Stochastic SIRS Epidemic Model Research Articles

Mean-square exponential stability of impulsive conformable fractional stochastic differential system with application on epidemic model

The main objective of this paper is to analyze the exponential stability of a new problem of an n-dimensional non-linear impulsive conformable stochastic differential system with the various fractional orders. By applying the conversion of the state variable, the solution of the aimed impulsive stochastic differential system is related to the corresponding solution of the non-impulsive system. Then the sufficient conditions for the exponential stability of the proposed impulsive system are accomplished by obtaining sufficient conditions for the exponential stability of the respective non-impulsive system. The novelty of this work is that the Razumikhin stability technique (without using the Lyapunov function) is used to achieve the stability conditions for the impulsive conformable stochastic system. The gained theoretical conclusions are confirmed by applying the conformable stochastic SIR epidemic model with impulsive immigration. Finally, the influence of fractional orders in the dynamical performance of the impulsive conformable stochastic SIR epidemic model is illustrated with graphical representations.

The threshold of a stochastic SIRS epidemic model with general incidence rate under regime-switching

In this paper, a stochastic SIRS epidemic model with general incidence rate under regime switching is investigated. First of all, by applying the Markov semigroups theory, we establish sufficient condition RT0*>0 for the densities of the distributions of solution converge in L1 to an invariant density. Then, we prove the extinction of disease under the condition of RT0*<0. Finally, numerical simulations for two special cases of response function are carried out to illustrate the theoretical results.

Optimal Vaccination Strategy in the Stochastic Epidemic Limited-Treatment Model.

The author analyzes the stochastic SIR epidemic model with vaccination and limited treatment. A method for obtaining the optimal vaccination strategy that minimizes the cost functional is proposed.

Bayesian Inference for SIR Epidemic Model with dependent parameters

Abstract This paper is concerned with the Bayesian inference for the dependent parameters of stochastic SIR epidemic model in a closed population. The estimation framework involves the introduction of m − 1 latent data between every pair of observations. Kibble’s bivariate gamma distribution is considered as a good candidate prior density of parameters, they give an appropriate frame to model the dependence between the parameters. A Markov chain Monte Carlo methods are then used to sample the posterior distribution of the model parameters. Simulated datasets are used to illustrate the proposed methodology.

A stochastic Sir epidemic evolution model with non-concave force of infection: Mathematical modeling and analysis

In this paper, we revisit the classical SIR epidemic model by replacing the simple bilinear transmission rate by a nonlinear one. Our results show that in the presence of environmental fluctuations represented by Brownian motion and that mainly act on the transmission rate, the generalized non-concave force of infection adopted here, greatly affects the long-time behavior of the epidemic. Employing the Markov semigroup theory, we prove that the model solutions do not admit a unique stationary distribution but converge to 0 in [Formula: see text]th moment for any [Formula: see text]. Furthermore, we prove that the disease extinguishes asymptotically exponentially with probability 1 without any restriction on the model parameters and we also determine the rate of convergence. This is an unexpected qualitative behavior in comparison with the existing literature where the studied epidemic models have a threshold dynamics behavior. It is also a very surprising behavior regarding the deterministic counterpart that can exhibit a rich qualitative dynamical behaviors such as backward bifurcation and Hopf bifurcation. On the other hand, we show by several numerical simulations that as the intensity of environmental noises becomes sufficiently small, the epidemic tends to persist for a very long time before dying out from the host population. To solve this problem and to be able to manage the pre-extinction period, we construct a new process in terms of the number of infected and recovered individuals which admits a unique invariant stationary distribution. Finally, we discuss the obtained analytical results through a series of numerical simulations.

Asymptotic behavior of a stochastic SIR model with general incidence rate and nonlinear Lévy jumps.

In this paper, we consider a stochastic SIR epidemic model with general disease incidence rate and perturbation caused by nonlinear white noise and Lacute{e}vy jumps. First of all, we study the existence and uniqueness of the global positive solution of the model. Then, we establish a threshold lambda by investigating the one-dimensional model to determine the extinction and persistence of the disease. To verify the model has an ergodic stationary distribution, we adopt a new method which can obtain the sufficient and almost necessary conditions for the extinction and persistence of the disease. Finally, some numerical simulations are carried out to illustrate our theoretical results.

Dynamics of a stochastic SIRS epidemic model with standard incidence and vaccination.

