Dynamic Transmission Models: the Impact of Behavioural Feedbacks And

Dynamic Transmission Models: The Impact of Behavioural Feedbacks and

Parametrization Methods on Disease Intervention Effectiveness

Michael Andrews

A Thesis presented to The University of Guelph

In partial fulﬁlment of the requirements for the degree of Doctor of Philosophy in Mathematics

Guelph, Ontario, Canada

c Michael Andrews, October, 2016 ABSTRACT

DYNAMIC TRANSMISSION MODELS: THE IMPACT OF BEHAVIOURAL

FEEDBACKS AND PARAMETRIZATION METHODS ON DISEASE

INTERVENTION EFFECTIVENESS

Michael Andrews Advisor: University of Guelph, 2016 Chris T. Bauch

Infectious diseases impose signiﬁcant health and economic burdens across the world, continuously threatening human quality of life. Mathematical models of infectious disease epidemiology can help to gain insight on potential health outcomes of a population that is vulnerable to disease spread. Models that incorporate decision-making mechanisms can furthermore capture how behaviour-driven aspects of transmission such as vaccination choices and the use of non-pharmaceutical interventions (NPIs) interact with disease dynamics.

In this thesis, we present three models of disease spread using the dynamic transmission and behaviour-disease modelling frameworks. Firstly, we investigate an age-stratiﬁed compartmental model of inﬂuenza transmission and the impact of estimating this model’s parameters using different surveillance data has on population level outcomes. In the latter two models, we remove random mixing assumptions and incorporate an individual-based approach. We also simultaneously integrate an individual’s decision making processes for utilizing two fundamental disease interventions: vaccination and NPIs. In the past, models have only considered the use of these interventions separately. All of our approaches focus on examining health outcomes of populations that are exposed to acute self-limited diseases, and offering insight on the effectiveness of disease mitigation strategies. iv

To my parents, who have always been supportive of my academic endeavours. v

Table of Contents

List of Tables viii

List of Figures x

1 Introduction 1 1.1 Infectious Disease Burdens ...... 1 1.2 Infectious Disease Modelling ...... 3 1.2.1 Homogeneous Models ...... 4 1.2.2 Network Models and Heterogeneous Contact Patterns ...... 6 1.2.3 Deterministic and Stochastic Modelling ...... 9 1.3 Disease Interventions ...... 10 1.3.1 Vaccination ...... 11 1.3.2 Non-Pharmaceutical Interventions ...... 12 1.4 Behavioural Epidemiology of Infectious Diseases ...... 13 1.4.1 Example of a Behaviour-Disease System ...... 16 1.5 Overview and Objectives ...... 19

2 Parameter Estimation in a Dynamic Model of Influenza Transmission Using Laboratory Confirmed Influenza Cases 21 2.1 Chapter Abstract ...... 21 2.2 Introduction ...... 22 2.3 Methods ...... 24 2.3.1 Population Demographics ...... 25 2.3.2 Influenza Incidence Data and Epidemiology ...... 25 2.3.3 Vaccination ...... 27 2.3.4 Model Structure ...... 28 2.3.5 Parameter Fitting ...... 30 Longitudinal Method ...... 31 Cross-Sectional Method ...... 33 2.4 Results ...... 35 vi

2.4.1 Parameter Fitting Comparison ...... 35 2.4.2 Projected Impact of Expanded Vaccination Coverage ...... 38 2.5 Discussion ...... 41

3 The Impacts of Simultaneous Disease Intervention Decisions on Epidemic Out- comes 45 3.1 Chapter Abstract ...... 45 3.2 Introduction ...... 47 3.3 Methods ...... 52 3.3.1 Disease Dynamics ...... 52 3.3.2 Contact Network ...... 53 3.3.3 Non-Pharmaceutical Interventions and Vaccination ...... 53 3.4 Results ...... 57 3.4.1 Baseline Dynamics ...... 57 3.4.2 Transmission Rate ...... 61 3.4.3 Vaccine Efﬁcacy ...... 65 3.4.4 Pairwise Correlations ...... 67 3.5 Discussion ...... 70 3.6 Supporting Information ...... 75 3.6.1 Asymptomatic Cases ...... 75 3.6.2 Network Types ...... 78 3.6.3 Pairwise Analysis ...... 84

4 Disease Interventions Can Interfere With One Another Through Disease- Be- haviour Interactions 86 4.1 Chapter Abstract ...... 86 4.2 Introduction ...... 88 4.3 Model ...... 91 4.3.1 Vaccination ...... 91 4.3.2 Non-Pharmaceutical Interventions ...... 97 4.3.3 Transmission Dynamics ...... 99 4.3.4 Model Calibration ...... 101 4.4 Results ...... 103 4.4.1 Baseline Scenario ...... 103 4.4.2 Interventions Can Interfere With One Another ...... 104 4.4.3 Determining Which Interventions Interfere Most Strongly . . . . . 109 Impact of Interference on Intervention Uptake Rates ...... 109 Impact of Interference on Inﬂuenza Incidence ...... 110 4.4.4 Understanding What Drives Different Levels of Interference for Different Interventions ...... 114 4.5 Discussion ...... 117 vii

5 Conclusion 122 5.1 Conclusions and Future Work ...... 122

References 125 viii

List of Tables

2.1 Parameter Descriptions ...... 34 2.2 Best fitting parameter values (mean and standard deviation) for the longitudinal method...... 36 2.3 Best fitting parameter values (mean and standard deviation) for the cross- sectional method...... 37 2.4 Mean number of cases for influenza strains A and B under different vaccination scenarios...... 40

3.1 Baseline Parameter Values...... 57 3.2 Epidemic final sizes (symptomatic cases only) with delayed vaccine availability. β = 0.004 ...... 76 3.3 Epidemic final sizes (symptomatic cases only) with delayed vaccine availability. β = 0.005 ...... 76 3.4 Epidemic final sizes (symptomatic cases only) with delayed vaccine availability. β = 0.006 ...... 77 3.5 Epidemic final sizes corresponding to vaccine efficacy (symptomatic cases only)...... 77 3.6 Population vaccine uptake corresponding to vaccine efficacy...... 77 3.7 Epidemic final sizes with delayed vaccine availability (random network). β = 0.00485 ...... 79 3.8 Epidemic final sizes with delayed vaccine availability (random network). β = 0.00585 ...... 80 3.9 Epidemic final sizes with delayed vaccine availability (random network). β = 0.00685 ...... 80 3.10 Epidemic final sizes corresponding to vaccine efficacy (random network). . 81 3.11 Population vaccine uptake corresponding to vaccine efficacy (random network)...... 81 3.12 Epidemic final sizes with delayed vaccine availability (power law network). β = 0.055 ...... 82 ix

3.13 Epidemic final sizes with delayed vaccine availability (power law network). β = 0.075 ...... 83 3.14 Epidemic final sizes with delayed vaccine availability (power law network). β = 0.095 ...... 83 3.15 Epidemic final sizes corresponding to vaccine efficacy (power law network). 83 3.16 Population vaccine uptake corresponding to vaccine efficacy (power law network)...... 84

4.1 Model parameters with baseline values and sources...... 96 4.2 Sampling ranges for parameters used to obtain 100 baseline sets...... 103 4.3 Acceptance ranges for simulation averages across 30 seasons...... 103 x

List of Figures

1.1 Time series of an epidemic...... 18

2.1 Diagram of the age-stratified SIRS compartmental model with vaccination. 31 2.2 Time series of confirmed influenza cases...... 35 2.3 Age-stratified cumulative cases for influenza compared to empirical targets. 38 2.4 Time series of confirmed influenza A and B cases in our model with different vaccination scenarios...... 40 2.5 Age stratified cumulative cases in our model for influenza A and B with different vaccination scenarios...... 42

3.1 Time series of an epidemic, 95% confidence intervals shown every 10 days around the mean of 500 realizations...... 60 3.2 Time series of infection prevalence with the vaccine-only scenario, the NPI-only scenario, and the combined scenario...... 62 3.3 Epidemic final sizes with respect to when vaccination is made available. . . 64 3.4 Epidemic measures with respect to transmission rate...... 66 3.5 Effects of vaccine efficacy between scenarios with and without NPIs. . . . . 68 3.6 Time series of epidemics over different values of σ, the weighting for global versus local information...... 71 3.7 Frequency of node degrees in the empirically based network...... 78 3.8 Pairwise sensitivity analysis of parameters λ and γ...... 85

4.1 Vaccination and NPI decisions...... 92 4.2 Seasonal time series...... 94 4.3 Impact of vaccine introduction...... 105 4.4 The effects of social parameters on interventions...... 107 4.5 The effects of infection and vaccination costs on interventions...... 108 4.6 Interference between vaccination and NPIs...... 112 4.7 Interference between vaccination and NPIs...... 113 4.8 The effects of NPI efﬁcacy...... 115 xi

4.9 The effects of vaccine efﬁcacy...... 116 1

Chapter 1

Introduction

1.1 Infectious Disease Burdens

Infectious diseases impose signiﬁcant health and economic burdens across the world, continuously threatening human quality of life (Klein et al., 2007). Throughout history, infectious disease epidemics have negatively impacted human societies causing signiﬁcant morbidity and mortality. For example, the “Black Death” in the mid 1300’s caused millions of deaths, eliminating a considerable portion of the human population at the time (Kelly,

2006). Also, the cholera epidemic in the 19th century in Europe, and eventually spreading to North America, killed hundreds of thousands. In fact, this disease still affects developing countries today (Barua, 1992). Cholera in particular is an important historical disease, as a physician named John Snow made major groundbreaking contributions to the ﬁeld of epidemiology through his studies of cholera throughout his medical career (Timmreck, 2

2002).

More recently, the 20th century was also marred by disease outbreaks. Beginning in

1918, an influenza pandemic swept across the world, killing millions (Crosby, 1989). This particular flu was unordinary in that many of the deaths occurred amongst young adults- an unusual characteristic of influenza (Taubenberger et al., 2000). Two more influenza pandemics also occurred in 1957 (the Asian flu) and 1968 (the Hong Kong flu). These viruses also spread worldwide, with infants and the elderly being the most likely to suffer severe complications (Hsieh et al., 2006). Finally, in the late 1970’s and early 1980’s, patients with symptoms such as fever, weight loss, and swollen lymph nodes began to appear in California and New York. These cases would soon be identified as HIV/AIDS, and the HIV virus has since been a worldwide pandemic responsible for the deaths of millions (Grmek, 1990).

The 21st century has been affected by several disease epidemics as well. In 2003, an outbreak of severe acute respiratory syndrome (SARS) was reported in China, prompting health agencies to issue global alerts. Fortunately, due to a fast response of surveillance and patient isolation, there were relatively few fatalities worldwide due to this virus (Pear- son et al., 2003). Yet another inﬂuenza pandemic in 2009 occurred when an inﬂuenza A

(H1N1) virus emerged and spread worldwide. Again, although the number of deaths directly caused by this pandemic are relatively low compared to the previously mentioned inﬂuenza pandemics, the economic impacts of an extensive disease spread such as this can be large. For example, the impact of H1N1 on Mexico was estimated to be upwards 3 of 3.2 billion (Girard et al., 2010). More recently, the Middle East respiratory syndrome

(MERS-CoV) saw cases ﬁrst develop in June 2012, with the ﬁrst clusters occurring in Jor- dan (Balkhair et al., 2013). This virus is particularly deadly to the elderly or those with weakened immune systems, and can cause severe pneumonia and organ dysfunction, leading to a high case fatality rate (Balkhair et al., 2013). Finally, the 2014 outbreak of the

Ebola virus in West Africa quickly became the largest Ebola epidemic in history (WHO

Ebola Response Team, 2014). The most impacted countries faced extreme challenges in taking measures to control the disease, and the high case fatality rate caused international concern (WHO Ebola Response Team, 2014).

Infectious diseases not only negatively impact societies economically, but also the livelihood of their inhabitants. Efforts of researchers continuously advance our knowledge on how diseases spread, how to limit the transmission of said diseases, and how to treat individuals who may become or who have become infected. However, even with these advances, the spread of infectious diseases is difﬁcult to predict and can depend on many factors. Thus, researchers have also developed theoretical methods to help with predicting disease transmission, which we discuss in the next section.

1.2 Infectious Disease Modelling

Mathematical models of infectious disease epidemiology can help to gain insight on potential health outcomes of a population that is vulnerable to disease spread. Speciﬁcally, 4 we will discuss compartmental models, pioneered by Kermack and McKendrick (Kermack and McKendrick, 1927). In these models, the health statuses of members in a population are divided into different classes, or compartments, depending on their state with respect to a disease. For example, some members may be susceptible to becoming infected, often denoted as S, they may be currently infected, often denoted as I, or they may be recovered from infection, often denoted as R. Having these distinct compartments allows us to build a robust framework for infectious disease dynamics, and can allow us to better understand how a disease spreads over time.

1.2.1 Homogeneous Models

Homogeneous models of infectious disease dynamics are those that assume members of a population mix uniformly and randomly with each other (Bansal et al., 2007). Many traditional models use this mixing assumption, and thus they share the property that every susceptible member has an identical risk of becoming infected. One of the most fundamental compartmental models uses the 3 states discussed above: the susceptible-infected- recovered (SIR) model (Keeling and Rohani, 2008). The most basic form of this model can be written as follows:

dS = −βSI (1.1) dt dI = βSI − γI (1.2) dt dR = γI. (1.3) dt 5

Here, S is the fraction of the population that is susceptible, I is the fraction of the population that is infected/infectious, R is the fraction of the population that has recovered from infection, β is the disease transmission rate, and infected individuals recover in an average time of 1/γ.

The system above models a single epidemic spreading through a population over time.

Although simplistic, it can offer powerful insight on diseases with an SIR natural history, such as measles, mumps, or a single outbreak of inﬂuenza. Modiﬁed versions of this system can also create more complex scenarios that include birth and death rates, furthering the realism of the observed dynamics offered by the model (Keeling and Rohani, 2008).

Additionally, immunity from some diseases may only be temporary, causing recovered individuals to become susceptible again after a period of time (Keeling and Ro- hani, 2008). This phenomenon is called waning immunity, and can be modelled with a susceptible-infected-recovered-susceptible (SIRS) compartmental model, an example of which is shown below:

dS = −βSI + ρR (1.4) dt dI = βSI − γI (1.5) dt dR = γI − ρR. (1.6) dt

The parameter ρ allows individuals to move from the recovered compartment to the susceptible compartment after a time 1/ρ, allowing for a disease to potentially become endemic 6

in a population.

To expand on this, we introduce one of the fundamental measures in a model of disease

transmission, the basic reproduction number, denoted R0. R0 is deﬁned as the number

of secondary infections an infected individual will produce in an otherwise susceptible

population (Keeling and Rohani, 2008). In the case of the system (1.4)-(1.6), R0 can be

expressed as β R = . (1.7) 0 γ

The value of R0 can be an important determinant of disease outcomes in the population.

For example, if R0 < 1, the disease is not sufﬁciently infectious to invade the population successfully. However, if R0 > 1, an outbreak will occur and result in a disease with an

SIRS natural history to become endemic.

In Chapter 2 of this thesis, we make use of a variation of the compartmental models discussed above.

1.2.2 Network Models and Heterogeneous Contact Patterns

Networks (also called graphs) are a collection of objects (nodes) that are connected to each other by links (edges) (Ore, 1965). The number of these edges stemming from a given node is called that node’s degree, which can be used to measure the node’s importance on a network. Networks can be used to represent many physical phenomena, including the transmission of infectious diseases (Moore and Newman, 2000; Kuperman and Abramson,

2001; Pastor-Satorras and Vespignani, 2001; Sander et al., 2002; Sattenspiel and Simon, 7

1988; Kretzschmar and Morris, 1996; Newman, 2002; Keeling and Eames, 2005). In this case, the nodes on a network are individuals making up a population, and the edges between them represent pathways allowing for potential disease transmission. Under this assumption, contact patterns amongst the population become heterogeneous. Typically, networks used in an infectious disease model are undirected networks. That is, transmission between two connected nodes can happen in both directions.

There are many different formations a network can take. With the types of infectious disease models that we will discuss in this thesis, networks can be static like those mentioned above, where edges on the network do not change, or dynamic, where network edges may change over time due to nodes adapting to the presence of the disease around them

(Gross and Sayama, 2009; Gross et al., 2006; Shaw and Schwartz, 2008, 2010). Depend- ing on the network used, the distribution of the node degrees may vary. In this context, one common network type is a random network, where edges between nodes are created at random. One method of creating a random network is to sample a node’s degree from a Poisson distribution and create edges at random, then, the probability of a node having degree k is k¯ke−k¯ P (k) = , (1.8) k! where k¯ is the average node degree. In this type of network, the early growth rate of a disease and ﬁnal epidemic size are reduced in comparison to the homogeneous mixing models discussed in the previous section (Keeling and Eames, 2005). Other types of networks often seen in the mathematical epidemiology literature are scale-free networks. In these 8 networks, the node degree distribution follows a power law (Keeling and Eames, 2005), with the probability of a node having degree k being

P (k) ∝ k−α. (1.9)

This results in many nodes having a low degree, with few having a high degree. This type of topology has been empirically observed, for example, in human sexual contact networks

(Liljeros et al., 2001).

When a disease spreads on a network, susceptible nodes can only get infected by their infectious neighbours, which can be expressed by the probability 1 − (1 − p)Ninf , where

Ninf is the number of infectious neighbours an individual has, and p is the probability the disease is transmitted across the edges of the network. Thus, a node with a high degree, also called a “hub” node which are often seen in scale-free networks, experiences a higher risk of becoming infected, and infecting others (Keeling and Eames, 2005). Due to this me- chanic of disease spread, a network’s topology is a powerful determinant of disease spread.

For example, a network with a low node degree and a high amount of clustering, where clustering is determined using the number of triangles in the network, and a triangle is a set of edges that connects 3 nodes to each other (Keeling and Eames, 2005), can effectively decrease R0 (Keeling, 2005). 9

Speciﬁcally, R0 in a network can be approximated by

! ¯ kvar R0 = T k − 1 + (1.10) k¯

(Meyers, 2006; Anderson, 1997; Miller, 2007), where kvar is the variance in degrees on

the network and T is the average transmissibility, 1 − e−pγ, where p is the probability of

transmission along an edge per day and γ is the infectious period (Meyers, 2006; Miller,

2007).

Network based epidemiological modelling has undergone extensive research in the past.

For examples, see Refs. (Klovdahl, 1985; Moore and Newman, 2000; Kuperman and

Abramson, 2001; Pastor-Satorras and Vespignani, 2001; Sander et al., 2002; Sattenspiel and Simon, 1988; Kretzschmar and Morris, 1996; Newman, 2002; Gross et al., 2006; Shaw and Schwartz, 2008, 2010; Funk et al., 2008; Rocha et al., 2011; Wells et al., 2013; Tully et al., 2013). In this thesis, we utilize empirically based networks that are similar to the types discussed in this section to attempt to capture realistic disease dynamics within a population.

1.2.3 Deterministic and Stochastic Modelling

The differential equation based models discussed above can be described as deterministic models. These classes of models are deﬁned by the characteristic that their output is determined only by model parameters and initial conditions (Trottier and Philippe, 2000; 10

Keeling and Eames, 2005; Hethcote, 2000). This property thus leads to deterministic mod-

els being very robust in their solutions and predictions particularly for large populations

(Trottier and Philippe, 2000). However, ﬂuctuations in individual behaviour or chance

occurrences of disease transmission may also exist in populations. To help model these

aspects, stochastic models are also often used in infectious disease modelling.

Due to their random nature, stochastic models are not as robust as their deterministic counterparts. Although they are able to capture certain features of real phenomena that deterministic models do not, many varying outcomes can occur from the same set of parameters and initial conditions (Bartlett, 1960). Thus, the underlying dynamics are complex, and may not always be fully understood.

In this thesis, we make use of both deterministic and stochastic models of disease spread. Using the former approach, we aim to create a model for estimating key transmission parameters of both A and B strains of inﬂuenza. Using the latter approach, we will attempt to capture important heterogeneities such as behaviourally based decisions which can be instrumental factors in disease spread.

