Beyond Ebola: Lessons for Mitigating Future Pandemics

Carlos Castillo-Chavez1

Regents Professor, Director

Simon A. Levin Mathematical, Computational and Modeling Sciences Center Tempe, AZ 85287-1904, USA

Graduate School of Public Health University of Pittsburgh

Monday, November 9, 2015

1 Draft [email protected];: https://twitter.com/mcmsc01 Overview Big Data Risk factors and recent outbreaks Modeling disease dynamics using a Lagrangian perspective Spatial spread of infectious diseases

Steven Strogatz. The Real Scientiﬁc Hero of 1953. New York Times (1923-Current ﬁle), page 1, 2003.

Three ways of doing science brought by James D. Watson and Francis Crick & the inventors of the computer experiment: Enrico Fermi, John Pasta and Stanislaw Ulam.

The computer experiment offered a third way of doing science.

Data Science (Big Data) is the fourth way of doingDraft science Beyond Ebola: Lessons learned ... Carlos Castillo-Chavez Simon A. Levin Modeling Center 2 / 65 Overview Big Data Risk factors and recent outbreaks Modeling disease dynamics using a Lagrangian perspective Spatial spread of infectious diseases

Enrico Fermi, John Pasta and Stanislaw Ulam – in 1953 invented the concept of a "computer experiment.

"... the most important lesson ...is how feeble even the best minds are at grasping the dynamics of large, nonlinear systems. Faced with a thicket of interlocking feedback loops, where everything affects everything else, our familiar ways of thinking fall apart. To solve the most important problems of our time, we’re going to have to change the way we do science." NYT 2003, S. Strogatz Draft Beyond Ebola: Lessons learned ... Carlos Castillo-Chavez Simon A. Levin Modeling Center 3 / 65 Overview Big Data Risk factors and recent outbreaks Modeling disease dynamics using a Lagrangian perspective Spatial spread of infectious diseases

Training of Mathematical Scientists for the 21st Century

“... As Fermi and his colleagues taught us, a complex system like cancer can’t be understood merely by cataloging its parts and the rules governing their interactions. The nonlinear logic of cancer will be fathomed only through the collaborative efforts of molecular biologists.” (Strogatz, 2003). The world’s ability to train 21st century mathematical scientists must rely on models of learning and thinking embedded within interdisciplinary educational research/mentorship models. Mathematical scientists must become proﬁcient on multiple models of doing science including the systematic use of computer experiments and in data science. Draft Beyond Ebola: Lessons learned ... Carlos Castillo-Chavez Simon A. Levin Modeling Center 4 / 65 Overview Big Data Risk factors and recent outbreaks Modeling disease dynamics using a Lagrangian perspective Spatial spread of infectious diseases

Source:%hlp://www.datasciencecentral.comDraft/forum/topics/theh3vshthathdeﬁnehbighdata% Beyond Ebola: Lessons learned ... Carlos Castillo-Chavez Simon A. Levin Modeling Center 5 / 65 Overview Big Data Risk factors and recent outbreaks Modeling disease dynamics using a Lagrangian perspective Spatial spread of infectious diseases

OakRidgeNaTonalLaboratory-June6,2013 Cray-madeTitan–thefastestcomputerintheworld Chinaannouncesfastercomputer Milky–Way2onJune17,2013

hl p://www.voanews.com/content/china-boasts-worlds- fastest-computer/1683465.html Draft Beyond Ebola: Lessons learned ... Carlos Castillo-Chavez Simon A. Levin Modeling Center 6 / 65 Overview Big Data Risk factors and recent outbreaks Modeling disease dynamics using a Lagrangian perspective Spatial spread of infectious diseases

A Brief History Block of Mathematical Epidemiology

The ﬁeld was the result of the work of medical doctors - mathematical scientists Daniel Bernoulli (1700–1782) Sir Ronald Ross (1857–1932) Anderson G. McKendrick (1876–1943) William O. Kermack (1898–1970)

www.fameimages.com/daniel-bernoulli

www.nobelprize.org/nobel_prizes/medicine/laureates/1902/ross-bio.html

www.york.ac.uk/depts/maths/histstat/people/

Draft Photograph courtesy of Godfrey Argent Studios Beyond Ebola: Lessons learned ... Carlos Castillo-Chavez Simon A. Levin Modeling Center 7 / 65 Overview Big Data Risk factors and recent outbreaks Modeling disease dynamics using a Lagrangian perspective Spatial spread of infectious diseases

The mathematical theory of infectious diseases started by medical doctors

Sir Ronald Ross (1857–1932)

www.nobelprize.org/nobel_prizes/medicine/laureates/1902/ross-bio.html NobelDraft Laureate 1902 Beyond Ebola: Lessons learned ... Carlos Castillo-Chavez Simon A. Levin Modeling Center 8 / 65 Overview Big Data Risk factors and recent outbreaks Modeling disease dynamics using a Lagrangian perspective Spatial spread of infectious diseases

Basic Malaria Model The Life–cycle of malaria parasites Ross-Macdonald Model x : Proportion of infected humans y : Proportion of infected mosquitoes dx = ab M y (1 x) rx dt N dy = ax (1 y) µy dt Parameter Deﬁnition Units M N Number of female mosquitoes per human host 1 a Biting rate on a human per mosquito day b Infected mosquito to human transmission efﬁciency 1 r Per capita human recovery rate day 1 µ Per capita mortalityDraft rate of mosquitos day Beyond Ebola: Lessons learned ... Carlos Castillo-Chavez 9 / 65 Table: The parameters for the VL modelSimon and A. their Levin Modeling dimensions Center Overview Big Data Risk factors and recent outbreaks Modeling disease dynamics using a Lagrangian perspective Spatial spread of infectious diseases

Basic Reproductive Number for Malaria

2 Biological interpretation 2 ma b 2 is the number of secondary cases = R0 0 µr of infection on hosts or vectors R generated by a single infective host or infective vector.

a – number of bites per unit time b – infected bites that produce an infection M m = N – number of female mosquitoes per human host 1 r – duration of infection in human 1 – lifetime of a mosquito µ Draft Beyond Ebola: Lessons learned ... Carlos Castillo-Chavez Simon A. Levin Modeling Center 10 / 65 Overview Big Data Risk factors and recent outbreaks Modeling disease dynamics using a Lagrangian perspective Spatial spread of infectious diseases

Holistic Perspective on Malaria

How can we use what we learned from person to vector to person or vector to host to vector transmission at higher levels of organization?

