This article proposes a diffused hepatitis B virus （HBV） model with CTL immune response and nonlinear incidence for the control of viral infections. By means of different Lyapunov functions, the global asymptotical ...This article proposes a diffused hepatitis B virus （HBV） model with CTL immune response and nonlinear incidence for the control of viral infections. By means of different Lyapunov functions, the global asymptotical properties of the viral-free equilibrium and immune-free equilibrium of the model are obtained. Global stability of the positive equilibrium of the model is also considered. The results show that the free diffusion of the virus has no effect on the global stability of such HBV infection problem with Neumann homogeneous boundary conditions.展开更多
This paper investigates a nonlocal dispersal epidemic model under the multiple nonlocal distributed delays and nonlinear incidence effects.First,the minimal wave speed c*and the basic reproduction number Ro are define...This paper investigates a nonlocal dispersal epidemic model under the multiple nonlocal distributed delays and nonlinear incidence effects.First,the minimal wave speed c*and the basic reproduction number Ro are defined,which determine the existence of traveling wave solutions.Second,with the help of the upper and lower solutions,Schauder's fixed point theorem,and limiting techniques,the traveling waves satisfying some asymptotic boundary conditions are discussed.Specifically,when Ro>1,for every speed c>c^(*) there exists a traveling wave solution satisfying the boundary conditions,and there is no such traveling wave solution for any 0<c<c^(*) when R_(0)>1 or c>0 when R_(0)<1.Finally,we analyze the effects of nonlocal time delay on the minimum wave speed.展开更多
Throughout the progress of epidemic scenarios,individuals in different health classes are expected to have different average daily contact behavior.This contact heterogeneity has been studied in recent adaptive models...Throughout the progress of epidemic scenarios,individuals in different health classes are expected to have different average daily contact behavior.This contact heterogeneity has been studied in recent adaptive models and allows us to capture the inherent differences across health statuses better.Diseases with reinfection bring out more complex scenarios and offer an important application to consider contact disaggregation.Therefore,we developed a nonlinear differential equation model to explore the dynamics of relapse phenomena and contact differences across health statuses.Our incidence rate function is formulated,taking inspiration from recent adaptive algorithms.It incorporates contact behavior for individuals in each health class.We use constant contact rates at each health status for our analytical results and prove conditions for different forward-backward bifurcation scenarios.The relationship between the different contact rates heavily in-fluences these conditions.Numerical examples highlight the effect of temporarily recov-ered individuals and initial conditions on infected population persistence.展开更多
In this paper,a discrete-time SIR epidemic model with nonlinear incidence and recovery rates is obtained by using the forward Euler’s method.The existence and stability of fixed points in this model are well studied....In this paper,a discrete-time SIR epidemic model with nonlinear incidence and recovery rates is obtained by using the forward Euler’s method.The existence and stability of fixed points in this model are well studied.The center manifold theorem and bifurcation theory are applied to analyze the bifurcation properties by using the discrete time step and the intervention level as control parameters.We discuss in detail some codimension-one bifurcations such as transcritical,period-doubling and Neimark–Sacker bifurcations,and a codimension-two bifurcation with 1:2 resonance.In addition,the phase portraits,bifurcation diagrams and maximum Lyapunov exponent diagrams are drawn to verify the correctness of our theoretical analysis.It is found that the numerical results are consistent with the theoretical analysis.More interestingly,we also found other bifurcations in the model during the numerical simulation,such as codimension-two bifurcations with 1:1 resonance,1:3 resonance and 1:4 resonance,generalized period-doubling and fold-flip bifurcations.The results show that the dynamics of the discrete-time model are richer than that of the continuous-time SIR epidemic model.Such a discrete-time model may not only be widely used to detect the pathogenesis of infectious diseases,but also make a great contribution to the prevention and control of infectious diseases.展开更多
On the multilingual online social networks of global information sharing,the wanton spread of rumors has an enormous negative impact on people's lives.Thus,it is essential to explore the rumor-spreading rules in m...On the multilingual online social networks of global information sharing,the wanton spread of rumors has an enormous negative impact on people's lives.Thus,it is essential to explore the rumor-spreading rules in multilingual environment and formulate corresponding control strategies to reduce the harm caused by rumor propagation.In this paper,considering the multilingual environment and intervention mechanism in the rumor-spreading process,an improved ignorants–spreaders-1–spreaders-2–removers(I2SR)rumor-spreading model with time delay and the nonlinear incidence is established in heterogeneous networks.Firstly,based on the mean-field equations corresponding to the model,the basic reproduction number is derived to ensure the existence of rumor-spreading equilibrium.Secondly,by applying Lyapunov stability theory and graph theory,the global stability of rumor-spreading equilibrium is analyzed in detail.In particular,aiming at the lowest control cost,the optimal control scheme is designed to optimize the intervention mechanism,and the optimal control conditions are derived using the Pontryagin's minimum principle.Finally,some illustrative examples are provided to verify the effectiveness of the theoretical results.The results show that optimizing the intervention mechanism can effectively reduce the densities of spreaders-1 and spreaders-2 within the expected time,which provides guiding insights for public opinion managers to control rumors.展开更多