1.3 Disease Interventions

The spread of infectious diseases can be limited when a population utilizes disease interventions. In this section, we discuss the main types of disease interventions that are incorporated into models seen in this thesis. 11

1.3.1 Vaccination

Vaccines are an extremely effective tool for reducing disease spread. Not only are those that receive vaccination protected from infection, those who remain unvaccinated become indirectly protected as well - a phenomenon called herd immunity (Fine, 1993). Although powerful, vaccines are not necessarily fully effective. A vaccine’s efﬁcacy is the percentage reduction of disease amongst vaccinated individuals compared to their unvaccinated counterparts (Weinberg and Szilagyi, 2010). It is important for a vaccine to not only be ef-

ﬁcacious, but also safe. Drops in vaccine uptake amongst communities have occurred due to groups or individuals having safety concerns over receiving vaccinations (Omer et al.,

2009). For example, parents may refuse to vaccinate their children over concerns of negative reactions to vaccines (Brown et al., 2010), which includes beliefs that vaccines may cause autism in children (Gross, 2009).

Thus far, the only disease to be eradicated by vaccination efforts is smallpox (Hen- derson, 2009). Although this is a great triumph for medicine, to date, efforts to eliminate other vaccine preventable diseases have failed. For example, developing countries do not generally have the resources for high quality health care (Jamison et al., 2006), causing diseases like polio and cholera to remain endemic in several regions (World Health Organi- zation, 2015b,a). Moreover, vaccines are often voluntary in many health jurisdictions and thus individuals can choose whether or not they receive vaccination, or whether or not their children get vaccinated. As a result of this, vaccine refusal has led to a recent increase in the frequency of measles cases, where outbreaks often stem from unvaccinated individuals 12

(Omer et al., 2009). Furthermore, vaccine trends for inﬂuenza have been below targeted

values for years (Nichol, 2001; Kwong et al., 2007).

Maintaining high vaccine uptake amongst a population is important for reducing preva-

lence and outbreak frequency of vaccine preventable diseases. However, vaccine refusal

or lack of health care quality can detriment vaccination efforts. Thus, understanding this

often complex vaccination behaviour is an important aspect of predicting disease trends.

1.3.2 Non-Pharmaceutical Interventions

Like vaccination, non-pharmaceutical interventions (NPIs) are also used to inhibit the

spread of diseases. NPIs are a broad term referring to non-drug related actions taken by

individuals to reduce their risk of becoming infected. In this section, we will discuss some

of the main NPIs that are typically used to limit disease spread.

Two of the easiest NPIs one can practice are strict respiratory etiquette and hand hy-

giene (Centers for Disease Control and Prevention, 2015e,b). To help prevent the transmis-

sion of respiratory infections such as inﬂuenza, individuals can cover their mouths when

they cough or sneeze. In addition, hand washing with soap and water or alcohol based

sanitizer helps to limit disease spread (Ryan et al., 2001; Schurmann´ and Eggers, 1983).

Due to the general belief that respiratory infections are transmitted through water droplets

(Bridges et al., 2003), these simple preventive measures can be very effective.

Another common NPI often used is the isolation of infected individuals or those who may be exposed to infection (Centers for Disease Control and Prevention, 2015c). This is 13 a simple yet effective practice as the isolation of possibly infectious individuals effectively eliminates transmission pathways. Quarantine has been used throughout history, in the

Middle Ages to protect against plague (Centers for Disease Control and Prevention, 2015d), to more recently with health care workers returning from Ebola stricken countries (Drazen et al., 2014).

Finally, the term social distancing can be used for a broad category of self protective measures that can be taken against disease spread. These can include school closures, general avoidance of infected individuals, quarantine-like methods such as staying home while sick, or practices such as serosorting which can be seen among HIV positive individuals.

Practices such as these can either be enforced, such as closure of public areas, or taken on voluntarily by members of a community to help reduce their risk of becoming infected.

NPIs are a very commonly practiced intervention strategy used against the spread of infectious diseases. Along with vaccination, individuals can decide how and when they want to protect themselves from infection. This behaviour is an important aspect to capture and attempt to understand, as it can often play an instrumental role in infectious disease spread in a human population.

1.4 Behavioural Epidemiology of Infectious Diseases

Human behaviour can have a large impact on the spread of infectious diseases (Funk et al., 2010; Ferguson, 2007). People have been observed to change their regular social 14 routines in response to an epidemic, in order to reduce their risk of becoming infected (Lau et al., 2005; Philipson, 1996; Ahituv et al., 1996). Mathematical models of infectious disease epidemiology incorporate behavioural aspects of a population to help gain additional insight on realistic disease dynamics. Behavioural aspects that are typically modelled include the activation of NPIs amongst a population (Del Valle et al., 2005; Reluga, 2010;

Funk et al., 2008; Rizzo et al., 2014; Bagnoli et al., 2007; Fenichel et al., 2011; Poletti et al., 2009, 2012) and decision making in regards to vaccinating (Fine and Clarkson, 1986;

Bauch, 2005; Bauch and Earn, 2004; Fu et al., 2010; Perisic and Bauch, 2009, 2008; Zhang et al., 2010; Reluga et al., 2006; Salathe´ and Bonhoeffer, 2008). Decision making in regards to these interventions depends not only on the prevalence of a disease, but also social inﬂuence and personal beliefs (Streeﬂand, 1999; Cummings et al., 1979; Sturn et al., 2005).

Given that human behaviour and decision making is complex and can depend on many factors, researchers have used several methods to attempt to capture the effects of this behaviourally-based decision making on disease spread. For example, some models have used disease prevalence as the primary driver for the activation of intervention decisions

(Rizzo et al., 2014; Bagnoli et al., 2007; Del Valle et al., 2005; Epstein et al., 2008; Sahneh et al., 2012; Perra et al., 2011; Kiss et al., 2010; Misra et al., 2011; Tanaka et al., 2002). Un- der this rule, it has been observed that because increases in prevalence causes higher intervention use, and higher intervention use causes decreases in prevalence, in turn decreasing intervention use, disease cycles may occur. Another method of modelling behaviour that has been used is the concept of imitation (Poletti et al., 2009, 2012; Bauch, 2005; Fu et al., 15

2010; Salathe´ and Bonhoeffer, 2008; Reluga et al., 2006; Xia and Liu, 2013). Imitation is commonly referred to as the process in which individuals copy the behaviour of others who are performing better than them. In the context of infectious disease dynamics, those who perform “better” than a given individual means other individuals who are putting in less effort to protect themselves from a disease (i.e. not vaccinating or practicing NPIs) and still avoid becoming infected. Similarly to the prevalence-based examples, models using imitation dynamics have also observed disease cycles occurring in populations. Further approaches to capture the complexity of decision making are models that include the ability for individuals to have memory. This has been done by allowing individuals to have memory of past prevalence (Vardavas et al., 2007; d‘Onofrio et al., 2007), or experiences with a vaccine (Wells and Bauch, 2012). A different method to model the change in behaviour of individuals during an epidemic are models utilizing adaptive networks (Gross et al., 2006;

Shaw and Schwartz, 2008, 2010). Here, network edges are removed and recovered, thus mimicking the use of NPIs by eliminating transmission pathways. Finally, economically- based methods have been used where individuals seek to maximize their utility, which is derived from engaging in social contact (Geoffard and Philipson, 1996; Kremer, 1996;

Auld, 2003; Chen et al., 2011; Fenichel et al., 2011). This can also include using a game theory framework, where certain payoffs are associated with different outcomes, such as vaccinating, practicing NPIs, or becoming infected (Bauch and Earn, 2004; Bhattacharyya and Bauch, 2011; Reluga, 2010; Chen, 2012; Reluga, 2013).

There are many different approaches used to model the effects human behaviour on 16 infectious disease dynamics. In this thesis, we will use methods similar to those discussed above, and compare our ﬁndings to the results obtained by previous models.

1.4.1 Example of a Behaviour-Disease System

Thus far, behaviour-disease models have largely only considered decision making for one of NPIs or vaccination separately. One of the goals in this thesis is to explore the dynamics of models that simultaneously incorporate individual decision making for both of these interventions. In this section, we will brieﬂy study a motivating example of a differential equation model that has the property of simultaneous intervention effects.

Consider a population that responds to the rising prevalence of a disease during an epidemic. Firstly, members of the population can vaccinate to protect themselves from infection. Vaccinated individuals move to the vaccinated compartment, where they stay for the remainder of the epidemic. Secondly, members can practice NPIs. If they do so, they move to the alternate susceptible class, where their susceptibility is reduced due to the self protective actions they have taken. We can model this with the following system of differential equations:

dS 1 = −S IγR − δS I − φS I + ηS (1.11) dt 1 0 1 1 2 dS 2 = −S IγR + δS I − φS I − ηS (1.12) dt 2 0 1 2 2 dI = S IγR + S IγR − γI (1.13) dt 1 0 2 0 dV = φI(S + S ), (1.14) dt 1 2 17

and R = 1 − S1 − S2 − I − V . Here, we use the relation (1.7), where γ is the recovery rate,

δ is the rate for non NPI users (S1) to move to the NPI adopter compartment (S2), φ is the

rate susceptibles become vaccinated, is the transmission reduction to NPI users, and η is

the rate at which NPI users discontinue their self-protective measures. In this model, NPI

and vaccination rates increase linearly with disease prevalence.

To recover the original SIR model (1.1), we can set the parameters δ, φ, and η to

0. Similarly, we can directly compare 3 different scenarios where the population utilizes

interventions: the scenario where both interventions are used, the scenario where there is

only vaccination used, and the scenario where there are only NPIs used (Fig 1.1). We can

see that the dynamics of the epidemic are altered when one of the interventions is removed,

or when both are included. For example, with the parameter values used, ﬁnal size is

highest and epidemic length is greatest when no vaccine is available (Fig 1.1A). Also, NPI

use is higher in the population as well (Fig 1.1B). Finally, vaccine uptake decreases when

the transmission reduction through adopted NPIs are taken into consideration (Fig 1.1C).

It is clear that including both intervention types into a model will offer different dynam-

ics from similar models that only include one of NPIs or vaccination. In Chapters 3 and 4

of this thesis, we explore this change in dynamics further, and seek to gain insight on how

individual decision making can affect epidemic outcomes. 18

0.7 A Two Interventions 0.6 Vaccination Only NPIs Only 0.5

0.4

0.3

0.2 Fraction Recovered

0.1

0 0 20 40 60 80 100 120 140 160 180 200 Day

0.9

0.8 B

0.7

0.6

0.5

0.4

0.3 Fraction NPI Use 0.2

0.1

0 0 20 40 60 80 100 120 140 160 180 200 Day

0.7 C 0.6

0.5

0.4

0.3

0.2 Fraction Vaccinated

0.1

0 0 20 40 60 80 100 120 140 160 180 200 Day

Figure 1.1: Time series of an epidemic A) Percent of population recovered, B) Percent of population practicing NPIs C) Vaccine coverage. Parameter values are R0 = 2.25, γ = 0.2 = 0.6, δ = 100, φ = 0.75, η = 0.1. 19

1.5 Overview and Objectives

In Chapter 2, we present an age-stratiﬁed compartmental model of seasonal inﬂuenza.

With this model, we attempt to estimate key transmission parameters of inﬂuenza A and

B strains by ﬁtting to conﬁrmed cases data in Canada over a 5 year timespan. We also

use our model to test various vaccination program strategies. This model, which does not

incorporate behavioural decisions of the population or explore the effects these have on

disease transmission, leads into the later chapters which drop this simplifying assumption.

In Chapter 3, we introduce a behaviour-disease model that includes NPI and vaccination

decisions amongst a population who are exposed to an epidemic of a self limiting disease.

With this model, we explore how NPIs and vaccination interact with each other, as well

as the impact of various contact networks imposed on the population. In Chapter 4, we

create another behaviour-disease model incorporating NPI and vaccination decisions. In

this model, we consider multi-year horizons for a population’s health outcomes with respect

to seasonal inﬂuenza, an endemic disease. To ﬁt the models in the previous three chapters,

we utilize probabilistic sampling of the parameter spaces and discard the parameter sets

that are not sufﬁciently close to the target data. In Chapter 5, we discuss conclusions for

our work and offer possible avenues for future research.

Chapters 3 and 4 are largely based on the publications “The Impacts of Simultaneous

Disease Intervention Decisions on Epidemic Outcomes”. Journal of Theoretical Biology, vol. 395, pages 1-10, and “Disease Interventions can Interfere With One Another Through

Disease-Behaviour Interactions”. PLOS Computational Biology, vol. 11(6): e1004291, 20 respectively. 21

Chapter 2

Parameter Estimation in a Dynamic

Model of Inﬂuenza Transmission Using

Laboratory Conﬁrmed Inﬂuenza Cases

2.1 Chapter Abstract

Dynamic transmission models of influenza are often used to evaluate vaccination strategies and potential health and economic burdens. Our goal is to use laboratory confirmed influenza cases to fit transmission parameters in a dynamic model of influenza for both A and B strains. We fit these parameters using two types of data: weekly time series case data and age-stratified cumulative case data. These two approaches will allow us to compare model outcomes when using the different types of data various regions may have available. 22

Surprisingly, we find that the longitudinal time series fitting method provides best fitting parameter sets that have a higher variance between the respective parameters in each set than the age-stratified cumulative case method. Also, introducing hypothetical vaccination scenarios to the models utilizing each type of data provide different outcomes-particularly for influenza A. Simulations using parameter sets obtained from fitting the model with time series data predict that vaccinating younger age groups yields greater declines in attack rates than vaccinating older age groups, whereas simulations using parameter sets obtained from the age-stratified cumulative case data predict the opposite. These results show that the type of data used to fit a dynamic transmission model can produce very different outcomes.

Thus, thorough analysis must be performed and caution must be exercised when making conclusions about a predictive model’s results.

2.2 Introduction

Seasonal influenza imposes significant health burdens each year, threatening quality of life for many across the globe (Simonsen, 1999). Although often considered a generally mild illness typically causing school or workplace absenteeism, influenza can cause significant complications for vulnerable populations - such as the elderly or those with weakened immune systems. In order to combat influenza, health jurisdictions may implement vaccination programmes (such as the Universal Influenza Immunization Program in Ontario,

Canada) that may target certain age groups, professions, or make vaccines widely available 23 to the public.

Dynamic transmission models can be used to evaluate the effectiveness of control strategies for seasonal inﬂuenza, such as targeted vaccinations and vaccine types (Dushoff et al.,

2007; Alexander et al., 2004; Glasser et al., 2010; Hsieh, 2010; Baguelin et al., 2013;

Thommes et al., 2014). For example, a frequent problem addressed in the literature is finding an optimal approach to distributing vaccines. Some research has found that targeting younger age groups produces the most benefit in limiting influenza spread and improving health outcomes across a population (Reichert et al., 2001; Monto et al., 1969; Piedra et al.,

2005; Weyecker et al., 2005; Halloran et al., 2002; Longini and Halloran, 2005). However, other research has also shown this may not be the case in all circumstances (Beutels et al.,

2013; Dushoff et al., 2007). In the past, influenza transmission models have either chosen parameters without a fitting process (Vynnycky and Edmunds, 2008; Pitman, 2012), have been fitted to a single year using cross-sectional cumulative cases for that season

(Poletti et al., 2011; Wu et al., 2014), to weekly time series longitudinal data (Goeyvaerts et al., 2015; Baguelin et al., 2013), to influenza like illness (ILI) data, assuming ILI incidence follows the same patterns as seasonal influenza (Poletti et al., 2011; Goeyvaerts et al., 2015), or incorporating data of laboratory confirmed influenza cases (Baguelin et al.,

2013; Wu et al., 2014). Here, we create an age stratiﬁed dynamic transmission model of seasonal inﬂuenza following similar approaches by Goeyvaerts et al. (Goeyvaerts et al.,

2015) and Thommes et al. (Thommes et al., 2014), and use positive inﬂuenza specimen tests for parameter estimation. 24

Our research questions are (1) to determine whether a dynamic transmission model can be fitted to longitudinal (multi-year) time series data of laboratory confirmed influenza cases, and (2) to compare the resulting fit and model predictions of the impact of vaccination to the case where the model is fitted to (non-longitudinal) data on age-stratified cumulative attack rates instead. This comparison will help determine how the quality and type of data can impact a model’s outcomes, as some regions have more complete surveillance than others. In our model, we seek to directly estimate the key parameters of influenza transmission for both A and B strains. Previous models have not solely used laboratory confirmed case data for both A and B strains of seasonal influenza, or used the same fitting process to directly compare results when fitting to longitudinal time series data and cross-sectional cumulative case data over multi-year time spans.

2.3 Methods

Our model is a compartmental age structured model (Anderson and May, 1992), and we will fit important transmission parameters to longitudinal influenza case data, as well as cross-sectional age stratified case data. Incorporating age structure is a critical factor, as population contact patterns, and therefore influenza transmission, depend on age. Details of the model development are given in the following sections. 25

2.3.1 Population Demographics

Our model uses age compartments 1 year in size, starting from 0-1 years, and ending at 99+ years (Goeyvaerts et al., 2015; Thommes et al., 2014). The population size and age structure are modelled after the province of Ontario, Canada, to remain consistent with our data on influenza incidence. When we age the population, we use yearly population projections given by Ontario’s Ministry of Finance, which are based on census data, birth/death rates, immigration, and emigration (Ontario Ministry of Finance, 2011, 2013, 2015), or census data (for the model’s 2011 population) (Statistics Canada, 2012). Due to the model’s high age resolution, we are able to specify age dependent contact rates. These contact rates play a crucial role in influenza transmission, and we use a contact matrix which specifies the mean daily duration of contact time in minutes between age groups (Zagheni et al.,

2008). These contact data are based on studies conducted in the United States, and thus we are making the assumption that contact rates in the region we are modelling are similar.

2.3.2 Inﬂuenza Incidence Data and Epidemiology

Data on confirmed influenza cases are available for the province of Ontario, Canada from the years 2010 to 2015 (Government of Canada Publications, 2015). The data give the weekly number of confirmed cases in the province for the specified years. For fitting our model to age stratified cumulative cases, we use the years 2011 to 2016 due to these years having the required data available. The age categories used in the fitting are 0-19,

19-65, and 65+. In our model, we will consider inﬂuenza cases caused by both the A and 26

B strains.

The inﬂuenza virus in our model has a susceptible-infected-recovered-vaccinated nat-

ural history. For transmission, we use the contact hypothesis (Zagheni et al., 2008) where

our contact matrix C taken from Table 1 in Zagheni et al (Zagheni et al., 2008) deﬁnes the average daily time of contact between age groups. We deﬁne βi to be the probability that

an individual in age group i becomes infected after being in contact with an infectious indi-

vidual, which in our case is constant across age groups. The time varying force of infection

for age group i is given by

100 ! X Ij λi(t) = βi Cij , (2.1) j=1 Nj

where Ij is the number of infected individuals in age group j and Nj is the size of age group

j. Additionally, inﬂuenza incidence shows a prominent annual recurrence in the winter

months, which has been thought to be caused by a variety of factors such as temperature,

humidity, and changes in contact patterns (Dushoff et al., 2004; Fuhrmann, 2010; Shaman

and Kohn, 2009). To ensure this seasonal variation in our model, we use a sinusoidal

function (Truscott et al., 2011) and multiply the force of infection by

2π(t + δ)! 1 + A cos , (2.2) 365

where A is the amplitude of the seasonality function which determines the variation of

the basic reproductive number R0, and δ determines on what day the maximum value 27

of the seasonality occurs (δ = 0 corresponds to January 1). This formulation is similar to previous work modelling the same dynamic (Goeyvaerts et al., 2015; Thommes et al.,

2014; Vynnycky and Edmunds, 2008), and we use the derivation found in Thommes et al.

(Thommes et al., 2014) to relate βi to R0.

Finally, infected individuals recover at a constant rate γ. Also, to model the antigenic

drift of the inﬂuenza virus (Cox and Bender, 1995), we force individuals that have been

infected to lose their immunity at a constant rate. In our model, natural immunity loss

occurs at rate ρN .

2.3.3 Vaccination

In Ontario, the primary types of vaccines used are the trivalent inactivated vaccine,

the quadrivalent inactivated vaccine, and the quadrivalent live-attenuated vaccine (Ontario

Ministry of Health and Long-Term Care, 2015). In this region, the recommended individ-

uals to receive vaccination are those aged 6 months and older, and especially individuals in

high risk groups or those who may directly transmit to high risk groups (Ontario Ministry

of Health and Long-Term Care, 2015).