Holistic view - Public good: Can malaria be controlled at higher levels of organization?

Problem across scales: how do we use knowledge at the individual level to understand phenomena at the population level?

Validation of proposed control policies via the existence of a threshold: the prestige of mathematics and mathematical modeling Power of abstraction, can weDraft use this framework elsewhere: STDs Beyond Ebola: Lessons learned ... Carlos Castillo-Chavez Simon A. Levin Modeling Center 11 / 65 Overview Big Data Risk factors and recent outbreaks Modeling disease dynamics using a Lagrangian perspective Spatial spread of infectious diseases

Gonorrhea: Transmission Dynamics and Control

Herbert Hethcote Kenneth Cooke James A. Yorke

University of Iowa Pomona College University of Maryland, College Park

Herbert Hethcote and Jim Yorke changed health policy with their work on gonorrhea via their concept of Core Group

Ken Cooke and Jim Yorke expanded signiﬁcantly the work of Ross in 1970s with their work on GonorrheaDraft transmission and control Beyond Ebola: Lessons learned ... Carlos Castillo-Chavez Simon A. Levin Modeling Center 12 / 65 Overview Big Data Risk factors and recent outbreaks Modeling disease dynamics using a Lagrangian perspective Spatial spread of infectious diseases

Ross’ "students" Kermack and McKendrick

William Ogilvy Kermack Anderson Gray McKendrick (1898–1970) (1876–1943)

Photograph courtesy of Godfrey Argent Studios Draftwww.york.ac.uk/depts/maths/histstat/people/ Beyond Ebola: Lessons learned ... Carlos Castillo-Chavez Simon A. Levin Modeling Center 13 / 65 Overview Big Data Risk factors and recent outbreaks Modeling disease dynamics using a Lagrangian perspective Spatial spread of infectious diseases

The basic reproduction number for the SIR model without vital dynamics

T(4) T(3)

T(2)

Infected T(1) Index case Susceptible T(0) Infected Index case R0=2 Susceptible No Transmission Transmission

The basic reproduction number, , deﬁned as the number of secondary cases R0 generated by a typical infectious individual during its period of infectiousness in an entirely susceptible population. Draft Beyond Ebola: Lessons learned ... Carlos Castillo-Chavez Simon A. Levin Modeling Center 14 / 65 Overview Big Data Risk factors and recent outbreaks Modeling disease dynamics using a Lagrangian perspective Spatial spread of infectious diseases

System of equations without vital dynamics - single outbreak

S β IS I γ R

dS SI States Meaning = , dt N S # of Susceptible dI SI = I, I # of Infectecd dt N R # of Recovered dR = I, dt Parameters Meaning Transmission coefﬁcient N = S + I + R. Draft Per-capita recovery rate Beyond Ebola: Lessons learned ... Carlos Castillo-Chavez Simon A. Levin Modeling Center 15 / 65 Overview Big Data Risk factors and recent outbreaks Modeling disease dynamics using a Lagrangian perspective Spatial spread of infectious diseases

On the basic reproduction number R0

= R0 1 depends on number of contacts and probability of transmission (both R0 quantities captured by ) and the infectious period (1/). 2 If < 1 then the infection dies out. R0 3 If > 1 then an epidemic ensues R0 4 Accurate estimation of the value of the reproductive number are central in the planning of control of intervention efforts. The goal of public health interventions can often be reduced to that of ! bringing below 1. R0 Draft Beyond Ebola: Lessons learned ... Carlos Castillo-Chavez Simon A. Levin Modeling Center 16 / 65 Overview Big Data Risk factors and recent outbreaks Modeling disease dynamics using a Lagrangian perspective Spatial spread of infectious diseases

Model ﬁtting and predictions using Inﬂuenza outbreak data Estimate parameters from data using the SIR model

1978 UK Boarding School 300 Data Prevalence of an inﬂuenza Best fit outbreak in a boys boarding 250 Parameter estimates and standard

200 school, in the UK, 1978. error: 150 Total population size: N = 763, 1 ˆ = 1.6682 0.0294 days 100 ± Initial # of susceptible: S0 = 762, 1 50 ˆ = 0.4417 0.0177 days ± Initial # of infectives: I = 1. 0 0 0 2 4 6 8 10 12 14 SIR epidemic model simulated with estimated parameters

S − I Phase Plane Portrait 0.4

1 (I) 0.3

Parameter values: 0.2 Infective 0.8 0.1

= 1.6682, = 0.4417 Infective Fraction (I) N=S+I 0 0.6 0 5 10 15 20 25 Time (Days)

