Analysis of an SI Epidemic Model with Nonlinear Incidence Rate in an Environmentally-driven Infectious Disease

Abstract

In this paper, a new model for the linking of within- and between-host with nonlinear incidence rate is constructed and investigated. We postulate that the infection process depend on the size of the infective inoculum that susceptible hosts may acquire by interacting with a contaminated environment. Because the dynamical processes associated with the within- and between-host occur on different time scales, the model behaviors can be analyzed by using a singular perturbation argument, which allows us to decouple the full model by separating the fast- and slow-systems. For the fast system, it is shown that the infection-free equilibrium Uο is locally asymptotically stable if Rvο < 1; whereas Uο is unstable if Rvο > 1, and a unique interior equilibrium  exists and it is locally asymptotically stable. For the slow system, under the condition that the fast system has a stable interior equilibrium ( i.e., Rvο > 1), there exists at least one endemic equilibrium W if Rh > 1 and βN < 1. We also prove that the endemic equilibrium W for the slow system is locally asymptotically stable if R*hο > 1. Some numerical simulations are provided to verify the theoretical results. In addition, some simple discussion is also presented at the end of the article.