district colonisation events, χ, where t > φ, for banana pests targeted by an early detection surveillance program in Cairns, Queensland. (a,b) at times t = l = 4.9 years and (c) at t = 6.0 years with no further surveillance. Contours display the prior or posterior point probabilities of infestation, Pr(Z(, t) = 1), resulting from colonisation and subsequent spread. 60 CHAPTER 3. EARLY DETECTION SURVEILLANCE

Table 3.4: Sensitivity of probability of colonisation and estimated area of infestation if colonised. Scenario 1 is the original model. Prior values for other scenarios are shown only where they diﬀer from scenario 1.

Scenario prior values V¯l α¯l 1 λ(10,100) υ(0.5,5) β(3,1) η(0.75,0.9) σ2(0.1) 0.09(0.29) 6.0( 9.2) 2 λ(20,100) ...... 0.17(0.38) 6.1( 9.4) 3 . . . υ(0.05,5) ...... 0.11(0.31) 4.6( 8.0) 4 . . . υ(0.5,10) ...... 0.07(0.26) 11.9(17.6) 5 ...... β(3,1.4) ...... 0.10(0.29) 8.1(13.4) 6 ...... η(0.5,0.9) ... 0.09(0.29) 6.2( 9.5) 7 ...... η(0.5,0.7) ... 0.09(0.29) 6.2( 9.1) 8 ...... σ2(0.05) 0.09(0.28) 6.1( 9.2) 9 ...... σ2(2) 0.09(0.28) 5.5( 8.1)

more profound eﬀect in the outlying regions where there is a low prior probability of colonisation. However, at a program management level, the changes in estimates were not severe enough to alter the general patterns in deployment of surveillance resources.

The model is relatively insensitive to changes in the variance for the rate of change

2 in detectability, σβ. Doubling the variance caused a slight increase in the estimated posterior extent of the pest. Changing the prior range for maximum detectability η from 0.75-0.9 to 0.5-0.9 and 0.5-0.7 had little impact upon the estimate of extent. For this application, where the observability is a function of the number of plants examined, the probability of failing to detect given a maximum detectability of 0.5 or 0.9 for a single plant will be negligible.

Changes to the visit specific random effects, σ2, produced unexpected results. Where variation in detectability due to random effects is high, there is a slight decrease in the estimate of posterior extent. It was expected that the additional uncertainty at the visit level would absorb some of the information in the absence 3.5. DISCUSSION 61

0 .0 0 2 0 0.0 .004 02 0 .012 0 0 0 . . 0 0 . 0 0 0 8 6 0 1 . 0 0 6 8 00 0.

. 0

0 4 0 0 .

0 0

2 3 4

0 .0 02 0 0.0 .00 06 0 4 .0 0 0 0 0 . .0 0 8 .0 0 0 . 0 1 0 8 0 6 . 0 0 1 2 0 . 6 0 .00 0 0 4

. 0

2 0 0 . 0

0 .

0 5 7 9

Figure 3.5: Posterior point probability of infestation for the central part of the domain for sensitivity analysis scenarios. For comparison, the posterior for scenario 1 is displayed (grey dashed). Note diﬀerent contour interval on 2. data so that there would be a higher probability of recording a false absence. How- ever, by imposing more variation, it is a priori more likely that the product of the likelihood of missing the pest over all observations will be lower than if there was little variation. In the scenarios examined, these competing attributes lead to a model that is insensitive to the random eﬀects due to local environmental variation.

3.5 Discussion

Hierarchical Bayesian models can estimate invasion status over space and time when given prior information on the invasion processes and space-time referenced surveillance absence data. By interpreting estimates of incursion extent as a liability 62 CHAPTER 3. EARLY DETECTION SURVEILLANCE for eradication costs, typical early detection programs can demonstrate a substantial reduction in the estimated ﬁnancial liability for eradication. This is achieved through a lower probability that a district has been colonised and a much reduced estimate of the infested area within the district. By developing these models as routine management tools for early detection surveillance, decisions can be made on when and where to most eﬀectively deploy surveillance as new data arrive.

Inference on the posterior extent of an invasion is dependent on the prior information embedded in the model. Model results are sensitive to the selection of priors for exposure rates and spread rates but are consistent enough to deliver the same conclusions at a program management level. Furthermore, we feel that the process of defining this prior information with respect to the program goals provides a more rigorous foundation for implementing an early detection surveillance program. Once these goals are set, regulators can manage liabilities by weighing up the costs and benefits of conducting surveillance to address their a priori defined threats. While we have not carried out a specific cost analysis, this could be done on a simple areal basis (Bogich et al., 2008) or by more detailed spatial analysis of expected surveillance and response costs (Hauser and McCarthy, 2009).

Identifying the potential pathways for the introduction of exotic plant pests is critical for assessing the probability of colonisation (Colunga-Garcia et al., 2009). Here we assume no prior information about which particular property in a sub-district might be the point of introduction. Within any given district, additional site information could be incorporated into the prior, based on likely pathways for introduction (Hulme, 2006). When exposure pathways are poorly understood, additional uncertainty about the spatial exposure rates should be included (Stohlgren and Schnase, 2006). As pathway uncertainty increases, model estimates will suggest that surveillance resources need to be spread more evenly, and hence more thinly for a given cost, over the area of interest.

As with all models, inference should not be extended to ecological systems beyond 3.5. DISCUSSION 63 the specification of the model structure (Barry and Elith, 2006). Incursion spread rates are known to be influenced by population abundance and this can be particularly important immediately following introduction (Liebhold and Tobin, 2008). It is also common for species to have multiple dispersal mechanisms that operate at quite different scales (Hastings et al., 2005). Short and long distance natural dispersal, as well as human assisted modes, may be present in both insects and pathogens (Kawasaki et al., 2006). Pests of interest to early detection surveillance programs almost always have a human assisted pathway, either on propagating material, produce, or as hitch-hikers. As the area of an incursion increases, there will be a proportional increase in the potential for human assisted modes of spread. Without additional information on multi-modal dispersal, it can be difficult to identify the contribution of each process to the evolution of the incursion (Cook et al., 2007a). Our simple deterministic model of spread imposes a strong spatial connectivity between sites. Incorporating human assisted or long distance spread will break down spatial connectivity, resulting in a loss of inferential strength (Dybiec et al., 2004). That is, the model may fail to adequately predict the extent of an incursion. We acknowledge the inability to delimit extent as a very real issue for eradication campaigns (Panetta and Lawes, 2005). The model presented is limited to assimilating information about pests early in the incursion phase when they are locally dispersed by natural diffusive processes. Models with additional dispersal modes warrant attention to determine whether pests with these characteristics are eradicable and should therefore be of interest to early detection surveillance programs.

For a specific high priority pest, the model could easily be parameterised to evaluate a targeted early detection program. The population dynamics of many high priority pests have been researched in their native or invaded habitats, providing reasonable prior information on life history parameters. Incursion process parameters should however reflect the uncertainty about the pest’s ecology in a novel environment (Fitzpatrick et al., 2007). Upon detection of a particular pest by a non-specific 64 CHAPTER 3. EARLY DETECTION SURVEILLANCE early detection surveillance program, reparameterisation of the model should allow existing absence data to fit seamlessly into an incursion response. Incursion status maps could then be generated to direct surveillance for delimiting the pest extent. Where a high degree of uncertainty about invasion process and observation parameters exists, experimental work can be undertaken in conjunction with surveillance. Empirical studies that aim to reduce uncertainty in model parameters will strengthen inference on extent and aid management decisions.

The flexibility of hierarchical models to manage data and ecological knowledge from many sources makes them ideal for biosecurity applications where decision making is required in the presence of considerable uncertainty. In contrast to traditional design prevalence methods that set a fixed acceptable level of infestation (Cannon, 2002), the onus is shifted to one of defining uncertainty in the invasion and observation processes. Regulators and ecologists will have a range of views on the subjective model specification and parameter uncertainty, but these views can be reconciled by focusing on the management consequences of the models proposed (Clark et al., 2003). We argue that the hierarchical Bayesian approach provides greater insight into the ecology of potential invasions and more tangible inference to support the management of plant pest incursions. Perhaps most importantly, the hierarchical Bayesian framework offers a transparent, formal language that should encourage ecological experts and regulators to discuss their prior assumptions about incursion processes and management objectives for more effective early detection surveillance programs.

3.6 Summary

Early detection surveillance programs aim to ﬁnd invasions of exotic plant pests and diseases before they are too widespread to eradicate. However, the value of these programs can be diﬃcult to justify when no positive detections are made. 3.6. SUMMARY 65

To demonstrate the value of pest absence information provided by these programs, we use a hierarchical Bayesian framework to model estimates of incursion extent with and without surveillance. A model for the latent invasion process provides the baseline against which surveillance data are assessed. Ecological knowledge and pest management criteria are introduced into the model using informative priors for invasion parameters. Information from sparse presence absence data, collected over space and time, is assimilated using an observation model for imperfect detection to generate a posterior distribution of pest extent. When applied to an early detection program operating in Queensland, Australia, the framework demonstrates that this typical surveillance regime provides a modest reduction in the estimate that a surveyed district is infested. More importantly, the model suggests that early detection surveillance programs can provide a dramatic reduction in the putative area of incursion and therefore offer a substantial benefit to incursion management. By mapping spatial estimates of the point probability of infestation, the model identifies where future surveillance resources can be most effectively deployed. 66 CHAPTER 3. EARLY DETECTION SURVEILLANCE Chapter 4

Spiralling Whiteﬂy Observation Model and Natural Spread

The early detection surveillance model proposed in the previous chapter inferred the latent extent of an incursion given absence only data. Information about the potential extent of the pest was provided using informed priors for the parameters of a mechanical dispersal model. In Chapters 4 to 6, case studies of the spiralling white- ﬂy invasion are examined. In this chapter, a local scale model is used to estimate natural dispersal parameters. In the previous chapter, parameters that accounted for uncertainty in the observation process were not identiﬁable from the absence data. The provision of presence data, incorporated within a structured observation model containing covariates, allows detectability parameters to be estimated and used to develop surveillance strategies.

4.1 Introduction

On detection of a plant pest incursion, governments need to delimit the extent of the invasion to arrive at appropriate management decisions (Leung et al., 2010;

67 68 CHAPTER 4. WHITEFLY DETECTION AND NATURAL SPREAD

Panetta and Lawes, 2005). Intensive surveillance of host plants is employed to collect information on the extent of the incursion, and this information is interpreted in light of what is known of the pest ecology. An important aspect of delimiting an incursion is understanding the spread mechanisms of the pest. Most pests arrive through some human-mediated process (Colunga-Garcia et al., 2010a) followed by spread from natural processes and further human-mediated spread. While there can be considerable complexity in the evolution of an invasive process (Hastings et al., 2005), natural spread processes are commonly described by diﬀusion process at some spatio-temporal scale (Kareiva, 1983; Wikle, 2003). Estimates of natural spread for a well delimited incursion can help provide the time frame over which the human-mediated spread may have operated. This information is vital to the management of control programs for trace forward of potential pest movements to outside the known infested area.

Hierarchical modelling can assimilate information from both surveillance data and pest ecology by breaking down the analysis into an observation and an invasion process model (Cook et al., 2007a; Wikle, 2003; Wikle and Berliner, 2007). In eﬀect, the model distributes information from the surveillance data amongst the observation and invasion process parameters. Redistribution of this information can be directed through an ecological model structure, into parameters and hidden variables that have some pre-deﬁned uncertainty. This prior uncertainty can then be updated by the data to generate posterior estimates of the parameters for observation and invasion processes.

For a plant pest incursion, the observation model defines the relationship between the presence/absence data and the true pest status of the host plant. Many invasive plant pests have a wide host range, but usually have preferred hosts on which pest numbers can build up to obviously damaging levels. In terms of a delimiting surveillance program, absence information from favourable hosts provides stronger evidence of pest absence in the vicinity. In addition to the number of pests present on particular hosts, the architecture of different host species also has a bearing on 4.2. PEST INFORMATION AND SURVEILLANCE PROGRAM 69 the ability of an observer to detect a pest if it were present (Bulman et al., 1999; Gambley et al., 2009). When an incursion is detected, it is common for inspectors to be quickly trained in identifying an unfamiliar pest, before being sent out over a wide area to conduct delimiting surveillance. Despite the impact that a poor understanding of detectability may have on estimating population sizes (Delaney and Leung, 2010; Royle and Link, 2006; Wintle et al., 2005), there have been few attempts to incorporate the effectiveness of inspectors into the biases that may exist in visual inspection data (Fitzpatrick et al., 2009). By explicitly modelling the effect of different hosts and inspectors on the detectability of a pest within a hierarchical Bayesian framework, it is possible to accommodate uncertainty about these parameters into the predictions of extent, as well as to learn more about the values of these observation parameters as data are collected.

By framing hierarchical models in terms of parameters and latent variables that have meaningful biological or operational interpretations, priors for model parameters can be readily defined by expert opinions. These may be further tested empirically to strengthen the model inference (Kuhnert et al., 2010). Here we incorporate ecological information on population growth from the literature and inspectors’ expert knowledge about observations, into an analysis for spiralling whitefly, Aleurodicus dispersus, (SW), that invaded Cairns, Queensland, Australia in 1998. The effect of host plants and individual inspectors as observational components are analysed and the colonisation time and rate of natural spread of the pest estimated.

4.2 Pest Information and Surveillance Program

Spiralling whiteﬂy, Aleurodicus dispersus Russell (Hemiptera: Aleyrodidae), is a sap sucking pest with over 100 recorded host plants (Lambkin, 1999). Originally from the Caribbean, it now has an almost pan-tropical distribution (CABI, 2006) (see Chapter 6, Figure 6.1). Prior to its detection in Australia, the pest was reported 70 CHAPTER 4. WHITEFLY DETECTION AND NATURAL SPREAD to pose a threat to major commercial horticultural industries, including bananas, citrus, mangoes and a range of solanaceaous crops (Waterhouse and Norris, 1989). Spiralling whiteﬂy was known to be present in Queensland in the Torres Strait Islands in 1991, and Weipa on the Cape York Peninsula since 1995, before being detected 600 kilometres southeast in Cairns in March 1998. The incursion in Cairns was suspected to be due to accidental movement of the pest from these remote northern areas.

In response to the detection in Cairns, an intensive delimiting surveillance program was instigated to determine whether the pest could be eradicated. Most of the inspectors carrying out the survey had at least general insect surveillance skills and were given a brief training session to identify the pest. After the ﬁrst four weeks, intensive surveillance was wound down and continued with a small number of experienced inspectors. The program was supplemented by a strong community awareness program that led to public reports of many of the conﬁrmed new detections. Around 1 per cent of Cairns residents called to report the pest between March 1998 and February 1999. Further structured surveillance continued sporadically to identify further spread around the known infested area, and to support domestic quarantine restrictions on nursery plants in the area.

A subset of the spiralling whitefly data is examined from 18 March 1998 to 16 December 1998 for all observations within 2.5 km of the suspected initial incursion site in Cairns. During this period, there were a number of examples of suspected human-mediated dispersal to satellite areas outside of this domain. Data were selected based on the area that superficially presented an expanding contiguous range that did not overlap with any other satellite populations. While human- mediated dispersal within the spatio-temporal domain is likely, it is assumed that the effect on the estimate of natural spread is limited. The presence / absence data set contains 2640 records from 687 distinct sites and is presented in Figure 4.1.

Data consist of a presence/absence assessment, ysth, for each host, h, examined 4.2. PEST INFORMATION AND SURVEILLANCE PROGRAM 71

Month 1 Month 2 - 3

Legend

Suspected Entry Surveillance this period 0 0.5 1 2 Kilometres Positive Negative Previous Surveillance Positive Negative Roads 2.5 km

Month 3 - 6 Month 6 - 9

Figure 4.1: Spiralling whiteﬂy surveillance data for periods following the detection on 18 March 1998. 1/06/1999 72 CHAPTER 4. WHITEFLY DETECTION AND NATURAL SPREAD at a visit time, t, to a site, s. Some sites were visited more than once over the period. Sites were predominantly suburban yards and were spatially identiﬁed from a cadastral database with coordinates accurate to within approximately 50 m. The inspector conducting the visit is recorded, but the data do not contain any information on the search intensity at a visit. To reduce computation times, host type with fewer than 9 records were amalgamated into a “plants” group and some plant species were amalgamated at the genus level.

4.3 Models

4.3.1 Observation model

An observation model is first formulated for the probability of an inspector detecting a single spiraling whitefly on a plant. This is broken down into two components. The first describes the proportion of the plant inspected by an “ideal” inspector due to the architecture of the host plant, ηh, where h is an index of the host taxon. The second component is the individual inspector’s skill at detecting the pest, θi, where i is an index for the inspector at st so that the effective proportion of the plant examined is ηhθi. By framing the model around the detection of a single whitefly, prior estimates of detectability due to plant architecture were elicited. Experienced inspectors were asked to construct informed priors for the proportion of a plant searched for each plant taxa.

Priors for an inspector’s skill level, as a proportion of an ideal inspector, were included as a vague Beta(1,1) prior. Assuming that SW are distributed independently over the plant, the probability of detecting spiralling whiteﬂy on each host can be modelled in relation to the number of spiralling whiteﬂy, Nsth, on the host, 4.3. MODELS 73

Nsth ysth ∼ Bernoulli(1 − (1 − ηhθi) ). (4.1)

Next, the population relationship between spiralling whiteflies and their hosts is defined. Hosts are considered to be those plants that allow the development of a pest from egg to adult. Egg spirals can be present on non-hosts, and even non-plant material, particularly when infestation levels on host plants in the vicinity are high (Waterhouse and Norris, 1989). When spiralling whitefly is present at a site, it is not necessarily on all potential host taxa at the site.

The population on a host is modelled as an intensity,

Nsth = Nst × γh × sth (4.2)

where Nst, is the potential number of spiralling whiteﬂies on a host at the site, γh is a host-dependent coeﬃcient and sth is an error term. The potential population parameter Nst is based on the growth model proposed in the following section. Note that it is possible for Nst and Nsth to be less than one. A vague prior distribution is adopted for the host suitability, γh,

γh ∼ Beta(1, 1), (4.3)

To accommodate random population eﬀects in the vicinity of the site, as well as individual variation in suitability within the taxa, the prior for the error, sth is set to,

logit(sth) ∼ Normal(0, 1), (4.4) to provide for some reasonable uncertainty. 74 CHAPTER 4. WHITEFLY DETECTION AND NATURAL SPREAD

4.3.2 Growth rates

Growth of the pest population at the site increases the expression of detectable symptoms on plants over time. For a simple model of growth, it is assumed that all plants at a site are colonised at the same time and that the growth rate includes increases from immigration as well as reproduction within the site. Furthermore, it is assumed that there is a constant maximum intrinsic rate of increase, β, operating across the domain.

For an observation made at time, t, the period for which the site has been colonised is,

Cst = t − φs, (4.5)

where φs is the colonisation time for the site.

p The pest population on a plant at (s, t), is denoted as Nst and modelled as proportional to a carrying capacity K. A logistic growth model, implemented as an inverse logit function, deterministically models the pest population at some time after the site is colonised,  0 if C ≤ 0, p  st Nst = (4.6) −1 logit (βCst + logit(1/K)) otherwise,

p so that as Cst increases, Nst will approach one. To relate growth back to site population level at the observation time given in Equation 4.2, we use,

p Nst = Nst × 4K, (4.7) where the carrying capacity is multiplied by 4 to rescale for the prior mean of 0.5, (logit(0)), in Equation 4.4, and to reintroduce the male whiteﬂies excluded from the calculation of the prior for β that follows. 4.3. MODELS 75

Prior information about the parameters for spiralling whiteﬂy growth are obtained by modelling results taken from the literature. Wen et al. (1994), investigated the growth rates, fecundity and survival of spiralling whiteﬂy on Canna indica over a range of laboratory controlled temperatures. A summary of their data is reproduced in Table 4.1. Development from egg to adult takes 23 days at 26◦C with egg laying peaking 6 days later.

Table 4.1: Temperature dependent survival and fecundity rates for spiralling whiteﬂy from Wen et al. (1994).

Temperature ◦C 15 20 25 30 32

Mean eggs laid 17 25 28 19 10 SD eggs laid 9 12.5 14.5 11 7.5 Survival 0.21 0.46 0.65 0.38 0.08

A stochastic simulation model was constructed to sample reproductive rates across the range of temperatures experienced in Cairns during the study period. A sine function between the daily maximum and minimum temperatures recorded by the Bureau of Meteorology was used to generate a temperature proﬁle for the study area. The simulation model took into account that only females contribute to reproduction, but males are reinstated in Equation 4.7. The mean daily intrinsic rate of increase over the period was estimated to be about 0.03 based on the laboratory data above.

