Repetitive Infection Spreading and Directed Evolution in the Susceptible-Infected-Recovered-Susceptible Model
Journal of the Physical Society of Japan(2024)
摘要
We study two simple mathematical models of the epidemic. At first, we study
the repetitive infection spreading in a simplified SIRS model including the
effect of the decay of the acquired immune. The model is an intermediate model
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 δof the immune is sufficiently small, the multiple
infection spreading occurs in spikes. The model equation can be reduced to be a
map when the decay rate δis sufficiently small, 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
variation. The nonlinear phenomena are reproduced by the map. Next, we study
coupled SIRS equations for the directed evolution where the mutation is
expressed with a diffusion-type term. A kind 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.
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