Fish-hook bifurcation branch in a spatial heterogeneous epidemic model with cross-diffusion. (August 2016) - Cai, Yongli; Wang, Weiming
Abstract: In this paper, we consider the following strongly coupled epidemic model in a spatially heterogeneous environment with Neumann boundary condition: Δ S + b S − ( m + k ( S + I ) ) S − β ( x ) S I = 0, x ∈ Ω, Δ ( ( 1 + c θ ( x ) S ) I ) + ρ b I − ( m + k ( S + I ) ) I − δ I + β ( x ) S I = 0, x ∈ Ω, ∂ n S = ∂ n I = 0, x ∈ ∂ Ω, where Ω ⊂ R n is a bounded domain with smooth boundary ∂ Ω ; b, m, k, c and δ are positive constants; β ( x ) ∈ C ( Ω ̄ ) and θ ( x ) is a smooth positive function in Ω ̄ within ∂ n θ ( x ) = 0 on ∂ 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 − m of S is sufficiently small, the disease-induced death rate δ is slightly small, and the cross-diffusion coefficient c is sufficiently large, we show that the model admits a bounded branch Γ of positive solutions, which is a monotone S -type or fish-hook-shaped curve with respect to the bifurcation parameter δ . 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.,