1 Initial conditions: 0.4

Infective Fraction Susceptible ( )= ( )= 0.5 S 0 762, I 0 1, 0.2 Recovered

R(0)=0. Population Size S (25)= 0. 02 0 0 0 0.2 0.4 0.6 0.8 1 0 5 10 15 20 25 SusceptibleDraft Fraction (S) Time (Days) Beyond Ebola: Lessons learned ... Carlos Castillo-Chavez Simon A. Levin Modeling Center 17 / 65 Overview Big Data Risk factors and recent outbreaks Modeling disease dynamics using a Lagrangian perspective Spatial spread of infectious diseases

Computational Epidemiology: The case of EpiSims

Sara del Valle

MTBI alumni – now a leading scientist Research Interests: Develop mathematical and computational models to help mitigate he spread of infectious diseases. (EpiSims slides courtesy of Sara delDraft Valle et al.) Beyond Ebola: Lessons learned ... Carlos Castillo-Chavez Simon A. Levin Modeling Center 18 / 65 Overview Big Data Risk factors and recent outbreaks Modeling disease dynamics using a Lagrangian perspective Spatial spread of infectious diseases

Typical Family Day- Families using Same Locations

P. Stroud, S. Del Valle, S. Sydoriak, J. Riese, S. Mniszewski, Spatial dynamics of pandemic influenza in a massive artificial society, Journal of Artificial Societies and SocialDraft Simulation, 10, (4) (2007) 9. Beyond Ebola: Lessons learned ... Carlos Castillo-Chavez Simon A. Levin Modeling Center 19 / 65 Overview Big Data Risk factors and recent outbreaks Modeling disease dynamics using a Lagrangian perspective Spatial spread of infectious diseases

Mobility in a Network Simulation

G. Chowell, J. M. Hyman, S. Eubank, and C. Castillo-Chavez. Scaling laws for the movement of people between locations in a large city. Phys. Rev. E, 68, 066102 – Published 15 December 2003

Draft Beyond Ebola: Lessons learned ... Carlos Castillo-Chavez Simon A. Levin Modeling Center 20 / 65

(EpiSims slides courtesy of Sara del Valle et al.) Overview Big Data Risk factors and recent outbreaks Modeling disease dynamics using a Lagrangian perspective Spatial spread of infectious diseases

Portland-Simulated Flu Pandemic-75% stay at home

S. Eubank, H. Guclu, V. S. Anil Kumar, M. V. Marathe, A. Srinivasan, Z. Toroczkai, N. Wang, Modelling disease outbreaks in realistic urban social networks, Nature 429 (6988) (2004) 180–184.

Draft Beyond Ebola: Lessons learned ... Carlos Castillo-Chavez Simon A. Levin Modeling Center 21 / 65 Overview Big Data Risk factors and recent outbreaks Modeling disease dynamics using a Lagrangian perspective Spatial spread of infectious diseases

Heterogeneous Spread

Draft Beyond Ebola: Lessons learned ... Carlos Castillo-Chavez Simon A. Levin Modeling Center 22 / 65 Overview Big Data Risk factors and recent outbreaks Modeling disease dynamics using a Lagrangian perspective Spatial spread of infectious diseases

Characteristics of the 2014 Ebola epidemic

G Chowell, N W Hengartner, C Castillo-Chavez, P W Fenimore, and J M Hyman. The basic reproductive number of Ebola and the effects of public health measures: the cases of Congo and Uganda J Theor Biol, 229(1):119–126, Jul 2004.

The causative Ebola strain in west Africa is closely related to a strain associated with past outbreaks in Central Africa Likely common reservoir: Fruit bats Epidemiological characteristics include: 2.0 (substantial uncertainty) R0 ⇠ Incubation period 11 days Serial interval 15 days Case fatality ratio 70.8% (95% IC 68.6 72.8%) Photograph courtesy of AP Photo/Abbas Dulleh High mortality rate (50-90% in previous outbreaks, 70% currently) Bodily ﬂuids are highly infectious,Draft as are the unburied dead Beyond Ebola: Lessons learned ... Carlos Castillo-Chavez Simon A. Levin Modeling Center 23 / 65 Overview Big Data Risk factors and recent outbreaks Modeling disease dynamics using a Lagrangian perspective Spatial spread of infectious diseases

2014 West African Outbreak

Index case Dec 2013, 2 year old girl in mountainous region of Guinea Porous borders and trafﬁcking have likely aided the spread of the disease (distribution as of Oct 6th., 2014 Image source: bbc.com) Draft Beyond Ebola: Lessons learned ... Carlos Castillo-Chavez Simon A. Levin Modeling Center 24 / 65 Overview Big Data Risk factors and recent outbreaks Modeling disease dynamics using a Lagrangian perspective Spatial spread of infectious diseases

What is unique about this epidemic?

Affected region lacked experience with EVD outbreaks Substantial delays in detection and implementation of control interventions in a region characterized by porous borders and high population movement Limited public health infrastructure in affected region including epidemiological surveillance systems and diagnostic testing Unsafe burials and health-care settings contributed to seeding the epidemic in multiple districts and across borders Draft Beyond Ebola: Lessons learned ... Carlos Castillo-Chavez Simon A. Levin Modeling Center 25 / 65 Overview Big Data Risk factors and recent outbreaks Modeling disease dynamics using a Lagrangian perspective Spatial spread of infectious diseases

http://www.bbc.com/news/world-africa-28755033 Draft Beyond Ebola: Lessons learned ... Carlos Castillo-Chavez Simon A. Levin Modeling Center 26 / 65 Overview Big Data Risk factors and recent outbreaks Modeling disease dynamics using a Lagrangian perspective Spatial spread of infectious diseases

Example 1: Exponential rise in new cases

Dr. Sherry Towers (Faculty, ASU)

