Spatial-temporal basic reproduction number and dynamics for a dengue disease diffusion model

A reaction-diffusion system with free boundary is proposed to describe the transmission of the dengue disease from mosquitoes to humans. In addition to the classical basic reproduction number R-0, the spatial-temporal basic reproduction number R0F(t) is introduced to determine the persistence and eradication of the disease. Some sufficient conditions for the disease vanishing or spreading are obtained. The disease will go extinct under one of the conditions: the classical basic reproduction number R(0)1 and the spatial-temporal basic reproduction number R0 spread of the disease in the whole area is possible if for some t0. Numerical simulations are also given to illustrate the theoretical results.