In our model, we specify a proportion of individuals in each age group to become

vaccinated each year. At the time of vaccination, a vaccinated individual in age group i

receives vaccine induced immunity according to the vaccine’s efﬁcacy with probability i, and remains susceptible with probability 1 − i. Vaccine efﬁcacy is set to 65% for ages

< 65 and 55% for older age groups (Simpson et al., 2012; Widgren et al., 2013; Breteler 28

et al., 2013). We also assume there is no partial immunity conferred with an inefﬁcacious

vaccination (Goeyvaerts et al., 2015; Thommes et al., 2014). For vaccination coverage

rates, we use data from the studies by (Campitelli et al., 2012; Moran et al., 2009; Kwong

et al., 2010, 2008), and based on the age ranges given, we use linear interpolation to restore

our yearly age resolution. The baseline coverages are 0-1 years: 3.7%, 1-2 years: 7.4%,

2-11 years: 29.48%, 12-19 years: 36%, 20-49 years: 25.5%, 50-64 years: 48%, 65-74

years: 73%, 75-84 years: 84%, and 85+ years: 82% (Campitelli et al., 2012; Moran et al.,

2009; Kwong et al., 2008, 2010). Much like natural immunity, vaccine acquired immunity

wanes at a constant rate of ρV . In our model, we choose ρV to be a ﬁtted parameter rather

than choosing it as a ﬁxed value or assuming it to be equal to ρN , as was used in previous studies (Goeyvaerts et al., 2015; Thommes et al., 2014; Vynnycky and Edmunds, 2008).

Finally, those who become infected regardless of vaccinating will not show a reduction in infectiousness.

2.3.4 Model Structure

Our system of differential equations consists of susceptible Si(t), infected Ii(t), recovered Ri(t), and vaccinated Vi(t) individuals where i denotes the respective age class an individual belongs to. 29

dS i = −λ (t) + ρ R + ρ V (2.3) dt i N i V i dI i = λ (t) − γI (2.4) dt i i dR i = γI − ρ R (2.5) dt i N i dV i = −ρ V (2.6) dt V i

The system is integrated with a time step of one day allowing for precise calculation the the daily force of infection as well as sufficient numerical solution accuracy. We use the MATLAB package ODE4 to fulfill our fixed time step requirement. In addition to the 5 year time period for which we have historical influenza incidence data, we run our model with a 10 year burn in period. During the burn in period, we use the 2010 population demographics and maintain the same vaccine uptake rates that were used during our period of interest.

Each year we choose a day near the end of summer (August 31), to age the population

(Vynnycky and Edmunds, 2008; Goeyvaerts et al., 2015; Thommes et al., 2014). Individ- uals are moved to the next age class in one time step, and those in the 99+ age category remain. Then, the population is scaled to match the demographics of the next year’s population, as projected by Statistics Canada and Ontario Ministry of Finance. If these more in depth metrics are not available, population birth and death rates may simply be used.

Newborns entering the ﬁrst age category all populate the S0 compartment. 30

Next, vaccination occurs on October 1 of each year because in our selected region the majority of vaccination occurs in the fall. In our model, we make the approximation that vaccination of the population occurs before each inﬂuenza epidemic begins. Then, at a point tseed we add an external value λext to the force of infection for the remainder of the inﬂuenza season. This is a hybrid between models by Goeyvaerts et al. (Goeyvaerts et al.,

2015) and Thommes et al. (Thommes et al., 2014) as we ﬁnd this small addition to the force of infection grants a smoother transition into each new inﬂuenza season as opposed to infecting a bulk amount of individuals all at once on tseed. In addition, for the period of time between seasons, we remove λext so no additional new cases arise. A diagram showing the primary transitions of our model is shown in Figure 2.1. Vertical arrows represent aging, and on the day of the year the population is aged, members in each compartment are added to the corresponding compartment in the next age group.

2.3.5 Parameter Fitting

We compared two methods of fitting our model’s parameters: fitting the parameters to longitudinal (multi-year) data (we will call this the ”longitudinal method”) and fitting the parameters to cross-sectional age-stratified data that lack a temporal variable (we call this the ”cross-sectional method”). 31

Figure 2.1: Diagram of the age-stratiﬁed SIRS compartmental model with vaccination.

Longitudinal Method

We aim to fit the parameters of our model to multi-year longitudinal time series data in a similar manner to Goeyvaerts et al. (Goeyvaerts et al., 2015). However, we use laboratory confirmed influenza specimen cases instead of ILI incidence data used by previous models which are based on reported influenza-like symptoms rather than laboratory confirmed cases.

In order to quantify the goodness of fit for a given parameter set, we use a least squares approach: historical weekly incidence of the number of positive influenza cases is compared to our model’s corresponding output. We define the number of historical reported

H cases in week w to be Iw , and the number of cases given by the model in week w to be

M Iw . In order to directly compare the two quantities, we use the parameter α introduced by 32

Goeyvaerts et al. (Goeyvaerts et al., 2015) to scale the model incidence. Here, α captures the probability that an infected individual is symptomatic, visits a medical practitioner and gets tested for the inﬂuenza virus which returns a positive result. The sum of squares error is then

X H M 2 Iw − (αIw ) . (2.7) ∀w

To evenly sample the parameter space, we use Latin hypercube sampling (Blower and

Dowlatabadi, 1994) to generate 35,000 parameter combinations. Parameter descriptions and ﬁtting ranges (that is, Latin hypercube sampling ranges) are given in Table 2.1. We then determine each parameter set’s sum of squares score over a simulation run. Next, we utilize

MATLAB’s GlobalSearch algorithm to search for optimal parameter combinations using the parameter sets that offered the lowest sum of squares values. GlobalSearch attempts to

find a function’s global minimum, and initializes its search over the parameter space from a user defined start point. In our case, the function we are seeking to minimize is the sum of squares score of our system of differential equations. The input points are used by the solver to determine an initial estimate for a basin of attraction, and the algorithm also generates a set of trial points to be used in finding the minimum. Additionally, upper and lower bounds may be specified for each parameter, which we define as the same bounds used in the Latin hypercube sampling. Any number of runs of the GlobalSearch algorithm may be performed, using a different starting point corresponding to the parameter sets obtained from the Latin hypercube samples for each run. Moreover, maximum runtimes may be specified as well. Due to the stochastic nature of the process, more runs may result in lower 33

least squares ﬁts, and the available computational resources will be a determinant of how many initial points, and therefore runs, of GlobalSearch are used. In our analysis, we use the 50 best performing parameter sets obtained from the Latin hypercube sampling to use as initial points for the GlobalSearch algorithm. We also tested a group of random initial points gathered from the top 15% of parameter sets from the Latin hypercube sampling, but they did not provide better results (lower sum of squares) than the aforementioned top

50 sets.

Cross-Sectional Method

For ﬁtting age-stratiﬁed cumulative cases over the 5-year period, the data available is

Canada wide. Thus, we scale the cases by the proportion of the Canadian population that lives in Ontario in order to remain consistent with the longitudinal method’s ﬁts.

The ﬁtting for the cross-sectional method was identical to the longitudinal method, except we did not use a weekly difference of squares from the model output to the historical data. Instead, the difference of squares was of the total cases over the entire 5-year period.

Also, the model output of each age category (ages 0-19, 19-65, and 65+) was separately compared to the corresponding historical data. 34 peak to fall between November and January 0 R ´ enez-Jorge et al. (2013). Assumption*. Assumption*; forces Truscott et al. (2009), peakinfluenza range strains also White encompasses et estimates al. of (2009); pandemic Chowell et al. (2006). Assumption.* Encompasses the range used byand similar based models on Thommes research et deducingJim al. that (2014); antibodies Goeyvaerts may et wane al. near (2015), the end of a season Table 2.1: Parameter Descriptions 1 - 90 1-120** -10-45 -60-10** 1.0-2.5 years 1.0-4.0 years** 0.01-0.2 Assumption. Amplitude of seasonality functionDays after vaccination when infectedseeded are into the population 0 - 1.0 Maximum range. Timing of seasonality function peak (days) Average basic reproduction number 1.0-2.5 Mean latent plus infectious periodNatural waning immunity rate 4 days (fixed) Vynnycky et al. (2008) Vaccine conferred waning immunity rate 0.5-1.5 years Infections originating from an outside(Value source. added to the force of infection) Scaling factor of model incidence 0.0005-0.15 Estimate based on Hayward et al. (2014). *Also based on preliminary Latin**Ranges hypercube used sampling. for Wider influenza ranges B. were originally used, but the best results were contained within ranges shown above. 0 ext N V seed Parameter Description Fitting Range Source A t δ R γ ρ ρ λ α 35

2.4 Results

2.4.1 Parameter Fitting Comparison

Time series of the best parameter combinations resulting from the the ﬁtting processes

for inﬂuenza A and B for the longitudinal method are shown in Figure 2.2 and Table 2.2.

The plotted results are compared to the historical laboratory conﬁrmed cases over the time

period. We used a separate ﬁtting process for each strain, although we assume that the

vaccine efﬁcacy and the infectious periods are the same for both. The largest differences in

our model emanate from the 2012 season for inﬂuenza B. Most parameter sets undershoot

the peak in this season, although some achieve much closer ﬁts.

1500 A Model 1000 Historical

# Cases 500

0 2011 2012 2013 2014 2015

400 B 300 200 # Cases 100 0 2011 2012 2013 2014 2015

Figure 2.2: Time series of confirmed influenza cases (black) and our model’s fits (grey). Shaded region represents 95% confidence intervals. (A) Influenza A, (B) Influenza B.

Simulations using the cross-sectional method produce the parameter combinations shown 36

Table 2.2: Best ﬁtting parameter values (mean and standard deviation) for the longitudinal method. Parameter Mean Value for A Strain Std. Dev. Mean Value for B Strain Std. Dev. A 0.6304 0.2211 0.3115 0.1269 tseed (day) 35.43 23.89 71.50 32.06 δ (day) 21.15 10.85 -25.14 19.26 R0 1.424 0.3195 1.322 0.3302 ρN (days) 593 237 1408 133 ρV (days) 468 88 427 169 λext 0.0965 0.0731 0.1108 0.0731 α 0.002608 0.0008535 0.004065 0.006953

in Table 2.3, with the results for each age category shown in Figure 2.3 A,B. Age-stratiﬁed

results from the longitudinal model are included for comparison in panels C and D. In the

cross-sectional method results, the variance of total cases produced by the parameter sets

for inﬂuenza A (Figure 2.3A) in each age category are much lower than that of inﬂuenza

B (Figure 2.3B). This could stem from the ﬁtting process, and how Globalsearch attempts

to ﬁnd optimal parameter combinations to match the age-stratiﬁed data. In the case of in-

ﬂuenza A, each search has parameters converge to very similar values, whereas the ﬁnal

values for inﬂuenza B have much higher variance in comparison. This could be due to how

the infections are spread out across each age category. For example, inﬂuenza A has many

more cases in the ages 19+ than the ages 0-18. However, inﬂuenza B’s cases are evenly

spread across all ages.

For inﬂuenza A, the primary differences in parameters for the two ﬁtting methods stem

from the parameters A (seasonality amplitude), R0 (average basic reproduction number), 37

Table 2.3: Best ﬁtting parameter values (mean and standard deviation) for the cross- sectional method. Parameter Mean Value for A Strain Std. Dev. Mean Value for B Strain Std. Dev. A 0.4405 0.06101 0.4904 0.1131 tseed (day) 46.16 26.14 59.84 36.14 δ (day) 12.02 14.64 -25.70 20.69 R0 1.003 0.01747 1.044 0.09240 ρN (days) 416 68 464 238 ρV (days) 548 11 550 0 λext 0.06387 0.03514 0.1411 0.06227 α 0.0935 0.02440 0.004088 0.0008010

and α (incidence scaling factor). For the seasonality amplitude, we notice that when not

required to meet multiple varying seasonal peaks as we did in the longitudinal method,

the average amplitude is lower with less variance in the cross-sectional method ﬁts than

in its longitudinal counterparts. Similarly, the average R0 value amongst the parameter

sets follows the same pattern: in the cross-sectional method’s ﬁts, the average value and

variance amongst the sets is lower than the values seen in the longitudinal method’s ﬁts.

Finally, α is much larger in the cross-sectional method’s ﬁts.

For inﬂuenza B, the biggest differences in parameters for the two ﬁtting methods stem

from R0 and the waning immunity rates ρN and ρV . Similarly to inﬂuenza A, the average

basic reproduction number is smaller and has less variance amongst the sets in the cross-

sectional method compared to the longitudinal method’s parameters. Also, the average

natural waning immunity rate is smaller as well. An interesting note is that the vaccine

conferred waning immunity rate takes on its maximum allowed value in all of the sets for

the cross-sectional method. A similar, but less extreme, shift towards the maximum ρV 38

4 x 10 1.6 A B 1.4 X X 6000 X 1.2 X 4000

1 # Cases # Cases X 0.8 X 2000 0.6 Ages 0−19 Ages 19−65 Ages 65+ Ages 0−19 Ages 19−65 Ages 65+

4 4 x 10 x 10 8 2 C D 6 1.5 4

# Cases 1 # Cases 2 0.5 0 Ages 0−19 Ages 19−65 Ages 65+ Ages 0−19 Ages 19−65 Ages 65+

Figure 2.3: Age-stratified cumulative cases for influenza A and B compared to empirical targets in our model. Target number of cases from the emprical data are given by X’s where applicable. (A) Cross-sectional method for influenza A. (B) Cross-sectional method for influenza B. (C) Longitudinal method with age-stratified results for influenza A. (D) Longitudinal method with age-stratified results for influenza B. From bottom to top, each line in each boxplot shows the following information: minimum value, first quartile, median, third quartile, maximum value. Red crosses are considered outliers.

value occurs in the inﬂuenza A parameter sets as well.

2.4.2 Projected Impact of Expanded Vaccination Coverage

These results may be further tested by observing the impact of implementing different

vaccination scenarios or strategies for vaccine allocation approaches. Here, we test the

changes in outcomes of our model with a targeted vaccine allocation in a scenario where a

health jurisdiction is expanding their inﬂuenza vaccination program. 39

When expanding vaccination coverage, an important consideration is targeted distribution of vaccines. The two main strategies are to target children, who are believed to be responsible for the majority of transmission, or to target high risk individuals and age groups, such as the elderly. To test these two scenarios, we will increase the vaccine uptake of younger aged age groups in our model (ages 0-18) by 30% for each age, a strategy which has been believed to indirectly protect other age groups as well (Longini and Hal- loran, 2005; Halloran et al., 2002; Weyecker et al., 2005). Then, we compare these results to increasing the total number of vaccines administered by the same amount in older age groups instead, which in our population is the ages 55+.

With the longitudinal fitting method for influenza A, vaccinating younger age groups produces a 24.53% drop in total cases on average from baseline vaccination (Figure 2.4A and Figure 2.5C). When targeting older age groups, we see an average reduction in total cases of 13.86%. Total mean confirmed cases and their 95% CIs are found in Table 2.4.

Thus, the vaccination program aimed at the younger age classes provides a small benefit in total case reduction on average compared to a similar program targeting older ages. This stems from the low baseline vaccine uptake in children and their high contact rates with each other as well as middle aged adults. In the case of influenza B, targeting the younger ages gives an average 19.52% drop in total cases, whereas targeting the older age groups gives an average 24.27% drop in the mean (Figure 2.4B and Figure 2.5D). Total mean confirmed cases and their 95% CIs are found in Table 2.4. In this case, vaccinating older age groups produces a small but largely negligible average reduction in the mean of total 40

cases across parameter sets used.

Table 2.4: Mean number of cases for inﬂuenza strains A and B under different vaccination scenarios. Strain Baseline (95% CI) Vac. Prog. 0-18 (95% CI) Vac. Prog. 55+ (95% CI)

Longitudinal Method Inﬂuenza A 28,787 (± 2,652) 21,725 (±2, 392) 24,797 (±3, 134) Inﬂuenza B 7,483 (±914) 6,022 (±935) 5,667 (±1, 072)

Cross-Sectional Method Inﬂuenza A 35,833 (±390) 24,173 (±597) 10,947 (±607) Inﬂuenza B 12,861 (±785) 8,760 (±505) 6,410 (±646)

1000 A Baseline Vaccination Program 0−18 500 Vaccination Program 55+ # Cases

0 2011 2012 2013 2014 2015

200 B

100 # Cases

0 2011 2012 2013 2014 2015

Figure 2.4: Time series of confirmed influenza A and B cases in our model with different vaccination scenarios. (A) Influenza A. (B) Influenza B.

Using the cross-sectional method, inﬂuenza A results differ from the longitudinal method. 41

In this case, vaccinating older age groups results in the best case reduction (Figure 2.5A with comparison to the longitudinal model in Figure 2.5C). When increasing vaccination rates in the ages 55+, we see less than half the total cases than when expanding vaccination amongst ages 0-18. Total mean conﬁrmed cases from the simulations are found in

Table 2.4. For influenza B, vaccinating the younger age groups yields a 31.89% reduction in mean cases compared to baseline, and vaccinating older age groups provides a 50.16% reduction (Figure 2.5B with comparison to the longitudinal model in Figure 2.5D). Total mean confirmed cases from the simulations are found in Table 2.4. In general, the cross- sectional method’s fitting predicts a much larger decrease in total cases for any vaccination expansion strategy than the longitudinal time series fitting method. These results reveal that the varying types of data that can be used to fit a predictive model of influenza transmission can produce very different results.

2.5 Discussion

We have designed and implemented an age-stratified dynamic transmission model of seasonal influenza for both A and B strains. The model parameters were fit to laboratory confirmed influenza cases from the years 2010-2015 in the province of Ontario, Canada, as well as age-stratified cumulative case data from the years 2011-2016 in Canada. We also used this model to evaluate vaccine expansion strategies which target certain age groups.

Using the cross-sectional method, the variance amongst the respective parameters in 42

4 x 10 A 15000 B 3 10000 2 # Cases # Cases 5000 1

Baseline Vac. Prog. 0−18 Vac. Prog. 55+ Baseline Vac. Prog. 0−18 Vac. Prog. 55+

4 x 10 10000 4 C D 8000 3.5 3 6000 # Cases

# Cases 2.5 4000 2 1.5 2000 Baseline Vac. Prog. 0−18 Vac. Prog. 55+ Baseline Vac. Prog. 0−18 Vac. Prog. 55+

Figure 2.5: Cross-sectional method results of our model for influenza A and B with different vaccination scenarios including comparison to the longitudinal method’s corresponding results. (A) Influenza A, (B) Influenza B, (C) Longitudinal method with age-stratified results for influenza A, (D) Longitudinal method with age-stratified results for influenza B. From bottom to top, each line in each boxplot shows the following information: minimum value, first quartile, median, third quartile, maximum value. Red crosses are considered outliers.

each of the 50 best sets is generally smaller than that of the variance amongst the param-

eters found using the longitudinal method. Also, when introducing vaccination scenarios

targeting different age groups, outcomes from using parameters derived from the two types

of data differ- particularly for inﬂuenza A. For example, the cross-sectional method’s data

predicts much larger decreases in total cases from baseline vaccination coverage than the

time series data. Additionally, those simulations show that vaccinating older age groups

will provide the most benefit in reducing the total number of cases in the population. Us- 43 ing the longitudinal method, results show that vaccinating younger age groups provides a moderate total case reduction for influenza A, and vaccinating older age groups provides a slight total case reduction for influenza B.

Our model makes some simplifying assumptions. For example, the parameter α, which represents the rate at which an infected individual is symptomatic and visits a physician who in turn administers a laboratory test for influenza which returns positive, is constant across all age groups. In reality, this may not be the case as some age groups may be more likely to visit a physician after becoming ill, or physicians may be more likely to administer tests for certain age groups. We also assume that the laboratory confirmed case data is a consistently uniform sample of all influenza cases. However, physicians may send in more tests depending on the time of year or when they perceive the prevalence of influenza is higher. Finally, we assume that vaccine efficacies are the same for both A and B strains, and that the infectious period is the same for both as well (Thommes et al., 2014).

There are some differences in the data used which hinder direct comparisons. The age- stratiﬁed cumulative case data gives country wide cases, whereas the time series data is for the province of Ontario. Although we scale the number of cases country wide by the proportion of Canada that lives in Ontario, the cases will still not be directly comparable.

Moreover, the age-stratiﬁed cumulative case data available covers a one year difference from the time series data, causing some discrepancy in the number of cases over each 5 year span.

