Asymptotic speed of spread and traveling waves for a nonlocal epidemic model
The authors consider the following reaction-diffusion system, which describes the dynamics of a nonlocal epidemic disease: \[ \begin{cases} \partial_t u_1 (t,x) = d\Delta u_1 (t,x) - a_{11}u_1 (t,x) + \int_{\mathbb{R}^n} K(x-y) u_2 (t,y)\,dy, \\ \partial_t u_2 (t,x) = - a_{22}u_2 (t,x) + g(u_1 (t,x)),\\ u_1 (0,x) = \phi_1 (x)\geq 0,\;\;\;u_2 (0,x) = \phi_2 (x)\geq 0, \;\;x\in {\mathbb{R}^n}, \end{cases}\tag{1} \] where \(u_1 (t,x), u_2 (t,x)\) are, respectively, the spatial density for infectious agents and infective human population at time \(t \geq 0. \) \( d, a_{11},\) and \(a_{22}\) are positive constants, \(g\) is the force of infection, and \( K \) is an isotropic kernel. The authors used the existing theory to obtain the asymptotic speeds of spread for solutions and minimal wave speeds for monotone traveling waves of problem (1). Furthermore, numerical simulations are presented.
scientific article; zbMATH DE number 5001894