## The unbeatable ESS level of virulence

We will now look at the problem in a slightly different manner, from the perspective of invasions. Recall that the dynamics of the infected

Figure 5.7. Marginal value construction used to find the optimal level of virulence.

irulence m

Figure 5.7. Marginal value construction used to find the optimal level of virulence.

irulence m individuals are dl/dt = blS — (v + m)1 from which we conclude that the steady state level of susceptibles is S(m) = (v + m)/b(m). Now let us consider an invader, which is rare and which uses an alternative level of virulence m. Because the invader is rare, we assume that it has no effect on the steady state level of the susceptible population, and we ask "when will the invader increase?''. Under these assumptions, if I denotes the number of invaders, the dynamics of the invader are

and we now substitute for the steady state level of susceptibles and factor out the number of infecteds to obtain

and the invader will spread if the term in brackets is greater than 0. This is true when b(m~)((v + m)/b(m)) > (v + m~), which is, of course, the same as b(m~)/(v + m~) > b(m)/(v + m). We thus conclude that the strategy that maximizes b(m)/(v + m) is unbeatable because it cannot be invaded. This is exactly the same condition that arises in the maximization of R0. In other words, the strategy that optimizes the basic reproductive rate is also unbeatable and cannot be invaded. This is a very interesting result, in part because optimality and ESS analyses may

Figure 5.8. The infection process modeled by Koella and Restif (2001) in their study of the coevolution of virulence and host age at maturity. The host becomes infected by disease propagules (such as spores) independent of the density of other infected individuals.

ortality rate ß

Figure 5.8. The infection process modeled by Koella and Restif (2001) in their study of the coevolution of virulence and host age at maturity. The host becomes infected by disease propagules (such as spores) independent of the density of other infected individuals.

 Susceptible individuals nfection rate nfected individuals
0 0