## The coevolution of virulence and host response

As the virulence of the parasite evolves, the host response may also change. Thus, we have a case of coevolution of parasite virulence and host response. Here, we develop, in a slightly different manner, a model due to Koella and Restif (2001) and I encourage you to seek out and read the original paper.

For the host, we assume a semelparous organism following von Bertalanffy growth with growth rate k, asymptotic size Lm, disease independent mortality and allometric parameter fl connecting size at maturity and reproductive success. With these assumptions, we know from Chapter 2 that if age at maturity is t, then an appropriate measure of fitness is F(t) / e—^ (1 — e -ktand we also know from Chapter 2 that the optimal age at maturity is t^ = (1/k) log [(u + flk) .

For the disease, we assume horizontal transmission between disease propagules and susceptible hosts at rate l that is independent of the number of infected individuals (think of a disease transmitted by propagules such as spores). The virulence of the disease can be characterized by an additional level of host mortality a, so that the mortality rate of infected hosts is ^ + a. (Figure 5.8). We then immediately predict that hosts that are infected will reproduce at a different age, given by tm =1 log(

Determine the corresponding values for size at maturity.

Our first prediction is that if there are no constraints acting on age at maturity, then infected individuals will mature at earlier age (and smaller size) than non-infected individuals. However, suppose all individuals are forced to use the same age at maturity (e.g. the physiological machinery required for maturity is slow to develop, so that the age of maturity has to be set long in advance of potential infection). We could then ask, as do Koella and Restif (2001), what is the best age at maturity, taking into account the potential effect of infection on the way to maturation.

In that case, we allow the age at maturity to be different from either of the values determined above and proceed as follows. First, we will determine the optimal level of virulence for the pathogen, given that the age of maturity is tm. This optimal level of virulence can be denoted as a*(tm.). Given the optimal level of virulence in response to an age at maturity, we then allow the host to determine the best age at maturity. This procedure, in which the age at maturity is fixed, the pathogen's optimal response to an age at maturity is determined, and then the host's choice of optimal age at maturity is then determined is a special form of dynamic game theory called a leader-follower or Stackelberg game (Basar and Olsder 1982). The general way that these games are approached is to first find the optimal response of the follower (here the parasite), given the response of the leader (here the host), and then find the optimal response of the leader, given the optimal response of the follower. So, let's begin.

If hosts mature and reproduce at age tm, then they may become infected at any time r between 0 and tm. Horizontal transmission of the disease will then be determined by transmission rate l and the length of time that that individual is infected. To find the latter, we set

D(tm,r) = Eflength of time an individual is alive, given infection at r}

This interval is composed of two kinds of individuals: those who survive to reproduction (and thus whose remaining lifetime is tm — r) and those who die before reproduction. We thus conclude

+ E{lifetime|death before tm, infection at r}

Since the mortality rate of an infected individual is p + a, the probability that an individual dies before age s is 1 — exp(— (p + a)s) and the probability density for the time of death is (p + a)exp(— (p + a)s). Consequently, the expected lifetime of individuals who die before tm and who are infected at age r is (p + a) J^111 r^ te—(p+a)idt. The integral in this expression can be evaluated using integration by parts (or the 1 — F(z) trick mentioned in Chapter 3).

Evaluate the integral, and combine it with the term corresponding to individuals who survive to reproduction to show that

Now this equation is conditioned on the time at which an individual becomes infected, so to find the average duration of the disease, we need to average over the distribution of the time of infection. Since the rate of horizontal transmission is 1, the probability that an individual is infected in the interval (t, t + dT) is 1e-lTdT. Consequently, the average duration of infection, when individuals reproduce at age tm is

To analyze the evolution of virulence, Koella and Restif separate transmission of disease propagules by contact between susceptible and infected hosts (with rate 1) and the efficiency of the transmission, which they denote by e(a) and which is assumed to have the same kind of form as b(m) that we encountered previously: e(a) = emaxa/(a + a0), where £max is the maximum efficiency and a0 is the level of virulence at which half of this efficiency is reached. We then combine Eq. (5.26) with the efficiency to obtain a measure of the success of horizontal transmission when the host matures at age tm and the level of virulence is a:

and we assume that natural selection has acted on virulence to maximize H(tm, a) with respect to the level of virulence a.

In Figure 5.9, I show the optimal level of virulence (i.e. that maximizes H(tm, a)) as a function of the age at which the host reproduces. The results accord with the intuition that we have developed thus far: slowly developing hosts select for reduced virulence in parasites because there is more time for the transmission of the disease. Let us denote the curve in Figure 5.9 by a*(tm), to remind ourselves that it is the optimal level of parasite virulence when the hosts mature at age tm.

We now turn to the computation of the optimal age of maturity for the hosts. Since we have assumed a semelparous host, the appropriate measure of fitness is expected lifetime reproductive success. Imagine a cohort of hosts, with initial population size N, and in which all individuals begin susceptible. At a later time, the population will consist of

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