Translating growth rate estimates to the field must account for uncertainty about population responses to a range of factors that may positively or negatively impact upon the rate of increase (e.g. Peterson et al., 2009). Variation in temperature over the seasons, as well as spatial variation in microhabitats, will affect the reproductive success of the pest. Lower field survivorship is likely due to weather, predation or failure to select suitable hosts for oviposition. Spiralling whitefly populations are reported to be adversely affected by heavy rain (Aishwariya et al., 2007) and high humidity (Mani and Krishnamoorthy, 2000), both features of the tropical 76 CHAPTER 4. WHITEFLY DETECTION AND NATURAL SPREAD weather experienced around the time of the detection. As part of the response to the incursion, a parasitoid wasp, Encarsia sp., was released to control the population the second week after the initial detection. In addition, stochastic variability in reproductive success is likely to lead to lower mean rates of increase over the course of many generations, particularly in the presence of local extinctions (Foley, 2000).

To incorporate this uncertainty, prior information about the intrinsic rate of increase must relate to how the observation information is returned through the model. One of the assumptions of the observation model is that SW are distributed independently over the search area. In fact, eggs are usually laid in groups on a single leaf and so the population, in relation to the observation model, will be appreciably smaller. Given this uncertainty, we adopt a prior speciﬁcation for the growth rate parameter to be normally distributed,

β ∼ N(0.03, 0.01). (4.8)

4.3.3 Spread

The suspected point of colonisation, χ, in Cairns used for the analysis was based on anecdotal evidence gathered by regulators during the initial investigation into the incursion. The prior for the colonisation time for the domain was modelled relative to the initial detection time as,

φa ∼ Uniform(−365, −50). (4.9)

It is assumed that the colonising front moves at a constant rate of, υ m /day, but with some variation in the time of arrival at a site. A uniform prior for the velocity of spread was adopted, 4.3. MODELS 77

υ ∼ Uniform(3, 20). (4.10)

Site colonisation time is then modelled as,

φs = φa + ks − χk /υ + s, (4.11)

where s ∼ Normal(0, 20) denotes variation in the arrival time. The variation provided by s accommodates some of the landscape heterogeneity that could impede or promote transmission of the incursion.

4.3.4 Analysis

A Markov chain Monte Carlo (MCMC) model was coded in OpenBUGS and initialised using the built-in OpenBUGS routine. Convergence for each parameter was assessed based upon the mixing of two chains and the Gelman-Rubin statistic (Brooks and Gelman, 1998b). Due to the large number of parameters, the model required a long burn in period of 150 000 iterations before convergence was acceptable. There was a high degree of autocorrelation within the Markov chains of some parameters. To estimate parameters from a reasonable number of eﬀective samples, 100 000 iterations were used to generate the posterior distributions. Due to memory constraints, two small chains of 10 000 iterations were used to evaluate the additional variation at a site, s.

The code is provided in Appendix B. 78 CHAPTER 4. WHITEFLY DETECTION AND NATURAL SPREAD

4.4 Results

Prior densities and posterior densities of the global parameters, φa, β and υ were estimated by MCMC and are presented in Figure 4.2. Colonisation time and spread rate estimates are highly correlated, leading to long convergence times. The presence of autocorrelation within chains is evident in the posterior density of υ. 0.04 200 1.2 1.0 0.03 150 0.8 100 0.02 0.6 Density 0.4 50 0.01 0.2 0 0.0 0.00

−180 −160 −140 −120 0.015 0.020 0.025 0.030 0.035 0.040 0.045 5.5 6.0 6.5 7.0 7.5 8.0

Colonisation Time φa Growth Rate β Spread m/day υ

Figure 4.2: Estimates of global parameters (red dashed - prior, black solid - posterior).

The posterior mean estimate of growth rate, β, was slightly lower than the prior estimate of 0.03 obtained from the simulation model based on laboratory data, Table 4.2.

Table 4.2: Posterior estimates of global parameters for the incursion.

Node mean sd 0.025 median 0.975

φa -139.8 10.4 -161.6 -139.7 -120.5 β 0.026 0.0023 0.022 0.026 0.031 υ 6.78 0.31 6.17 6.78 7.39

There is little shrinkage in the estimate of detectability in the posterior distribution shown in Figure 4.3. The priors for detectability are formulated as the probability of a single whitefly being detected, while host suitability is modelled as a proportion 4.4. RESULTS 79 of the carrying capacity. Detectability and suitability are only partially identifiable through the different structures of the equations 4.1 and 4.2. Many of the data are collected from sites that are expected to have been occupied for a relatively long period. It appears from Figure 4.4 that most of the observation information is being assimilated into the estimate of host suitability due to the vague priors. 3.0 4 4 2 1.5 2 0 0 0.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 Acalypha Bauhinia Canna 3.0 3.0 1.5 1.5 1.5 0.0 0.0 0.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 Capsicum Carica papaya Citrus 3.0 2.0 3.0 1.5 1.0 1.5 0.0 0.0 0.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 Colocasia esculenta Euphorbia Euphorbia pulcherrima 40 1.5 1.5 0 0.0 0.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 Hibiscus Lycopersicon esculentum Mangifera indica 3.0 3 2 1.5 1.5 1 0 0.0 0.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 Manihot esculenta Morus nigra Musa 3.0 3.0 3.0 1.5 1.5 1.5 0.0 0.0 0.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 Plantae Plumeria Psidium 6 3.0 3.0 4 1.5 1.5 2 0 0.0 0.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 Rosa Solanum melongena Terminalia

Figure 4.3: Detectability, ηh, as the eﬀective proportion of hosts searched (red dashed - prior, black solid - posterior).

A better description of the search eﬀectiveness on diﬀerent hosts arises from the joint probability of the host-related detection parameters. In Figure 4.5, estimates 80 CHAPTER 4. WHITEFLY DETECTION AND NATURAL SPREAD 2.0 1.0 1.0 1.0 0.0 0.0 0.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 Acalypha Bauhinia Canna 2.0 4 4 2 2 1.0 0 0 0.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 Capsicum Carica papaya Citrus 3.0 3.0 1.0 1.5 1.5 0.0 0.0 0.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 Colocasia esculenta Euphorbia Euphorbia pulcherrima 3.0 3.0 1.0 1.5 1.5 0.0 0.0 0.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 Hibiscus Lycopersicon esculentum Mangifera indica 2.0 1.2 8 1.0 4 0.6 0 0.0 0.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 Manihot esculenta Morus nigra Musa 3 1.0 2 1.5 1 0 0.0 0.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 Plantae Plumeria Psidium 4 1.0 1.0 2 0 0.0 0.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 Rosa Solanum melongena Terminalia

Figure 4.4: Posterior estimates of suitability, γh, as a proportional rate of increase on host plants. 4.4. RESULTS 81 of the increase in the probability of detecting the pest on a host over time are presented. It should be noted that this ﬁgure does not include additional error variation due to host visit eﬀects, sth, which for display are set at the mean, 0.5. The model describes the increase in detectability as the number of pests in the vicinity approaches K. 0.8 0.8 0.8 0.4 0.4 0.4 0.0 0.0 0.0 0 50 100 150 200 0 50 100 150 200 0 50 100 150 200 Acalypha Bauhinia Canna 0.8 0.8 0.8 0.4 0.4 0.4 0.0 0.0 0.0 0 50 100 150 200 0 50 100 150 200 0 50 100 150 200 Capsicum Carica papaya Citrus 0.8 0.8 0.8 0.4 0.4 0.4 0.0 0.0 0.0 0 50 100 150 200 0 50 100 150 200 0 50 100 150 200 Colocasia esculenta Euphorbia Euphorbia pulcherrima 0.8 0.8 0.8 0.4 0.4 0.4 0.0 0.0 0.0 0 50 100 150 200 0 50 100 150 200 0 50 100 150 200 Hibiscus Lycopersicon esculentum Mangifera indica 0.8 0.8 0.8 0.4 0.4 0.4 0.0 0.0 0.0 0 50 100 150 200 0 50 100 150 200 0 50 100 150 200 Manihot esculenta Morus nigra Musa 0.8 0.8 0.8 0.4 0.4 0.4 0.0 0.0 0.0 0 50 100 150 200 0 50 100 150 200 0 50 100 150 200 Plantae Plumeria Psidium 0.8 0.8 0.8 0.4 0.4 0.4 0.0 0.0 0.0 0 50 100 150 200 0 50 100 150 200 0 50 100 150 200 Rosa Solanum melongena Terminalia

Figure 4.5: Estimated probability of detection on hosts at days after colonisation given the joint posterior probability of host detectability and suitability, inspector eﬀectiveness of 0.3 and intrinsic rate of increase of 0.026 (median - solid lines, 0.05 and 0.95 quantiles - grey dashed).

Inspector efficiency in relation to the “ideal” inspector is remarkably variable, as 82 CHAPTER 4. WHITEFLY DETECTION AND NATURAL SPREAD shown in Figure 4.6. The model structure weights the inspector’s efficiency in relation to the inspector who found SW at sites where detectability was expected to be low given the invasion model and the hosts inspected. Posterior estimates of efficiency for some inspectors remain vague, reflecting those people deployed to areas where the pest was not expected to be present and/or where only poor hosts were examined. 6 6 2.0 2.0 4 4 1.0 1.0 2 2 0 0 0.0 0.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 n= 103 n= 872 n= 70 n= 223 120 6 12 0.8 80 8 4 0.4 4 2 40 0 0 0 0.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 n= 127 n= 39 n= 32 n= 26 3.0 2.0 0.8 1.0 1.5 1.0 0.4 0.0 0.0 0.0 0.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 n= 135 n= 27 n= 54 n= 185 6 3.0 1.5 0.8 4 2.0 1.0 0.4 2 1.0 0.5 0 0.0 0.0 0.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 n= 40 n= 54 n= 36 n= 50 5 8 1.5 4 0.8 6 3 1.0 4 2 0.4 0.5 2 1 0 0 0.0 0.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 n= 33 n= 22 n= 61 n= 78 5 3.0 4 4 0.8 3 2.0 2 2 0.4 1.0 1 0 0 0.0 0.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 n= 57 n= 73 n= 101 n= 40

Figure 4.6: Estimated relative eﬀectiveness of selected inspectors θi. Only estimates for inspectors who surveyed more than 20 sites are shown.

Spread rates were estimated to be approximately 7 m a day, but it is also of interest 4.5. DISCUSSION 83 to examine the spatial variation in spread rates as captured by the colonisation time errors. Figure 4.7a displays a cluster of sites, (red), that were colonised earlier than predicted by the mean dispersal rate, υ. While this may suggest a human-mediated mechanism of spread, the “early” colonised sites are also quite close to some “late” colonised sites so this eﬀect appears inconsistent. Another feature of the estimated colonisation time errors is that late colonisation times are well represented close to the suspected initial colonisation point. These sites are typically those with absence data that are not reconciled by other observation model parameters. Therefore the model attempts to accommodate the absence information through a reduced period of colonisation.

In cases where the observation model provides strong information about the colonisation time of a site, variance in the error estimates is low (Figure 4.7b). Conse- quently, these observations provide most of the information for estimating spread velocity.

4.5 Discussion

The hierarchical Bayesian model provides estimates of spread and observation parameters given a large set of surveillance data collected from an invasion. The veracity of the estimates depends largely on how well the observation model and invasion model reflect the process of visual inspection and the underlying incursion dynamics. The model presented offers one analytical view of the informal wisdom that is typically used to interpret incursion data with respect to ecology (Plant Health Australia, 2010b). It is prudent to examine the results in relation to potential model inadequacies before using the estimates for making biosecurity decisions. This exploratory approach can suggest model modifications that could be adopted for more robust inference. 84 CHAPTER 4. WHITEFLY DETECTION AND NATURAL SPREAD

Legend Suspected Entry Colonisation time error MEDIAN -63 - -40 -40 - -14 -14 - -7 -7 - 7 7 - 14 14 - 31 Roads 2.5 km Qld

Legend Suspected Entry Colonisation time error SD 7- 11 11 - 14 14 - 17 17 - 21 Roads 2.5 km Qld

0 0.5 1 Kilometres b

Figure 4.7: Random eﬀects error in site colonisation time s. a) Median b) Standard deviation. 4.5. DISCUSSION 85

Understanding variable detectability in visual inspections is a critical part of surveillance (Fitzpatrick et al., 2009; Royle and Link, 2006; Tyre et al., 2003; Wintle et al., 2004). Hierarchical analysis implemented during an incursion can accommodate uncertainty about observations that may lead to false absence records. More importantly, it can estimate detection parameters for covariates such as hosts and inspectors that can be used to improve the information content of surveillance data for stronger inference. The advantage conferred by the spatio-temporal model structure is that it allows for the marginal posterior probabilities of parameters to be estimated. As information is analysed, the hosts targeted for surveillance can be altered in favour of those with higher detection probabilities and inspectors with low (or abnormally high) eﬀectiveness can be identiﬁed.

From a biosecurity operational point of view, the model provides evidence that the major horticultural crops, citrus, bananas (Musa) and mangoes (Mangifera indica) are relatively unsuitable hosts of SW. As model estimates are constrained to constant growth rates for a host over time, within taxa variability and temporal effects are not able to be identified. The model displayed poor convergence on frangipani (Plumeria sp.), possibly due to temporal differences in host status for this deciduous plant. Some host taxa contained species or cultivars known to have quite distinct suitability as spiralling whitefly hosts. For such taxa, mixture modelling may be a suitable option if host types are not distinguished in the covariate data. Zero- inflated binomial models offer one potential method to address variable responses (Hall, 2000; Martin et al., 2005b). By adopting a fixed rate of increase and carrying capacity, the expected probability of detection approaches one within a period of a couple of hundred days, even given the visit variability on the logit scale. An alternative modelling approach would be to include additional uncertainty using a beta-binomial probability of detection to evaluate the likelihood of the data. The analysis suggests some sources of variation that could be modelled differently, however, the operational decisions to be made are likely to be based on the ranking of parameters for hosts and inspectors. While the model specification could be further 86 CHAPTER 4. WHITEFLY DETECTION AND NATURAL SPREAD developed for more reliable estimates, the overall impact on the decisions is likely to be negligible.

Here, the invasion model is fixed on a spatio-temporal intercept for colonisation time and a coefficient for spread rate and therefore represents a stationary process. The estimate of colonisation time provides a useful starting point for investigating the source of the introduction and the period over which the pest may have been trans- ported elsewhere. Spread in the early stages of an incursion can be slow until the population builds up to a local carrying capacity (Frappier et al., 2003). Estimates of colonisation time based on constant spread rates should consider some additional period required for the population to establish. By incorporating a source of variability in the spread rate, spatial anomalies in the expression of spread through the study area were identified that suggest refinements to the model specification. To adequately predict spread over a broader area would require models that incorporate human-mediated spread, (e.g. Leung et al., 2004; Pysek et al., 2008) and the influence of landscape heterogeneity, (e.g. Pitt et al., 2009). As invasion processes are dynamic, in that the process is driven by the progression of dispersal events, more complex statistical models to incorporate spatio-temporal variability must be sought.

In developing a Bayesian model, the analyst must deal with the balance between the information content of the data and the expert opinion used to inform prior distributions for parameters (Kuhnert et al., 2010). Hierarchical models can include deterministic or stochastic modelling components that describe the relationships between parameters, latent variables and data. The model speciﬁcation itself is a form of prior information and the components should be scrutinised to determine if the ecological system is adequately represented in terms of the management decisions required. The modelling process itself should be integral to managing an incursion by providing a formal description of the system as it is understood at the time. Without incorporating this structure, the occurrence of events that are outside the realm of informal wisdom concerning the pest can go unnoticed. Tools 4.6. SUMMARY 87 to interpret our understanding of invasion over time should be employed during the early stages of an incursion while information is gathered (D’Evelyn et al., 2008). By approaching ecological modelling as an exploratory process and inference on parameters and latent variables as a progressive task, the full value of the surveillance data to delimit extent can be realised.

4.6 Summary

The early warning surveillance model has been extended to an application for a delimiting survey for an invading pest, spiralling whitefly. Further observational parameters of operational interest have been incorporated into the modelling framework. The model demonstrates how parameters for individual inspector efficiency, the ability to search different hosts and host suitability for population growth can be estimated so that they can be used to improve surveillance strategies. An invasion model that allows for spatial variability around a constant spread rate suggests that dynamic models for incursions are required for inference on extent at a local scale. The roles of ecological and statistical modelling in the context of the early stages of incursion management are discussed. 88 CHAPTER 4. WHITEFLY DETECTION AND NATURAL SPREAD Chapter 5

Hierarchical Bayesian Analysis of Plant Pest Invasions with Human-Mediated Dispersal

In Chapter 4, the local dynamics of spiralling whiteﬂy were investigated and the rate of natural spread estimated in continuous space and time. The application has a limited spatial and temporal scope once human-mediated dispersal begins to drive the progress of the incursion. As the model describes a stationary spatio-temporal process based on the district colonisation time and constant velocity of spread, it is not possible to extend its form to deal with a dynamic process. In this chapter, we examine a reliability framework to capture the dynamic nature of the spread process from potentially many dispersal mechanisms. The reliability framework imposes greater ecological and computational complexity, but also provides the ﬂexibility to adapt to more realistic incursion management scenarios.

89 90CHAPTER 5. HUMAN-MEDIATED DISPERSAL RELIABILITY ANALYSIS

5.1 Introduction

The distribution of an invading plant pest cannot be eﬀectively delimited by surveillance without understanding where the pest may have moved over time. While most pests can disperse by natural means, it is the human-mediated dispersal of invading organisms that can have the most profound impact on the rates of spread (Brock- mann et al., 2006; Cacho et al., 2010; Hastings et al., 2005). As regulation of human-mediated pathways is the primary tool available to contain pests, modelling approaches that can inform managers about these spread mechanisms oﬀer a valu- able contribution to invasion management (Colunga-Garcia et al., 2010b; Harwood et al., 2009; Venette et al., 2010; Yemshanov et al., 2010). To predict the intensity of spread along pathways, an underlying model for the dispersal mechanism (or mechanisms) is required to relate the invasion process to the observed spread pattern (Carrasco et al., 2010c).

Hierarchical Bayesian modelling provides a statistical framework, based on the conditional probabilities of components, to combine the observation process and the invasion process (Cressie et al., 2009; Hooten et al., 2007). By incorporating an internal ecological structure for processes such as spread and growth, hierarchical models provide the ﬂexibility to assimilate information from the pattern of spatio-temporal surveillance data to estimate parameters for a dynamic process. Ecological processes can be structured within the statistical framework through mechanical models (Fabre et al., 2010; Hooten and Wikle, 2010; Soubeyrand et al., 2009). Mechanical models used for naturally invading pests have drawn on a range of dispersal kernels including Gaussian, Cauchy and negative exponential distributions (Carrasco et al., 2010b,c; Clark et al., 2001; Leung et al., 2010). Gravity models provide another mechanical structure to model human-mediated movement of invading organisms between discrete locations at discrete time periods (Bossen- broek et al., 2007; Carrasco et al., 2010c; Leung et al., 2004). Further extensions to estimate colonisation times have been implemented using a geometric distribution 5.1. INTRODUCTION 91 of discrete time Bernoulli outcomes (Jerde and Lewis, 2007). Recently, Sahlin et al. (2010) used a Weibull hazard function to model colonisation times for crayﬁsh in lakes across Sweden. While analyses elsewhere have used maximum likelihood estimates, it has been suggested that Bayesian analyses of these problems may provide more stable estimates (Leung and Delaney, 2006).

