Abstract
In this study, we build a stochastic SIR epidemic model with vertical infection and nonlinear incidence. The influence of the fluctuation of disease transmission parameters and state variables on the dynamic behaviors of the system is the focus of our study. Through the theoretical analysis, we obtain that there exists a unique global positive solution for any positive initial value. A threshold R 0 s is given. When R 0 s < 1 , the diseases can be extincted with probability one. When R 0 s > 1 , we construct a stochastic Lyapunov function to prove that the system exists an ergodic stationary distribution, which means that the disease will persist. Then, we obtain the conditions that the solution of the stochastic model fluctuates widely near the equilibria of the corresponding deterministic model. Finally, the correctness of the results is verified by numerical simulation. It is further found that the fluctuation of disease transmission parameters and infected individuals with the environment can reduce the threshold of disease outbreak, while the fluctuation of susceptible and recovered individuals has a little effect on the dynamic behavior of the system. Therefore, we can make the disease extinct by adjusting the appropriate random disturbance.
Highlights
At the beginning of 2020, a sudden epidemic (COVID-19) has disrupted people’s normal life
We are mainly concerned about the influence of stochastic factors on the behavior of the epidemic SIR model with a nonlinear incidence and vertical transmission
The stochastic factors mainly include the disease transmission coefficient and state variables affected by white noise
Summary
At the beginning of 2020, a sudden epidemic (COVID-19) has disrupted people’s normal life. The process of disease transmission is inevitably affected by random factors In this way, the stochastic differential model is more suitable than the deterministic model. Ere are many literatures on the stochastic differential model of infectious diseases [7,8,9,10,11,12,13,14,15,16,17,18,19,20]. In literature [8], Aadil and Omari built the SIRS stochastic differential model with parameter perturbation, vaccination of recruitment susceptible, and nonlinear incidence rate. Zhou et al [16] studied an SIR model with the bilinear infection rate and stochastic perturbation of parameter and state variables and got the conditions of survival and stationary distribution. By Ito’s formula [24], dV(x, t) LV(x, t)dt + Vx(x, t)g(x, t)dB(t). (19)
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