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 delayed eco‐epidemiological model with Holling type‐III functional response

In this paper, an eco‐epidemiological model with Holling type‐III functional response and a time delay representing the gestation period of the predators is investigated. In the model, it is assumed that the predator population suffers a transmissible disease. The disease basic reproduction number is obtained. By analyzing the corresponding characteristic equations, the local stability of each of feasible equilibria and the existence of Hopf bifurcations at the disease‐free equilibrium and the endemic‐coexistence equilibrium are established, respectively. By using the persistence theory on infinite dimensional systems, it is proved that if the disease basic reproduction number is greater than unity, the system is permanent. By means of Lyapunov functionals and LaSalle's invariance principle, sufficient conditions are obtained for the global stability of the endemic‐coexistence equilibrium, the disease‐free equilibrium and the predator‐extinction equilibrium of the system, respectively. Copyright © 2013 John Wiley & Sons, Ltd.     

Global stability and Hopf bifurcation of a delayed eco‐epidemiological model with Holling type II functional response

In this paper, a delayed eco‐epidemiological model with Holling type II functional response is investigated. By analyzing corresponding characteristic equations, the local stability of each of the feasible equilibria and the existence of Hopf bifurcations at the disease‐free equilibrium, the susceptible predator‐free equilibrium and the endemic‐coexistence equilibrium are established, respectively. By means of Lyapunov functionals and LaSalle's invariance principle, sufficient conditions are derived for the global stability of the endemic‐coexistence equilibrium, the disease‐free equilibrium, the susceptible predator‐free equilibrium and the predator‐extinction equilibrium of the system, respectively. Numerical simulations are carried out to illustrate the theoretical results. Copyright © 2014 John Wiley & Sons, Ltd.     

Global dynamics of a delayed SEIS infectious disease model with logistic growth and saturation incidence

In this paper, a delayed Susceptible‐Exposed‐Infectious‐Susceptible (SEIS) infectious disease model with logistic growth and saturation incidence is investigated, where the time delay describes the latent period of the disease. By analyzing corresponding characteristic equations, the local stability of a disease‐free equilibrium and an endemic equilibrium is discussed. The existence of Hopf bifurcations at the endemic equilibrium is established. By using the persistence theory for infinite dimensional dynamic systems, it is proved that if the basic reproduction number is greater than unity, the system is permanent. By means of suitable Lyapunov functionals and LaSalle's invariance principle, sufficient conditions are obtained for the global stability of the disease‐free equilibrium and the endemic equilibrium, respectively. Numerical simulations are carried out to illustrate the theoretical results. Copyright © 2015 John Wiley & Sons, Ltd.     

Stability and Hopf bifurcation of a delayed epidemic model with stage structure and nonlinear incidence rate

In this paper, a stage‐structured SI epidemic model with time delay and nonlinear incidence rate is investigated. By analyzing the corresponding characteristic equations, the local stability of an endemic equilibrium and a disease‐free equilibrium, and the existence of Hopf bifurcations are established. By comparison arguments, it is proved that if the basic reproduction number is less than unity, the disease‐free equilibrium is globally asymptotically stable. If 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. Copyright © 2013 John Wiley & Sons, Ltd.     

Within‐host and synaptic transmissions: contributions to the spread of HIV infection

We study the contributions of within‐host (virus‐to‐cell) and synaptic (cell‐to‐cell) transmissions in a mathematical model for human immunodeficiency virus epidemics. The model also includes drug resistance. We prove the local and global stability of the disease‐free equilibrium and the local stability of the endemic equilibrium. We analyse the effect of the cell‐to‐cell transmission rate on the value of the reproduction number, R₀. Moreover, we show evidence of a qualitative change in the models' dynamics, subjected to the value of the drug efficacy. In the end, important inferences are drawn. Copyright © 2016 John Wiley & Sons, Ltd.     

Asymptotic profiles of the endemic equilibrium of a diffusive SIS epidemic system with saturated incidence rate and spontaneous infection

A susceptible‐infected‐susceptible (SIS) epidemic reaction‐diffusion model with saturated incidence rate and spontaneous infection is considered. We establish the existence of endemic equilibrium by using a fixed‐point theorem. The global asymptotic stability of the constant endemic equilibrium is discussed in the case of homogeneous environment. We mainly investigate the effects of diffusion and saturation on asymptotic profiles of the endemic equilibrium. When the saturated incidence rate tends to infinity, the susceptible and infective distributes in the habitat in a nonhomogeneous way; this result is in strong contrast with the case of no spontaneous infection, where the endemic equilibrium tends to the disease free equilibrium. Our analysis shows that the spontaneous infection can enhance the persistence of an infectious disease and may provide some useful implications on disease control.     

