http://www.slcn.ac.jp/ http://university.luke.ac.jp MATHEMATICAL MODELLING OF INFECTIOUS DISEASES Dr Zoie Shui-Yee Wong, BSc, PhD Associate Professor, Center for Clinical Epidemiology, Graduate School of Public Health Planning Office, St Luke’s International University, Phone: +81-3-5550-4101 (Ext 4397); Fax: +81-3-5550-4114 Email: zoiewong@luke.ac.jp; zoiesywong@gmail.com Copyright © St Luke’s International University All rights reserved http://www.slcn.ac.jp/ http://university.luke.ac.jp Overview • • • • Self-introduction Modelling overview Building models Estimating reproduction numbers for epidemic outbreaks • Opportunities for epidemic models in public health Copyright © St Luke’s International University All rights reserved http://www.slcn.ac.jp/ http://university.luke.ac.jp Graduate School of Public Health Planning Office, St Luke’s International University Copyright © St Luke’s International University All rights reserved http://www.slcn.ac.jp/ Copyright © St Luke’s International University All rights reserved http://www.slcn.ac.jp/ Copyright © St Luke’s International University All rights reserved http://www.slcn.ac.jp/ Copyright © St Luke’s International University All rights reserved http://www.slcn.ac.jp/ http://university.luke.ac.jp Something about me Zoie Shui-Yee WONG, PhD • Associate Professor, Center for Clinical Epidemiology, Graduate School of Public Health Planning Office, St Luke’s International University, Japan (2016- Present) • Postdoctoral Research Fellow, School of Public Health and Community Medicine, University of New South Wales, Australia (2015-2016) • Senior Manager (Research and Operation), Centre for Systems Informatics Engineering, City University of Hong Kong (2013-2015) • JSPS Postdoctoral Fellow, Policy Alternatives Research Institute (PARI), The University of Tokyo, Japan (2011-2013) Cross disciplinary research in health domain Research Interests: Health Informatics, Infectious disease modelling, Simulation method Copyright © St Luke’s International University All rights reserved http://www.slcn.ac.jp/ http://university.luke.ac.jp Modelling overview Copyright © St Luke’s International University All rights reserved http://www.slcn.ac.jp/ http://university.luke.ac.jp Emergence/re-emergence of human pathogens •An infectious disease is a clinically evident illness resulting from the presence of pathogenic microbial agents with potential of transmission from one person or species to another Human lives •Spanish Flu (1918): 10 - 50 M deaths •SARS (2003): 811 deaths in months •H1N1/09 virus: ~14,000 as of Jan2010 (ECDC,2010) •Ebola west Africa: 11,310 (as of 10 Jan 2016) Economics •SARS: Worldwide: approx US$ 50B •Avian Flu: approx US$ 30B •Foot & Mouth Disease: US$ 20B Travel restrictions and alerts Copyright © St Luke’s International University All rights reserved http://www.slcn.ac.jp/ http://university.luke.ac.jp Epidemiologic Triangle • • • • An epidemic of a communicable disease is an interplay among the pathogen, and the host, & the environment Transmissibility of a communicable disease depends – Pathogen (disease epidemiology), – Characteristics of the host population, • E.g contact patterns and immunity among individuals – Environmental factors • E.g climate conditions and animal reservoirs Complicated interplay – what you observe (data) may not reflect what is happening – E.g hidden disease compartments may result Ebola flare-up cases (Abbate, Murall et al 2016) Computational epidemiology aims to model the establishment and spread of pathogens – Allows us to examine some assumptions – Provides flexibility to model interrelated attributes and interventions Host Disease Agent Environment Copyright © St Luke’s International University All rights reserved http://www.slcn.ac.jp/ http://university.luke.ac.jp Difficulties in determining R0 Ideally • Full information about who infected whom (closely monitored epidemics) – to construct an infection network - cases are connected if one person infected the other • Estimation of R involves simply counting the number of secondary infections per case Practically • Only the epidemic curve is observed • No information about – who infected whom, – no contact information (when and how), – missing cases • When only times of symptom onset are available, we approximate R – by assuming an exponential increase / growth rate in the number of cases over time • These counts increase exponentially in the initial phase of an epidemic – by fitting a specific model that summarizes assumptions about the epidemiology of the disease Copyright © St Luke’s International University All rights reserved http://www.slcn.ac.jp/ http://university.luke.ac.jp Estimating R Copyright © St Luke’s International University All rights reserved http://www.slcn.ac.jp/ http://university.luke.ac.jp Epidemic models • A epidemic model categorizes the population of hosts according to their infection status SIR model • The rate of leaving the infectious stage is denoted by b • The duration of a generation interval is specified as an exponential distribution with mean Tc=1/b (duration of the infectious period) provided that r>-b SEIR model • The rate of leaving the exposed stage is b1, the rate of leaving the infectious stage is b2 and both rates are assumed constant • Compose M(a) from the generating functions of duration of each stage • Convolution of two exponential distributions with a mean Tc=1/b1+1/b2, • Such a distribution has a long right tail (short dashed line) r>min(-b1,-b2) • Quadratic increasing curve provided that Generation interval distributions g(a) with identical mean Tc and increasing coefficient of variation s/Tc from 0.25 (long dashes), 0.5 (long and short dashes), 0.75 (short dashes) to (dots) (Source: Wallinga and Lipsitch 2007) Copyright © St Luke’s International University All rights reserved http://www.slcn.ac.jp/ http://university.luke.ac.jp Estimating R using growth rate r • • • • R can be inferred analytically from the observed exponential epidemic growth rate r • the per capita change in number of new cases per unit of time Derive from Lotka–Euler equation (Wallinga and Lipsitch 2007) Laplace transform of the function generation interval distribution g(a) Estimating the moment generating function, M(z) of the generation time distribution g(a) Generation interval distribution • R can be obtained by: Epidemic growth rate A relationship between r and R uniquely characterizes the shape of the g(a) and, vice versa Copyright © St Luke’s International University All rights reserved http://www.slcn.ac.jp/ http://university.luke.ac.jp Generation intervals shape the relationship between growth rates and reproductive numbers • • The shape of the generation interval distribution determines which equation is appropriate for inferring the reproductive number from the observed growth rate R can be obtained by: Table: The relationships between R and r (Source: Wallinga and Lipsitch 2007) Copyright © St Luke’s International University All rights reserved http://www.slcn.ac.jp/ http://university.luke.ac.jp Estimation of time dependent reproduction numbers using likelihood-based estimation • • • • • Possible to infers “who infected whom” from the observed dates of symptom onset - as provided by the epidemic curve Require consideration of all possible infection networks Computational burden • For a small outbreak of 50 cases there are > 7000 possible infection networks A likelihood-based estimation procedure (Wallinga and Teunis 2004) There are other estimation methods (Cintron-Arias, Castillo-Chavez et al 2009), such as • Branching Process (Nishiura 2007) • Maximum likelihood estimates (MLE) method (White and Pagano 2008) • Markov chain Monte Carlo and Monte Carlo sampling method (Cauchemez, Boelle et al 2006) • Sequential Bayesian Method (Bettencourt and Ribeiro 2008) Generatio n Initial phase of epidemic (R0 = 3) Copyright © St Luke’s International University All rights reserved http://www.slcn.ac.jp/ http://university.luke.ac.jp Estimation of time dependent reproduction numbers using sample 1918 influenza pandemic data • • Data is taken from Nishiura’ study for key transmission parameters of an institutional outbreak during 1918 influenza pandemic in Germany (Nishiura 2007) Temporal distribution of Spanish flu cases in Prussia, Germany, from 1918-19 Copyright © St Luke’s International University All rights reserved http://www.slcn.ac.jp/ http://university.luke.ac.jp Opportunities for epidemic models in public health Copyright © St Luke’s International University All rights reserved http://www.slcn.ac.jp/ http://university.luke.ac.jp What models can and cannot do? Can • Models have two distinct roles - prediction and understanding • A high degree of accuracy from any predictive model, • Transparency is an important quality - models used to improve our understanding • Models provide epidemiologists with a ideal world in which individual factors can be examined in isolation • Every facet of the disease spread is recorded in perfect detail Cannot • Impossible to build a fully accurate model; there will always be some element unknown or even unknowable • Unable to predict the precise course of an epidemic, or which people will be infected Copyright © St Luke’s International University All rights reserved http://www.slcn.ac.jp/ http://university.luke.ac.jp What is a good model? What constitutes a good model is context dependent! A model should be suited to its purpose • As simple as possible, but no simpler – appropriate balance of accuracy, transparency, and flexibility • concentrate on the characteristics that are of interest while simplifying all others • A model built for accurate prediction should – – • provide a comprehensive picture of the full dynamics include all the relevant features of the disease and host Note: To determine which factors are relevant and which may be safely ignored is a complex and skilled process A model should be parameterizable from available data • A predictive model requires the inclusion of many features – • • parameterized from available data At the start of an emerging (novel) epidemic— impossible to produce a good predictive model ODEs: add component may double the parameters used, imagine challenge of stochastic model with heterogeneity Copyright © St Luke’s International