Hierarchical Bayesian models for surveillance data, adequately positioned within space and time, allow the invasion process parameters to be derived from the invasion pattern. Bayesian applications have been applied to epidemic models for the transmission of foot and mouth disease by treating the connections between properties as time-inhomogenous Poisson processes (Jewell et al., 2008). Gibson et al. (2006) demonstrate a Bayesian approach to capturing emergent behaviour of a small scale invasion system based on a percolation model for plant disease within a field. Their approach models the difference between colonisation times of neighbours on a lattice as an exponential function and is implemented by evaluating the likelihoods of colonisation times drawn in order from potential infectious sources. Here a novel reliability analysis within a hierarchical Bayesian framework is developed to estimate the colonisation times of areas from multiple geographical sources, and potentially multiple dispersal mechanisms. The implementation of multiple dispersal mechanisms requires a large amount of discriminating information to identify the model parameters (Leung and Delaney, 2006). While the framework could potentially estimate parameters from multiple spread mechanisms, a gravity function is used to demonstrate the implementation of the model, with analysis performed by Markov chain Monte Carlo (MCMC) simulations. The model is motivated by the estimation of human-mediated spread parameters and the extent of a spiralling whitefly (SW, Aleurodicus dispersus) invasion in the Cairns district, Queensland, Australia from 1998 to 2002. 92CHAPTER 5. HUMAN-MEDIATED DISPERSAL RELIABILITY ANALYSIS

5.2 General Incursion Model

5.2.1 Overview

Our aim is to use space-time indexed surveillance data, Y, to estimate the hidden incursion process as a vector of colonisation times, φ, across an irregular lattice with M nodes. To predict future spread, we are also interested in the dispersal parameters, γ, for a mechanistic model that drives the space-time dynamics of the invasion system. Dispersal across the district is captured at a low resolution on a lattice of cells. Observation process parameters, θ, are not estimated, but used to incorporate uncertainty about the detectability of the pest. The general approach is to break down the observation and spread processes into a series of hierarchical conditional probability statements,

[φ, γ|Y] ∝ [Y|φ, θ][φ|γ][θ][γ]. (5.1)

The probability statements are structured around mechanistic models that describe components of the pest’s ecology. Parameters for the model are deﬁned by probability distributions that represent prior uncertainty about their true values. For some parameters, such as increase in detectability over time, evidence from experimental studies can be used to augment the data. Parameters for the dispersal function are estimated from the data using a statistical model overlaid on a mechanical model. The statistical model is analytically intractable but by employing MCMC simulation, the posterior distributions of the model parameters can be estimated.

The model consists of a district divided into geographical areas called cells, indexed

2 i = {1, 2,...,M}, where the cell, Ii ⊂ R , has an area of Ai. The cells may or may not be contiguous. Some cells that contribute to the invasion process may contain no observation data. Sites within a cell are inspected for pest presence / absence on a given date and their geographic coordinates recorded. While site level information at a high resolution is recorded, here the site records are aggregated 5.2. GENERAL INCURSION MODEL 93

into monthly totals of positive detections, Yim, out of Vim inspections for a cell so as to reduce computation time.

5.2.2 Reliability framework for colonisation

Each target cell of interest, i, could potentially be colonised by any of a combination of dispersal modes and geographical source areas. Each combination of dispersal type and source area can be termed a pathway, so that the modelled set of pathways for a target cell, i, is Si. Individual pathways from source to target cell are indexed,

Sij : j = 1, 2,...,Ji. For example, pests may establish in the target cell after being carried by people from a source that is outside the district, Ai, or by being carried from other cells in the district, Gi. Pests may also colonise cell i by natural spread from neighbouring cells in Ni. Interest is in estimating the colonisation time, φi, of each cell from all sources Si ∈ {Ai, Gi, Ni}, as well as the parameters for the diﬀerent spread functions. The propensity for establishment in a cell from each pathway is commonly termed the propagule pressure in the invasion literature (Lockwood et al., 2005; Simberloﬀ, 2009). While the model proposed is general, human-mediated spread pathways within a district are investigated here to demonstrate the model functionality.

Consider a single target cell exposed to propagule pressure from a single source cell. Adopting the terminology of survival or reliability analysis, the failure time (colonisation time) of a random variable, W , can be described by some probability density function, f(w). Common models for failure time are relative to a reference time, τ, which denotes the real time at which the exposure process initially applies. For example, if the cell is exposed to a constant propagule pressure from a single source, j, beginning at time τj = 0, a suitable model for colonisation time could be the exponential distribution fij(w) = λ exp(−λw). The cumulative distribution function for an exponential model is, Fij(w) = P (W ≤ w) = 1 − exp(−λw). The reliability function for a random variable is the probability that the cell is not 94CHAPTER 5. HUMAN-MEDIATED DISPERSAL RELIABILITY ANALYSIS

colonised by time W , and is therefore the complement of Fij(w), Rij(w) = P (W > w) = 1 − Fij(w). The instantaneous propensity to fail at time w, given that i has not already failed, is known as the hazard function,

hij(w) = fij(w)/Rij(w), (5.2) which, for the example, would be the constant, λ.

As the target cell i “fails” when it is ﬁrst colonised from any of j ∈ Si sources, a serial component failure model (Hamada et al., 2008, p. 146) can be adopted for the probability density of the time of colonisation, φi. The probability density function for the cell’s colonisation time from all pathways is thus,

J ( J ) Xi Yi fi(φi) = fij(φi) Ril(φi) (5.3) j=1 l6=j

Ji Ji Y X fij(φi) = R (φ ) ij i R (φ ) j=1 j=1 ij i J J Yi Xi = Rij(φi) hij(φi). (5.4) j=1 j=1

For computational reasons, Equation 5.4 is used. Given some functional form for the dispersal mechanisms between cells, the joint probability distribution of all cell colonisation times can be given by,

M ( J J ) Y Yi Xi f(φ) ∝ Rij(φi) hij(φi) . (5.5) i=1 j=1 j=1

Returning to the speciﬁc application, the propagule pressure model for a human- mediated mode of dispersal is developed for use within a hierarchical Bayesian reliability analysis. 5.2. GENERAL INCURSION MODEL 95

5.2.3 Gravity model

Gravity models have been used to express the invasion dynamics of pests that are moved by people between discrete habitat patches such as lakes (Leung and Delaney,

2006). By analogy to gravity, they consider a force, λij, to denote the propagule pressure between two masses ri and rj at Euclidean distance dij. Here the masses are considered to be the number of residential properties within the source cell, rj, and the target cell, ri. The standard gravity model can be given by,

−ω λij = ψdij rirj, (5.6) where the parameter ψ is the scaling factor and ω is the distance coeﬃcient. This function deterministically models the rate of movement of some material that could carry pests from j and i. Let λij provide the rate parameter for a Poisson process between a source cell, j, and a target cell, i.

Leung et al. (2004) recognised that the population density of zebra mussels in a source lake would aﬀect the propagule pressure to the target lake but did not implement a growth phase in their model. As the source cell may not be colonised, or may not be fully colonised, a variable, αj(t), is introduced to denote the proportion of cell j that is infested at time, t. After the source cell, j is colonised at φj, it is assumed that the pest spreads radially at velocity, υ, so that the area of j covered

2 at time, t, is π(υ(t − φj)) . Disregarding the shape of the cell, the time taken to reach the full extent of a circular cell of area Aj will be

q Ej = (Aj/π)/υ. (5.7)

Consider the proportion of the source cell infested at a potential colonisation time 96CHAPTER 5. HUMAN-MEDIATED DISPERSAL RELIABILITY ANALYSIS

for a target cell, φi, to be,  0 φ < φ ,  i j  2 αj(φi) = π(υ(φi − φj)) /Aj φj < φi < (φj + Ej), (5.8)   1 φi > (φj + Ej).

Consequently, the hazard function for the gravity neighbour, Gij, at some potential colonisation time, φi, will be,

Gij h (φi) = λijαj(φi). (5.9)

As there is a squared increase in the hazard function over time while the pest spreads in the source cell, a Weibull distribution with a shape parameter of a = 3 can be used to model the reliability function during this phase (Rinne, 2008). To calculate the Weibull scale parameter, bij, it can be seen that when the cell is completely covered at time φj + Ej, the hazard rate function will be λij. Therefore, the scale parameter will be,

2 1/3 bij = (3Ej /λij) . (5.10)

For the second case in Equation 5.8, the colonisation times will be distributed,

[φi − φj|bij, φj] ∼ Weibull(a, bij), (5.11) during the internal spread phase of the source cell.

The reliability function will be,

Gij 3 Rwei(φi) = exp (−((φi − φj)/bij) ), (5.12)

and will operate until the source cell reaches full capacity at time, φj + Ej, when it will be, 5.2. GENERAL INCURSION MODEL 97

Gij 3 Rwmax = exp (−(Ej/bij) ). (5.13)

For the third case in Equation 5.8, conditional on the cell, i, not being colonised previously, colonisation time will be distributed,

[φi − (φj + Ej)|λij, φj] ∼ Exponential(λij). (5.14)

The reliability function for Equation 5.14 is conditional on the colonisation process in Equation 5.11 not occurring. Therefore, the post-capacity reliability function after time φj + Ej is,

Gij Gij Rexp(φi) = Rwmax exp(−λij(φi − (φj + Ej))), (5.15) so that the full model becomes,

 1 φ < φ ,  i j  RGij (φ ) Gij (5.16) i Rwei(φi) φj < φi < (φj + Ej),   Gij Rexp(φi) φi > (φj + Ej).

Example plots of the probability density functions of these compound distributions are shown in Figure 5.1. The plots are characterised by an increasing probability of colonisation while spread is occurring within the source cell. For a particular gravity function, if spread within the source cell is relatively slow, there is a higher probability that a target cell will be colonised before the pest has reached its full extent (Figure 5.1 a). On reaching the full extent of the source cell, the hazard function becomes constant, leading to an exponential decline over the remaining proportion of the cumulative distribution function. 98CHAPTER 5. HUMAN-MEDIATED DISPERSAL RELIABILITY ANALYSIS 0.20 0.2 0.10 Density Density 0.0 0.00 0 2 4 6 8 10 0 2 4 6 8 10

a) Time b) Time 0.6 0.6 0.4 0.4 Density Density 0.2 0.2 0.0 0.0 0 2 4 6 8 10 0 2 4 6 8 10

c) Time d) Time

Figure 5.1: Combined Weibull (solid) and exponential (dashed) model of the distribution of time to colonisation from a unit cell j where λij = 0.7 and υ = (.08, 0.2, 0.7, 2). Dotted line displays

Ej, the period of colonisation at which source cell j is completely covered. 5.3. DATA AND OBSERVATION MODEL 99

The reliability model for human-mediated spread now provides the stochastic framework around which parameters for the mechanical gravity model can be estimated in conjunction with the latent invasion process. Information from the surveillance data can now be assimilated with the model using a hierarchical Bayesian analysis.

5.3 Data and Observation Model

5.3.1 Spiralling whiteﬂy data

Pest surveys were generally conducted on suburban residential properties, where a range of hosts were examined. Spiralling whitefly surveillance data were selected from the wet tropics coast surrounding Cairns from 18 March 1998 to 31 December 2002 (see Figure 5.2). The data consist of 10143 inspection records of which 1882 were positive. This period saw the spread of SW from Cairns through to Daintree and Cardwell. Surveillance was intensive for the first few months, particularly in areas around the initial detection site. After 2002, spiralling whitefly was present across most inspected areas so the data contain little further information on the incursion process of interest. As temperature is known to limit establishment and spread overseas (Cherry, 1979), analysis is restricted to the coastal plain where the July average minimum temperature is above 12◦C. Spiralling whitefly spread to towns further south during the study period but these have dissimilar climates and will be addressed in Chapter 6.

The district is broken up into hexagonal cells of 10 km2. As SW is not common in natural or agricultural areas, only those 53 cells within the district that had 100 or more properties were considered. A subset of 26 cells in the central Cairns area was selected to examine the impact of scale on parameter estimation.

Natural spread had been estimated in Chapter 4 to be around 7 m a day. Adopting this as a constant rate for internal spread, full coverage of each cell is expected 255 100CHAPTER 5. HUMAN-MEDIATED DISPERSAL RELIABILITY ANALYSIS

Legend Legend Mass First Detection Properties year 119 - 362 1998 Mossman 363 - 762 Mossman 1999 Port Douglas 763 - 2144 Port Douglas 2000 2145 - 4265 2001 4266 - 8852 2002 Major Roads Coastal Coastal Major Towns Major Towns Cairns Mareeba Mareeba

Gordonvale Gordonvale

Atherton Atherton Babinda Babinda

Innisfail Innisfail

Mission Beach Mission Beach Tully Tully

Kilometres Cardwell Cardwell 0 25 50 100 a.) b.)

Figure 5.2: a.) Study area with number of residential properties used to generate the mass values for the gravity model. The central Cairns cells are boxed. b.) Year that spiralling whiteﬂy was ﬁrst detected in cells. days after colonisation.

5.3.2 Observation model

Surveillance data have been condensed into monthly records of positive and negative outcomes recorded on hosts in each cell. For each cell, i, and month, m, there are

Vim visits to hosts, of which yim are positive. Some of these hosts may be at the same site or from sites close by. Here the observations yim are modelled as being selected at random from a point within the cell.

To retain consistency with the analysis from Chapter 4, the same model of growth is used but with several of the parameters previously estimated set as constants. 5.3. DATA AND OBSERVATION MODEL 101

The pest population, Nst, on a plant at a site, s, at time m, is modelled as a logistic function,

 0 if Csm ≤ 0, Nsm = (5.17) −1 logit (βCsm + logit(1/K)) × 2K × b otherwise.

where β is the rate of increase, Csm is the period that the site has been colonised and K is the carrying capacity which is set at a nominal value of 100 females. The posterior mean of the growth rate parameter was estimated in the previous chapter as 0.026 and is set here as a constant. The parameter b denotes the host type and other factors that may aﬀect population size and is implemented as a constant estimated from other studies. A constant detectability due to an inspector’s ability to inspect a whole plant, p = 0.3, is assumed so that the probability of detection at a site within a cell, Psm, will be,

Nsm Psm = 1 − (1 − p) . (5.18)

As all other parameters for Equations (5.17 & 5.18) are constants, the probability of detection at a site can be denoted as a function of the period for which the site has been colonised g(Csm) ≡ Psm.

The data for observation outcomes are aggregated spatially at the cell resolution and temporally into one month periods. Given a colonisation time for the cell, φi, the distribution of Csm at time, m, across the cell can be derived from the radial spread within a cell. The distribution has support over the range max(0, m−φi−Ei) to m − φi where Ei is the time taken to spread across the target cell (see Equation

5.7). The within-cell probability of detection, Pim, is therefore distributed,

Pim ∼ g(Csm). (5.19) 102CHAPTER 5. HUMAN-MEDIATED DISPERSAL RELIABILITY ANALYSIS

The observation data carry some further sources of uncertainty. There are examples of hidden over-reporting in the data. Detections based on public submissions provide positive records but ignore the implicit absence data. Given the large variation in detectability and reportability across the cell, we model detection uncertainty, η, using a Beta(2,2) distribution, so that the observation model becomes,

η ∼ Beta(2, 2) (5.20)

Qim = Pimη (5.21)

yim ∼ Binomial(Vim,Qim). (5.22)

The likelihood for the data given the joint distribution of observer uncertainty and population size uncertainty is obtained by numerically integrating Qim over the range of deterministically modelled population sizes and the plausible detectabilities for the cell. Note that this models the likelihood of all observations at a time, m, as being drawn from a single place within the cell at random.

Likelihood

Likelihood Likelihood

Days colonised Days colonised Days colonised

Beta Prob Det Beta Prob Det Beta Prob Det

Figure 5.3: Likelihood function for detectability yim|Qim,Vim given a Beta(2,2) distribution of observer detectability and population-based detection probabilities for potential site colonisation times across the cell. From left to right, the periods of cell colonisation m − φi are 30, 90, 365 with yim=10 and Vim=50. 5.4. SIMULATIONS AND COMPUTATION 103

A vague prior was adopted for the gravity model scaling parameter with,

log(ψ) ∼ N(0.1, 1), (5.23) to provide a plausible range of values. For two adjacent cells with average number of residential properties of 1500, the 5%, 50% and 95% quantiles of the prior distribution equate to expected colonisation events at 263, 50 and 10 days when the source cell is at capacity with a gravity model distance coeﬃcient, ω = 2 (Equation 5.6). The value of ω was ﬁxed to provide a distance-squared model for the initial analysis. For subsequent models, a non-informative prior for ω over the positive real line was used.

5.4 Simulations and Computation

Computing posterior estimates of φ, ψ and ω given Y and the other fixed parameters requires the exploration of a high dimensional space with a complex dependence structure between the individual φi. For initial models, the distance coefficient, ω, was fixed at 2. To determine the joint distribution of,

[ψ, φ|Y ] ∝ [Y |φ][φ|ψ][ψ], (5.24) the individual φi need to be conditioned on all previously colonised cells.

Two algorithms were developed, one to simulate independent blocks of colonisation times (SB) given the value of other parameters, and the other to individually propose colonisation times for each cell (PI).

Given a value of ψ, the simulated block algorithm (SB) progressively draws φi for all uncolonised cells, with simulated draws from the probability density function of the gravity model for all colonised cells. The cell with the earliest proposed colonisation time is then considered the next in the series to be colonised. The process continues until a block of colonisation times, φ∗, has been proposed. A Metropolis algorithm is then used on the likelihood of [Y |φ∗] and the block of φ∗ accepted or rejected. 104CHAPTER 5. HUMAN-MEDIATED DISPERSAL RELIABILITY ANALYSIS

∗ The alternative algorithm (PI) was developed to propose each φi individually and evaluate the likelihood of the block of φ∗ by using a Metropolis-Hastings algorithm to sample from [Y |φ][φ|ψ].

Both algorithms then proceed to use the Metropolis-Hastings algorithm to sample from the posterior distributions of other parameters.

Detailed algorithms for the MCMC simulation are given in Appendix C.

Models were developed using the R statistical software (R Development Core Team, 2008). Both algorithms were implemented on a small six cell data set to check for consistent results. For each algorithm, two MCMC chains were run with 10 000 burn-in iterations to overcome initialisation eﬀects and a further 50 000 iterations kept for analysis. For models consisting of more than 12 cells, the SB algorithm became stuck in local minima. To investigate the likelihood structure further, 100 000 values of φ were simulated for a range of set ψ and the likelihoods evaluated.

Due to the poor performance of SB on larger data sets, all further models were run using the PI method only. The MCMC model was run on data from both the 26 central cells and the 53 district cells. For each model run, two parallel chains were used with a 10 000 burn-in and 50 000 iterations kept as estimates of the posterior distribution. An extension to the MCMC algorithm above was made to also sample posterior estimates of ω using an additional Metropolis step. The central and district models were run again with the same number of iterations.

Convergence for all models was checked by visually inspecting the chains and calculating the Gelman-Rubin statistic (Brooks and Gelman, 1998b). 5.5. RESULTS 105

5.5 Results

5.5.1 Likelihoods

The likelihood structure shown in Figure 5.4 demonstrates the computational dif- ﬁculty faced by the model. Within the probability space of the joint distribution of [φ, ψ|Y ], there exist areas with very high likelihoods for some values of φ, but over all the potential realisation for a given ψ, they have a very high probability of being zero. When estimating within a Gibbs sampler, the model occasionally draws a realisation of φ with late colonisation times that have a high likelihood. The following MCMC step tends to draw a correspondingly low gravity model parameter of ψ. As the low gravity parameter encourages a block of late colonisation times to be drawn on subsequent iterations, it becomes unlikely that all φ draws precede the initial detection times on all cells. The MCMC simulation therefore becomes stuck in one local area of the high dimensional space for φ.

● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●

● 0.8 ● ● −560 ● ●

● 0.4 ● valid Pr. ● ● ●

Mean LL of valid ● ●

−620 ● ● ● ● 0.0

−4 −2 0 2 −4 −2 0 2

Log ψ Log ψ a) b)

Figure 5.4: a) Mean likelihoods of valid draws of 100 000 simulated realisations of the spread process given the observation data within the central cells. b) Probability of the draw being valid for a range of values of ψ. Note that the ﬁrst mean likelihood estimate is based upon only two valid draws in 100 000. 106CHAPTER 5. HUMAN-MEDIATED DISPERSAL RELIABILITY ANALYSIS

5.5.2 Comparison of algorithms

Models were run for each algorithm on 6 and 26 cell data sets to compare the estimation of φ and ψ. Posterior estimates for the colonisation times and the scaling factor for the six cell model, shown in Figure 5.5, demonstrate that the two algorithms produce similar results. Both algorithms converged but the simulated block algorithm displayed a high degree of autocorrelation due to low acceptance rates of around 3.6%. In contrast, acceptance rates for the PI algorithm could be tuned to around 25%. While the SB algorithm computation was 20% faster than the PI algorithm, the higher autocorrelation led to an eﬀective size of the samples for SB parameters of only 10% to 40% of the PI method.

Further runs of the two algorithms were conducted on the 26 central cells in the Cairns district. After the burn-in phase, the simulated block algorithm produced only 7 accepted values for φ in 200 000 iterations. All further results pertain to models run using the individual proposal algorithm.

