Rutgers University, Piscataway, NJ

**Organizers:****Carlos Castillo-Chavez,**, Cornell University, Biometrics, cc32@cornell.edu**Herbert Hethcote**, University of Iowa, Mathematics, herbert-hethcote@uiowa.edu**Pauline van den Driessche**, University of Victoria, Mathematics and Statistics, pvdd@math.uvic.ca

1. Three Basic Epidemiology Models I Herbert Hethcote University of Iowa This introductory lecture develops the three basic types of epidemiological models for infectious diseases, formulates them as systems of nonlinear differential equations, identifies the thresholds and equilibrium points, and describes their dynamical behavior by analyzing local and global stability of these models. The goal is to introduce notation, terminology, and results that will be generalized in later lectures on more advanced models. Intuitive interpretations will be given for the basic reproduction number, the contact number, and the infective replacement number.

2. Three Basic Epidemiology Models II Herbert Hethcote University of Iowa This continuation of the previous introductory lecture looks at applications of the basic epidemiology models. The basic reproduction number is estimated for many directly transmitted diseases, herd immunity is defined, and the fraction that must be vaccinated to obtain herd immunity is determined for diseases such as measles, rubella, mumps, chickenpox, influenza, polio, and smallpox. These diseases are compared with each other and the modeling results are compared with national and international experience with these diseases.

3. Epidemiology Models with Immigration Pauline van den Driessche University of Victoria Some models of disease transmission that include immigration of infectives and variable population size are constructed and analyzed. Threshold-like results are given, but in the presence of immigration of infectives there is no disease-free equilibrium. In a model for a disease that confers no immunity, the number of infectives is shown to tend to an endemic value. In a simple model for HIV transmission in a prison system, a considerable reduction of infectives is suggested by the incorporation of screening and quarantining of infectives. Mathematical tools are presented as needed in the analysis.

4. Thresholds and the Basic Reproduction Number R_{O}Pauline van den Driessche University of Victoria A general compartmental disease transmission model is formulated as a system of ordinary differential equations, and a precise definition of the basic reproduction number R_{O}is presented. This parameter acts as a threshold, with the disease-free equilibrium being locally stable if R_{O}<1, but unstable if R_{O}>1. The results are illustrated by several specific models, for example, a treatment model for tuberculosis, a staged progression model for HIV/AIDS and a model that includes vaccination.

5. Epidemiology Models with Variable Population Size Herbert Hethcote University of Iowa Basic epidemiology models often assume that births balance deaths, so that the total population size is constant. But populations may be growing or decreasing significantly due to differences in the natural birth and death rates, an excess disease-related mortality, or disease-related decreases in reproduction. In models with a variable total population size, the persistence of the infectious disease may slow the growth rate of a naturally growing population, lead to a lower equilibrium population size, or even reverse the population growth to a decay to extinction. Variable population size models of SIS and SIR type are formulated and analyzed in order to demonstrate the effects of the disease on the population size and the effects of the population structure on the disease dynamics.

6. Age Structured Models Herbert Hethcote University of Iowa When modeling a disease in which vaccinations are given at different ages, it is necessary to include both age and time as independent variables. SEIR and MSEIR models are formulated with either continuous age or discrete age groups. Expressions for the basic reproduction number are derived and Liapunov functions used to prove stability below the thresholds. Values of the basic reproduction number and contact number are estimated for various diseases including measles in Niger, Africa and pertussis in the United States.

7. Dynamical Models of Tuberculosis Transmission and Control Carlos Castillo-Chavez Cornell University The reemergence of Tuberculosis (TB) in the United States and elsewhere has generated intense research on its transmission dynamics and on potential methods of control. These tutorial lectures will focus on a review of recent tuberculosis models and some of the questions that they addressed. A discsussion of the epidemiology of Tuberculosis is followed by a quick review of historical models for TB transmission before a simple framework for modeling TB dynamics is introduced. Specific models built on this framework are used to study the impact of various factors on TB dynamics and its control (further discussion of these topics will continue on the second lecture). 1. Demographic effects. The modeling framework includes density-dependent recruitment rates of various types that are used to illustrate the impact of some demographic regimes on TB dynamics. 2. Fast and slow TB. The impact of long and variable periods of latency and progression on TB dynamics are studied. 3. Exogenous reinfections. The impact of exogenous reinfections on TB dynamics is addressed. 4. Age structure and vaccination strategies. A model with age structured is instroduced and used to define the concept of optimal vaccination strategy. Some recent results are presented and discussed.

