A Study on Numerical Solutions of Epidemic Models

Transcription

1 A Study on Numerical Solutions of Epidemic Models Md. Samsuzzoha A thesis submitted in fulfilment of requirements for the Degree of Doctor of Philosophy Mathematics Discipline Faculty of Engineering and Industrial Sciences Swinburne University of Technology Australia 2012

2 Dedicated To my wife and daughter, my source of strength and inspiration i

3 Abstract This thesis deal with three different types of deterministic epidemic models that explore the transmission dynamics of influenza. Models are represented by ordinary differential equations and reaction-diffusion equations. First two models are based on basic Susceptible-Exposed-Infectious-Recovered (SEIR) model. Third model represents an attempt on modeling the effects of vaccination on the SEIR epidemic model. First the SEIR type model is discussed. The SEIR model equations with and without diffusion have been solved numerically using different initial conditions. The effect of diffusion on the influenza transmission is investigated. The effects of some intervention strategies have also been investigated. It is shown that diffusion and initial population distribution play crucial role for disease transmission. Numerical analysis is provided to explain the multiple steady states and their bifurcation. In order to represent the transmission dynamics of influenza, SEIRS model is proposed. Sustained and damped oscillations are obtained for different type of disease transmission rate. To help controlling the influenza transmission by vaccination, vaccinated type (SV EIRS) epidemic model is proposed. First the model of SV EIRS is discussed to describe the behavior of an epidemic disease when a vaccination policy is in effect. The stability analysis of steady states and vaccinationreduced basic reproduction number is investigated. In order to determine the most influential parameters to the initial disease transmission and equilibrium disease prevalence, the sensitivity indices of the basic reproduction number, (R vac ) and the endemic point of equilibrium, (P ) to the parameters in the model are calculated. Sensitivity analysis of basic reproduction number is performed with two different approaches: (i) through local derivative on R vac, where R vac is estimated by specific input parameter values; (ii) sampling-based approaches, where sensitivity indices are calculated considering the effects of uncertainty in the parameter estimation. Results are obtained and compared for random and Latin hypercube sampling approaches with 1000 sample sizes. The SV EIRS model parameters are estimated afresh using the available field data. The SV EIRS model equations are solved numerically using new estimated parameters value. To investigate the vaccination ii

4 policy to the diseases transmission, different vaccination rate, vaccine efficacy and initial vaccine coverage are introduced. The analysis of this model suggests that vaccination program can control the spread of the disease. iii

5 Acknowledgements I am deeply grateful to my principal supervisor Dr. Manmohan Singh for introducing me to the subject of Mathematical Biology and for his invaluable guidance and continual encouragement throughout the research work. His ability to quick understanding the outlines of a problem and provide suggestions for a solution was very helpful at various points throughout my PhD project. This thesis would not have been possible otherwise. I am also thankful to my associate supervisor Dr. David Lucy for his constant assistance and invaluable suggestions. I appreciate his constructive criticism and helpful suggestions during this research work. I would like to express my deep appreciation to Professor Geoffrey Brooks, Head of Mathematics Discipline, Faculty of Engineering and Industrial Sciences, for his kind help, advice and warm encouragement throughout my stay at Swinburne University of Technology. I wish to express my gratitude to my parents, parents-in-law, brothers and sisters for their love and encouragement. I find no words to express my feeling to my wife, Sanjeeda Islam and daughter, Zarin Binte Zoha for their patience, inspiration and moral support which they have given to me during the period of this study. I am grateful to Faculty of Engineering and Industrial Sciences, Swinburne University of Technology for providing all technical support. This research was generously supported by a Swinburne University Postgraduate Research Award (SUPRA). iv

6 Declaration The candidate hereby declares that the work in this thesis, presented for the Degree of Doctorate in Mathematics submitted in the faculty of Engineering and Industrial Sciences, Swinburne University of Technology: 1. is that the candidate alone and has not been submitted previously, in whole or in part, in respect of any other academic award and has not been published in any form by other person except where due references are given, and 2. has been carried out during the period from February 2008 to January 2012 under the supervision of Dr. Manmohan Singh and Dr. David Lucy. Md. Samsuzzoha Date: January, 2012 v

7 Contents 1 Summary of the Thesis Research overview Structure of the thesis Literature Survey Introduction History of mathematical epidemic model The story of influenza Influenza virus Transmission route Models of influenza Prevention Surveillance Impact of influenza on public-health Basic mathematical epidemic model Numerical Study of an Influenza Epidemic Model with Diffusion Introduction The SEIR model Equations Initial conditions Numerical scheme Stability of equilibria of the system Disease-free equilibrium and stability analysis Stability of endemic equilibrium without diffusion vi