5.5.3 Scalability of models

Posterior estimates of the scaling factor, ψ, for the central 26 cell model and the full 53 cell district model with ﬁxed ω are shown in Figure 5.6. There is almost a four-fold increase in the mean estimate of ψ between the central and district models. While the pattern of spread observed in the central model is best explained by low propagule pressure between nearby and heavily populated areas, the full district model requires high propagule pressure to explain the colonisation of some of the less populated, outlying areas.

Estimates of the colonisation times for the cells after adopting a non-informative prior distribution for ω are shown in Figure 5.7. Cells close to the initial colonisation cell in Whitﬁeld are frequently estimated to have been colonised, despite absence 5.5. RESULTS 107

White Rock Bayview Heights Lake Placid Caravonica 4 4 2.0 3 3 2 2 1.0 Density Density Density 1 1 0 0 0.0 1998.0 1999.0 2000.0 1998.0 1999.0 2000.0 1998.0 1999.0 2000.0

φ1 φ2 φ4 Holloways Beach Palm Cove Psi 3.0 12 2.0 2.0 8 1.0 Density Density Density 4 1.0 0 0.0 0.0 1998.0 1999.0 2000.0 1998.0 1999.0 2000.0 0.00 0.10 0.20

φ5 φ6 ψ

Figure 5.5: Comparison of the posterior estimates from the two algorithms, blue solid - simulated block φ, red dashed - individual proposal. Months with absence data are indicated by grey dashed lines while months with presence data are given by black dashed lines. Cell 3 was set as the initialising cell with a known time of colonisation. 40 0.08 ψ 20 Density 0.02 0

0 100 200 300 400 500 0.02 0.04 0.06 0.08

a) Index b) ψ 15 0.20 ψ 5 Density 0 0.05 0 100 200 300 400 500 0.05 0.10 0.15 0.20 0.25

c) Index d) ψ

Figure 5.6: Trace and posterior density for estimates of ψ for central cells (a, b) and the full set of district cells (c,d). Trace plots are thinned to show every 100th iteration. 108CHAPTER 5. HUMAN-MEDIATED DISPERSAL RELIABILITY ANALYSIS data being collected there for several months. In contrast, there is little information to ascertain the colonisation times of the cells further aﬁeld, as indicated by the wide credible intervals. Cells further from the initial colonisation point generally have lower numbers of residential properties that lead to low propagule pressure, λ, and therefore relatively ﬂat distributions from the gravity model. To compound their low information status from the gravity model, these cells lack surveillance data earlier in the incursion process that would have helped reduce the variance in posterior estimates.

Cell colonisation times are estimated from the likelihood of observing the data and the propagule pressures, λij, operating over the course of the incursion. It is therefore expected that the two gravity model parameters, the scaling factor, ψ, and the distance coefficient, ω, that provide λij are highly correlated. The marginal and joint distributions of these parameters, shown in Figure 5.8, demonstrate that estimates of the influence of distance on propagule pressure differ between the two sets of data. For the central cells, the reduction in propagule pressure over distance is estimated to be greater than for the district as a whole.

5.6 Discussion

Hierarchical Bayesian approaches to analysing invasions provide pest managers with the capacity to incorporate underlying invasion structure into a statistical framework (Cook et al., 2007a; Fabre et al., 2010; Gibson et al., 2006; Heisey et al., 2010; Wikle, 2003). The use of gravity models to provide a structure for human-mediated movement is particularly useful as it allows inference to be made on a major pathway for spread (Bossenbroek et al., 2001; Carrasco et al., 2010c; Jerde and Lewis, 2007; Leung et al., 2006). Understanding human-mediated movement is particularly important, not only because it can have the greatest impact on the progress of an incursion but also because these pathways are the most readily regulated (Carrasco 5.6. DISCUSSION 109

Daintree ● ●● ● Wonga ● ● ● ● Cooya Beach ● ● ● Mossman ●● ● ● ● ● Wharf Street Port Douglas ● ● ●● ● Bonnie Doon ● ● ● ● ● Davidson Street Port Douglas ●●● ●● ● ● ● ● Baler Street Port Douglas ● ● Palm Cove ●●●● ●● ● ● ● Kewarra Beach ●● ●●●●●●● ● ● ● ● ● ● Trinity Beach ●●● ●● ●●● ● ● ● Yorkeys Knob ●● ●●●●●●●● ● ● ● Hicks Close Kewarra Beach ●● ●● ● Dump Smithfield ●●●●●● ● ● ● ● ● ● Holloways Beach ●●● ● ● ● ● ● ● ● Langsat Close Smithfield ●● ● ● ● ● ● ● Machans Beach ●●●● ● ● ● ● ● ● ● Kalyan Close Caravonica ● ● ● ● Marett Street Stratford ● ●● ●● ●● ●● ●● ● ● ● ●● Lake Placid Caravonica ●●●●●● ● ● ● ● ● ● ● Aeroglen ● ●●●●●●● ●●● ● ● ● ● Freshwater ● ●●●●● ● ●● ● ● ● ● Whitfield ● ●●●●●● ●●●● ● ● ●● Parramatta Park ● ●●●●●●●●●●● ●● ● ● ● ● Crystal Cascades ●●●●●●● ● ● ● ● ● Kenny Street Portsmith ● ●●●●●● ●● ● ● Mooroobool ● ●●●●●● ●●●● ●●● ● ● ● ●● Redden Street Portsmith ●●●●●●●●●● ● ● ● ● Redlynch ●●● ● ● Bayview Heights ●●●●●●● ●●● ● ● ● ● White Rock ●●●● ● ● ● ● ● ● Mount Sheridan ●●●● ● ● ● ● Stewart Street Edmonton ●●● ● ● ● ● ● Noble Close Edmonton ●● ●●●●●● ●● ● ● Cairns Road Gordonvale ●●●●● ● ● ● ● ● Gordonvale 2 ● ● ● ● ● Gordonvale 1 ● ● Fishery Falls ● ● Deeral ● ● Bramston Beach ● ● Babinda ● ●● ● Nth Miriwinni ● Sth Miriwinni ● Webb Innsifail ● ● Goondi Bend Innisfail ● ● ● ● Mighell Innisfail ●● ● ● ●● Kurrimine ● El Arish ● Mission Beach ● Wongaling Beach ● South Mission ● ● Tully ● ● ● ● Cardwell ●● ● ● ●

1998 1999 2000 2001 2002

Year

Figure 5.7: Mean posterior estimate of colonisation time (black circle) and estimated 90% range of colonisation times from the individual proposal algorithm (lines). Cells are arranged roughly north to south. Positive observation data are displayed as red circles and absence records are displayed as green circles. The ﬁrst detection within the cell is indicated by a red triangle. 110CHAPTER 5. HUMAN-MEDIATED DISPERSAL RELIABILITY ANALYSIS 2.5 0.2 3.0 2.4

0.6 6 2.5

2.3 0.8

1.2 2.0 2.2 4 ω

1.4 1.5 Density Density 2.1

1 1.0 2.0 2

0.4 0.5 1.9 0 1.8 0.0

−5 −4 −3 −2 −1 0 0.0 0.2 0.4 0.6 0.8 1.0 1.8 2.0 2.2 2.4 2.6

a) log ψ b) ψ c) ω 2.0 4 100

1.9 0.6 80 3 1.8 1.2

1.4 60 ω

1.6 1.7 Density Density 2

1 40 1.6

0.8 1 20 1.5 0.4

0.2 0 0 1.4

−7 −6 −5 −4 −3 0.00 0.04 0.08 1.4 1.6 1.8 2.0 2.2

d) log ψ e) ψ f) ω

Figure 5.8: Posterior density estimates of gravity model parameters for central and district models. Central 26 cells - a) joint distribution of the distance coeﬃcient, ω, and the scaling factor, ψ. b) marginal distribution of ψ, c) marginal distribution of ω. District 53 cells - d) joint distribution of the distance coeﬃcient, ω, and the scaling factor, ψ. e) marginal distribution of ψ, f) marginal distribution of ω. 5.6. DISCUSSION 111 et al., 2010c).

The reliability framework lends itself to inference on multiple modes of dispersal, however its capacity to identify parameters for multiple dispersal components is yet to be tested. For any incursion pattern, there are many plausible permutations of the colonisation order of cells. The capacity for the reliability framework to infer multiple spread modes and parameters will depend largely on the complexity of the spread ecology. Strong inference may require strong prior information, radically different parametric forms for each component and/or large quantities of data. The interdependencies of colonisation times within the gravity process alone are computationally challenging. Without additional information on the spread processes, it would be difficult to identify the parameters for different modes of dispersal, or even the degree to which different modes are contributing (Cook et al., 2007a). Nevertheless, the model developed here provides a novel statistical framework for investigating multiple spread mechanisms that are critical to biosecurity management.

The results draw attention to the danger of ignoring the underlying invasion process when assessing pest absence based on surveillance data. Estimates of colonisation times identify a considerable number of cells that were likely to have contained false absence records. Given the assumptions of independence in the observation model and high prior uncertainty, the absence data are modelled as containing little information. In the absence of strong surveillance information, the model is instead driven by attempts to ﬁnd consistent parameter estimates for the gravity model that do not violate the presence data. An analysis of the model sensitivity to the diﬀerent sources of information is warranted. Despite this, the modelling approach does illustrate that area freedom estimation over space and time requires a cohesive process model to describe the invasion in progress.

Hierarchical Bayesian models, developed and implemented early in an incursion, would allow parameter estimates for the underlying invasion process to be used to 112CHAPTER 5. HUMAN-MEDIATED DISPERSAL RELIABILITY ANALYSIS infer the likely locations of pests. In this application, the reliability analysis on the central cells estimated much lower rates of spread than at the district scales. Predictive inference from gravity models based on smaller data sets therefore needs to be undertaken with caution. Differences in parameter estimates for the gravity model parameters may be due to inconsistent geographical patterns in the movement of the infested material. For example, if SW is being moved on host plants, there may be higher per capita rates of plant movement between more rural areas than in more densely populated city areas. Another alternative is that spread of the pest was initially inhibited by low population levels, or that temporal variability in population growth affected the observation and spread process in the first two years. Many sources of variation only become apparent as the ecology of an invasive pest is observed in its new environment (Simberloff, 2003). Hierarchical modelling of pest invasions as they are unfolding can provide a statistical platform to learn about these critical parameters for spread and to identify the sources of uncertainty that must be accommodated to arrive at sound management decisions (Epanchin-Niell and Hastings, 2010).

Analysis of the data was based on monthly aggregate figures within a cell as a whole. By collapsing the data, information on spread processes at a fine scale results in a loss of space-time information. Ideally, a finer scale model would be used but such large space-time data sets present computational difficulties. One solution, provided by Banerjee et al. (2008) and Latimer et al. (2009), is to use predictive process models to anchor the spatial process at a smaller number of points and use spatial prediction to link the latent process back to the sample locations. Such methods can be implemented as a fully specified statistical model for even large spatial data sets. The development of these techniques within a hierarchical Bayesian modelling environment warrants further attention to overcome the computational hurdles of invasion modelling.

A critical assumption of the model is that all of the information about the internal cell dynamics is encapsulated in the estimated colonisation time. The intensity of 5.7. SUMMARY 113 some of the gravity function connections suggest that multiple incursions into cells would be assured, and that these may have a considerable eﬀect on the observation model. In contrast, there may be considerable stochastic variability surrounding the colonisation process, including Allee eﬀects, that may result in local extinctions or long lead times until the populations become stable (Carrasco et al., 2010d; Dennis, 2002; Drake and Lodge, 2006). However, at a coarse scale, the model provides a useful starting point for discussing the invasion process within the limitations of the particular dispersal function chosen.

While there can be a long lead time to develop hierarchical Bayesian models for particular geographic, pest and biosecurity policy scenarios, there is value in biosecurity regulators adopting a modelling toolbox approach. Development of a suite of biosecurity focussed ecological process models to describe invasions would assist in the rapid development of statistical models that take advantage of the ﬂexibility of hierarchical approaches.

5.7 Summary

A novel hierarchical Bayesian model set in a reliability analysis framework has been developed to accommodate multiple dispersal modes and to estimate human- mediated dispersal parameters for an invasive species. Management of human- mediated dispersal is a key component to the containment of pests. The model infers the probability of colonisation times on discrete space and estimates connectivity parameters between discrete spatial units. Local natural spread within the spatial units is implicitly included in the model based on estimates from Chapter 4. The local scale spread describes the interaction with the observation process, while the large scale incursion dynamics are based on a gravity model structure. A novel approach to accommodate the local pest build-up phase is proposed by using a Weibull distribution modiﬁcation for the standard gravity model. A hierarchical 114CHAPTER 5. HUMAN-MEDIATED DISPERSAL RELIABILITY ANALYSIS

Bayesian model of the observation and ecological components captures the low resolution dynamics of the invasion. Two MCMC computational approaches are developed to estimate the complex joint distribution of colonisation times for a dynamical process. The modelling approach developed aligns well with biosecurity applications for regulating pest movement through trade. Chapter 6

Has the Spiralling Whiteﬂy Invasion Reached its Limit? A Hierarchical Bayesian Approach to Risk Analysis.

Hierarchical Bayesian models set in a reliability framework were demonstrated in Chapter 5 to be a promising tool for analysing dynamic invasion processes. In this chapter, the model is extended to a broader scale to estimate the ﬁnal extent of spiralling whiteﬂy in Queensland. A mechanistic model for temperature stress is developed to estimate climatic limits of the pest. Results are interpreted in terms of their impact on current biosecurity policy. The use of hierarchical Bayesian models as an analytical framework to progressively inform management decisions as data is gathered is examined.

115 116 CHAPTER 6. PREDICTING SPIRALLING WHITEFLY DISTRIBUTION

6.1 Introduction

Inferring the environmental limits of invading plant pests is a challenge for biosecurity regulators, both for defining areas at risk and for directing delimiting surveillance. Access to markets can be restricted if trade pathways exist between a potentially infested source area and a potentially suitable target area (IPPC, 2006). Climate matching, using software such as CLIMEX, is a commonly adopted first approach to predicting the potential extent of arthropod and disease pests (MacLeod et al., 2002; Sutherst and Bourne, 2009; Sutherst and Maywald, 1985). However, when pests enter a new environment, uncertainty about their ecology should be associated with the genetic provenance of the pest and unknown predator and environmental relationships (Fitzpatrick et al., 2007). This additional uncertainty makes it difficult to confidently translate extent from one region to another. Of- ten it can take many years before it is acknowledged that a pest has reached the extremities of its range. From a regulatory perspective, this can cause extended restrictions to markets for which there may be no significant risk.

Spiralling whiteﬂy, (Aleurodicus dispersus, Hemiptera: Aleyrodidae) entered Aus- tralia in 1991 and has now spread through most coastal tropical areas in the state of Queensland. Due to its wide host range across ornamental and horticulturally important plants, access to interstate markets has been restricted for some nursery stock and produce. Despite unrestricted intrastate trade, spiralling whiteﬂy (SW) does not appear to be colonising south east Queensland or cooler inland areas. If it could be established that SW has reached its climatic limits, it may be possible to relax trade restrictions based on the inability of the pest to establish in the export market area, or to certify pest free source areas where there is a negligible risk of the pest being present. 6.1. INTRODUCTION 117

Hierarchical Bayesian approaches to species distribution modelling have been demonstrated for purely spatial applications using conditional autoregressive models (La- timer et al., 2006). However, such models are difficult to implement for non- equilibrium systems, particularly for invasions that are dynamic spatio-temporal processes. Standard approaches using logistic regression to model the presence / absence data based on covariates are inadequate, as absence data may reflect yet to be invaded areas (Sutherst and Bourne, 2009). Without incorporating some dispersal measure into the analysis, it is difficult to meaningfully interpret absence data (Vaclavik and Meentemeyer, 2009). Mechanical-statistical modelling, or state-space modelling, allows quite complex ecological behaviour over space and time to be structured within the analyses (Buckland et al., 2007; Royle and Kery, 2007; Soubeyrand et al., 2009). Such models can impose an underlying spatio- temporal structure for the observation data. Stochastic spatio-temporal modelling approaches, when combined with Bayesian modelling, have been useful in predicting the distribution of invasive species (Cook et al., 2007a).

Two components to be addressed in developing a hierarchical Bayesian model are the development of the observation process model and the invasion process model (Wikle, 2003). Observation models relate the probability of surveillance outcomes to the latent state of the observation unit. At the site inspection scale, there can be considerable differences in the ability of inspectors to detect pests (Christy et al., 2010) as well as differences in the suitability of different hosts to support pests. When determining species boundaries, it is necessary to accommodate the poor detectability of species at low population levels. By modelling population growth relative to the colonisation time of the observation unit, Bayesian models can provide tractable solutions.

Hierarchical Bayesian invasion process models have addressed incursions on continuous discrete space (Wikle, 2003), but here we consider a discontinuous discrete space model for a pest spreading through human-mediated long distance dispersal. Whether long distance movement is associated with the movement of nursery stock 118 CHAPTER 6. PREDICTING SPIRALLING WHITEFLY DISTRIBUTION or simply from hitchhiking adults, gravity models provide a suitable description of exposure between one area and another (Bossenbroek et al., 2001; Leung et al., 2006; Leung and Delaney, 2006). In gravity models, movement rates between one area and another are considered to be a function of the population of the two areas and the distance apart. Such models that can accommodate the amount of ﬂow along pathways are ideally suited to biosecurity applications (Jerde and Lewis, 2007).

While pests may reach a location, they may not establish a permanent population if environmental conditions are unfavourable. A hierarchical approach to analysing the data can be adopted to formalise the relationship between surveillance, climate- dependent population growth and the rate of exposure between towns. Here, a hierarchical Bayesian statistical model is applied to a mechanical process model of a human-mediated invasion to allow both extent and other parameters of interest to be estimated. The model draws on information gained about movement between towns in the favourable part of the range, to learn about temperature limits in another part of the range. After reviewing the distribution of spiralling whiteﬂy globally and in Australia, a mechanical model for the stress related population growth and dispersal is proposed in Section 6.3. The hierarchical Bayesian statistical model for observation and the invasion process is then developed in Section 6.4. The value of hierarchical Bayesian analysis as a modelling tool for estimating the risk of conducting trade is discussed.

6.2 Distribution and Data

6.2.1 Distribution of spiralling whiteﬂy

Before embarking on the modelling process, it is fruitful to investigate the range limiting factors elsewhere in the world. Spiralling whiteﬂy, a native of Central 6.2. DISTRIBUTION AND DATA 119

America and the Caribbean, has spread through the Paciﬁc Islands (1978-) to Asia (1982-1987), Africa (1992) and Papua New Guinea (1987). Its global distribution is restricted to the tropics and some subtropical coastal climates such as the southern part of Florida in the U.S.A., the west African Macaronesian islands and Taiwan (Figure 6.1).

47°0'0"N 47°0'0"N

23°30'0"N 23°30'0"N

0°0'0" 0°0'0"

23°30'0"S 23°30'0"S

47°0'0"S 47°0'0"S Legend

Detected

Not Detected

Figure 6.1: Countries where spiralling whiteﬂy has been detected. Administrative regions within some countries are shown when documented. Source CABI (2006); Monteiro et al. (2005). Personal communications (J.H. Martin, 2008, B.M. Waterhouse, 2008).

Climate matching has a well established history in biosecurity applications (Elith and Leathwick, 2009; MacLeod et al., 2002; Sutherst, 1991; Sutherst and May- wald, 1985). To estimate the potential range of exotic pests across continents, software such as CLIMEX can translate temperature and other meteorological variables into environmental favourability indices. Given the near global length of the discriminating meteorological boundary available for SW, it would be expected that a reasonable estimate of the spiralling whitefly range in Australia could be obtained. However, as the range fitting process is manual, the choice and coefficients of meteorological parameters is open to interpretation which can lead to differing results. CLIMEX models to estimate the potential distribution of SW were developed independently by the Department of Agriculture Western Australia, Stefano 120 CHAPTER 6. PREDICTING SPIRALLING WHITEFLY DISTRIBUTION

De Faveri from the Department of Employment, Economic Development and In- novation in Queensland and the Commonwealth Scientiﬁc and Industrial Research Organisation. Two of these models return similar predictions for establishment across Northern Australia (from Canarvon in Western Australia to Gladstone in Queensland) but the other predicted a range as far south as Coﬀs Harbour in New South Wales.

