Global stability of delay multigroup epidemic models with group mixing and nonlinear incidence rates

In this paper, we present a new delay multigroup SEIR model with group mixing and nonlinear incidence rates and investigate its global stability. We establish that the global dynamics of the models are completely determined by the basic reproduction number R₀. It is shown that, if R₀?1, then the disease free equilibrium is globally asymptotically stable and the disease dies out; if R₀>1, there exists a unique endemic equilibrium that is globally asymptotically stable and thus the disease persists in the population. Finally, a numerical example is also discussed to illustrate the effectiveness of the results.     

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.     

On global stability of the intra-host dynamics of malaria and the immune system

In this paper we consider an intra-host model for the dynamics of malaria. The model describes the dynamics of the blood stage malaria parasites and their interaction with host cells, in particular red blood cells (RBC) and immune effectors. We establish the equilibrium points of the system and analyze their stability using the theory of competitive systems, compound matrices and stability of periodic orbits. We established that the disease-free equilibrium is globally stable if and only if the basic reproduction number satisfies R₀?1 and the parasite will be cleared out of the host. If R₀>1, a unique endemic equilibrium is globally stable and the parasites persist at the endemic steady state. In the presence of the immune response, the numerical analysis of the model shows that the endemic equilibrium is unstable.     

A differential equation model of HIV infection of CD4T-cells with cure rate

A differential equation model of HIV infection of CD4⁺T-cells with cure rate is studied. We prove that if the basic reproduction number R₀<1, the HIV infection is cleared from the T-cell population and the disease dies out; if R₀>1, the HIV infection persists in the host. We find that the chronic disease steady state is globally asymptotically stable if R₀>1. Furthermore, we also obtain the conditions for which the system exists an orbitally asymptotically stable periodic solution. Numerical simulations are presented to illustrate the results.     

Global stability of multigroup epidemic model with group mixing and nonlinear incidence rates

In this paper, we introduce a basic reproduction number for a multigroup SEIR model with nonlinear incidence of infection and nonlinear removal functions between compartments. Then, we establish that global dynamics are completely determined by the basic reproduction number R₀. It shows that, the basic reproduction number R₀ is a global threshold parameter in the sense that if it is less than or equal to one, the disease free equilibrium is globally stable and the disease dies out; whereas if it is larger than one, there is a unique endemic equilibrium which is globally stable and thus the disease persists in the population. Finally, two numerical examples are also included to illustrate the effectiveness of the proposed result.     

Stochastic Nonlinear Complementarity Problem and?Applications to?Traffic Equilibrium under?Uncertainty

The expected residual minimization (ERM) formulation for the stochastic nonlinear complementarity problem (SNCP) is studied in this paper. We show that the involved function is a stochastic R ₀ function if and only if the objective function in the ERM formulation is coercive under a mild assumption. Moreover, we model the traffic equilibrium problem (TEP) under uncertainty as SNCP and show that the objective function in the ERM formulation is a stochastic R ₀ function. Numerical experiments show that the ERM-SNCP model for TEP under uncertainty has various desirable properties. This work was partially supported by a Grant-in-Aid from the Japan Society for the Promotion of Science. The authors thank Professor Guihua Lin for pointing out an error in Proposition 2.1 on an earlier version of this paper. The authors are also grateful to the referees for their insightful comments.     

SEIQRS model for the transmission of malicious objects in computer network

Susceptible (S) – exposed (E) – infectious (I) – quarantined (Q) – recovered (R) model for the transmission of malicious objects in computer network is formulated. Thresholds, equilibria, and their stability are also found with cyber mass action incidence. Threshold R_cq determines the outcome of the disease. If R_cq ? 1, the infected fraction of the nodes disappear so the disease die out, while if R_cq > 1, the infected fraction persists and the feasible region is an asymptotic stability region for the endemic equilibrium state. Numerical methods are employed to solve and simulate the system of equations developed. The effect of quarantine on recovered nodes is analyzed. We have also analyzed the behavior of the susceptible, exposed, infected, quarantine, and recovered nodes in the computer network.     