8 3.4.3 Bifurcation value of transmission coefficient β without diffusion for Case Bifurcation value of recovery rate for clinically ill γ without diffusion for Case Stability of equilibrium with diffusion Bifurcation value of transmission coefficient β with diffusion for Case Bifurcation value of recovery rate for clinically ill γ with diffusion for Case The solutions Solutions of SEIR model without diffusion (Case 1) Solutions of SEIR model with diffusion (Case 1) Other cases Control strategies Discussion and conclusion Numerical Study of a Diffusive Epidemic Model of Influenza with Variable Transmission Coefficient Introduction Model Initial conditions Numerical scheme Point of equilibrium analysis Derivation of basic reproduction number Disease-free equilibrium and stability analysis Stability of endemic equilibrium Transmission coefficient β Bifurcation value of transmission coefficient β Numerical solutions Case A : Numerical solutions of the model without diffusion i.e. d 1 = d 2 = d 3 = d 4 = vii

9 4.7.2 Case B : Numerical solutions of the model with diffusion i.e. d 1 = 0.050, d 2 = 0.025, d 3 = and d 4 = Discussion and conclusion A Numerical Study on an Influenza Epidemic Model with Vaccination and Diffusion Introduction Model formulation Initial conditions Points of equilibrium Derivation of basic reproduction number Stability analysis Stability of disease-free equilibrium Stability of endemic equilibrium The Routh-Hurwitz stability condition Bifurcation value of transmission coefficient β Numerical scheme Numerical solutions Numerical solutions of the model with initial condition (i) and without diffusion Numerical solutions of the model with initial condition (ii) and without diffusion Numerical solutions of the model with initial condition (iii) and without diffusion Numerical solutions of the model with initial condition (ii) and with diffusion Numerical solutions of the model with initial condition (iii) and with diffusion Other cases Discussion and conclusion Uncertainty and Sensitivity Analysis of the Basic Reproduction viii

10 Number of a Vaccinated Epidemic Model of Influenza Introduction Model R vac for influenza Sensitivity analysis of R vac Sensitivity indices of R vac based on perturbation of fixed point estimations Sensitivity indices of R vac when parameter estimation is uncertain Sensitivity analysis of the point of endemic equilibrium Discussion and conclusion Parameter Estimation of Influenza Epidemic Model Introduction Data Models Model I : SEIRS model Model II : SV EIRS model Method of parameter estimation Differential equation solver Runge-Kutta methods Embedded methods for error estimation Optimization algorithm Non-linear least squares method The Gradient Descent method The Gauss-Newton method The Levenberg-Marquardt method Parameter estimation Model validation Graphical method Numerical method Model uncertainty ix

11 7.9 Vaccination program on the influenza epidemic Impact of vaccination rate on the influenza epidemic Impact of vaccine efficacy to the influenza epidemic Impact of initial vaccine coverage on the influenza epidemic Discussion and conclusion Conclusions 189 Bibliography 193 Appendix 213 List of Publications 231 x

12 List of Figures 2.1 The structure of influenza virus [199] The structure of the influenza virus [200] Influenza spread by droplet [200] The flow diagram of the SIR model The flow diagram of the SEIR model Initial conditions (i) (iv) Determination of first excited mode with β as an unknown parameter Solutions with initial condition (i) and without diffusion Solutions with initial condition (ii) and without diffusion Solutions with initial condition (iii) and without diffusion Solutions with initial condition (iv) and without diffusion Solutions with initial condition (ii) and with diffusion Solutions with initial condition (iii) and with diffusion Solutions with initial condition (iv) and with diffusion The flow diagram of the SEIRS model Initial conditions (i) (iii) Determination of first excited mode with β as an unknown parameter Graphs for disease transmission coefficient, β Solutions with initial condition (i) and without diffusion Solutions with initial condition (ii) and without diffusion Solutions with initial condition (iii) and without diffusion Solutions with initial condition (i) and without diffusion Solutions with initial condition (ii) and without diffusion xi

13 4.10 Solutions with initial condition (iii) and without diffusion Solutions with initial condition (i) and without diffusion Solutions with initial condition (ii) and without diffusion Solutions with initial condition (iii) and without diffusion Solutions with initial condition (iii) and with diffusion Solutions with initial condition (iii) and with diffusion Solutions with initial condition (iii) and with diffusion The flow diagram of the SV EIRS model Initial conditions (i) (iii) Determination of first excited mode with β as an unknown parameter Bifurcation value of β Solutions with initial condition (i) and without diffusion for Cases 1(a) 1(c). Continuous, dotted and mashed lines represent the solution for Cases 1(a), 1(b) and 1(c) respectively Solutions with initial condition (iii) and without diffusion for Cases 1(a) 1(c) at x = 1.0. Continuous, dotted and mashed lines represent the solution for Cases 1(a), 1(b) and 1(c) respectively Solutions with initial condition (ii) and with diffusion for Cases 1(a) 1(c) at x = 0.0. Continuous, dotted and mashed lines represent the solution for Cases 1(a), 1(b) and 1(c) respectively Solutions with initial condition (ii) and with diffusion for Cases 1(a) 1(c) at x = 1.0. Continuous, dotted and mashed lines represent the solution for Cases 1(a), 1(b) and 1(c) respectively Solutions with initial condition (iii) and with diffusion for Cases 1(a) 1(c) at x = 0.0. Continuous, dotted and mashed lines represent the solution for Cases 1(a), 1(b) and 1(c) respectively Solutions with initial condition (iii) and with diffusion for Cases 1(a) 1(c) at x = 1.0. Continuous, dotted and mashed lines represent the solution for Cases 1(a), 1(b) and 1(c) respectively xii

