Fish hook bifurcation branch in a spatial heterogeneous epidemic model with cross diffusion
Abstract In this paper, we consider the following strongly coupled epidemic model in a spatially heterogeneous environment with Neumann boundary condition: D S b S m k S I S b x S I 0 x O D 1 c th x S I r b I m k S I I d I b x S I 0 x O n S n I 0 x O where O R n is a bounded domain with smooth boundary O b m k c and d are positive constants; b x C O and th x is a smooth positive function in O within n th x 0 on O The main result is that we have derived the set of positive solutions (endemic) and the structure of bifurcation branch: after assuming that the natural growth rate a b m of S is sufficiently small, the disease induced death rate d is slightly small, and the cross diffusion coefficient c is sufficiently large, we show that the model admits a bounded branch G of positive solutions, which is a monotone S type or fish hook shaped curve with respect to the bifurcation parameter d One of the most interesting findings is that the multiple endemic steady states are induced by the cross diffusion and the spatial heterogeneity of environments together.