Different regions will often have varying types of data available for influenza attack 44 rates. In this work, we have considered weekly time series confirmed influenza cases and age-stratified cumulative cases, but other research has utilized ILI incidence as well (Goey- vaerts et al., 2015; Baguelin et al., 2013; Axelson et al., 2014). We have shown that when using an identical fitting process, these different types of data used to fit the model can produce varying results. Thus, when fitting a dynamic transmission model for influenza, the quality of case notification data used is an important aspect that impacts model outputs. 45

Chapter 3

The Impacts of Simultaneous Disease

Intervention Decisions on Epidemic

Outcomes

3.1 Chapter Abstract

Mathematical models of the interplay between disease dynamics and human behavioural dynamics can improve our understanding of how diseases spread when individuals adapt their behaviour in response to an epidemic. Accounting for behavioural mechanisms that determine uptake of infectious disease interventions such as vaccination and non-pharmaceutical interventions (NPIs) can signiﬁcantly alter predicted health outcomes in a population. How-

This chapter is based on the article in the Journal of Theoretical Biology (2016), vol. 395, pages 1-10. Authored by Michael A. Andrews and Chris T. Bauch. 46 ever, most previous approaches that model interactions between human behaviour and disease dynamics have modelled behaviour of these two interventions separately. Here, we develop and analyze an agent based network model to gain insights into how behaviour toward both interventions interact adaptively with disease dynamics (and therefore, indirectly, with one another) during the course of a single epidemic where an SIRV infection spreads through a contact network. In the model, individuals decide to become vaccinated and/or practice NPIs based on perceived infection prevalence (locally or globally) and on what other individuals in the network are doing. We find that introducing adaptive NPI behaviour lowers vaccine uptake on account of behavioural feedbacks, and also decreases epidemic final size. When transmission rates are low, NPIs alone are as effective in reducing epidemic final size as NPIs and vaccination combined. Also, NPIs can compensate for delays in vaccine availability by hindering early disease spread, decreasing epidemic size significantly compared to the case where NPI behaviour does not adapt to mitigate early surges in infection prevalence. We also find that including adaptive NPI behaviour strongly mitigates the vaccine behavioural feedbacks that would otherwise result in higher vaccine uptake at lower vaccine efficacy as predicted by most previous models, and the same feedbacks cause epidemic final size to remain approximately constant across a broad range of values for vaccine efficacy. Finally, when individuals use local information about others’ behaviour and infection prevalence, instead of population-level information, infection is controlled more efficiently through ring vaccination, and this is reflected in the time evolution of pair correlations on the network. This model shows that accounting for both 47 adaptive NPI behaviour and adaptive vaccinating behaviour regarding social effects and infection prevalence can result in qualitatively different predictions than if only one type of adaptive behaviour is modelled.

3.2 Introduction

Infectious disease outbreaks have the potential to cause unexpected burdens and panic in societies. For example, the outbreak of severe acute respiratory syndrome (SARS) in

2003 caused signiﬁcant economic impacts across the world, despite lasting only six months

(Lee et al., 2002). Occurring unexpectedly, outbreaks such as the aforementioned SARS outbreak (Lee et al., 2002; Pearson et al., 2003), the Middle East respiratory syndrome outbreak in 2012 (Balkhair et al., 2013), Ebola outbreak in 2014 (WHO Ebola Response

Team, 2014), or an inﬂuenza pandemic, which has happened as recently as 2009 (Girard et al., 2010), can be difﬁcult to predict and can spread locally or globally and last anywhere from months to years.

Human behaviour can have a large impact on the spread of infectious diseases (Funk et al., 2010). People have been observed to change their regular social routines in response to an epidemic, in order to reduce their risk of becoming infected (Lau et al., 2005; Philip- son, 1996; Ahituv et al., 1996). The infection prevalence or incidence of a disease in a community serves to drive these behavioural changes, as an individual’s perceived susceptibility generally rises along with these disease measures (Funk et al., 2010; Durham and 48

Casman, 2012; de Zwart et al., 2009; Koh et al., 2005). There are two primary self protective intervention strategies susceptible members of a population can utilize to reduce their chances of contracting a disease. These are pharmaceutical interventions, such as vaccination, and non-pharmaceutical interventions (NPIs), such as social distancing and increased hand washing (Centers for Disease Control and Prevention, 2012). The usage of these intervention strategies are voluntary in many health jurisdictions, and so perceived risks play an important role in how often they are utilized (Chapman and Coups, 1999b).

Coupled disease-behaviour models combine human decision making behaviour with traditional transmission dynamics, helping to capture an additional, and often important, aspect of disease spread (Funk et al., 2010; Bauch et al., 2013; Wang et al., 2015). Be- haviourally based models that incorporate NPIs and social distancing during an outbreak show that these practices can lower the attack rate of a disease (Del Valle et al., 2005;

Reluga, 2010; Funk et al., 2008; Rizzo et al., 2014; Bagnoli et al., 2007; Fenichel et al.,

2011; Poletti et al., 2009, 2012). Suppressing an outbreak using these means can be very critical, as vaccines may not always be immediately available to the general population

(Check, 2005). Modelling how NPIs are utilized can be approached in various ways by mathematical models. For example, Funk et al (Funk et al., 2008) allow an individual’s level of awareness to the presence of a disease shape their usage of self-protective measures. Rizzo et al. model a population where susceptible individuals base their activity rates on the infection prevalence of a disease in the population or the infection incidence over a time step (Rizzo et al., 2014). Similarly, Bagnoli et al. (Bagnoli et al., 2007) and 49

Del Valle et al. (Del Valle et al., 2005) have individuals lower their susceptibility according to the proportion of their contacts in a transmission network that are infectious, and to the infection prevalence, respectively. Poletti et al. (Poletti et al., 2009, 2012) incorporate imitation dynamics to model the behavioural changes of the population. Finally, Fenichel et al.

(Fenichel et al., 2011) and Chen et al. (Chen et al., 2011) study models where individuals derive utility from engaging in social contact, but raise their risk of infection when doing so. In these aforementioned models, each individual’s behaviour is shaped by the information they gather about the disease status of those around them. Thus, in these models, transmission dynamics depend heavily on the perceived risks that drive contact patterns.

Further approaches to mathematical models that integrate self protective behaviour into disease transmission utilize adaptive and multiplex networks. An adaptive network is a network whose edges between contacts change dynamically over time. Using these, Gross et al. (Gross et al., 2006), Shaw and Schwartz (Shaw and Schwartz, 2008, 2010) and

Zanette and Risau-Gusman (Zanette and Risau-Gusman, 2008) allow susceptible nodes to rewire their existing connections away from infectious nodes at a given rate. The approach of multiplex networks helps to model the many types of social networks individuals may use to acquire information, and Granell et al. (Granell et al., 2013) and Cozzo et al. (Cozzo et al., 2013) use these to study the impact of different information ﬂows on the spread of epidemics. On the other hand, Glass et al. (Glass et al., 2006) and Kelso et al. (Kelso et al.,

2009) use contact networks which include families, schools, and workplaces to study the effects of various NPIs such as school closures and staying at home while infectious. 50

Additionally, vaccines (if available) play a major role in reducing infection rates during an epidemic. Some mathematical models have shown that under voluntary vaccination, populations may not reach sufﬁcient uptake levels to stop an epidemic (Vardavas et al.,

2007; Bauch et al., 2003). However, under voluntary policies in a network, Zhang et al. demonstrate that nodes with high degree can help to suppress disease spread through their increased desires to vaccinate (Zhang et al., 2010). During an outbreak, complications may arise when there are delays in vaccination. As a result of a delay, epidemic ﬁnal size can increase signiﬁcantly (Yang et al., 2009), especially as the delay lengthens (Gojovic et al.,

2009). When considering the efﬁcacy of a vaccine, Wu et al. suggest through their model that a less effective vaccine causes vaccine uptake to increase (to an effectiveness of about

50%), especially for more serious diseases (Wu et al., 2011). Insights from the models discussed above, as well as more empirically based research (Brewer et al., 2007), have shown that perceived risks play an important factor in an individual’s decision to protect themselves through vaccination. These risks include perceived susceptibility to the illness and perceived risks associated with vaccinating (due to potential side effects) (Roberts et al., 1995; Streeﬂand, 2001). Much like NPIs, members of a population will base their vaccination decisions on information they are able to gather about the disease during an outbreak.

Perceived risks surrounding a disease play a crucial role in vaccination and NPI decisions. Information that shapes these perceptions is gathered by individuals in a population and may be derived from local information (Ahituv et al., 1996; Philipson, 1996; Klein 51 et al., 2007) (such as social contact networks), or through global information such as media reports about the population as a whole (Klein et al., 2007; Berry et al., 2007). We note that disease-behaviour models like those discussed above do not typically consider the intervention strategies of vaccination and NPIs simultaneously. However, it is clear that both are important factors in the spread of a disease. Andrews and Bauch (Andrews and Bauch,

2015) have studied the interactions of these two disease interventions with a utility based decision framework model in the context of seasonal influenza. In contrast to our previous work that considers long-term, year-to-year dynamics, here we develop a disease-behaviour individual based network simulation model to study interactions between vaccinating behaviour and NPI behaviour and their impact on health outcomes during the course of a single, and sudden, epidemic outbreak of a novel, self limiting infection, where perceived risks and social influence serve as the primary drivers of individual behaviour. Moreover, we include parameters that allow controlling the relative influence of local versus global information on behaviour. Our main objective is to compare how our model predictions differ from predictions of models that capture behaviour for only one of the two interventions, under various assumptions for (1) transmission probabilities, (2) timing of vaccine introduction, and (3) vaccine efficacy, and how efficacy influences vaccine uptake. Further- more, we explore how the utilization of local versus global information regarding disease spread and vaccine uptake can alter network wide outcomes. 52

3.3 Methods

3.3.1 Disease Dynamics

We consider a disease with a susceptible - infectious - recovered - vaccinated (SIRV)

natural history. Susceptible individuals may become infected by their infectious neigh-

NInf bours with probability P (NInf ) = 1 − (1 − β) per day, where NInf is the number of

infectious network neighbours, and β is the transmission rate. Infectious individuals move to a recovered (and immune) state for the remainder of the epidemic in a number of days sampled from a Poisson distribution with a mean of 7 days. Finally, susceptible individuals may choose to vaccinate and thus become immune for the duration of the epidemic.

Baseline parameter values were calibrated to obtain epidemic ﬁnal size and vaccine uptake trends within the plausible ranges of the corresponding measures in the United States for the 2009 H1N1 inﬂuenza pandemic (Shrestha et al., 2011; Centers for Disease Control and

Prevention, 2015a), although we emphasize that we are not modelling inﬂuenza in particular, but rather we intend our disease represent a hypothetical self-limiting, acute infection where individuals only lose natural immunity on a time scale of years. We also assume that this is a novel strain of a disease, and individuals have no prior immunity - either natural or vaccine conferred. Full details regarding network structure, transmission dynamics, and decision modelling appear in the following subsections. 53

3.3.2 Contact Network

The disease is transmitted on a network consisting of 10,000 nodes which was constructed by sampling from a large contact network derived from empirical contact data in

Portland, Oregon (Network Dynamics and Simulation Science Laboratory, 2008a). The network we used can be found in the supplementary material in (Andrews and Bauch,

2015). Previous research has shown that the subnetwork is a good approximation to the full network (Wells and Bauch, 2012). This network’s structure (see Supplementary Infor- mation (SI) Figure 1) remains static throughout an epidemic, and we assume that the edges in the network provide sufﬁcient contact between individuals to allow potential disease transmission. We also run simulations testing our primary results on two other types of networks: random networks and power law networks. For details regarding these results, we direct the reader to the SI.

3.3.3 Non-Pharmaceutical Interventions and Vaccination

Susceptible individuals in the population may engage in self-protective behaviour in response to a growing epidemic. Their self-protecting activity is governed by both the presence and fear of the disease itself (Sadique et al., 2007; Henrich and Holmes, 2011; an Lori

Uscher-Pines and Harris, 2010; Brown et al., 2010; Harvard School of Public Health, 2009) and by the social inﬂuence of their contacts and the population as a whole (Poletti et al.,

2009; Bauch, 2005). To model this intervention use, we begin by allowing an individual

−(Φ+ΓNPI ) to reduce their susceptibility to βNPI = (e )β. Firstly, Φ is an individual’s risk 54

perception of the disease, given by

I I ! Φ = σf λ Net + (1 − σ)f λ P op , (3.1) k NP op

where INet is the number of a given individual’s contacts that have been infected, k is the node degree of the individual on the network, IP op is the number of individuals in the population that have been infected, NP op is the population size, and σ dictates how members

of the population weigh information gathered from their contacts and the population as a

whole. Finally, f is a function that determines an individual’s response level to increasing infection incidence, given by 1

f(x, y) = 1 − exp(−x), (3.2)

where x is a proportionality constant that governs the response dynamic (λ in (3.1)). Since perceived risks only increase with the number of infected individuals in our model (due to the relatively small timespan of one epidemic), we use this functional form. Also, it is an increasing function bounded between 0 and 1 whose shape (or response of increasing perceived risk to incidence) can be governed by a single parameter. Similar functions have been used in the literature surrounding disease spread and self-protective behaviour, for

example, see (Funk et al., 2008). Secondly, ΓNPIj measures an individual j’s imitation of

1In the article appearing in the Journal of Theoretical Biology, Function 3.2 is instead written as a function of two variables. 55

others who are utilizing self-protective NPI practices, given by

 V uln  NP op  P 1 − exp(−(Φ + Γ )) P i NPIi  1 − exp(−(Φi + ΓNPI )) i∈N V uln  i   j   i6=j=1  ΓNPIj = σf γ V uln +(1−σ)f γ V uln  ,  k   N − 1  j  P op  (3.3)

V uln where NP op is the number of susceptible or vaccinated (potentially vulnerable) individ-

V uln uals in the population, kj is the number of susceptible or vaccinated neighbours in- kV uln j P 1−exp(−(Φi+ΓNPIi )) dividual j has, i=1 is the average amount of transmission rate reduc- kV uln tion caused by self-protective behaviour amongst an individual’s susceptible neighbours, NV uln P op P 1−exp(−(Φi+ΓNPIi )) i6=j=1 V uln is the similar average reduction induced by the susceptible pop- NP op −1 ulation as a whole, and γ is a parameter that governs the response strength of imitation

behaviour. Equation (3.3) captures how individuals reduce their probabilities of becoming

infected through observations of others doing the same. This includes imitation of both

network neighbours (σ), and imitation of how the entire population is behaving (1 − σ).

−2 Thus, e ≤ exp(−(Φ + ΓNPI )) ≤ 1 dictates how individuals lower their probabilities of

becoming infected as they gain awareness of the epidemic by witnessing the disease spread

throughout the population. In our simulations, NPI use for each individual is updated non-

synchronously in a random order at the beginning of each day. That is, an individual’s

NPI use will be updated using both information of infection levels and the NPI levels of

others from the end of the previous day. Also, we observed from the simulations we ran

that the transmission rate reduction through NPI use will typically be ≤ 50% for any given 56 individual, which is consistent with the available literature regarding the efﬁcacy of NPIs

(Larson et al., 2010; Sheehan et al., 2007).

If vaccines are available, members of the population may also choose to protect themselves from infection by receiving a vaccine. The decision to vaccinate becomes a more attractive option as vaccine uptake increases (Bhattacharyya and Bauch, 2011), and thus an individual’s vaccination decision will depend both on their perceived risk of the disease as well as the decisions of others to vaccinate. We represent this as

I V I ! V !! σ f λ Net + f γ Net + (1 − σ) f λ P op + f γ P op , (3.4) k k NP op NP op

where VNet and VP op are the numbers of a given individual’s contacts and total number

VNet of individuals that have been vaccinated, respectively. If we deﬁne ΓV = σf γ k + (1 − σ)f γ VP op , then (3.4) can simply be written as NP op

Φ + ΓV . (3.5)

Equation (3.5) combines an individual’s risk perception of becoming infected, which is based on local and global information of disease incidence, with an individual’s imitation of self protective behaviour, which also based on local and global information.

If on any day a susceptible individual’s preference towards vaccinating, which we set as 1 − exp(−(Φ + ΓV )), exceeds a given threshold, θ, then that individual will be trans- ferred to the vaccinated compartment. Otherwise, this is interpreted as an individual being 57

undecided, and they therefore remain susceptible. This process is similar to methods from decision ﬁeld theory (Busemeyer and Townsend, 1993), where individuals update their preferences towards making certain decisions based on available information. If their preference toward making an action reaches a pre-deﬁned level, a decision is then subsequently made.

3.4 Results

3.4.1 Baseline Dynamics

The baseline scenario of our model (Table 3.1) simulates an outbreak in a population

Table 3.1: Baseline Parameter Values. Parameter Description Value λ Constant governing awareness/risk perception of disease 1.5 γ Constant governing behaviour imitation 0.5 θ Vaccinating threshold 0.35 σ Weighting for global versus local information 0.8 Transmission rate β 0.005 (Probability of infection per infectious contact per day) NP op Population size 10, 000

I0 Initial number of infectious persons 20 η Mean infectious period, in days 7

whose individuals may protect themselves from infection using NPIs or vaccination. We call this the baseline scenario as it was calibrated to achieve plausible epidemic outcomes 58 under the realistic assumption that both vaccination and NPIs are available simultaneously.

Henceforth, we will refer to this scenario as the “combined scenario”, as both interventions may be used. If both intervention options are available, the ﬁnal size of the epidemic is lowest compared to when only one of the two interventions are used, as expected (Fig

3.1A). We also compare the epidemic time series of the combined scenario to hypothetical scenarios where there is no vaccine available over the course of the outbreak (“NPI-only scenario”), or self-protective behaviour is completely ineffective (“vaccine-only scenario”).

We note that the NPI-only scenario gives similar infection rates as the combined scenario for the first 3 weeks of the epidemic. This occurs because vaccine uptake in the combined scenario is close to zero in the first few weeks, due to low perceived risks of becoming infected while infection prevalence is still minimal, and therefore the differences between scenarios with and without vaccination are small during this period of time. The impli- cation of this is that delays in vaccine availability in the first few weeks of an epidemic may not hinder vaccine uptake under a voluntary vaccination policy. After this initial period, we observe consistently higher cumulative infected for the NPI-only scenario over the combined scenario for the remainder of the epidemic. The NPI-only simulations yield the greatest average cumulative infection incidence, as the response from solely NPIs amongst susceptible individuals cannot match the disease mitigation of a perfectly efficacious vaccine, in the long term (however, we note that the difference in cumulative incidence is relatively small). In the vaccine-only scenario, infection incidence spikes rapidly but the epidemic lasts a shorter amount of time than in the NPI-only scenario. The relatively rapid 59

early spike in total infected individuals is due to the lack of vaccine uptake in the first weeks of the epidemic, as vaccination decisions are not activated until the perceived threat of becoming infected is sufficiently high. In all these cases, self-protective behaviour serves to slow the spread of an epidemic, but does not successfully reduce the final attack rate as significantly compared to when it is aided by vaccination.

In the NPI-only scenario, NPI uptake amongst susceptible individuals is much more pronounced than when vaccination is also an option (Fig 3.1B). This occurs for two reasons.

