# Introduction

Infectious agents are no respecters of age. The population as a whole is their natural habitat, so a focus on the population therefore does not preclude, but rather facilitates, investigation of how aging and life span influence and are influenced by the public health impact of infections. Thus the population in any given epidemio-logical setting is the sensible unit to consider when using mathematical models to address issues relating to aging and infectious diseases.

The first attempts to model infections in populations appear to have been undertaken by Bernouilli in 1760

in relation to smallpox. A few instances occur also in the 19th century, but the true origins of today's mathematical modeling of infection transmission in populations can be found in the work of Hamer and of Ross in the early years of the 20th century on malaria. Further developed in the 1920s by Soper, and Kermack and McKendrick, this has led to a rapidly expanding literature on the dynamics of infection transmission in populations (see Anderson and May (1991) for an overview and a comprehensive introduction to the field). In essence, infectious disease transmission dynamics consists of the ebb and flow of infection within populations, as mediated by the distribution of immunity in the population and patterns of contacts leading to exposure. While a substantial proportion of the relevant literature is largely theoretical and to be found in mathematical journals, increasingly work involving transmission dynamics modeling is concerned with questions involving real data, future trends, and what may be expected from specific control interventions. Such work may be found with increasing frequency in journals with an orientation towards public health.

This chapter largely focuses on the use of a particular class of mathematical models to investigate the transmission dynamics of infectious diseases. Mathematical modeling, in general, is a very wide field from which many new approaches to the modeling of infection transmission have arisen, but the class of model considered here is one of the most generally useful. These models are known as deterministic compartmental differential equation models. In simple terms, differential equations describe the relationship between the value of a variable (e.g., number of infected individuals at a particular point in time) and the way in which that variable changes (e.g., how the rate of change increases or decreases as the number of infected individuals changes); if they are deterministic, their results are completely determined by the initial model parameters and initial conditions (i.e., there are no stochastic or random processes included in these models), and the models are described as compartmental because the population

Handbook of Models for Human Aging

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under consideration is divided between compartments representing different states of susceptibility, infection and disease. This chapter will describe how this initially simple approach can usefully be expanded to take into account, among other things, demographics, age-related processes of infection and immunity, patterns of contact and risk behaviors, and public health interventions. First, however, it may be helpful to say something about the ecological context in which such models operate, as many useful developments and insights from the field of modeling population ecology have found their way into that of modeling infections in human populations. Infectious agents are, after all, simply organisms which operate in a particular and specialized environment.

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