14 6.1 Histograms of the values obtained from random sampling using a sample size of 1000 for the six input parameters Histograms obtained from Latin hypercube sampling using a sample size of 1000 for the six input parameters Graphic representation of the basic reproduction number based on random sampling Graphic representation of the basic reproduction number based on Latin hypercube sampling Scatter plots for the basic reproduction number and six sampled input parameters values. These results were obtained from random sampling using a sample size of Scatter plots for the basic reproduction number and six sampled input parameters values. These results were obtained from Latin hypercube sampling using a sample size of Time plot of influenza incidence (first and second waves) data, Sydney Characteristic of influenza incidence (first and second waves) data, Sydney SEIRS model fits for the influenza incidence (first and second waves) data, Sydney Solid line depicts the model output for I, while the small circles show the data SV EIRS model fits for the influenza incidence (first and second waves) data, Sydney Solid line depicts the model output for I, while the small circles show the data Residuals analysis of SEIRS model for infected individuals (corresponding to first and second waves data) Residuals analysis of SV EIRS model for infected individuals (corresponding to first and second waves data) Correlation of residuals for the infected individuals of SEIRS model (corresponding to first and second waves data) xiii

15 7.8 Correlation of residuals for the infected individuals of SV EIRS model (corresponding to first and second waves data) Comparison of different vaccination rate Comparison of different vaccine efficacy Comparison of different initial vaccine coverage Solutions with initial condition (i) and without diffusion for Case Solutions with initial condition (ii) and without diffusion for Case Solutions with initial condition (iii) and without diffusion for Case Solutions with initial condition (iv) and without diffusion for Case Solutions with initial condition (ii) and with diffusion for Case Solutions with initial condition (iii) and with diffusion for Case Solutions with initial condition (iv) and with diffusion for Case Solutions with initial condition (i) and without diffusion for Case Solutions with initial condition (ii) and without diffusion for Case Solutions with initial condition (iii) and without diffusion for Case Solutions with initial condition (iv) and without diffusion for Case Solutions with initial condition (ii) and with diffusion for Case Solutions with initial condition (iii) and with diffusion for Case Solutions with initial condition (iv) and with diffusion for Case Solutions with initial condition (i) and without diffusion for Case Solutions with initial condition (ii) and without diffusion for Case Solutions with initial condition (iii) and without diffusion for Case Solutions with initial condition (iv) and without diffusion for Case Solutions with initial condition (ii) and with diffusion for Case Solutions with initial condition (iii) and with diffusion for Case Solutions with initial condition (iv) and with diffusion for Case Solutions with initial condition (i) and without diffusion for Case Solutions with initial condition (ii) and without diffusion for Case Solutions with initial condition (iii) and without diffusion for Case Solutions with initial condition (iv) and without diffusion for Case Solutions with initial condition (ii) and with diffusion for Case xiv

16 27 Solutions with initial condition (iii) and with diffusion for Case Solutions with initial condition (iv) and with diffusion for Case xv

17 List of Tables 2.1 Top ten leading causes of death, 2001 [121] Influenza outbreaks Biological meaning of all parameters and state variables Interpretation of parameters Parameters used in the numerical solution Characteristic of equilibria without diffusion Characteristic of equilibria with diffusion Bifurcation value of β Bifurcation value of γ The combined scenarios of effectiveness of interventions strategies (without diffusion) The combined scenarios of effectiveness of interventions strategies (with diffusion) Result summary (IC denotes initial condition) Parameters used in the numerical solution Characteristic of the point of equilibrium Values of basic reproduction number, R Bifurcation value of β Model parameters and their interpretations Points of equilibrium Characteristic of the point of equilibrium (S, V, E, I, R ) Bifurcation value of β Proportion of infected population in the absence of diffusion at x = xvi

18 5.6 Proportion of infected population in the presence of diffusion at x = Sensitivity indices of R vac Estimates of the basic reproduction number (R vac ) PRCCs for R vac and six input parameters and corresponding sensitivity index Sensitivity indices of the point of equilibrium Influenza incidence data, Sydney Biological meaning of all parameters and state variables Butcher tableau Butcher extended tableau Butcher extended (Dormand-Prince method) tableau Estimated parameters value for SEIRS model Estimated parameters value for SV EIRS model Akaike s information criterion Summary of covariance Different vaccination rate Different vaccine efficacy Different initial vaccine coverage xvii

