Global Stability Of Equilibria Research Articles

Global stability of discrete virus dynamics models with humoural immunity and latency

This paper studies the global stability of discrete-time viral infection models with humoural immunity. We consider both latently and actively infected cells. We study also a model with general production and clearance rates of all compartments as well as general incidence rate of infection. We use nonstandard finite difference method to discretize the continuous-time models. The positivity and boundedness of solutions of the discrete models are established. We establish by using Lyapunov method, the global stability of equilibria in terms of the basic reproduction number and the humoural immune response activation number . The results signify that the infection dies out if . Moreover, the infection persists with inactive immune response if and with active immune response if . We illustrate our theoretical results by using numerical simulations.

Analysis of HIV Transmission of Commercial Sex Wokers and Their Clients with Condom Use Treatment

A model of the HIV/AIDS epidemic among sex workers and their clients is discussed to study the effects of condom use in the prevention of HIV transmission. The model is addressed to determine the existence of equilibrium states, and then analyze the global stability of disease free and endemic equilibrium states. The global stability of equilibria depends on the vales of the basic reproduction ratio derived from the next generation matrix of the model. The endemic equilibrium state is globally stable when the ratio exceeds unity. The simulation results are presented to discuss the effect of condom use treatment in preventing the spread of HIV/AIDS among sex workers and their clients. The results show that the effectiveness level in using condoms in sexual intercourse corresponds to the decreasing level of the spread of HIV/AIDS. We use Maple and Matlab software to simulate the impact of condom use.

A Fractional Order Model for Viral Infection with Cure of Infected Cells and Humoral Immunity

In this paper, we study the dynamics of a viral infection model formulated by five fractional differential equations (FDEs) to describe the interactions between host cells, virus, and humoral immunity presented by antibodies. The infection transmission process is modeled by Hattaf-Yousfi functional response which covers several forms of incidence rate existing in the literature. We first show that the model is mathematically and biologically well-posed. By constructing suitable Lyapunov functionals, the global stability of equilibria is established and characterized by two threshold parameters. Finally, some numerical simulations are presented to illustrate our theoretical analysis.

Dynamics of a diffusive virus model with general incidence function, cell-to-cell transmission and time delay

In this paper, we revisit a diffusive virus dynamics model with general incidence function and time delay. One novelty of our model is that we introduce cell-to-cell transmission via formation of virological synapses to reflect the fact that it may play a more important role in virus spreading in addition to virus-to-cell infection. We justify the well-posedness of the model and identify the basic reproduction number ℜ0 for the model to be a sharp threshold value. The global stability of equilibria is determined by constructing suitable Lyapunov functionals in the sense that: the infection-free equilibrium is globally asymptotically stable if ℜ0≤1, and when ℜ0>1, the global asymptotic stability of infection equilibrium implies that the infection will persist. A significant impact of the cell-to-cell transmission is that they increase the basic reproduction number. If one neglects either the cell-to-cell transmission or virus-to-cell infection, the basic reproduction number of the model that is under-evaluated. Last, we perform numerical simulation to support our theoretic results. We set the domain of the viruses to be a two-dimensional square domain with the homogeneous Neumann boundary conditions to reflect the spatial spreading.

Global stability of an information-related epidemic model with age-dependent latency and relapse

We study the problem of information-related contact rate for an SEIR epidemic model with age-dependent latency and relapse. The contact pattern includes an information variable which is a negative feedback of susceptible individuals related to the memory of past and current states of infectious disease. We prove the asymptotic smoothness of solutions and the existence of equilibria in the positive invariant set. And we show uniform persistence of the system by comparing with a system of Volterra type integro-differential equations. Importantly, it follows from appropriate conditions that the local stability and global stability of equilibria can be concluded by the methods of corresponding characteristic equations and proper Lyapunov functionals, respectively. Finally, we give numerical simulations to illustrate the influence of the information on changing the coordinate of endemic equilibrium by increasing susceptible population and decreasing infectious population. It is remarkable to find that information coverage and the length of disease-relate memory would work effectively on the progression of disease.

