Global dynamics in a multi-group epidemic model for disease with latency spreading and nonlinear transmission rate

In this paper, we investigate a class of multi-group epidemic models with general exposed distribution and nonlinear incidence rate. Under biologically motivated assumptions, we show that the global dynamics are completely determined by the basic production number $R_0$. The disease-free equilibrium is globally asymptotically stable if $R_0\leq1$, and there exists a unique endemic equilibrium which is globally asymptotically stable if $R_0>1$. The proofs of the main results exploit the persistence theory in dynamical system and a graph-theoretical approach to the method of Lyapunov functionals. A simpler case that assumes an identical natural death rate for all groups and a gamma distribution for exposed distribution is also considered. In addition, two numerical examples are showed to illustrate the results.     

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.     

Dynamics of a delayed epidemic model with non-monotonic incidence rate

A delayed epidemic model with non-monotonic incidence rate which describes the psychological effect of certain serious on the community when the number of infectives is getting larger is studied. The disease-free equilibrium is globally asymptotically stable when R₀<1 and is globally attractive when R₀=1 are derived. On the other hand, The disease is permanent when R₀>1 is also obtained. Numerical simulation results are given to support the theoretical predictions.     

Global behavior and permanence of SIRS epidemic model with time delay

In this paper an autonomous SIRS epidemic model with time delay is studied. The basic reproductive number R₀ is obtained which determines whether the disease is extinct or not. When the basic reproductive number is greater than 1, it is proved that the disease is permanent in the population, and explicit formula are obtained by which the eventual lower bound of the fraction of infectious individuals can be computed. Throughout the total paper, we mainly use the technique of Lyapunov functional to establish the global stability of the infection-free equilibrium and the local stability of the endemic equilibrium but need another sufficient condition.     

Analysis of epidemic spreading of an SIRS model in complex heterogeneous networks

In this paper, we study the spreading of infections in complex heterogeneous networks based on an SIRS epidemic model with birth and death rates. We find that the dynamics of the network-based SIRS model is completely determined by a threshold value. If the value is less than or equal to one, then the disease-free equilibrium is globally attractive and the disease dies out. Otherwise, the disease-free equilibrium becomes unstable and in the meantime there exists uniquely an endemic equilibrium which is globally asymptotically stable. A series of numerical experiments are given to illustrate the theoretical results. We also consider the SIRS model in the clustered scale-free networks to examine the effect of network community structure on the epidemic dynamics.     

Global stability of a stage-structured epidemic model with a nonlinear incidence

In this paper, a stage-structured epidemic model with a nonlinear incidence with a factor S^p is investigated. By using limit theory of differential equations and Theorem of Busenberg and van den Driessche, global dynamics of the model is rigorously established. We prove that if the basic reproduction number R₀ is less than one, the disease-free equilibrium is globally asymptotically stable and the disease dies out; if R₀ is greater than one, then the disease persists and the unique endemic equilibrium is globally asymptotically stable. Numerical simulations support our analytical results and illustrate the effect of p on the dynamic behavior of the model.     

Global stability of multi-group epidemic models with distributed delays

We investigate a class of multi-group epidemic models with distributed delays. We establish that the global dynamics are completely determined by the basic reproduction number R₀. More specifically, we prove that, if R₀?1, then the disease-free equilibrium is globally asymptotically stable; if R₀>1, then there exists a unique endemic equilibrium and it is globally asymptotically stable. Our proof of global stability of the endemic equilibrium utilizes a graph-theoretical approach to the method of Lyapunov functionals.     

Global stability of an SIS epidemic model with delays

In this paper, we consider the global dynamics of the S(E)IS model with delays denoting an incubation time. By constructing a Lyapunov functional, we prove stability of a disease‐free equilibrium E₀ under a condition different from that in the recent paper. Then we claim that R₀≤1 is a necessary and sufficient condition under which E₀ is globally asymptotically stable. We also propose a discrete model preserving positivity and global stability of the same equilibria as the continuous model with distributed delays, by means of discrete analogs of the Lyapunov functional.     

