On the Dynamics of a Diffusive Foot-and-Mouth Disease Model with Nonlocal Infections

Source 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 $\mathcal{R}_0$: if $\mathcal{R}_0<1$, then the disease-free equilibrium $E_0$ is globally asymptotically stable; otherwise $E_0$ is unstable, and there exists an endemic equilibrium $E^\ast$. Numerical simulations confirm the theoretical results and suggest that reducing the direct contact rate $\beta_1$ and the indirect contact rate $\beta_2$ is important in relieving FMD outbreaks. More importantly, we obtain the effect of diffusion on the time from initial values to steady state when $\mathcal{R}_0>1$: 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 $\mathcal{R}_0$ 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 $\mathcal{R}_0 (>1)$ and the equilibrium value of the infectious individuals $I^\ast$ in terms of $\beta_1$ and $\beta_2,$ it is found that the $(\beta_1, \beta_2)$-plane is divided into two regions by the intersection of two parameter-related surfaces; the sensitivity of $\mathcal{R}_0$ and $I^\ast$ varies when $\beta_1$ and $\beta_2$ belong to different regions. When the values of both $\beta_1$ and $\beta_2$ are very large or very small, $\beta_1$ 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.