Persistence and periodic orbits for an sis model in a polluted environment

Usually, man is infected with some kinds of epidemic disease since they live in a polluted environment [1–8], such as air pollution (e.g., pulmonary tuberculosis), or water pollution (e.g., snail fever). These kinds of toxicant are generated by polluted biological (e.g., degradation of forests, Creutzfeldt-Jakob disease result from bovine spongiform encephalopathy (BSE)), physical (e.g., nuclear radiation, syndrome of the Gulf War), or chemical environment (e.g., petroleum leaking, dioxin event in Belgium). As we know, the environmental pollution has been a very serious global problem, which may influence the spread of infectious diseases, and hence, has big effects on human health. In this paper, we study an SIS (susceptible/infected/susceptible) epidemic model with toxicology, using the Brouwer fixed-point theorem we show the existence of periodic solution of such a system, we also prove the global attraction of this solution, and we obtain the threshold between extinction and weakly persistent for the infected class.