Oscillation and global asymptotic stability in a discrete epidemic model

In this paper, we study the oscillation, global asymptotic stability, and other properties of the positive solutions of the difference equation

     

Global stability of an SAIRS epidemic model with vaccinations,transient immunity and treatment

We consider an SAIRS epidemic model with vaccinations and treatment, where asymptomatic and symptomatic infectious individuals are considered in the transmission of the disease. We found the basic reproduction number, ${ℛ}_0$ and using ${ℛ}_0$, we conducted global stability analysis. We proved when ${ℛ}_0<1$, the disease-free equilibrium is globally stable. If ${ℛ}_0>1$, the disease-free equilibrium in unstable and a unique endemic equilibrium exists. We explored the global stability of the endemic equilibrium and noticed it is globally stable under certain conditions. Moreover, we then considered a special case of the SAIRS model, the SAIR model. We proved the disease-free equilibrium is globally stability when ${ℛ}_0<1$ and the endemic equilibrium is globally stable when ${ℛ}_0>1$. Next, we numerically simulated our analytical results and plotted these for various cases. Finally, we performed sensitivity analysis to tell us how each parameter in the system affects disease transmission.     

Global dynamics of a discretized SIRS epidemic model with time delay

We derive a discretized SIRS epidemic model with time delay by applying a nonstandard finite difference scheme. Sufficient conditions for the global dynamics of the solution are obtained by improvements in discretization and applying proofs for continuous epidemic models. These conditions for our discretized model are the same as for the original continuous model.     

Global dynamics of a cell mediated immunity in viral infection models with distributed delays

In this paper, we investigate global dynamics for a system of delay differential equations which describes a virus-immune interaction in vivo. The model has two distributed time delays describing time needed for infection of cell and virus replication. Our model admits three possible equilibria, an uninfected equilibrium and infected equilibrium with or without immune response depending on the basic reproduction number for viral infection R₀ and for CTL response R₁ such that R₁<R₀. It is shown that there always exists one equilibrium which is globally asymptotically stable by employing the method of Lyapunov functional. More specifically, the uninfected equilibrium is globally asymptotically stable if R₀?1, an infected equilibrium without immune response is globally asymptotically stable if R₁?1<R₀ and an infected equilibrium with immune response is globally asymptotically stable if R₁>1. The immune activation has a positive role in the reduction of the infection cells and the increasing of the uninfected cells if R₁>1.     

Global asymptotic stability and a property of the SIS model on bipartite networks

In this paper, we study an SIS model on bipartite networks, in which the network structure and a connectivity-dependent infection scheme are considered. Applying the theory of the multigroup model, we prove the existence and the asymptotic stability of the endemic equilibrium. And then we examine the ratio between the densities of infected female and male individuals on the bipartite networks. In particular, we find that when the scale exponent (γ_F) of females is equal to and that of males (γ_M), the ratio is only determined by the scale exponents and the proportion between the infection rates of females and males (λ_F/λ_M). We also present a result for the ratio by numerical simulations when γ_F≠γ_M.     

Local and global stability analysis of a two prey one predator model with help

In this paper we propose and study a three dimensional continuous time dynamical system modelling a three team consists of two preys and one predator with the assumption that during predation the members of both teams of preys help each other and the rate of predation of both teams are different. In this work we establish the local asymptotic stability of various equilibrium points to understand the dynamics of the model system. Different conditions for the coexistence of equilibrium solutions are discussed. Persistence, permanence of the system and global stability of the positive interior equilibrium solution are discussed by constructing suitable Lyapunov functional. At the end, numerical simulations are performed to substantiate our analytical findings.     

Complete global analysis of an SIRS epidemic model with graded cure and incomplete recovery rates

In this paper, applying two types of Lyapunov functional techniques to an SIRS epidemic model with graded cure and incomplete recovery rates, we establish complete global dynamics of the model whose threshold parameter is the basic reproduction number _R0$R_0$ such that the disease-free equilibrium is globally asymptotically stable when _R0?1$R_0?1$, and the endemic equilibrium is globally asymptotically stable when _R0>1$R_0>1$.     

Complete global stability for an SIRS epidemic model with generalized non-linear incidence and vaccination

This paper deals with global dynamics of an SIRS epidemic model for infections with non permanent acquired immunity. The SIRS model studied here incorporates a preventive vaccination and generalized non-linear incidence rate as well as the disease-related death. Lyapunov functions are used to show that the disease-free equilibrium state is globally asymptotically stable when the basic reproduction number is less than or equal to one, and that there is an endemic equilibrium state which is globally asymptotically stable when it is greater than one.     

Global exponential stability for a class of retarded functional differential equations with applications in neural networks

In this paper, the global exponential stability and asymptotic stability of retarded functional differential equations with multiple time-varying delays are studied by employing several Lyapunov functionals. A number of sufficient conditions for these types of stability are presented. Our results show that these conditions are milder and more general than previously known criteria, and can be applied to neural networks with a broad range of activation functions assuming neither differentiability nor strict monotonicity. Furthermore, the results obtained for neural networks with time-varying delays do not assume symmetry of the connection matrix.