Spiralling whitefly has been known from southern Florida since 1957. Data on collections of spiralling whitefly in Florida were provided by the Florida State Col- lection of Arthropods. A map of the distribution is shown in Figure 6.2, superim- posed on a climate surface generated using the ArcView, Kriging interpolation tool on temperature data from 106 sites in Florida, generated from the U.S. Climate Normals. There are only single records of collection in Orange, Osceola and Levy, with the Levy collection being taken from a nursery plant recently imported from southern Florida. In Fort Lauderdale, Florida, summer survival in field trials averaged 63% and winter survival averaged 20%, though in January and February only 1 in 251 eggs survived to 4th instar (Cherry, 1979). In 50 years, it has only been collected rarely outside the southern counties, supporting Cherry’s proposition that cold stress would limit the expansion outside of southern Florida.

In the Canary Islands, SW has been present since 1965 and has not been found at cooler altitudes over 500m above sea level despite the presence of suitable hosts (E. Hernandez-Suarez, pers. comm. 2008). In Hawaii, it has only been found in low numbers and, after the introduction of a parasitoid, at elevations above 120 m (Kumashiro et al., 1983). In the twenty years since its detection there, it has rarely been found at elevations over 350 m (B. Kumashiro pers. comm. 2008).

Environmental effects other than cold temperature limits are reported to affect SW populations. Heavy rain is suspected to cause a temporary reduction in population on Pacific Islands (Waterhouse and Norris, 1989). Ramani et al. (2002), in a review of spiralling whitefly in India, reports a variable response of population size to 6.2. DISTRIBUTION AND DATA 121

Figure 6.2: Spiralling whiteﬂy records from Florida State Collection of Arthropods in relation to January average minimum monthly temperature records.

weather in diﬀerent regions with some areas having high populations in summer and others in winter or after the rainy season (Palaniswami et al., 1995). These additional covariates could be included into a process model if further reﬁnement was needed.

6.2.2 Surveillance for spiralling whiteﬂy in Queensland

Spiralling whiteﬂy was ﬁrst detected in Australia in the Torres Strait (1991) and spread to the Northern tip of the Cape York Peninsula (1995), Weipa (1997) and Cairns (1998). By 2007, it was present in most coastal towns south to Bundaberg. 122 CHAPTER 6. PREDICTING SPIRALLING WHITEFLY DISTRIBUTION

The pest was ﬁrst detected in Darwin, Northern Territory in March 2006.

Surveillance for spiralling whitefly was conducted at varying intensities from 1998 to 2010 across Queensland. For the first year, surveillance was directed at delimiting the extent of the Cairns incursion to validate the integrity of a quarantine zone. By this time it became apparent that long-distance dispersal was occurring and that the pest was only causing superficial damage to commercial hosts. Effort was therefore reduced to opportunistic surveillance during plant health surveys, responding to community reports and delimiting incursions in new areas. Many of the detections in new areas were notifications by the general public. Targeted surveillance to verify the boundaries of the invasion was undertaken in 2008 and 2010 during February - April when populations would be expected to peak. A map of the presence / absence outcomes of surveillance for SW in Queensland over the study period is shown in Figure 6.3a.

6.2.3 Data

Formal surveillance was conducted by plant health inspectors and scientiﬁc staﬀ from Biosecurity Queensland. At each site, the geographic coordinates, date, host plants present and presence / absence of SW was recorded. Search intensity at each site was not recorded although the number of host plants examined was recorded during later surveys.

For the analysis, a selection of 19 major built-up zones, from climatically different areas have been chosen. Zones were chosen to represent major transport points, areas with surveillance data from a variety of potentially limiting climates and areas that have some significance for market access. Details about the zones are given in Table 6.1 and their locations are shown in Figure 6.3b. A 10 km by 10 km grid was placed over the state of Queensland in ArcMap using a transverse Mercator projection from the Map Grid of Australia Zone 55 using Geodetic Datum 6.2. DISTRIBUTION AND DATA 123 a.) Presence and absence records of spiralling whitefly in Queensland in relation to minimum July temperatures b.) Zones used for the climate model. Figure 6.3: 124 CHAPTER 6. PREDICTING SPIRALLING WHITEFLY DISTRIBUTION of Australia 1994. The number of land parcels in each zone were counted from the Queensland Digital Cadastral Database, June 2004. As the vast majority of the land parcels in the zones are in built up areas, we refer to these as residential properties for convenience. Interpolated daily temperature data for the centroids of zones were retrieved from the Queensland Climate Change Centre of Excellence data drill set (http://www.longpaddock.qld.gov.au/silo), (Jeffrey et al., 2001).

Individual spiralling whiteﬂy observation records were aggregated into monthly counts of positive records and total number of observations. Excluding the Cairns zone, there are 259 combinations of month of survey and zone comprising 604 presence and 6427 absence records from 1878 distinct sites.

6.3 Mechanistic Models

6.3.1 Temperature stress on growth

Mortality due to temperature stress can be modelled as a function of the time and degree of exposure below critical temperatures using a cosine curve between daily maximum and minimum temperatures (Sutherst and Maywald, 1985). Ideally, the real time weather stresses to which individual cohorts are exposed would be used to model survival and growth (Yonow et al., 2004). Here, a model is proposed that is based on the average daily stress accumulation over the study period.

The model calculates the stress accrued when insects are exposed to temperatures below a threshold, τ, and translates this to a reduced rate of pest increase within the zone. Using standard meteorological temperature data for maximum, Tmax, and minimum, Tmin, we calculate the mean temperature Tm = (Tmax + Tmin)/2 and the half range of the temperature Ta = Tmax − Tm. Temperature is assumed to follow a cosine curve between the maximum and minimum temperature for a half day, 6.3. MECHANISTIC MODELS 125

Table 6.1: Details of the zones, arranged roughly north to south. Date of ﬁrst detection is the date of ﬁrst collection from the zone, followed by whether it was a public report or a structured survey. Missing values indicate the pest is not known to occur. The total number of observations on all hosts and visits is given for the study period.

Zone Name Residential Area (km2) 1st by Number Properties Detected Observations 1 Cairns 62973 670 19-Mar-98 Public 10728 2 Mareeba 4341 200 08-Jun-00 Survey 416 3 Atherton 4648 200 476 4 Innisfail 7375 200 22-Mar-00 Survey 371 5 Ingham 3256 100 18-Feb-04 Survey 28 6 Townsville 58653 530 30-Jun-99 Public 2196 7 Burdekin 8102 300 26-Aug-05 Survey 25 8 Charters Towers 4793 200 02-May-06 Public 1 9 Bowen 4693 100 18-Jun-04 Public 20 10 Whitsunday 8084 250 03-Mar-02 Survey 225 11 Mackay 29805 200 15-Apr-02 Public 1302 12 Yeppoon 9443 72 28-Apr-08 Survey 40 13 Rockhampton 33952 400 57 14 Gladstone 17767 200 07-Jun-06 Public 96 15 Bundaberg 33541 450 02-Apr-08 Survey 287 16 Maryborough 14009 200 59 17 Hervey Bay 21245 250 91 18 Nambour 91677 1000 964 19 Brisbane 600235 1700 377 126 CHAPTER 6. PREDICTING SPIRALLING WHITEFLY DISTRIBUTION

Tt = Tm + Ta cos(u). (6.1) where u is the proportion of the half day elapsed (i.e. scaled to 0 to π). For a given temperature threshold, the time of day intercept, c, will be,

 0 τ ≤ T  min  −1 c = cos ((τ − Tm)/Ta) Tmin < τ < Tmax (6.2)   π τ ≥ Tmax.

Stress, S, is calculated by integrating Equation 6.1 to obtain the area of the curve under τ,

S = (τ − Tm)(π − c) + Ta sin(c))/(2π). (6.3)

The stress function is illustrated in Figure 6.4 and the daily temperature stresses calculated for all of the zones over the 13 years of the study period are provided in Figure 6.5.

Consider a zone, i, to be colonised at time, φi. In the absence of any environmental stress, let the maximum intrinsic rate of increase be β. The zone speciﬁc temperature stress, Si is calculated as the mean of the stresses from Equation 6.3 based on the daily temperature data for each zone over the whole study period. For a mean temperature stress, Si, the intrinsic rate of increase, βi, for the ith zone is modelled as,

βi = β − δ × Si, (6.4) where δ is a coeﬃcient for the rate of accumulation of stress. 6.3. MECHANISTIC MODELS 127 20 15

S= 0.64 Temperature 10

5 c

0.0 0.1 0.2 0.3 0.4 0.5

Day

Figure 6.4: Half day measure of cold temperature stress accumulation for τ=10◦C as the area below a threshold of a cos curve between the maximum and minimum temperature.

These zone growth rates, βi, will have an impact on the detectability of the pest at any particular site within a zone given the colonisation time of the site as well as the rate of spread within a zone.

6.3.2 Movement within zones

Spiralling whitefly nymphs are sedentary and adults are poor fliers that generally restrict flights to times when the wind is light (Waterhouse and Norris, 1989). Natural movement of the spiralling whitefly front through the urban environment of Cairns was previously estimated in Chapter 4 to be around 7 m a day. However, dispersal to suburbs several kilometres outside the contiguous range was apparent early in the surveillance program, suggesting a strong role for human-mediated dispersal in the local spread dynamics.

Human-mediated dispersal of whiteﬂies is usually associated with the movement of infested plant material (Cacliagi, 2007). However, movement of adult SW on vehicles and ships was considered a mode of spread within Hawaii (Waterhouse 128 CHAPTER 6. PREDICTING SPIRALLING WHITEFLY DISTRIBUTION

◦ Figure 6.5: Daily temperature stresses, Si, given a temperature threshold, τ = 10 C for centroids of each zone from January 1997 to June 2010. 6.3. MECHANISTIC MODELS 129 and Norris, 1989) and in west Africa where spread was noted mainly along roads (M’Boob and Van Oers, 1994; Neuenschwander, 1994). Further anecdotal evidence from the Queensland incursion suggests that some infestations in new towns were associated with vehicle movement rather than infested plants.

At the within-zone scale, a model of movement is adopted that is functionally similar to diﬀusive spread in relation to the observation data used. From reaction- diﬀusion models, it is known that the rate of areal increase will be proportional to the intrinsic rate of increase (Kot et al., 1996). Based on the results in Chapter 5 for the Cairns zone, a maximum velocity through a zone that accounts for natural spread when augmented by human-mediated movement, υ, was estimated to be 20 m/day. While human-mediated spread is implicitly included within a zone, it is still assumed that the proportional relationship between areal increase and the intrinsic rate of increase for the zone, βi, is retained in the context of the data model presented in Section 6.4.1. Following the approach of Pitt et al. (2009), consider that the zone is circular with known area, Ai, so that for a zone, i, colonised at time φi, the proportion of the zone covered at time, t, will be,

∗ 2 βi π (υ(t − φi)) αit = (6.5) βAi where αit = min(1, αit) and αit = 0 if t < φi. To prevent negative spread rates, ∗ βi = max(βi, 0).

Spread at a constant velocity will result in a zone colonised at φi being completely covered at time φi + Ei where,

p Ai/πβ Ei = . (6.6) υβi

A number of simplifying ecological and modelling assumptions have been made about spread rates within a zone and the mode of stress accumulation. However, it 130 CHAPTER 6. PREDICTING SPIRALLING WHITEFLY DISTRIBUTION is worth noting that their impact on the ﬁnal inference will be somewhat ameliorated by the data in the statistical model. The relationship between the spread within a zone and the observation model is developed later in Section 6.4.1.

6.3.3 Movement between zones

The development of the gravity model is given in the previous chapter in Section 5.2.3 but is repeated here for completeness.

Gravity models consider the connection between two areas to be analogous to gravity, where a force, λij, exists between two masses ri and rj at distance dij. These models have been applied to human-mediated pest movement between areas in discrete space and time (Bossenbroek et al., 2001; Leung et al., 2004). Assuming that movement of whiteﬂies is proportional to the human population in zones and mod- erated by the distance between zones, we let the mass of a zone, ri, be the number of residential properties in zone i. The expected propagule pressure between a source zone, j, and a target zone, i, at some time t can be deterministically modelled as,

−ω λijt = ψdij αjtrirj. (6.7)

The scaling factor, ψ, and the distance coeﬃcient, ω, describe the intensity of movement of potentially infested material between the zones. Note that an additional variable, αjt, is introduced to the standard gravity model to reﬂect the proportion of the area infested in the source zone j at time t. For M zones, the total propagule pressure into i from all sources is,

M Z t X −ω λit = ψ dij rirj αjt dt (6.8) j=1,i6=j 0

While Equation 6.8 provides a model for introductions into i, the full model requires the instantaneous pressure into a zone from all potential source zones. Considering 6.3. MECHANISTIC MODELS 131 introduction of a pest to be a Poisson process, colonisation time can be modelled for a constant propagule pressure, λij, using an exponential distribution. However, here we are dealing with propagule pressure that changes over time from each source zone.

Reliability analysis provides a solution to modelling the colonisation times in this dynamic setting. For pressure emanating from a single zone that has reached its full extent at time Ej, the probability density function for colonisation times will be,

E φi − Ej ∼ Exponential(λij); φi ≥ Ej, (6.9)

E where λij is Equation 6.7 with αjt = 1.

While the pest is still spreading within source zone j, the area covered is propor-

2 tional to (t−φj) . It follows that the hazard function, rather than being constant as for the exponential distribution, will increase as a squared function of time. For a propagule pressure with a power coeﬃcient of 2, a Weibull distribution with shape parameter 3 provides a compatible extension to the exponential model (Rinne,

2008). The scale parameter of the Weibull distribution, bij, will be,

2 E 1/3 bij = (3Ej /λij) , (6.10) so that,

φi − φj ∼ Weibull(3, bij); φi < Ej. (6.11)

We now need to consider the colonisation time, conditional on all sources operating simultaneously. Using a reliability framework (Hamada et al., 2008), the hazard function, hij, for instantaneous propagule pressure from the source zones will be the probability density functions of Equations 6.9 and 6.11 divided by their reliability functions. Reliability functions quantify the probability that the target zone has not already been colonised by one of the other source zones. These reliability functions,

Rij, are the complement of the cumulative distribution functions. It was shown in 132 CHAPTER 6. PREDICTING SPIRALLING WHITEFLY DISTRIBUTION

Section 5.2.2 that the probability of colonisation of zone, i, at time, φi, from any of, Ji, potential source zones can be expressed by,

J J Yi Xi p(φi) = Rij(φi) hij(φi). (6.12) j=1 j=1

The joint distribution of colonisation times over all zones is therefore,

M ( J J ) Y Yi Xi p(φ) ∝ Rij(φi) hij(φi) . (6.13) i=1 j=1 j=1

The mechanical model of spread and growth processes for the pest, both within and between zones, contains deterministic and stochastic components. Taken together, the model describes the joint distribution of the parameters and latent variables within a formal structure. To complete the hierarchical Bayesian implementation, the assimilation of information from the surveillance observations and the process model is now examined.

6.4 Hierarchical Model

The mechanistic model provides the structure for local growth and dispersal within and between zones at the chosen scale. Surveillance evidence is then used to estimate the parameters and latent variables of the model within a hierarchical Bayesian framework. Parameter estimates are sought for zone colonisation times (φi), temperature threshold and stress accumulation, (τ, δ), and the gravity model parameters (ψ, ω).

Surveillance data have been collected over a period of thirteen years at different intensities in different zones. While there is information value in the fine scale spatial and temporal data within each zone, it is computationally convenient to amalgamate data by zones and months given the scale of the analysis. Consider 6.4. HIERARCHICAL MODEL 133

the number of positive detections, yim, to be binomially distributed from zone, i, in month, m, where the time of the observation, tm, is taken as the mid-point of the month.

Ultimately, it is the model combination of the threshold for stress accumulation,

τc, and the rate of stress accumulation, δ, that would deﬁne the ﬁnal boundary conditions for the extent of SW. Mean temperature stress values, Si, calculated for each zone at τ = 6, 8, 10, 12 and 14◦C, were very highly correlated leaving little opportunity to identify the contribution from each parameter. Instead, τ is set to 10◦C, which the literature suggests to be critical for survival (Wen et al., 1994), and the model formulated to estimate δ.

Correlation of parameters in the gravity model (Equation 6.7) can make ψ and ω difficult to identify individually (see Bossenbroek et al., 2007). Given the limited number of zones defined, preliminary investigations suggested that there was not enough information to estimate ω. In analyses of zebra mussel movement by boaters, distance coefficients between 2 and 3 have been estimated (Bossenbroek et al., 2007; Leung et al., 2006). In Section 5.5, estimates of ω of around 2 were obtained for district scale models. For this analysis, ψ is estimated with ω fixed at 2, so that connection strengths between zones are a function of the inverse-squared distance.

While the model could be set up with propagule pressure from outside of the zones, inspection of the data indicate that the ﬁrst incursion into a major population centre was in Cairns at a previously estimated time of October 1997 (see Section

4.4). The model is thus set relative to a Cairns colonisation time of φ1 = 0.

As an overview, the model seeks to estimate the joint distribution of parameters and latent variables, namely the zone colonisation times, φ, the gravity model scaling factor, ψ, and the temperature stress accumulation rate, δ. Given the underlying deterministic structure, the hierarchical model uses MCMC to sample the parameters, conditional on the covariate and observation data, Figure 6.6. 134 CHAPTER 6. PREDICTING SPIRALLING WHITEFLY DISTRIBUTION

Gravity Growth Φ Y Model Model

X Ψ δ βi Si

Figure 6.6: Overview of hierarchical model components. The deterministic gravity model is deﬁned by the covariate data for the zones, X, and the scaling factor ψ. The joint colonisation times, φ, are deﬁned by the gravity model and the intrinsic growth rate, βi, that moderates the rate of internal spread through each zone. βi is determined by the stress accumulation rate, δ, and the temperature data, Si. The growth model provides the detection probabilities for the observation data, Y , collected at times T .

Representing all of the observation and covariate data by the vector, Z, the hierarchical Bayesian model for the joint distribution of parameters of interest breaks down into a simple series,

[φ, ψ, δ|Z] ∝ [Z|φ, δ][φ|ψ, δ][ψ][δ]. (6.14)

The ﬁrst term, [Z|φ, δ], is for the likelihood of observing the data given the stress accumulation parameter and the zone colonisation times and is dealt with in Section 6.4.1. The approach to modelling the joint distribution of the other parameters is explained in Section 6.4.2. 6.4. HIERARCHICAL MODEL 135

6.4.1 Observation model

As described in Section 6.2.3, surveillance data have been aggregated into monthly records of positive and negative outcomes recorded on hosts in each zone. For each zone, i, and month, m, we have Vim visits to hosts of which yim are positive. Some of these hosts may be at the same site or from sites close by. However, here it is assumed that they are chosen at random from a point within the zone. The observation outcomes are therefore modelled as,

yim ∼ Binomial(Vim, qim) (6.15)

where qim is the probability of detection for each observation. The value of qim must encompass variability in detectability that is expected across the zone at the observation time, particularly in terms of the pest population and the uncertainty surrounding the observers and their ability to search a plant. To incorporate this uncertainty, the model draws upon the structure and results of the local scale model developed in Chapter 4.

The pest population on a plant at a site, s, in zone, i, is mechanistically modelled as a logistic function over time,

 0 if Csm ≤ 0, Nst = (6.16) −1 logit (βiCsm + logit(1/K)) × 2K × b otherwise. where βi is the rate of increase within the zone, Csm is the period for which the site has been colonised and K is the carrying capacity which is set at 100. Note that depending upon the particular value of the temperature stress parameter, βi can be negative indicating a failure to colonise. To maintain consistency with the model from Chapter 4, the parameter b denotes the host type and other factors that may aﬀect population size and is implemented as a constant. A constant detectability due to an inspector’s ability to inspect a whole plant, p = 0.3, is also based on 136 CHAPTER 6. PREDICTING SPIRALLING WHITEFLY DISTRIBUTION results from the local scale model. At the observation time m, the probability of detection at a site within a zone, Psm, is thereby modelled as,

Nsm Psm = 1 − (1 − p) . (6.17)

As the probability of detection at a site can be considered a function of the period of colonisation, we denote this as g(Csm).

Spread rates described in Equation 6.5 are used to translate g(Csm) from a probability of detection at a site into the observation units of zones and months. For some observation time, m, the distribution of the period of colonisation across the zone i has support over the range max(0, m−φi −Ei) to m−φi. The distribution of the number of pests on plants, Nim, across zone, i, can be obtained by integrating over the spread (Equation 6.5) with respect to Csm, to arrive at a distribution for the probability of detection, Pim, within the zone at observation time, m.