19 Chapter 1 Summary of the Thesis 1.1 Research overview Influenza is an infectious disease caused by the influenza virus. It is transmitted among humans mainly through direct contact with infected individuals as well as by contact with contaminated objects and inhalation of aerosols that contain virus particles. Influenza has been a major cause of morbidity and mortality among human populations. Millions of people suffer or die every year because of influenza. During the twentieth century, there were influenza pandemics in 1918, 1957 and About one third of the world population was infected during the period with flu pandemic. In most of the affected places, the death rate was 2.3% 5%. Although different control and prevention strategies are available to control transmission of influenza, still influenza has been a major cause of morbidity and mortality among humans all over world. Comparative knowledge of the effectiveness and efficacy of different control strategies is necessary to have a desirable programs to control influenza epidemics. Mathematical modeling of influenza has played an important role in improving understanding of the spread of influenza among the world population. This has helped in comparing the effects of different control strategies developed for influenza [51, 66, 129, 193, 196]. The aim of this thesis is to study the transmission dynamics of influenza leading to investigation of different control strategies. In order to investigate the effect of transmission dynamics of influenza on the initial population distribution differ- 1

20 ent initial conditions have been used. To investigate the impact of diffusion on the transmission of influenza, differential equations have been solved numerically with and without the inclusion of diffusion in the system. The bifurcation values of transmission coefficient, β of the point of endemic equilibrium for all the models considered have been obtained. With a view to investigate different intervention strategies, various cases have been taken into consideration. In order to investigate the formation of periodic behavior, it is essential to know the nature of susceptible-infectious dynamics. In view of that, the existence of sustained and damped oscillations have been obtained using different modes of transmission coefficient, initial population distribution and diffusion. In order to study the effect of vaccination on the SEIRS system, the vaccinated class has been introduced into the differential equations governing the system. Vaccination plays a crucial role in reducing the transmission of influenza. Different values of the vaccination rate and vaccine efficacy have been used in the investigation of effects on transmission of influenza. The most common threshold parameter R 0, the basic reproduction number has also been derived for all of the models considered. With a view to estimate the SV EIRS model parameters, field data from the Sydney epidemic of 1919 has been used. All epidemiological parameters have been estimated. The effect of vaccination strategies also has been investigated. Main aims of this thesis are: To investigate the effect of model parameters, initial populations distribution and diffusion on the periodic behavior of influenza transmission dynamics. To investigate the effect of intervention (medical and nonmedical) on the spread of influenza epidemic. To investigate the effect of vaccination coverage on the influenza epidemic. To demonstrate uncertainties and sensitivity analysis of the model parameters to the basic reproduction number. To demonstrate the sensitivity analysis of the model parameters to the point of endemic equilibrium. 2

21 Estimation of parameters involved in epidemic models. Analysis of the main characteristic of an influenza epidemic that occurred in Sydney, In achieving these aims, I have investigated three different types of epidemic models. MATHEMATICA and MATLAB have been used to obtain numerical solutions. 1.2 Structure of the thesis This thesis is divided into eight chapters and an appendix. Chapter 1. Summary of the Thesis In this chapter a brief introduction to the work done in this thesis is given. Chapter 2. Literature Survey In this chapter a brief historical background of infectious diseases and Mathematical models in epidemiology is given. A review of influenza epidemic and mathematical models that represent the transmission dynamics of influenza has also been included. Chapter 3. Numerical Study of an Influenza Epidemic Model with Diffusion In this chapter a diffusive epidemic model is investigated with a view to describing the transmission of influenza as an epidemic. The equations are solved numerically using the splitting method under different initial distribution of population density. It is shown that the initial population distribution and diffusion play an important role for the spread of disease. It is also shown that interventions (medical and nonmedical) significantly slow down the spread of disease. Stability of equilibria of the numerical solutions are also established. This work has been published in the Journal of Applied Mathematics and Computation. 3

22 Chapter 4. Numerical Study of a Diffusive Epidemic Model of Influenza with Variable Transmission Coefficient In this chapter a diffusive epidemic model for influenza is formulated with a view to gain basic understanding of the virus behavior. All newborns are assumed to be susceptible. The mortality rate for infective individuals in the population is assumed to be greater than the natural mortality rate. Latent, infectious and immune periods are assumed to be constant throughout this study. The numerical solutions of this model are carried out under three different initial population distributions. In order to investigate the effect of the disease transmission coefficient on the spread of disease, β is taken to be constant as well as a function of seasonally varying time t and a function of spatial variable x. The threshold quantity (R 0 ) that governs the disease dynamics has also been derived. This work has been published in the Journal of Applied Mathematical Modelling. Chapter 5. A Numerical Study on an Influenza Epidemic Model with Vaccination and Diffusion In this chapter a vaccinated diffusive compartmental epidemic model is developed to explore the impact of vaccination as well as diffusion on the transmission dynamics of influenza. The basic reproduction numbers with and without vaccination are obtained. Stability analysis of the points of equilibrium has been investigated. Using the combined effect of the vaccine efficacy and vaccination rate, the model is analysed to determine criteria for control of influenza epidemics. The roles of vaccine efficacy and vaccination rate are also compared. This work has been submitted for publication in a refereed journal. Chapter 6. Uncertainty and Sensitivity Analysis of the Basic Reproduction Number of a Vaccinated Epidemic Model of Influenza In this chapter the sensitivity analysis of the basic reproduction number and the point of endemic equilibrium has been investigated. The sensitiv- 4

