Global dynamics of alcoholism epidemic model with distributed delays.

This paper aims to investigate the global dynamics of an alcoholism epidemic model with distributed delays. The main feature of this model is that it includes the effect of the social pressure as a factor of drinking. As a result, our global stability is obtained without a "basic reproduction number" nor threshold condition. Hence, we prove that the alcohol addiction will be always uniformly persistent in the population. This means that the investigated model has only one positive equilibrium, and it is globally asymptotically stable independent on the model parameters. This result is shown by proving that the unique equilibrium is locally stable, and the global attraction is shown using Lyapunov direct method.

Threshold dynamics of difference equations for SEIR model with nonlinear incidence function and infinite delay.

In this research, we explore the global conduct of age-structured SEIR system with nonlinear incidence functional (NIF), where a threshold behavior is obtained. More precisely, we will analyze the investigated model differently, where we will rewrite it as a difference equations with infinite delay by the help of the characteristic method. Using standard conditions on the nonlinear incidence functional that can fit with a vast class of a well-known incidence functionals, we investigated the global asymptotic stability (GAS) of the disease-free equilibrium (DFE) using a Lyapunov functional (LF) for R 0 ≤ 1 . The total trajectory method is utilized for avoiding proving the local behavior of equilibria. Further, in the case R 0 > 1 we achieved the persistence of the infection and the GAS of the endemic equilibrium state (EE) using the weakly ρ -persistence theory, where a proper LF is obtained. The achieved results are checked numerically using graphical representations.

Global stability for a class of functional differential equations with distributed delay and non-monotone bistable nonlinearity.

The present work is devoted to the global stability analysis for a class of functional differential equations with distributed delay and non-monotone bistable nonlinearity. First, we characterize some subsets of attraction basins of equilibria. Next, by Lyapunov functional and fluctuation method, we obtain a series of criteria for the global stability of equilibria. Finally, we illustrate our results by applying them to a problem with Allee effect.

Global stability of an age-structured epidemic model with general Lyapunov functional.

In this paper, we focus on the study of the dynamics of a certain age structured epidemic model. Our aim is to investigate the proposed model, which is based on the classical SIR epidemic model, with a general class of nonlinear incidence rate with some other generalization. We are interested to the asymptotic behavior of the system. For this, we have introduced the basic reproduction number R0 of model and we prove that this threshold shows completely the stability of each steady state. Our approach is the use of general constructed Lyapunov functional with some results on the persistence theory. The conclusion is that the system has a trivial disease-free equilibrium which is globally asymptotically stable for R0 < 1 and that the system has only a unique positive endemic equilibrium which is globally asymptotically stable whenever R0 > 1. Several numerical simulations are given to illustrate our results.