Global analysis of a multi-group animal epidemic model with indirect infection and time delay

The transmission mechanism of some animal diseases is complex because of the multiple transmission pathways and multiple-group interactions, which lead to the limited understanding of the dynamics of these diseases transmission. In this paper, a delay multi-group dynamic model is proposed in which time delay is caused by the latency of infection. Under the biologically motivated assumptions, the basic reproduction number $R_0$ is derived and then the global stability of the disease-free equilibrium and the endemic equilibrium is analyzed by Lyapunov functionals and a graph-theoretic approach as for time delay. The results show the global properties of equilibria only depend on the basic reproductive number $R_0$: the disease-free equilibrium is globally asymptotically stable if $R_0\leq 1$; if $R_0>1$, the endemic equilibrium exists and is globally asymptotically stable, which implies time delay span has no effect on the stability of equilibria. Finally, some specific examples are taken to illustrate the utilization of the results and then numerical simulations are used for further discussion. The numerical results show time delay model may experience periodic oscillation behaviors, implying that the spread of animal diseases depends largely on the prevention and control strategies of all sub-populations.     

Stability and bifurcation analysis of a viral infection model with delayed immune response

In this paper, we study a viral infection model with an immunity time delay accounting for the time between the immune system touching antigenic stimulation and generating CTLs. By calculation, we derive two thresholds to determine the global dynamics of the model, i.e., the reproduction number for viral infection $R_{0}$ and for CTL immune response $R_{1}$. By analyzing the characteristic equation, the local stability of each feasible equilibrium is discussed. Furthermore, the existence of Hopf bifurcation at the CTL-activated infection equilibrium is also studied. By constructing suitable Lyapunov functionals, we prove that when $R_{0}\leq1$, the infection-free equilibrium is globally asymptotically stable; when $R_{0}>1$ and $R_{1}\leq1$, the CTL-inactivated infection equilibrium is globally asymptotically stable; Numerical simulation is carried out to illustrate the main results in the end.     

Global results for an HIV/AIDS model with multiple susceptible classes and nonlinear incidence

In this paper, an HIV/AIDS epidemic model is proposed in which there are two susceptible classes. Two types of general nonlinear incidence functions are employed to depict the scenarios of infection among cautious and incautious individuals. Qualitative analyses are performed, in terms of the basic reproduction number $\R_0$, to gain the global dynamics of the model: the disease-free equilibrium is of global asymptotic stability when $\R_0\leq 1$; a unique endemic equilibrium exists and globally asymptotically stable $\R_0> 1$. The introduction of cautious susceptible and the resulting multiple transmission functions has positive effect on HIV/AIDS prevalence. Numerical simulations are carried out to illustrate and extend the obtained analytical results.     

Global analysis for a delayed SIV model with direct and environmental transmissions

In this paper, we propose a new SIV epidemic model with time delay, which also involves both direct and environmental transmissions. For such model, we first introduce the basic reproduction number $\mathscr{R}$ by using the next generation matrix. And then global stability of the equilibria is discussed by means of Lyapunov functionals and LaSalle''s invariance principle for delay differential equations, which shows that the infection-free equilibrium of the system is globally asymptotically stable if $\mathscr{R}<1$ and the epidemic equilibrium of the system is globally asymptotically stable for $\m     

Stability and Hopf Bifurcation of a Virus Infection Model with a Delayed CTL Immune Response

In this paper, a virus infection model with standard incidence rate and delayed CTL immune response is investigated. By analyzing corresponding characteristic equations, the local stability of each of feasible equilibria and the existence of Hopf bifurcations at the CTL-activated infection equilibrium are established, respectively. By means of comparison arguments, it is verified that the infection-free equilibrium is globally asymptotically stable if the basic reproduction ratio is less than unity. By using suitable Lyapunov functional and LaSalle＇s invariance principle, it is shown that the CTL-inactivated infection equilibrium of the system is globally asymptotically stable if tile immune response reproduction ratio is less than unity and the basic reproduction ratio is greater than unity. Numerical simulations are carried out to illustrate the theoretical result.     

