Spatial dynamics for an SIRE epidemic model with diffusion and prevention in contaminated environments

Abstract

In this paper, a reaction–diffusion SIRE epidemic model in contaminated environments is proposed, in which the effect of protection for susceptible individuals is included by the nonlinear incidence functions b ( S ) E $b(S)E$ and g ( S ) I $g(S)I$ . When the space is heterogeneous, the basic reproduction number R 0 $\mathcal {R}_{0}$ is derived, by which we find that if R 0 ≤ 1 $\mathcal {R}_{0}\le 1$ , the disease-free steady state is globally asymptotically stable, while R 0 > 1 $\mathcal {R}_{0}>1$ , the disease is uniform persistent. Furthermore, when R 0 > 1 $\mathcal {R}_{0}>1$ and additional conditions hold, the global asymptotic stability of special endemic steady state is obtained in homogeneous space. Finally, the theoretical results are validated by numerical simulations, some open questions are illustrated.