Firstly, if vaccination can occur, those that practice the strongest self protective behaviour due to having high levels of perceived risk will be amongst the first to vaccinate. In turn, this will lower the average NPI uptake amongst the remaining susceptible population. Secondly, if members of the population are vaccinating, the spread of the disease will be suppressed causing perceived risks of becoming infected to be lower. Thus, resulting NPI use will be less prominent. In the absence of vaccination, transmission reduction through NPI use simply continues to rise along with the infection incidence seen in Figure 3.1A. In the final vaccine-only scenario, total vaccine uptake is increased on average (Fig 3.1C). Moreover, vaccine coverage begins to rise earlier in response to the rapid spike in infection incidence that is observed when no transmission reduction is present through NPIs. Thus, when NPI effects are not considered, predicted vaccine uptake is significantly higher. 60

0.25 A

0.2

0.15

0.1 Combined Scenario NPI−Only Scenario Cumulative Infected 0.05 Vaccine−Only Scenario

0 0 20 40 60 80 100 120 Day

0.4 B 0.35

0.3

β 0.25 /

NPI 0.2 β

1− 0.15

0.1

0.05

0 0 20 40 60 80 100 120 Day

C 0.5

0.4

0.3

0.2 Vaccine Uptake

0.1

0 0 20 40 60 80 100 120 Day

Figure 3.1: Time series of an epidemic, 95% conﬁdence intervals shown every 10 days around the mean of 500 realizations. (A) Cumulative infection incidence (B) Transmission rate reduction due to self-protective behaviour (NPIs) amongst the susceptible population, (C) Cumulative vaccine coverage. 61

3.4.2 Transmission Rate

Time series of infection prevalence corresponding to different transmission rates can help us understand epidemic spread in our 3 scenarios (Fig 3.2). When the transmission rate is low, NPIs alone are relatively effective at hindering the growth of the epidemic, lowering the peak infection prevalence compared to the vaccine-only case (Fig 3.2A). For a transmission rate of β = 0.00493 per infectious contact per day, simulations that utilize

NPIs only or vaccination only result in the same epidemic ﬁnal size (Fig 3.2B). Although the peak infection prevalence in this scenario is larger for vaccine-only simulations, the epidemic dies out more quickly compared to the NPI-only scenario, resulting in the same cumulative infection incidence over the epidemic duration. For higher transmission rates, the vaccine only scenario outperforms the NPI only scenario (Fig 3.2C). Although NPIs delay the peak of the epidemic, infection prevalence dies out more slowly than when the population uses solely vaccination instead. However, this highlights the importance of

NPIs in epidemics where vaccination may not be immediately available. Considering the combined scenario data in Fig 3.2C, which indicates infection prevalence in simulations utilizing both NPIs and vaccination, the initial disease spread is very similar to that of the

NPI-only scenario. Only when individuals begin to vaccinate does the infection prevalence in the combined scenario show quantitative difference to the infection prevalence in the

NPI-only scenario. Thus, as long as a vaccine is made available within a given time frame, the ﬁnal size can be expected to be the same due to the early activation of NPIs. 62

0.025 A Vaccine−Only Scenario NPI−Only Scenario 0.02 Combined Scenario

0.015

0.01

Infection Prevalence 0.005

0 0 20 40 60 80 100 120 140 Day

0.045

0.04 B

0.035

0.03

0.025

0.02

0.015

Infection Prevalence 0.01

0.005

0 0 20 40 60 80 100 120 140 Day

0.07 C 0.06

0.05

0.04

0.03

0.02 Infection Prevalence 0.01

0 0 10 20 30 40 50 60 70 80 90 100 Day

Figure 3.2: Time series of infection prevalence with the vaccine-only scenario, the NPI- only scenario, and the combined scenario. 95% conﬁdence intervals shown every 10 days around the mean of 500 realizations. (A) β = 0.004 (B) β = 0.00493 (C) β = 0.006. 63

Vaccine timing plays a critical role in the health outcomes of the population during an

epidemic, across a range transmission rates (Fig 3.3). In the combined scenario (Fig 3.3A),

a vaccine can be introduced up to 20 days after the start of an epidemic for the epidemic ﬁ-

nal size to be roughly the same as the scenario when a vaccine is available from day one, for

baseline transmission rates. If we disregard the use of NPIs (Fig 3.3B), the vaccine must be

made available within 15 days before we begin to observe larger epidemic ﬁnal sizes. This

effect is similar for β = 0.006 per infectious contact per day. In the combined scenario, a

vaccine must be made available within 15 days before epidemic sizes increase. However,

in the vaccine-only case, vaccine availability must occur within just 10 days. Finally, for

lower disease transmission, vaccine introduction timing has little impact on infection inci-

dence in the combined scenario. However, in the vaccine-only scenario, we see ﬁnal sizes

begin to increase when availability occurs after 20 days. From these results, we also notice

that the rate of increase of epidemic ﬁnal sizes corresponding to the timing of vaccine intro-

duction are much greater. For example, given the baseline transmission rate, the difference

in infection incidence between immediate vaccine availability and availability beginning

on day 60 is ≈ 20% in the vaccine-only case. However, the same measure in the combined case is only ≈ 4%. Thus, the prediction from these two modelling approaches of epidemic

size induced by vaccine timing introduction differs by about 16% of the entire population

size.

Finally, we also consider measures for epidemic ﬁnal size (Fig 3.4). When the trans-

mission rate is low, the ﬁnal size is the same for the combined scenario as for the NPI-only 64

0.5 β = 0.006 0.45 A β = 0.005 β = 0.004 0.4

0.35

0.3

0.25 Final Size 0.2

0.15

0.1

0.05 0 10 20 30 40 50 60

0.5

0.45 B

0.4

0.35

0.3

0.25 Final Size 0.2

0.15

0.1

0.05 0 10 20 30 40 50 60 Day Vaccine Introduced

Figure 3.3: Epidemic final sizes with respect to when vaccination is made available. (A) With NPIs. (B) Without NPIs scenario, but much higher for the vaccine-only scenario (Fig 3.4A). Hence, for low transmission rates, NPIs on their own can reduce final size as much as combined use of NPIs and vaccines, although the same is not true for vaccines on their own. This is due to individuals promptly adopting NPIs, which are targeted at the leading edge of the epidemic and quick to implement, curbing disease spread immediately. Also, vaccine uptake is much larger in the vaccine-only scenario than in the combined scenario (Fig 3.4B). In contrast, when the transmission rate is high, the final size is almost (but not quite) the same for 65

the combined scenario as for the vaccine-only scenario, but much higher for the NPI-only scenario. Moreover, vaccine uptake is also nearly the same for both of the scenarios that include vaccination. Hence, for high transmission rates, vaccines on their own can reduce

ﬁnal size almost as much as combined use of NPIs and vaccines, although the same is not true for NPIs on their own. In the case of NPIs, the NPI uptake amongst susceptible individuals does not change for β >∼ 0.0045, due to the adoption of vaccination (Fig 3.4C).

However, in the NPI-only scenario, susceptibility reduction through NPIs continues to rise along with the transmission rate.

In summary, when transmission rates are sufﬁciently low, NPIs alone can be almost as effective as having both vaccines and NPIs (whereas vaccination alone is relatively less effective), but when transmission rates are sufﬁciently high, vaccines alone can be almost as effective as having both interventions (whereas NPIs alone are relatively less effective).

3.4.3 Vaccine Efﬁcacy

Vaccines are never 100% efﬁcacious. For less effective vaccines, we can expect infection incidence and vaccination coverage to change as individuals in the population adapt to the quality of interventions available to them. Thus, we explore the dynamics under various vaccine efﬁcacies (denoted ), and how they relate to vaccine coverage and epidemic

ﬁnal size with and without the additional impacts of NPIs (Fig 3.5). We note that in our simulations, vaccines give full protection with probability , and no additional protection with probability 1 − . 66

A 0.3

0.25

0.2

0.15 Final Size NPI−Only Scenario 0.1 Vaccine−Only Scenario Combined Scenario 0.05

0 3.5 4 4.5 5 5.5 6 6.5 β −3 x 10

0.6 B

0.5

0.4

0.3

0.2 Vaccine Uptake

0.1

0 3.5 4 4.5 5 5.5 6 6.5 β −3 x 10

0.45

0.4 C

0.35

0.3 β

/ 0.25 NPI

β 0.2

1 − 0.15

0.1

0.05

0 3.5 4 4.5 5 5.5 6 6.5 β −3 x 10

Figure 3.4: Epidemic measures with respect to transmission rate. (A) Epidemic ﬁnal size (ﬁnal fraction infected). (B) Vaccine uptake. (C) Transmission rate reduction amongst susceptible individuals. 67

As vaccine efficacy decreases, the vaccine-only scenario overestimates the amount of vaccine uptake demanded by up to 16.5% of the population size relative to the combined scenario. We also observe that as vaccine efficacy decreases, the subsequent increase in vaccine coverage of the population is larger when NPI effects are not incorporated. For example, between efficacies of 100% to 50%, the combined scenario of the model predicts

∼ 3.5% more of the population vaccinating, whereas with the vaccine-only scenario, simulations predict an ∼ 8% increase. This effect is also seen with epidemic ﬁnal size (Fig

3.5A). Across all efficacies, final size increase is only ∼ 1.5% of the entire population size with combined NPI and vaccine utilization, and ∼ 5.5% with only vaccination. Thus, we see that disregarding the impact of NPIs may lead to an overestimation of the population’s vaccine demand and final epidemic size. Moreover, the increases in vaccine uptake and final size with decreasing vaccine efficacy may be less significant than what previous predictions which disregard NPI effects show Wu et al. (2011). Finally, when incorporating vaccination decisions and self protective behaviour simultaneously into the model, we observe that predicted levels of vaccine uptake are much smaller than when no NPIs are implemented (Fig 3.5B).

3.4.4 Pairwise Correlations

As an epidemic unfolds across a network, the status of the nodes will develop while the disease spreads and intervention decisions are made. As a result, the spatial structure of infected and susceptible individuals on the networks will evolve over time as well. The cor- 68

0.26 A 0.24

0.22 Combined Scenario Vaccine−Only Scenario 0.2 Final Size

0.18

0.16 1.0 0.95 0.9 0.85 0.8 0.75 0.7 0.65 0.6 0.55 0.5 ε

0.65 B 0.6

0.55

0.5 Vaccine Uptake

0.45

0.4 1.0 0.95 0.9 0.85 0.8 0.75 0.7 0.65 0.6 0.55 0.5 ε

Figure 3.5: Effects of vaccine efﬁcacy between scenarios with and without NPIs on (A) Final epidemic size (ﬁnal fraction infected), and (B) Vaccine uptake. relation between these pairs can offer insight on the vulnerability of the network to disease spread and how individuals react to infection prevalence according to the information available to them, which we control with the parameter σ. To measure the correlation between node pairs, we follow Keeling Keeling and Eames (2005):

NP op [AB] CAB = , (3.6) kavg [A][B] 69

where kavg is the average node degree on the network. With this formulation, an increase in

CAB indicates an increase in correlation as the number of [AB] pairs in the network relative

to the number of type [A] nodes and type [B] nodes also increases. A value of CAB = 1

indicates no correlation Keeling and Eames (2005).

Considering the correlation between susceptible-infected ([SI]) pairs (Fig 3.6C), we

observe a rapid initial spike in the network. This early increase is due to the ﬁrst infected

individuals spreading the disease to their network contacts, enabling more opportunities

for transmission. As infection prevalence begins to peak (Fig 3.6A), infected individuals

have a higher probability of being connected to a non-susceptible node, which results in

the decline of CSI in the network, as distinct clusters of infected and other infected, recovered or vaccinated individuals develop. However, the correlation rises again as infectious nodes recover and only a ﬁnal few clusters of infected and susceptible nodes remain, more so for lower σ as those who vaccinated are less likely to be connected to an infectious node. Correlations of vaccinated nodes with nodes that are or have been infected, [VI] and [VR], respectively, also show how network dynamics respond to different levels of σ

(Fig 3.6D,E). When individuals base their decisions on local information, that is, on the

basis of the number of infectious neighbours (σ = 1.0), CVI and CVR are higher. This

indicates successful ring vaccination occurring in proximity to the infectious individuals.

Under strong inﬂuence of local information, neighbours of infectious individuals develop

a high perceived risk and decide to vaccinate earlier. Then, social inﬂuences reinforce this

vaccinating behaviour, resulting in clusters of vaccinated individuals around infectious in- 70

dividuals. However, when decisions are made more strongly based on population level

infection prevalence (σ = 0.5), [VI] and [VR] pairs become less common in proportion

to all vaccinated and infected/recovered nodes since vaccinations do not always occur on

the epidemic front. The cumulative vaccine uptake under global information is higher than

under local information, however, vaccine uptake increases more rapidly in the early stages

of the epidemic under local information (Fig 3.6B). This reﬂects the efﬁciency of targeted

vaccination under local information, where vaccines are administered to the contacts of

infectious individuals so that infection spread is efﬁciently prevented. Finally, as the epi-

demic dies out and infectious nodes become rare, [VI] and [VR] correlations across varying

levels of σ converge to similar values.

As σ decreases in our model, [SS] pairs become more common relative to the total

number of susceptible nodes towards the end of an epidemic, increasing CSS (Fig 3.6F).

On the other hand, with higher σ, ﬁnal values of CSS continue to decrease. However,

we note that during an epidemic, the opposite is true, albeit to a lesser extent. When

σ = 0.5, vaccination occurs in locations other than the epidemic front, in turn decreasing

CSS compared to higher values of σ. Nonetheless, the ring vaccination observed with

increased σ is more efﬁcient than the more random vaccine allocation seen when σ = 0.5, for example, due to the disease only being able to spread along the network edges. 71

0.035 σ 0.5 0.03 A = 0.5 B σ = 0.6 0.025 0.4 σ = 0.7 0.02 σ = 0.8 0.3 0.015 σ = 0.9 0.2 0.01 σ = 1.0

Vaccine Uptake 0.1 0.005 Infection Prevalence

0 0 0 20 40 60 80 100 120 0 20 40 60 80 100 120 Day Day

2.2 2 C 8 D 1.8 6 1.6 SI VI

C 1.4 C 4

1.2 2 1

0.8 0 0 50 100 150 200 0 20 40 60 80 100 120 Day Day

10 1.1 E 1 F 8 0.9

6 0.8 SS VR C C 4 0.7 0.6 2 0.5

0 0.4 0 20 40 60 80 100 120 0 20 40 60 80 100 120 Day Day

Figure 3.6: Time series of epidemics over different values of σ, the weighting for global versus local information. (A) Infection prevalence, (B) Vaccine uptake, (C) Correlation between SI pairs, (D) VI pairs, (E) VR pairs, and (F) SS pairs. Lines show the average values over 500 realizations.

3.5 Discussion

We have developed and analyzed a model that simulates a population’s adaptive self

protective behaviour (use of NPIs and vaccination) in the face of a disease outbreak, in

contrast to most previous approaches that model only vaccinating behaviour or only NPI

behaviour. We allow an individual’s actions to depend both on their perceived risk of 72 infection developed from their experiences with the disease on the network (both from their network neighbours and from the population as a whole), as well as imitation of the behaviour of others in the population.

Surprisingly, when transmission rates are low, the NPI-only scenario offers comparable disease mitigation effectiveness to the combined scenario, while the vaccine-only scenario results in relatively larger epidemic sizes than either the NPI-only scenario or the combined scenario. For higher transmission rates, the opposite becomes true. That is, the vaccine-only scenario is almost as effective as the combined scenario for reducing infection incidence, but the NPI-only scenario fares worse. If a vaccine is not available immediately to the population at the start of an epidemic, epidemic mitigation through adaptive

NPI behaviour can curb the growth of an epidemic. Thus, if vaccination is made available to the population within a given time frame, health outcomes will be very similar to situations where a vaccine was always available. If, however, the effects of NPIs are not incorporated, then these time frames are comparatively shorter. Moreover, the increases in infection incidence for the vaccine-only scenarios are signiﬁcantly higher the later the vaccine is introduced, resulting in increasingly higher predictions of epidemic ﬁnal size.

Finally, the impact of varying vaccine efficacy on both vaccine uptake and epidemic final size varies significantly between scenarios with and without adaptive NPI behaviour. The increases in both final size and vaccine uptake when vaccine efficacy is decreased are much higher for the vaccine-only scenario than the combined scenario. Hence, a model of adaptive vaccinating behaviour that does not also account for adaptive NPI behaviour will make 73 very different predictions than a model that accounts for adaptive behaviour toward both interventions. This again highlights the positive benefits of epidemic mitigation through adaptive NPI behaviour.

From a network perspective, individuals basing their decisions to practice NPIs or become vaccinated based on the infection prevalence and behaviour in their infection contact network leads to the most effective disease control. Pairwise correlations between vaccinated and infected nodes are highest when this information gathering is possible, as those that vaccinate are typically connected to infected nodes. We also tested the main results with two additional types of networks: random networks and power law networks (see SI).

While the dynamics are qualitatively the same, the amount of change in epidemic final size or vaccine uptake with differing vaccine delays or vaccine efficacies can depend on the specific network type. Assumptions about network structure and transmission are an important consideration - particularly when modelling a specific disease. For example, a transmission network for influenza likely has a different structure than one that would be used to model HIV transmission.

In the combined scenario, epidemic ﬁnal size is suppressed most effectively compared to when only single interventions are possible. Also, when the effects of NPIs are not considered in our vaccine-only scenario (an assumption which is common in previous behaviour-disease models focusing on vaccinating behaviour), vaccine uptake predictions are higher compared to when these effects are considered by our model, on account of counteractive feedbacks from NPI behaviour. 74

Our model includes some simplifying assumptions about behaviour-disease dynamics.

For example, NPI efﬁcacy is poorly quantiﬁed in the epidemiological literature, and it is not always known in what situations individuals may practice them most often. Thus, we assume that NPIs for the spreading disease are not used initially, but in reality there may be some baseline level of NPIs used in the population due to other circulating diseases.

Moreover, we do not model the effects of NPI practices that infectious individuals may utilize, such as self isolation. Instead, we make the assumption that infectious NPI use is absorbed into the transmission rate. Also, the network we used in our simulations could be extended to distinguish family, friend, and work structures, where transmission rates to an individual can vary depending on what category certain network contacts fall in. Similarly, age structure can be introduced into the model. As children will be much less likely to effectively practice NPIs, disease transmission in these groups may be more rapid than our model predicts. Finally, we did not include the impact of asymptomatic infections, and assumed all cases were identifiable in our main results. However, we also considered a scenario where 50% of cases were asymptomatic (see SI), and the primary results regarding vaccine efficacy and vaccine availability delays across various transmission rates are qualitatively the same. Although the main results are similar, in future work that aims to model a specific disease, accounting for asymptomatic infections is an important factor.

Through these experiments, we see that predictions of health outcomes and vaccine uptake in a population can vary signiﬁcantly when NPI use is, or is not, considered. It is important for behaviourally based epidemiological models to incorporate the effects of 75

transmission reduction through this adaptive behaviour, as perceived risks of a disease will

in turn be shaped by them - subsequently altering the outcomes of an epidemic. The same is

also true of models that focus on modelling NPI behaviour, in populations where adaptive

vaccinating behaviour could signiﬁcantly alter model predictions of NPI practices.

3.6 Supporting Information

3.6.1 Asymptomatic Cases

Asymptomatic infections can be prevalent in many diseases, and we thus explore the

impact of asymptomatic cases in our model. In these analyses, we use an asymptomatic

probability of 50% for infectious cases, and assume an asymptomatic individual is as infec-

tious as a symptomatic one. We also recalibrate parameters to maintain epidemic outcomes

similar to the baseline scenario while maintaining the baseline transmission rate of 0.005.

In this scenario, we use parameter values of λ = 2.85, γ = 0.75, θ = 0.403, and the remaining parameters at baseline values. This will allow us to directly compare the impact of asymptomatic infections.

With low transmission rate, vaccination delay does not impact epidemic size when NPIs are used. (Table 4.2). When NPIs are not used, epidemic size is minimally affected (≈

2.4% difference between no vaccine delay and a 60 day vaccine delay.) With baseline

transmission rate, the difference becomes larger (Table 4.3). With NPIs, ﬁnal size changes

by under 1 percentage point, but without NPIs, the difference is ≈ 8.5%. Finally, with a 76

higher transmission rate of β = 0.006 (Table 3.4), NPIs cause the ﬁnal size to change ≈

2.4% across all vaccine delays, whereas without NPI effects, ﬁnal size changes by ≈ 12%.

Thus, we observe similar effects of NPIs on epidemic final size when there are vaccine delays with asymptomatic cases as we saw without. That is, changes in final size between no delay and longer delays are much smaller when NPIs are incorporated than when they are not. Table 3.2: Epidemic final sizes (symptomatic cases only) with delayed vaccine availability. β = 0.004 Delay Final Size (with NPIs) ±95% CI Final Size (without NPIs) ±95% CI 0 0.03839 ± 0.0025 0.0932 ± 0.0013 10 0.0381 ± 0.0026 0.0924 ± 0.0014 20 0.0395 ± 0.0024 0.0933 ± 0.0012 30 0.0397 ± 0.0024 0.0940 ± 0.0023 40 0.04086 ± 0.0025 0.1010 ± 0.0019 50 0.04105 ± 0.0023 0.1108 ± 0.0024 60 0.03953 ± 0.0025 0.1173 ± 0.0029

Table 3.3: Epidemic ﬁnal sizes (symptomatic cases only) with delayed vaccine availability. β = 0.005 Delay Final Size (with NPIs) ±95% CI Final Size (without NPIs) ±95% CI 0 0.0816 ± 0.0016 0.1107 ± 0.0006127 10 0.0826 ± 0.0013 0.1111 ± 0.0006301 20 0.0821 ± 0.0012 0.1145 ± 0.0007419 30 0.0817 ± 0.0013 0.1313 ± 0.0022 40 0.0838 ± 0.0014 0.1648 ± 0.0031 50 0.0847 ± 0.0018 0.1855 ± 0.0031 60 0.0864 ± 0.0015 0.1958 ± 0.0023

Considering the effects of differing vaccine efﬁcacy on epidemic outcomes, we see sim- 77

Table 3.4: Epidemic ﬁnal sizes (symptomatic cases only) with delayed vaccine availability. β = 0.006 Delay Final Size (with NPIs) ±95% CI Final Size (without NPIs) ±95% CI 0 0.0987 ± 0.0006992 0.1236 ± 0.0006546 10 0.0985 ± 0.0006611 0.1225 ± 0.0006434 20 0.0994 ± 0.0006425 0.1373 ± 0.0016 30 0.1059 ± 0.0009648 0.1898 ± 0.0038 40 0.1130 ± 0.0013 0.2288 ± 0.0027 50 0.1202 ± 0.0013 0.2407 ± 0.0017 60 0.1228 ± 0.0014 0.2451 ± 0.0011

ilar results to the case with no asymptomatic infections. With NPIs used in the population,

symptomatic epidemic size and vaccine uptake do not show a large change across vaccine

efﬁcacies (≈ 0.3% and 2%, respectively) compared to when NPIs are not included (2.4% difference for ﬁnal size and 4.5% difference for vaccine uptake, see Tables 3.5 and 3.6).