Dr. Oscar Patterson Draft (Postdoc, Harvard Univ) Beyond Ebola: Lessons learned ... Carlos Castillo-Chavez Simon A. Levin Modeling Center 27 / 65 Overview Big Data Risk factors and recent outbreaks Modeling disease dynamics using a Lagrangian perspective Spatial spread of infectious diseases

Example 1: Exponential rise in new cases

Draft Beyond Ebola: Lessons learned ... Carlos Castillo-Chavez Simon A. Levin Modeling Center 28 / 65 Overview Big Data Risk factors and recent outbreaks Modeling disease dynamics using a Lagrangian perspective Spatial spread of infectious diseases

Exponential rise in new cases

www.newfluwiki2.com Early in an epidemic, when the population is predominantly naive, initial rise in cases is exponential, When everybody is susceptible the reproduction number is called , R0 The Effective Reproduction Number measures the changes in exponential growth as a function of time ( (t)), Reff (t) is the average number of new cases one case will cause over the course Reff of their infection at time t. Draft Beyond Ebola: Lessons learned ... Carlos Castillo-Chavez Simon A. Levin Modeling Center 29 / 65 Overview Big Data Risk factors and recent outbreaks Modeling disease dynamics using a Lagrangian perspective Spatial spread of infectious diseases

Reproduction number

In closed populations, (t) usually declines as an epidemic progresses Reff (fewer and fewer naïve people available to infect) As an epidemic progresses social distancing due to fear can also cause additional reductions in (t) Reff But poorly designed control strategies can, unfortunately, potentially do more harm than good. Draft Beyond Ebola: Lessons learned ... Carlos Castillo-Chavez Simon A. Levin Modeling Center 30 / 65 Overview Big Data Risk factors and recent outbreaks Modeling disease dynamics using a Lagrangian perspective Spatial spread of infectious diseases

Temporal variations in (t) Reff

Temporal variations in the effective reproduction number of the 2014 West Africa Ebola outbreak

Rate of exponential rise, in conjunction with a mathematical model of the spread of the disease, can be used to determine (t). Reff For instance, SEIRD modelDraft for Ebola. Beyond Ebola: Lessons learned ... Carlos Castillo-Chavez Simon A. Levin Modeling Center 31 / 65 Overview Big Data Risk factors and recent outbreaks Modeling disease dynamics using a Lagrangian perspective Spatial spread of infectious diseases

Temporal variations in (t) contd... Reff Developed a simple, model-independent way to determine if the relative transmission rate of a disease in a closed population appears to increase or decrease in time; Method applies piece-wise exponential ﬁts to the time series of cases in outbreak to determine if the rate of exponential rise in cases increases over time (evidence that effective (t) is Reff increasing). Draft Beyond Ebola: Lessons learned ... Carlos Castillo-Chavez Simon A. Levin Modeling Center 32 / 65 Overview Big Data Risk factors and recent outbreaks Modeling disease dynamics using a Lagrangian perspective Spatial spread of infectious diseases

Example 2: Research question

Volume 15, Issue 2, February 2015, Pages 148–149

Modelling the effect of early detection of Ebola

Diego Chowella, b, Carlos Castillo-Chaveza, Sri Krishnab, Xiangguo Qiuc, Karen S Andersonb,

a Simon A Levin Mathematical, Computational and Modeling Sciences Center, Biodesign Institute, Arizona State University, Tempe, AZ, USA b Center for Personalized Diagnostics, Biodesign Institute, Arizona State University, Tempe, AZ, USA c National Laboratory for Zoonotic Diseases and Special Pathogens, Public Health Agency of Canada, Winnipeg, MB, Canada

DiegoDraft Chowell (PhD student) Beyond Ebola: Lessons learned ... Carlos Castillo-Chavez Simon A. Levin Modeling Center 33 / 65 Overview Big Data Risk factors and recent outbreaks Modeling disease dynamics using a Lagrangian perspective Spatial spread of infectious diseases

• Twenty'four,“asymptoma2c”,individuals, exposed,to,Ebola,were,tested,using,P.C.R.,,

• Eleven,of,the,exposed,pa2ents,eventually, developed,the,infec2on.,,

• Seven,of,the,11,tested,posi2ve,for,the, DraftP.C.R.,assay;,none,of,the,other,13,did., Beyond Ebola: Lessons learned ... Carlos Castillo-Chavez Simon A. Levin Modeling Center 34 / 65 Overview Big Data Risk factors and recent outbreaks Modeling disease dynamics using a Lagrangian perspective Spatial spread of infectious diseases

Modeling the effect of early detection of Ebola

The polymerase chain reaction (PCR) can detect Ebola virus in both human beings and non-human primates in the pre-symptomatic stage. We evaluated the potential effect of early diagnosis of pre-symptomatic individuals in west Africa. We used a simple mathematical model calibrated to the transmission dynamics of Ebola virus in west Africa. The baseline model includes the effects of contact tracing and effective isolation of infectious individuals in health-care settings. Members of the European Mobile Laboratory Project use PCR tests in Guéckédou, Guinea. G. Vogel, Testing new Ebola tests, Science Draft(2014). Beyond Ebola: Lessons learned ... Carlos Castillo-Chavez Simon A. Levin Modeling Center 35 / 65 Overview Big Data Risk factors and recent outbreaks Modeling disease dynamics using a Lagrangian perspective Spatial spread of infectious diseases Model of transmission dynamics of Ebola infection incorporating diagnosis of infected and pre-symptomatic individuals