A delayed computer virus propagation model and its dynamics

In this paper, we propose a delayed computer virus propagation model and study its dynamic behaviors. First, we give the threshold value R₀ determining whether the virus dies out completely. Second, we study the local asymptotic stability of the equilibria of this model and it is found that, depending on the time delays, a Hopf bifurcation may occur in the model. Next, we prove that, if R₀ = 1, the virus-free equilibrium is globally attractive; and when R₀ < 1, it is globally asymptotically stable. Finally, a sufficient criterion for the global stability of the virus equilibrium is obtained.     

Global stability for an HIV/AIDS epidemic model with different latent stages and treatment

An HIV/AIDS epidemic model with different latent stages and treatment is constructed. The model allows for the latent individuals to have the slow and fast latent compartments. Mathematical analyses establish that the global dynamics of the spread of the HIV infectious disease are determined by the basic reproduction number under some conditions. If R₀ < 1, the disease free equilibrium is globally asymptotically stable, and if R₀ > 1, the endemic equilibrium is globally asymptotically stable for a special case. Some numerical simulations are also carried out to confirm the analytical results.     

Hopf bifurcation in epidemic models with a latent period and nonpermanent immunity

In this paper, some SEIRS epidemiological models with vaccination and temporary immunity are considered. First of all, previously published work is reviewed. In the next section, a general model with a constant contact rate and a density-dependent death rate is examined. The model is reformulated in terms of the proportions of susceptible, incubating, infectious, and immune individuals. Next the equilibrium and stability properties of this model are examined, assuming that the average duration of immunity exceeds the infectious period. There is a threshold parameter R_o and the disease can persist if and only if R_o exceeds one. The disease-free equilibrium always exists and is locally stable if R_o < 1 and unstable if R_o > 1. Conditions are derived for the global stability of the disease-free equilibrium. For R_o > 1, the endemic equilibrium is unique and locally asymptotically stable.For the full model dealing with numbers of individuals, there are two critical contact rates. These give conditions for the disease, respectively, to drive a population which would otherwise persist at a finite level or explode to extinction and to cause a population that would otherwise explode to be regulated at a finite level. If the contact rate β(N) is a monotone increasing function of the population size, then we find that there are now three threshold parameters which determine whether or not the disease can persist proportionally. Moreover, the endemic equilibrium need no longer be locally asymptotically stable. Instead stable limit cycles can arise by supercritical Hopf bifurcation from the endemic equilibrium as this equilibrium loses its stability. This is confirmed numerically.     

Analysis of a delayed HIV/AIDS epidemic model with saturation incidence

In this paper, a delayed HIV/AIDS epidemic model with saturation incidence is proposed and analyzed. The equilibria and their stability are investigated. The model exhibits two equilibria, namely, the disease-free equilibrium and the endemic equilibrium. It is found that if the threshold R ₀<1, then the disease-free equilibrium is globally asymptotically stable, and if the threshold R ₀>1, the system is permanent and the endemic equilibrium is asymptotically stable under certain conditions.     

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.     

Analysis of an SVEIS epidemic model with partial temporary immunity and saturation incidence rate

In this paper, an SVEIS epidemic model for an infectious disease that spreads in the host population through horizontal transmission is investigated. The role that temporary immunity (natural, disease induced, vaccination induced) plays in the spread of disease, is incorporated in the model. The total host population is bounded and the incidence term is of the Holling-type II form. It is shown that the model exhibits two equilibria, namely, the disease-free equilibrium and the endemic equilibrium. The global dynamics are completely determined by the basic reproduction number R₀. If R₀<1, the disease-free equilibrium is globally stable which leads to the eradication of disease from population. If R₀>1, a unique endemic equilibrium exists and is globally stable in the feasible region under certain conditions. Further, the transcritical bifurcation at R₀=1 is explored by projecting the flow onto the extended center manifold. We use the geometric approach for ordinary differential equations which is based on the use of higher-order generalization of Bendixson’s criterion. Further, we obtain the threshold vaccination coverage required to eradicate the disease. Finally, taking biologically relevant parametric values, numerical simulations are performed to illustrate and verify the analytical results.     

