Propagation dynamics of a nonautonomous epidemic system with nonlocal diffusion

Abstract

This paper investigates the spreading speeds and generalized traveling wave solutions for a time-dependent epidemic system with nonlocal dispersal. In this nonautonomous epidemic system, both the nonlocal dispersal kernel and the coefficients in reaction terms are general time heterogeneous and possess uniform mean values. To characterize the spreading speed of the system, we first establish the spreading speed of a scalar nonautonomous KPP equation. Then, using a derived key pointwise estimate, we compare the solution of the system with that of the scalar KPP equation and thus obtain the spreading speed of the system. Combining with the spreading speed results, we demonstrate the nonexistence of generalized traveling wave solutions with small wave speeds in average sense. By constructing proper super and sub solutions, we establish the existence of generalized traveling wave solutions and explore the limiting behavior of the wave profile using the uniform persistence theory.