ON THE DYNAMICS OF A DIFFUSIVE FOOT-AND-MOUTH DISEASE MODEL WITH NONLOCAL INFECTIONS

Abstract

Foot-and-mouth disease (FMD) is an acute and highly contagious infectious disease of cloven-hoofed animals. In order to reveal the transmission dynamics and explore effective control measures of FMD, we formulate a diffusive FMD model with a fixed latent period and nonlocal infections. The threshold dynamics of the FMD model are determined by using the basic reproduction number R0: if R0 < 1, then the disease-free equilibrium E0 is globally asymptotically stable; otherwise E0 is unstable, and there exists an endemic equilibrium E*. Numerical simulations confirm the theoretical results and suggest that reducing the direct contact rate β1 and the indirect contact rate β2 is important in relieving FMD outbreaks. More importantly, we obtain the effect of diffusion on the time from initial values to steady state when R0 > 1: at a low infection level, the faster the infectious individuals and virus diffuse, the faster the disease reaches the steady state. However, at a high infection level (i.e., the value of R0 is relatively large), the influence of diffusion on time from initial values to steady state is more complicated, but at least it is certain that the time will be shortened overall. By carrying out some sensitivity analysis of R0(> 1) and the equilibrium value of the infectious individuals I* in terms of β1 and β2, it is found that the (β1, β2)-plane is divided into two regions by the intersection of two parameter-related surfaces; the sensitivity of R0 and I* varies when β1 and β2 belong to different regions. When the values of both β1 and β2 are very large or very small, β1 plays a more significant role in the transmission of FMD. These results indicate that stamping out the infected individuals and blocking the epidemic spots and areas are effective in preventing and controlling the spread of FMD.