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 epidemic—an outbreak of a non-fatal infection directly transmitted between hosts in a closed population, without host demographic changes (May, 1980):
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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