Global analysis of a periodic epidemic model on cholera in presence of bacteriophage

We propose and analyze a recurrent epidemic model of cholera in the presence of bacteriophage. The model is extended by general periodic incidence functions for low‐infectious bacterium and high‐infectious bacterium, respectively. A general periodic shedding function for two infected class (phage‐positive and phage‐negative) and a generalized contact and intrinsic growth function for susceptible class are also considered. Under certain biological assumptions, we derive the basic reproduction number (R₀) in a periodic environment for the proposed model. We also observe the global stability of the disease‐free equilibrium, existence, permanence, and global stability of the positive endemic periodic solution of our proposed model. Finally, we verify our results with specific functional form. Copyright © 2016 John Wiley & Sons, Ltd.     

Effect of emergent carrying capacity in an eco‐epidemiological system

The paper explores an eco‐epidemiological model of a predator–prey type, where the prey population is subject to infection. The model is basically a combination of S‐I type model and a Rosenzweig–MacArthur predator–prey model. The novelty of this contribution is to consider different competition coefficients within the prey population, which leads to the emergent carrying capacity. We explicitly separate the competition between non‐infected and infected individuals. This emergent carrying capacity is markedly different to the explicit carrying capacities that have been considered in many eco‐epidemiological models. We observed that different intra‐class and inter‐class competition can facilitate the coexistence of susceptible prey‐infected prey–predator, which is impossible for the case of the explicit carrying capacity model. We also show that these findings are closely associated with bi‐stability. The present system undergoes bi‐stability in two different scenarios: (a) bi‐stability between the planner equilibria where susceptible prey co‐exists with predator or infected prey and (b) bi‐stability between co‐existence equilibrium and the planner equilibrium where susceptible prey coexists with infected prey; have been discussed. The conditions for which the system is to be permanent and the global stability of the system around disease‐free equilibrium are worked out. Copyright © 2015 John Wiley & Sons, Ltd.     

Stability and bifurcation in epidemic models describing the transmission of toxoplasmosis in human and cat populations

A five‐dimensional ordinary differential equation model describing the transmission of Toxoplamosis gondii disease between human and cat populations is studied in this paper. Self‐diffusion modeling the spatial dynamics of the T. gondii disease is incorporated in the ordinary differential equation model. The normalized version of both models where the unknown functions are the proportions of the susceptible, infected, and controlled individuals in the total population are analyzed. The main results presented herein are that the ODE model undergoes a trans‐critical bifurcation, the system has no periodic orbits inside the positive octant, and the endemic equilibrium is globally asymptotically stable when we restrict the model to inside of the first octant. Furthermore, a local linear stability analysis for the spatially homogeneous equilibrium points of the reaction diffusion model is carried out, and the global stability of both the disease‐free and endemic equilibria are established for the reaction–diffusion system when restricted to inside of the first octant. Finally, numerical simulations are provided to support our theoretical results and to predict some scenarios about the spread of the disease. Copyright © 2017 John Wiley & Sons, Ltd.     

Feedback control variables to restrain the Babesiosis disease

In this paper, we complete the study of the dynamics of a recognized continuous‐time model for the Babesiosis disease. The local and global asymptotic stability of the endemic state are established theoretically and experimentally. In addition, to restrain the disease in the original model when the endemic state exists, we propose and study the continuous model with feedback controls. The global stability of the boundary‐equilibrium point of this model is analyzed by means of rigorous mathematical methods. As an important consequence of this result, we propose a strategy to select feedback control variables in order to restrain the disease in the original model. This strategy allows us to make the disease vanish completely. In other words, the feedback controls are specially effective for restraining disease in the model. The validity of the established theoretical result is supported by a set of numerical simulations.     

Dynamics analysis of a quarantine model in two patches

A new deterministic model for assessing the impact of quarantine on the transmission dynamics of a communicable disease in a two‐patch community is designed. Rigorous analysis of the model shows that the imperfect nature of quarantine (in the two patches) could induce the phenomenon of backward bifurcation when the associated reproduction number of the model is less than unity. For the case when quarantined susceptible individuals do not acquire infection during quarantine, the disease‐free equilibrium of the model is shown to be globally asymptotically stable when the associated reproduction number is less than unity. Furthermore, the model has a unique Patch i‐only boundary equilibrium (i = 1,2) whenever the associated reproduction number for Patch i is greater than unity. The unique Patch i‐only boundary equilibrium is locally asymptotically stable whenever the invasion reproduction number of Patch 3 ? i is less than unity (and the associated reproduction number for Patch i exceeds unity). The model has at least one endemic equilibrium when its reproduction number exceeds unity (and the disease persists in both patches in this case). It is shown that adding multi‐patch dynamics to a single‐patch quarantine model (which allow the quarantine of susceptible individuals) in a single patch does not alter its quantitative dynamics (with respect to the existence and asymptotic stability of its associated equilibria as well as its backward bifurcation property). Copyright © 2014 John Wiley & Sons, Ltd.     