Global dynamics in a multi-group epidemic model for disease with latency spreading and nonlinear transmission rate

In this paper, we investigate a class of multi-group epidemic models with general exposed distribution and nonlinear incidence rate. Under biologically motivated assumptions, we show that the global dynamics are completely determined by the basic production number $R_0$. The disease-free equilibrium is globally asymptotically stable if $R_0\leq1$, and there exists a unique endemic equilibrium which is globally asymptotically stable if $R_0>1$. The proofs of the main results exploit the persistence theory in dynamical system and a graph-theoretical approach to the method of Lyapunov functionals. A simpler case that assumes an identical natural death rate for all groups and a gamma distribution for exposed distribution is also considered. In addition, two numerical examples are showed to illustrate the results.     

Dynamics analysis of a delayed viral infection model with immune impairment

In this paper, the dynamical behavior of a delayed viral infection model with immune impairment is studied. It is shown that if the basic reproductive number of the virus is less than one, then the uninfected equilibrium is globally asymptotically stable for both ODE and DDE model. And the effect of time delay on stabilities of the equilibria of the DDE model has been studied. By theoretical analysis and numerical simulations, we show that the immune impairment rate has no effect on the stability of the ODE model, while it has a dramatic effect on the infected equilibrium of the DDE model.     

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.     

总人口规模变化的年龄结构SEIR流行病模型的稳定性

运用泛函分析中的谱理论和非线性发展方程的齐次动力系统理论,讨论了总人口规模变化情况下的年龄结构的SEIR流行病模型．得到了与总人口增长指数λ*有关的再生数R0的表达式,证明了当R0<1时,系统存在唯一局部渐近稳定的无病平衡态;当 R0>1时,无病平衡态不稳定,此时存在地方病平衡态,并在一定条件下证明了地方病平衡态是局部渐近稳定的．     

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 dynamics of a reaction and diffusion model for an HTLV-I infection with mitotic division of actively infected cells

This paper is concerned with the global dynamics of a reaction and diffusion model for an HTLV-I infection with mitotic division of actively infected cells and CTL immune response. The well posedness of the proposed model is investigated. In the case of a bounded spatial domain, we establish the threshold dynamics in terms of the basic reproduction number $\mathcal{R}_0$ for the spatially heterogeneous model. Also, by means of different Lyapunov functions, the global asymptotic properties of the steady states for the spatially homogeneous model are studied. In the case of an unbounded spatial domain, there are no travelling wave solutions connecting the infection-free steady state with itself when $\mathcal{R}_0 < 1$. Finally, numerical simulations and conclusions are given.     

Analysis of an HIV model with post-treatment control

Recent investigation indicated that latent reservoir and immune impairment are responsible for the post-treatment control of HIV infection. In this paper, we simplify the disease model with latent reservoir and immune impairment and perform a series of mathematical analysis. We obtain the basic infection reproductive number $R_{0}$ to characterize the viral dynamics. We prove that when $R_{0}<1$, the uninfected equilibrium of the proposed model is globally asymptotically stable. When $R_{0}>1$, we obtain two thresholds, the post-treatment immune control threshold and the elite control threshold. The model has bistable behaviors in the interval between the two thresholds. If the proliferation rate of CTLs is less than the post-treatment immune control threshold, the model does not have positive equilibria. In this case, the immune free equilibrium is stable and the system will have virus rebound. On the other hand, when the proliferation rate of CTLs is greater than the elite control threshold, the system has stable positive immune equilibrium and unstable immune free equilibrium. Thus, the system is under elite control.     

Dynamics of a Deterministic and Stochastic Susceptible-exposed-infectious-recovered Epidemic Model

We investigate a susceptible-exposed-infectious-recovered (SEIR) epidemic model with asymptomatic infective individuals. First, we formulate a deterministic model, and give the basic reproduction number $\mathcal{R}_{0}$. We show that the disease is persistent, if $\mathcal{R}_{0}>1$, and it is extinct, if $\mathcal{R}_{0}<1$. Then, we formulate a stochastic version of the deterministic model. By constructing suitable stochastic Lyapunov functions, we establish sufficient criteria for the extinction and the existence of ergodic stationary distribution to the model. As a case, we study the COVID-19 transmission in Wuhan, China, and perform some sensitivity analysis. Our numerical simulations are carried out to illustrate the analytic results.     

