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In this article, we develop a Hepatitis B virus model with six compartments affected by environmental fluctuations since the Hepatitis B virus produces serious liver infections in the human body, putting many people at high risk. The existence of a global positive solution is shown to prove the positivity of solutions. We demonstrate that the system experiences the extinction property for a specific parametric restriction. Besides that, we obtain the stochastic stability region for the proposed model through the stationary distribution. To determine the appearance and disappearance of infection in the population, we find and analyze the reproduction ratio R0S$$ {R}_0^S $$. In addition, we have verified the condition of the reproduction ratio through the graphical simulations. [ABSTRACT FROM AUTHOR]

A stochastic epidemic model with random noise transmission is taken into account, describing the dynamics of the measles viral infection. The basic reproductive number is calculated corresponding to the stochastic model. It is determined that, given initial positive data, the model has bounded, unique, and positive solution. Additionally, utilizing stochastic Lyapunov functional theory, we study the extinction of the disease. Stationary distribution and extinction of the infection are examined by providing sufficient conditions. We employed optimal control principles and examined stochastic control systems to regulate the transmission of the virus using environmental factors. Graphical representations have been offered for simplicity of comprehending in order to further verify the acquired analytical findings. [ABSTRACT FROM AUTHOR]

The coronavirus disease (COVID-19) is a dangerous pandemic and it spreads to many people in most of the world. In this paper, we propose a COVID-19 model with the assumption that it is affected by randomness. For positivity, we prove the global existence of positive solution and the system exhibits extinction under certain parametric restrictions. Moreover, we establish the stability region for the stochastic model under the behavior of stationary distribution. The stationary distribution gives the guarantee of the appearance of infection in the population. Besides that, we find the reproduction ratio R 0 S for prevail and disappear of infection within the human population. From the graphical representation, we have validated the threshold conditions that define in our theoretical findings. [ABSTRACT FROM AUTHOR]

Brucellosis is one of the major infective and contagious bacterial diseases among animals in pastoral areas of some countries. In this paper, we introduce the effect of environment white noise in the spatial propagation process of brucellosis, and consider a stochastic two-patch brucellosis model. On one hand, we get existence and uniqueness of the global positive solution to the stochastic systems. On the other hand, by using the stochastic Lyapunov function theory we obtain a series of stochastic threshold dynamics results, incorporating extinction of the disease, existence of a unique ergodic stationary distribution of the positive solutions to systems in both patch 1 and patch 2. Furthermore, we find that stochastic perturbation is contribute to extinction of the disease to some extent by numerical simulations. [ABSTRACT FROM AUTHOR]

Abstract In this paper, a stochastic SIRV epidemic model with general nonlinear incidence and vaccination is investigated. The value of our study lies in two aspects. Mathematically, with the help of Lyapunov function method and stochastic analysis theory, we obtain a stochastic threshold of the model that completely determines the extinction and persistence of the epidemic. Epidemiologically, we find that random fluctuations can suppress disease outbreak, which can provide us some useful control strategies to regulate disease dynamics. In other words, neglecting random perturbations overestimates the ability of the disease to spread. The numerical simulations are given to illustrate the main theoretical results.

Abstract A stochastic susceptible–infectious–recovered epidemic model with temporary immunity and media coverage is proposed. The effects of Lévy jumps on the dynamics of the model are considered. A unique global positive solution for the epidemic model is obtained. Sufficient conditions are derived to guarantee that the epidemic disease is extinct and persistent in the mean. The threshold behavior is discussed. Numerical simulations are given to verify our theoretical results.

In this paper, a stochastic SIRV epidemic model with general nonlinear incidence and vaccination is investigated. The value of our study lies in two aspects. Mathematically, with the help of Lyapunov function method and stochastic analysis theory, we obtain a stochastic threshold of the model that completely determines the extinction and persistence of the epidemic. Epidemiologically, we find that random fluctuations can suppress disease outbreak, which can provide us some useful control strategies to regulate disease dynamics. In other words, neglecting random perturbations overestimates the ability of the disease to spread. The numerical simulations are given to illustrate the main theoretical results. [ABSTRACT FROM AUTHOR]

