Stationary distribution and density function analysis of SVIS epidemic model with saturated incidence and vaccination under stochastic environments

Abstract

In this article, we study the dynamical properties of susceptible-vaccinated-infected-susceptible (SVIS) epidemic system with saturated incidence rate and vaccination strategies. By constructing the suitable Lyapunov function, we examine the existence and uniqueness of the stochastic system. With the help of Khas'minskii theory, we set up a critical value [Formula: see text] with respect to the basic reproduction number [Formula: see text] of the deterministic system. A unique ergodic stationary distribution is investigated under the condition of [Formula: see text]. In the epidemiological study, the ergodic stationary distribution represents that the disease will persist for long-term behavior. We focus for developing the general three-dimensional Fokker-Planck equation using appropriate solving theories. Around the quasi-endemic equilibrium, the probability density function of the stochastic system is analyzed which is the main theme of our study. Under [Formula: see text], both the existence of ergodic stationary distribution and density function can elicit all the dynamical behavior of the disease persistence. The condition of disease extinction of the system is derived. For supporting theoretical study, we discuss the numerical results and the sensitivities of the biological parameters. Results and conclusions are highlighted.

Keywords: Density function analysis; Ergodic stationary distribution; Extinction; Fokker–Planck equation; Stochastic SVIS epidemic model.

Fig. 1

Population trajectories in the system of (i) Deterministic system (ii) Stochastic system under the noise intensities ${\tau }_1=0.08,{\tau }_2=0.02,{\tau }_3=0.01.$

Fig. 2

Extinction of the compartment of $I\left(t\right)$ under the noise ${\tau }_1=0.08,{\tau }_2=0.02,{\tau }_3=0.01$ and the other parameters $\left(p,q,\lambda ,k_1,\psi ,{\alpha }_1,{\alpha }_2,{\beta }_1,{\beta }_2\right)=\left(2,50,0.00009,0.65,0.02,0.9,0.03,0.2,0.05\right)$

Fig. 3

Profile of the compartment $S\left(t\right),V\left(t\right),I\left(t\right)$ in deterministic system and stochastic system with intensities ${\tau }_1=0.08,{\tau }_2=0.02$, ${\tau }_3=0.01$ and other parameters are $\left(p,q,\lambda ,k_1,\psi ,{\alpha }_1,{\alpha }_2,{\beta }_1,{\beta }_2\right)=\left(2,50,0.9,0.65,0.02,0.9,0.03,0.2,0.05\right).$

Fig. 4

Effect of the noise intensities ${\tau }_1,{\tau }_2,{\tau }_3$ on the compartment of $V\left(t\right)$

Fig. 5

Effect of the noise intensities ${\tau }_1,{\tau }_2,{\tau }_3$ on the compartment of $I\left(t\right)$

Fig. 6

Effect of all the compartment $\left(S\left(t\right),V\left(t\right),I\left(t\right)\right)$ for the parameter $q$

Fig. 7

Impact of all the compartment for the parameter $\lambda$

Fig. 8

Impact of all the compartment for the parameter $k_1$