S E1 E2 I J P

Class Description Parameter Value S Susceptible 1/k2 4 days

E1 Latent undetectable 1/k2 3 days

E2 Latent detectable 1/↵ 3 days I Infectious and symptomatic 1/ 6 days

J Isolated 1/r 7 days P Recovered and Dead Draft Beyond Ebola: Lessons learned ... Carlos Castillo-Chavez Simon A. Levin Modeling Center 36 / 65 Overview Big Data Risk factors and recent outbreaks Modeling disease dynamics using a Lagrangian perspective Spatial spread of infectious diseases Predictions on the effect of diagnosing pre-symptomatic individuals on the Ebola epidemic attack rate

100 Effectiveness of isolation 50% Effectiveness of isolation 60% 90 Effectiveness of isolation 65% 80 70 60 50 40 Attack rate (%) Attack rate 30 20 10 0 0 5 10 15 20 25 30 35 40 45 50 Proportion of latent individuals diagnosed before onset of symptoms (%)

We can make now two observations: The effect of early Ebola detection is a function of existing public health measures and resources. There is a tipping point, where early diagnosis of high risk individuals, combined with adequate isolation, can lead to rapid reduction in Ebola transmission. Draft Beyond Ebola: Lessons learned ... Carlos Castillo-Chavez Simon A. Levin Modeling Center 37 / 65 Overview Big Data Risk factors and recent outbreaks Modeling disease dynamics using a Lagrangian perspective Spatial spread of infectious diseases Example 3: A disease model framework under virtual dispersal and varying residence times Difficulties in defining a contact in the context of communicable diseases. Notion of contact is well-defined in STDs and vector-born diseases.

Estimate the average risk of acquiring http://pilgrimagetoindia.com/gallery/26.html TB or ﬂu to individuals spending 3 hours on the average per day, in public transportation.

Contacts or variable environmental risks?

From differential susceptibility to local environmental risk infectivity. http://www.livetradingnews.com

Overall Question: How does environmental pathogen risk deﬁned by risk/transmission vector B and patch residence time distribution P =(pij) impact disease transmission dynamics and control. Formulation of epidemic models (host parasite dynamics)Draft where risk of infection (parasitism) is a function of local residence times. Beyond Ebola: Lessons learned ... Carlos Castillo-Chavez Simon A. Levin Modeling Center 38 / 65 Overview Big Data Risk factors and recent outbreaks Modeling disease dynamics using a Lagrangian perspective Spatial spread of infectious diseases

Two approaches to incorporate heterogeneity

Eulerian Approach: Patch 1 Patch 2 p Focus on the whole population stratiﬁed 12 into different patches. Individuals take the identity of the host p patch after leaving their residence patch. 21 More realistic for long term dispersal.

Lagrangian Approach: Patch 1 Patch 2 p Keep track of individuals for population 12 in each patch through time and space. Focus on the patch level of population p21 interacting between different patches . More realistic for short scale movements. Draft Beyond Ebola: Lessons learned ... Carlos Castillo-Chavez Simon A. Levin Modeling Center 39 / 65 Overview Big Data Risk factors and recent outbreaks Modeling disease dynamics using a Lagrangian perspective Spatial spread of infectious diseases

Model formulation

p12 is the proportion of times residents of Patch 1 spend in Patch 2. p21 is the proportion of timesDraft residents of Patch 2 spend in Patch 1. Beyond Ebola: Lessons learned ... Carlos Castillo-Chavez Simon A. Levin Modeling Center 40 / 65 Overview Big Data Risk factors and recent outbreaks Modeling disease dynamics using a Lagrangian perspective Spatial spread of infectious diseases

Model formulation

A multi-group epidemic framework under virtual dispersal with the per patch risk of infection is a function of Local environmental risk vector deﬁned as =( , ,..., )t where B 1 2 n i is a measure of the risk per susceptible per unit of time while in residence in Patch i.

The residence times matrix P whose entry pij is proportion of time that i-resident spends as a visitor in j-patch. If we apply this Lagrangian approach to a n-patch SIS model, we get that

S˙ = b d S + I n (S infected in Patch j) i i i i i i j=1 i I˙ = n (S infected in Patch j) I d I (1) 8 i j=1 i P i i i i N˙ = b d N . < i Pi i i : Draft Beyond Ebola: Lessons learned ... Carlos Castillo-Chavez Simon A. Levin Modeling Center 41 / 65 Overview Big Data Risk factors and recent outbreaks Modeling disease dynamics using a Lagrangian perspective Spatial spread of infectious diseases

Residence-times heterogeneity contributes to infection.

[Si infected in Patch j]=

p S j ⇥ ij i the risk of infection in Patch j Susceptible from Patch i who are currently in Patch j |{z} |{z} n p I k=1 kj k . ⇥ n Pk=1 pkjNk ProportionP of infected in Patch j And so, (1) becomes | {z }

n n bi k=1 pkjIk I˙i = jpij Ii (i + di)Ii i = 1, 2,...,n. (2) d n bk i k=1 pkj d ! Xj=1 ✓ ◆ P k whose basic reproduction number canP be computed as a function of the risk vector and the residence R0 B times matrix P.

