Dynamics and stationary distribution of a stochastic SIRS epidemic model with a general incidence and immunity

Abstract

Infected individuals often obtain or lose immunity after recovery in medical studies. To solve the problem, this paper proposes a stochastic SIRS epidemic model with a general incidence rate and partial immunity. Through an appropriate Lyapunov function, we obtain the existence and uniqueness of a unique globally positive solution. The disease will be extinct under the threshold criterion. We analyze the asymptotic behavior around the disease-free equilibrium of a deterministic SIRS model. By using the Khasminskii method, we prove the existence of a unique stationary distribution. Further, solutions of the stochastic model fluctuate around endemic equilibrium under certain conditions. Some numerical examples illustrate the theoretical results.

Keywords: General incidence; Partial immunity; SIRS epidemic model; Stationary distribution.

Figure 1

Dynamical curves of compartments: (a) S, (b) I, and (c) R under $\mathcal{R}<1$ and ${\mathcal{R}}^s<1$

Figure 2

Dynamical curves of compartments: (a) S, (b) I, and (c) R under $\mathcal{R}>1$ and ${\mathcal{R}}^s<1$

Figure 3

(a) Time-series phases of solutions ($S(t)$, $I(t)$, $R(t)$) for model (2); (b) Time-series phases of solutions ($S(t)$, $I(t)$, $R(t)$) for model (3). The parameters are taken from Example 2

Figure 4

Dynamical curves of compartments: (a) S, (b) I, (c) R. The frequency histograms and marginal density functions of compartments: (d) S, (e) I, (f) R

Figure 5

The effect of partial immunity on compartment I: (a) model (2) and (b) model (3)

Figure 6

The effect of partial immunity p with respect to time t for compartments: (a) S, (b) I, and (c) R in model (2)

Figure 7

The effect of partial immunity p with respect to time t for compartments: (a) S, (b) I, and (c) R in model (3)