A stochastic SIRS epidemic model with vaccination is discussed. A new stochastic threshold $ R_0^s $ is determined. When the noise is very low ($ R_0^s < 1 $), the disease becomes extinct, and if $ R_0^s > 1 $, the disease persists. Furthermore, we show that the solution of the stochastic model oscillates around the endemic equilibrium point and the intensity of the fluctuation is proportional to the intensity of the white noise. Computer simulations are used to support our findings.

Long time behavior of an two diffusion stochastic sir epidemic model with nonlinear incidence and treatment

In this paper we propose a stochastic SIR epidemic model to evaluate effect of the randomness on treatment and nonlinear incidence rate. More precisely, we perturb both nonlinear incidence and treatment rates in deterministic SIR model with Gaussian white noise and obtain two diffusion stochastic model. For the model, we theoretically prove that it?s solution is positive and global, and then, we obtain the conditions under which we can claim the existence of the stationary distribution. Also, by constructing suitable Lyapunov functions, weestablish sufficient conditions for p-th moment and almost sure exponential stability of disease-free equilibrium. Conditions for disease extinction are obtained, as well. We close the paper by presenting numerical simulations to verify our theoretical results. For that purpose we use real-life data for spread of cholera in the Department of Artibonite in Haiti, as well as for influenza A H1N1 in Guangdong Province, China.

Asymptotic behavior of the solutions for a stochastic SIRS model with information intervention.

In this paper, a stochastic SIRS epidemic model with information intervention is considered. By constructing an appropriate Lyapunov function, the asymptotic behavior of the solutions for the proposed model around the equilibria of the deterministic model is investigated. We show the average in time of the second moment of the solutions of the stochastic system is bounded for a relatively small noise. Furthermore, we find that information interaction response rate plays an active role in disease control, and as the intensity of the response increases, the number of infected population decreases, which is beneficial for disease control.

The long-time behavior of a stochastic SIR epidemic model with distributed delay and multidimensional Lévy jumps

This paper reports novel theoretical and analytical results for a perturbed version of a SIR model with Gamma-distributed delay. Notably, our epidemic model is represented by Itô–Lévy stochastic differential equations in order to simulate sudden and unexpected external phenomena. By using some new and ameliorated mathematical approaches, we study the long-run characteristics of the perturbed delayed model. Within this scope, we give sufficient conditions for two interesting asymptotic properties: extinction and persistence of the epidemic. One of the most interesting results is that the dynamics of the stochastic model are closely related to the intensities of white noises and Lévy jumps, which can give us a good insight into the evolution of the epidemic in some unexpected situations. Our work complements the results of some previous investigations and provides a new approach to predict and analyze the dynamic behavior of epidemics with distributed delay. For illustrative purposes, numerical examples are presented for checking the theoretical study.

The Dynamics of a Stochastic SIR Epidemic Model with Nonlinear Incidence and Vertical Transmission

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 &lt; 1 , the diseases can be extincted with probability one. When R 0 s &gt; 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.

Dynamics of a stochastic SIR epidemic model driven by Lévy jumps with saturated incidence rate and saturated treatment function

In this article, we consider a stochastic SIR model with a saturated incidence rate and saturated treatment function incorporating Lévy noise. First, we prove the existence of a unique global positive solution to the model. We investigate the stability of the free equilibria E 0 by using the Lyapunov method. We give sufficient conditions for the persistence in the mean. We show the dynamic properties of the solution around endemic equilibria point of the deterministic model. Moreover, we display some numerical results to confirm our theoretical results.

Ergodic property, extinction and density function of a stochastic SIR epidemic model with nonlinear incidence and general stochastic perturbations

Focusing on the unpredictability of person-to-person contacts and the complexity of random variations in nature, this paper will formulate a stochastic SIR epidemic model with nonlinear incidence rate and general stochastic noises. First, we derive a stochastic critical value R0S related to the basic reproduction number R0. Via our new method in constructing suitable Lyapunov function types, we obtain the existence and uniqueness of an ergodic stationary distribution of the stochastic system if R0S>1. Next, via solving the corresponding Fokker-Planck equation, it is theoretically proved that the stochastic model has a log-normal probability density function when another critical value R0H>1. Then the exact expression of the density function is obtained. Moreover, we establish the sufficient condition R0C<1 for disease extinction. Finally, several numerical simulations are provided to verify our analytical results. By comparison with other existing results, our developed theories and methods will be highlighted to end this paper.