Robust dynamics when P is irreducible, i.e., patches are strongly connected: If 1 the DFE is GAS while if > 1 then there exists a unique interior EE which is GAS. R0  R0 Draft Beyond Ebola: Lessons learned ... Carlos Castillo-Chavez Simon A. Levin Modeling Center 42 / 65 Overview Big Data Risk factors and recent outbreaks Modeling disease dynamics using a Lagrangian perspective Spatial spread of infectious diseases

Patch-speciﬁc dynamics

We deﬁne a Patch-speciﬁc basic reproduction number:

bi n pij i i j di (P)= pij . 0 0 ⇣n ⌘bk R R ⇥ i 0 pkj 1 Xj=1 ✓ ◆ k=1 dk @ P A If i (P) > 1 then the disease persists in Patch i. R0 i If pkj = 0 for all k = 1,..,n, and k = i, whenever pij > 0 and (P) < 1, then the disease dies 6 R0 out in Patch i. Remarks: This results also include the non-strongly connected patches case. • The connectivity of Patch i to other patches may promote or suppress endemicity: • Via the presence of high risk patches. For instance, if there exists a patch j such that j is large i enough. In this last case Patch j is actually a source and Patch i a sink. Whenever individuals spend more timeDraft in high risk patches than in low risk patches. Beyond Ebola: Lessons learned ... Carlos Castillo-Chavez Simon A. Levin Modeling Center 43 / 65 Overview Big Data Risk factors and recent outbreaks Modeling disease dynamics using a Lagrangian perspective Spatial spread of infectious diseases

Findings

We have used residence times in heterogeneous environments as a proxy for “effective” contacts over a certain window in time; We have formulated general multi-patch SIS epidemic models with residence times that provide conditions for extinction or persistence of the disease at global level and patch-speciﬁc level.

Derdei Bichara, Yun Kang, Carlos Castillo-Chavez, Richard Horan, and Charles Perrings. SIS and SIR Epidemic Models Under Virtual Dispersal. pages 1–31, 2015.

Joint work with:

Derdei Bichara Yun Kang DraftCharles Perrings Richard Horan Beyond Ebola: Lessons learned ... Carlos Castillo-Chavez Simon A. Levin Modeling Center 44 / 65 Overview Big Data Risk factors and recent outbreaks Modeling disease dynamics using a Lagrangian perspective Spatial spread of infectious diseases

Heterogeneity challenges

A cordon sanitaire is a medieval control strategy that involves creating boundaries around an area experiencing an epidemic of disease. This strategy was used in 1821 in the border of France and Spain to avert the spread of a deadly fever. Prior 2014, it was last used in 1918 at the Polish-Russian border in an attempt to stop the spread of typhus.

cordons sanitaire were used in 2014 in some of the Ebola-stricken countries. The effectiveness of cordons sanitaire is controversial and debatable.

Soldiers enforcing the cordon securitaire in Kanema, Sierra Leone (via the NYT) Research question: How does the movement of individuals between a low and high risk area promote, mitigate or suppress disease dynamics?

A Lagrangian conceptual framework that models the movement of individuals across different areas keeping track of residence times is used on theDraft Ebola outbreak. Beyond Ebola: Lessons learned ... Carlos Castillo-Chavez Simon A. Levin Modeling Center 45 / 65 Overview Big Data Risk factors and recent outbreaks Modeling disease dynamics using a Lagrangian perspective Spatial spread of infectious diseases

×104 Infected class patch 1 Infected class patch 2 4.5 12000 p12=0 4 p12=0 p12=0.15 p12=0.15 10000 p12=0.3 3.5 p12=0.3 p12=.0.45 p12=0.45 3 8000

2.5 6000 2

1.5 4000

1 2000 0.5

0 0 0 500 1000 1500 0 500 1000 1500 Time Time

Figure: Dynamics of incidence in each patch for p21 = 0 and varying p12. Parameter values: "D = 1,1 = 0.35, 2 = 0.1, fdeath = 0.8, k = 1/24, ↵ = 0, ⌫ = 1/2 and = 1/5.6. The blue graph is the case where the patches are isolated. The disease reaches its highest pick in the high risk Patch 1 whereas it dies off without a major outbreak in the low risk patch 2.

Besides the isolated case, we note that the disease prevalence decreases in both patches as p12 increases. While the results in Patch 1 are expected, the results are counterintuitive in Patch 2, as we expect more infections as the ﬂow of individuals from the high risk area increases. Draft Beyond Ebola: Lessons learned ... Carlos Castillo-Chavez Simon A. Levin Modeling Center 46 / 65 Overview Big Data Risk factors and recent outbreaks Modeling disease dynamics using a Lagrangian perspective Spatial spread of infectious diseases

Figure: Dynamics of the cumulative incidence in each patch for p21 = 0 and varying p12.

The predicted dynamics of the overall infected individuals is similar to those in Patch 1 since it is assumed that the high risk Patch 1 is overpopulated when compared to the low risk Patch 2. Draft Beyond Ebola: Lessons learned ... Carlos Castillo-Chavez Simon A. Levin Modeling Center 47 / 65 Overview Big Data Risk factors and recent outbreaks Modeling disease dynamics using a Lagrangian perspective Spatial spread of infectious diseases

Conclusion

A conceptual framework to test the effectiveness of control strategies aimed at limiting the movements of individuals across different risk areas has been introduced. It was observed, for example, that increases in the time that residents of high risk areas spend in low risk areas do not necessarily generates notable prevalence increases in low risk areas. Joint work with:

Baltazar Victor Derdei Draft Beyond Ebola: Lessons learned ... Carlos Castillo-Chavez Simon A. Levin Modeling Center 48 / 65 Overview Big Data Risk factors and recent outbreaks Modeling disease dynamics using a Lagrangian perspective Spatial spread of infectious diseases

Mass Media and contagion of fear

Sherry Towers, Shehzad Afzal, Gilbert Bernal, Nadya Bliss, Shala Brown, Baltazar Espinoza, , et al. Mass media and the contagion of fear: The case of ebola in america. PLoS ONE, 10(6):e0129179 EP –, 06 2015.

Ebola related searches and tweets originating in the U. S. during the outbreak tied in to public interest or panic.

Of interest, how would public interest, curiosity, or panic on certain topic affects social media and Internet search dynamics?

