Threshold dynamics of the stochastic epidemic model with jump-diffusion infection force

This paper formulates a stochastic SIR epidemic model by supposing that the infection force is perturbed by Brown motion and L\''{e}vy jumps. The globally positive and bounded solution is proved firstly by constructing the suitable Lyapunov function. Then, a stochastic basic reproduction number $R_0^{L}$ is derived, which is less than that for the deterministic model and the stochastic model driven by Brown motion. Analytical results show that the disease will die out if $R_0^{L}<1$, and $R_0^{L}>1$ is the necessary and sufficient condition for persistence of the disease. Theoretical results and numerical simulations indicate that the effects of L\''{e}vy jumps may lead to extinction of the disease while the deterministic model and the stochastic model driven by Brown motion both predict persistence. Additionally, the method developed in this paper can be used to investigate a class of related stochastic models driven by L\''{e}vy noise.     

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.     

Birds movement impact on the transmission of West Nile virus between patches

Spatial heterogeneity plays an important role in the distribution and persistence of many infectious disease. In the paper, a multi-patch model for the spread of West Nile virus among $n$ discrete geographic regions is presented that incorporates a mobility process. In the mobility process, we assume that the birds can move among regions, but not the mosquitoes based on scale-space. We show that the movement of birds between patches is sufficient to maintain disease persistence in patches. We compute the basic reproduction number $R_{0}$. We prove that if $R_{0}<1$, then the disease-free equilibrium of the model is globally asymptotically stable. When $R_{0}>1$, we prove that there exists a unique endemic equilibrium, which is globally asymptotically stable on the biological domain. Finally, numerical simulations demonstrate that the disease becomes endemic in both patches when birds move back and forth between two regions.     

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.     

具常数输入及饱和发生率的脉冲接种SIQRS传染病模型

研究了具有常数输入及饱和发生率的脉冲接种SIQRS传染病模型,得到了疾病消除与否的阈值R_0=1.证明了当R_01时,系统存在全局渐近稳定的无病周期解;当R_01时,系统一致持久.     

Threshold dynamics in a stochastic SIRS epidemic model with nonlinear incidence rate

We discuss the dynamic of a stochastic Susceptible-Infectious-Recovered-Susceptible (SIRS) epidemic model with nonlinear incidence rate.The crucial threshold $\tilde{R}_0$ is identified and this will determine the extinction and persistence of the epidemic when the noise is small. We also discuss the asymptotic behavior of the stochastic model around the endemic equilibrium of the corresponding deterministic system. When the noise is large, we find that a large noise intensity has the effect of suppressing the epidemic, so that it dies out. Finally, these results are illustrated by computer simulations.     

Dynamical Analysis of a Delayed SIQS Epidemic Model on Scale-free Networks

In this paper, we establish a novel delayed SIQS epidemic model on scale-free networks, where time delay represents the average quarantine period. Through mathematical analysis, we present the basic reproduction number $R_{0}$. Then, we provide the global asymptotical stability of the disease-free equilibrium and the local asymptotical stability of the endemic equilibrium. Finally, we perform numerical simulations to verify the correctness of the main results and analyze the sensitivity of parameters. Our research shows that when $R_0>1$, lengthening the quarantine period can slow the spread of the disease and reduce the number of infected individuals.     

Modelling and stability of epidemic model with free-living pathogens growing in the environment

To understand the impact of free-living pathogens (FLP) on the epidemics, an epidemic model with FLP is constructed. The global dynamics of our model are determined by the basic reproduction number $R_0$. If $R_0<1$, the disease free equilibrium is globally asymptotically stable, and if $R_0>1$, the endemic equilibrium is globally asymptotically stable. Some numerical simulations are also carried out to illustrate our analytical results.     

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.     

