## The optimal level of virulence

Recall that we closed the section on the SIR model with a discussion of the basic reproductive rate for a disease when the disease related mortality rate is m and recovery rate is v

v + m where I have made explicit the dependence of the contagion on the virulence, still assumed to have the shape as in Figure 5.4. How might natural selection act on the reproductive rate of a disease? A reasonable starting point is to assume that the disease strain that spreads the fastest (i.e. has the greatest value of R0(m)) will be the most prevalent. If we accept this assumption as a starting point, we then ask for the value ofm that maximizes R0(m) given by Eq. (5.20).

Now you should compare Eq. (5.20) with Eq. (1.6). They are essentially the same equation: a saturating function of a variable divided by that variable plus a constant. Thus, from the marginal value construction in Chapter 1, we instantly know how to find the optimal level of virulence. First, we plot b(m) versus m. Second, we draw the tangent line from (—v, 0) to the curve b(m). Third, we read the predicted optimal level of virulence from the intersection of the tangent line and the x-axis (Figure 5.7). Thus, the marginal value theorem, developed for foraging in patchy environments, is also useful here.

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