23 ity analysis of the basic reproduction number based on mathematical as well as statistical techniques has been obtained to determine the importance of the epidemic model parameters. Sensitivity analysis of the point of endemic equilibrium based on parameters involved in the system has also been done. This work has been submitted for publication in a referred journal. Chapter 7. Parameter Estimation of Influenza Epidemic Model In this chapter all parameters of the SEIRS and SV EIRS epidemic models that have been developed in Chapter 4 and Chapter 5 respectively, are estimated afresh using the field data Sydney, 1919 (first and second waves data) of reported cases. The least squares method has been applied to estimate the unknown parameters for both models. Graphical as well as numerical methods have been used to validate these models. The dynamics of all waves fit these models very well. It is shown that both models considerably reflect the dynamical behavior of the influenza epidemic data used. Some important picture of the disease dynamics has also been developed. Vaccine efficacy and level of vaccination coverage have also been investigated. The results presented here suggest that the severity of influenza epidemic could be controlled with the help of the expansion of the vaccination programs. This work has been submitted for publication in a refereed journal. Chapter 8. Conclusions This chapter presents the conclusions and suggests opportunities for future developments. 5

24 Chapter 2 Literature Survey 2.1 Introduction With the development of antibiotics and vaccines, improved living conditions including health care and surveillance systems led to impressive declines in the morbidity and mortality of many infectious diseases throughout the last century. However, infectious diseases still continue to be major causes of human suffering and mortality, both in developing and developed countries. In 2001 alone, about 56.2 million people died with different causes of infection. Among these deaths, approximately 26% deaths were related to infectious diseases. In Table 2.1 is given some leading causes of death in year As shown in Table 2.1, the impact of human mortality attributed to infectious diseases among the developed countries is less then that of under developed countries. Six out of ten deaths, particularly in countries with low and middle-income groups, are infectious diseases related deaths. Among the infectious diseases group, acute respiratory infectious diseases are the most common causes of death. The cause of death by infectious diseases are 26.1% and those by HIV/AIDS are 17.5%, by diarrheal diseases are 12.2%, by tuberculosis are 10.9%, by vaccine preventable childhood diseases are 9.3%, by malaria are 8.2%, by sexually transmitted diseases excluding HIV/AIDS are 1.2%, by meningitis are 1.2%, by hepatitis B and C are 1.0%, by tropical-cluster diseases are 0.8% and other infectious diseases contributed to approximately 11.4%. Particularly in countries with low and middle-income population, one in two deaths is 6

25 preventable [121, 139]. Table 2.1: Top ten leading causes of death, 2001 [121] High per capita income countries Low and middle per capita income countries Cause Total deaths (%) Cause Total deaths (%) Ischemic heart diseasease 17.3 Ischemic heart dis Cerebrovascular disease 9.9 Cerebrovascular disease 9.5 Trachea, bronchus 5.8 Lower respiratory infection 7.0 and lung cancer Lower respiratory infection 4.4 HIV/AIDS 5.3 Chronic obstructive 3.8 Perinatal conditions 5.1 pulmonary disease Colon and rectal cancers 3.3 Chronic obstructive 4.9 pulmonary disease Alzheimer s and 2.6 Diarrheal diseases 3.7 other dementias Diabetes mellitus 2.6 Tuberculosis 3.3 Breast cancer 2.0 Malaria 2.5 Stomach cancer 1.9 Road traffic accidents 2.2 The successful implementation of infectious disease control or prevention strategies depends on a good understanding of epidemiological aspects of a disease. An understanding of various aspects of infectious diseases such as the clinical and biological understanding of infection agent, is the key to the model building process. Also the impact of previous outbreak of infectious diseases to our health system need to be studied for better understanding of their behavior. It will then lead to better preparedness for any pandemic. This chapter aims to give an overall view of 7

26 epidemic modeling of infectious diseases describing underlying mechanisms of the influenza transmission dynamics. 2.2 History of mathematical epidemic model The study of epidemic modeling of infectious diseases is over three centuries old. In 1760, a mathematical model was developed by Daniel Bernoulli with a view to evaluating the effectiveness of variolation of healthy people affected with the smallpox virus. This was the first application of mathematics to the study of any infectious diseases. After a long pause, William Farr fitted a normal curve based on the smallpox deaths in England and Wales in the year In 1906, John Brownlee published a paper entitled Statistical studies in immunity; the theory of epidemic in which he described and fitted Pearson frequency distribution curve based on smallpox deaths in England and Wales over the period Hamer [95, 96] developed a simple discrete-time mathematical model in order to understand the regular recurrence of measles epidemic and investigated the properties of this model. He introduced the mass action principle that represents the course of an epidemic depending on the rate of contact between susceptible and infectious populations. Sir Ronald Ross [164] formulated a continuous-time mathematical model to explore the transmission dynamics of malaria. He explained the relationship between numbers of mosquitoes and the incidence of malaria. He also investigated the effectiveness of various intervention strategies for malaria infection. The concepts developed by Hamer and Ross were extended by Kermack and McKendrick [116, 117, 118] and Soper [180]. Kermack and McKendrick clarified why an epidemic ended before all susceptibles were infected. They established the threshold theory and explained how a few infectious individuals into a community of susceptibles would not give rise to an epidemic outbreak unless the density or number of susceptibles was above a certain critical value. Soper [180] explored the underlying mechanisms responsible for the often-observed periodicity of epidemics [12, 107, 121]. All of these models were based on principle of mass action elaborated later on by Ross [165], Kermack and McKendrick [116] and Soper [180]. Another 8