8. TB Cluster Models, Time Scales and Relations to HIV Carlos Castillo-Chavez Cornell University The focus of this lecture is on a review of recent tuberculosis models and some of the questions that they addressed. Specific models built on the framework discussed in the first lecture on TB are modified to study the impact of some additional factors on TB dynamics and its control. 1. TB and HIV. A model for TB dynamics that incorporates the demography of the US population and the recent impact of HIV is considered. The possibility of reaching national goals on the reduction of TB prevalence are discussed using this model. 2. Close and Casual Contacts. Models that incorporate local and individual interactions are introduced in the context of the transmission dynamics of tuberculosis (TB). The multi-level contact structure implicitly assumes that individuals who are at risk of infection from close contacts in generalized household (clusters) as well as from casual (random) contacts in the general population. Epidemiological time scales can be used to reduce the dimensionality of the model and singular perturbation methods can be used to corroborate the results of time-scale approximations. It can be shown that quasi-steady assumptions (diabetic elimination of variables) are valid. The concept optimal average cluster or generalized household size and its impact on TB dynamics is discussed.

9. Vaccination Strategies: Rubella Herbert Hethcote University of Iowa Rubella is a mild infectious disease, but children of women who get rubella in the first trimester of pregnancy can have congenital rubella syndrome. One method of controlling rubella is to vaccinate young girls before they reach childbearing age. Another method is to routinely vaccinate young children for measles, mumps and rubella. The effects of these two strategies are compared in a cost-benefit analysis using an age structured model. Then rubella vaccination strategies are compared in some countries that vaccinate for rubella. About half of the world including China, India, and Africa do not currently vaccinate against rubella. A model for rubella vaccination in China is presented.

10. Vaccination Strategies: Chickenpox Herbert Hethcote University of Iowa Two possible dangers of an extensive varicella vaccination program are more varicella (chickenpox) cases in adults, when the complication rates are higher, and an increase in cases of zoster (shingles). An age-structured epidemiologic-demographic model with vaccination is developed for varicella and zoster. Parameters are estimated from epidemiological data. This mathematical and computer simulation model is used to evaluate the effects of varicella vaccination programs.

11. Epidemiology Models with Delay Pauline van den Driessche University of Victoria In the dynamics of epidemics, time delays can be used to model some mechanisms. for example, the infectious period. A model can then be formulated in terms of delay differential and/or integral equations. Such models are more difficult to analyse than analogous ordinary differential equation models, but can exhibit richer behavior. Some of the mathematical tools needed are explained, and the ideas are illustrated with an SIS model in a variable size population with delay corresponding to a constant infectious period

12. Modeling the Spread and the Evolution of Influenza Carlos Castillo-Chavez Cornell University This tutorial lecture first gives a quick review of some of the basic epidemiology of influenza at the population level. A review of models for the spread of influenza and the questions that they adressed is presented. The impact of population structure and crossimmunity on the evolution of influenza is addressed.

13. Vaccination Strategies: Pertussis Herbert Hethcote University of Iowa Both disease-acquired and vaccine-acquired immunity to pertussis (whooping cough) wane with time, so that several infections can occur in an individual's lifetime. The severity of a pertussis infection depends on how low the immunity has declined since the previous vaccination, infection, or exposure. In the United States five DTP or DTaP (diphtheria-tetanus-pertussis) vaccinations are recommended at ages 2, 4, 6, 15-18 months, and 4-6 years. The new acellular pertussis vaccine (aP) has fewer side effects, so that it is safe for adults. New strategies for reducing pertussis incidence include: 1) combining the aP vaccine with the current Td (tetanus-diphtheria) booster that is now recommended every ten years, 2) giving the aP vaccine to adolescents at age 12 years, 3) giving the aP vaccine to young adults at age 20 years, and 4) giving the aP vaccine to adults at age 50 years. The effects of these new vaccination strategies are analysed using an age-structured model.