In this paper,a reaction-diffusion SIRS epidemic model with nonlinear incidence rate and partial immunity in a spatially heterogeneous environment is proposed.The well-posedness of the solution is firstly established....In this paper,a reaction-diffusion SIRS epidemic model with nonlinear incidence rate and partial immunity in a spatially heterogeneous environment is proposed.The well-posedness of the solution is firstly established.Then the basic reproduction number R0 is defined and a threshold dynamics is obtained.That is,when R_(0)<1,the disease-free steady state is locally stable,which implies that the disease is extinct,when R_(0)>1,the disease is permanent,and there exists at least one positive steady state solution.Finally,the asymptotic profiles of the positive steady state solution as individuals disperse at small and large rates are investigated.Furthermore,as an application of theoretical analysis,a numerical example involving the spread of influenza is discussed.Based on the numerical simulations,we find that the increase of transmission rate and spatial heterogeneity can enhance the risk of influenza propagation,and the increase of diffusion rate,saturation incidence for susceptible and recovery rate can reduce the risk of influenza propagation.Therefore,we propose to reduce the flow of people to lower the effect of spatial hetero-geneity,increase the transfer of infected individuals to hospitals in surrounding areas to increase the diffusion rate,and increase the construction of public medical resources to increase the recovery rate for controlling influenza propagation.展开更多
In this paper,a stochastic epidemic system with both switching noise and white noise is proposed to research the dynamics of the diseases.Nonlinear incidence and vaccination strategies are also considered in the propo...In this paper,a stochastic epidemic system with both switching noise and white noise is proposed to research the dynamics of the diseases.Nonlinear incidence and vaccination strategies are also considered in the proposed model.By using the method of stochastic analysis,we point out the key parameters that determine the persistence and extinction of the diseases.Specifically,if R0^s is greater than 0,the stochastic system has a unique ergodic stationary distribution;while if R ^* is less than 0,the diseases will be extinct at an exponential rate.展开更多
A heroin model with nonlinear incidence rate and age structure is investigated. The basic reproduction number is determined whether or not a heroin epidemic breaks out. By employing the Lyapunov functionals, the drug-...A heroin model with nonlinear incidence rate and age structure is investigated. The basic reproduction number is determined whether or not a heroin epidemic breaks out. By employing the Lyapunov functionals, the drug-free equilibrium is globally asymptotically stable if R0 ≤1; while the drug spread equilibrium is also globally asymptotically stable if R0≤ 1. Our results imply that improving detected rates and drawing up the efficient prevention play more important role than increasing the treatment for drug users.展开更多
A SIR model of epidemiological dynamics with stage-structure and a type of nonlinear incidence rate is considered under the assumption that the susceptible individual satisfy the logistic equation. The global attracti...A SIR model of epidemiological dynamics with stage-structure and a type of nonlinear incidence rate is considered under the assumption that the susceptible individual satisfy the logistic equation. The global attractivity of the model is studied using Lyapunov functions and LaSalle＇s invariance principle. By the uniform persistence theories, the permanence of the system and the existence of the positive equilibrium are obtained. Moreover, by the normal form theory and the center manifold presented by Hassard, a stability and Hopf bifurcation analysis of the system around positive equilibrium from a local perspective are performed. Numerical simulation is carried out to illustrate our results.展开更多
基金supported by the National Natural Science Foundation of China(10971166,10901131)the National High Technology Research and Development Program of China(863 Program,2009AA01A135)the Natural Science Foundation of Xinjiang Province(2010211B04)
文摘This article proposes a diffused hepatitis B virus （HBV） model with CTL immune response and nonlinear incidence for the control of viral infections. By means of different Lyapunov functions, the global asymptotical properties of the viral-free equilibrium and immune-free equilibrium of the model are obtained. Global stability of the positive equilibrium of the model is also considered. The results show that the free diffusion of the virus has no effect on the global stability of such HBV infection problem with Neumann homogeneous boundary conditions.
