A statistical model for the transmission on infectious diseases

A STATISTICAL MODEL FOR THE TRANSMISSION OF INFECTIOUS DISEASES WANG WEI NATIONAL UNIVERSITY OF SINGAPORE 2007 A STATISTICAL MODEL FOR THE TRANSMISSION OF INFECTIOUS DISEASES WANG WEI (B.Sc. University of Auckland, New Zealand) A THESIS SUBMITTED FOR THE DEGREE OF MASTER OF SCIENCE DEPARTMENT OF STATISTICS AND APPLIED PROBABILITY NATIONAL UNIVERSITY OF SINGAPORE 2007 ACKNOWLEDGEMENTS I would like to express my deep and sincere gratitude to Associate Prof. Xia Yingcun, my supervisor, for his valuable advices and guidance, endless patience, kindness and encouragements. I appreciate all the time and efforts he has spent in helping me to solve the problems I encountered. I have learned many things from him, especially regarding academic research and character building. I also would like to give my special thanks to my husband Mi Yabing for his love and patience during my graduate period. I feel a deep sense of gratitude for my parents who teach me the things that really matter in life. Furthermore, I would like to attribute the completion of this thesis to other members of the department for their help in various ways and providing such a pleasant working environment, especially to Ms. Yvonne Chow and Mr. Zhang Rong. Finally, it is a great pleasure to record my thanks to my dear friends: to Mr. Loke Chok Kang, Mr. Khang Tsung Fei, Ms. Zhang Rongli, Ms. Zhao Wanting, Ms. Huang Xiaoying, Ms. Zhang Xiaoe, Mr. Li Mengxin, Mr. Jia Junfei and Mr. Wang Daqing, who have given me much help in my study. Sincere thanks to all my friends who help me in one way or another for their friendship and encouragement. Wang Wei July 2007 ii CONTENTS Acknowledgements ii Summary vi List of Tables viii List of Figures ix Chapter Introduction 1.1 Epidemiological background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Main objectives of this thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Organization of this thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chapter Classical Epidemic Models 2.1 Susceptible-infective-removed models (SIR) . . . . . . . . . . . . . . . . . . . . . . 2.2 The assumptions for epidemic models . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 Assumptions about the population of hosts . . . . . . . . . . . . . . . . . . . 2.2.2 Assumptions about the disease mechanism . . . . . . . . . . . . . . . . . . . 10 2.3 Deterministic models. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.4 Stochastic models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.5 Some terminology. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 2.5.1 Basic reproductive rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 2.5.2 Important time scales in epidemiology . . . . . . . . . . . . . . . . . . . . . . 19 iii Chapter Statistical Epidemic Models 21 3.1 The time series – susceptible – infected – recovered model (TSIR) . . . . 21 3.1.1 Check the validation of TSIR model . . . . . . . . . . . . . . . . . . . . . . . 22 3.1.2 The relationship between the parameters in TSIR model and the 3.1.2 deterministic SIR model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 3.1.3 Impact of aggregated data on the model fit . . . . . . . . . . . . . . . . . . 27 3.2 The cumulative alertness infection model (CAIM) . . . . . . . . . . . . . . . . . 33 3.2.1 Development of CAIM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 3.2.2 Extension of CAIM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 3.2.2.1 Change of alertness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 3.2.2.2 Long term epidemics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 3.2.3 Data denoising and model estimation . . . . . . . . . . . . . . . . . . . . . . . . 39 Chapter Application of CAIM to Real Epidemiological Data 41 4.1 Foot and Mouth Disease (FMD) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 4.2 Severe Acute Respiratory Syndrome (SARS) . . . . . . . . . . . . . . . . . . . . . . 45 4.2.1 SARS in Hong Kong . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 4.2.2 SARS in Singapore and Ontario, Canada . . . . . . . . . . . . . . . . . . . . . 51 4.3 Measles. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 4.4 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 BIBLIOGRAPHY 59 Appendix 65 Programme Codes Programme to produce the realization of deterministic SIR model with 4.4 different parameters in (2.3) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 iv Programme to produce the realization of deterministic SIR model with 4.4 different R0 in (2.5.1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 Programme to check the validation of TSIR model in (3.1.1) . . . . . . . . . . . 68 Programme to investigate the relationship between the parameters in TSIR 4.4 model and R0 in (3.12) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 Programme to check the impact of aggregated data on the TSIR model fit in 4.4 (3.1.3) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 Programme to investigate the relationship between the three parameters in 4.4 CAIM and R0 in (3.2.1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 Programme to apply CAIM to fit 2001 FMD data in UK in (4.1) . . . . . . . . . 