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.     

Persistence and global stability for n-species ratio-dependent predator-prey system with time delays

A n-species ratio-dependent predator-prey food-chain model with time delays is investigated. It is shown that the system is permanent under some appropriate conditions, and sufficient conditions are obtained for the global stability of the positive equilibrium of the system.     

Stability and Hopf bifurcation in a viral infection model with nonlinear incidence rate and delayed immune response

A viral infection model with nonlinear incidence rate and delayed immune response is investigated. It is shown that if the basic reproduction ratio of the virus is less than unity, the infection-free equilibrium is globally asymptotically stable. By analyzing the characteristic equation, the local stability of the chronic infection equilibrium of the system is discussed. Furthermore, the existence of Hopf bifurcations at the chronic infection equilibrium is also studied. By means of an iteration technique, sufficient conditions are obtained for the global attractiveness of the chronic infection equilibrium. Numerical simulations are carried out to illustrate the main results.     

Periodic solution and global stability for a nonautonomous competitive Lotka-Volterra diffusion system

The periodic solution and global stability for a nonautonomous competitive Lotka-Volterra diffusion system is considered in this paper. By using of Brouwer fixed point theorem and constructing a suitable Liapunov function, under some appropriate conditions, the system has a unique periodic solution which is globally stable.     

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 stability of humoral immunity virus dynamics models with nonlinear infection rate and removal

In this paper, we investigate the dynamical behavior of two nonlinear models for viral infection with humoral immune response. The first model contains four compartments; uninfected target cells, actively infected cells, free virus particles and B cells. The intrinsic growth rate of uninfected cells, incidence rate of infection, removal rate of infected cells, production rate of viruses, neutralization rate of viruses, activation rate of B cells and removal rate of B cells are given by more general nonlinear functions. The second model is a modification of the first one by including an eclipse stage of infected cells. We assume that the latent-to-active conversion rate is also given by a more general nonlinear function. For each model we derive two threshold parameters and establish a set of conditions on the general functions which are sufficient to determine the global dynamics of the models. By using suitable Lyapunov functions and LaSalle’s invariance principle, we prove the global asymptotic stability of the all equilibria of the models. We perform some numerical simulations for the models with specific forms of the general functions and show that the numerical results are consistent with the theoretical results.     

Global stability and periodic solution of the viral dynamics

It is well known that the mathematical models provide very important information for the research of human immunodeficiency virus-type 1 and hepatitis C virus (HCV). However, the infection rate of almost all mathematical models is linear. The linearity shows the simple interaction between the T cells and the viral particles. In this paper, we consider the classical mathematical model with saturation response of the infection rate. By stability analysis we obtain sufficient conditions on the parameters for the global stability of the infected steady state and the infection-free steady state. We also obtain the conditions for the existence of an orbitally asymptotically stable periodic solution. Numerical simulations are presented to illustrate the results.     

Boundedness and global exponential stability for delayed differential equations with applications

The boundedness of solutions for a class of n-dimensional differential equations with distributed delays is established by assuming the existence of instantaneous negative feedbacks which dominate the delay effect. As an important by-product, some criteria for global exponential stability of equilibria are obtained. The results are illustrated with applications to delayed neural networks and population dynamics models.     

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

     

On the use of the geometric approach to global stability for three dimensional ODE systems: A bilinear case

In this paper, we consider a general bilinear three dimensional ODE system, whose structure generalizes many mathematical models of biological interest, including many from epidemics. Our main goal is to find sufficient conditions, expressed in terms of the parameters of the system, ensuring that the geometric approach to global stability analysis, due to [M.Y. Li, J.S. Muldowney, A geometric approach to global-stability problems, SIAM J. Math. Anal. 27 (4) (1996) 1070-1083], may be successfully applied. We completely determine the dynamics of the general system, including thresholds and global stability of the nontrivial equilibrium. The obtained result is applied to several epidemic models. We further show how the role of new parameters on stability of well-established models may be emphasized.     

On the global offensive alliance number of a graph

An offensive alliance in a graph Γ=(V,E) is a set of vertices S⊂V where for each vertex v in its boundary the majority of vertices in v’s closed neighborhood are in S. In the case of strong offensive alliance, strict majority is required. An alliance S is called global if it affects every vertex in V?S, that is, S is a dominating set of Γ. The global offensive alliance numberγ_o(Γ) is the minimum cardinality of a global offensive alliance in Γ. An offensive alliance is connected if its induced subgraph is connected. The global-connected offensive alliance number, γ_co(Γ), is the minimum cardinality of a global-connected offensive alliance in Γ.In this paper we obtain several tight bounds on γ_o(Γ) and γ_co(Γ) in terms of several parameters of Γ. The case of strong alliances is studied by analogy.     

