Coexistence of a cross-diffusive dengue fever model in a heterogeneous environment

A spatio-temporal SIS-SI dengue model with cross-diffusion is formulated as the movements of human and mosquitoes have been intricately linked with the spread of dengue fever. To highlight the impacts of cross-diffusion on the dynamical processes, we focus on the nonnegative steady-state solutions of the dengue model, that is, the coexistence of the corresponding strongly-coupled elliptic system. By means of the relevant eigenvalue problem, we investigate some properties of the basic reproduction number to the model, and further present the existence of coexistence solutions. Our results imply that the virus carried in human and that in mosquitoes can coexist if the basic reproduction number is greater than one and the extent of cross-diffusion is small enough. The final numerical simulations and epidemiological explanations make our analytical findings easier to understand.