The observation data carry some further sources of uncertainty. Some areas which were known to be infested for a long period appear to be over-represented by absence data. Once spiralling whitefly has been present in a zone for a long period, the motivation for inspectors to report presence data can diminish. As the model specified above cannot reconcile population decreases after colonisation, under-reporting would lead to instability in the parameter estimates. Another source of uncertainty not included in the model so far is that the detectability of the pest is likely to be lower in those areas where its survival is marginal. In marginal areas, even after long periods of colonisation, the “carrying capacity” may be considerably lower. There are also examples of hidden over-reporting in the data. Detections based on public submissions provide positive records but ignore the implicit absence data associated with no public reports. Given the large variation in detectability and reportability across the zone, additional detection uncertainty, η, is imposed via a non-informative Jeffreys prior on a Beta distribution (Gelman et al., 2004), so that the observation model becomes, 6.4. HIERARCHICAL MODEL 137

η ∼ Beta(0.5, 0.5) (6.18)

pim ∼ g(Csm) (6.19)

qim = pimη. (6.20)

Examples of the shape of qim with respect to zone colonisation time and uncertainty in detectability are shown in Figure 6.7. The plots illustrate that early in the colonisation phase of the zone (Figure 6.7a), the highest contribution to the likelihood comes from the potential for high probabilities of detection from the distribution of η. As the period of colonisation increases (Figure 6.7b), there is potential for the detections to be due to sampling sites with low pest prevalence with high detectability or sites with higher populations at lower detectabilities. Once the zone has been colonised for a long period and reached a carrying capacity, the distribution of likelihoods reﬂect only the proportion of the sites colonised at diﬀerent times on one axis and a fairly constant probability of detection on the other axis.

Likelihood

Days colonised Days colonised Days colonised

Beta Prob Det Beta Prob Det Beta Prob Det a) b) c)

Figure 6.7: Likelihood function for detectability yim|Qim,Vim given a Beta(0.5,0.5) distribution of observer detectability and population-based detection probabilities for potential site colonisation times across the zone. From left to right, the periods of zone colonisation are a) 30 days, b) 365 days and c) 730 days, with yim=10 and Vim=30.

The likelihood for this beta-binomial distribution can be obtained by numerically 138 CHAPTER 6. PREDICTING SPIRALLING WHITEFLY DISTRIBUTION

integrating qim over the range of plausible population sizes and detectabilities for the zone to evaluate Equation 6.15.

6.4.2 Incorporating the movement model

Now that the likelihood model for observing the data given the zone colonisation times and temperature stresses has been formulated, we develop the model for the second term in the overview Equation 6.14.

The joint distribution of the process model consists of the parameters of interest, ψ and δ, and the complex latent process of colonisation times, φ.

[φ, ψ, δ] ∝ [φ|ψ, δ][ψ][δ] (6.21)

As the zone colonisation times, φi, have a complicated dependence structure, their likelihoods cannot be estimated individually. However, it is possible to estimate their likelihood as a block when given ψ and δ and φ1 (for the Cairns zone). To generate samples from the distribution of φ, each φi is proposed in turn and the density of φ calculated from the reliability and hazard functions. Note that while

φi for each zone is proposed individually, the reliability framework in Equation 6.13 requires the likelihood to be calculated for the whole block of φ. Therefore, the Metropolis algorithm is used to calculate the conditional posterior of,

[φ|ψ, δ, Z] ∝ [Z|δ, φ][ψ][δ]. (6.22)

Conditional posterior probabilities for the gravity model scaling parameter, ψ, are calculated in turn by sampling,

[ψ|φ] ∝ [φ|ψ, δ][ψ][δ]. (6.23) 6.4. HIERARCHICAL MODEL 139

As the stress accumulation parameter, δ, determines βi, which in turn aﬀects both the rate of areal increase and the detectability, it is necessary to sample,

[δ|φ, Z] ∝ [Z|δ, φ][φ|δ, ψ][δ]. (6.24)

A log-normal prior for ψ with a mean of log(0.1) and a standard deviation of 1 was chosen to provide a plausible range of values for the gravity model scaling factor. The prior for δ was chosen as uniform over the positive real line.

A detailed algorithm for implementing the model within an MCMC simulation is given in Appendix C.

6.4.3 Computation

The model was developed within the R statistical software (R Development Core Team, 2008). Convergence was assessed by visually inspecting the chains and calculating the Gelman-Rubin statistic (Brooks and Gelman, 1998b) to determine a suitable burn-in period for the sampler. Due to a high degree of autocorrelation, three parallel Markov chains were initialised with diﬀerent starting values for the parameters and run for 400 000 iterations after discarding a burn-in of 10 000 iterations.

Further MCMC simulations were run with the same model, but using the data that would have been progressively available over one year intervals from 1998 to 2008. For these models, a single chain of 100 000 iterations with a burn-in of 10 000 was used for each of the 11 years. 140 CHAPTER 6. PREDICTING SPIRALLING WHITEFLY DISTRIBUTION

6.5 Results

The posterior distribution of the stress accumulation parameter, δ, is fairly uniform between 2.5 and 4.9 (Figure 6.8). The upper limit for stress accumulation is determined by those zones with positive detections that must therefore have a positive

βi. The lower limit of δ is determined by the failure to detect SW in zones that have been exposed. The uniformity of the posterior distribution reflects the lack of discriminating information in the data about different growth and spread rates within colonised zones over time. The lack of information is partly due to the aggregated model used for within-zone spread, but is compounded by the uncertainty due to variability in surveillance practices. Estimates of the scaling factor for the gravity model are skewed, (Figure 6.8), reflecting the potential for weak gravity model connections to still act as a source of colonisation. Traces of the MCMC simulations illustrate the necessity for long simulations to achieve adequate convergence (Figure 6.9). 10 8 0.4 6 4 Density Density 0.2 2 0 0.0

2.5 3.0 3.5 4.0 4.5 5.0 0.0 0.2 0.4 0.6 0.8

δ ψ

Figure 6.8: Estimates of the posterior densities of the stress accumulation parameter, δ, and the scaling factor for the gravity model, ψ.

Posterior estimates for the zone colonisation times are shown in Figures 6.10 & 6.11 in relation to the observation data. Those zones that are marginal for colonisation, have small residential populations and few surveillance data for the early 6.5. RESULTS 141

Figure 6.9: MCMC chains for the scaling factor, ψ, thinned to show every 100th draw.

years converged slowly. Of note is the bimodal distribution of some zones, in particular Bundaberg, for which the surveillance information is unable to provide stable estimates of colonisation times.

Figure 6.12 demonstrates the translation of the stress accumulation parameter into the estimates of rates of increase, βi, within each zone. As the quantity of surveillance data statewide provides strong evidence that the pest has not established, given the connectivity between towns, there is a strong demarcation between the sites where the pest is, and is not, known to occur. For zones with βi values close to zero, it should be considered that the within year variation of the pest population, including potential extinctions, would not be adequately represented by a model run on long term averages. Therefore, it is important that this model is only interpreted as a description of the long term establishment potential of SW.

The rate of exchange between zones over a ﬁxed period can be estimated by the gravity model using posterior estimates of ψ. Table 6.2 provides estimates of the mean annual exchange of propagules, assuming that zones are infested at capacity, that could establish if conditions were favourable.

While the model is based on over a decade of surveillance, it is of interest to know 142 CHAPTER 6. PREDICTING SPIRALLING WHITEFLY DISTRIBUTION

Mareeba Atherton Innisfail 1.0 1.5 2.0 0.8 1.5 1.0 0.6 1.0 Density Density Density 0.4 0.5 0.5 0.2 0.0 0.0 0.0

1998 2002 2006 1998 2002 2006 1998 2002 2006

Introduction time Introduction time Introduction time Ingham Townsville Burdekin 0.8 2.0 0.6 0.6 1.5 0.4 0.4 1.0 Density Density Density 0.2 0.2 0.5 0.0 0.0 0.0

1998 2002 2006 1998 2002 2006 1998 2002 2006

Introduction time Introduction time Introduction time Charters Towers Bowen Whitsunday 0.5 0.6 1.5 0.4 0.4 0.3 1.0 Density Density Density 0.2 0.2 0.5 0.1 0.0 0.0 0.0

1998 2002 2006 1998 2002 2006 1998 2002 2006

Introduction time Introduction time Introduction time

Figure 6.10: Posterior density of estimates for the time of introduction into northern zones for three MCMC chains. Black dashed line indicates the time of ﬁrst detection where applicable. Black +, positive detection for the month. Grey ×, absence record for the month. 6.5. RESULTS 143

Mackay Yeppoon Rockhampton 0.4 3.0 0.3 0.20 2.0 0.2 Density Density Density 0.10 1.0 0.1 0.0 0.0 0.00 1998 2002 2006 2010 1998 2002 2006 2010 1998 2002 2006 2010

Introduction time Introduction time Introduction time Gladstone Bundaberg Maryborough 0.4 0.20 0.6 0.3 0.15 0.4 0.2 0.10 Density Density Density 0.2 0.1 0.05 0.0 0.0 0.00 1998 2002 2006 2010 1998 2002 2006 2010 1998 2002 2006 2010

Introduction time Introduction time Introduction time Hervey Bay Nambour Brisbane 0.4 0.8 0.20 0.3 0.6 0.2 0.4 Density Density Density 0.10 0.1 0.2 0.0 0.0 0.00 1998 2002 2006 2010 1998 2002 2006 2010 1998 2002 2006 2010

Introduction time Introduction time Introduction time

Figure 6.11: Posterior density of estimates for the time of introduction into southern zones for three MCMC chains. Black dashed line indicates the time of ﬁrst detection where applicable. Black +, positive detection for the month. Grey ×, absence record for the month. 144 CHAPTER 6. PREDICTING SPIRALLING WHITEFLY DISTRIBUTION

● ● ● ● ● ● ● ● ● ● ● 0.02 ● ● 0.00 ● ●

i ● β ● −0.02

●

● −0.06 2010 2006

i ● φ ● ● ● ●

● ● ● ● 2002 ● ● ● ● ● ● ● ● ● ● 1998 Cairns Bowen Ingham Innisfail Mackay Atherton Yeppoon Mareeba Burdekin Brisbane Nambour Townsville Gladstone Bundaberg Hervey Bay Hervey Whitsunday Maryborough Rockhampton Charters Towers

Figure 6.12: Mean posterior estimates and 90 % credible intervals for intrinsic rates of increase,

βi and zone introduction times, φi. Estimates of βi less than zero suggest that colonisation will not occur. 6.5. RESULTS 145 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 Mean of potential propagules introduced from source to target zones per annum if source cells were completely covered. Note that these 1. Cairns2. Mareeba3. Atherton4. Innisfail ... 12.435. Ingham ... 5.776. 12.43 Townsville7. 0.83 5.77 3.49 Burdekin 0.83 3.498. ... Charters 0.20 0.17 0.17 Towers 1.76 0.209. 0.01 0.09 Bowen 0.01 0.28 0.28 1.76 0.11 0.1610. 0.11 Whitsunday ... 0.01 0.02 0.16 0.02 0.15 0.0111. 0.01 Mackay 0.15 0.09 0.06 0.01 0.37 0.06 0.01 0.0812. 0.01 0.06 0.01 ... Yeppoon 0.02 0.06 0.87 0.37 0.00 0.0113. 0.08 0.01 0.03 0.02 Rockhampton ... 0.03 0.01 0.00 0.87 0.00 0.20 0.04 0.2014. 1.09 0.01 0.02 0.01 Gladstone 0.11 0.04 0.03 0.01 0.03 2.98 2.98 0.00 0.09 0.0115. 0.01 0.01 0.00 Bundaberg 0.02 0.02 0.01 ... 1.09 0.01 ... 0.00 0.11 0.01 0.0116. 0.02 0.01 0.01 Maryborough 0.00 0.05 0.34 0.01 0.01 0.33 0.03 0.00 0.09 0.0517. 0.03 0.01 0.02 0.06 0.33 0.34 Hervey 0.00 Bay 0.01 0.01 0.00 0.02 0.10 0.00 0.00 0.15 0.06 0.02 0.02 0.0218. 0.62 0.15 0.00 Nambour 0.00 0.01 0.02 0.00 0.00 0.10 0.00 0.00 0.02 0.62 0.00 0.01 0.05 0.06 0.0219. 0.00 0.04 0.21 0.41 Brisbane 0.00 0.06 0.00 0.14 0.01 0.14 0.01 ... 0.00 0.04 0.01 0.01Total 0.04 ... 0.21 Entering 0.00 0.01 0.01 0.00 0.00 0.01 0.00 0.05 0.13 0.00 0.01 0.02 0.02 0.13 0.00 0.00 0.01 0.08 0.41 0.08 0.21 0.01 0.00 0.00 0.86 0.04 0.00 0.00 0.01 0.74 0.03 0.00 0.01 0.10 0.01 0.01 0.21 0.10 0.86 25.4 0.00 0.02 0.04 0.74 0.07 0.01 0.01 0.00 0.00 0.01 0.02 0.05 0.01 0.02 0.05 0.01 0.01 0.04 ... 0.00 0.01 13.7 0.46 0.07 0.02 0.00 0.01 0.02 0.13 0.01 0.00 0.02 0.06 0.05 0.00 0.03 0.06 0.05 10.73 0.01 0.13 7.2 0.01 0.01 ... 0.02 0.02 0.00 0.10 0.01 0.00 0.01 ... 0.01 0.19 0.03 0.10 0.19 0.14 0.03 0.01 0.46 4.6 0.05 0.00 0.01 0.02 0.01 0.13 0.03 10.73 0.70 0.01 0.05 1.08 0.01 0.04 0.03 0.14 1.08 1.3 2.59 0.00 0.19 0.02 0.70 0.09 0.17 0.01 2.59 0.05 0.01 0.13 0.68 10.4 0.02 0.19 0.01 0.17 0.68 0.09 0.06 0.04 0.17 ... 0.17 0.04 4.0 0.04 0.05 0.12 0.07 0.02 0.90 0.06 0.19 0.26 0.17 1.6 0.22 0.07 0.90 0.22 ... 0.26 0.16 2.74 0.12 0.19 1.05 0.60 0.17 1.4 0.16 0.27 ... 0.60 2.74 0.78 0.27 1.05 2.3 4.98 2.91 0.78 0.48 4.98 2.91 15.71 15.71 0.48 4.4 2.60 2.60 2.18 ... 2.81 2.81 13.0 2.18 9.07 9.07 7.14 3.34 7.14 18.9 3.34 ... 9.22 7.6 9.22 323.52 21.6 323.52 ... 28.9 34.1 334.2 358.6 Table 6.2: estimates are based on connectivity only and therefore include pressure from zones that are not predicted to contain any pests. 146 CHAPTER 6. PREDICTING SPIRALLING WHITEFLY DISTRIBUTION how the predictions would have changed as the invasion proceeded. As surveillance data are progressively collated and modelled, more knowledge can be gained about both the environmental constraints and the connection strengths between different zones. In Figure 6.13, it can be seen that the estimates of φi are variable until a sufficient number of discriminating colonisation events are detected. Similarly, in Figure 6.14, the evidence to support the stress accumulation parameter estimate firms as absence data is collected for zones that are predicted to have heavy propagule pressure. Mean estimates of ψ remain fairly constant but the variance shrinks with the accumulation of data.

6.6 Discussion

From a biosecurity risk perspective, the model provides substantial evidence that south east Queensland is exposed to pest pressure, yet spiralling whiteﬂy has been unable to establish widely. The pest could colonise some areas in warmer months followed by winter extinction. Alternatively, it could persist through winter around restricted microclimates such as glasshouses. However, if such marginal persistence does occur, the opportunity for spread in favourable times would be limited and the probability of movement on any particular consignment of nursery plants would be very low. If the pest is being moved as an oﬀ-host hitchhiker, then the value of restricting plant trade given a coincident pathway becomes less relevant as a risk mitigation function. There are currently restrictions on movements into the more temperate areas of Australia. It is apparent that there is a low probability of the pest establishing in natural environments in these target areas.

Estimation of species boundaries has been demonstrated for an invading pest in a dynamic setting with human-mediated dispersal. The model could be further re- ﬁned. There are two prominent issues to be addressed when choosing the modelling scale. Firstly, the internal dynamics of spread within a zone must be adequately 6.6. DISCUSSION 147

Mareeba Atherton Innisfail 2006 2006 2006 i i i φ φ φ

● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ●● ●● ●● ●● ●● ●● ●● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●

1998 2000 2002 2004 2006 2008 1998 2000 2002 2004 2006 2008 1998 2000 2002 2004 2006 2008 data to year data to year data to year Ingham Townsville Burdekin

●

● 2006 2006 2006 i i i

● φ φ φ ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●●

1998 2000 2002 2004 2006 2008 1998 2000 2002 2004 2006 2008 1998 2000 2002 2004 2006 2008 data to year data to year data to year

Charters Towers ● Bowen Whitsunday

●

● ● ● ● ● 2006 2006 2006 i i ● i ● ● ● ● ● ● ● ● φ φ φ ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ●● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●

1998 2000 2002 2004 2006 2008 1998 2000 2002 2004 2006 2008 1998 2000 2002 2004 2006 2008 data to year data to year data to year Mackay Yeppoon Rockhampton ● ● ● ● ● ● ● ● ● ● ● ● ● ● 2006 ● 2006 2006

i i i ● ● ● ● ● ● ● ● φ φ ● φ ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ●● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●

1998 2000 2002 2004 2006 2008 1998 2000 2002 2004 2006 2008 1998 2000 2002 2004 2006 2008 data to year data to year data to year Gladstone Bundaberg Maryborough ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 2006 2006 2006

i i ● i ● φ φ φ ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●

1998 2000 2002 2004 2006 2008 1998 2000 2002 2004 2006 2008 1998 2000 2002 2004 2006 2008 data to year data to year data to year Hervey Bay Nambour Brisbane

● ●

● ● ● ● ● ● ● ● ● ● ● ● 2006 2006 2006 i i ● ● ● ● i ● ● ● φ ● ● ● ● ● ● ● φ φ ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●

1998 2000 2002 2004 2006 2008 1998 2000 2002 2004 2006 2008 1998 2000 2002 2004 2006 2008 data to year data to year data to year

Figure 6.13: Progressive posterior estimates of the time of introduction into zones for data collected up to a particular year. Solid lines - posterior means, grey dashed - 90 % credible intervals. Red dashed indicates the time of ﬁrst detection for the twelve known colonised zones. 148 CHAPTER 6. PREDICTING SPIRALLING WHITEFLY DISTRIBUTION 25 20 0.8 15 δ ψ 10 0.4 5 0 0.0

2000 2002 2004 2006 2008 2000 2002 2004 2006 2008

data to year data to year

Figure 6.14: Progressive posterior estimates of the stress accumulation parameter δ, and the scaling factor, ψ, for data collected up to a particular year. Solid lines - posterior means, grey dashed - 90 % credible intervals. captured in relation to the observation model. In particular, the relationship between the expression of spread within a zone and the observation process deserves attention. For this study, the internal dynamics of the zone are simplistically represented in the model, but the necessary inclusion of severe observational uncertainty results in a considerable loss of power. However, as the surveillance evidence for the climate boundary is compelling, this loss of power does not greatly affect the final inference. Secondly, the resolution of the model must be fine enough to suit the management decisions that are required. The rather coarse spatial scale of this model was adopted for ease of computation on a small number of zones. However, the gross predictions of the model are not expected to differ significantly at the scale at which management decisions are made.

The inverse-squared distance function for the gravity model is considered reasonable for the human-mediated movement of the pest. By applying a particular model form, even with one of the parameters estimated from the data, there is a danger that the model specification may lead to errors in inference (Barry and Elith, 2006). Instability in some of the model results may reflect that the form of the model should be investigated further. Gravity models provide an interesting start to investigating 6.6. DISCUSSION 149 biosecurity problems, but what may be required are new trade-related metrics for topological distances (Ferrari and Lookingbill, 2009). While the model could be used to predict the arrival times in other centres around the country, it would be necessary to consider the type of pathways available with respect to the pest before extending the model beyond its inferential bounds. If road or rail transport of off- host adults was implicated, then transport times to distant centres may exceed the survival time. The development of methods to include prior information on pathways, particularly those associated with trade, into hierarchical Bayesian models would provide a useful starting point for interpreting biosecurity risk in relation to surveillance activity.