82 Programme to apply CAIM to fit 2003 SARS data in HK in (4.2.1) . . . . . . . 84 Programme to apply CAIM to fit 2003 SARS data in Singapore in (4.2.2) . . 86 10 Programme to apply CAIM to fit 2003 SARS data in Ontario in (4.2.2) . . . 89 11 Programme to apply CAIM to fit measles data in China in (4.3) . . . . . . . . . 92 v SUMMARY Infectious diseases are the diseases which can be transmitted from one host of humans or animals to other hosts. The impact of the outbreak of an infectious disease on human and animal is enormous, both in terms of suffering and in terms of social and economic consequences. In order to make predictions about disease dynamics and to determine and evaluate control strategies, it is essential to study their spread, both in time and in space. To serve this purpose, mathematical modeling is a useful tool in gaining a better understanding of transmission mechanism. There are two classic mathematical models: deterministic model and stochastic model that have been applied to study infectious diseases and the concepts derived from such models are now widely used in the design of infection control programmes. Although the mathematical models are elegant, there are still some reasons that more practical statistical models should be developed. During the outbreak of an infectious disease, what we can observe in the first place is a time series of the number of cases. It is very important to an instantaneous analysis of the available time series and provide useful suggestions. However, most existing mathematical models are based on a system of differential equations with lots of unknown parameters which are difficult to estimate statistically. Furthermore, these models need the effective number of susceptibles, which is also difficult to calculate and define. In this thesis, we first propose the time series-susceptible-infected-recovered (TSIR) model based on the compartmental SIR mechanism. The validation of TSIR model was checked by simulation. The result showed that TSIR model performed well. Then we develop a vi more practical statistical model, the cumulative alertness infection model (CAIM) based on the TSIR model, which only requires the reported number of cases. The parameters in the CAIM have been interpreted and some extensions of CAIM have been discussed. We also apply the CAIM to fit the real data of several infectious diseases: foot-andmouth disease in UK in 2001, SARS in Hong Kong, Singapore and Ontario, Canada in 2003 and measles in China from 1994 to 2005. Our findings showed that the CAIM could mimic the dynamics of these diseases reasonably well. The results indicate that the CAIM may be helpful in making predictions about infectious disease dynamics. vii List of Tables Table 2.1 Estimated values of R0 for various infectious diseases ………………. 19 Table 2.2 Incubation, latent and infectious periods (in days) for some infectious diseases . . . . . . . . . . . . . . . . . . . . . . . . ………………………………… 20 viii List of Figures Figure 2.1 Plot of a typical deterministic realization of an epidemic SIR model with N=100, β=0.005, γ=0.1 for change in the number of infectives ……. 13 Figure 2.2 Plot of a typical deterministic realization of an epidemic SIR model with N=100, β=0.005, γ=0.1 for change in the number of susceptibles … 13 Figure 2.3 Plot of a typical deterministic realization of an epidemic SIR model with N=100, β=0.015, γ=0.1 for change in the number of infectives ……. 14 Figure 2.4 Plot of a typical deterministic realization of an epidemic SIR model with N=100, β=0.015, γ=0.1 for change in the number of susceptibles …. 14 Figure 2.5 Plot of a realization from the differential SIR model with R0 = 0.8, N=100 and γ =0.1 ………………………………………………………… 17 Figure 2.6 Plot of a realization from the differential SIR model with R0 = 1.5, N=100 and γ =0.1 ………………………………………………………… 18 Figure 2.7 Plot of a realization from the differential SIR model with R0 = 5, N=100 and γ =0.1 ………………………………………………………… 18 Figure 2.8 Diagrammatic illustration of the relationship between the incubation, latent and infectious periods for a hypothetical microparastic infection …. 20 Figure 3.1 Plot of a deterministic realization of the estimated TSIR model and the SIR model with R0 = 2.0, γ= 1/5, N = 5000000 ……………………… 23 Figure 3.2 Plot of a deterministic realization of the estimated TSIR model and the SIR model with R0 = 2.0, γ= 1/10, N = 5000000 …………………… 23 Figure 3.3 Plot of a deterministic realization of the estimated TSIR model and the SIR model with R0 = 7.0, γ= 1/5, N = 5000000 ……………………… 24 ix % Define the function of the deterministic model function A = sir(beta, v, N) options = odeset ('RelTol',1e-5,'AbsTol',1e-5); [T,Y] = ode45(@sirsys1,[0:1:200],[N 0],options,beta,v); A=[T,Y]; function dy = sirsys1(t,y,beta,v) dy = zeros (3,1); dy(1)=-beta*y(1)*y(2); dy(2)=beta*y(1)*y(2)-v*y(2); dy(3)=v*y(2); % Estimate the coefficients of TSIR model R0=2:0.1:20; N=5000000; gamma=1/3; % or gamma =1/5, 1/7, 1/10 l1=size(gamma); l=l1(2); beta=gamma*R0/N; k1=size(beta); k=k1(2); R3=zeros(l,k); r3=zeros(l,k); % or R5, R7, R10 corresponding to gamma =1/5, 1/7, 1/10 % or r5, r7, r10 corresponding to gamma =1/5, 1/7, 1/10 alpha3=zeros(l,k); %alpha5, alpha7, alpha10 corresponding to gamma =1/5, 1/7, 1/10 tol = 1e-35; maxit = 25; 80 [beta_final,gamma_final]=meshgrid(beta,gamma); for i = 1:l for j = 1:k A=SIR(beta_final(i,j),gamma_final(i,j),N); a=size(A); n=a(1); check=[A(1:(n-1),2)-A(2:n,2) (1:1:(n-1))']; check1=check(check(:,1)[...]