Global dynamics of an epidemic model with relapse and nonlinear incidence

On the basis of a basic SIR epidemic model, we propose and study an epidemic model with nonlinear incidence. The model also incorporates many features of the recovered such as relapse and with/without immunity. A threshold dynamics is established, which is completely determined by the basic reproduction number. The global stability of the disease‐free equilibrium is proved by means of the fluctuation lemma. To prove the global stability of the endemic equilibrium, we need some novel techniques including the transformation of variables, the construction of a new type of Lyapunov functions, and the seeking of an appropriate positively invariant set of the model.     

Global behaviour of an SIR epidemic model with time delay

We study the stability of a delay susceptible–infective–recovered epidemic model with time delay. The model is formulated under the assumption that all individuals are susceptible, and we analyse the global stability via two methods—by Lyapunov functionals, and—in terms of the variance of the variables. The main theorem shows that the endemic equilibrium is stable. If the basic reproduction number ?₀ is less than unity, by LaSalle invariance principle, the disease‐free equilibrium E_s is globally stable and the disease always dies out. By applying the integral averaging theory, we also investigate the stability in variance of the model. Copyright © 2006 John Wiley & Sons, Ltd.     

考虑直销的多期供应链网络均衡模型

研究了直销供应链网络动态均衡模型,分别对制造商、零售商和需求市场的决策者的多期独立决策行为及其相互作用进行了分析,利用变分不等式构建了各层均衡模型以及系统均衡模型,得到了系统达到均衡的条件,给出了具体算例并进行了求解.构建的供应链网络动态均衡模型具有更好的适用性及普遍性,为供应链的各层成员保持供应链的稳定和长期均衡提供决策依据和决策方法.     

Stability preserving NSFD scheme for a multi‐group SVIR epidemic model

A discrete multi‐group SVIR epidemic model with general nonlinear incidence rate and vaccination is investigated by utilizing Mickens' nonstandard finite difference scheme to a corresponding continuous model. Mathematical analysis shows that the global asymptotic stability of the equilibria is fully determined by the basic reproduction number by constructing Lyapunov functions. The results imply that the discretization scheme can efficiently preserves the global asymptotic stability of the equilibria for corresponding continuous model, and numerical simulations are carried out to illustrate the theoretical results. Copyright © 2017 John Wiley & Sons, Ltd.     

Global stability of a multi‐group model with distributed delay and vaccination

A delayed multi‐group SVEIR epidemic model with vaccination and a general incidence function has been formulated and studied in this paper. Mathematical analysis shows that the basic reproduction number plays a key role in the dynamics of the model: the disease‐free equilibrium is globally asymptotically stable when , while the endemic equilibrium exists uniquely and is globally asymptotically stable when . For the proofs, we exploit a graph‐theoretical approach to the method of Lyapunov functionals. Our results show that distributed delay has no impact on the global stability of equilibria, and the results improve and generalize some known results. Copyright © 2016 John Wiley & Sons, Ltd.     

Disease‐selective predation may lead to prey extinction

In real world bio‐communities, predational choice plays a key role to the persistence of the prey population. Predator's ‘sense’ of choice for predation towards the infected and noninfected prey is an important factor for those bio‐communities. There are examples where the predator can distinguish the infected prey and avoids those at the time of predation. Based on the examples, we propose two mathematical models and observe the dynamics of the systems around biologically feasible equilibria. For disease‐selective predation model there is a high risk of prey extinction. On the other hand, for non‐disease selective predation both populations co‐exist. Local stability analysis and global stability analysis of the positive interior equilibrium are performed. Moreover, conditions for the permanence of the system are obtained. Finally, we conclude that strictly disease‐selective predation may not be acceptable for the persistence of the prey population. Copyright © 2005 John Wiley & Sons, Ltd.     

Permanence of a delayed SIR epidemic model with general nonlinear incidence rate

In this paper, a SIR model with two delays and general nonlinear incidence rate is considered. The local and global asymptotical stabilities of the disease‐free equilibrium are given. The local asymptotical stability and the existence of Hopf bifurcations at the endemic equilibrium are also established by analyzing the distribution of the characteristic values. Furthermore, the sufficient conditions for the permanence of the system are given. Some numerical simulations to support the analytical conclusions are carried out. At last, some conclusions are given. Copyright © 2014 John Wiley & Sons, Ltd.     

Global Stability for a Dynamic model of Hepatitis B with Antivirus Treatment

An epidemic model on the basis of therapy of chronic Hepatitis B with antivirus treatment was introduced in this paper. By applying a comparison theorem and analyzing the corresponding characteristic equations, we obtain sufficient conditions on the parameters for the global stability of the disease-free state. It's proved that if the basic reproduction number \(R_0 < 1\) , the disease-free equilibrium is globally asymptotically stable. If \(R_0 > 1\), the disease-free equilibrium is unstable and the disease is uniformly permanent. Moreover, if \(R_0 > 1\), sufficient conditions are obtained for the global stability of the endemic equilibrium.