The models presented use long-term averaged climate data aggregated over space. Spatial variance in microclimates coupled with dispersal within the zone would be expected to buffer the direct effects of climates on populations. In contrast to spatial buffering, there will be inescapable temporal variation in extreme weather events. These would result in local extinctions that could be modelled using geometric means for the rates of increase. However, incorporating daily weather into the model would require stress parameters to be recalculated across the model for each new proposed colonisation time. It is of interest that SW was detected in Bundaberg in 2008 but surveillance since then has not detected the pest. A within- generation state-space model (e.g. Royle and Kery, 2007) may be more suitable to allow for local extinctions, but the computational time would increase markedly. A more computationally efficient alternative would be to implement a reversible jump algorithm for each time step (Jewell et al., 2009a). More recently, agent based invasion models have been proposed to build up the process from the level of the individual (Hooten and Wikle, 2010). Detailed small-scale models for marginal zones could be constructed to address issues at a local scale, based on what is learnt at a larger scale. One of the advantages with the Bayesian approach is that it allows information from separate studies to be incorporated into models so that computation can be targeted at more specific problems (Cressie et al., 2009). 150 CHAPTER 6. PREDICTING SPIRALLING WHITEFLY DISTRIBUTION

Bayesian modelling is well suited to incorporating multiple observation sources to estimate the latent extent of pests (Clark, 2005). Surveillance by the general public is continual but undocumented, yet, with the provision of suitable priors, this information could be incorporated into the modelling framework. For some new detections in zones, delimiting surveillance was deployed around the known infested area and stopped once a “suitable” buﬀer of absence data was obtained. A cursory examination of zones that were delimited internally after a public report suggest detectability for a member of the public of roughly between 1 in 1000 and 1 in 10000 residential properties for a one month period. As passive detection programs can be more eﬀective than structured surveillance for exotic pests (Cacho et al., 2010), methods to incorporate prior information about the observation value into estimating pest extent are warranted. For structured surveillance, consistent data collection methods, properly audited to provide empirical support for models, would provide for stronger and more reliable inference.

Entrenched in biosecurity practices for trade restrictions is the notion of applying buﬀer zones around known detections. Predictions from this model demonstrate that there can be considerable lead time between the introduction and detection of pests given routine surveillance practices conducted by regulatory agencies. There is rarely any regulatory requirement for a level of surveillance to be maintained for pests. This is partly due to resource constraints, but also due to a lack of statistical tools to understand the information that is being collected. Hierarchical Bayesian models that incorporate spread and detectability into a spatio-temporal analytical framework allow these risks to be quantiﬁed in relation to surveillance so that trading parties can have a basis for negotiating their appropriate levels of protection. 6.7. SUMMARY 151

6.7 Summary

In this chapter, solutions for two impediments to assessing biosecurity risks have been modelled. Firstly, estimation of the probability of human-mediated movement between distant areas is demonstrated, based on observed patterns of spread. By borrowing strength across the spatio-temporal data, these estimates can be used to predict the propagule pressure into other zones of interest. Secondly, the model provides a progressive way to identify range restricting environmental covariates, such as temperature, by using the predictive power of a spread model in a non- equilibrium system. The model adds considerable value to the large amounts of data collected by providing an inferential framework based on an underlying mechanistic model of ecological processes. 152 CHAPTER 6. PREDICTING SPIRALLING WHITEFLY DISTRIBUTION Chapter 7

Modelling the Invasion Ecology of Red Banded Mango Caterpillar

Incursion management, and in particular eradication, requires decisions about the pest status of areas to be made at fine spatio-temporal scales. Inference at spatial scales in the order of a kilometre and temporal scales of less than a month may be desired from data collected in areas with low pest densities. A functional limitation of the reliability analysis presented in Chapters 5 & 6 is that the dynamic invasion process is considered to be entirely described by the colonisation time of sub-areas. In the presence of temporal variability, both in covariates and population expression, such models can struggle to capture complex behaviour within sub-areas. One particular deficiency is their inability to model local extinction probabilities. Local extinctions not only play a major part in the dynamics of pests over fragmented landscapes, but also are the aim of eradication campaigns. In previous chapters, mechanistic models provided a simplified structure for the ecological process. This chapter uses exploratory modelling to investigate some issues of ecological complexity that may be faced by an analyst developing hierarchical Bayesian models for the tactical control of invasive pests. As a case study, the ecology of red banded mango caterpillar is reviewed and this knowledge examined in the context of hierarchical

153 154 CHAPTER 7. MANGO CATERPILLAR INVASION ECOLOGY

Bayesian modelling and surveillance for biosecurity programs.

7.1 Introduction

To eradicate invading pests, it is critical that the boundary between infested and uninfested areas can be reliably delimited to define control areas. Pest extent can be delimited by conducting surveillance and estimating the probability of presence (or absence) over a spatial domain at a particular time. Surveillance for one such invader, red banded mango caterpillar (RBMC), Deanolis sublimbalis Snellen (Lepidoptera: Pyralidae), led to its detection for the first time on the Australian mainland in 2001 in the remote Northern Peninsula Area (NPA) of Queensland (Foulis et al., 2001). Eradication was attempted in 2002, but the pest was soon detected outside of the control area. By 2007, it had been found across the district up to 40 km away (Royer, 2009). The breach of the control area highlighted the need to understand both the incursion ecology of RBMC and the sensitivity of surveillance, so that observations could be translated into effective management responses (Trinca and Foulis, 2002).

Observation models to infer the plant pest status at a particular place and time must account for the imperfect visual inspection of plants. In particular, when pest densities are low there is a high probability of obtaining false absence records (Barrett et al., 2009; Cacho et al., 2010; Carrasco et al., 2010b; Kery et al., 2006; Royle and Dorazio, 2006), making it diﬃcult to estimate boundaries for control areas. An alternative to investing more resources in detection is to borrow pest population information from surrounding sites to better estimate the extent (Royle et al., 2007; Wikle and Hooten, 2006). The hidden population at a particular place and time is the result of a dynamic invasion process that induces spatio- temporal autocorrelation in population density. By formulating ecological models of population growth and dispersal, hierarchical Bayesian approaches can incorporate 7.1. INTRODUCTION 155 prior information about these autocorrelated population responses into the estimate of extent.

Successful predictive modelling must be able to generalise the dynamic processes of invasion from a single realisation of an incursion process in order for prediction to be applied in habitats with novel spatial structures. Spread models describe the interaction between the pest population and the spatial features of the environment into which it is introduced (Hastings et al., 2005). Two sources of information are available to help make inference on pest dispersal and therefore strengthen estimates of extent at a particular time. Firstly, there is the imperfect surveillance data that records presence / absence of the pest based on visual inspections that provide a pattern of spread. Secondly, there is ecological information available from the literature and empirical studies that can be used to inform the predictive parameters of population dynamics models. Both sources of information, with their incumbent uncertainty, can be brought together within a hierarchical Bayesian model to estimate the status of invasive pests over space and time (Cook et al., 2007a; Hooten et al., 2007; Hooten and Wikle, 2008; Jewell et al., 2009b; Wikle and Hooten, 2006). These models break down the observation and dynamic invasion process into simpler component models to make inference on the extent of the pest, as well as to estimate the parameters that drive the incursion processes (Clark, 2005; Ellison, 2004).

Modelling complex ecology is challenging when the prior information about the pest is vague, not only for model parameters, but for the model specification as well. As for many invading pests, there are some empirical data on the life history of RBMC under laboratory conditions but there is little quantitative information on reproduction, mortality and dispersal in the field. In this chapter, information gleaned from the literature, and supplemented by personal observations, is used to develop an ecologically reasonable model specification for a stochastic simulation model of RBMC invasion. Craigmile et al. (2009) provide an excellent guide to the development of hierarchical modelling components through exploratory analysis and 156 CHAPTER 7. MANGO CATERPILLAR INVASION ECOLOGY prior knowledge. The aim of this chapter is to explore some of the ecological and data components that would be needed to develop a hierarchical Bayesian model to help manage an RBMC invasion in another area. Part of this exploration is to identify the potential impact of parameter assumptions and model choices on inference for invasive pests with complex ecology. Recent approaches by Hooten and Wikle (2010) have examined such agent-based models in a hierarchical Bayesian setting to accommodate the spatial interaction of a pest with its environment. While a full hierarchical Bayesian model is not developed here, it is the intention to seek the intrinsic ecological and behavioural characteristics of red banded mango caterpillar in relation to its invasion history in the NPA. The investigation considers the prospects for developing an analytical model that would estimate ecological parameters that could be used for the management of the pest in the event of an incursion in a commercial mango producing area.

7.2 Overview

Red banded mango caterpillar, Deanolis sublimbalis(=Noorda albizonalis) is classi- ﬁed as an emergency plant pest by Plant Health Australia and is therefore a potential candidate for government cost shared eradication campaign (Plant Health Australia, 2010a). Its host range is restricted to mangoes, Mangifera indica, and a few closely related hosts which are rare in Australia (Krull, 2004). Larvae feed within the generally unripe fruit causing the fruit to rot before it has ripened.

The general lifecycle of RBMC consists of an egg stage that is usually found near the fruit stem, ﬁve larval stages that feed in the fruit, pre-pupal and pupal stages found in the bark that are dormant in the non-fruiting season, and an adult stage that usually rests in the tree canopy (Waterhouse, 1998). The lifecycle takes approximately 4-6 weeks from egg to adult. More detailed information on the biology and ecology of the pest is introduced in the relevant modelling sections. 7.2. OVERVIEW 157

Pest spread through the NPA has been monitored by at least annual surveillance since 2001 (Royer, 2009). The annual history of the furthest extent detected is shown in Figure 7.1. Domestic quarantine is in place to prevent movement of the pest out of the NPA and into the mango production areas 800 km to the south.

Indonesia 5°S

N o Cape York Papua New Guinea t D e t Punsand Bay 10°S

Heathlands Roko Island 1 Roma Flat 0 Weipa 0 Somerset 2

Coen

15°S

MareebaCairns Lockerbie 2 Northern Territory Queensland 200 Australia 20°S Loyalty Beach 140°E 145°E 150°E 3 200 Seisia 2004 New Mapoon Bamaga Legend Umagico Mangoes Injinoo Extent of Detection Muttee Heads 5 200 Place Names Roads

2006 0 2.5 5 10 Kilometres 2007

Figure 7.1: Furthest extent of red banded mango caterpillar detections and distribution of mango trees within the Northern Peninsula Area. Extent contours are hand-drawn to delimit mango trees with detections up to that year. The shape of the contours is immaterial except in relation to the host distribution.

Given the pattern of spread in the NPA, the desired modelling goal is to infer the extent of RBMC over time and to estimate the parameters for a dispersal model that may be used if the pest establishes in another region. Using the hierarchical framework, an overview of the spatio-temporal model can be expressed as,

[φ, χ, γ|Y ] ∝ [Y |φ][φ, χ|γ][φ][χ][γ] (7.1) 158 CHAPTER 7. MANGO CATERPILLAR INVASION ECOLOGY where Y is the surveillance data, φ is the infested fruit status, χ is the pest population and γ are parameters that model the intrinsic biological characteristics such as growth and dispersal. The model therefore consists of some spatio-temporal data and two latent spatio-temporal variables for fruit and population. Each of these latent variables is a joint distribution represented within a spatio-temporal structure. It should also be noted that the fruit infested, φ, and the pest population, χ, contain mutual feedback processes and so their distribution must be estimated jointly. The model would also include additional mechanistic models to link the information from the data to the parameters of interest. In summary, Equation 7.1 identifies the model components required. The first term on the right hand side identifies the need for an model to describe the likelihood of observing the surveillance outcomes given the fruit infestation status over space and time. The second term identifies the model required for interpreting the probability of fruit infestation status and population status over space and time given the biological parameters. Finally the hierarchical Bayesian framework identifies the prior uncertainty about the parameters.

Rather than tackling this large problem from the statistical perspective and developing a simpliﬁed hierarchical Bayesian model, the invasion ecology components are examined to gain a greater understanding of the data and dynamic complexity. In the previous chapters, observation models assumed detectability to be a function of colonisation time and uncertainty surrounding the interaction between observer and pest. To calculate the likelihood for the RBMC pest status of an area, given some surveillance data, the observation model needs to accommodate seasonally changing relationships between the pest, its host and observers. Inherent in these relationships is considerable uncertainty over how observations for fruit damage relate to the population as a whole. In Section 7.3, the surveillance data are examined and a set of potential observation models for the likelihood component, [Y |φ], are proposed and discussed.

Not only is the RBMC invasion complex in terms of the observational model, but 7.3. DATA AND OBSERVATION MODEL 159 there is also a great deal of uncertainty about its biology. The approach taken here is to develop a stochastic population matrix model (Caswell, 2001), to incorporate the known complexity and uncertainty and to examine the results as they apply to hierarchical statistical modelling. This approach allows the behaviour of individual insects that make up the extent to be examined in light of the observational data. In Section 7.4, a local simulation model is developed to incorporate known sources of variation, including the complex relationships with fruiting phenology. The intention is to understand the uncertainty within [φ, χ] at a local level before considering a full spatial model. An analysis of Allee eﬀects in relation to resource quantity is conducted to examine one of the critical components for estimating extent, the probability of colonisation given introduction.

In Section 7.5, potential dispersal mechanisms for RBMC are reviewed and a dispersal model for the immigration and emigration components is proposed and discussed.

In each section, the exploratory results for that component are discussed in terms of the potential to develop hierarchical Bayesian models that could be transferred to a novel environment to estimate pest extent. The chapter concludes with a more general discussion to tie together the elements and examine more general issues of hierarchical Bayesian model development for invasive pest management.

7.3 Data and Observation Model

Surveillance data are not analysed here but they are described in order to convey the amount of information available and to develop the rationale for potential observation models. Data for each site visit include the location, date, number of trees examined and whether the pest was present or absent. In the field, sites were arbitrarily defined as a collection of mango trees on an identifiable land parcel with roughly less than a 50 m gap between trees. Spatial resolution of the data is at 160 CHAPTER 7. MANGO CATERPILLAR INVASION ECOLOGY

GPS accuracy (usually < 30 m except in rainforest areas). Due to the imprecise field definition of a site and the GPS accuracy in some areas, site data have been attributed to a grid of hexagons with a width of 100 m which are termed cells. This resolution is considered to adequately reflect the quality of the spatial data while retaining sufficient resolution for population modelling.

A map has been constructed of all mangoes that could contribute to the hidden invasion process throughout the fragmented landscape. In the study area, mango trees are commonly grown in backyards, park areas and bush within the ﬁve NPA indigenous community areas of Injinoo, Umagico, Bamaga, New Mapoon and Seisia. Mango trees introduced into abandoned European settlements northeast of the com- munities have since spread along rainforested areas along roads. In the opinion of people who frequent the rainforest, there are no large unrecorded stands of feral mangoes within the less accessible areas (James Bond, Scott Templeton pers. comm. 2007). Surveys had not covered every tree in the area and so not all mango locations were recorded. Aerial photos and satellite images of the area were used to supple- ment the surveillance data and estimate the number of mango trees in each cell. In 2002, an 8 km control zone was placed around the initial detection at Somerset. Trees in the control zone were felled or radically pruned to prevent fruit set for two years in an attempt to eradicate the pest (Trinca and Foulis, 2002). These trees would not contribute to the population process during this time.

A total of M = 228 cells were identiﬁed in the study area as providing the highest reasonable spatial resolution given the data recording. Presence or absence was recorded at 323 site visits after inspecting and cutting fruit from 2441 trees including revisits. Data are available on the number of fruit cut in some locations. Only 134 of the 228 cells contain any surveillance data.

Evidence of presence / absence comes primarily from visual inspections of fruit and cutting of fruit. The ﬁrst instar larvae have a head capsule width of around 0.5 mm and usually burrow into the top part of the fruit (Golez, 1986). When fruit 7.3. DATA AND OBSERVATION MODEL 161 are small, or there are multiple larvae in a fruit, caterpillars will silk to other fruit on the tree once the food supply is exhausted (Waterhouse, 1998). For each of the four larval moults, the head capsule width increases by approximately 0.5 mm so that a ﬁfth instar larva entering a new fruit leaves a hole with a diameter greater than 2.5 mm. Holes left by the caterpillars soon begin to rot, with the damaged larger fruit being particularly conspicuous.

When a site is visited, inspectors initially assess fruit on the tree and on the ground for symptomatic damage. Accessible fruit displaying suspicious symptoms are removed from the tree and examined for larvae. Most fruit cutting is carried out on fallen fruit as they are easier to reach and more likely to be infested.

Surveillance data on fruit symptoms provides the information to estimate the population of larvae within the cell. Consider the observation data to be discretised into two-week time periods, t. Let the number of trees in a cell, i, be Mi and the number of trees inspected at a visit at time t be mit. The number of uninfested fruit in the cell is denoted, Fit, and the number of infested fruit, Dit. On some survey visits, the number of fruit inspected or cut, xit, is recorded. For each visit the outcome, yit ∈ {0, 1}, (absent, present) is recorded.

Two observation models are suggested by the data, assuming that the infested fruit are sampled independently across the cell. For the case where the number of trees examined is known, but the number of fruit examined or cut is not, suppose that the proportion of fruit that can be reliably inspected on a tree is η, so that the eﬀective area of fruit searched on trees in the cell is ηmit/Mi; then the outcome is distributed,

Dit yit ∼Bernoulli(1 − (1 − ηmit/Mi) ). (7.2)

For the case where the number of fruit inspected is recorded, an alternative observation model is,

xit yit ∼Bernoulli(1 − (1 − ηDit/(Fit + Dit)) ) (7.3) 162 CHAPTER 7. MANGO CATERPILLAR INVASION ECOLOGY

To accommodate uncertainty about the proportion of fruit inspected or cut, an overdispersed model with η being beta distributed with hyperparameters aη, bη could be chosen. As the season progresses and fruit grow, damage on individual fruit becomes more pronounced. Consideration should therefore be given to modelling this component of detectability over the season, to more accurately reﬂect the diﬀerent weights of the observation information.

If the number of infested and uninfested fruit were counted, then a standard beta- binomial model (Gelman, 2006) would make stronger inference on the true proportion of infested fruit. However, these data were sacriﬁced in the ﬁeld in favour of greater spatial coverage.

In addition to fruit inspection and cutting, trapping for adult males has been conducted sporadically to trial a pheromone that was developed for the pest (Gibb et al., 2007). A Poisson observation model could be constructed with a trap attrac-

A tancy parameter, θ for adult males, Nit , in the population, so that,

A yit ∼Poisson(θNit ) (7.4) or a negative-binomial model could be adopted if the data were thought to be overdispersed. Further spatial embellishments to models for insect trap attractancy can be found in Barclay and Humble (2009) and Byers (2009).

Hierarchical Bayesian models provide an excellent platform to incorporate multiple data sources into the estimation of parameters (Clark, 2005). Each of these data sources, with their respective models, is capable of contributing information about the population status within the hierarchical structure through the construction of conditionally independent components. Further models that include the eﬀect of community reporting of pests could also be accommodated as further evidence for the absence of the pest in an area (Cacho et al., 2010).

The base models presented carry some important caveats. Inspection of a tree involves a selective process that violates the assumption of independence for the 7.3. DATA AND OBSERVATION MODEL 163 models suggested. The motivation for inspectors is to ﬁnd as many infested sites as possible in a limited time. Therefore it is common sense on approaching a tree to scan it for signs of damage, and to select the most obviously damaged fruit if there is one present. Positive outcomes recorded against inspection of a handful of fruit, or even a single fruit, are not uncommon. While absence data, with their complementary search intensity information, provide likelihood information that contradicts high infestation rates, presence data from apparently low intensity searches cannot be readily interpreted as contradicting low infestation rates. This of course has ﬂow on implications for the population process model.

The visual inspection scenario presented illustrates that uncertainty is difficult to standardise for observation models. Consistent procedures for data collection and searching would alleviate the need to define different models but, notwithstanding this, variation between observers and tree architecture still contribute greatly to the uncertainty in what the data say about the underlying population. Empirical testing of observer effectiveness may offer one solution to standardising the information collected. Collection of covariate data on the tree architecture and fruiting status in more structured formats may provide for more tractable statistical modelling. However, the cost of collecting these data may not outweigh the potential information from new detections that could be made, given the intuitive selective searching by observers. It is perhaps more effective to cultivate these pest targeting skills in observers and to understand the information in order to bring additional power into the inferential framework.

A choice of basic observation models has been presented to calculate the likelihood of surveillance outcomes given the infestation status. These base models could be further developed to better model the interactions between observers and infestation. Some of these extensions may be informed by additional knowledge about the pest population that is treated in the following section. 164 CHAPTER 7. MANGO CATERPILLAR INVASION ECOLOGY

7.4 Local Ecological Process Modelling

The relationship between the number of infested fruit and the RBMC pest population involves a mutual feedback process. In this section, a local stochastic simulation model is constructed to represent the population dynamics on mango trees within a cell over the season in the absence of dispersal. The purpose of constructing a within cell model is to diﬀerentiate the population changes that can be attributed to local exploratory movements of adults from those due to longer distance dispersal events (Van Dyck and Baguette, 2005). A bimonthly time-step is adopted to allow lifestage transitions within a cell to be modelled with a biologically reasonable resolution while keeping computation manageable.