27 model called the Reed-Frost formulation was derived by Reed and Frost [1, 141]. In the Reed-Frost model, the transmission of infection is defined in terms of a probability of effective contact rather than mass action principle [173]. According to Sattenspiel [173] Both the mass-action and Reed-Frost approaches are common in mathematical epidemiology. Generally, the more mathematically inclined researchers prefer the mass-action approach, whereas more statistically inclined researchers prefer the Reed-Frost approach. A Reed-Frost approach is probably better for small populations because of the random effects built into the model, but Reed-Frost models are also more difficult to analyze. The ultimate choice of model invariably will depend on both the modelers background and the focus of the model. All epidemic models developed earlier were quite simple, mainly designed to explore some general qualitative behavior. The development of realistic epidemic models need to incorporate more epidemiologic detail such as passive immunity, the response to mass immunizations and the possibilities of interrupting transmission along with stages of infection, vertical transmission, age structure, social and sexual mixing groups, spatial spread, age differences, latent period, demographic effects, vaccination, quarantine and chemotherapy. Special models have been formulated for diseases such as influenza, measles, rubella, chickenpox, whooping cough, diphtheria, smallpox, malaria, rabies, gonorrhea, syphilis and HIV/AIDS [107]. In the middle of the 20th century, a large number of mathematical models were developed in epidemiology. In his classical review on The Mathematical Theory of Infectious Diseases, Bailey [16] referred to 539 research articles on mathematical epidemiology written between 1900 and Among these articles, 62% were published between 1964 and Measles is a serious disease among children. Before the introduction of vaccination program, one million children were dying worldwide each year. Mathematical model of the transmission dynamics of measles was first developed in 1906 by Hamer. It was then studied extensively by Soper in Bartlett in 1957 and 1960, London and Yourk in 1973, Dietz and Schenzle in 1985 and Anderson and May in 1985 and 1991, analyzed the transmission dynamics of measles infection 9

28 and investigated the effects of heterogeneities in transmission on the basis of age and seasonal variation of the host population. They also investigated the effects of seasonality in measles transmission. Dietz in 1976 and Anderson and May in 1982 investigated the effects of vaccination on measles epidemics and estimated the threshold level of vaccination required to eradicate the infection from the community. Damped oscillations in modelling were first mentioned by Soper in Schenzle in 1984 using an age-structured model, successfully produced the stable biennial cycle. Schaffer in 1985, Schaffer and Kot in 1985, Olsen et al. in 1988 and Olsen and Schaffer in 1990 have identified fluctuations in measles incidence for large populations. Apparently, they have shown chaotic behavior in measles epidemic. All recent work on epidemic modeling of measles have incorporated the effect of demographic as well as epidemic variations on measles epidemics [150]. Malaria is a common vector-borne (mosquito-borne) infectious disease of humans and its transmission can be reduced by preventing mosquito bites. The use of mathematical approaches to explain the transmission of malaria has a long history starting from early 1900 s. Bruce-Chwatt [27] presents a review of early development of malaria models on mathematical epidemiology and discussed the basic epidemiological principles associated with malaria modeling. A variety of models have been formulated that analyzed the effects of maternal immunity, the delayed appearance of infective gametocytes, age structure and the loss of immunity. Elderkin et al. [68] used an age-structured model to analyse the stability conditions for malaria. Dietz et al. [54], Molineaux et al. [148] and Molineaux and Gramiccia [149] developed a malaria transmission model that incorporated the partial immunity to the disease. They have estimated the value of the prevalence of the parasite, including variation by age, season, place and different control measures. Muirhead-Thomson in 1951, Carnevale et al. in 1978, Shidrawi et al. in 1974 and Nedelmanin in 1984 have proposed the dynamics of malarial transmission models that incorporates variability of the biting rates of mosquitoes with age of the human. Several studies have suggested that mosquitoes do not feed randomly with respect to host infection. Edman et al. in 1985 and Kingsolver in 1987 have developed and analyzes the dynamics of a malarial transmission model that incorporates 10

