Global stability of a delayed SEIRS epidemic model with saturation incidence rate

In this paper, an SEIRS epidemic model with a saturation incidence rate and a time delay describing a latent period is investigated. By analyzing the corresponding characteristic equations, the local stability of an endemic equilibrium and a disease-free equilibrium is established. When the basic reproduction number is greater than unity, by means of an iteration technique, sufficient conditions are obtained for the global asymptotic stability of the endemic equilibrium. By comparison arguments, it is proved that if the basic reproduction number is less than unity, the disease-free equilibrium is globally asymptotically stable. Numerical simulations are carried out to illustrate the main theoretical results.     

Periodic solutions of an epidemic model with saturated treatment

Based on the fact that many infectious diseases exhibit periodic fluctuations and there is a saturated phenomenon during disease treatment, we study an SIR model with periodic incidence rate and saturated treatment function. Firstly, we find that the basic reproduction number less than 1 cannot insure the global stability of disease-free equilibrium and it needs to add other conditions. Moreover, we establish sufficient conditions for the multiplicity of positive periodic solutions. We also apply the numerical method to confirm theoretical results and show the stability of the periodic solutions. We observe that there are two periodic solutions in the system where one is stable and the other one is unstable. These results will provide some guidance for control measures of disease.     

GLOBAL STABILITY ANALYSIS IN CELLULAR NEURAL NETWORKS WITH UNBOUNDED TIME DELAYS

Without assuming the boundedness and differentiability of the activation functions, the conditions ensuring existence, uniqueness, and global asymptotical stability of the equilibrium point of cellular neural networks with unbounded time delays and variable delays were studied. Using the idea of vector Liapunov method, the intero-differential inequalities with unbounded delay and variable delays were constructed. By the stability analysis of the intero-differential inequalities, the sufficient conditions for global asymptotic stability of cellular neural networks were obtained.     

Global Analysis of Two Tuberculosis Models

Two models for tuberculosis (TB) that include treatment of latent and infective individuals are considered. The first model assumes constant recruitment with a fixed fraction entering each class, having the consequences that TB never dies out and that the usual threshold condition does not apply. The unique endemic equilibrium is locally asymptotically stable for all parameter values and is shown to be globally asymptotically stable under certain parameter restrictions. The second model has a general recruitment function, but all recruitment is into the susceptible class. Three threshold parameters determine the existence and local stability of equilibria. The disease-free equilibrium is globally asymptotically stable when the basic reproduction number is less than or equal to one. The endemic equilibrium, when it exists, is shown to be globally asymptotically stable under certain parameter restrictions. Global stability results for the endemic equilibria are proved using the geometric approach of Li and Muldowney.     

Dynamic analysis of mathematical model of ethanol fermentation with gas stripping

In this paper, a mathematical model for ethanol fermentation with gas stripping is investigated. Firstly, the model with continuous substrate input is taken. We study the existence and local stability of two equilibrium points. According to Poincare–Bendixson Theorem, the sufficient condition for the globally asymptotical stability of positive equilibrium point is obtained, which implies that we can get stable ethanol product. Secondly, we study the model with impulsive substrate input and obtain the sufficient condition for the local stability of cell-free periodic solution by using the Floquet’s theory of impulsive differential equation and small-amplitude perturbation skills. In a certain limiting case, it is shown that a nontrivial periodic solution emerges via a supercritical (subcritical) bifurcation. Finally, our results are confirmed by means of numerical simulation.     

Periodic solution of a turbidostat model with impulsive state feedback control

In this paper, a turbidostat model with impulsive state feedback control is considered. We obtain sufficient conditions of the global asymptotical stability of the system without impulsive state feedback control. We also obtain that the system with impulsive state feedback control may have order one periodic solution, and the sufficient condition for existence and stability of order one periodic solution is gotten as well. For some special cases, it is shown that in the system an order two periodic solution may exist. Our results show that the control measure is effective and reliable.     

Stability analysis and estimation of domain of attraction for the endemic equilibrium of an SEIQ epidemic model

This paper mainly addressed the stability analysis and the estimation of domain of attraction for the endemic equilibrium of a class of susceptible-exposed-infected-quarantine epidemic models. Firstly, we discuss the global stability of the disease-free equilibrium and the local stability of the endemic equilibrium in the feasible region D of the epidemic model, respectively. Secondly, we use a geometric approach to investigate the global stability of the endemic equilibrium in a positive invariant region \(D_s(\subset D)\). Furthermore, we estimates the domain of attraction for the endemic equilibrium via sum of squares optimization method, and obtain the optimal estimation by solving an semidefinite programming problem with sum of squares polynomial constraints. Finally, numerical simulation is examined to demonstrate the feasibility and effectiveness of the research results.     

