## Microparasite models the simple epidemic

The essential functional feature of microparasites is that they reproduce directly within their hosts (Anderson and May, 1991). Thus, for many diseases, we can capture the essential dynamics via a compartmental model, where hosts are divided between different infection categories. The most studied form is the family of susceptible ! infected! recovered (SIR) models for strongly immunizing infections. Individuals are recruited into a previously uninfected susceptible class, S (often via births). They then become infected (I) by contact with infected individuals, before finally moving into an immune-recovered class, R (Bartlett, 1956; May, 1980; Mollison, 1995). The simplest realization of the SIR model describes the dynamics of the simple epidemicâ€”an outbreak of a non-fatal infection directly transmitted between hosts in a closed population, without host demographic changes (May, 1980):

dR dt

The model assumes a well-mixed population, with homogeneous random mixing between individuals. As with the simplest predator-prey models (see Chapter 3 in this volume), this leads to a bilinear (I x S) interaction term for the infection process, controlled by the infection parameter, p. This form assumes that transmission depends upon the density of susceptible and infectious individuals (as discussed further below; Begon et al., 2002). For simplicity (eg Bjornstad and Grenfell (2002)) we allow b to subsume the rate at which susceptible and infectious individuals make sufficiently close contact to allow a chance of transmission, and the probability of transmission of the infectious agent given such a contact. After infection, individuals move into the recovered class, R, at the recovery rate, g.

A key epidemiological parameter leaps out of this model: consider an epidemic sparked by a single infected individual introduced into a total susceptible population of density N. At the start of the epidemic (when S ffi N) the level of infection, I, can only increase (dI/dt > 0) if > g; this leads naturally to a definition of the basic reproduction ratio or basic reproductive number of infection:

Since 1/ g is the average duration of infection, R0 can be interpreted as the total number of secondary cases produced by one infected individual when introduced into a population of N susceptibles. As the epidemic proceeds, and more individuals recover and are therefore immune, the effective reproduction ratio, R, declines with the proportion of susceptibles, s = S/N; that is, R = R0s = R0S/N. The basic and effective reproduction ratios provide a powerful framework for exploring the dynamics and control of epidemics. In particular, an infectious agent can only invade a susceptible population if R0 > 1; if this criterion is satisfied the subsequent epidemic will only continue to increase while R > 1; when the proportion of susceptibles is reduced by the epidemic below 1/R0, the number of cases must start to decline. An endemic infection, steadily maintained at a constant level, will have an effective reproduction ratio R = 1; but R = R0s*, where s* is the equilibrium fraction who are susceptible, whence R0 can be estimated as R0 = 1/s*.

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