A mathematical model has been employed to simulate the potential inﬂuence of Ebola-related news videos on peoples’ tendency to perform Ebola-related Internet searches or tweets. Fits of the news media contagion model, and a simple linear regression model, to the sources of data used in this study. Overall Goal: Determine if news coverage was a signiﬁcant factor on the temporal patterns in Ebola-relatedDraft Internet and Twitter data. Beyond Ebola: Lessons learned ... Carlos Castillo-Chavez Simon A. Levin Modeling Center 49 / 65 Overview Big Data Risk factors and recent outbreaks Modeling disease dynamics using a Lagrangian perspective Spatial spread of infectious diseases

News on Ebola

Significant evidence of contagion was found, with each Ebola-related news video inspiring tens of thousands of Ebola-related tweets and Internet searches. There is no significant evidence of contagion due to effects other than news videos. There is no statistically significant evidence that people return to the susceptible class after recovery. In all cases, the contagion model had a better predictive power than the linear regression model. There is no statistically significant evidence that Ebola-related Internet searches and tweets Granger-cause temporal patterns in Ebola-related news videos, but there is evidenceDraft in several cases that the reverse is true. Beyond Ebola: Lessons learned ... Carlos Castillo-Chavez Simon A. Levin Modeling Center 50 / 65 Overview Big Data Risk factors and recent outbreaks Modeling disease dynamics using a Lagrangian perspective Spatial spread of infectious diseases

Beyond Ebola: lessons to mitigate future pandemics

Open Access

Draft Beyond Ebola: Lessons learned ... Carlos Castillo-Chavez Simon A. Levin Modeling Center 51 / 65 Overview Big Data Risk factors and recent outbreaks Modeling disease dynamics using a Lagrangian perspective Spatial spread of infectious diseases

SIR with diffusion

SIR no diffusion

S˙ = SI I˙ = SI ↵I

Standard model with diffusion

S = SI + D S t s xx I = SI ↵I + D I t Draft i xx Beyond Ebola: Lessons learned ... Carlos Castillo-Chavez Simon A. Levin Modeling Center 52 / 65 Overview Big Data Risk factors and recent outbreaks Modeling disease dynamics using a Lagrangian perspective Spatial spread of infectious diseases

Fisher-Kolmogorov in Epidemics

Loss of Immunity

S = SI + ↵I + DS t xx I = SI ↵I + DI . t xx N = S + I

Amplitud 1 1 R0 It =( ↵)I(1 I)+DIxx ↵ 5 1 Velocity p D(1 ) 6 R0 q growth rate ↵ 24D Width ↵ carrying capacity 1 R0 Draft q Beyond Ebola: Lessons learned ... Carlos Castillo-Chavez Simon A. Levin Modeling Center 53 / 65 Overview Big Data Risk factors and recent outbreaks Modeling disease dynamics using a Lagrangian perspective Spatial spread of infectious diseases

Results: Traveling Waves

Model Minimum speed of propagation c⇤

Kendall: Contact Distribution 2 2( ↵) p Rabies: Reaction Diffusion 2 D( ↵) p Fisher 2 Df 0(0) p SIS Exact Solution Speed: 5 D( ↵) p6 Draftp Beyond Ebola: Lessons learned ... Carlos Castillo-Chavez Simon A. Levin Modeling Center 54 / 65 Overview Big Data Risk factors and recent outbreaks Modeling disease dynamics using a Lagrangian perspective Spatial spread of infectious diseases

Two type of infectious

Edgar Diaz Herrera. Diffusive instability and aggregation in epidemics. PhD thesis, Arizona State University, Tempe, Arizona, July 2010.

Model

@S @2S Two kind of infected: = SI1 + ↵I2 + DS 2 @t 1 + I2 @ x I2 with symptoms. 2 @I1 @ I1 I1 with no = SI1 I1 + DI1 2 @t 1 + I2 @ x symptoms. @I @2I 2 = I ↵I + D 2 @t 1 2 I2 @2x S = 1 I I Steady state: 1 2 ↵( ) ( ) ¯I = and ¯I = 1 ↵ + + Draft2 2 ↵ + + 2 Beyond Ebola: Lessons learned ... Carlos Castillo-Chavez Simon A. Levin Modeling Center 55 / 65 Overview Big Data Risk factors and recent outbreaks Modeling disease dynamics using a Lagrangian perspective Spatial spread of infectious diseases

Turing’s method

What is needed @c @2c 1 = a c + a c + D 1 1 Two or more densities. @t 11 1 12 2 1 @x2 2 2 Different rates of diffusion for the @c @ c 2 = a c + a c + D 2 participants. @t 21 1 22 2 2 @x2 @ 3 Interactions between the two Ri ai,j = ci,cj . densities @Cj |

Instructions @C @2C 1 = R (C , C )+D 1 Positive spatial steady state @t 1 1 2 1 @x2 2 ¯ ¯ @C @ C Ri(C1, C2)=0 2 = R (C , C )+D 1 @t 2 1 2 2 @x2 DraftLinearization Beyond Ebola: Lessons learned ... Carlos Castillo-Chavez Simon A. Levin Modeling Center 56 / 65 Overview Big Data Risk factors and recent outbreaks Modeling disease dynamics using a Lagrangian perspective Spatial spread of infectious diseases

Necessary and sufﬁcient conditions for Aggregation

Superposition and Instability t 2 ci(x, t)=↵i cos(qx)e @c1 @ c1 = a11c1 + a12c2 + D1 values of q s.t. () > 0 @t @2x < 2 @c2 @ c2 = a21c1 + a22c2 + D2 2 ↵ ( a + D q2) ↵ a = 0 @t @ x 1 11 1 2 12 ↵ a + ↵ ( a + D q2)=0 1 21 2 22 2 Necessary and Sufﬁcient Conditions

1 a11 + a22 < 0 2 a a a a > 0 11 22 12 21 3 a11D2 + a22D1 > 2 D D (a a a a ) 1 2 11 22 12 21 Draftp Beyond Ebola: Lessons learned ... Carlos Castillo-Chavez Simon A. Levin Modeling Center 57 / 65 Overview Big Data Risk factors and recent outbreaks Modeling disease dynamics using a Lagrangian perspective Spatial spread of infectious diseases

RESULT

K. E. Yong, E. Díaz Herrera, and C. Castillo-Chavez. From bee species aggregation to models of disease avoidance: The Ben Hur effect. ArXiv e-prints, October 2015.

