Analysis of a Local Diffusive SIR Model with Seasonality and Nonlocal Incidence of Infection

Abstract

For infectious diseases such as influenza and brucellosis, the susceptibility of a susceptible highly depends on the distance from each adjacent infectious individual. Such a propagation mechanism is often modeled by a nonlocal incidence with a kernel function $K(x)$, whose support determines the effective infection area. This nonlocal incidence of infection, together with important seasonal factors, leads to a periodic kernel function $K(t,x)$ and periodic parameters in a diffusive susceptible-infectious-recovered (SIR) model equipped with homogeneous Neumann boundary conditions. We first study the global well-posedness and the dissipativity of solutions. This is followed by the investigation of the global dynamics in terms of the basic reproduction number $\mathcal{R}_0$ defined to be the spectral radius of the next infection operator. The following results are shown rigorously: (1) If $\mathcal{R}_0<1$, the disease-free periodic solution is globally asymptotically stable. (2) If $\mathcal{R}_0>1$, the model is uniformly persistent and admits an endemic periodic solution. When the support size of $K(t,x)$ tends to 0, the basic reproduction number takes the $\mathcal{R}_0$ of the local infection model as the limit. Without seasonality, $\mathcal{R}_0$ and the endemic size of $I(t)$ (i.e., the total number of infected individuals at time $t$ in the studied area) both decrease when the support size of $K(x)$ increases; however, there is no uniform result for the value and time of the disease outbreak peak. We also explore the integrated impact of seasonality and the infection rate $\beta$ on $I(t)$. In addition, when the support size of $K(t,x)$ increases, the time difference between the first peak and the last peak of $I(t,x)$ decreases, and when the support size of $K(t,x)$ is large enough, the disease outbreak occurs almost simultaneously in the whole region. Moreover, due to seasonality not all locations experience a major disease outbreak in the early stage of disease transmission when the support size of $K(t,x)$ is relatively small.