Influence of cross-diffusion on the fecally-orally epidemic model with spatial heterogeneity

A strongly coupled cooperative parabolic system, which describes fecally-orally epidemic model with cross-diffusion in a heterogeneous environment, was formulated and analyzed. The basic reproduction number R0 , which serves as a threshold parameter that predicts whether the coexistence will exist or not, is introduced by the next infection operator and the related eigenvalue problems. By applying upper and lower solutions method, we present the sufficient conditions for the existence of the coexistence solution. The true positive solutions can also be obtained by monotone iterative method. Our results imply that the fecally-orally epidemic model with cross-diffusion admits at least one coexistence solution when the basic reproduction number exceeds one and the cross-diffusion coefficient is sufficiently small, while no coexistence exists when the basic reproduction number is smaller than one or the cross-diffusion coefficient is large enough. Finally, some numerical simulations are exhibited to confirm our analytical findings.