Threshold Dynamics in Stochastic SIRS Epidemic Models with Nonlinear Incidence and Vaccination

In this paper, the dynamical behaviors for a stochastic SIRS epidemic model with nonlinear incidence and vaccination are investigated. In the models, the disease transmission coefficient and the removal rates are all affected by noise. Some new basic properties of the models are found. Applying these properties, we establish a series of new threshold conditions on the stochastically exponential extinction, stochastic persistence, and permanence in the mean of the disease with probability one for the models. Furthermore, we obtain a sufficient condition on the existence of unique stationary distribution for the model. Finally, a series of numerical examples are introduced to illustrate our main theoretical results and some conjectures are further proposed

A stochastic SIRI epidemic model with Lévy noise

Some diseases such as herpes, bovine and human tuberculosis exhibit relapse in which the recovered individuals do not acquit permanent immunity but return to infectious class. Such diseases are modeled by SIRI models. In this paper, we establish the existence of a unique global positive solution for a stochastic epidemic model with relapse and jumps. We also investigate the dynamic properties of the solution around both disease-free and endemic equilibria points of the deterministic model. Furthermore, we present some numerical results to support the theoretical work

A stochastic SIRI epidemic model with relapse and media coverage

This work is devoted to investigate the existence and uniqueness of a global positive solution for a stochastic epidemic model with relapse and media coverage. We also study the dynamical properties of the solution around both disease-free and endemic equilibria points of the deterministic model. Furthermore, we show the existence of a stationary distribution. Numerical simulations are presented to confirm the theoretical results.Fondo Europeo de Desarrollo RegionalMinisterio de Economía y CompetitividadConsejería de Innovación, Ciencia y Empresa (Junta de Andalucía)Faculty of Sciences (Ibn Tofail University

Analysis of a stochastic distributed delay epidemic model with relapse and Gamma distribution kernel

In this work, we investigate a stochastic epidemic model with relapse and distributed delay. First, we prove that our model possesses and unique global positive solution. Next, by means of the Lyapunov method, we determine some sufficient criteria for the extinction of the disease and its persistence. In addition, we establish the existence of a unique stationary distribution to our model. Finally, we provide some numerical simulations for the stochastic model to assist and show the applicability and efficiency of our results.Ministerio de Ciencia, Innovación y Universidades (MICINN). EspañaEuropean Commission (EC). Fondo Europeo de Desarrollo Regional (FEDER

Lyapunov stability of an SIRS epidemic model with varying population

In this paper we consider an SIRS epidemic model under a general assumption of density-dependent mortality. We prove the global stability of the disease-free equilibrium and propose a Lyapunov function that allows to demonstrate the global stability of the (unique) endemic state under broad conditions.Comment: 7 page

Lyapunov Stability of an SIRS epidemic model with varying population: ecological VS physical approaches

We consider a general SIRS epidemiological transmission model for an endemic infection with variable population including a general assumption of density-dependent mortality and vertical transmission. We analyse global stability by the Lyapunov approach, proposing two alternative Lyapunov functions. The first one is based on traditional ecological arguments, while the second one is a 'physical' Lyapunov function based on the energy of an appropriate second order equation. Both functions allow to prove the global stability of the (unique) endemic state under broad - not equivalent - parametric conditions. This suggests that using different Lyapunov functions provide useful complementary information on the parametric regions where global stability prevails