Branching Processes : Variation , Growth , and Extinction of Populations 7 . 6 Modeling Measles Outbreaks

Epidemiology is one of the areas in biology to which mathematical modeling has been applied most successfully. It was through the mathematical concept of basic reproductive number that Ronald Ross gained the insight that malaria could be controlled if the mosquito population was sufficiently suppressed (Heesterbeek 2002). Later, similar concepts were applied to the eradication of smallpox and the control of many viral diseases through vaccination campaigns (Anderson and May 1991). To date, mathematical models remain an important tool. Epidemiology studies the distribution of a disease within a host population. Most models used for infectious diseases are based on partitioning the host population into different classes that correspond to the different stages of the infectious process (e.g., susceptible, infectious, and recovered) and the transitions between these classes. Such models work particularly well for diseases caused by microparasites, a class of pathogens that comprises viruses and bacteria and that are characterized by an infection normally accomplished with a single dose that consists of a relatively small number of infective particles (Anderson and May 1991). Mathematical epidemiology has concentrated on cases for which the disease is endemic or in which large epidemics occur. In this context the reproductive number is an important parameter. It is defined as the average number of secondary infections caused by an infected host over the lifetime of the infection. In a completely susceptible population, this number is known as the basic reproductive number, R0. If the reproductive number is larger than one, a single infection can lead to a chain reaction of infections and, eventually, to an epidemic or an endemic state. If the reproductive number is smaller than one, large epidemics do not occur and the pathogen is bound to disappear from the population. If the basic reproductive number is less than one, the numbers of infected individuals hardly ever become truly large and stochastic effects prevail. If this is the case the disease manifests itself in the form of outbreaks that follow the introduction of the disease into the population. The size and duration of an outbreak can vary enormously through chance events. To capture these dynamics a stochastic formalism is needed. In this section we illustrate how the theory of branching processes can be used to describe the epidemiology of pathogens with a reproductive number smaller than one, using the epidemiology of measles as an example. We use this to explain the distribution of disease outbreaks in small island populations, and the effect of a recent decline in vaccination in the UK after a scare about vaccine safety.