Propagation dynamics for a time-periodic reaction–diffusion SI epidemic model with periodic recruitment

Abstract

The paper is devoted to the study of the asymptotic speed of spread and traveling wave solutions for a time-periodic reaction–diffusion SI epidemic model which lacks the comparison principle. By using the basic reproduction number $$R_0$$\n of the corresponding periodic ordinary differential system and the minimal wave speed $$c^*$$\n , the spreading properties of the corresponding solution of the model are established. More precisely, if $$R_{0} \\leqslant 1$$\n , then the solution of the system converges to the disease-free equilibrium as $$t \\rightarrow \\infty $$\n and if $$R_0 > 1$$\n , the disease is persistent behind the front and extinct ahead the front. On the basis of it, we then analyze the full information about the existence and nonexistence of traveling wave solutions of the system involved with $$R_0$$\n and $$c^*$$\n .