@S @2S = SI1 + ↵I2 + DS 2 (3) @t 1 + I2 @x 2 @I1 @ I1 = SI1 I1 + DI1 2 @t 1 + I2 @x @I @2I 2 = I ↵I + D 2 @t 1 2 I2 @2x

Theorem (Diffusive Instability in Epidemics) The linearization of the system (3) satisﬁes the necessary and sufﬁcient conditions for instability if and only if > 1 and > 1 Draft ↵ Beyond Ebola: Lessons learned ... Carlos Castillo-Chavez Simon A. Levin Modeling Center 58 / 65 Overview Big Data Risk factors and recent outbreaks Modeling disease dynamics using a Lagrangian perspective Spatial spread of infectious diseases

Different diffusion rates: linear model

time = 0 time = 6 time = 13 time = 0 time = 6 time = 13 0 0 0 0 0 0

20 20 20 20 20 20

40 40 40 40 40 40

60 60 60 60 60 60

80 80 80 80 80 80 0 20 40 60 80 0 20 40 60 80 0 20 40 60 80 0 20 40 60 80 0 20 40 60 80 0 20 40 60 80

time = 19 time = 25 time = 31 time = 19 time = 25 time = 31 0 0 0 0 0 0

20 20 20 20 20 20

40 40 40 40 40 40

60 60 60 60 60 60

80 80 80 80 80 80 0 20 40 60 80 0 20 40 60 80 0 20 40 60 80 0 20 40 60 80 0 20 40 60 80 0 20 40 60 80

time = 38 time = 44 time = 50 time = 38 time = 44 time = 50 0 0 0 0 0 0

20 20 20 20 20 20

40 40 40 40 40 40

60 60 60 60 60 60

80 80 80 80 80 80 0 20 40 60 80 0 20 40 60 80 0 20 40 60 80 0 20 40 60 80 0 20 40 60 80 0 20 40 60 80

D D D > D 2 1 Draft2 1 Beyond Ebola: Lessons learned ... Carlos Castillo-Chavez Simon A. Levin Modeling Center 59 / 65 Overview Big Data Risk factors and recent outbreaks Modeling disease dynamics using a Lagrangian perspective Spatial spread of infectious diseases

Different diffusion rates: nonlinear model

1 0.5 2 u 0 0 −2 150 140

120 140 100 100 120 150 80 100 60 80 50 100 40 60 40 50 20 20 0 0 0 0 Fast aggregation DraftSlow aggregation Beyond Ebola: Lessons learned ... Carlos Castillo-Chavez Simon A. Levin Modeling Center 60 / 65 Overview Big Data Risk factors and recent outbreaks Modeling disease dynamics using a Lagrangian perspective Spatial spread of infectious diseases

Mathematical and Theoretical Biology Institute, 1996

Draft Beyond Ebola: Lessons learned ... Carlos Castillo-Chavez Simon A. Levin Modeling Center 61 / 65 Overview Big Data Risk factors and recent outbreaks Modeling disease dynamics using a Lagrangian perspective Spatial spread of infectious diseases

Presidential award for excellence in Sciences, Math and Engineering mentoring, 2011 – Given in recognition to the Mathematical and Theoretical Biology Institute – MTBI

Draft Beyond Ebola: Lessons learned ... Carlos Castillo-Chavez Simon A. Levin Modeling Center 62 / 65 Overview Big Data Risk factors and recent outbreaks Modeling disease dynamics using a Lagrangian perspective Spatial spread of infectious diseases

MCMSC Center Renaming Ceremony in Honor of Simon A. Levin

Draft Beyond Ebola: Lessons learned ... Carlos Castillo-Chavez Simon A. Levin Modeling Center 63 / 65 Overview Big Data Risk factors and recent outbreaks Modeling disease dynamics using a Lagrangian perspective Spatial spread of infectious diseases

Grad students and researchers involved in the Ebola research at SAL-MCMSC - all MTBI alumni

Dr. Kamuela Yong: Ecology, Dr. Derdei Bichara: Epidemiology, Diffusion Infectious Diseases

Baltazar Espinoza: Epidemiology, Victor Moreno: Epidemiology, Game theory, Economics Graph theory

Diego Chowell: Evolutionary biology, Cancer, Computational Claudia Rodriguez: Math education modeling and policy, Retention of underrepresented students

Maryam Khan: Epidemiology, Social Sciences, Environment Dr. Edgar Herera Diaz: Epidemiology, Diffusion

Kamal Barley: Epidemiology, Health Dr. Anuj Mubayi: Mathematical disparities Epidemiology, Quantitative Social DraftSciences Beyond Ebola: Lessons learned ... Carlos Castillo-Chavez Simon A. Levin Modeling Center 64 / 65 Overview Big Data Risk factors and recent outbreaks Modeling disease dynamics using a Lagrangian perspective Spatial spread of infectious diseases

Draft Beyond Ebola: Lessons learned ... Carlos Castillo-Chavez Simon A. Levin Modeling Center 65 / 65