Table 3.5: Epidemic final sizes corresponding to vaccine efficacy (symptomatic cases only). Efficacy Final Size (With NPIs) ±95% CI Final Size (Without NPIs) ±95% CI 0.5 0.0845 ± 0.0012 0.1350 ± 0.0008755 0.6 0.0835 ± 0.0014 0.1291 ± 0.0007565 0.7 0.0819 ± 0.0019 0.1231 ± 0.0007093 0.8 0.0819 ± 0.0013 0.1186 ± 0.0008193 0.9 0.0826 ± 0.0013 0.1145 ± 0.0006082 1.0 0.0814 ± 0.0014 0.1111 ± 0.0022

In general, the main qualitative results are similar to when there are asymptomatic in-

fections to when there are not. When NPIs are included, vaccine availability delays and

changes in vaccine efﬁcacy do not change epidemic ﬁnal size and vaccine uptake as signif-

icantly compared to scenarios where NPIs are not incorporated. 78

Table 3.6: Population vaccine uptake corresponding to vaccine efﬁcacy. Efﬁcacy Vaccine Uptake (With NPIs) ±95% CI Vaccine Uptake (Without NPIs) ±95% CI 0.5 0.4662 ± 0.018 0.7124 ± 0.0037 0.6 0.4604 ± 0.0201 0.7051 ± 0.0036 0.7 0.4449 ± 0.0223 0.6938 ± 0.0037 0.8 0.4408 ± 0.019 0.6850 ± 0.0053 0.9 0.4503 ± 0.019 0.6784 ± 0.0036 1.0 0.4478 ± 0.02 0.6676 ± 0.013

3.6.2 Network Types

3500

3000

2500

2000

1500 Frequency 1000

500

0 0 50 100 150 200 250 Node Degree

Figure 3.7: Frequency of node degrees in the empirically based network. The majority of nodes have a degree < 75, whereas fewer nodes have higher degrees. The average node degree is 38.645.

Different network types will have an impact on epidemic outcomes due to contact structures playing a pivotal role in disease transmission. In our model, we initially used an empirically based network. Here, we will look at results stemming from a random network and a power law network. For these two types of networks, we recalibrate the parameters to 79

achieve similar baseline scenarios to the outcomes corresponding to the empirical network.

For the random networks, we use the same average node degree seen in the empiri-

cally based network (Figure 3.7), and we generate new networks each simulation. The

parameters used to obtain the same epidemic outcomes as the original baseline scenario are

λ = 1.75, γ = 0.25, θ = 0.28, β = 0.00585, and the remaining parameters at baseline values. We note that the transmission rate must be higher with this network structure to achieve the same epidemic ﬁnal size, and in turn the same vaccine uptake, and NPI use as the empirical network. When either increasing or decreasing the transmission rate, we still use intervals of size 0.001.

With low transmission rate, vaccination delay does not signiﬁcantly impact epidemic size when NPIs are used. (Table 3.7). With a transmission rate of β = 0.00585, the

difference becomes larger (Table 3.8). With NPIs, ﬁnal size changes by under 1 percentage

point, but without NPIs, the difference is ≈ 18%, and the difference grows larger the later

the vaccine is made available. Finally, with a higher transmission rate of β = 0.00685

(Table 3.9), NPIs cause the ﬁnal size to change ≈ 10% across all vaccine delays, whereas

without NPI effects, ﬁnal size changes by ≈ 41%.

Considering the effects of differing vaccine efﬁcacy on epidemic outcomes in the random networks, we see similar results to the case with an empirically based network. With

NPIs used in the population, epidemic size and vaccine uptake show changes across vaccine efﬁcacies of ≈ 0.55% and 16%, respectively, compared to when NPIs are not included 80

Table 3.7: Epidemic ﬁnal sizes with delayed vaccine availability (random network). β = 0.00485 Delay Final Size (with NPIs) ±95% CI Final Size (without NPIs) ±95% CI 0 0.0608 ± 0.0047 0.1543 ± 0.0044 10 0.05964 ± 0.0047 0.1527 ± 0.0049 20 0.06297 ± 0.0047 0.1525 ± 0.0054 30 0.0602 ± 0.0049 0.1500 ± 0.0063 40 0.06353 ± 0.0045 0.1545 ± 0.0052 50 0.0602 ± 0.0046 0.1523 ± 0.005 60 0.0602 ± 0.0048 0.1539 ± 0.0049

Table 3.8: Epidemic ﬁnal sizes with delayed vaccine availability (random network). β = 0.00585 Delay Final Size (with NPIs) ±95% CI Final Size (without NPIs) ±95% CI 0 0.1592 ± 0.0015 0.1821 ± 0.0005972 10 0.1591 ± 0.0016 0.1818 ± 0.0005030 20 0.1592 ± 0.0014 0.1816 ± 0.00053807 30 0.1610 ± 0.0013 0.1823 ± 0.0007169 40 0.1602 ± 0.0017 0.2074 ± 0.0062 50 0.1609 ± 0.002 0.2911 ± 0.011 60 0.1649 ± 0.0027 0.3667 ± 0.012

Table 3.9: Epidemic ﬁnal sizes with delayed vaccine availability (random network). β = 0.00685 Delay Final Size (with NPIs) ±95% CI Final Size (without NPIs) ±95% CI 0 0.1779 ± 0.0005021 0.1927 ± 0.0009735 10 0.1781 ± 0.0004927 0.1932 ± 0.001 20 0.1776 ± 0.0004817 0.1933 ± 0.0009566 30 0.1798 ± 0.0012 0.2634 ± 0.0091 40 0.2116 ± 0.0038 0.4638 ± 0.011 50 0.2524 ± 0.0042 0.5780 ± 0.0063 60 0.2739 ± 0.0033 0.6100 ± 0.0042

(3.4% difference for ﬁnal size and only 1% difference for vaccine uptake, see Tables 3.10 and 4.1). An interesting dynamic occurs in the random network as although ﬁnal size dif- 81

ference for the range of vaccine efﬁcacy given in tables 3.10 and 4.1 with NPIs is much

smaller than when there are no NPIs, vaccine uptake increases much more. However, with

NPIs, vaccine uptake amongst the population is approximately 20-40% lower than with no

NPIs, and produces smaller epidemic sizes.

Table 3.10: Epidemic final sizes corresponding to vaccine efficacy (random network). Efficacy Final Size (With NPIs) ±95% CI Final Size (Without NPIs) ±95% CI 0.5 0.1644 ± 0.001 0.2158 ± 0.0009870 0.6 0.1616 ± 0.0014 0.2059 ± 0.0007589 0.7 0.1617 ± 0.0011 0.1978 ± 0.0006078 0.8 0.1609 ± 0.001 0.1904 ± 0.0005643 0.9 0.1603 ± 0.001 0.1858 ± 0.0004162 1.0 0.1589 ± 0.001 0.1821 ± 0.0004704

Table 3.11: Population vaccine uptake corresponding to vaccine efﬁcacy (random network). Efﬁcacy Vaccine Uptake (With NPIs) ±95% CI Vaccine Uptake (Without NPIs) ±95% CI 0.5 0.5940 ± 0.039 0.8176 ± 0.0003564 0.6 0.5505 ± 0.041 0.8159 ± 0.0003212 0.7 0.5286 ± 0.042 0.8137 ± 0.0003098 0.8 0.4762 ± 0.042 0.8112 ± 0.0002534 0.9 0.4546 ± 0.042 0.8089 ± 0.0002167 1.0 0.4323 ± 0.043 0.8067 ± 0.0002177

For the power law networks, we use a Barabasi-Albert´ algorithm with three initial con-

nected nodes to create a contact network for each simulation. The parameters used to ob-

tain the same epidemic outcomes as the original baseline scenario are λ = 1.25, γ = 0.5,

θ = 0.35, β = 0.075, and the remaining parameters at baseline values. We note that the transmission rate must be higher with this network structure to achieve the same epidemic 82

ﬁnal size, and in turn the same vaccine uptake, and NPI use as the empirical network. When

either increasing or decreasing the transmission rate, we will use intervals of size 0.02.

With a transmission rate of 0.0055, vaccination delay impacts epidemic size by ≈ 4% when NPIs are used and ≈ 11 when they are not (Table 3.12). With a transmission rate of β = 0.075, the difference becomes larger (Table 3.13). With NPIs, ﬁnal size changes by about 9 percentage points, but without NPIs, the difference is ≈ 23%. For both cases,

the difference grows larger in the early stages of the epidemic. Finally, with a higher

transmission rate of β = 0.0095 (Table 3.14), NPIs cause the ﬁnal size to change ≈ 15%

across all vaccine delays, whereas without NPI effects, ﬁnal size changes by ≈ 31%. Again,

NPIs reduce epidemic sizes, as well as decrease the change in ﬁnal size induced by delays in vaccine availability. In the scale free networks we created, the majority of epidemic incidence increase occurs before the 30 day mark of vaccination absence.

Table 3.12: Epidemic ﬁnal sizes with delayed vaccine availability (power law network). β = 0.055 Delay Final Size (with NPIs) ±95% CI Final Size (without NPIs) ±95% CI 0 0.1231 ± 0.0014 0.1736 ± 0.000806 10 0.1282 ± 0.0028 0.1805 ± 0.0012 20 0.1411 ± 0.0033 0.2242 ± 0.0038 30 0.1623 ± 0.0017 0.2644 ± 0.0043 40 0.1607 ± 0.0021 0.2822 ± 0.006 50 0.1628 ± 0.0018 0.2869 ± 0.0027 60 0.1638 ± 0.002 0.2875 ± 0.0028

Considering the effects of differing vaccine efﬁcacy on epidemic outcomes in the power 83

Table 3.13: Epidemic ﬁnal sizes with delayed vaccine availability (power law network). β = 0.075 Delay Final Size (with NPIs) ±95% CI Final Size (without NPIs) ±95% CI 0 0.1685 ± 0.0008614 0.2053 ± 0.0013 10 0.1755 ± 0.0011 0.2253 ± 0.0023 20 0.2318 ± 0.0029 0.3602 ± 0.0057 30 0.2520 ± 0.002 0.4276 ± 0.0028 40 0.2565 ± 0.002 0.4306 ± 0.0024 50 0.2576 ± 0.002 0.4344 ± 0.0029 60 0.2599 ± 0.0021 0.4330 ± 0.0023

Table 3.14: Epidemic ﬁnal sizes with delayed vaccine availability (power law network). β = 0.095 Delay Final Size (with NPIs) ±95% CI Final Size (without NPIs) ±95% CI 0 0.1921 ± 0.0011 0.2316 ± 0.0023 10 0.2124 ± 0.0023 0.2823 ± 0.0057 20 0.3136 ± 0.0031 0.4975 ± 0.0044 30 0.3346 ± 0.0025 0.5418 ± 0.0026 40 0.3427 ± 0.0022 0.5502 ± 0.0026 50 0.3401 ± 0.0021 0.5444 ± 0.0024 60 0.3426 ± 0.002 0.5442 ± 0.0026

law networks, we see similar results to the case with an empirically based network. With

NPIs used in the population, epidemic size and vaccine uptake show changes across vaccine efﬁcacies of ≈ 3.6% and 2%, respectively, compared to when NPIs are not included (8.8% difference for ﬁnal size and 4.4% difference for vaccine uptake, see Tables 3.10 and 4.1).

We have seen with these two alternate types of networks that the main results consistently hold. However, each network has unique dynamics that must be taken into account in certain scenarios. Thus, assumptions about network structure and transmission are an important consideration particularly when modelling a speciﬁc disease. For example, a 84

Table 3.15: Epidemic final sizes corresponding to vaccine efficacy (power law network). Efficacy Final Size (With NPIs) ±95% CI Final Size (Without NPIs) ±95% CI 0.5 0.2040 ± 0.0014 0.2941 ± 0.0017 0.6 0.1982 ± 0.0011 0.2751 ± 0.0016 0.7 0.1922 ± 0.0011 0.2557 ± 0.0014 0.8 0.1816 ± 0.0009980 0.2381 ± 0.0014 0.9 0.1762 ± 0.0008868 0.2219 ± 0.0014 1.0 0.1682 ± 0.0007904 0.2060 ± 0.0013

Table 3.16: Population vaccine uptake corresponding to vaccine efﬁcacy (power law network). Efﬁcacy Vaccine Uptake (With NPIs) ±95% CI Vaccine Uptake (Without NPIs) ±95% CI 0.5 0.4504 ± 0.003 0.5416 ± 0.0025 0.6 0.4405 ± 0.0027 0.5321 ± 0.002 0.7 0.4493 ± 0.0029 0.5223 ± 0.0022 0.8 0.4439 ± 0.0028 0.5213 ± 0.0022 0.9 0.4303 ± 0.0025 0.4987 ± 0.0019 1.0 0.4302 ± 0.0026 0.4972 ± 0.0017

transmission network for inﬂuenza likely has a different structure than one that would be

used to model HIV transmission.

3.6.3 Pairwise Analysis

We conducted pairwise sensitivity analysis for the behaviour response parameters λ

and γ (Figure 3.8). In total, 81 combinations were used and results of each combination are given as the averages over 500 realizations. We ﬁnd that increasing λ has the most

beneﬁcial effect on decreasing epidemic ﬁnal size and increasing vaccination rates. Also,

increasing γ alongside λ can complement these results decreasing incidence or increasing

vaccination further. Increasing both of these parameters can also decrease the NPI use 85

amongst susceptible individuals at the end of an epidemic. This seems counter intuitive, but

since more individuals are vaccinating efﬁcaciously and the amount of infected individuals

becomes smaller, many in the population do not need to practice strong NPI use. However,

increased γ with smaller values of λ can promote NPIs in the population, increasing the

use of these self protective measures when an epidemic is more widespread. Finally, an

increase in λ and γ will shorten the duration the an epidemic, given that far fewer people are becoming infected.

0.9 0.9 0.3 0.4 0.7 0.25 0.7 0.2 γ 0.5 γ 0.5 0.15 0.2 0.3 0.3 0.1 0.1 0.1 0 0.5 1.0 1.5λ 2.0 2.5 0.5 1.0 1.5λ 2.0 2.5

(a) (b)

0.9 0.35 0.9 100 0.7 0.3 0.7 90

γ 0.5 0.25 γ 0.5 80 0.3 0.2 0.3 70 0.1 0.1 0.15 0.5 1.0 1.5λ 2.0 2.5 0.5 1.0 1.5λ 2.0 2.5

Figure 3.8: Pairwise sensitivity analysis of parameters λ and γ. (A) cumulative incidence, (B) cumulative vaccination, (C) transmission rate reduction amongst susceptible population, (D) epidemic length. Results averaged over 500 trials. 86

Chapter 4

Disease Interventions Can Interfere

With One Another Through Disease-

Behaviour Interactions

4.1 Chapter Abstract

Theoretical models of disease dynamics on networks can aid our understanding of how infectious diseases spread through a population. Models that incorporate decision- making mechanisms can furthermore capture how behaviour-driven aspects of transmission such as vaccination choices and the use of non-pharmaceutical interventions (NPIs) interact with disease dynamics. However, these two interventions are usually modelled separately.

This chapter is based on the article in PLOS Computational Biology (2015), vol. 11(6): e1004291. Authored by Michael A. Andrews and Chris T. Bauch. 87

Here, we construct a simulation model of inﬂuenza transmission through a contact network, where individuals can choose whether to become vaccinated and/or practice NPIs. These decisions are based on previous experience with the disease, the current state of infection amongst one’s contacts, and the personal and social impacts of the choices they make.

We find that the interventions interfere with one another: because of negative feedback between intervention uptake and infection prevalence, it is difficult to simultaneously increase uptake of all interventions by changing utilities or perceived risks. However, on account of vaccine efficacy being higher than NPI efficacy, measures to expand NPI practice have only a small net impact on influenza incidence due to strongly mitigating feedback from vaccinating behaviour, whereas expanding vaccine uptake causes a significant net reduction in influenza incidence, despite the reduction of NPI practice in response. As a result, measures that support expansion of only vaccination (such as reducing vaccine cost), or measures that simultaneously support vaccination and NPIs (such as emphasizing harms of influenza infection, or satisfaction from preventing infection in others through both interventions) can significantly reduce influenza incidence, whereas measures that only support expansion of NPI practice (such as making hand sanitizers more available) have little net impact on influenza incidence. (However, measures that improve NPI efficacy may fare better.) We conclude that the impact of interference on programs relying on multiple interventions should be more carefully studied, for both influenza and other infectious diseases. 88

4.2 Introduction

Infectious diseases continue to threaten human health throughout the world (Laxmi- narayan et al., 2006; Klein et al., 2007). In order to help alleviate these impacts, researchers have utilized mathematical models to improve our understanding of infectious disease dynamics (Hethcote, 2000). In many cases, these models assume that human behaviour does not change over time or respond to disease dynamics in epidemiologically relevant ways.

However, individual behaviour often does both inﬂuence–and evolve in response to–disease dynamics. For example, when vaccination is not mandatory, the prevalence of an infectious disease can depend on individual decisions of whether or not to vaccinate (Brewer et al.,

2007; Galvani et al., 2007). Other behavioural practices that impact the spread of a disease include non-pharmaceutical interventions (NPIs) (Bootsma and Ferguson, 2007; He et al.,

2013; Del Valle et al., 2005; Kelso et al., 2009; Centers for Disease Control and Prevention,

2012). For susceptible individuals, NPIs can include hand washing or general avoidance of infectious individuals. For infectious individuals, these can include reducing contact with susceptible contacts, hand washing, or strict respiratory etiquette.

Mathematical models of the behavioral epidemiology of infectious diseases capture interplay between disease dynamics and individual behaviour (Bauch et al., 2013) (we will refer to these as “disease-behaviour” models hereafter). These and similar types of models have focussed on vaccinating decisions where individual decision-making occurs according to a strategic environment or is determined by some other utility-based or rule-based mechanism (Fine and Clarkson, 1986; Bauch, 2005; Bauch and Earn, 2004; Fu et al., 2010; 89

Perisic and Bauch, 2009, 2008; Zhang et al., 2010; Reluga et al., 2006; Salathe´ and Bon- hoeffer, 2008). Using such frameworks, Bauch (Bauch, 2005), Fu et al. (Fu et al., 2010),

Salathe´ and Bonhoeffer (Salathe´ and Bonhoeffer, 2008), and Reluga et al. (Reluga et al.,

2006) use models with imitation dynamics to predict potential vaccine uptake in populations. Similarly, Xia and Liu (Xia and Liu, 2013) base vaccination decisions not only on minimization of the associated costs, but also the impact that social inﬂuence has on each individual. d’Onofrio et al. (d‘Onofrio et al., 2007) use an information dependent model where vaccination decisions are based on private and public information gathered about a disease. Further approaches by Vardavas et al. (Vardavas et al., 2007) incorporate memory of past disease prevalence, and Wells and Bauch (Wells and Bauch, 2012) include memory of previous infections to study the effect of these factors on vaccinating behaviour. Other research has explored the impact of increased individual sexual risk behaviour on disease incidence, in response to the introduction of a hypothetical HIV/AIDS vaccine (Francis et al., 2003; Smith and Blower, 2004), or how risk perception, HIV prevalence and sexual behaviour interact with one another in a core group population (Tully et al., 2013).

Other disease-behaviour models incorporate social distancing and other NPI-related behaviours. For example, Reluga (Reluga, 2010) analyzes a differential game in which individuals choose a daily investment in social distancing in order to reduce the risk of infection. Funk et al. (Funk et al., 2010) allow information of a disease to spread over a network, and individuals protect themselves according to the quality of information they possess. Gross et al. (Gross et al., 2006) and Shaw and Schwartz (Shaw and Schwartz, 90

2008, 2010) study adaptive networks, where susceptible nodes rewire their connections from infectious to non-infectious nodes at a certain rate. Along the same vein, Zanette and Risau-Gusman (Zanette and Risau-Gusman, 2008) allow susceptible nodes to either permanently sever a connection with an infectious node, or rewire to another randomly chosen (and possibly infectious) node. Del Valle et al. (Del Valle et al., 2005) assume some individuals lower their contact rates once an epidemic is detected, whereas Glass et al. (Glass et al., 2006) and Kelso et al. (Kelso et al., 2009) use complex contact networks which include families, schools, and workplaces to test differing social distancing methods such as school closures and the effects of staying at home while infectious.

