Stochastic SIS Epidemic Model Research Articles

Dynamics of a stochastic multi-strain SIS epidemic model driven by Lévy noise

A stochastic multi-strain SIS epidemic model is formulated by introducing Lévy noise into the disease transmission rate of each strain. First, we prove that the stochastic model admits a unique global positive solution, and, by the comparison theorem, we show that the solution remains within a positively invariant set almost surely. Next we investigate stochastic stability of the disease-free equilibrium, including stability in probability and pth moment asymptotic stability. Then sufficient conditions for persistence in the mean of the disease are established. Finally, based on an Euler scheme for Lévy-driven stochastic differential equations, numerical simulations for a stochastic two-strain model are carried out to verify the theoretical results. Moreover, numerical comparison results of the stochastic two-strain model and the deterministic version are also given. Lévy noise can cause the two strains to become extinct almost surely, even though there is a dominant strain that persists in the deterministic model. It can be concluded that the introduction of Lévy noise reduces the disease extinction threshold, which indicates that Lévy noise may suppress the disease outbreak.

Stationary distribution of stochastic SIS epidemic model with vaccination under regime switching

This paper addresses a stochastic SIS epidemic model with vaccination under regime switching. The stochastic model in this paper includes white and color noises. By constructing stochastic Lyapunov functions with regime switching, we establish sufficient conditions for the existence of a unique ergodic stationary distribution.

Le modèle stochastique SIS pour une épidémie dans un environnement aléatoire.

The stochastic SIS epidemic model in a random environment. In a random environment that is a two-state continuous-time Markov chain, the mean time to extinction of the stochastic SIS epidemic model grows in the supercritical case exponentially with respect to the population size if the two states are favorable, and like a power law if one state is favorable while the other is unfavorable.

Persistence and extinction for a class of stochastic SIS epidemic models with nonlinear incidence rate

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 R̂0 is introduced, and it is proved that if R̂0>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.

The Dynamical Behaviors in a Stochastic SIS Epidemic Model with Nonlinear Incidence.

A stochastic SIS-type epidemic model with general nonlinear incidence and disease-induced mortality is investigated. It is proved that the dynamical behaviors of the model are determined by a certain threshold value . That is, when and together with an additional condition, the disease is extinct with probability one, and when , the disease is permanent in the mean in probability, and when there is not disease-related death, the disease oscillates stochastically about a positive number. Furthermore, when , the model admits positive recurrence and a unique stationary distribution. Particularly, the effects of the intensities of stochastic perturbation for the dynamical behaviors of the model are discussed in detail, and the dynamical behaviors for the stochastic SIS epidemic model with standard incidence are established. Finally, the numerical simulations are presented to illustrate the proposed open problems.

The asymptotic behavior of a stochastic SIS epidemic model with vaccination

In this paper, we discuss a stochastic SIS epidemic model with vaccination. We investigate the asymptotic behavior according to the perturbation and the reproduction number $R_{0}$ . When the perturbation is large, the number of infected decays exponentially to zero and the solution converges to the disease-free equilibrium regardless of the magnitude of $R_{0}$ . Moreover, we get the same exponential stability and the convergence if $R_{0}<1$ . When the perturbation and the disease-related death rate are small, we derive that the disease will persist, which is measured through the difference between the solution and the endemic equilibrium of the deterministic model on average in time if $R_{0}>1$ . Furthermore, we prove that the system is persistent in the mean. Finally, the results are illustrated by computer simulations.

Persistence-time estimation for some stochastic SIS epidemic models

In this paper, we study two stochastic SIS epidemic models: the first one with a constant population size, and the second one with a death factor. We analyze persistence and extinction behaviors for these models. The persistence time depends on the initial population size and satisfies a stationary backward Kolmogorov differential equation, which is a linear second-order partial differential equation with variable degenerate coefficients. We solve this equation numerically using a classical finite element method. We give computational evidence that the importance of understanding the dynamics of both the deterministic and the stochastic epidemic models is due to the numerical approximations to the mean persistence time. This can give more information about the model and may perhaps explain strange behaviors, such as the differences between the deterministic model and the stochastic one for long times.

Dynamics of a novel nonlinear stochastic SIS epidemic model with double epidemic hypothesis

In this paper, we propose new mathematical models with nonlinear incidence rate and double epidemic hypothesis. Then we dedicate to develop a method to obtain the threshold of the stochastic SIS epidemic model. To this end, first, we investigate the stability of the equilibria of the deterministic system and obtain the conditions for the extinction and the permanence of two epidemic diseases. Second, we explore and obtain the threshold of a stochastic SIS system for the extinction and the permanence in mean of two epidemic diseases. The results show that a large stochastic disturbance can cause infectious diseases to go to extinction, in other words, the persistent infectious disease of a deterministic system can become extinct due to the white noise stochastic disturbance. This implies that the stochastic disturbance is conducive to epidemic diseases control. To illustrate the performance of the theoretical results, we present a series of numerical simulations of these cases with respect to different noise disturbance coefficients.

A stochastic SIS epidemic model incorporating media coverage in a two patch setting

In this paper, we investigate the stochastic disease dynamics of an SIS epidemic model on two patches incorporating media coverage. We give the global existence and positivity of the solutions, and the sufficient conditions for almost surely exponentially stability of the disease-free equilibrium, which means that the disease will be stochastic extinction. Furthermore, we perform some numerical simulations to validate the analytical finding.

The threshold of stochastic SIS epidemic model with saturated incidence rate

This paper considers a stochastic SIS model with saturated incidence rate. We investigate the existence and uniqueness of the positive solution to the system, and we show the condition for the infectious individuals to be extinct. Moreover, we prove that the system has the ergodic property and derive the expression for its invariant density. The simulation results are illustrated finally.

