In this article, we establish the global asymptotic stability of a disease-free equilibrium and an endemic equilibrium of an SIRS epidemic model with a class of nonlin- ear incidence rates and distributed delays. By u...In this article, we establish the global asymptotic stability of a disease-free equilibrium and an endemic equilibrium of an SIRS epidemic model with a class of nonlin- ear incidence rates and distributed delays. By using strict monotonicity of the incidence function and constructing a Lyapunov functional, we obtain sufficient conditions under which the endemic equilibrium is globally asymptotically stable. When the nonlinear inci- dence rate is a saturated incidence rate, our result provides a new global stability condition for a small rate of immunity loss.展开更多
This paper considers two differential infectivity（DI） epidemic models with a nonlinear incidence rate and constant or varying population size. The models exhibits two equilibria, namely., a disease-free equilibrium ...This paper considers two differential infectivity（DI） epidemic models with a nonlinear incidence rate and constant or varying population size. The models exhibits two equilibria, namely., a disease-free equilibrium O and a unique endemic equilibrium. If the basic reproductive number σ is below unity,O is globally stable and the disease always dies out. If σ〉1, O is unstable and the sufficient conditions for global stability of endemic equilibrium are derived. Moreover,when σ〈 1 ,the local or global asymptotical stability of endemic equilibrium for DI model with constant population size in n-dimensional or two-dimensional space is obtained.展开更多
In a polluted environment, considering the biological population infected with a kind of disease and hunted by human beings, we formulate a nonautonomous SIR population-epidemic model with time-varying impulsive relea...In a polluted environment, considering the biological population infected with a kind of disease and hunted by human beings, we formulate a nonautonomous SIR population-epidemic model with time-varying impulsive release and general nonlinear incidence rate and investigate dynamical behaviors of the model. Under the reasonable assumptions, the sufficient conditions which guarantee the globally attractive of the disease-free periodic solution and the permanence of the infected fish are established, that is, the infected fish dies out if , whereas the disease persists if . To substantiate our theoretical results, extensive numerical simulations are performed for a hypothetical set of parameter values.展开更多
In this paper,an SEIR model with nonlinear incidence rates are studied.The basic reproduction number R_0 characterizes the disease transmission dynamics: if R_0≤ 1,the disease-free equilibrium is globally asymptotica...In this paper,an SEIR model with nonlinear incidence rates are studied.The basic reproduction number R_0 characterizes the disease transmission dynamics: if R_0≤ 1,the disease-free equilibrium is globally asymptotically stable and the disease always dies out,if R_0> 1 then there is a unique endemic equilibrium which is globally asymptotically stable and the disease persists.展开更多
In this paper, a susceptible-vazcinated-exposed-infectious-recovered epidemic model with waning immunity and continuous age structures in vaccinated, exposed and infectious classes has been formulated. By using the Fl...In this paper, a susceptible-vazcinated-exposed-infectious-recovered epidemic model with waning immunity and continuous age structures in vaccinated, exposed and infectious classes has been formulated. By using the Fluctuation lemma and the approach of Lyapunov functionals, we establish a threshold dynamics completely determined by the basic reproduction number. When the basic reproduction number is less than one, the disease-free steady state is globally asymptotically stable, and otherwise the endemic steady state is globally asymptotically stable.展开更多
This paper presents a study on a new rumor propagation model with nonlinear propagation rate and secondary propagation rate. We divide the total population into three groups, the ignorant, the spreader and the aware. ...This paper presents a study on a new rumor propagation model with nonlinear propagation rate and secondary propagation rate. We divide the total population into three groups, the ignorant, the spreader and the aware. The nonlinear incidence rate describes the psychological impact of certain serious rumors on social groups when the number of individuals spreading rumors becomes larger. The main contributions of this work are the development of a new rumor propagation model and some results of deterministic and stochastic analysis of the rumor propagation model. The results show the influence of nonlinear propagation rate and stochastic fluctuation on the dynamic behavior of the rumor propagation model by using Lyapunov function method and stochastic related knowledge. Numerical examples and simulation results are given to illustrate the results obtained.展开更多