University All rights reserved http://www.slcn.ac.jp/ http://university.luke.ac.jp Model assumptions and limitations • • • • Simple models can provide complementary insights to the dynamics of an outbreak and development of disease transmission – Unable to address all complicated scenarios, may oversimplify the situation, and incorporate inaccurate assumptions Underreporting and data bias issue (e.g H1N1 cases – asymptomic, effect of change of control strategies over the course of epidemic) The introduction of realistic heterogeneity, behavioral patterns, contact networks and traveling patterns - improve the accuracy of these prediction model outcomes – Challenging due to parameterization/lack of detailed data Sensitivity analysis – Identifying the kinds of parameters that potentially affect the model outcomes – Providing insights to a focused data collection efforts Copyright © St Luke’s International University All rights reserved http://www.slcn.ac.jp/ http://university.luke.ac.jp Potential questions that can be answered by models • To mitigate Ebola outbreak, should we prioritise medical Ebola treatment unit (ETU)? • How much should we expect a 70% efficacious vaccine against seasonal influenza to achieve? • Should we close schools in the event of an pandemic influenza and when and how should we implement? • To mitigate long term care facility institutional outbreak, should we vaccinate the elderly or staff? ã Copyright â St Luke’s International University All rights reserved http://www.slcn.ac.jp/ http://university.luke.ac.jp Selected References • • • • • • • Wong ZSY, Bui C, Chughtai A, MacIntyre R A Systematic Review of Early Modeling Studies of Ebola Virus Disease in West Africa Epidemiology and Infection 2017 Accepted for publication This paper compares methods and outputs of published phenomenological and mechanistic modelling studies pertaining to the 2013 - 2016 Ebola virus disease (EVD) epidemics in four West African countries Wong ZSY, Goldsman D, Tsui KL Economic Evaluation of Individual School Closure Strategies: The Hong Kong 2009 H1N1 Pandemic PloS one 2016;11(1):e0147052 This study used data from the 2009 H1N1 pandemic in Hong Kong to develop a simulation model of an influenza pandemic with a localised population structure to provide scientific justifications for and economic evaluations of individual-level school closure strategies ShenXB, Wong ZSY, Ling MH, Goldsman D, Tsui KL (2016) Comparison of disease transmission algorithms Journal of Simulation, 1-10 Doi: 10.1057/s41273-016-0003-3 This study aims to compare the computational efficiency of two competitive disease transmission algorithms under variation in population dynamics Ling MH, Wong SY, Tsui KL 2015 Efficient Heterogeneous Sampling for Stochastic Simulation with an Illustration in Healthcare Applications', Communications in Statistics-Simulation and Computation, http://dx.doi.org/10.1080/03610918.2014.977914 This study proposed a new efficient algorithm for sampling the disease transmission in a subset of the heterogeneous population The algorithm is validated through a stochastic model of pandemic influenza in a Canadian city Shu L; Ling MH; Wong ZSY; Tsui KL, 2014, 'Spatial Clustering in Public Health: Advances and Challenges', IEEE Intelligent Systems, vol 29, no 3, pp 65 – 68 Tsui K, Wong ZSY, Goldsman D, Edesess M Tracking infectious disease spread for global pandemic containment Intelligent Systems, IEEE 2013;28(6):60-4 This study addresses research issues on how to improve future global pandemic containment with the help of advanced artificial intelligence and simulation methods Tsui K-L, Wong ZSY, Jiang W, Lin C-J Recent Research and Developments in Temporal and Spatiotemporal Surveillance for Public Health IEEE Transactions on Reliability 2011;60(1):49-58 This study provides recent studies on temporal and spatiotemporal surveillance methods and compared surveillance algorithms using male thyroid cancer cases in New Mexico Copyright © St Luke’s International University All rights reserved http://www.slcn.ac.jp/ http://university.luke.ac.jp Acknowledgement Dr Wong’s research was supported by the Research Grants Council Theme-Based Research Scheme (Ref.: T32-102/14N) and Collaborative Research Fund (Ref CityU8/CRF/12G: Syndromic Surveillance and Modeling for Infectious Diseases) Copyright © St Luke’s International University All rights reserved http://www.slcn.ac.jp/ http://university.luke.ac.jp Thank you for your attention! Copyright © St Luke’s International University All rights reserved ... Concepts of building a mathematical modelling • Compartmental models consists of two types of objects: – Compartments: • Individuals in different stage of infection • State variables – keep track of. .. Probability of transmission when an infectious individual contacts a susceptible (per-contact): p • Proportion of population infectious = I/N • Rate of contacting infectious individual = c*I/N • Rate of. .. of contacts are discarded ,S Transmission events Deaths of I Infectious, I Recoveries Deaths of R Removed, R ௗௌ ௗ௧ • • • = Transmission events – Recoveries Deaths of I = Recoveries - Deaths of