Fish-hook bifurcation branch in a spatial heterogeneous epidemic model with cross-diffusion

In this paper, we consider the following strongly coupled epidemic model in a spatially heterogeneous environment with Neumann boundary condition: { Delta s + bs - (m + k (S + I))S - beta(x)SI = 0, x epsilon Omega, Delta((1+c theta(x)S)I) + rho bI - (m + k (S + I))I - delta I + beta (x)SI = 0, x epsilon Omega, partial derivative S-n = partial derivative I-n = 0, x epsilon partial derivative Omega, where Omega subset of R-n is a bounded domain with smooth boundary partial derivative Omega; b, m, k, c and delta are positive constants; beta(x) epsilon C((Omega) over bar) and theta(x) is a smooth positive function in ((Omega) over bar) within partial derivative(n)theta(x) = 0 on partial derivative Omega. 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 delta is slightly small, and the cross-diffusion coefficient c is sufficiently large, we show that the model admits a bounded branch Gamma of positive solutions, which is a monotone S-type or fish-hook-shaped curve with respect to the bifurcation parameter delta. 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. (C) 2015 Elsevier Ltd. All rights reserved.