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...

In this paper, we establish the global asymptotic stability of an endemic equilibrium for an SIRS epidemic model with distributed time delays. It is shown that the global stability holds for any rate of immunity loss, if the basic reproduction number is greater than 1 and less than or equals to a critical value. Otherwise, there is a maximal rate of immunity loss which guarantees the global sta...

A differential electronic Susceptible-Infectious-RemovedSusceptible (e-SIRS) epidemic model of virus and worms in a computer network has been formulated. Latent period, immune period and time for self replication have been considered. Stability of the result is stated in terms of the threshold parameter. We have derived an explicit formula for the reproductive number and have shown that the vir...

A discrete-time deterministic epidemic model is proposed with the aim of reproducing the behaviour observed in the incidence of real infectious diseases, such as oscillations and irregularities. For this purpose we introduce, in a naïve discrete-time SIRS model, seasonal variability in the loss of immunity and in the infection probability, modelled by sequences of kicks. Restrictive assumptions...

The study of recurrent epidemic outbreaks has been attracting great attention for decades, but its underlying mechanism is still under debate. Based on a large number of real data from different cities, we find that besides the seasonal periodic outbreaks of influenza, there are also non-periodic outbreaks, i.e. non-seasonal or non-annual behaviors. To understand how the non-periodicity shows u...

The seasonality of respiratory diseases (common cold, influenza, etc.) is a well-known phenomenon studied from ancient times. The development of predictive models is still not only an actual unsolved problem of mathematical epidemiology but also is very important for safety of public health. Here we show that SIRS (Susceptible-Infected-Recovered-Susceptible) model accurately enough reproduces r...

This paper is concerned with a delayed SIRS epidemic model with a nonlinear incidence rate. The main results are given in terms of local stability and Hopf bifurcation. Sufficient conditions for the local stability of the positive equilibrium and existence of Hopf bifurcation are obtained by regarding the time delay as the bifurcation parameter. Further, the properties of Hopf bifurcation such ...