Threshold theorem for an epidemic model

The article discusses a probability model of the spread of an epidemic in which the elimination of sick persons (through death, immunity, or isolation) is taken into account. The authors find a limit distribution for the magnitude of the epidemic,v, on the assumption that n, where n is the original number of susceptible persons, and , where and are the coefficient of infection and the coefficient of elimination, respectively.Translated from Matematicheskie Zametki, Vol. 3, No. 2, pp. 179–185, February, 1968.     

Threshold dynamics in a stochastic SIRS epidemic model with nonlinear incidence rate

We discuss the dynamic of a stochastic Susceptible-Infectious-Recovered-Susceptible (SIRS) epidemic model with nonlinear incidence rate.The crucial threshold $\tilde{R}_0$ is identified and this will determine the extinction and persistence of the epidemic when the noise is small. We also discuss the asymptotic behavior of the stochastic model around the endemic equilibrium of the corresponding deterministic system. When the noise is large, we find that a large noise intensity has the effect of suppressing the epidemic, so that it dies out. Finally, these results are illustrated by computer simulations.     

Threshold dynamics of the stochastic epidemic model with jump-diffusion infection force

This paper formulates a stochastic SIR epidemic model by supposing that the infection force is perturbed by Brown motion and L\''{e}vy jumps. The globally positive and bounded solution is proved firstly by constructing the suitable Lyapunov function. Then, a stochastic basic reproduction number $R_0^{L}$ is derived, which is less than that for the deterministic model and the stochastic model driven by Brown motion. Analytical results show that the disease will die out if $R_0^{L}<1$, and $R_0^{L}>1$ is the necessary and sufficient condition for persistence of the disease. Theoretical results and numerical simulations indicate that the effects of L\''{e}vy jumps may lead to extinction of the disease while the deterministic model and the stochastic model driven by Brown motion both predict persistence. Additionally, the method developed in this paper can be used to investigate a class of related stochastic models driven by L\''{e}vy noise.     

Threshold conditions for a non-autonomous epidemic model with vaccination

In this article, we wish to investigate the dynamical behaviour of an SIRVS epidemic model with time-dependent coefficients. Under the quite weak assumptions, we give some new threshold conditions which determine whether or not the disease will go to extinction. The permanence and extinction of the infectious disease is studied. When the system degenerates into periodic or almost periodic system, the corresponding sharp threshold results are obtained for permanent endemicity versus extinction in terms of asymptotic time. In order to illustrate our theoretical analysis, some numerical simulations are also included in the end.     

Threshold dynamics for a cholera epidemic model with periodic transmission rate

A cholera epidemic model with periodic transmission rate is presented. The basic reproduction number is defined. It is shown that the disease-free equilibrium is globally asymptotically stable and the cholera eventually disappears if the basic reproduction number is less than one. And if the basic reproduction number is greater than one, there exists a positive periodic solution which is globally asymptotically stable. Numerical simulations are provided to illustrate analytical results.     

Optimal immunization rules for an epidemic with recovery

The classical Kermack-McKendrick model for the spread of an epidemic through a closed population has recently been extended by Billard to allow for the recovery and possible reinfection of infective cases. In this paper, we study the optimal control of such an epidemic through immunization of susceptibles when costs are proportional to the area under the infectives trajectory plus the total number of immunizations. When the immunization rate is bounded, optimal controls are of bang-bang type and are characterized by switching curves in the epidemic state space. Explicit expressions for these curves are obtained in the case of deterministic dynamics. When the epidemic is described by a Markov chain, numerical solutions for the switching curve are easy to obtain by dynamic programming, and useful analytic approximations to them are described. The results include those for the so-called general epidemic in which no recovery is allowed.The author is grateful to the referees for their detailed and constructive criticism of an earlier version of this paper.     

Threshold behaviour of a stochastic SIR model

In this paper, we investigate the threshold behaviour of a susceptible-infected-recovered (SIR) epidemic model with stochastic perturbation. When the noise is small, we show that the threshold determines the extinction and persistence of the epidemic. Compared with the corresponding deterministic system, this value is affected by white noise, which is less than the basic reproduction number of the deterministic system. On the other hand, we obtain that the large noise will also suppress the epidemic to prevail, which never happens in the deterministic system. These results are illustrated by computer simulations.     

Some limit theorems for a general stochastic model of epidemics

In a general stochastic model of epidemics it is assumed that the initial number of patients is finite, and that with increasing size of the population the control parameter approaches a constant. Under these conditions we study the properties of the limiting distribution of the size of an epidemic.Translated from Matematicheskie Zametki, Vol. 13, No. 5, pp. 709–716, May, 1973.     