14. Modeling HIV/AIDS Carl Simon University of Michigan To model the spread of HIV, one must include stages of infection, disease-related death, and non-random mixing formulations. We examine such models and compute R_{o}for certain non-random mixing patterns. We use these models to shed light on the importance of the primary infection period and to describe how an HIV vaccine might work.

15. Epidemics on Attractors Abdul-Aziz Yakubu Howard University Discrete-time S-I-S epidemic models are capable of generating complex (chaotic) dynamics, a property not shared by classical continuous-time epidemic models. Models for epidemic processes on attractors are presented. Thresholds for disease persistence are computed and used in the study of global behavior of solutions of simple epidemic processes. The potential role of delayed recruitment (age-structure) on disease is explored via a simple model that differentiates between adults and juveniles.

16. Epidemiology Models for Heterogeneous Populations Herbert Hethcote University of Iowa The spread of infectious diseases is greatly influenced by the patterns of encounters between people in the population, particularly for sexually transmitted diseases. The overview here of heterogeneous mixing in deterministic models includes discussions of the threshold result, proportionate mixing and preferred mixing. It is shown how data on seropositivity in socially-defined groups can be used to estimate mixing matrices. Relationships are given between the basic reproduction numbers for heterogeneously mixing populations and "equivalent" homogeneously mixing populations. The mixing matrices between age-related groups have often been based on ad hoc assumptions, but proportionate or preferred mixing matrices could also be used. Proportionate or preferred mixing formulations have the advantage that the activity levels and mixing matrices can often be estimated from epidemiological data.

17. Modeling Peer Influence by Epidemiological Approaches Carlos Castillo-Chavez Cornell University Epidemiological modeling approaches can be used to study the dynamics of processes whose spread depends strongly on one-to-one contacts between distinct types of individuals. Dynamical models for the spread of social "diseases" such as cigarettes smoking, ecstasy use or for the spread of fanatic ideologies are presented. The concept of core group introduced by Hethcote and Yorke plays a fundamental role in the study of these socially-driven processes. Some surprising resulst are found. For example, it is shown that peer pressure can drive a sudden increase in the growth of the "epidemic." In fact, a small group of "infectious" is capable of supporting dramatic rapid growth on disease prevalence. The analysis of these type of models demonstrate how difficulty it is to eliminate social diseases once a "considerable" number of people have become infected.

18. Mathematical Models Coupling Epidemics and Population Genetics Zhilan Feng Purdue University Several ODE models will be presented to demonstrate the interaction between the prevalence of an infectious disease and the genetic structure of a population. The dynamics of such models can usually be separated into two time-scales with a faster time-scale for the epidemics and a slower time-scale for the change in gene frequencies. A particular model for malaria epidemics and sickle-cell gene dynamics will be discussed.

19. Disease Control Strategies for HIV in a Population with Genetic Heterogeneity Christopher Kribs-Zaleta University of Texas, Arlington Recent studies have identified two alleles which may provide some level of genetic resistance to HIV infection. This paper discusses joint work which investigates the interplay between two levels of genetic resistance and three control strategies -- vaccination, treatment, and education -- in a homosexually active population, with a focus on their effects in the infection's basic reproductive number.

20. Deterministic and Stochastic SIR Epidemic Models Linda J.S. Allen Texas Tech University The deterministic ODE models for SIS and SIR epidemics will be put in a stochastic framework. SIS and SIR epidemics will be formulated as discrete-time Markov chains, continuous-time Markov chains and stochastic differential equations. The expected duration of an epidemic, occurrence of an epidemic, and size of an epidemic will be discussed in terms of each of these formulations. In particular, in models with variable host population size, S+I+R not constant, it will be shown that the results depend on the particular form of the birth and death rates.

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Document last modified on June 21, 2002.