A stochastic SIS epidemic model with heterogeneous contacts

A stochastic model for the spread of an SIS epidemic among a population consisting of N individuals, each having heterogeneous infectiousness and/or susceptibility, is considered and its behavior is analyzed under the practically relevant situation when N is small. The model is formulated as a finite time-homogeneous continuous-time Markov chain X. Based on an appropriate labeling of states, we first construct its infinitesimal rate matrix by using an iterative argument, and we then present an algorithmic procedure for computing steady-state measures, such as the number of infected individuals, the length of an outbreak, the maximum number of infectives, and the number of infections suffered by a marked individual during an outbreak. The time till the epidemic extinction is characterized as a phase-type random variable when there is no external source of infection, and its Laplace–Stieltjes transform and moments are derived in terms of a forward elimination backward substitution solution. The inverse iteration method is applied to the quasi-stationary distribution of X, which provides a good approximation of the process X at a certain time, conditional on non-extinction, after a suitable waiting time. The basic reproduction number R0 is defined here as a random variable, rather than an expected value.

Threshold Behavior in a Stochastic SIS Epidemic Model with Standard Incidence

In this paper, we consider a stochastic SIS epidemic model with perturbed disease transmission coefficient. We discuss the threshold behavior in the sense that if \(\mathcal {R}_0^s 1\), the densities of the distributions of the solution can converge in \(L^1\) to an invariant density.

On the stochastic SIS epidemic model in a periodic environment.

In the stochastic SIS epidemic model with a contact rate a, a recovery rate b < a, and a population size N, the mean extinction time τ is such that (log τ)/N converges to c = b/a - 1 - log(b/a) as N grows to infinity. This article considers the more realistic case where the contact rate a(t) is a periodic function whose average is bigger than b. Then log τ/N converges to a new limit C, which is linked to a time-periodic Hamilton-Jacobi equation. When a(t) is a cosine function with small amplitude or high (resp. low) frequency, approximate formulas for C can be obtained analytically following the method used in Assaf et al. (Phys Rev E 78:041123, 2008). These results are illustrated by numerical simulations.

The threshold of a stochastic SIS epidemic model with vaccination

This paper is concerned with the long time behavior of a stochastic SIS epidemic model with vaccination. Sufficient conditions for extinction and persistence in mean are obtained. We find a threshold of the stochastic model which determines the outcome of the disease in case the white noises are small. At last, some numerical simulations are carried out to support our results.

Parameter estimation for the stochastic SIS epidemic model

In this paper we estimate the parameters in the stochastic SIS epidemic model by using pseudo-maximum likelihood estimation (pseudo-MLE) and least squares estimation. We obtain the point estimators and \(100 (1-\alpha )\%\) confidence intervals as well as \(100 (1-\alpha )\%\) joint confidence regions by applying least squares techniques. The pseudo-MLEs have almost the same form as the least squares case. We also obtain the exact as well as the asymptotic \(100 (1-\alpha )\%\) joint confidence regions for the pseudo-MLEs. Computer simulations are performed to illustrate our theory.

Dynamics of a Stochastic SIS Epidemic Model with Saturated Incidence

We introduce stochasticity into the SIS model with saturated incidence. The existence and uniqueness of the positive solution are proved by employing the Lyapunov analysis method. Then, we carry out a detailed analysis on both its almost sure exponential stability and itspth moment exponential stability, which indicates that thepth moment exponential stability implies the almost sure exponential stability. Additionally, the results show that the conditions for the disease to become extinct are much weaker than those in the corresponding deterministic model. The conditions for the persistence in the mean and the existence of a stationary distribution are also established. Finally, we derive the expressions for the mean and variance of the stationary distribution. Compared with the corresponding deterministic model, the threshold value for the disease to die out is affected by the half saturation constant. That is, increasing the saturation effect can reduce the disease transmission. Computer simulations are presented to illustrate our theoretical results.

Stationary distribution of a stochastic SIS epidemic model with vaccination

In this paper, we consider a stochastic SIS epidemic model with vaccination. We prove that the densities of the distributions of the solution can converge in L1 to an invariant density under appropriate conditions. Also we find the support of the invariant density.

The extinction and persistence of the stochastic SIS epidemic model with vaccination

In this paper, we discuss the dynamics of a stochastic SIS epidemic model with vaccination. When the noise is large, the infective decays exponentially to zero regardless of the magnitude of R0. When the noise is small, sufficient conditions for extinction exponentially and persistence in the mean are established. The results are illustrated by computer simulations.

Extinction and Ergodic Property of Stochastic SIS Epidemic Model with Nonlinear Incidence Rate

We investigate a stochastic SIS model with nonlinear incidence rate. We show that there exists a unique nonnegative solution to the system, and condition for the infectious individualsI(t)to be extinct is given. Moreover, we prove that the system has ergodic property. Finally, computer simulations are carried out to verify our results.

Survival and Stationary Distribution in a Stochastic SIS Model

The dynamics of a stochastic SIS epidemic model is investigated. First, we show that the system admits a unique positive global solution starting from the positive initial value. Then, the long-term asymptotic behavior of the model is studied: whenR0≤1, we show how the solution spirals around the disease-free equilibrium of deterministic system under some conditions; whenR0&gt;1, we show that the stochastic model has a stationary distribution under certain parametric restrictions. In particular, we show that random effects may lead the disease to extinction in scenarios where the deterministic model predicts persistence. Finally, numerical simulations are carried out to illustrate the theoretical results.