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.     

GLOBAL ANALYSIS OF SIS EPIDEMIC MODEL WITH A SIMPLE VACCINATION AND MULTIPLE ENDEMIC EQUILIBRIA

An SIS epidemic model with a simple vaccination is investigated in this article. The efficiency of vaccine, the disease-related death rate and population dynamics are also considered in this model. The authors find two threshold R0 and Rc (Rc may not exist). There is a unique endemic equilibrium for R0 > 1 or Rc = R0; there are two endemic equilibria for Rc < R0 < 1; and there is no endemic equilibrium for R0 < Rc < 1. When Rc exists, there is a backward bifurcation from the disease-free equilibrium for R0 = 1. They analyze the stability of equilibria and obtain the globally dynamic behaviors of the model. The results acquired in this article show that an accurate estimation of the efficiency of vaccine is necessary to prevent and controll the spread of disease.     

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.     

Global behavior of a multi-group SIS epidemic model with age structure

We study global dynamics of a system of partial differential equations. The system is motivated by modelling the transmission dynamics of infectious diseases in a population with multiple groups and age-dependent transition rates. Existence and uniqueness of a positive (endemic) equilibrium are established under the quasi-irreducibility assumption, which is weaker than irreducibility, on the function representing the force of infection. We give a classification of initial values from which corresponding solutions converge to either the disease-free or the endemic equilibrium. The stability of each equilibrium is linked to the dominant eigenvalue s(A), where A is the infinitesimal generator of a “quasi-irreducible” semigroup generated by the model equations. In particular, we show that if s(A)<0 then the disease-free equilibrium is globally stable; if s(A)>0 then the unique endemic equilibrium is globally stable.     

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.     

Assessing the impact of the environmental contamination on the transmission of Ebola virus disease (EVD)

This paper deals with the following biological question: how influential is the environmental contamination on the transmission of EVD? Based on the works in (Bibby et al., Environ Sci Technol Lett 2:2–6, 2015; Leroy et al., Nature 438: 575–576, 2005; World Health Organization. Unprecedented number of medical staff infected with Ebola), we design a new mathematical model to address this question by assessing the effect of the Ebola virus contaminated environment on the dynamical transmission of EVD. The formulated model captures two infection pathways through both direct human-to-human transmission and indirect human-to-environment-to-human transmission by incorporating the environment as a transition and/or reservoir of Ebola viruses. We compute the basic reproduction number \({\mathcal {R}}^{env}_0\) for the model with environmental contamination and prove that the disease-free equilibrium is globally asymptotically stable (GAS) whenever \({\mathcal {R}}^{env}_0 \le 1\). When \({\mathcal {R}}^{env}_0 > 1\), we show that the said model has a unique endemic equilibrium which is GAS. Similar results hold for the free environmental contamination sub-model (without the incorporation of the indirect transmission). More precisely, for the latter model, calculate the corresponding basic reproduction number \({\mathcal {R}}^{h}_0\) and establish the GAS of the disease-free and endemic equilibria, whenever \({\mathcal {R}}^{h}_0 \le 1\) and \({\mathcal {R}}^{h}_0 > 1\), respectively. At the endemic level, we show that the number of infected individuals for the full model with the environmental contamination is greater than the corresponding number for the free environmental contamination sub-model. In conjunction with the inequality \({\mathcal {R}}^{h}_0 < {\mathcal {R}}^{env}_0\), our finding suggests a negative answer to the biological question under investigation, i.e. the contaminated environment plays a detrimental role on the transmission dynamics of EVD by increasing the endemic level and/or the severity of the outbreak. Therefore, it is natural to implement a control strategy which aim at reducing the severity of the disease by providing adequate hygienic living conditions, educate populations at risk to follow rigorously those basic hygienic conditions as well as ask them avoid contact with suspected contaminated objects. Further, we perform numerical simulations to support the theory.     

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 SIS epidemic model with diffusion

The aim of this paper is to study the dynamics of an SIS epidemic model with diffusion. We first study the well-posedness of the model. And then, by using linearization method and constructing suitable Lyapunov function, we establish the local and global stability of the disease-free equilibrium and the endemic equilibrium, respectively. Furthermore, in view of Schauder fixed point theorem, we show that the model admits traveling wave solutions connecting the disease-free equilibrium and the endemic equilibrium when R_0 1 and c c~*. And also, by virtue of the two-sided Laplace transform, we prove that the model has no traveling wave solution connecting the two equilibria when R_0 1 and c ∈ [0, c~*).     

Global stability of a SEIR epidemic model with infectious force in latent, infected and immune period

In this paper, we studied the global dynamics of a SEIR epidemic model in which the latent and immune state were infective. The basic reproductive rate, R₀, is derived. If R₀  1, the disease-free equilibrium is globally stable and the disease always dies out. If R₀ > 1, there exists a unique endemic equilibrium which is locally stable. Furthermore, we proved the global stability of the unique endemic equilibrium when ₁ = ₂ = 0 and the disease persists at an endemic equilibrium state if it initially exists.     