Dynamics of a stochastic SIRS epidemic model with standard incidence under regime switching

The goal of this paper is to introduce and initiate a study of a stochastic SIRS epidemic model with standard incidence which is perturbed by both white and telegraph noises. We first show persistence in the mean and then establish the sufficient conditions for extinction of the disease. Moreover, in the case of persistence, we obtain sufficient conditions for the existence of positive recurrence of the solutions by means of structuring suitable stochastic Lyapunov function with regime switching. Meanwhile, the threshold between persistence in the mean and extinction of the stochastic system is also obtained. Finally, we test our theory conclusion by simulations.

Stochastic SIR Lévy Jump Model with Heavy-Tailed Increments

This paper considers a general stochastic SIR epidemic model driven by a multidimensional Lévy jump process with heavy-tailed increments and possible correlation between noise components. In this framework, we derive new sufficient conditions for disease extinction and persistence in the mean. Our method differs from previous approaches by the use of Kunita’s inequality instead of the Burkholder–Davis–Gundy inequality for continuous processes, and allows for the treatment of infinite Lévy measures by the definition of new threshold \(\mathcal {\bar{R}}_0\). An SIR model driven by a tempered stable process is presented as an example of application with the ability to model sudden disease outbreak, illustrated by numerical simulations. The results show that persistence and extinction are dependent not only on the variance of the processes increments, but also on the shapes of their distributions.

Threshold dynamics for a class of stochastic SIRS epidemic models with nonlinear incidence and Markovian switching

In this paper, we consider a stochastic SIRS epidemic model with nonlinear incidence and Markovian switching. By using the stochastic calculus background, we establish that the stochastic threshold ℜswt can be used to determine the compartment dynamics of the stochastic system. Some examples and numerical simulations are presented to confirm the theoretical results established in this paper.

Threshold dynamics and sensitivity analysis of a stochastic semi-Markov switched SIRS epidemic model with nonlinear incidence and vaccination

&lt;p style='text-indent:20px;'&gt;In this paper, a stochastic SIRS epidemic model with nonlinear incidence and vaccination is formulated to investigate the transmission dynamics of infectious diseases. The model not only incorporates the white noise but also the external environmental noise which is described by semi-Markov process. We first derive the explicit expression for the basic reproduction number of the model. Then the global dynamics of the system is studied in terms of the basic reproduction number and the intensity of white noise, and sufficient conditions for the extinction and persistence of the disease are both provided. Furthermore, we explore the sensitivity analysis of &lt;inline-formula&gt;&lt;tex-math id="M1"&gt;\begin{document}$ R_0^s $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt; with each semi-Markov switching under different distribution functions. The results show that the dynamics of the entire system is not related to its switching law, but has a positive correlation to its mean sojourn time in each subsystem. The basic reproduction number we obtained in the paper can be applied to all piecewise-stochastic semi-Markov processes, and the results of the sensitivity analysis can be regarded as a prior work for optimal control.&lt;/p&gt;

Global dynamics of a novel deterministic and stochastic SIR epidemic model with vertical transmission and media coverage

In this paper, we study a novel deterministic and stochastic SIR epidemic model with vertical transmission and media coverage. For the deterministic model, we give the basic reproduction number R_{0} which determines the extinction or prevalence of the disease. In addition, for the stochastic model, we prove existence and uniqueness of the positive solution, and extinction and persistence in mean. Furthermore, we give numerical simulations to verify our results.

Ergodicity of a Nonlinear Stochastic SIRS Epidemic Model with Regime-Switching Diffusions

In this paper, taking both white noises and colored noises into consideration, a nonlinear stochastic SIRS epidemic model with regime switching is explored. The threshold parameter R s is found, and we investigate sufficient conditions for the existence of the ergodic stationary distribution of the positive solution. Finally, some numerical simulations are also carried out to demonstrate the analytical results.

Dynamics of a stochastic periodic SIRS model with time delay

Dynamical behaviors of a stochastic periodic SIRS epidemic model with time delay are investigated. By constructing suitable Lyapunov functions and applying Itô’s formula, the existence of the global positive solution and the property of stochastically ultimate boundedness of model (1.1) are proved. Moreover, the extinction and the persistence of the disease are established. The results are verified by numerical simulations.