Global analysis of a vector-host epidemic model with nonlinear incidences

In this paper, an epidemic model with nonlinear incidences is proposed to describe the dynamics of diseases spread by vectors (mosquitoes), such as malaria, yellow fever, dengue and so on. The constant human recruitment rate and exponential natural death, as well as vector population with asymptotically constant population, are incorporated into the model. The stability of the system is analyzed for the disease-free and endemic equilibria. The stability of the system can be controlled by the threshold number R₀. It is shown that if R₀ is less than one, the disease free equilibrium is globally asymptotically stable and in such a case the endemic equilibrium does not exist; if R₀ is greater than one, then the disease persists and the unique endemic equilibrium is globally asymptotically stable. Our results imply that the threshold condition of the system provides important guidelines for accessing control of the vector diseases, and the spread of vector epidemic in an efficient way can be prevented. The contribution of the nonlinear saturating incidence to the basic reproduction number and the level of the endemic equilibrium are also analyzed, respectively.     

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.     

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 analysis of an environmental disease transmission model linking within-host and between-host dynamics

In this paper, a multi-scale mathematical model for environmentally transmitted diseases is proposed which couples the pathogen-immune interaction inside the human body with the disease transmission at the population level. The model is based on the nested approach that incorporates the infection-age-structured immunological dynamics into an epidemiological system structured by the chronological time, the infection age and the vaccination age. We conduct detailed analysis for both the within-host and between-host disease dynamics. Particularly, we derive the basic reproduction number R₀ for the between-host model and prove the uniform persistence of the system. Furthermore, using carefully constructed Lyapunov functions, we establish threshold-type results regarding the global dynamics of the between-host system: the disease-free equilibrium is globally asymptotically stable when R₀ < 1, and the endemic equilibrium is globally asymptotically stable when R₀ > 1. We explore the connection between the within-host and between-host dynamics through both mathematical analysis and numerical simulation. We show that the pathogen load and immune strength at the individual level contribute to the disease transmission and spread at the population level. We also find that, although the between-host transmission risk correlates positively with the within-host pathogen load, there is no simple monotonic relationship between the disease prevalence and the individual pathogen load.     

An epidemic model of a vector-borne disease with direct transmission and time delay

This paper considers an epidemic model of a vector-borne disease which has direct mode of transmission in addition to the vector-mediated transmission. The incidence term is assumed to be of the bilinear mass-action form. We include both a baseline ODE version of the model, and, a differential-delay model with a discrete time delay. The ODE model shows that the dynamics is completely determined by the basic reproduction number R₀. If R₀?1, the disease-free equilibrium is globally stable and the disease dies out. If R₀>1, a unique endemic equilibrium exists and is locally asymptotically stable in the interior of the feasible region. The delay in the differential-delay model accounts for the incubation time the vectors need to become infectious. We study the effect of that delay on the stability of the equilibria. We show that the introduction of a time delay in the host-to-vector transmission term can destabilize the system and periodic solutions can arise through Hopf bifurcation.     

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.     

Analysis of an epidemic model with awareness programs by media on complex networks

In this paper, we propose an epidemic disease model about the effect of awareness programs on complex networks, where the contacts between nodes are treated as a social network. Two forms on increasing rate of awareness programs, which are a constant and the change with the number of infected individuals, are analyzed. Through dynamical analysis, we obtain the basic reproduction number R₀ and prove the stability of disease-free equilibrium and endemic equilibrium. Furthermore, numerical simulations about the model are taken to reach that, on the one hand, the two forms, which are increasing rate of awareness programs, respectively, have advantages and disadvantages on preventing and controlling diseases, and they are complementary; on the other hand, awareness programs have more effects on nodes with smaller degrees.     

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 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 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 dynamics of a multistage SIR model with distributed delays and nonlinear incidence rate

In this paper, a multistage susceptible‐infectious‐recovered model with distributed delays and nonlinear incidence rate is investigated, which extends the model considered by Guo et al. [H. Guo, M. Y. Li and Z. Shuai, Global dynamics of a general class of multistage models for infectious diseases, SIAM J. Appl. Math., 72 (2012), 261–279]. Under some appropriate and realistic conditions, the global dynamics is completely determined by the basic reproduction number R₀. If R₀≤1, then the infection‐free equilibrium is globally asymptotically stable and the disease dies out in all stages. If R₀>1, then a unique endemic equilibrium exists, and it is globally asymptotically stable, and hence the disease persists in all stages. The results are proved by utilizing the theory of non‐negative matrices, Lyapunov functionals, and the graph‐theoretical approach. Copyright © 2016 John Wiley & Sons, Ltd.     

Qualitative analysis of the SICR epidemic model with impulsive vaccinations

Control of epidemic infections is a very urgent issue today. To develop an appropriate strategy for vaccinations and effectively prevent the disease from arising and spreading, we proposed a modified Susceptible‐Infected‐Removed model with impulsive vaccinations. For the model without vaccinations, we proved global stability of one of the steady states depending on the basic reproduction number R₀. As typically in the epidemic models, the threshold value of R₀ is 1. If R₀ is greater than 1, then the positive steady state called endemic equilibrium exists and is globally stable, whereas for smaller values of R₀, it does not exist, and the semi‐trivial steady state called disease‐free equilibrium is globally stable. Using impulsive differential equation comparison theorem, we derived sufficient conditions under which the infectious disease described by the considered model disappears ultimately. The analytical results are illustrated by numerical simulations for Hepatitis B virus infection that confirm the theoretical possibility of the infection elimination because of the proper vaccinations policy. Copyright © 2012 John Wiley & Sons, Ltd.     

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.     

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.     

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.     

Dynamics of an epidemic model with host migration

Host migration among discrete geographical regions is demonstrated as an important factor that brings about the diffusion and outbreak of many vector-host diseases. In the paper, we develop a mathematical model to explore the effect of host migration between two patches on the spread of a vector-host disease. Analytical results show that the reproduction number R₀ provides a threshold condition that determines the uniform persistence and extinction of the disease. If both the patches are identical, it is shown that an endemic equilibrium is locally stable. It is also shown that a unique endemic equilibrium, which exists when the disease cannot induce the death of the host, is globally asymptotically stable. Finally, two examples are given to illustrate the effect of host migration on the spread of the vector-host disease.     

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 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.     

Two-group SIR epidemic model with stochastic perturbation

A stochastic two-group SIR model is presented in this paper.The existence and uniqueness of its nonnegative solution is obtained,and the solution belongs to a positively invariant set.Furthermore,the globally asymptotical stability of the disease-free equilibrium is deduced by the stochastic Lyapunov functional method if R0 ≤ 1,which means the disease will die out.While if R0 1,we show that the solution is fluctuating around a point which is the endemic equilibrium of the deterministic model in time average.In addition,the intensity of the fluctuation is proportional to the intensity of the white noise.When the white noise is small,we consider the disease will prevail.At last,we illustrate the dynamic behavior of the model and their approximations via a range of numerical experiments.     

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.