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 SEIR epidemic models with distributed delays and nonlinear transmission

The dynamics of multi-group SEIR epidemic models with distributed and infinite delay and nonlinear transmission are investigated. We derive the basic reproduction number R₀ and establish that the global dynamics are completely determined by the values of R₀: if R₀≤1, then the disease-free equilibrium is globally asymptotically stable; if R₀>1, then there exists a unique endemic equilibrium which is globally asymptotically stable. Our results contain those for single-group SEIR models with distributed and infinite delays. In the proof of global stability of the endemic equilibrium, we exploit a graph-theoretical approach to the method of Lyapunov functionals. The biological significance of the results is also discussed.     

Global dynamics of a Vector‐Borne disease model with two delays and nonlinear transmission rate

In this paper, we investigate a Vector‐Borne disease model with nonlinear incidence rate and 2 delays: One is the incubation period in the vectors and the other is the incubation period in the host. Under the biologically motivated assumptions, we show that the global dynamics are completely determined by the basic reproduction number R₀. The disease‐free equilibrium is globally asymptotically stable if R₀≤1; when R₀>1, the system is uniformly persistent, and there exists a unique endemic equilibrium that is globally asymptotically. Numerical simulations are conducted to illustrate the theoretical results.     

Dynamical behavior of computer virus on Internet

In this paper, we presented a computer virus model using an SIRS model and the threshold value R₀ determining whether the disease dies out is obtained. If R₀ is less than one, the disease-free equilibrium is globally asymptotically stable. By using the time delay as a bifurcation parameter, the local stability and Hopf bifurcation for the endemic state is investigated. Numerical results demonstrate that the system has periodic solution when time delay is larger than a critical values. The obtained results may provide some new insight to prevent the computer virus.     

Epidemic spreading with time delay on complex networks

In this paper, we propose a susceptible-infected-susceptible (SIS) model on complex networks, small-world (WS) networks and scale-free (SF) networks, to study the epidemic spreading behavior with time delay which is added into the infected phase. Considering the uniform delay, the basic reproduction number R ₀ on WS networks and \(\bar R_0\) on SF networks are obtained respectively. On WS networks, if R ₀ ≤ 1, there is a disease-free equilibrium and it is locally asymptotically stable; if R ₀ > 1, there is an epidemic equilibrium and it is locally asymptotically stable. On SF networks, if \(\bar R_0 \leqslant 1\), there is a disease-free equilibrium; if \(\bar R_0 > 1\), there is an epidemic equilibrium. Finally, we carry out simulations to verify the conclusions and analyze the effect of the time delay τ, the effective rate λ, average connectivity 〈k〉 and the minimum connectivity m on the epidemic spreading.     

Global dynamics of a class of SEIRS epidemic models in a periodic environment

In this paper, we study a class of periodic SEIRS epidemic models and it is shown that the global dynamics is determined by the basic reproduction number R₀ which is defined through the spectral radius of a linear integral operator. If R₀<1, then the disease free periodic solution is globally asymptotically stable and if R₀>1, then the disease persists. Our results really improve the results in [T. Zhang, Z. Teng, On a nonautonomous SEIRS model in epidemiology Bull. Math. Biol. 69 (8) (2007) 2537-2559] for the periodic case. Moreover, from our results, we see that the eradication policy on the basis of the basic reproduction number of the time-averaged system may overestimate the infectious risk of the periodic disease. Numerical simulations which support our theoretical analysis are also given.     

Global analysis of virus dynamics model with logistic mitosis,cure rate and delay in virus production

In this paper, we study a virus dynamics model with logistic mitosis, cure rate, and intracellular delay. By means of construction of a suitable Lyapunov functionals, obtained by linear combinations of Volterra—type functions, composite quadratic functions and Volterra—type functionals, we provide the global stability for this model. If R₀, the basic reproductive number, satisfies R₀ ≤ 1, then the infection‐free equilibrium state is globally asymptotically stable. Our system is persistent if R₀ > 1. On the other hand, if R₀ > 1, then infection‐free equilibrium becomes unstable and a unique infected equilibrium exists. The local stability analysis is carried out for the infected equilibrium, and it is shown that, if the parameters satisfy a condition, the infected equilibrium can be unstable and a Hopf bifurcation can occur. We also have that if R₀ > 1, then the infected equilibrium state is globally asymptotically stable if a sufficient condition is satisfied. We illustrate our findings with some numerical simulations. Copyright © 2014 John Wiley & Sons, Ltd.