Repetitive Infection Spreading and Directed Evolution in the Susceptible–Infected–Recovered–Susceptible Model

We study two simple mathematical models of an epidemic. First, we study the repetitive infection spreading in a simplified susceptible–infected–recovered–susceptible (SIRS) model including the effect of the decay of the acquired immunity. The model is an intermediate of the SIRS model including the recruitment and death terms and the SIR model in which the recovered population is assumed to be never infected again. When the decay rate δ of the acquired immunity is sufficiently low, multiple infection spreading occurs in spikes. The model equation can be reduced to a map when the decay rate δ is sufficiently low, and the spike-like multiple infection spreading is reproduced in the mapping. The period-doubling bifurcation and chaos are found in the simplified SIRS model with seasonal variations. The nonlinear phenomena are reproduced using the map. Next, we study coupled SIRS equations for the directed evolution where the mutation is expressed with a diffusion-type term. A type of reaction–diffusion equation is derived by the continuum approximation for the infected population I. The reaction–diffusion equation with the linear dependence of infection rate on the type space has an exact Gaussian solution with a time-dependent average and variance. The propagation of the Gaussian pulse corresponds to the successive transitions of the dominant variant.