29 non-random feeding behavior by the mosquito. They have shown that non random feeding may occur at three different stages: attraction and penetration, probing and the location of blood, and blood intake. Struchiner et al. [183] and Halloran et al. [89] have developed malaria transmission models that focus on the effects of vaccination programs for malaria transmission [173]. Age-structured epidemiology models are essential to explore the transmission of some infectious diseases. The characteristic of some infectious diseases depend on the age-related mixing behavior. Kermack and McKendrick [116, 117, 118] incorporated continuous age structure in their early epidemiology models [107]. According to Hethcote [107] Modern mathematical analysis of age-structured models appears to have started with Hoppensteadt [109], who formulated epidemiology models with both continuous chronological age and infection class age (time since infection), showed that they were well posed, and found threshold conditions for endemicity. Expressions for R 0 for models with both chronological and infection age were obtained by Dietz and Schenzle [57]. In age-structured epidemiology models, proportionate and preferred mixing parameters can be estimated from age-specific force of infection data [104]. Dietz [55, 56], Anderson and May [10, 11] and Rouderfer et al. [166] have used continuous age-structured models for the evaluation of vaccination strategies of measles and rubella. Tudor [185] has used continuous agestructured model to evaluate the threshold conditions for measles. Halloran et al. [91], Ferguson et al. [76], and Schuette and Hethcote [175] have used age-structured models to study the effects of vaccination programs for chickenpox. Grenfell and Anderson [86] and Hethcote [105, 106] have used age-structured models to study the effects of vaccination programs for pertussis (whooping cough) [173]. Sexually transmitted diseases (STDs) spread from humans to humans through sexual activities. The history of models for sexually-transmitted diseases begins with the gonorrhea transmission model introduced by Cooke and Yorke in 1973 [48]. They developed an one-sex gonorrhea transmission model. Lajmanovich and Yorke in 1976, developed a deterministic model for gonorrhea in a nonhomogeneous population. The general goal of both models of gonorrhea transmission was to obtain insight into the structure of the transmission process. Since then there 11

30 has been development of a large number mathematical model of the transmission dynamics of STDs with the inclusion of demographic, biological and behavioral parameters. The main goal of most of this research was to improve the prevention strategies with a view to promote the use of sexual devices to control the spread of the disease [12, 107]. Among sexually transmitted diseases is another disease called AIDS (Acquired Immune Deficiency Syndrome). AIDS was first identified in 1981 and since then number of HIV/AIDS infected people are increasing. At present AIDS is one of the most important public-health problems in developing countries. The epidemiological characteristics of HIV/AIDS is very complex. HIV/AIDS is known as a sexually transmitted disease but it can be transferred in different ways such as from infected mothers to their babies and with the sharing of infected syringes etc. Mathematical models based on the underlying transmission mechanism of HIV/AIDS can help to have better understanding of the disease transmission. Mathematical models can also assist in seeing how changes in the various assumptions and parameter values affect the dynamics of the disease. Such models that described the transmission dynamics of HIV/AIDS epidemic have been studied since the first case was recognized. Any development of epidemic model of HIV/AIDS is very challenging as many different factors affect the transmission of HIV/AIDS. Mathematical models for the spread of HIV/AIDS, usually require a lot more epidemiologic detail as compared to other sexually transmitted diseases [107]. According to Sattenspiel [173] There are several specific aspects of the natural history of the disease and the behavior of the host population that models can address: 1) the effect that variability in infectivity throughout the course of the disease in an individual has on the spread of the infection through the population; 2) the role of long incubation periods in the dynamics of the disease; 3) the effect of level of sexual activity on an individuals risk for disease; 4) the effect of assumptions about mixing between groups on both individual risk and transmission throughout a population; 5) the consequences of changes in sexual behavior; and 6) the demographic consequences of the epidemic, particularly in those parts of the world, such as Africa and parts of the Caribbean, with very high rates of infection. 12

31 According to Krämer et al. [121] Preventing and reducing the spread of infectious disease among humans is an essential function of public health. Epidemiology is often called the core science of public health, which studies the distribution and determinants of disease risk in human populations. Starting in the middle of the 19th century, infectious disease epidemiology applies the fundamentals of epidemiology to study infectious diseases and deals with questions about conditions for disease emergence, spread and persistence. It describes the prevalence and incidence of infectious diseases through which the epidemiological trends can be characterized for different world regions. The importance of epidemic models can be summarized as [121, 153]: Epidemic models can lead to better understanding of the natural history of the disease. Epidemic models can play an important role to compute and estimate parameters that are important for understanding the transmission dynamics. Once a model has been formulated that captures the main features of the progression and transmission of a particular disease, it can be used to predict the effectiveness of different strategies for disease control. Epidemic models give more understanding about thresholds properties. Epidemic models can compute exact number of steady states (endemic or diseases-free states) and are able to analyse their stability. Also it can determine bifurcation values where there is a qualitative change in population dynamics. Epidemic models can be used to simulate outbreaks and to find out the effectiveness of different prevention and intervention strategies. Epidemic models can provide information about accuracy of the existing surveillance system of any specific infectious diseases and propose more effective surveillance methods. 13