Hence, disease-behaviour models studying either vaccinating behaviour or NPI behaviour separately from one another are relatively abundant, but models incorporating both types behaviour are rare, to our knowledge. However, for many infectious diseases (such as influenza) both NPIs and vaccines are part of infection control strategies, and both also respond to disease dynamics (Brewer et al., 2007; Chapman and Coups, 1999a). Under these circumstances, it becomes important to study disease-behaviour interactions in populations where both vaccinating behaviour and NPI behaviour respond to disease dynamics. The effectiveness of one type of intervention may interfere with the effectiveness of the other intervention, through the mediator of disease dynamics. The objective of our research is to explore the interplay between individual decision-making (which is driven by methods from decision field theory (Busemeyer and Townsend, 1993)), regarding vaccines, NPIs, and disease dynamics in the context of influenza transmission and control, and to study the 91

implications for disease mitigation strategies.

4.3 Model

4.3.1 Vaccination

We model the vaccination decision process using random walk subjective expected util-

ity (SEU) theory, an intermediate stage in the mathematical derivation of decision ﬁeld

theory (Busemeyer and Townsend, 1993). This approach allows us to capture the decision-

making process of individuals in an uncertain environment. In the case of vaccination, the

uncertainty lies in the chance of becoming infected in a given inﬂuenza season, depending

on whether or not the individual is vaccinated and how effective the vaccine is.

For the vaccinating decision, each susceptible individual compares the possible out-

comes stemming from the “Yes” branch versus the possible outcomes stemming from the

“No” branch (Fig 4.1A). The difference in these expected utilities, or ‘valence’ (VY (t) −

VN (t)), updates an individual’s preference, P (t), towards choosing one of these actions.

The preference state of each individual is updated daily according to the rule:

P (t) = P (t − 1) + [VY (t) − VN (t)]. (4.1)

If P (t) reaches a speciﬁed threshold, θ, then an individual decides to become vaccinated in that inﬂuenza season. Conversely, if P (t) reaches −θ, the individual decides not to become vaccinated that season. Intermediate values −θ < P (t) < θ can be interpreted 92

as an individual being undecided regarding whether or not to vaccinate. If a choice is

made, an individual’s preference state remains constant at P (t) = θ or P (t) = −θ until the beginning of the next season, when it then resets to (1−s)Pend where Pend is the preference

at the end of the last inﬂuenza season, and s is a memory decay factor.

A E + w E w1 w3 E + E I 5 H No Yes V S Vaccinate 0 w2 w4 EV + VN (t)

B E w5 w7 (E )(N )+ E H No Yes D T ot S NPI 0 w6 w8 (ED)(NT ot)+ VN (t)

C E w9 w7 (E )(N ) I No Yes D Inf NPI 0 w10 w8 (ED)(NInf )+ VN (t)

Figure 4.1: (A) Diagram representing the vaccination choice problem. (B) Diagram representing the infectious NPI choice problem. (C) Diagram representing the susceptible NPI choice problem.

Let us now deﬁne the social utility parameters associated with the vaccinating decision.

The quantity EI < 0 is the negative utility received (cost incurred) when an individual gets infected; EV < 0 is the negative utility received (cost incurred) when an individual vaccinates; ES > 0 is the positive utility received when an individual takes actions that they perceive will inhibit the spread of infection and therefore saves others from becoming infected; and EH < 0 is the negative utility received (cost incurred) when an individual be- 93

lieves they are responsible for harming a neighbour by infecting them (Shim et al., 2012).

The baseline values for these and other parameters can be found in Table 4.1. If an individ-

ual chooses to not vaccinate during a season, they may become infected that season. If an

individual instead chooses to vaccinate, the vaccine is effective for that season with prob-

ability V (the vaccine efﬁcacy) and otherwise the individual remains susceptible for the

remainder of the season. The wi parameters (0 ≤ wi ≤ 1) represent the ‘subjective prob-

ability weights’ that determine the possible outcomes that are considered on a given day.

In the case of vaccination, w1 is the probability an individual perceives of being infected if they do not vaccinate in a particular season. We assume w1 depends on how many of an individual’s neighbours have been infected in the current season, as well as their memory of the cumulative incidence from previous seasons:

w1 = σMH (Xn) + (1 − σ)ξn−1, (4.2)

where σ controls the relative importance of incidence from current versus past seasons, Xn is the cumulative number of neighbours who have become infected in the current inﬂuenza

−κH x season n, MH (x) = 1 − e where κH is a proportionality constant controlling the perceived chance of becoming infected in a season, and ξn−1 is an individual’s memory of incidence from past seasons:

ξn = σMH (Yn) + (1 − σ)ξn−1, (4.3) 94

where Yn is the cumulative number of an individual’s neighbours, including themselves, that have been infected by the end of season n. In this way, the memory of past inﬂuenza incidence declines with time according to σ.

Individuals may vaccinate during certain days of the year 280 (October 6) < t or t < 40

(February 9) (Fig 4.2), where t = 0 is January 1st, and the influenza season from the previous year is considered to end on day t = 285. We use this constraint because it reflects the typical timing of public influenza vaccination programs in fall and winter in many northern hemisphere countries, such as the United States and Canada. At the end of an influenza season, we set Yn = Xn and then incorporate Yn into the individual’s memory.

0.4

Cumulative Vaccine Uptake 0.35

Cumulative Incidence 0.3

0.25

0.2

0.15

Mean Over 100 Parameter Sets 0.1

0.05

0 Nov Dec Jan Feb

Figure 4.2: Example time series of vaccination (blue) and incidence (green) over a season. When a vaccine becomes available in early October, uptake increases in anticipation of the upcoming inﬂuenza season. 95% conﬁdence intervals over the 100 parameter sets included (see Model Calibration section). 95

The second subjective probability weight we will discuss is

w5 = MC (Nsusc + (1 − V )Nvac), (4.4)

−κC x where MC (x) = 1 − e , Nvac and Nsusc are the number of currently vaccinated and susceptible neighbours, respectively, and κC controls the perceived probability of infection. We interpret w5 as an individual’s perceived probability of infecting one or more neighbours, and the term w5EH captures the future outcome of an individual potentially infecting his/her neighbours that season after becoming infectious themselves, ultimately leading to a negative utility. To complete the outcomes of this branch, we note that the perceived probability of not becoming infected when choosing to not vaccinate is simply w2 = 1 − w1. This outcome leads to a utility of 0.

On the ‘Yes’ branch, we define w3 = V as the perceived probability that an individual is efficaciously vaccinated, thus w4 = 1 − V . In both cases, an individual knows that they must absorb the vaccine cost, EV . In the case of efficacious vaccinating, a positive utility

ES is also gained by protecting their neighbours for the remainder of the influenza season, which serves a similar function to the w5EH term stemming from the ‘No’ branch where an individual is considering future outcomes. In the case of inefficacious vaccination, an individual assumes that they may still become infected with a probability that increases with past and current disease incidence. This is represented by the term VN (t), the valence of the ‘No’ branch. 96 Source (Network Dynamics and Simulation Science Laboratory, 2008a,b,c) Calibrated Calibrated Calibrated Calibrated Calibrated Calibrated Calibrated (Truscott et al., 2011); Calibrated (Wells and Bauch, 2012) (Wells and Bauch, 2012) Calibrated Calibrated Calibrated (Bridges et al., 2000; Demicheli et al., 2007) (Larson et al., 2010; Sheehan et al., 2007) (Earn et al., 2002; Longini et al., 2000; Cox(Ambrose et et al., al., 2004) 2008) (Lee et al., 2002; Carrat et al., 2007; EarnCalibrated et al., 2002; Nichol et al., 2010) 6 4 4 − − − 10 10 10 1 1 2 7 4 5 000 25 ...... 0055 0015 5 045 . , . . 20 0.9 0.5 . 0 0 0 0 0 0 × × × 0.25 0.05 0 0 0 0 Value 10 6 5 9 − − . − 6 − R to state , per season , per season I S S to State to State R V Table 4.1: Model parameters with baseline values and sources. Description Number of Individuals in Network Preference state threshold for vaccinating Preference state threshold for distancing Memory decay rate, per season Weight assigned to present state of infection in neighbours Proportionality constant of perceived seasonal infection risk Proportionality constant of perceived daily infection risk Average transmission rate Change in seasonality amplitude Cost incurred from becoming infected Cost incurred from vaccination Cost incurred from distancing a neighbour Payoff for saving a neighbour from infection Cost incurred for infecting a neighbour(s) Vaccine efficacy NPI efficacy Probability of moving from state Probability of moving from state Average number of days to move from state Total number of exogenous infections per season I β S 0 V D H C H V D V s ρ η λ σ ω P op β θ θ E κ E κ NPI E ∆ E E N Parameter 97

4.3.2 Non-Pharmaceutical Interventions

We model the NPI decision process using sequential SEU theory, a method similar to

random walk SEU theory, but excludes the possibility that the preference state may start

from a non-neutral initial value (Busemeyer and Townsend, 1993). We use sequential SEU

theory because we assume individuals make social distancing decisions on a day-to-day

basis, dependent only on the current state of infection amongst their respective neighbours,

whereas in the case of vaccination, the tendency to vaccinate or not can be carried over from

one season to the next. Each infectious individual decides whether or not to practice NPIs

to protect their neighbours for the duration of their illness (Fig 4.1B), and each susceptible

individual decides whether to practice NPIs to protect themselves from their infectious

neighbours that day (Fig 4.1C).

On the ‘Yes’ branch for the infectious NPI decision (Fig 4.1B), we have the probability

of efﬁcaciously using NPIs, w7 = NPI , or inefﬁcaciously using them, w8 = 1 − NPI .

If NPIs are efﬁcacious, they receive a positive utility ES for saving susceptible neighbours from infection. However, if NPIs are inefﬁcacious, they receive the valence VN (t) of the

‘No’ decision, on the branch associated with w8, representing that the outcome is the same as if they had never practiced NPIs to begin with. An individual believes that their use of interventions during their illness will be either fully effective or ineffective on all of their neighbours. Regardless of whether NPIs work or not, the infectious individual who practices NPIs pays a cost (ED)(NT ot) for having to utilize NPIs to protect their NT ot neighbours, where ED < 0 is the negative utility received (cost incurred) per neighbour. 98

For the ‘No’ branch where the individuals decides not to practice NPIs, they may infect a neighbour with probability w5, receiving a negative utility (cost incurred) of EH < 0 due to feeling responsible for spreading the infection. On the other hand, they infect no neighbours with probability w6 = 1 − w5, thereby receiving a utility of zero.

NPI decisions are made by susceptible individuals who seek to protect themselves from their infectious neighbours in a similar way (Fig 4.1C). On the ‘Yes’ branch, an individual believes their NPIs will be efﬁcacious with probability w7, receiving utility zero. If the

NPIs are not efficacious, they receive the valence VN (t) from the ‘No’ branch. In either case, they pay a cost ED in order to practice NPIs. On the ‘No’ branch, an individual who does not practice NPIs that day becomes infected with probability w9 = MC (NInf ), and receive a negative infection utility EI . With probability 1 − w9, they believe they will not become infected and receive utility zero. We do not apply social utilities EH and ES in the susceptible NPI decision process because we assume as a first-order approximation that their decision focuses on short term outcomes (NPIs are only effective for the duration of infection, and may have to be repeated several times in the season, whereas a one-time vaccination decision will protect their neighbours from infection for the duration of the season). Also, if an individual is vaccinated, their perceived probability of becoming infected that day is reduced by 1−V . This reflects the fact that these individuals will believe themselves to have less chance of becoming infected than those who have not vaccinated.

We note that infectious individuals practice NPIs on all of their neighbours, whereas susceptible individuals only practice them on their infectious neighbours. Infectious persons 99

may stay home from work, and their hand washing beneﬁts all susceptible persons with

whom they come in contact with. In contrast, susceptible persons can be selective about

avoiding infectious persons, or hand-washing after contact with infectious persons.

In our model, individuals are not aware of their own or their neighbours’ true suscep-

tibility statuses. That is, they will make their intervention decisions based only on their

own acquired knowledge which assumes everyone is susceptible at the beginning of each

inﬂuenza season. This is in contrast to the true state of the system, which incorporates

factors such as waning immunity. Moreover, the data we present on susceptible NPI rates

only reports for those who are truly susceptible.

4.3.3 Transmission Dynamics

The vaccination and NPI decision-making processes are embedded in an agent-based

simulation model of inﬂuenza transmission through a static contact network. The contact

network consists of 10,000 nodes through which inﬂuenza is transmitted, and was con-

structed by sampling a subnetwork from a larger contact network derived from census data

from Portland, Oregon (Network Dynamics and Simulation Science Laboratory, 2008a).

Previous research has conﬁrmed that the subnetwork is a good approximation to the full

network (Wells et al., 2013). The full network can be found in the supplementary material

in (Andrews and Bauch, 2015).

We assume a Susceptible-Infectious-Recovered-Vaccinated-Susceptible (SIRVS) natural history. Individuals move from the susceptible state S to the infectious state I with 100

Ninf probability P r(t, Ninf ) = 1−(1−β(t)) per day, where Ninf is the number of infectious

neighbours around the susceptible person on day t, and β is the transmission probability

2πt which varies seasonally according to β(t) = β0(1 + ∆βcos( 365 )) (Truscott et al., 2011).

If either the susceptible person or the infected person has opted to practice NPIs that day,

then NPIs are effective with probability NPI , and that infected neighbour is not included in Ninf for the purposes of computing P r(t, Ninf ). Infectious individuals recover (move

from state I to state R) in a number of days sampled from a Poisson distribution with mean

λ days. Individuals who have been efﬁcaciously vaccinated with probability V are moved

to the V state, 14 days after being vaccinated (Centers for Disease Control and Prevention,

2013). Both recovered and vaccinated individuals become susceptible again at the begin-

ning of each new season (day 285 of each year) with probabilities ρ and ω, respectively. In

order to capture seasonal case importation, 5 randomly chosen susceptible individuals are

made infectious every 10 days, from day 330 to 360. We deﬁne the total number of such

cases as eta.

Each day, the following sequence of events occurs: (1) each susceptible individual de-

cides whether or not to practice NPIs on that day; (2) the following occur in a random order

for each randomly chosen individual in the population: (i) If an individual is susceptible,

they update their vaccination preference, (ii) if an individual is susceptible, they may be-

come infected and make an infectious NPI decision, (iii) if an individual is infectious, they

may recover. 101

4.3.4 Model Calibration

To construct a baseline scenario, we calibrated the transmission probability β, ampli-

tude of seasonality ∆β, and case importation rate η to the targets: (1) a cumulative seasonal incidence of approximately 15% to 20% in the absence of vaccination, and (2) infection

prevalence that peaked, on average, between December and January of each year (Wells

and Bauch, 2012; World Health Organization, 2014; Bridges et al., 2000; Keitel et al.,

1997). These estimates come from North American (primarily, United States) populations.

The calibrated value of ∆β was constrained on the interval (0.15, 0.3) (Truscott et al.,

2011).

We also calibrated the preference state threshold for vaccinating θV , per season memory decay rate s, and proportionality constant for the perceived chance of becoming infected in an inﬂuenza season κH to the targets: (1) vaccine coverage of 30% to 40% per season,

with (2) the majority of vaccinations occurring in October and November. These targets

provide disease dynamics very similar to seasonal inﬂuenza. The utilities EI and EV were set according to Ref. (Wells and Bauch, 2012), based in turn on Ref. (Bridges et al., 2000;

Keitel et al., 1997; Lee et al., 2002; Nichol, 2001; Meltzer et al., 1999; Bureau of Labor

Statistics, 2014; Keech et al., 1998). Finally, the social utilities ED, ES, and EH were calibrated to the targets: (1) an infectious individual practices NPIs with 87% probability, and (2) a susceptible individual practices NPIs with a 66% probability (Mitchell et al.,

2011).

After obtaining this baseline scenario, we conducted three-point estimation Monte Carlo 102

probabilistic sampling using triangular distributions to obtain sets of parameter values that yielded outputs within acceptable ranges. The triangular distributions were deﬁned around the most uncertain baseline parameter values. Very broad ranges were used for the most uncertain parameters, to reduce model ﬁtting issues due to having more parameters than calibration targets (generally, for each set of calibrated parameter values described above, there was one less target data point than the number of parameters to be calibrated) (Table

4.2). Parameters with less intuitive ranges had their ranges chosen based on results from preliminary sampling. We repeatedly sampled parameter values from these distributions, and ran simulations using the sampled parameter sets. We discarded any parameter sets that yielded outcomes outside of a feasible range for vaccine uptake and NPI practice rates

(Table 4.3). We accepted a larger range of NPI practice rates than for vaccine uptake, due to the greater uncertainties about the frequency of NPI practice for seasonal inﬂuenza

(Mitchell et al., 2011). In total, 2250 simulations were tested, providing a target number of 100 parameter sets yielding feasible outcomes. All simulations ran for 50 seasons with an initial population of susceptible individuals having no preference towards vaccinating and no perceived probabilities of becoming infected. For our results, the ﬁrst 20 seasons of each simulation were discarded to remove transient effects.

The data shows the average value per season over all 100 parameter sets, for vaccine coverage, infection incidence, probability of susceptible individuals practicing NPIs given that they encounter one or more infectious individuals on a given day (“susceptible NPI 103

Table 4.2: Sampling ranges for parameters used to obtain 100 baseline sets. Parameter Sampling Limits Sampling Mode θV [0.01, 0.2] 0.1 θD [0.01, 0.2] 0.1 s [0.8, 0.99] 0.9 σ [0.3, 0.7] 0.5 κH [0.1, 0.4] 0.25 κC [0.036, 0.064] 0.05 −6 −6 ED [0.0, −12 × 10 ] −6 × 10 −4 −4 ES [0.0, 13 × 10 ] 6.5 × 10 −4 −4 EH [0.0, −18 × 10 ] −9 × 10

Table 4.3: Acceptance ranges for simulation averages across 30 seasons. Intervention Measure Acceptance Range Vaccine Uptake (% of population) [30, 40] Susceptible NPI Probability (%) [50, 75] Infectious NPI Probability (%) [70, 90]

practice”), and probability of infectious individuals practicing NPIs while ill (“infectious

NPI practice”).

4.4 Results

4.4.1 Baseline Scenario

When vaccination is first introduced to the population, vaccine coverage climbs rapidly and peaks in the first few years after the vaccine becomes available, as members of the population adopt vaccination to avoid infection (Fig 4.3). As a result, seasonal influenza incidence decreases, which in turn causes a decrease in the probability that susceptible per- 104

sons practice NPIs if they have an infected neighbour. This occurs because the reduced infection incidence due to the vaccine reduces the perceived infection risk among susceptible individuals, and thus makes them less willing to practice NPIs. After the initial peak in vaccine coverage and the corresponding dip in NPI practice, the vaccine uptake, infection incidence, and NPI practice rates equilibrate. In contrast to susceptible NPI practice, the infectious NPI practice rate stays almost constant during the whole period, due to the relative stability of the utilities found in the decision branches regarding this decision (Fig

4.1B). For example, an infectious individual deciding to use NPIs will always look to protect all of their neighbours, and this does not depend strongly on population-level incidence of infection that season. This is in contrast to a susceptible individual’s decision whether to use NPIs, which depends on how many of their neighbours are perceived to be infected.

4.4.2 Interventions Can Interfere With One Another

Next, we conducted a univariate sensitivity analysis, evaluating the impact of changes in baseline parameter values corresponding to measures that public health might take in order to improve outcomes. Increasing the utility for saving others from infection (ES) causes a signiﬁcant reduction in infection incidence (Fig 4.4A,B). It also causes a slight decrease in NPI practice by susceptible individuals, but this is outweighed by large increases in both vaccine uptake and NPI practice by infectious persons that are sufﬁcient to cause a net decline in infection incidence. These results illustrate a tradeoff whereby vaccine uptake, NPI practice among infectious individuals, and NPI practice among susceptible individuals can- 105

Figure 4.3: Example of a baseline scenario of our model where vaccination becomes available in season 10, causing a change in the vaccine coverage each season (blue), the seasonal infection incidence (green), the probability that a susceptible individual practices NPIs given that they encounter one or more infectious individuals on a given day (black), and the probability that an infectious individual practices NPIs while ill (red). 95% conﬁdence intervals over the 100 parameter sets included.

not be simultaneously increased by changing ES. A reduced incidence due to expanding any one of these interventions will reduce the perceived infection risk and make individuals incrementally more complacent about preventing infection, which in turn reduces the uptake rates for the other interventions. Hence, each intervention tends to interfere with the other. Our focus in the remainder of this paper is to determine the conditions under which the interference between the two intervention types is strongest, which model parameters are most subject to interference, and how to bring about the greatest net reductions in infection incidence, despite interference. How interference plays out over time has already been described (Fig 4.3).

