Coevolutionary Dynamics of Host Immune and Parasite Virulence Based on an Age-Structured Epidemic Model

Hosts can activate a defensive response to clear the parasite once being infected. To explore how host survival and fecundity are affected by host-parasite coevolution for chronic parasitic diseases, in this paper, we proposed an age-structured epidemic model with infection age, in which the parasite transmission rate and parasite-induced mortality rate are structured by the infection age. By use of critical function analysis method, we obtained the existence of the host immune evolutionary singular strategy which is a continuous singular strategy (CSS). Assume that parasite-induced mortality begins at infection age [Formula: see text] and is constant v thereafter. We got that the value of the CSS, [Formula: see text], monotonically decreases with respect to infection age [Formula: see text] (see Case (I)), while it is non-monotone if the constant v positively depends on the immune trait c (see Case (II)). This non-monotonicity is verified by numerical simulations and implies that the direction of immune evolution depends on the initial value of immune trait. Besides that, we adopted two special forms of the parasite transmission rate to study the parasite's virulence evolution, by maximizing the basic reproduction ratio [Formula: see text]. The values of the convergence stable parasite's virulence evolutionary singular strategies [Formula: see text] and [Formula: see text] increase monotonically with respect to time lag L (i.e., the time lag between the onset of transmission and mortality). At the singular strategy [Formula: see text] and [Formula: see text], we further obtained the expressions of the case mortalities [Formula: see text] and how they are affected by the time lag L. Finally, we only presented some preliminary results about host and parasite coevolution dynamics, including a general condition under which the coevolutionary singular strategy [Formula: see text] is evolutionarily stable.