Evolutionary dynamics in an SI epidemic model with phenotype-structured susceptible compartment

We present an SI epidemic model whereby a continuous variable captures variability in proliferative potential and resistance to infection among susceptibles. The occurrence of heritable, spontaneous changes in these phenotype and the presence of a fitness trade-off between resistance to infection and proliferative potential are incorporated into the model. The model comprises an ODE for the number of infected individuals that is coupled with a partial integrodifferential equation for the population density of susceptibles through an integral term. The expression for the basic reproduction number $\mathcal{R}_0$ is derived, the disease-free and endemic equilibrium of the model are characterised and a threshold theorem is proved. Analytical results are integrated with numerical simulations of a calibrated version of the model based on the results of artificial selection experiments in a host-parasite system. The results of our mathematical study disentangle the impact of different evolutionary parameters on the spread of infectious diseases and the consequent phenotypic adaption of susceptible individuals. In particular, these results provide a theoretical basis for the observation that infectious diseases exerting stronger selective pressures on susceptibles and being characterised by higher infection rates are more likely to spread. Moreover, our results indicate that spontaneous phenotypic changes in proliferative potential and resistance to infection can either promote or prevent the spread of diseases depending on the strength of selection acting on susceptible individuals prior to infection. Finally, we demonstrate that, when an endemic equilibrium is established, higher levels of resistance to infection and lower degrees of phenotypic heterogeneity are to be expected in the presence of infections which are characterised by lower rates of death and exert stronger selective pressures.