基金supported by a grant from the Young Scientist Funds of Natural Science Foundation of Xinjiang Uygur Autonomous Region(No.2022D01C63)Natural Science Foundation of China(No.12271421).
文摘This paper investigates a nonlocal dispersal epidemic model under the multiple nonlocal distributed delays and nonlinear incidence effects.First,the minimal wave speed c*and the basic reproduction number Ro are defined,which determine the existence of traveling wave solutions.Second,with the help of the upper and lower solutions,Schauder's fixed point theorem,and limiting techniques,the traveling waves satisfying some asymptotic boundary conditions are discussed.Specifically,when Ro>1,for every speed c>c^(*) there exists a traveling wave solution satisfying the boundary conditions,and there is no such traveling wave solution for any 0<c<c^(*) when R_(0)>1 or c>0 when R_(0)<1.Finally,we analyze the effects of nonlocal time delay on the minimum wave speed.
基金support from the Research Center in Pure and Applied Mathematics and the Department of Mathematics at Universidad de Costa Rica.
文摘Throughout the progress of epidemic scenarios,individuals in different health classes are expected to have different average daily contact behavior.This contact heterogeneity has been studied in recent adaptive models and allows us to capture the inherent differences across health statuses better.Diseases with reinfection bring out more complex scenarios and offer an important application to consider contact disaggregation.Therefore,we developed a nonlinear differential equation model to explore the dynamics of relapse phenomena and contact differences across health statuses.Our incidence rate function is formulated,taking inspiration from recent adaptive algorithms.It incorporates contact behavior for individuals in each health class.We use constant contact rates at each health status for our analytical results and prove conditions for different forward-backward bifurcation scenarios.The relationship between the different contact rates heavily in-fluences these conditions.Numerical examples highlight the effect of temporarily recov-ered individuals and initial conditions on infected population persistence.
基金supported by the NSF of Shandong Province(ZR2021MA016,ZR2019MA034,ZR2018BF018)the China Postdoctoral Science Foundation(2019M652349)the Youth Creative Team Sci-Tech Program of Shandong Universities(2019KJI007).
文摘In this paper,a discrete-time SIR epidemic model with nonlinear incidence and recovery rates is obtained by using the forward Euler’s method.The existence and stability of fixed points in this model are well studied.The center manifold theorem and bifurcation theory are applied to analyze the bifurcation properties by using the discrete time step and the intervention level as control parameters.We discuss in detail some codimension-one bifurcations such as transcritical,period-doubling and Neimark–Sacker bifurcations,and a codimension-two bifurcation with 1:2 resonance.In addition,the phase portraits,bifurcation diagrams and maximum Lyapunov exponent diagrams are drawn to verify the correctness of our theoretical analysis.It is found that the numerical results are consistent with the theoretical analysis.More interestingly,we also found other bifurcations in the model during the numerical simulation,such as codimension-two bifurcations with 1:1 resonance,1:3 resonance and 1:4 resonance,generalized period-doubling and fold-flip bifurcations.The results show that the dynamics of the discrete-time model are richer than that of the continuous-time SIR epidemic model.Such a discrete-time model may not only be widely used to detect the pathogenesis of infectious diseases,but also make a great contribution to the prevention and control of infectious diseases.