The simulation model contains many parameters for both the fruiting and population components. Values for these parameters have been chosen based on the pest literature and personal observations in the study area. An extensive sensitivity analysis is outside the scope of this investigation, but the implications of choosing this particular parameterisation are discussed in terms of the additional uncertainty within a hierarchical Bayesian setting. Table 7.6 on p.181 provides a description of all the parameters, variables and constants used in this section.

7.4.1 Mango fruit development

Both the ecology of RBMC and its observation in the field are inextricably linked to the changing fruiting status over the season. The onset and suspension of pre-pupal dormancy, oviposition preferences for fruit of different sizes and larval mortality are all suspected to be influenced by the fruiting state of the tree (Krull, 2004). To model the population ecology of RBMC, particularly in relation to the observed symptoms, it is necessary to model the pest’s seasonal interaction with its food resource. 7.4. LOCAL ECOLOGICAL PROCESS MODELLING 165

Spatial variability in environmental conditions and proximity to the equator contribute to a poorly synchronised fruiting across the NPA. Year to year variation in the crop timing and fruit load is also high. In the study area, fruit that could be infested with RBMC may be present from July to February the following year, with a range of fruit sizes present at a given time. As the surveillance data contain no information on the fruiting status of the trees, a model for variable fruiting phenology is proposed.

Consider fruit maturity classes, c = {1,...,C}, that may be present within a cell at a particular time. Over the season, the fruit class distribution is required at bimonthly intervals, w = {1,...,W }, so that the fruit class distribution of the population over a season is given by the matrix, Fc,w. The fruit class c = 1 is a nominal state that holds all of the potential fruit that will progressively form over the season. Oviposition on mango fruit can occur from the time they reach marble size (10-15 mm, 2g) (Krull, 2004) and so the class c = 2 is used to represent this ﬁrst susceptible class for the pest. Subsequent classes from c = 3, 4,...,C − 1 represent bimonthly progression in fruit sizes. The ﬁnal class, c = C represents the number of fruit that have fallen to the ground. Let the potential number of fruit set by a tree that reach marble size be n, so that the number of potential fruit for a cell, i, in a season with Mi trees is initialised with F1,1 = nMi. The number of fruit of class c, at each time-step, w is contained in,

  F11 F12 ...F1W    .   F21 F22 ... .  F =   , (7.5)  . . .. .   . . . .    FC1 FC2 ...FCW where the notation F•w is used to represent the fruit class distribution at time w. A C × C transition matrix, G, deﬁnes the process that advances fruit cohorts at each time period. The parameters that determine transition of fruit classes are the probability that susceptible fruit are initiated at a seasonal time, Aw, and the 166 CHAPTER 7. MANGO CATERPILLAR INVASION ECOLOGY

fruit drop rate for the diﬀerent fruit classes, Bc. Values for these parameters have been derived from published studies and personal observations in the NPA and are discussed later in this section.

The proportion of fruit moving from the pre-susceptible stage, c = 1, to being available for oviposition, c = 2, for each time period, w, is denoted as Aw. Susceptible fruit may either advance to the next fruit class, or fall from the tree, at a time step.

Letting the probability of a fruit remaining on the tree from class c to c + 1 be Bc for c = {2,...,C − 2}, the transition probabilities for each time w = 1, 2,...,W are,

  (1 − Aw) 0 0 ... 0 0 0    .   Aw 0 0 . 0 0 0     .   0 B2 0 . 0 0 0     .  G(w) =  0 0 B3 . 0 0 0  . (7.6)    ......   ......       0 0 0 ...BC−2 0 0    0 (1 − B2) (1 − B3) ... (1 − BC−2) 1 0

A deterministic model for fruiting status could be implemented as,

F•w+1 = G(w)F•w . (7.7)

However, to incorporate variability in fruiting status, the transition matrix for each

0 year is simulated by stochastically drawing a transition matrix, G(w), from a Dirich- let distribution with values,

0 G(w)•c ∼ Dirichlet(10G(w)•c), (7.8) where the variability produced by the value of 10 was considered to produce a reasonable expression of fruiting status between seasons. 7.4. LOCAL ECOLOGICAL PROCESS MODELLING 167

Further stochastic variability is modelled by sampling the transition of the fruit to the next class from a multinomial distribution. For a time step, w, the model is implemented for each current fruit class, c, and summing the samples,

C X 0 F•w+1 ∼ Multinomial(Fc,w, G(w)•c). (7.9) c=1

As the number of infested fruit is also required for the observation model, a separate matrix for infested fruit, D, is maintained. Infested fruit have their own transition matrix, H, developed along the same lines as for G, but with slightly diﬀerent parameters discussed in the following section.

Parameterisation

Fruit drop in the early stages of mango development in managed orchards is severe and generally only 10-20% of fruit make it from marble size to the mature ripe stage (Catchpole and Bally, 1991; Chadha and Singh, 1964). In the NPA, further attrition due to birds and bats results in perhaps 50-100 ripe fruit per tree surviving to maturity. Most of the accessible trees in the area are mature with an expectation of around 800 marble-sized fruit per tree. Reasonable values for fruit induction over the season, Aw, were gleaned from expert opinion and are given in Table 7.1.

Table 7.1: Proportion of fruit induced over the season, Aw, where month subscripts refer to the 1st or 2nd half of the month.

Time Jun2 Jul1 Jul2 Aug1 Aug2 Sep1 Sep2 Oct1

Aw 0.01 0.04 0.2 0.3 0.2 0.1 0.1 0.05

Infestation by RBMC is reported to cause fruit to drop prematurely from the tree (Leefmans and van der Vecht, 1930; Sahoo and Jha, 2009), although Krull (2004) suggests the eﬀect is not as noticeable on young fruit. To accommodate the eﬀect of caterpillars on fruit drop, the number of infested fruit classes on the tree at a time, 168 CHAPTER 7. MANGO CATERPILLAR INVASION ECOLOGY

D•w, is modelled in a similar fashion to the uninfested fruit F•w (i.e. Equation 7.9). However, the transition of fruit is sampled using the slightly modiﬁed fruit drop rates shown in Table 7.2.

Table 7.2: Probability of fruit advancing to the next fruiting class, Bc, for infested and uninfested fruit and the expected number entering the class.

Class (c) 2 3 4 5 5 7 Weight (g) 2 15 60 120 200 350 Uninfested Fruit (G) 0.40 0.50 0.75 0.85 0.90 0.00 Infested Fruit (H) 0.35 0.45 0.65 0.70 0.60 0.00 Expected No. Entering Class 800 320 160 120 102 92

Simulations of the fruiting phenology model for uninfested fruit, shown in Figure 7.2, illustrate the seasonal variability that the model expresses between years. Some realisations display the bimodal ﬂushes of fruit that are commonly noted in the NPA.

7.4.2 RBMC life history model

The population process model is based on the sparse, and somewhat conﬂicting, literature on RBMC biology, as well as personal observations made during surveillance visits from 2001 - 2005. Life history information, provided in Table 7.3, has been taken from the work of Golez (1991); Sahoo and Jha (2009); Sujatha and Za- heruddeen (2002); Tenakanai et al. (2006) and Sarker et al. (2007). The information presented is of varying quality and highlights the uncertainty around parameter estimates that must be introduced into the model. As Papua New Guinea (PNG) is the most likely genetic source for the NPA incursion, the lifecycle model is more heavily weighted towards this information.

Two generations throughout the fruiting season have been reported in Papua New Guinea where there are often two distinct ﬂowerings a year (Tenakanai et al., 2006). In India, three to four generations have been reported (Sujatha and Zaheruddeen, 7.4. LOCAL ECOLOGICAL PROCESS MODELLING 169 No. fruit No. fruit No. fruit

time time time fruit class fruit class fruit class No. fruit No. fruit No. fruit

time time time fruit class fruit class fruit class

Figure 7.2: Nine independent realisations from a model of temporal variability in fruiting status Fc,w, with smallest fruit class at the back and mature fruit at the front. Seasonal time is represented as bi-monthly classes. 170 CHAPTER 7. MANGO CATERPILLAR INVASION ECOLOGY

Table 7.3: Lifestage length (days) and fecundity reported in the literature for red banded mango caterpillar in Papua New Guinea (PNG), the Philippines (PHL), Andhra Pradesh, India (APR), West Bengal, India (WBN), Bangladesh (BGD).

PNG PHL APR WBN BGD Egg 8-12 3-4 2-3 7-8 Larva 11-21 14-20 12-18 11-13 18-21 Pre-pupa 4-14 2-3 3 5-6 Pupa 5-20 9-14 11-15 9-11 10-12 Pre-oviposition 3-5 2 2-3 Eggs female−1 6-100 46-163 4-38 10-21

2002). Cohort data is not available from these studies. A six week lifecycle from egg to egg has been adopted and is modelled on a two week time-step.

Modelled lifestages

The lifecycle is broken down into three active stages and a dormant stage, with the population in each stage denoted as; N E12 - eggs, ﬁrst and second larval instars, N L35 - third to ﬁfth larval instars, N PPA - prepupae, pupae and adults and N DPA - dormant prepupae, pupae and adults. Transitions between lifestages and mortality and dormancy factors at each time step are shown in Figure 7.3. Throughout the season, egg and larval stages are assigned to a susceptible fruiting class, c, so that

E12 at a time, w, the number of fruit infested by lifestage classes are given by Nc,w L35 and Nc,w . Dormant pupae and adults are not assigned to a fruiting class and are therefore only indexed by the time w.

Emergence

Each season starts with the emergence of moths that have spent the non-fruiting season in a pre-pupal resting stage in the bark of the tree. Emergence of overwin- tering moths is believed to be triggered by the physiological state of the tree (Krull, 7.4. LOCAL ECOLOGICAL PROCESS MODELLING 171

3 NE12 1 NL35 1 Fruit drop

2 Seasonal 1 1 3 Fruit available 3 2 2 4 Distance

2 3

NDPA NPPA

3 3 Local

Dispersing

Figure 7.3: Lifestage transitions between time steps and influencing factors. The numbers in each lifestage are for eggs and the first two larval instars (N E12), third to fifth instars (N L35), dormant pre-pupae (N DPA), and pupae and adults (N PPA). Influencing factors, given in the numbered legend, are indicated on the transition lines between stages. 172 CHAPTER 7. MANGO CATERPILLAR INVASION ECOLOGY

2004) and is suspected to be staggered over the early part of the season. Pheromone trapping in the NPA suggests that the number of adults looking for mates begins to drop oﬀ in November once the fruit are mature, when resources for full development of larvae are less assured (Yarrow and Chandler, 2007). Prior estimates of

e the probability of pre-pupae emerging from dormancy, δw, and entering dormancy, d δw for time, w, are shown in Table 7.4.

Table 7.4: Probability of a pre-pupa emerging from and a larva entering a dormant state over the season.

Emerge Jul1 Jul2 Aug1 Aug2 Sep1 -- δe 0.05 0.2 0.6 0.9 1 - -

Dormant Oct2 Nov1 Nov2 Dec1 Dec2 Jan1 Jan2 δd 0.4 0.5 0.6 0.8 1 1 1

Those moths emerging from dormancy at time w are denoted N AE and are modelled as,

AE DPA e Nw+1 ∼ Binom(Nw , δw). (7.10)

Moths in this temporary category are then removed from N DPA in its transition from w to w + 1 and added to the existing adult population for calculating reproduction.

Oviposition

Oviposition behaviour is introduced here in terms of the local model and is later revisited in conjunction with dispersal. Golez (1986) observed caged adults laying 5-9 clusters with cluster sizes of 6-18 eggs. In PNG ﬁeld studies, Krull (2004) found eggs most commonly laid in pairs with 90% of clutches containing between 1 and 6 eggs. In India, adults laying 1-10 eggs in 2 to 6 clusters are reported (Sujatha and Zaheruddeen, 2002).

Egg clusters are laid by moths emerging from the dormant pre-pupal state (N AE) 7.4. LOCAL ECOLOGICAL PROCESS MODELLING 173 and by moths that did not enter the dormant state (N PPA). The number of egg clusters available for laying at time w + 1 is modelled as,

e AE PPA Rw+1 ∼ Poisson(ρ(δwNw+1 + Nw )), (7.11) where ρ is a parameter for the number of egg clusters per female. Based on the life history data, ρ = 7 is used for the simulation.

It is apparent from the laboratory fecundity information presented in Table 7.3 that there is potential for rapid population growth. However, this is unlikely to be realised in the ﬁeld. Surveillance in the NPA has rarely found infestation rates greater than 30%, which is comparable to infestation rates reported in Papua New Guinea (Krull, 2004; Waterhouse, 1998) and in the Philippines (Golez, 1991). Infestation rates between trees in the same area can be highly variable, especially during a colonisation period (Krull, 2004).

Density dependence is incorporated into the model by assuming that fruit infestation is limited by an a priori availability of suitable oviposition sites near the fruit. A limit is imposed on the maximum proportion of fruit that may be infested,

K, here taken as 25%. Assuming that infested fruit, Dc,w, are also unsuitable, the number of fruit that are suitable for oviposition in each class / time, qc,w, is,

qc,w = (Fc,w + Dc,w) K − Dc,w : qc,w ≥ 0, (7.12)

where Fc,w and Dc,w are the numbers of uninfested and infested fruit.

Populations within newly colonised orchards in PNG are reported to build up on individual trees before the pest spreads out to other trees (Krull, 2004), suggesting that local resources are likely to be used first. In Cactoblastis cactorum, a well studied pyralid moth, there is evidence that newly mated moths explore their local 174 CHAPTER 7. MANGO CATERPILLAR INVASION ECOLOGY area for suitable oviposition sites before dispersing further afield (Myers et al., 1981). The selection of oviposition sites by moth species may be determined by the composition of fruit volatiles (Sidney et al., 2008) or influenced by the density of nearby larvae or other pest species (Harmon et al., 2003). While the cues for selecting oviposition sites are unknown, the important characteristic of the RBMC population process is to model the probability that eggs are laid locally. It is assumed that moths emerging are immediately mated and search fruit in their local cell at random. Let the probability that a moth lands on a suitable fruit, Qc,w+1, in class c is,

q Q = c,w (7.13) c,w PC−1 c=2 (Fc,w + Dc,w)

If the fruit is unsuitable for oviposition, it tests another, up to a total of ν fruit. Here, ν is set to 15 oviposition attempts before the adult leaves the local area. Marble size fruit of 10-20mm are reported to be the most commonly attacked by RBMC in the ﬁeld in Papua New Guinea (Krull, 2004) while in cage trials in the Philippines, oviposition is reported to be most common on medium size fruit (75 to 85 days after induction) (Golez, 1991). As there is evidence that moths preferentially prefer smaller fruit, let the probability that it would lay if it ﬁnds a suitable fruit in class c to be ωc. Parameter values selected for ωc are shown in Table 7.5.

Table 7.5: Probability of ovipositing on fruit of each size class after testing.

Weight class (g) 2 15 60 120 200 350

ωc 0.2 0.3 0.2 0.15 0.05 0

For each egg cluster, the probability of ν unsuccessful attempts at ovipositing at a

0 time, Pw, is calculated, 7.4. LOCAL ECOLOGICAL PROCESS MODELLING 175

C !ν 0 X Pw = 1 − (Qc,wωc) . (7.14) c=1

The probability of successfully ovipositing on each class is,

ωcQc,w 0 Pc,w = P (1 − Pw). (7.15) (ωcQc,w)

0 For convenience, the probability of failing to oviposit Pw, can be assigned to the ∗ pre-susceptible fruit class D1,w. The fruit class selected for each cluster, Dc , can be sampled from,

∗ D• ∼ Multinomial(1,P•w+1), (7.16) and the number of infested fruit and eggs updated,

∗ Dc,w = Dc,w + Dc , (7.17)

E12 E12 ∗ Nc,w = Nc,w + Dc . (7.18)

As the success of each attempted egg lay changes the probability of success for subsequent egg lays, the model simulates oviposition on a fruit class iteratively for each of the Rw+1 clusters through Equations 7.12 - 7.18.

Those egg clusters that are not deposited in the local cell, (i.e. D1,w) are considered to be available to emigrating moths and are examined in Section 7.5.4.

Eggs and larval mortality

Eggs are generally laid in clusters on the fruit stalk under dried sepals or in crevices (Golez, 1991; Sahoo and Jha, 2009; Sujatha and Zaheruddeen, 2002). Egg para- sitism has been reported but seems to be uncommon (Krull, 2004). Eggs hatch from the egg mass together and enter the fruit at the same time. The distribution of early instar numbers observed in fruit in the NPA corresponds well with the 176 CHAPTER 7. MANGO CATERPILLAR INVASION ECOLOGY numbers reported in egg masses. As this suggests that mortality factors are likely to aﬀect the whole egg mass, a cohort of eggs is treated as the smallest unit in the model. Mortality within early larval stages for Lepidoptera in general are poorly understood. However, the egg and larval stages for RBMC are well protected from predators and have a stable environment within the fruit, therefore mortality may be comparatively low for those that remain on the tree (Zalucki et al., 2002).

RBMC larvae are frequently found in fruit on the ground where they are susceptible to predators (Golez, 1991). Vespid wasps have been observed to prey on the larvae as they leave the fruit to pupate (Waite, 2002). Krull (2004) speculates that un- observed night time predation may also play an important role in larval mortality. Pupation on the ground (Golez, 1991) and pupation in fruit (Sarker et al., 2007) have both been reported, however, an intensive soil sifting investigation failed to ﬁnd evidence of this in the NPA (Trinca and Foulis, 2002). For the purposes of modelling, it is assumed that larvae in fallen fruit fail to successfully pupate.

Half of the egg and larval stage, N E12, will be spent on the fruit stalk which is retained on the tree, leaving larvae available to ﬁnd other fruit on hatching. The number surviving the stage to become late instar larvae, N L35, is modelled as a

E12 binomial probability of the N being in retained infested fruit, from Dc,w to

Dc+1,w+1, so that,

L35 E12 Nc+1,w+1 ∼ Binom Nc,w , (Dc+1,w+1 − Dc,w) /2Dc,w . (7.19)

Note that as the fruit drop and mortality components occur within the model before the oviposition components, the probability for Equation 7.19 is always valid.

Late instar stages that survive either become pre-pupae that directly develop into

d adult moths, or become dormant pre-pupae with a probability, δw, given in Table P 7.4. Let the total number of survivors from the late instar stage be Nw+1, so that, 7.4. LOCAL ECOLOGICAL PROCESS MODELLING 177

C P X L35 Nw+1 ∼ Binom Nc,w , (Dc+1,w+1 − Dc,w) /Dc,w (7.20) c=1 DPA P d Nw+1 ∼ Binom(Nw+1, δw) (7.21)

PPA P DPA Nw+1 = Nw+1 − Nw+1 . (7.22)

7.4.3 Local population model simulations

Simulations of the model were run for a 16 year sequence for 10 trees initialised with a starting population of 100 dormant pupae. It can be seen from Figure 7.4 that the larval population generally shows two peaks over the season, although the timing and magnitude are quite variable. The dormant pre-pupal population between fruiting seasons ranges between 10-30 per tree. Variability in the distribution of adult moths over the season is particularly high.

Figure 7.5 portrays the simulated number of infested and uninfested fruit over the season, as well as the proportion of infested fruit. There is generally a dip in the proportion of infested fruit corresponding to that in the aforementioned larval populations. Ideally, the simulations would be used to construct a distribution of detectability over the season that could be incorporated into an observation model. However, the relationships over the season are weak compared to the between-season variation so it unlikely that they would provide substantial information to a future analytical model.

Other simulations, not presented here, showed that the model was highly sensitive to many of the parameter values. A particular feature of the simulation is the large number of eggs that cannot be laid over the season due to the lack of suitable oviposition sites. As there is no empirical information available to verify these sensitive relationships, any future analytical model would need to accept a great deal of uncertainty when relating observations of absence in fruit to absence of all life 178 CHAPTER 7. MANGO CATERPILLAR INVASION ECOLOGY

Year 1 Year 2 Year 3 Year 4

E12 E12 E12 E12 L35 L35 L35 L35 DPA DPA DPA DPA

300 PPA 300 PPA 300 PPA 300 PPA Number Number Number Number 100 100 100 100 0 0 0 0