Stability analysis of an epidemic model with diffusion and stochastic perturbation

In this paper, we investigate the stability of an epidemic model with diffusion and stochastic perturbation. We first show both the local and global stability of the endemic equilibrium of the deterministic epidemic model by analyzing corresponding characteristic equation and Lyapunov function. Second, for the corresponding reaction–diffusion epidemic model, we present the conditions of the globally asymptotical stability of the endemic equilibrium. And we carry out the analytical study for the stochastic model in details and find out the conditions for asymptotic stability of the endemic equilibrium in the mean sense. Furthermore, we perform a series of numerical simulations to illustrate our mathematical findings.     

Threshold dynamics and ergodicity of an SIRS epidemic model with semi-Markov switching

This work studies the threshold dynamics and ergodicity of a stochastic SIRS epidemic model with the disease transmission rate driven by a semi-Markov process. The semi-Markov process used in this paper for describing a randomly changing environment is a very large extension of the most common Markov regime-switching process. We define a basic reproduction number for the semi-Markov regime-switching environment and show that its position with respect to 1 determines the extinction or persistence of the disease. In the case of disease persistence, we give mild sufficient conditions for ensuring the existence and absolute continuity of the invariant probability measure. Under the same conditions, we also prove the global attractivity of the Ω-limit set of the system and the convergence in total variation norm of the transition probability to the invariant measure. Compared with the existing results in the Markov regime-switching environment, the results generalized require almost no additional conditions.     

Two-group SIR epidemic model with stochastic perturbation

A stochastic two-group SIR model is presented in this paper.The existence and uniqueness of its nonnegative solution is obtained,and the solution belongs to a positively invariant set.Furthermore,the globally asymptotical stability of the disease-free equilibrium is deduced by the stochastic Lyapunov functional method if R0 ≤ 1,which means the disease will die out.While if R0 1,we show that the solution is fluctuating around a point which is the endemic equilibrium of the deterministic model in time average.In addition,the intensity of the fluctuation is proportional to the intensity of the white noise.When the white noise is small,we consider the disease will prevail.At last,we illustrate the dynamic behavior of the model and their approximations via a range of numerical experiments.     

Dynamics of an impulsive stochastic SIR epidemic model with saturated incidence rate

In this paper, the dynamics of an impulsive stochastic SIR epidemic model with saturated incidence rate are analyzed. The existence and uniqueness of the global positive solution is proved by constructing the equivalent system without pulses. The threshold which determines the extinction and persistence of the disease is obtained. The global attraction of disease-free periodic solution is addressed. Sufficient condition for the existence of a positive periodic solution is established. These results are supported by computer simulations.     

潜伏期变化的随机流行病模型及其贝叶斯推断

本文讨论了潜伏期和传染期均服从威布尔分布、易感性随机变化的一类随机流行病模型,并利用M CM C算法对潜伏期、传染期的参数和易感性的超参数作了贝叶期推断.这种分析方法比以往各种方法更适用于各类疾病.     

The threshold of a stochastic SIRS epidemic model with saturated incidence

In this paper, we investigate the dynamics of a stochastic SIRS epidemic model with saturated incidence. When the noise is small, we obtain a threshold of the stochastic system which determines the extinction and persistence of the epidemic. Besides, we find that large noise will suppress the epidemic from prevailing.     

Nontrivial periodic solution of a stochastic epidemic model with seasonal variation

In this paper, we consider a stochastic SIR epidemic model with seasonal variation. First, we obtain the threshold of our model which determines whether the epidemic occurs or not. In the case of persistence, we prove that there is a nontrivial positive periodic solution.     

Dynamic behavior of a stochastic SIRS epidemic model with media coverage

The main purpose of this paper is to explore the global behavior of a stochastic SIRS epidemic model with media coverage. The value of this research has 2 aspects: for one thing, we use Markov semigroup theory to prove that the basic reproduction number can be used to control the dynamics of stochastic system. If , the stochastic system has a disease‐free equilibrium, which implies the disease will die out with probability one. If , under the mild extra condition, the stochastic differential equation has an endemic equilibrium, which is globally asymptotically stable. For another, it is known that environment fluctuations can inhibit disease outbreak. Although the disease is persistent when R₀ > 1 for the deterministic model, if , the disease still dies out with probability one for the stochastic model. Finally, numerical simulations were carried out to illustrate our results, and we also show that the media coverage can reduce the peak of infective individuals via numerical simulations.     

SVEIRS epidemic model with delays and partial immunization for internet worms

A delayed SVEIRS model for the transmission of worms in internet with partial immunization is proposed. The impact of the possible combination of the two delays on the model is investigated. By analyzing the corresponding characteristic equations and regarding the possible combination of the two delays as the bifurcation parameter, local stability of the endemic equilibrium and existence of local Hopf bifurcation at the viral equilibrium are addressed, respectively. Further, explicit formulas that determine direction and stability of the Hopf bifurcation are derived with the help of the normal form theory and the center manifold theorem. Finally, some numerical simulations are carried out to verify the obtained theoretical findings.     

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.