In contrast to signiﬁcant reductions in infection incidence caused by increasing ES, in- 106

creasing the cost for harming others (EH ) causes only small net reductions in incidence, because a large increase in the NPI practice among infectious persons is strongly offset by a modest decline in vaccine uptake, while the rate of NPI practice by susceptible persons remains relatively constant (Fig 4.4C,D). Similarly, attempting to reduce incidence by decreasing the perceived cost of practicing NPIs (ED) also causes only a small net reduction in incidence, since the resulting increases in NPI practice among infectious and susceptible persons are again offset by reductions in vaccine uptake (Fig 4.4E,F).

Decreasing the cost of vaccination (EV )–for instance by making the vaccine more easily accessible–results in signiﬁcant reductions in infection incidence, because the signiﬁcant increase in vaccine uptake is only partly offset by the resulting decline in NPI practice (Fig

4.5A,B). Likewise, increasing the perceived cost of infection (EI ) causes an increase in both vaccine uptake and susceptible NPI practice, although infectious NPI practice remains relatively unchanged. The effect is a signiﬁcant net decrease in incidence (Fig 4.5C,D). In summary, increasing the utility for saving others from infection (ES), decreasing the perceived cost of vaccination (EV ), or increasing the perceived cost of infection (EI ), are more effective in reducing infection incidence than changing perceived harms (EH ) or perceived cost of NPI (ED), despite interference. 107

Figure 4.4: Univariate sensitivity analysis for parameters ES, EH , and ED. The numbers on the horizontal axes correspond to multiples of the baseline values for ES, EH , and ED, hence 1.0 corresponds to the baseline value of the each parameter. (A),(C),(E) Average values across 30 seasons for vaccination coverage (blue) and incidence (green). (B),(D),(F) Average values across 30 seasons for NPI usage amongst susceptible individuals (black) and infectious individuals (red). Conﬁdence intervals represent two standard deviations of the mean of the 100 parameter sets across 30 simulated seasons. 108

Figure 4.5: Univariate sensitivity analysis for EV and EI . The numbers on the horizontal axes correspond to multiples of the baseline values for EV and EI , hence 1.0 corresponds to the baseline value of the each parameter. (A),(C) Average values across 30 seasons for vaccination coverage (blue) and incidence (green). (B),(D) Average values across 30 seasons for NPI usage amongst susceptible individuals (black) and infectious individuals (red). Conﬁdence intervals represent two standard deviations of the mean of the 100 parameter sets across 30 simulated seasons. 109

4.4.3 Determining Which Interventions Interfere Most Strongly

In order to better understand how NPIs interfere with vaccine uptake, we compute the

difference ∆V in vaccine uptake between the baseline scenario where individuals are free to practice susceptible and infectious NPIs versus a hypothetical scenario where they cannot practice either form of NPI. Similarly, to determine how vaccination interferes with susceptible (and infectious) NPI practice, we compute the difference ∆NS in susceptible

NPI practice rates (similarly, ∆NI for infectious NPI practice rates) between the baseline

scenario where individuals are free to choose vaccination versus a hypothetical scenario

where vaccination is not available. We also compute the difference in seasonal incidence

∆I between the baseline scenario and all of these hypothetical scenarios.

Impact of Interference on Intervention Uptake Rates

With respect to impacts of interference on intervention uptake rates, we observe that

in general, across a broad range of utilities for ES, EH , ED, EV , and EI , NPI practice

interferes signiﬁcantly with vaccine uptake (∆V usually ranging from 5% to 15%, Fig

4.6A,C,E and Fig 4.7A,C). Vaccination also interferes signiﬁcantly with susceptible NPI

practice (∆NS usually ranging from 5% to 15%, Fig 4.6B,D,F and Fig 4.7B,D), but not

as much with infectious NPI practice (∆NI is smaller, Fig 4.6B,D,F and Fig 4.7B,D).

Infectious NPI practice is not as strongly affected because infectious NPI practice depends

less on population prevalence than susceptible NPI practice does. 110

Interference of NPIs with vaccination is strongest at intermediate values of EV (Fig

4.7A). When EV is small, NPIs are not popular in the population and hence NPIs do not

interfere strongly with vaccination. Thus, when NPIs are removed, there is little impact ∆V on vaccination levels. When EV is large, vaccination becomes too costly for the population to adopt it widely. Therefore, vaccination levels do not increase by signiﬁcant amounts when NPIs are added or removed from the population. When vaccination is not present, we see that for small EV , NPIs are interfered with especially strongly. This is because high vaccination coverage seen in the baseline scenario disappears, and thus individuals choose to practice NPIs more often. For larger values of EV , NPIs are undermined by decreasing amounts since vaccination levels in the baseline scenarios are greatly reduced (Fig 4.7B).

Similar reasoning can explain trends in ∆V , ∆NI and ∆NS as other utilities are varied

(Fig 4.7C,D).

Impact of Interference on Inﬂuenza Incidence

Overall, when removing either vaccination or NPIs, we observe an increase in incidence, compared to baseline scenarios where both interventions are available (∆I > 0; green lines in Fig 4.6 and 4.7). This occurs because having both interventions as an option is better than just having one of them.

However, the increase in incidence caused by removing vaccination (Fig 4.6B,D,F and

4.7B,D) is much larger than the increase in incidence caused by removing NPIs (≈ 5−10% versus ≈ 1 − 2% Fig 4.6A,C,E and 4.7A,C). Hence, vaccination appears to be a much 111

stronger determinant of inﬂuenza incidence than NPI practice. When NPIs are introduced

to the population, vaccine uptake declines sufﬁciently such that the net change in incidence

∆I is small, thus vaccination responds to the introduction of NPIs in a way that strongly

mitigates the effectiveness of NPIs in reducing incidence (Fig 4.6A,C,E and 4.7A,C). How-

ever, when vaccination is introduced to the population, the resulting decline in NPI practice

is not sufﬁcient to prevent signiﬁcant changes in ∆I (Fig 4.6B,D,F and 4.7B,D), thus NPIs do not strongly mitigate the effectiveness of vaccination. In other words, vaccination interferes strongly with the effectiveness of NPI practice, whereas NPI practice interferes weakly with the effectiveness of vaccination.

As a result of interference, efforts to boost NPI practice should be partially counter- productive due to the mitigating response of vaccine uptake, whereas efforts to boost vaccine uptake should be productive because the mitigating responses of NPI practice are not strong enough to prevent a signiﬁcant decrease in inﬂuenza incidence. This trend is consistent across a broad range of parameter values. 112

Figure 4.6: Univariate analysis for social parameters ES, EH , and ED determining the amount that vaccination and NPIs interfere with each other in each scenario. (A),(C),(E) Average values across 30 seasons for change in vaccination coverage (blue) and change in incidence (green) between hypothetical scenarios without NPI usage and the baseline scenarios. (B),(D),(F) Average values across 30 seasons for change in NPI usage amongst susceptible (black) and infectious (red) individuals and change in incidence (green) between hypothetical scenarios without vaccine usage and the baseline scenarios. Conﬁdence intervals represent two standard deviations of the mean of the 100 parameter sets across 30 simulated seasons. 113

Figure 4.7: Univariate analysis for EV and EI determining the amount that vaccination and NPIs interfere with each other in each scenario. (A),(C) Average values across 30 seasons for change in vaccination coverage (blue) and change in incidence (green) between hypothetical scenarios without NPI usage and the baseline scenarios. (B),(D) Average values across 30 seasons for change in NPI usage amongst susceptible (black) and infectious (red) individuals and change in incidence (green) between hypothetical scenarios without vaccine usage and the baseline scenarios. Conﬁdence intervals represent two standard deviations of the mean of the 100 parameter sets across 30 simulated seasons. 114

4.4.4 Understanding What Drives Different Levels of Interference for

Different Interventions

To understand the source of this asymmetry between the two interventions, we vary the

NPI efficacy (NPI ) and the vaccine efficacy (V ) (Fig 4.8 and 4.9). As the NPI efficacy increases, the proportion of individuals practicing NPIs increases significantly (Fig 4.8B) and the vaccine uptake decreases in response, while the infection incidence also declines (Fig

4.8A). Similarly, when the NPI efficacy is very high, removing NPIs has a much larger impact on incidence than removing vaccination, the latter of which has almost no effect. But when the NPI efficacy is very low, the situation is reversed, and removing vaccination has a much bigger impact on incidence than removing NPIs (Fig 4.8C,D). These results show that feedback between interventions operates such that, if a less efficacious intervention is removed, the resulting increased uptake of the more efficacious intervention is sufficient to prevent a net increase in incidence. In contrast, if a more efficacious intervention is removed, the resulting increased uptake of the less efficacious intervention is not adequate to prevent an increase in incidence. Similar patterns hold when the vaccine efficacy (V )

is varied, for similar underlying reasons (Fig 4.9). However, a secondary factor working

in favour of vaccination is that vaccination–if efﬁcacious–protects individuals throughout

the inﬂuenza season, whereas NPI practice needs to be efﬁcacious every time there is an

infection in a network neighbour, in order for an individual to avoid infection throughout

the entire season. 115

Figure 4.8: (A),(B) Univariate sensitivity analysis for NPI efficacy, NPI . Data shows average values across 30 seasons for vaccination coverage (blue), incidence (green), NPI usage amongst susceptible individuals (black), and NPI usage amongst infectious individuals (red). (C),(D) Determining the amount that vaccination and NPIs interfere with each other for various NPI efficacies. Average values across 30 seasons for change in vaccination coverage (blue), change in incidence (green), and change in NPI usage amongst susceptible (black) and infectious (red) individuals are shown. Confidence intervals represent two standard deviations of the mean of the 100 parameter sets across 30 simulated seasons.

The difference in vaccine uptake with and without NPIs is highest for intermediate values of V . In general, this occurs because when intervention efﬁcacy is very low, individuals will not adopt it, regardless of whether there is an alternative or not. Therefore, even if the alternative intervention is removed, individuals will tend not to increase adoption of the inefﬁcacious intervention (Fig 4.9X, lowest values of V ). On the other hand, 116

Figure 4.9: (A),(B) Univariate sensitivity analysis for vaccine efficacy, V . Data shows average values across 30 seasons for vaccination coverage (blue), incidence (green), NPI usage amongst susceptible individuals (black), and NPI usage amongst infectious individuals (red). (C),(D) Determining the amount that vaccination and NPIs interfere with each other for various vaccine efficacies. Average values across 30 seasons for change in vaccination coverage (blue), change in incidence (green), and change in NPI usage amongst susceptible (black) and infectious (red) individuals are shown. Confidence intervals represent two standard deviations of the mean of the 100 parameter sets across 30 simulated seasons. if an intervention is significantly more effective than the alternative intervention, or not very costly, then it will continue to be used by a large proportion of the population, and will not experience significant interference from the less effective alternative intervention which does not significantly change incidence (Fig 4.9X, highest values of V ). 117

In summary, these results show that, the more efﬁcacious an intervention is, the less its

effectiveness will be compromised by the other intervention, but the more it will compro-

mise the effectiveness of other intervention. The central role of intervention efﬁcacy also

explains why the highest reductions in incidence occur when utilities that support vacci-

nation only are changed (e.g. the vaccine cost is reduced, Fig 4.5A,B), or when utilities

that support both vaccination and NPI practice are changed (e.g. when the payoff for sav-

ing others from infection is increased, Fig 4.4A,B, or the perceived cost of infection is

increased, Fig 4.5C,D). In contrast, decreasing the perceived cost of social distancing ED has little impact on incidence (Fig 4.4E,F), since this NPI practice is interfered with by the mitigating response of vaccine uptake.

4.5 Discussion

We have constructed a seasonal inﬂuenza transmission model that incorporates how behavioural decisions for both individual vaccinating decisions and individual NPI practice

(hand-washing, social distancing) respond to changes in infection incidence. Our population was distributed across an empirically-based network, and parameter values were based either on literature (Wells and Bauch, 2012; Truscott et al., 2011; Bridges et al., 2000; Kei- tel et al., 1997; Lee et al., 2002; Demicheli et al., 2007; Larson et al., 2010; Sheehan et al.,

2007; Earn et al., 2002; Longini et al., 2000; Cox et al., 2004; Ambrose et al., 2008; Carrat et al., 2007; Nichol et al., 2010), or were calibrated to typical inﬂuenza seasonal patterns 118 using a probabilistic sampling approach.

These results illustrate how vaccine uptake and NPI practice interfere with one another.

If vaccine coverage increases, the resulting change in transmission patterns causes a decrease in the practice of NPIs. This is especially true for susceptible individuals, since susceptible NPI practice is more sensitive to population incidence. Similarly, if NPI practice expands, vaccine coverage will decrease by a roughly similar amount.

Although susceptible NPI practice and vaccine coverage have similar impacts on each other’s uptake, the impact on incidence is highly asymmetrical between the two interventions: the effectiveness of NPI practice is strongly mitigated by the response of vaccine uptake, whereas the effectiveness of vaccination is only weakly mitigated by the response of NPI practice. This asymmetry is driven by the differing efficacies of the two types of interventions: the higher the efficacy of the intervention (V , NPI ), the less its effectiveness in terms of reducing influenza incidence will be mitigated by the other intervention, but the more it will mitigate the effectiveness of other intervention.

Because influenza vaccine efficacy is generally higher than NPI efficacy, these effects have potentially important implications for influenza mitigation strategies. Efforts to boost

NPI practice could be strongly counteracted by the resulting declines in vaccine coverage, hence boosting NPI practice could be counter-productive. However, boosting vaccine coverage can still be productive since the resulting response of NPI practice will not as strongly mitigate the effectiveness of expanded vaccine coverage. Because of the role of efﬁcacy, increasing NPI efﬁcacy through fostering better hand-washing techniques or respiratory 119

etiquette might more useful than only increasing NPI uptake rates.

As a result of this asymmetry, we observed that increasing the utility for saving others

from infection, ES was the most effective way of decreasing incidence because it sup-

ports both vaccination and NPI practice. From the standpoint of an advertising campaign,

this would mean highlighting the fact that saving friends and family from becoming ill,

both through vaccination and through NPIs, would be effective. In contrast, attempting

to expand NPI practice without simultaneously encouraging vaccine uptake (for example

through making hand sanitizer stations more widely available) could be counter-productive

since NPI efﬁcacy is not as high as vaccine efﬁcacy. Similarly, increasing the perceived

cost of infection, EI was also found to be an effective way to reduce incidence, since both vaccination and NPI practice are thereby supported.

The asymmetry also explains why decreasing the cost of vaccination, EV , was observed to be effective. The resulting expansion in vaccine uptake suppresses NPI practice to some extent, but because vaccine efﬁcacy is higher than NPI efﬁcacy, decreasing EV still causes

net reductions incidence. Hence, reducing the perceived cost of the vaccine by expanding

availability (through more seasonal inﬂuenza vaccine clinics) or decreasing its price will

reduce inﬂuenza incidence, despite interference.

Our model makes several simplifying assumptions with respect to decision making.

Firstly, we group all NPIs into two categories: those utilized by susceptible individuals,

and those utilized by infectious individuals. In reality, however, an infectious individual

may practice combinations, such as choosing to practice strict respiratory etiquette, but 120 not staying home to isolate themselves. Both forms of NPI likely have differing efﬁcacy, hence our grouping assumption could inﬂuence results. Similarly, we used a common cost parameter, ED, for all NPIs, but different forms of NPI would likely impose varying costs.

We also allowed our population to have knowledge of both vaccine and NPI efﬁcacies.

Additional factors that could impact the decision making processes are misinterpreta- tions of influenza-like illnesses (ILIs) as being cases of influenza. Often, individuals may mistake other respiratory illnesses for influenza, artificially inflating perceived infection numbers and impacting perceived vaccine efficacy. Similarly, we assume only a single strain of influenza, when in reality, there are often multiple circulating strains. The model could be improved in future research by adding further heterogeneity such as age structure, making perceived efficacy depend on individual experience with interventions, and introducing a probability of ILI being mistaken for influenza, or vice versa (Wells and Bauch,

2012). The contact network could also be modiﬁed to include family and work structures, which may in turn inﬂuence memories of previous infections and perceived infection risk.

For example, individuals may weigh the fact that they have had a family member who was recently infected more so than if a casual contact like a co-worker recently fell ill. More- over, individuals may be more inclined to practice NPIs around family members than their other contacts. Finally, personal infection history may be considered to be signiﬁcantly more important than neighbour infection history when evaluating perceived risks, which is not accounted for in the model.

Being a highly parametrized model, there are several drawbacks associated with cali- 121 brating the model to empirical targets in order to obtain a baseline parameter set. We took parameter values directly from estimates in the available literature whenever possible, but

(especially for NPIs) there is a dearth of information regarding intervention behaviour and impact for seasonal inﬂuenza (Larson et al., 2010), necessitating calibration. As a result, we had more calibrated parameters than calibration targets (see Methods), meaning that alternative baseline parameter sets could have matched the data almost as well as the baseline set that we used. Our adoption of very broad sampling intervals for the probabilistic uncertainty sampling partially addresses this since the broad intervals will include those parameter values, however, the resulting frequency distribution of outcomes could still vary depending on what baseline parameter set is used as the baseline for deﬁning the triangular distributions. Future work could explore alternative parameterization methods, such as

Latin hypercube sampling, which might help overcome this limitation.

In conclusion, interference stemming from feedbacks between interventions and disease dynamics can comprise the realized effectiveness of those interventions for reducing inﬂuenza incidence, depending on the clinical efﬁcacy of the interventions in individuals.

Health authorities and epidemiologists should further explore the potential for interference between different interventions for the same infectious disease, and formulate infection control strategies accordingly. 122

Chapter 5

Conclusion

5.1 Conclusions and Future Work

This thesis examines the dynamics of simulated populations which are exposed to acute, self-limited diseases. These investigations include comparing ﬁtting processes for a dynamic transmission model of inﬂuenza also used to analyze vaccination scenarios, as well as novel behaviour-disease models used to explore the interaction between common disease intervention techniques.

Chapter 2 introduces a homogeneously mixed dynamic transmission model for seasonal influenza. We estimate transmission parameters for both A and B strains of seasonal influenza as well as analyze potential vaccination scenarios and their effects on the population. To fit the model, two methods using data of laboratory confirmed influenza cases are used. The first method uses weekly time series cases, whereas the second method uses 123 age-stratified cumulative cases. Although using both of these types of data are viable approaches to fit a dynamic influenza model, each method can produce differing outcomes regarding the effectiveness of vaccine intervention scenarios. Due to this result, it is important to obtain the highest quality of data available for the specific region to be modelled.

Chapters 3 and 4 utilize individual based models with heterogeneous mixing to evaluate population dynamics and intervention effectiveness when behaviour based decisions are also incorporated. In these models, individuals in the population make both vaccination and non-pharmaceutical intervention (NPI) related decisions. Previously, behaviour- disease models such as these included decision making for either vaccination or NPIs, but not both simultaneously. We find that introducing a second intervention decision will lower disease incidence, but also lower the use of the opposite intervention amongst the population. In general, NPIs have a beneficial impact, particularly if vaccines are not immediately available in the beginning of an epidemic. However, measures to increase vaccine uptake may be more useful to improve health outcomes, on the account of vaccines being more efficacious than NPIs.

Common shortcomings of behaviour-disease models such as these include the lack of knowledge of how individuals make decisions and form contacts. For example, decisions can be made rationally or irrationally, and the mechanisms behind what causes individuals to behave in certain ways are, of course, not fully understood. There are many methods used to model human behaviour, and there are an abundance of opportunities for future research to incorporate them into models with similar goals to ours. It is also difﬁcult to capture the 124 complexities of human contact networks, particularly those which enable disease spread.

Our network models are based on empirical networks derived from research studies, but nonetheless have signiﬁcant room for improvement.

The secondary goal of the latter two models is to introduce to the area of behaviour- disease modelling the concept of an individual’s simultaneous decision making regarding the two major interventions. To do this, we use common modelling paradigms: SIR and

SIRS models. We show that when only considering one intervention, simulation outcomes differ from those when both are incorporated. Thus, we hope that future research in behaviour-disease dynamics recognizes the importance of simultaneous decision making regarding both major interventions, and strives to include this characteristic in future model designs. 125

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