Exponential and global stability of nonlinear dynamical systems relative to initial time difference

The exponential and global stability of nonlinear differential dynamical systems with different initial times are investigated. Several criteria for the stability of nonlinear dynamical systems relative to initial time difference are obtained by means of vector Lyapunov functions. The obtained criteria have been applied to a proposed differential dynamic system. The numerical simulation validates our conclusions.     

Permanence extinction and global asymptotic stability in a stage structured system with distributed delays

In this paper we consider a nonautonomous stage-structured competitive system of n-species population growth with distributed delays which takes into account the delayed feedback in both interspecific and intraspecific interactions. We obtain, by using the method of repeated replace, sufficient conditions for permanence and extinction of the species. The global attractivity of the unique positive equilibrium is proved in the autonomous case. Our results extend previous ones obtained by Liu et al. in [Nonlinear Anal. 51 (2002) 1347-1361; J. Math. Anal Appl. 274 (2002) 667-684].     

Persistence and global stability in discrete models of Lotka-Volterra type

In this paper, we establish new sufficient conditions for global asymptotic stability of the positive equilibrium in the following discrete models of Lotka-Volterra type:

     

Dynamics analysis of a delayed viral infection model with immune impairment

In this paper, the dynamical behavior of a delayed viral infection model with immune impairment is studied. It is shown that if the basic reproductive number of the virus is less than one, then the uninfected equilibrium is globally asymptotically stable for both ODE and DDE model. And the effect of time delay on stabilities of the equilibria of the DDE model has been studied. By theoretical analysis and numerical simulations, we show that the immune impairment rate has no effect on the stability of the ODE model, while it has a dramatic effect on the infected equilibrium of the DDE model.     

On the dynamics of a degenerate parabolic equation: global bifurcation of stationary states and convergence

We study the dynamics of a degenerate parabolic equation with a variable, generally non-smooth diffusion coefficient, which may vanish at some points or be unbounded. We show the existence of a global branch of nonnegative stationary states, covering both the cases of a bounded and an unbounded domain. The global bifurcation of stationary states, implies-in conjuction with the definition of a gradient dynamical system in the natural phase space-that at least in the case of a bounded domain, any solution with nonnegative initial data tends to the trivial or the nonnegative equilibrium. Applications of the global bifurcation result to general degenerate semilinear as well as to quasilinear elliptic equations, are also discussed. Mathematics Subject Classification (1991) 35B40, 35B41, 35R05     

A note on global stability of three-dimensional Ricker models

In the recent paper [E. C. Balreira, S. Elaydi, and R. Luís, J. Differ. Equ. Appl. 23 (2017), pp. 2037–2071], Balreira, Elaydi and Luís established a good criterion for competitive mappings to have a globally asymptotically stable interior fixed point by a geometric approach. This criterion can be applied to three dimensional Kolmogorov competitive mappings on a monotone region with a carrying simplex whose planar fixed points are saddles but globally asymptotically stable on their positive coordinate planes. For three dimensional Ricker models, they found mild conditions on parameters such that the criterion can be applied to. Observing that Balreira, Elaydi and Luís' discussion is still valid for the monotone region with piecewise smooth boundary, we prove in this note that the interior fixed point of three dimensional Kolmogorov competitive mappings is globally asymptotically stable if they admit a carrying simplex and three planar fixed points which are saddles but globally asymptotically stable on their positive coordinate planes. This result is much easier to apply in the application.     

Stability and bifurcations for the chronic state in Marchuk's model of an immune system

In the paper we present known and new results concerning stability and the Hopf bifurcation for the positive steady state describing a chronic disease in Marchuk's model of an immune system. We describe conditions guaranteeing local stability or instability of this state in a general case and for very strong immune system. We compare these results with the results known in the literature. We show that the positive steady state can be stable only for very specific parameter values. Stability analysis is illustrated by Mikhailov's hodographs and numerical simulations. Conditions for the Hopf bifurcation and stability of arising periodic orbit are also studied. These conditions are checked for arbitrary chosen realistic parameter values. Numerical examples of arising due to the Hopf bifurcation periodic solutions are presented.