THE MATHEMATICAL ANALYSIS OF THE MODEL OF ANTIBIOTIC-RESISTANT BACTERIA AND THE IMMUNE CELLS

Currently, World Health Organization (WHO) report confirms that the antibiotic-resistant infections are the greatest threat to health. Despite the new therapeutic strategies, the bacteria develop mechanisms to defend themselves against antibiotics. In this paper, we propose a mathematical model describing the dynamics of resistant bacteria, non-resistant bacteria and immune cells exposed to an antibiotic. The global stability of equilibria is performed by using Lyapunov functions.

Analysis of an SVIC model with age-dependent infection and asymptomatic carriers

In this paper,we formulated an age-dependent model for the transmission dynamics of HBV with vaccination. The class of acutely infectious individuals,asymptomatic carrier of host population is stratified by age. Mathematically, we established that basic reproduction number can govern the global stability of equilibria. Biologically, we verify the impacts of the asymptomatic carriers and the effectiveness of vaccination on the disease transmission through numerical simulation. Our results indicated that the more number of infectious individuals specific to frequently progressed to asymptomatic carriers, the more likely the disease can be eradicated by continuous vaccination strategies.

Cyclic Growth and Global Stability of Economic Dynamics of Kaldor Type in Two Dimensions

This article proposes nonlinear economic dynamics continuous in two dimensions of Kaldor type, the saving rate and the investment rate, which are functions of ecological origin verifying the nonwasting properties of the resources and economic assumption of Kaldor. The important results of this study contain the notions of bounded solutions, the existence of an attractive set, local and global stability of equilibrium, the system permanence, and the existence of a limit cycle.

Threshold dynamics of a nonlinear multi‐group epidemic model with two infinite distributed delays

The global stability of equilibria is investigated for a nonlinear multi‐group epidemic model with latency and relapses described by two distributed delays. The results show that the global dynamics are completely determined by the basic reproduction number under certain reasonable conditions on the nonlinear incidence rate. Moreover, compared with the results in Michael Y. Li and Zhisheng Shuai, Journal Differential Equations 248 (2010) 1–20, it is found that the two distributed delays have no impact on the global behaviour of the model. Our study improves and extends some known results in recent literature. Copyright © 2016 John Wiley & Sons, Ltd.

Global stability of a multi‐group model with distributed delay and vaccination

A delayed multi‐group SVEIR epidemic model with vaccination and a general incidence function has been formulated and studied in this paper. Mathematical analysis shows that the basic reproduction number plays a key role in the dynamics of the model: the disease‐free equilibrium is globally asymptotically stable when , while the endemic equilibrium exists uniquely and is globally asymptotically stable when . For the proofs, we exploit a graph‐theoretical approach to the method of Lyapunov functionals. Our results show that distributed delay has no impact on the global stability of equilibria, and the results improve and generalize some known results. Copyright © 2016 John Wiley & Sons, Ltd.

Dynamically consistent discrete metapopulation model

In this paper we construct nonstandard difference schemes, which are dynamically consistent with a metapopulation model formulated by Keymer et al. in 2000, i.e. preserve all dynamical properties of the differential equations of the model. These properties are: monotone convergence, boundedness, local asymptotic stability and especially, global stability of equilibria and non-periodicity of solutions. Numerical examples confirm the obtained theoretical results of the properties of the constructed difference schemes.

Bifurcation and Stability Analysis of a Food Web in a Chemostat

In this study, a classical model describing a food web in a chemostat involving three species competing for non-reproducing, growth rate-limiting nutrient in which one of the competitors predates on one of the other competitors is considered. Quantitative analyses of non-negativity and boundedness of solution trajectories, dissipativity, behavior around equilibria, global stability and persistence of the model equations are analyzed. We present the global stability of equilibria by constructing a Lyapunov function. Hopf bifurcation theory is applied.

Dynamical Analysis of a Parasite-Host Model within Fluctuating Environment

A parasite-host model within fluctuating environment is proposed. Firstly, the positivity and boundedness of solutions of the model within deterministic environment are discussed, and, then, the asymptotical stability and global stability of equilibria of deterministic model are investigated. Secondly, we show that the stochastic model has a unique global positive solution; furthermore, we show that the stochastic model has a stationary distribution under certain conditions. Finally, we give some numerical simulations to illustrate our analytical results.