The study of a ratio-dependent predator–prey model with stage structure in the prey

In this paper, a ratio-dependent predator–prey model with stage structure in the prey is constructed and investigated. In the first part of this paper, some sufficient conditions for the existence and stability of three equilibriums are obtained. In the second part, we consider the effect of impulsive release of predator on the original system. A sufficient condition for the global asymptotical stability of the prey-eradication periodic solution is obtained. We also get the condition, under which the prey would never be eradicated, i.e., the impulsive system is permanent. At last, we give a brief discussion.

     

A LMI-based approach to global asymptotic stability of neural networks with time varying delays

In this paper, the asymptotic stability of neural networks with time varying delay is studied by using the nonsmooth analysis, Lyapunov functional method and linear matrix inequality (LMI) technique. It is noted that the proposed results do not require smoothness of the behaved function and activation function as well as boundedness of the activation function. Several sufficient conditions are presented to show the uniqueness and the global asymptotical stability of the equilibrium point. Also, a high-dimensional matrix condition to ensure the uniqueness and the global asymptotical stability of equilibrium point can be reduced to a low-dimensional condition. The obtained results are easy to apply and improve some earlier works. Finally, we give two simulations to justify the theoretical analysis in this paper. This work was supported by the Natural Science Foundation of Jiangsu Province of China under Grant No. BK2006093.     

Codimension-two bifurcation analysis in two-dimensional Hindmarsh–Rose model

In this paper, we analyze the codimension-2 bifurcations of equilibria of a two-dimensional Hindmarsh–Rose model. By using the bifurcation methods and techniques, we give a rigorous mathematical analysis of Bautin bifurcation. The main result is that no more than two limit cycles can be bifurcated from the equilibrium via Hopf bifurcation; sufficient conditions for the existence of one or two limit cycles are obtained. This paper also shows that the model undergoes a Bogdanov–Takens bifurcation which includes a saddle-node bifurcation, an Andronov–Hopf bifurcation, and a homoclinic bifurcation. In some case, the globally asymptotical stability is discussed.     

An SIR model with infection delay and propagation vector in complex networks

Based on the SIR (Susceptible-Infected-Removed) model, we propose a novel epidemic model to investigate the impact of infection delay and propagation vector on the spreading behaviors in complex networks. Mean-field approximations and extensive numerical simulations indicate that the infection delay and propagation vector can largely reduce the critical threshold and promote the outbreak of epidemics, and even lead to the case that the infectious diseases transform from the disease-free state to endemic one. The current results are greatly instructive for us to further understand the epidemic spreading and design some effective prevention and containment strategies to fight the epidemics.     

Global stability in switched recurrent neural networks with time-varying delay via nonlinear measure

In this paper, based on switched systems and recurrent neural networks (RNNs) with time-varying delay, the model of switched RNNs is formulated. Global asymptotical stability (GAS) and global robust stability (GRS) for such switched neural networks are studied by employing nonlinear measure and linear matrix inequality (LMI) techniques. Some new sufficient conditions are obtained to ensure GAS or GRS of the unique equilibrium of the proposed switched system. Furthermore, the proposed LMI results are computationally efficient as it can be solved numerically with standard commercial software. Finally, three examples are provided to illustrate the usefulness of the results.     

Control problems of a mathematical model for schistosomiasis transmission dynamics

Drug treatment, snail control, cercariae control, improved sanitation and health education are the effective strategies which are used to control the schistosomiasis. In this paper, we formulate a deterministic model for schistosomiasis transmission dynamics in order to explore the role of the several control strategies. The basic reproductive number is computed. Sufficient conditions for the global asymptotic stability of the disease-free equilibrium are obtained. By using the Center Manifold Theory, we analyze the local stability of endemic equilibrium. Finally, numerical simulations support our analytical conclusions and the sensitive analysis on the basic reproductive number to the changes of control parameters are shown. Our results imply that snail-killing is the most effective way to control the transmission of schistosomiasis.     

