## The SIRS model of endemic diseases

We now modify the basic S/R model to assume that recovered individuals may lose resistance to the disease and thus become susceptible again, but continue to assume that the population is closed. Assuming that the rate at which resistance to the disease is lost isf the dynamics of susceptible, infected, and recovered individuals becomes

dt dR

One possible steady state for this system is / = R = 0 and S = N, in which case we conclude that the disease is extirpated from the population. If this is not the case, we then set R = N — S — / and work with the dynamics of susceptible and infected individuals:

The number of infected individuals is at a steady state if S = v/b. We then set dS/dt = 0 and solve for the steady state number of infected individuals (this is why the assumption of a closed population is such a nice one to make):

f (N — S) bS + f and if we evaluate this at the steady number of susceptible individuals, we obtain Figure 5.5. The phase plane for the SIRS model for the case in which the disease is predicted to be endemic.

so that we conclude the steady number of infecteds is positive if N> v/b (a quantity which should now be familiar). That is, we have determined a condition for endemicity of the disease, in the sense that the steady state number of infected individuals is greater than 0.

The next question concerns the dynamics of the disease. In Figure 5.5, I show the phase plane for the case in which the disease is predicted to be endemic. The phase plane suggests that we should, in general, expect oscillations in the case of an endemic disease - that is periodic outbreaks that are not caused by anything other than the fundamental population biology of the disease.

Furthermore, from this analysis we conclude that, although whether the disease is endemic or not depends only upon the ratio v/b and the size of the population N, the level of endemicity (determined by the steady state number of infected individuals) will also depend, as Eq. (5.17) shows us, upon the ratio v/f Through this analysis, we thus learn what critical parameters to measure in the study of an endemic disease.

A numerical example is found in the next section.

Figure 5.5. The phase plane for the SIRS model for the case in which the disease is predicted to be endemic.

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