Abstract. In this paper, we study a class of multi-group SIRS epidemic models with nonlinear incidence rate which have cross patch infection between different groups. The basic reproduction number R0 is calculated. By using the method of Lyapunov functions, LaSalle’s invariance principle, the theory of the nonnegative matrices and the theory of the persistence of dynamical systems, it is proved...

Social networks are an important infrastructure for information, viruses and innovations propagation. Since users’ behavior has influenced by other users’ activity, some groups of people would be made regard to similarity of users’ interests. On the other hand, dealing with many events in real worlds, can be justified in social networks; spreading disease is one instance of them. People’s manne...

In this work we introduce a new approach to Dynamical Monte Carlo methods to simulate markovian processes. We apply this approach to formulate and study an epidemic generalized SIRS model. The results are in excellent agreement with the fourth order Runge-Kutta method in a region of deterministic solution. Introducing local stochastic interactions, the Runge-Kutta method is no longer applicable...

Spreading of two different diseases in small world network, with restriction that an individual can be ill only with one disease in the same time, is investigated in the frame of SIRS model. It was found that in the special range of control parameters the presence of the second disease can significantly decrease the number of individuals, who passed the first disease. The speed of propagation o...

An SIRS age-structured epidemic model for a vertically as well as horizontally transmitted disease under vaccination is investigated when the fertility, mortality and removal rates depend on age and the force of infection of proportionate mixing assumption type, and vaccination wanes over time. We prove the existence and uniqueness of solution to the model equations, and show that solutions of ...

In this article, we establish the global asymptotic stability of a disease-free equilibrium and an endemic equilibrium of an SIRS epidemic model with a class of nonlinear incidence rates and distributed delays. By using strict monotonicity of the incidence function and constructing a Lyapunov functional, we obtain sufficient conditions under which the endemic equilibrium is globally asymptotica...

This paper studies the dynamics of a network-based SIRS epidemic model with vaccination and a nonmonotone incidence rate. This type of nonlinear incidence can be used to describe the psychological or inhibitory effect from the behavioral change of the susceptible individuals when the number of infective individuals on heterogeneous networks is getting larger. Using the analytical method, epidem...

In this paper, we consider a stochastic SIRS model with parameter perturbation, which is a standard technique in modeling population dynamics. In our model, the disease transmission coefficient and the removal rates are all affected by noise. We show that the stochastic model has a unique positive solution as it is essential in any population model. Then we establish conditions for extinction o...

In this paper we study global dynamics of a general SIRS epidemic model with a ratiodependent incidence rate and its corresponding stochastic differential equation version. For the deterministic model, we show that the disease reproduction number R0 determines whether there is an endemic outbreak or not: the disease free dynamics occurs if R0 ≤ 1 while the unique endemic steady state is globall...

An epidemic model with standard incidence rate and saturated treatment function of infectious individuals is proposed to understand the effect of the capacity for treatment of infective individuals on the disease spread. The treatment function in this paper is a continuous and differential function which exhibits the effect of delayed treatment when the rate of treatment is lower and the number...