Modelling effects of public health educational campaigns on drinking dynamics

ABSTRACTThis paper deals with the global property of a drinking model with public health educational campaigns. With the help of Lyapunov function, global stability of equilibria of the model is derived. The alcohol-free equilibrium is globally asymptotically stable and the alcohol problems are eliminated from population if . A unique alcohol present equilibrium is globally asymptotically stable if . Furthermore, the basic reproductive for the model is compared with the basic reproductive number for the absence of public health educational campaigns. We conclude that the public health educational campaigns of drinking individuals can slow down the drinking dynamics. Some numerical simulations are also given to explain our conclusions.

Global dynamics for viral infection model with Beddington-DeAngelis functional response and an eclipse stage of infected cells

In this paper, we formulate a viral infection model with Beddington-DeAngelis functional response and an eclipse stage of infected cells. The model can describe the in vivo infection dynamics of many viruses such as HIV-I, HCV, and HBV, where the infected cells of eclipse stage can revert to the uninfected class. Under certain parameters range, we establish that the global stability of equilibria is completely determined by the basic reproduction number $\mathfrak{R}_0$, which give us a complete picture on their global dynamics.

Global stability of an SEIR epidemic model with age-dependent latency and relapse

In this paper, an SEIR epidemic model for a disease with age-dependent latency and relapse is proposed. Such a model is very appropriate for tuberculosis and herpes. The basic reproduction number ℜ0 is defined and proved as a classic threshold parameter. Asymptotic smoothness of solutions and uniform persistence of the system are shown via reformulating the system as a system of Volterra integral equations. The local stability and global stability of equilibria are obtained by analyzing the corresponding characteristic equations and constructing the proper Volterra-type Lyapunov functionals, respectively. Finally, some special cases of the model as key functions reducing to constant are discussed in the last section.

The Dynamics of a Diffusive Nutrient-Algae Model Based upon the Sanyang Wetland

The stability and spatiotemporal dynamics of a diffusive nutrient-algae model are investigated mathematically and numerically. Mathematical theoretical studies have considered the positivity and boundedness of the solution and the existence, local stability, and global stability of equilibria. Turing instability has also been studied. Furthermore, a series of numerical simulations was performed and a complex Turing pattern found. These results indicate that the nutrient input rate has an important influence on the density and spatial distribution of algae populations. This may help us to obtain a better understanding of the interactions of nutrient and algae and to investigate plankton dynamics in aquatic ecosystems.

The dynamics of a time delayed epidemic model on a population with birth pulse

An epidemic model with time delay and impulse is proposed to describe if time delay plays an important role in the spread of an infectious disease. Mathematical analyses with regard to the local, global stability of equilibrium are performed. We give out the conditions for global attractivity of the disease-free equilibrium and permanence of the system, respectively.

The selection pressures induced non-smooth infectious disease model and bifurcation analysis

Mathematical models can assist in the design strategies to control emerging infectious disease. This paper deduces a non-smooth infectious disease model induced by selection pressures. Analysis of this model reveals rich dynamics including local, global stability of equilibria and local sliding bifurcations. Model solutions ultimately stabilize at either one real equilibrium or the pseudo-equilibrium on the switching surface of the present model, depending on the threshold value determined by some related parameters. Our main results show that reducing the threshold value to a appropriate level could contribute to the efficacy on prevention and treatment of emerging infectious disease, which indicates that the selection pressures can be beneficial to prevent the emerging infectious disease under medical resource limitation.

Global stability of a transport-related infection model with general incidence rate in two heterogeneous cities

To further understand the effects of travel on disease spread, a transport-related infection model with general incidence rate in two heterogeneous cities is proposed and analyzed. Some analytical results on the global stability of equilibria (including disease free equilibrium and endemic equilibrium) are obtained. The explicit formula for the basic reproduction number R0 is derived and it is proved to be a threshold for disease spread. To reveal how incidence rate and travel rate influence the disease spread, effects of general incidence rate and travel rate on the dynamics of system are shown via numeric simulations.