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We investigate a stochastic SIRS model with transfer from infectious to susceptible and nonlinear incidence rate. First, using stochastic stability theory, we discuss stochastic asymptotic stability of disease-free equilibrium of this model. Moreover, if the transfer rate from infectious to susceptible is sufficiently large, disease goes extinct. Then, we obtain almost surely exponential stability of disease-free equilibrium, which implies that noises can lead to extinction of disease. By the Lyapunov method, we give conditions to ensure that the solution of this model fluctuates around endemic equilibrium of the corresponding deterministic model in average time. Furthermore, numerical simulations show that the fluctuation increases with increase in noise intensity. Finally, these theoretical results are verified by numerical simulations. Hence, noises play a vital role in epidemic transmission. Our results improve and extend previous related results.

Stability of a Nonlinear Stochastic Epidemic Model with Transfer from Infectious to Susceptible