一类有迁移的SIS流行病模型

研究一类种群有迁移的流行病模型,得到了这类模型的基本再生数R0,证明了R0＜1无病平衡点是局部渐近稳定的,而当R0＞1时无病平衡点是不稳定的.进一步讨论了疾病持续存在与无病平衡点和地方病平衡点全局稳定的条件.     

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.     

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 dynamical analysis of a heroin epidemic model on complex networks

In this paper, a heroin epidemic model on complex networks is proposed. By the next generation matrix, the basic reproduction number $R_0$ is obtained. If $R_0<1$, then the drug-free equilibrium is globally asymptotically stable. If $R_0>1$, there is an unique endemic equilibrium and it is also globally asymptotically stable. Our results show that if the degree of the network is large enough, the drug transmission always spreads. Sensitivity analysis of the basic reproduction number with the various parameters in the model are carried out to verify the important effects for control the drug transmission. Some simulations illustrate our theoretical 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 analysis of an epidemic model with diffusion and stochastic perturbation

In this paper, we investigate the stability of an epidemic model with diffusion and stochastic perturbation. We first show both the local and global stability of the endemic equilibrium of the deterministic epidemic model by analyzing corresponding characteristic equation and Lyapunov function. Second, for the corresponding reaction–diffusion epidemic model, we present the conditions of the globally asymptotical stability of the endemic equilibrium. And we carry out the analytical study for the stochastic model in details and find out the conditions for asymptotic stability of the endemic equilibrium in the mean sense. Furthermore, we perform a series of numerical simulations to illustrate our mathematical findings.     

${R}_{0}$代数中的滤子格

Abstract In the present paper, some basic properties of MP filters of Ro algebra M are investigated. It is proved that（FMP（M）,包含,′∧^-∨^-,{1},M）is a bounded distributive lattice by introducing the negation operator ′, the meet operator ∧^-, the join operator ∨^- and the implicati on operator → on the set FMP（M） of all MP filters of M. Moreover, some conditions under which （FMP（M）,包含,′∨^-,→{1},M）is an Ro algebra are given. And the relationship between prime elements of FMP （M） and prime filters of M is studied. Finally, some equivalent characterizations of prime elements of .FMP （M） are obtained.     

关于杨全会和陈永高提出的一个问题

对任意的正整数与集合,令为解的个数.杨全会和陈永高证明了:若整数且,则不存在集合使得对所有充分大的整数成立,其中.对整数和,定义为满足对所有整数成立的集合的个数.杨全会和陈永高证明了是有限的,且.同时,他们问对任意整数,是否存在使得对所有整数成立.在本文中,我们给出了在时的准确公式.从而推出在时成立.     

Dynamics Analysis of an SIS Epidemic Model with the Effects of Awareness

In this paper, an SIS model incorporating the effects of awareness spreading on epidemic is analyzed. Four kinds of equilibria of the model are given, and a new method is used to prove the stability of the equilibria. The threshold of awareness is $R_{1}^{a}$, which measures whether awareness spreads. When awareness does not spread, the basic reproduction number of disease is $R_{1}^{d}$, it is $R_{2}^{d}$ when awareness spreads. The relationship among the three kinds of thresholds is discussed in details. Specially, the effects of various awareness parameters on epidemic are analyzed. Our theoretical results suggest that raising awareness can effectively reduce the basic reproduction number of disease and reduce the spread of disease. Furthermore, numerical simulations are performed to illustrate our results.     

基于完备BR0-代数的全蕴涵三I算法

研究了基础$BR_0$-代数的性质和基于完备基础$BR_0$-代数的全蕴涵三I算法,对一般蕴涵算子给出了三I算法解存在的一个充分条件,并将结果应用于$R_0$-单位区间$\overline{W}$,不但极大的简化了$R_0$-单位区间$\overline{W}$的$R_0$-型$\alpha$-三I算法结果的证明,而且使其证明过程与相应的模糊命题演算系统结合起来,说明了$R_0$-型三I算法是与$B{\cal L}^*$系统相匹配的模糊推理方法.