## The population biology of disease

We now turn to a study of the population biology of disease. We will consider both microparasites - in which populations increase in hosts by multiplication of numbers - and macroparasites - in which populations increase in hosts by both multiplication of numbers and by growth of individual disease organisms. The age of genomics and bioinformatics makes the material in this chapter more, and not less, relevant for three reasons. First, with our increasing ability to understand type and mechanism at a molecular level, we are able to create models with a previously unprecedented accuracy. Second, although biomedical science has provided spectacular success in dealing with disease, failure of that science can often be linked to ignoring or misunderstanding aspects of evolution, ecology and behavior (Schrag and Weiner 1995, de Roode and Read 2003). Third, there are situations, as is well known for AIDS but is true even for flu (Earn et al. 2002), in which ecological and evolutionary time scales overlap with medical time scales for treatment (Galvani 2003).

To begin, a few comments and caveats. At a meeting of the (San Francisco) Bay Delta Modeling Forum in September 2004, my colleague John Williams read the following quotation from the famous American jurist Oliver Wendell Holmes: ''I would not give a fig for simplicity this side of complexity, but I would give my life for simplicity on the other side of complexity". It could take a long time to fully deconstruct this quotation but, for our purposes, I think that it means that models should be sufficiently complicated to do the job, but no more complicated than necessary and that sometimes we have to become more complicated in order to see how to simplify. In this chapter, we will develop models of increasing complexity. The building-up feeling of the progression of sections is not intended to give the impression that more complicated models are better. Rather, the scientific question is paramount, and the simplest model that helps you answer the question is the one to aim for.

Furthermore, the mathematical study of disease is a subject with an enormous literature. As before, I will point you toward the literature in the main body of the chapter and in Connections. As you work through this material, you will develop the skills to read the appropriate literature. That said, there is a warning too: disease problems are inherently nonlinear and multidimensional. They quickly become mathematically complicated and there is a considerable literature devoted to the study of the mathematical structures themselves (very often this is described by the authors as "mathematics motivated by biology''). As a novice theoretical biologist, you might want to be chary of these papers, because they are often very difficult and more concerned with mathematics than biology.

There are two general ways of thinking about disease in a population. First, we might simply identify whether individuals are healthy or sick, with the assumption that sick individuals are able to spread infection. In such a case, we classify the population into susceptible (S), infected (I ) andrecoveredorremoved (R) individuals (more details on this follow). This classification is commonly done when we think of microparasites such as bacteria or viruses. An alternative is to classify individuals according to the parasite burden that they carry. This is typically done when we consider parasitic worms. We will begin with the former (classes of individuals) and move towards the latter (parasite burden).

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