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: {ΔS+bS−(m+k(S+I))S−β(x)SI=0,x∈Ω,Δ((1+cθ(x)S)I)+ρbI−(m+k(S+I))I−δI+β(x)SI=0,x∈Ω,∂nS=∂nI=0,x∈∂Ω, where Ω⊂RnΩ⊂Rn is a bounded domain with smooth boundary ∂Ω∂Ω; b,m,k,cb,m,k,c and δδ are positive constants; β(x)∈C(Ω̄) and θ(x)θ(x) is a smooth positive function in Ω̄ within ∂nθ(x)=0 on ∂Ω∂Ω. 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−ma:=b−m of SS is sufficiently small, the disease-induced death rate δδ is slightly small, and the cross-diffusion coefficient cc is sufficiently large, we show that the model admits a bounded branch ΓΓ of positive solutions, which is a monotone S-type or fish-hook-shaped curve with respect to the bifurcation parameter δδ. 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.