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 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 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 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.     

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 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.     

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.     

Modeling diseases with latency and nonlinear incidence rates: global dynamics of a multi‐group model

In this paper, we perform global stability analysis of a multi‐group SEIR epidemic model in which we can consider the heterogeneity of host population and the effects of latency and nonlinear incidence rates. For a simpler version that assumes an identical natural death rate for all groups, and with a gamma distribution for the latency, the basic reproduction number is defined by the theory of the next generation operator and proved to be a sharp threshold determining whether or not disease spread. Under certain assumptions, the disease‐free equilibrium is globally asymptotically stable if R₀≤1 and there exists a unique endemic equilibrium which is globally asymptotically stable if R₀>1. The proofs of global stability of equilibria exploit a matrix‐theoretic method using Perron eigenvetor, a graph‐theoretic method based on Kirchhoff's matrix tree theorem and Lyapunov functionals. Copyright © 2015 John Wiley & Sons, Ltd.     

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 stability of a virus dynamics model with intracellular delay and CTL immune response

In this paper, the global stability of a virus dynamics model with intracellular delay, Crowley–Martin functional response of the infection rate, and CTL immune response is studied. By constructing suitable Lyapunov functions and using LaSalles invariance principle, the global dynamics is established; it is proved that if the basic reproductive number, R₀, is less than or equal to one, the infection‐free equilibrium is globally asymptotically stable; if R₀ is more than one, and if immune response reproductive number, R⁰, is less than one, the immune‐free equilibrium is globally asymptotically stable, and if R⁰ is more than one, the endemic equilibrium is globally asymptotically stable. Copyright © 2014 John Wiley & Sons, Ltd.     

Global stability of epidemiological models with group mixing and nonlinear incidence rates

For a multigroup SEIR epidemiological model with nonlinear incidence rates, the basic reproduction number is identified. It is shown that, under certain group mixing patterns and nonlinearity and/or nonsmoothness in the incidence of infection, the basic reproduction number is a global threshold parameter in the sense that the disease free equilibrium is globally stable if the basic reproduction number is less than one and the endemic equilibrium is globally stable if the basic reproduction number is greater than one.     

Global analysis of an SIS model with an infective vector on complex networks

In this paper, a modified SIS model with an infective vector on complex networks is proposed and analyzed, which incorporates some infectious diseases that are not only transmitted by a vector, but also spread by direct contacts between human beings. We treat direct human contacts as a social network and assume spatially homogeneous mixing between vector and human populations. By mathematical analysis, we obtain the basic reproduction number R₀ and study the effects of various immunization schemes. For the network model, we prove that if R₀<1, the disease-free equilibrium is globally asymptotically stable, otherwise there exists an unique endemic equilibrium such that it is globally attractive. Our theoretical results are confirmed by numerical simulations and suggest a promising way for the control of infectious diseases.     

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 dynamics of a delayed SIRS epidemic model with a wide class of nonlinear incidence rates

In this paper, by constructing Lyapunov functionals, we consider the global dynamics of an SIRS epidemic model with a wide class of nonlinear incidence rates and distributed delays $\int^{h}_{0} p(\tau)f(S(t),I(t-\tau)) \mathrm{d}\tau$ under the condition that the total population converges to 1. By using a technical lemma which is derived from strong condition of strict monotonicity of functions f(S,I) and f(S,I)/I with respect to S??0 and I>0, we extend the global stability result for an SIR epidemic model if R ₀>1, where R ₀ is the basic reproduction number. By using a limit system of the model, we also show that the disease-free equilibrium is globally asymptotically stable if R ₀=1.     

Mathematical model for the dynamics of visceral leishmaniasis–malaria co‐infection

A mathematical model to understand the dynamics of malaria–visceral leishmaniasis co‐infection is proposed and analyzed. Results show that both diseases can be eliminated if R₀, the basic reproduction number of the co‐infection, is less than unity, and the system undergoes a backward bifurcation where an endemic equilibrium co‐exists with the disease‐free equilibrium when one of R_m or R_l, the basic reproduction numbers of malaria‐only and visceral leishmaniasis‐only, is precisely less than unity. Results also show that in the case of maximum protection against visceral leishmaniasis (VL), the disease‐free equilibrium is globally asymptotically stable if malaria patients are protected from VL infection; similarly, in the case of maximum protection against malaria, the disease‐free equilibrium is globally asymptotically stable if VL and post‐kala‐azar dermal leishmaniasis patients and the recovered humans after VL are protected from malaria infection. Numerical results show that if R_m and R_l are greater than unity, then we have co‐existence of both disease at an endemic equilibrium, and malaria incidence is higher than visceral leishmaniasis incidence at steady state. Copyright © 2016 John Wiley & Sons, Ltd.     

Analysis of a SEIV epidemic model with a nonlinear incidence rate

In this paper, a SEIV epidemic model with a nonlinear incidence rate is investigated. The model exhibits two equilibria, namely, the disease-free equilibrium and the endemic equilibrium. It is shown that if the basic reproduction number _R0<1${\mathfrak{R}}_0<1$, the disease-free equilibrium is globally asymptotically stable and in such a case the endemic equilibrium does not exist. Moreover, we show that if the basic reproduction number _R0>1${\mathfrak{R}}_0>1$, the disease is uniformly persistent and the unique endemic equilibrium of the system with saturation incidence is globally asymptotically stable under certain conditions.     

Stability of a delayed SIRS epidemic model with a nonlinear incidence rate

In this paper, an SIRS epidemic model with a nonlinear incidence rate and a time delay is investigated. By analyzing the corresponding characteristic equations, the local stability of an endemic equilibrium and a disease-free equilibrium is discussed. By comparison arguments, it is proved that if the basic reproductive number R₀<1, the disease-free equilibrium is globally asymptotically stable. If R₀>1, by means of an iteration technique, sufficient conditions are derived for the global asymptotic stability of the endemic equilibrium.     

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.     

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.