A stochastic susceptible–infectious–recovered epidemic model with temporary immunity and media coverage is proposed. The effects of Lévy jumps on the dynamics of the model are considered. A unique global positive solution for the epidemic model is obtained. Sufficient conditions are derived to guarantee that the epidemic disease is extinct and persistent in the mean. The threshold behavior is discussed. Numerical simulations are given to verify our theoretical results. [ABSTRACT FROM AUTHOR]

Taking lévy jumps into account, a Lotka–Volterra food chain chemostat model in random environment is proposed and investigated. We first prove the existence and uniqueness of the global positive solution. Then conditions for extinction of the microorganisms are derived in two cases. Furthermore, we establish sufficient conditions for persistence in the mean of the system. Theoretical analysis indicates that the dynamics of the considered model are determined by two threshold parameters R 0 s and R 1 s , and both white noise and lévy noise are disadvantageous to the system. Finally, numerical simulations are given to illustrate the results. • A stochastic Lotka–Volterra food chain chemostat model with both Brownian motion and lévy jumps is studied. • Conditions for extinction of the microorganisms are derived in two cases. • We establish sufficient conditions for persistence in the mean of the system. • Theoretical analysis indicates the dynamics of the model are determined by two threshold parameters. • Results show that both white noise and lévy noise are disadvantageous to the system. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the dynamical behavior for a stochastic SIS epidemic model with isolation which is as an important strategy for the elimination of infectious diseases. It is assumed that the stochastic effects manifest themselves mainly as fluctuation in the transmission coefficient, the death rate and the proportional coefficient of the isolation of infective. It is shown that the extinction and persistence in the mean of the model are determined by a threshold value R 0 S . That is, if R 0 S < 1 , then disease dies out with probability one, and if R 0 S > 1 , then the disease is stochastic persistent in the means with probability one. Furthermore, the existence of a unique stationary distribution is discussed, and the sufficient conditions are established by using the Lyapunov function method. Finally, some numerical examples are carried out to confirm the analytical results. [ABSTRACT FROM AUTHOR]

A class of SIR epidemic model with generalized nonlinear incidence rate is presented in this paper. Temporary immunity and stochastic perturbation are also considered. The existence and uniqueness of the global positive solution is achieved. Sufficient conditions guaranteeing the extinction and persistence of the epidemic disease are established. Moreover, the threshold behavior is discussed, and the threshold value R 0 is obtained. We show that if R 0 < 1 , the disease eventually becomes extinct with probability one, whereas if R 0 > 1 , then the system remains permanent in the mean. [ABSTRACT FROM AUTHOR]

In this paper, Brownian motion and L e ́ vy jumps are introduced to a SIS type epidemic model with nonlinear incidence rate. The dynamical behavior of the considered model is investigated. In order to reveal the extinction and permanence of the disease, two threshold values R ˜ 0 , R ̄ 0 are showed. We find that if R ˜ 0 < 1 , the disease may die out, and when R ̄ 0 > 1 , the disease may be persistent. Finally, the numerical simulations are presented to illustrate our mathematical results. [ABSTRACT FROM AUTHOR]

In this paper, a class of stochastic SIS epidemic models with nonlinear incidence rate is investigated. It is shown that the extinction and persistence of the disease in probability are determined by a threshold value R ˜ 0 . That is, if R ˜ 0 < 1 and an additional condition holds then disease dies out, and if R ˜ 0 > 1 then disease is weak permanent with probability one. To obtain the permanence in the mean of the disease, a new quantity R ̂ 0 is introduced, and it is proved that if R ̂ 0 > 1 the disease is permanent in the mean with probability one. Furthermore, the numerical simulations are presented to illustrate some open problems given in Remarks 1–3 and 5 of this paper. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, non-autonomous SIRS epidemic models with bilinear incidence and disease-induced mortality are studied. Under the quite weak assumptions, the sufficient and necessary conditions on the permanence and strong persistence of the disease and the sufficient condition on the extinction of the disease are established. Some new threshold values of the integral form , and are obtained. We prove that the disease is permanent if and only if , and if or , then the disease is extinct. As applications of the main results, we discuss the periodic and almost periodic models. The corresponding basic reproductive numbers are obtained. We show that if , then the disease is permanent and if , then the disease is extinct. [Copyright &y& Elsevier]