具有种群Logistic增长及饱和传染率的SIS模型的稳定性和Hopf分支

该文研究一类具有种群Logistic增长及饱和传染率的SIS传染病模型,讨论了平衡点的存在性及全局渐近稳定性,得到疾病消除的阈值就是基本再生数$R_{0}=1$. 证明了,当$R_{0}<1$ 时,无病平衡点全局渐近稳定;当$R_{0}>1$ 且$\alpha K\leq 1$ 时,正平衡点全局渐近稳定;当$R_{0}>1$ 且$\Delta ={0}$ 时,系统在正平衡点附近发生Hopf分支;当$R_{0}>1$ 且$\Delta <{0}$ 时,系统在正平衡点外围附近存在唯一稳定的极限环.     

Dynamics analysis of a delayed viral infection model with logistic growth and immune impairment

In this paper, the dynamics behavior of a delayed viral infection model with logistic growth and immune impairment is studied. It is shown that there exist three equilibria. By analyzing the characteristic equations, the local stability of the infection-free equilibrium and the immune-exhausted equilibrium of the model are established. By using suitable Lyapunov functional and LaSalle invariant principle, it is proved that the two equilibria are globally asymptotically stable. In the following, the stability of the positive equilibrium is investigated. Furthermore, we investigate the existence of Hopf bifurcation by using a delay as a bifurcation parameter. Finally, numerical simulations are carried out to explain the mathematical conclusions.     

与截面相关的Morrey空间与奇异积分

In this paper, we define the Morrey spaces M_F~(p,q) (Rn) and the Campanato spaces E_F~(p,q) (R~n) associated with a family F of sections and a doubling measure μ, where F is closely related to the Monge-Amp`ere equation. Furthermore, we obtain the boundedness of the Hardy-Littlewood maximal function associated to F, Monge-Amp`ere singular integral operators and fractional integrals on M_F~(p,q)(R~n). We also prove that the Morrey spaces M_F~(p,q) (R~n)and the Campanato spaces E_F~(p,q) (R~n) are equivalent with 1 ≤ q ≤ p ∞.     

Approximate Representation of Bergman Submodules

In the present paper, the author shows that if a homogeneous submodule M of the Bergman module L_a~2(B_d) satisfies P_M-sum from i to ( M_(zi)P_MM*_(zi))≤c/(N + 1)P_M for some number c 0, then there is a sequence {f_j } of multipliers and a positive number c such that c'P_M ≤sum from j to ( M_(fj)M*_(fj))≤ P_M, i.e., M is approximately representable. The author also proves that approximately representable homogeneous submodules are p-essentially normal for p d.     

Global stability analysis and permanence for an HIV-1 dynamics model with distributed delays

This paper mainly investigates the global asymptotic stabilities of two HIV dynamics models with two distributed intracellular delays incorporating Beddington-DeAngelis functional response infection rate. An eclipse stage of infected cells (i.e. latently infected cells), not yet producing virus, is included in our models. For the first model, it is proven that if the basic reproduction number $R_0$ is less than unity, then the infection-free equilibrium is globally asymptotically stable, and if $R_0 $ is greater than unity, then the infected equilibrium is globally asymptotically stable. We also obtain that the disease is always present when $R_0 $ is greater than unity by using a permanence theorem for infinite dimensional systems. What is more, a n-stage-structured HIV model with two distributed intracellular delays, which is the extensions to the first model, is developed and analyzed. We also prove the global asymptotical stabilities of two equilibria by constructing suitable Lyapunov functionals.     

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.     

一类具免疫应答和非线性感染函数的时滞HIV-1感染模型的全局稳定性

考虑到HIV-1感染过程中免疫反应和非线性感染函数,建立了一类具有三个分布时滞的HIV-1感染动力学模型.得到了关于病毒感染的基本再生数R0和CTLs免疫反应的基本再生数R1 ＜R0.通过构造Lyapunov泛函证明了系统具有阈值动力学性质,即当R0≤1时,系统存在全局渐近稳定的无感染平衡点;当R1≤1＜R0时,系统出...