Two types of condition for the global stability of delayed sis epidemic models with nonlinear birth rate and disease induced death rate

We study global asymptotic stability for an SIS epidemic model with maturation delay proposed by K. Cooke, P. van den Driessche and X. Zou, Interaction of maturation delay and nonlinear birth in population and epidemic models, J. Math. Biol. 39(4) (1999) 332352. It is assumed that the population has a nonlinear birth term and disease causes death of infective individuals. By using a monotone iterative method, we establish sufficient conditions for the global stability of an endemic equilibrium when it exists dependently on the monotone property of the birth rate function. Based on the analysis, we further study the model with two specific birth rate functions B 1(N) = be -aN and B 3(N) = A/N + c, where N denotes the total population. For each model, we obtain the disease induced death rate which guarantees the global stability of the endemic equilibrium and this gives a positive answer for an open problem by X. Q. Zhao and X. Zou, Threshold dynamics in a delayed SIS epidemic model, J. Math. Anal. Appl. 257(2) (2001) 282291

Global stability of SIR epidemic models with a wide class of nonlinear incidence rates and distributed delays

In this paper, we establish the global asymptotic stability of equi-libria for an SIR model of infectious diseases with distributed time delays gov-erned by a wide class of nonlinear incidence rates. We obtain the global prop-erties of the model by proving the permanence and constructing a suitable Lyapunov functional. Under some suitable assumptions on the nonlinear term in the incidence rate, the global dynamics of the model is completely deter-mined by the basic reproduction number R0 and the distributed delays do not influence the global dynamics of the model

Global stability of sirs epidemic models with a class of nonlinear incidence rates and distributed delays

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 asymptotically stable. When the nonlinear incidence rate is a saturated incidence rate, our result provides a new global stability condition for a small rate of immunity loss

On the global stability of an SIRS epidemic model with distributed delays

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 stability. By using an extension of a Lyapunov functional established by [C.C. McCluskey, Complete global stability for an SIR epidemic model with delay-Distributed or discrete, Nonlinear Anal. RWA. 11 (2010) 55-59], we provide a partial answer to an open problem whether the endemic equilibrium is globally stable, whenever it exists, or not

Monotone iterative techniques to SIRS epidemic models with nonlinear incidence rates and distributed delays

In this paper, for SIRS epidemic models with a class of nonlinear incidence rates and distributed delays of the forms βS(t)∫0hk(τ)G(I(t-τ) )dτ, we establish the global asymptotic stability of the disease-free equilibrium E0 for R0<1, and applying new monotone techniques, we obtain sufficient conditions which ensure the global asymptotic stability of the endemic equilibrium of the system. The obtained results improve those in Xu and Ma (2009) [Stability of a delayed SIRS epidemic model with a nonlinear incidence rate, Chaos Solitons Fractals 41 (2009) 23192325], and are very useful for a large class of SIRS models

Global stability of a delayed SIRS epidemic model with a non-monotonic incidence rate

In this paper, we investigate a disease transmission model of SIRS type with latent period τ≥0 and the specific nonmonotone incidence rate, namely, For the basic reproduction number R0>1, applying monotone iterative techniques, we establish sufficient conditions for the global asymptotic stability of endemic equilibrium of system which become partial answers to the open problem in [Hai-Feng Huo, Zhan-Ping Ma, Dynamics of a delayed epidemic model with non-monotonic incidence rate, Commun. Nonlinear Sci. Numer. Simul. 15 (2010) 459-468]. Moreover, combining both monotone iterative techniques and the Lyapunov functional techniques to an SIR model by perturbation, we derive another type of sufficient conditions for the global asymptotic stability of the endemic equilibrium

Global stability for a discrete SIS epidemic model with immigration of infectives

In this paper, we propose a discrete-time SIS epidemic model which is derived from continuous-time SIS epidemic models with immigration of infectives by the backward Euler method. For the discretized model, by applying new Lyapunov function techniques, we establish the global asymptotic stability of the disease-free equilibrium for R 0 ≤ 1 and the endemic equilibrium for R 0 > 1, where R 0 is the basic reproduction number of the continuous-time model. This is just a discrete analogue of a continuous SIS epidemic model with immigration of infectives

Lyapunov functional techniques for the global stability analysis of a delayed SIRS epidemic model

In this paper, we study the global dynamics of a delayed SIRS epidemic model for transmission of disease with a class of nonlinear incidence rates of the form βS(t)∫ 0 hf(τ)G(I(t-τ))dτ. Applying Lyapunov functional techniques in the recent paper [Y. Nakata, Y. Enatsu, Y. Muroya, On the global stability of an SIRS epidemic model with distributed delays, Discrete Contin. Dyn. Syst. Supplement (2011) 11191128], we establish sufficient conditions of the rate of immunity loss for the global asymptotic stability of an endemic equilibrium for the model. In particular, we offer a unified construction of Lyapunov functionals for both cases of R 0 ≤ 1 and R 0 > 1, where R 0 is the basic reproduction number

Permanence and global stability of a class of discrete epidemic models

In this paper we investigate the permanence of a system and give a sufficient condition for the endemic equilibrium to be globally asymptotically stable, which are the remaining problems in our previous paper (G. Izzo, Y. Muroya, A. Vecchio, A general discrete time model of population dynamics in the presence of an infection, Discrete Dyn. Nat. Soc. (2009), Article ID 143019, 15 pages. doi:10.1155/2009/143019.

Global dynamics of difference equations for SIR epidemic models with a class of nonlinear incidence rates

In this paper, by applying a variation of the backward Euler method, we propose a discrete-time SIR epidemic model whose discretization scheme preserves the global asymptotic stability of equilibria for a class of corresponding continuous-time SIR epidemic models. Using discrete-time analogue of Lyapunov functionals, the global asymptotic stability of the equilibria is fully determined by the basic reproduction number, when the infection incidence rate has a suitable monotone property