Dynamic behavior of a delay cholera model with constant infectious period

In this paper, a delay cholera model with constant infectious period is investigated. By analyzing the characteristic equations, the local stability of a disease-free equilibrium and an endemic equilibrium of the model is established. It is proved that if the basic reproductive number $\mathcal{R}_0>1$, the system is permanent. If $\mathcal{R}_0<1$, by means of an iteration technique, sufficient conditions are obtained for the global asymptotic stability of the disease-free equilibrium. If $\mathcal{R}_0>1$, also by means of an iteration technique, sufficient conditions are obtained for the global asymptotic stability of the endemic equilibrium. Numerical simulations are carried out to illustrate the main theoretical results.     

The analysis and application of an HBV model

A mathematical model is formulated to describe the spread of hepatitis B. The stability of equilibria and persistence of disease are analyzed. The results shows that the dynamics of the model is completely determined by the basic reproductive number ρ₀. If ρ₀ < 1, the disease-free equilibrium is globally stable. When ρ₀ > 1, the disease-free equilibrium is unstable and the disease is uniformly persistent. Furthermore, under certain conditions, it is proved that the endemic equilibrium is globally attractive. Numerical simulations are conducted to demonstrate our theoretical results. The model is applied to HBV transmission in China. The parameter values of the model are estimated based on available HBV epidemic data in China. The simulation results matches the HBV epidemic data in China approximately.     

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.     

Mathematical Modeling and Analysis of an Epidemic Model with Quarantine, Latent and Media Coverage

Epidemic models are very important in today''s analysis of diseases. In this paper, we propose and analyze an epidemic model incorporating quarantine, latent, media coverage and time delay. We analyze the local stability of either the disease-free and endemic equilibrium in terms of the basic reproduction number $\mathcal{R}_{0}$ as a threshold parameter. We prove that if $\mathcal{R}_{0}<1,$ the time delay in media coverage can not affect the stability of the disease-free equilibrium and if $\mathcal{R}_{0}>1$, the model has at least one positive endemic equilibrium, the stability will be affected by the time delay and some conditions for Hopf bifurcation around infected equilibrium to occur are obtained by using the time delay as a bifurcation parameter. We illustrate our results by some numerical simulations such that we show that a proper application of quarantine plays a critical role in the clearance of the disease, and therefore a direct contact between people plays a critical role in the transmission of the disease.     

一类具有时滞的流行病模型分析

讨论了一类带有时滞的SE IS流行病模型,并讨论了阈值、平衡点和稳定性.模型是一个具有确定潜伏期的时滞微分方程模型,在这里我们得到了各类平衡点存在条件的阈值R0;当R0<1时,只有无病平衡点P0,且是全局渐近稳定的;当R0>1时,除无病平衡点外还存在唯一的地方病平衡点Pe,且该平衡点是绝对稳定的.     

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.     

HIV/AIDS epidemic fractional-order model

In this paper a non-linear mathematical model with fractional order ?, 0 < ? ≤ 1 is presented for analyzing and controlling the spread of HIV/AIDS. Both the disease-free equilibrium E₀ and the endemic equilibrium E^* are found and their stability is discussed using the stability theorem of fractional order differential equations. The basic reproduction number R₀ plays an essential role in the stability properties of our system. When R₀ < 1 the disease-free equilibrium E₀ is attractor, but when R₀ > 1, E₀ is unstable and the endemic equilibrium (EE) E^* exists and it is an attractor. Finally numerical Simulations are also established to investigate the influence of the system parameter on the spread of the disease.     

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 Cholera model with age-of-immunity structure and reinfection

To understand V.Cholera transmission dynamics, in this paper, a mathematical model for the dynamics of cholera with reinfection is formulated that incorporates the duration time of the recovery individuals (age-of-immunity). The basic reproduction number $\Re_0$ for the model is identified and the threshold property of $\Re_0$ is established. By applying the persistence theory for infinite-dimensional systems, we show that the disease is uniformly persistent if the reproductive number $ \Re_0>1$. By constructing a suitable Lyapunov function, the global stability of the infection-free equilibrium in the system is obtained for $\Re_0<1$; the unique endemic equilibrium of the system is globally asymptotically stable for $\Re_0>1$.     

Curves of Positive Solutions for Supercritical Problems

For the Dirichlet problem on ball in R n with p < ( n + s )/( n m 2), we show existence of a critical u 0 > 0, such that there are two positive radial solutions for u ] (0, u 0 ), one at u = u 0 , and none for u > u 0 . In another direction we show that for the model problem with 1 h q < ( n + 2)/( n m 2) < p , on a bounded star-shaped domain in R n , there is a u * > 0 so that the problem has no positive solution for u < u *.     

一类具有细胞免疫和吸收效应的时滞病毒动力学模型(英文)

本文主要研究一类具有细胞免疫和吸收效应的时滞病毒动力学模型.通过构造Lyapunov泛函,证明当R0≤1,R1 ≤1相似文献