基金the National Natural Science Foundation of People’s Republic of China(Grant Nos.U1703262 and 62163035)the Special Project for Local Science and Technology Development Guided by the Central Government(Grant No.ZYYD2022A05)Xinjiang Key Laboratory of Applied Mathematics(Grant No.XJDX1401)。
文摘On the multilingual online social networks of global information sharing,the wanton spread of rumors has an enormous negative impact on people's lives.Thus,it is essential to explore the rumor-spreading rules in multilingual environment and formulate corresponding control strategies to reduce the harm caused by rumor propagation.In this paper,considering the multilingual environment and intervention mechanism in the rumor-spreading process,an improved ignorants–spreaders-1–spreaders-2–removers(I2SR)rumor-spreading model with time delay and the nonlinear incidence is established in heterogeneous networks.Firstly,based on the mean-field equations corresponding to the model,the basic reproduction number is derived to ensure the existence of rumor-spreading equilibrium.Secondly,by applying Lyapunov stability theory and graph theory,the global stability of rumor-spreading equilibrium is analyzed in detail.In particular,aiming at the lowest control cost,the optimal control scheme is designed to optimize the intervention mechanism,and the optimal control conditions are derived using the Pontryagin's minimum principle.Finally,some illustrative examples are provided to verify the effectiveness of the theoretical results.The results show that optimizing the intervention mechanism can effectively reduce the densities of spreaders-1 and spreaders-2 within the expected time,which provides guiding insights for public opinion managers to control rumors.
基金supported by the National Natural Science Foundation of China(Grant Nos.12271525,11871475)the Hunan Provincial Innovation Foundation for Postgraduate(Grant Nos.CX20200096)the Fundamental Research Funds for the Central Universities of Central South University(Grant Nos.2020zzts024).
文摘In this paper,a reaction-diffusion SIRS epidemic model with nonlinear incidence rate and partial immunity in a spatially heterogeneous environment is proposed.The well-posedness of the solution is firstly established.Then the basic reproduction number R0 is defined and a threshold dynamics is obtained.That is,when R_(0)<1,the disease-free steady state is locally stable,which implies that the disease is extinct,when R_(0)>1,the disease is permanent,and there exists at least one positive steady state solution.Finally,the asymptotic profiles of the positive steady state solution as individuals disperse at small and large rates are investigated.Furthermore,as an application of theoretical analysis,a numerical example involving the spread of influenza is discussed.Based on the numerical simulations,we find that the increase of transmission rate and spatial heterogeneity can enhance the risk of influenza propagation,and the increase of diffusion rate,saturation incidence for susceptible and recovery rate can reduce the risk of influenza propagation.Therefore,we propose to reduce the flow of people to lower the effect of spatial hetero-geneity,increase the transfer of infected individuals to hospitals in surrounding areas to increase the diffusion rate,and increase the construction of public medical resources to increase the recovery rate for controlling influenza propagation.
基金Z.Qiu is supported by the National Natural Science Foundation of China(NSFC)grant No.11671206X.Zhao is supported by the Scholarship Foundation of China Scholarship Council grant No.201906840072+2 种基金T.Feng is supported by the Scholarship Foundation of China Scholarship Council grant No.201806840120the Out-standing Chinese and Foreign Youth Exchange Program of China Association of Science and Technologythe Fundamental Research Funds for the Central Universities grant No.30918011339.
文摘In this paper,a stochastic epidemic system with both switching noise and white noise is proposed to research the dynamics of the diseases.Nonlinear incidence and vaccination strategies are also considered in the proposed model.By using the method of stochastic analysis,we point out the key parameters that determine the persistence and extinction of the diseases.Specifically,if R0^s is greater than 0,the stochastic system has a unique ergodic stationary distribution;while if R ^* is less than 0,the diseases will be extinct at an exponential rate.
文摘A heroin model with nonlinear incidence rate and age structure is investigated. The basic reproduction number is determined whether or not a heroin epidemic breaks out. By employing the Lyapunov functionals, the drug-free equilibrium is globally asymptotically stable if R0 ≤1; while the drug spread equilibrium is also globally asymptotically stable if R0≤ 1. Our results imply that improving detected rates and drawing up the efficient prevention play more important role than increasing the treatment for drug users.
文摘A SIR model of epidemiological dynamics with stage-structure and a type of nonlinear incidence rate is considered under the assumption that the susceptible individual satisfy the logistic equation. The global attractivity of the model is studied using Lyapunov functions and LaSalle＇s invariance principle. By the uniform persistence theories, the permanence of the system and the existence of the positive equilibrium are obtained. Moreover, by the normal form theory and the center manifold presented by Hassard, a stability and Hopf bifurcation analysis of the system around positive equilibrium from a local perspective are performed. Numerical simulation is carried out to illustrate our results.
