A stochastic toxoplasmosis spread model between cat and oocyst with jumps process

Abstract

In this paper, Epidemic models are inevitably influenced by environmental white noise which is an important component in realism, using stochastic models can provide an additional degree of realism in comparison to their deterministic counterparts. Furthermore, it is possible for the population to confront emergency or sudden environmental changes such like chemical leak, abnormal weather, natural disaster and pestilence. In this paper, a toxoplasmosis spread model between cat and oocyst populations with independent stochastic perturbations and a jump process is proposed, the existence of global positive solution is derived. By the method of stochastic Lyapunov functions, we study its asymptotic behavior. When the perturbations about the the susceptible and infective cats are sufficiently small, as well as magnitude of the reproductive number is less than one, the infective cats, recovered cats and population oocysts decay to zero whilst the susceptible components converge to a class of explicit stationary distributions regardless of the perturbations on the recovered cats and population oocysts. When all the perturbations are small and the reproductive number is larger than one, we construct a new class of stochastic Lyapunov functions to show the positive recurrence, and our results reveal some cycling phenomena of recurrent diseases. These results mean that stochastic system has the similar property with the corresponding deterministic system when the white noise is small.