... mathematical models for infectious diseases and some important terminology in epidemiological study In Chapter 3, we introduce two new statistical 5 models: TSIR and CAIM The validation of the models will be investigated and the parameters in the new models interpreted In the last chapter, Chapter 4, we apply the model, CAIM to the real data of some infectious diseases and demonstrate that the new model. .. inference and deterministic modeling, as well as some historical remarks Another new monograph by Diekmann and Heesterbeek [2000] is concerned with mathematical epidemiology of infectious diseases and their methods are also applied to real data Although the mathematical models are elegant, there are still some reasons that more practical statistical models should be developed During the outbreak of an infectious. .. and stochastic models, illustrates the use of a variety of the models using real outbreak data and provides us with a complete bibliography of the area The book that has received most attention recently is Anderson and May [1991] The authors model the spread of disease for several different situations and give many practical applications, but only focus on deterministic models A thematic semester at... is also difficult to calculate and define In this thesis, a simple and practical statistical model is proposed that only 4 considers the reported number of cases and the relationship with the classical mathematical models will be analyzed 1.2 Main objectives of this thesis We start with an introduction of two classical mathematical models for infectious diseases For simplicity, in this thesis we assume... host Most viral and bacterial parasites, and many protozoan and fungal parasites fall into the micro-parasitic category In addition, we assume that the disease is spread by a contagious mechanism so that contact between an infectious individual and a susceptible is necessary After an infectious contact, the infectious individual succeeds in changing the susceptible individual’s disease status 2.3 Deterministic... differential SIR model with R0 = 5, N=100 and γ =0.1 The estimated values of the basic reproductive rate, R0, for some common infectious diseases are listed in Table 2.1(Data from Anderson and May [1991], Wallinga and Teunis [2004]) 18 Table 2.1 Estimated values of R0 for various infectious diseases Disease Measles Mumps Rubella poliomyelitis HIV/AIDS SARS Location England and Wales Kansas, USA Ontario, Canada... propose the time series-susceptible-infectedrecovered (TSIR) model Then we check the validation of the TSIR model and investigate the relationship between the parameters in the TSIR model and the classical mathematical epidemic model Next, based on the TSIR model, we develop a complete case-driven model, the cumulative alertness infection model (CAIM) The parameters in the CAIM have been interpreted and... make assumptions about the population as follows: (a) the population structure: the population is a single group of homogeneous individuals who mix uniformly ; (b) the population dynamics: the population is closed so that it is a constant collection of the same set of individuals for all time; (c) a mutually exclusive and exhaustive classification of individuals according to their disease status: at... deterministic realization of the estimated model for the data of Measles in China ………………………………………………… 57 Figure 4.17 Plot of the deterministic realization of one-step ahead prediction based on the estimated model for the data of measles in China ………… 57 xii CHAPTER 1 Introduction 1.1 Epidemiological background In recent years, as the terms such as the Ebola virus, avian influenza and SARS frequently dominate news... any given time, an individual is either susceptible to the disease, or infectious with it, or a removed case by acquired immunity or isolation or death 2.2.2 Assumptions about the disease mechanism As far as the disease mechanism is concerned, we assume that we deal with microparasites, which are characterized by the fact that a single infection triggers an autonomous process in the host Micro-parasites . A STATISTICAL MODEL FOR THE TRANSMISSION OF INFECTIOUS DISEASES WANG WEI NATIONAL UNIVERSITY OF SINGAPORE 2007 A STATISTICAL MODEL FOR THE TRANSMISSION OF INFECTIOUS. real data. Although the mathematical models are elegant, there are still some reasons that more practical statistical models should be developed. During the outbreak of an infectious disease,. infection control programmes. Although the mathematical models are elegant, there are still some reasons that more practical statistical models should be developed. During the outbreak of an infectious