32 Epidemic models can help to interpret epidemiological data and recommend some evidence-based decisions for targeting interventions for effective disease control measures. Epidemic models can link to the transmission dynamics of infectious diseases to environmental phenomena on a global scale. 2.3 The story of influenza Influenza is a respiratory infection, commonly known as the flu, caused by RNA viruses. The viruses spread with the contact of respiratory secretions from an infected person who is coughing and sneezing. Contagious period is between 1 to 2 days before onset of symptoms and 3 to 10 days after onset of symptoms. Infected individuals are able to transmit the influenza virus even before the onset of the clinical disease and continue up to four/five days after the onset of symptoms. This time may be longer in some people, especially children and people with weak immune systems. The flu season occurs during winter. Pandemic influenza may occur at any time of the year, but conditions are the most favorable for rapid spread during regular flu season [121, 153] Influenza virus There are three main types of influenza viruses known as A, B and C based on the relatedness of the matrix and nucleoprotein antigens of influenza viruses. Influenza A virus was first isolated from chickens in Shope isolated the swine flu virus in Wilson Smith, Andrew and Laidlaw first isolated human influenza A virus in Influenza B virus was isolated by Francis in 1940 and type C virus was isolated by Taylor in Influenza virus types A and B contain 8 RNA genomic segments. On the other hand, virus type C contains 7 RNA genomic segments. All of these viruses can infect humans and other species, such as pigs and birds, whereas virus type A is responsible for all influenza pandemic which occurred in the last century. Influenza viral types B and C are mainly responsible for infecting animals. Although influenza viral types are distinguished on the basis of antigenic 14

33 differences between their nucleoproteins and matrix proteins, influenza virus type A is again further subdivided into different subtypes on the basis of antigenic variation of surface glycoproteins such as hemagglutinin (HA) and neuraminidase (NA). The hemagglutinin are spike shaped and the neuraminidase are mushroom shaped as shown in Figure 2.2. The hemagglutinin is responsible for the receptor binding and membrane fusion. The neuraminidase is responsible for destruction of receptors and the release of viral progeny. The function of protein (shown in Figure 2.2) as an ion channel for the acidification of the interior of the viral particle during viral infection. There are 16 subtypes of hemagglutinin (HA) and 9 subtypes of neuraminidase (NA). All combination of HA and NA are isolated from pigs and birds (apart from human). The Influenza virus A subtypes are labeled according to H number (for hemagglutinin) and N number (for neuraminidase). Among these combinations H1N1, H2N2, H3N2, H5N1, H7N7 and H9N2 subtypes are isolated from humans. These limited host range are apparently responsible for all influenza pandemic. Because of the great genetic variability of influenza viruses, our immune system can not recognize them at all. The structure of the influenza A viruses is shown in Figures 2.1 and 2.2 [36, 121]. Figure 2.1: The structure of influenza virus [199]. 15

34 Figure 2.2: The structure of the influenza virus [200] Transmission route The influenza virus is transmitted either directly among individuals or indirectly through the environment. Any infected individuals can spread the flu even before the symptoms start as well as 4 5 days after appearance of symptoms. Respiratory secretions of infected persons may contain up to 10 5 virus particles/ml. The flu virus can remain active on a hard surface for up to hours and on a soft surface for around 20 minutes [153]. The Influenza virus can be transmitted in the following ways [121, 153]: Contact transmission: It happens with the touching of items recently contaminated by a person with the flu virus. It can also happens with the touching of mouth and nose on something with flu viruses on it. By droplet spread: Flu virus can be spread by respiratory secretions from an infected person who is coughing and sneezing as shown in Figure 2.3. Aerosol spread: It is unusual but possible in very crowded conditions with the breathing in droplets produced when infected person talks/coughs/sneezes. 16

35 Figure 2.3: Influenza spread by droplet [200] Models of influenza In the last few decades a large number of models have been developed to describe the transmission dynamics of influenza. The classical Susceptible-Infectious- Recovered (SIR) model that could be used to describe an influenza epidemic was developed early in the 20th century by Kermack and McKendrick. This model has been discussed briefly in Section 2.4. According to Coburn et al. [43] The SIR model has been used as a basis for all subsequent influenza models. The simplest extension to the SIR model includes demographics; specifically, inflow and outflow of individuals into the population. Analysis of this demographic model shows that influenza epidemics can be expected to cycle, with damped oscillations, and reach a stable endemic level. By modifying the basic SIR model in a variety of ways (e.g., by including seasonality [64, 181]) influenza epidemics can be shown to have sustained cycles. The SIR model has also been extended so that it can be used to represent and/or predict the spatial dynamics of an influenza epidemic. The first spatial-temporal model of influenza was developed in the late 1960s by Rvachev [168]. He connected a series of SIR models in order to construct a network model of linked epidemics. He then modeled the geographic spread of influenza in the former Soviet Union by using travel data to estimate the degree of linkage between epidemics in major cities. In the 1980s, he and his colleagues Baroyan and Longini extended his network model and evaluated the effect of air travel on influenza pan- 17