Stationary distribution and extinction of a stochastic staged progression AIDS model with staged treatment and second-order perturbation

Transmission dynamics of symptom-dependent HIV/AIDS models

In this study, we proposed two, symptom-dependent, HIV/AIDS models to investigate the dynamical properties of HIV/AIDS in the Fujian Province. The basic reproduction number was obtained, and the local and global stabilities of the disease-free and endemic equilibrium points were verified to the deterministic HIV/AIDS model. Moreover, the indicators $ R_0^s $ and $ R_0^e $ were derived for the stochastic HIV/AIDS model, and the conditions for stationary distribution and stochastic extinction were investigated. By using the surveillance data from the Fujian Provincial Center for Disease Control and Prevention, some numerical simulations and future predictions on the scale of HIV/AIDS infections in the Fujian Province were conducted.

Stationary distribution and extinction of a stochastic HIV/AIDS model with nonlinear incidence rate

This paper studies a stochastic HIV/AIDS model with nonlinear incidence rate. In the model, the infection rate coefficient and the natural death rates are affected by white noise, and infected people are affected by an intervention strategy. We derive the conditions of extinction and permanence for the stochastic HIV/AIDS model, that is, if $ R_0^s < 1, $ HIV/AIDS will die out with probability one and the distribution of the susceptible converges weakly to a boundary distribution; if $ R_0^s > 1 $, HIV/AIDS will be persistent almost surely and there exists a unique stationary distribution. The conclusions are verified by numerical simulation.

Stationary distribution and probability density function analysis of a stochastic Microcystins degradation model with distributed delay

A stochastic Microcystins degradation model with distributed delay is studied in this paper. We first demonstrate the existence and uniqueness of a global positive solution to the stochastic system. Second, we derive a stochastic critical value $ R_0^s $ related to the basic reproduction number $ R_0 $. By constructing suitable Lyapunov function types, we obtain the existence of an ergodic stationary distribution of the stochastic system if $ R_0^s > 1. $ Next, by means of the method developed to solve the general four-dimensional Fokker-Planck equation, the exact expression of the probability density function of the stochastic model around the quasi-endemic equilibrium is derived, which is the key aim of the present paper. In the analysis of statistical significance, the explicit density function can reflect all dynamical properties of a chemostat model. To validate our theoretical conclusions, we present examples and numerical simulations.

Stationary distribution and probability density for a stochastic SEIR-type model of coronavirus (COVID-19) with asymptomatic carriers

In this paper, we propose a stochastic SEIR-type model with asymptomatic carriers to describe the propagation mechanism of coronavirus (COVID-19) in the population. Firstly, we show that there exists a unique global positive solution of the stochastic system with any positive initial value. Then we adopt a stochastic Lyapunov function method to establish sufficient conditions for the existence and uniqueness of an ergodic stationary distribution of positive solutions to the stochastic model. Especially, under the same conditions as the existence of a stationary distribution, we obtain the specific form of the probability density around the quasi-endemic equilibrium of the stochastic system. Finally, numerical simulations are introduced to validate the theoretical findings.

The Unique ergodic stationary distribution of two stochastic SEIVS epidemic models with higher order perturbation

Two types of susceptible, exposed, infectious, vaccinated/recovered, susceptible (SEIVS) epidemic models with saturation incidence and temporary immunity, driven by higher order white noise and telegraph noise, are investigated. The key aim of this work is to explore and obtain the existence of the unique ergodic stationary distribution for the above two models, which reveals whether the disease will be prevalent and persistent under some noise intensity assumptions. We also use meticulous numerical examples to validate the feasibility of the analytical findings. Finally, a brief biological discussion shows that the intensities of noises play a significant role in the stationary distributions of the two models.

Dynamics Analysis of a Class of Stochastic SEIR Models with Saturation Incidence Rate

In this article, a class of stochastic SEIR models with saturation incidence is studied. The model is a symmetric and compatible distribution family. This paper studies various properties of the system by constructing Lyapunov functions. First, the gradual properties of the systematic solution near the disease-free equilibrium of the deterministic model is studied, followed by the final behavior of the model, including stochastic persistence and final extinction. Finally, the existence conditions of the stationary distribution of the model are given, and then it is proved that it is traversed, and the corresponding conclusions are verified through numerical simulation.

Stationary distribution and density function expression for a stochastic SIQRS epidemic model with temporary immunity

Recently, considering the temporary immunity of individuals who have recovered from certain infectious diseases, Liu et al. (Phys A Stat Mech Appl 551:124152, 2020) proposed and studied a stochastic susceptible-infected-recovered-susceptible model with logistic growth. For a more realistic situation, the effects of quarantine strategies and stochasticity should be taken into account. Hence, our paper focuses on a stochastic susceptible-infected-quarantined-recovered-susceptible epidemic model with temporary immunity. First, by means of the Khas'minskii theory and Lyapunov function approach, we construct a critical value R 0 S corresponding to the basic reproduction number R 0 of the deterministic system. Moreover, we prove that there is a unique ergodic stationary distribution if R 0 S > 1 . Focusing on the results of Zhou et al. (Chaos Soliton Fractals 137:109865, 2020), we develop some suitable solving theories for the general four-dimensional Fokker-Planck equation. The key aim of the present study is to obtain the explicit density function expression of the stationary distribution under R 0 S > 1 . It should be noted that the existence of an ergodic stationary distribution together with the unique exact probability density function can reveal all the dynamical properties of disease persistence in both epidemiological and statistical aspects. Next, some numerical simulations together with parameter analyses are shown to support our theoretical results. Last, through comparison with other articles, results are discussed and the main conclusions are highlighted.