Analysis on stability of an autonomous dynamics system for sars epidemic

Introduction Severeacuterespiratorysyndrome(SARS)spreadsmostrapidlythroughthe23areasand countriesintheworldsincethefirstSARScasewasreportedinGuangdonginNovember,2002andreachedtheitsclimaxinApril May2003.Althoughtheperiodoftheepidemicwasover now,itwasnotclearaboutSARSorigin,themechanismoftransmission,andtheregularityof SARSemergingexceptthatSARSasanovelcoronaviruswasknown.ThestudiesofSARS weremadebyavarietyofwaysandinmanyfieldsinorderthatSARSwouldbewellknown andeffectivemeasuresofi…     

STABILITY ANALYSIS OF HOPFIELD NEURAL NETWORKS WITH TIME DELAY

ＩｎｔｒｏｄｕｃｔｉｏｎＩｎｒｅｃｅｎｔｙｅａｒｓｔｈｅｄｙｎａｍｉｃｂｅｈａｖｉｏｒｏｆｔｈｅｆｏｌｌｏｗｉｎｇＨｏｐｆｉｅｌｄｎｅｕｒａｌｎｅｔｗｏｒｋｓｍｏｄｅｌｗｉｔｈｔｉｍｅｄｅｌａｙｈａｖｅｂｅｅｎｉｎｖｅｓｔｉｇａｔｅｄｔｈｏｒｏｕｇｈｌｙＣｉ ｘｉ(ｔ) =-ｘｉＲｉ + ∑ｎｊ=1Ｔｉｊｆｊ(ｘｊ(ｔ-τｉｊ) ) +Ｉｉ　　 (ｉ=1 ,2 ,… ,ｎ) ,( 1 )ｗｈｅｒｅＲｉ,ＣｉａｎｄＩｉｒｅｐｒｅｓｅｎｔｒｅｓｉｓｔａｎｃｅ ,ｃａｐａｃｉｔａｎｃｅａｎｄｅｌｅｃｔｒｉｃｃｕｒｒｅｎｔ.Ｒｉ,Ｃｉａｒｅｐａ…     

Global stability of a virus dynamics model with Beddington–DeAngelis incidence rate and CTL immune response

In this paper, the global stability of virus dynamics model with Beddington–DeAngelis infection rate and CTL immune response is studied by constructing Lyapunov functions. We derive the basic reproduction number R ₀ and the immune response reproduction number R ⁰ for the virus infection model, and establish that the global dynamics are completely determined by the values of R ₀. We obtain the global stabilities of the disease-free equilibrium E ₀, immune-free equilibrium E ₁ and endemic equilibrium E ^∗ when R ₀≤1, R ₀>1, R ⁰>1, respectively.     

Global asymptotic stability for Hopfield-type neural networks with diffusion effects

The existence, uniqueness and global asymptotic stability for the equilibrium of Hopfield-type neural networks with diffusion effects are studied. When the activation functions are monotonously nondecreasing, differentiable, and the interconnected matrix is related to the Lyapunov diagonal stable matrix, the sufficient conditions guaranteeing the existence of the equilibrium of the system are obtained by applying the topological degree theory. By means of constructing the suitable average Lyapunov functions, the global asymptotic stability of the equilibrium of the system is also investigated. It is shown that the equilibrium (if it exists) is globally asymptotically stable and this implies that the equilibrium of the system is unique.     

Traveling Wave Phenomena in a Kermack–McKendrick SIR Model

We study the existence and nonexistence of traveling waves of a general diffusive Kermack–McKendrick SIR model with standard incidence where the total population is not constant. The three classes, susceptible S, infected I and removed R, are all involved in the traveling wave solutions. We show that the minimum wave speed of traveling waves for the three-dimensional non-monotonic system can be derived from its linearizaion at the initial disease-free equilibrium. The proof in this paper is based on Schauder fixed point theorem and Laplace transform. Our study provides a promising method to deal with high dimensional epidemic models.     

Global stability and Hopf bifurcation of a predator-prey model with stage structure and delayed predator response

A Holling type predator-prey model with stage structure for the predator and a time delay due to the gestation of the mature predator is investigated. By analyzing the characteristic equations, the local stability of a predator-extinction equilibrium and a coexistence equilibrium of the model is addressed and the existence of Hopf bifurcations at the coexistence equilibrium is established. By means of the persistence theory on infinite dimensional systems, it is proven that the system is permanent if the coexistence equilibrium is feasible. By using Lyapunov functionals and the LaSalle invariance principle, it is shown that the predator-extinction equilibrium is globally asymptotically stable when the coexistence equilibrium is not feasible, and sufficient conditions are derived for the global stability of the coexistence equilibrium. Numerical simulations are carried out to illustrate the main theoretical results.     

Dynamics of an inshore–offshore fishery model with impulsive pollutant input in inshore area

Concerning the polluted environment in inshore areas and the effects of unbalanced impulsive diffusion between inshore and offshore areas on the size of fish population, based on optimal harvesting policy, a logistic offshore fishery system with impulsive pollutant input in inshore areas and impulsive diffusion at different fixed time is formulated, and we investigate the dynamics of such system. The sufficient conditions are obtained for the existence of the positive periodic solution and the global asymptotic stability of both the trivial periodic solution and the positive periodic solution. Moreover, the maximum sustainable yield and the corresponding harvesting effort are obtained.