## The SI model

As always, it is best to begin with a simple and familiar story. Lest you think that this is too simple and familiar, it is motivated by the work of Pybus et al. (2001), published in Science in June 2001. Since this is our first example, we begin with something relatively simple.

Envision a closed population of size N and let S(t) and I(t) denote respectively the number of individuals who are susceptible to infection (susceptibles) and who are infected (infecteds) with the disease at time t. Since the population is closed, S(t) + I(t) = N, which we will exploit momentarily. New cases of the disease arise when an infected individual comes in contact with a susceptible individual. One representation of this rate of new infections is bSI, which is called the mass action formulation of transmission, and which we will discuss in more detail in the next section. Note that because the population is closed, the rate of

Figure 5.1. The solution of the SI model (Eq. (5.1)) is logistic growth if bN> vand decline of the number of infected individuals if bN < v. Parameters here are N = 500, v=0.1 and b = 2v/N or b = 0.95v/N.

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