In this paper,a reaction-diffusion SEI epidemic model with nonlinear incidence rate is proposed.The well-posedness of solutions is studied,including the existence of positive and unique classical solution and the exis...In this paper,a reaction-diffusion SEI epidemic model with nonlinear incidence rate is proposed.The well-posedness of solutions is studied,including the existence of positive and unique classical solution and the existence and the ultimate boundedness of global solutions.The basic reproduction numbers are given in both heterogeneous and homogeneous environments.For spatially heterogeneous environment,by the comparison principle of the diffusion system,the infection-free steady state is proved to be globally asymptotically stable if R_(0)<1,if R_(0)>1,the system will be persistent and admit at least one positive steady state.For spatially homogenous environment,by constructing a Lyapunov function,the infect ion-free steady state is proved to be globally asymptotically stable if,R_(0)<1,and then the unique positive steady state is achieved and is proved to be globally asymptotically stable if R_(0)>1.Finally,two examples are given via numerical simulations,and then some control strategies are also presented by the sensitive analysis.展开更多
In the spread of infectious diseases,intervention levels play a crucial role in shaping interactions between healthy and infected individuals,leading to a nonlinear transmission process.Additionally,the availability o...In the spread of infectious diseases,intervention levels play a crucial role in shaping interactions between healthy and infected individuals,leading to a nonlinear transmission process.Additionally,the availability of medical resources limits the recovery rate of infected patients,adding further nonlinear dynamics to the healing process.Our research introduces novelty by combining nonlinear incidence and recovery rates alongside waning immunity in an epidemic model.We present a modified SIRW-type model,examining the epidemic problem with these factors.Through analysis,we explore conditions for non-endemic and co-existing cases based on the basic reproduction ratio.The local stability of equilibria is verified using the Routh-Hurwitz criteria,while global stability is assessed using Lyapunov functions for each equilibrium.Furthermore,we investigate bifurcations around both non-endemic and co-existing equilibria.Numerically,we give some simulations to support our analytical findings.展开更多
This paper develops an SIBR cholera transmission model with general incidence rate. Necessary and sufficient conditions for local and global asymptotic stability of the equilibria are established by Routh Hurwitz crit...This paper develops an SIBR cholera transmission model with general incidence rate. Necessary and sufficient conditions for local and global asymptotic stability of the equilibria are established by Routh Hurwitz criterium, Lyapunov function, and the second additive composite matrix theorem. What is more, exploiting the DED is cover simulation tool, the parameter values of the model are estimated with the 1998-2021 cholera case data in China. Finally, we perform sensitivity analysis for the basic reproduction number to seek for effective interventions for cholera control. .展开更多
In this paper, we treat the spread of COVID-19 using a delayed stochastic SVIRS (Susceptible, Infected, Recovered, Susceptible) epidemic model with a general incidence rate and differential susceptibility. We start wi...In this paper, we treat the spread of COVID-19 using a delayed stochastic SVIRS (Susceptible, Infected, Recovered, Susceptible) epidemic model with a general incidence rate and differential susceptibility. We start with a deterministic model, then add random perturbations on the contact rate using white noise to obtain a stochastic model. We first show that the delayed stochastic differential equation that describes the model has a unique global positive solution for any positive initial value. Under the condition R<sub>0</sub> ≤ 1, we prove the almost sure asymptotic stability of the disease-free equilibrium of the model.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
In this paper, the sharp threshold properties of a （2n ＋ 1）-dimensional delayed viral infection model are investigated. This model combines with n classes of uninfected tar- get cells, n classes of infected cells a...In this paper, the sharp threshold properties of a （2n ＋ 1）-dimensional delayed viral infection model are investigated. This model combines with n classes of uninfected tar- get cells, n classes of infected cells and nonlinear incidence rate h（x,v）. Two kinds of distributed time delays are incorporated into the model to describe the time needed for infection of uninfected target cells and virus replication. Under certain conditions, it is shown that the basic reproduction number is a threshold parameter for the existence of the equilibria, uniform persistence, as well as for global stability of the equilibria of the model.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 in part by JSPS Fellows,No.237213 of Japan Society for the Promotion of Science to the first authorthe Grant MTM2010-18318 of the MICINN,Spanish Ministry of Science and Innovation to the second authorScientific Research (c),No.21540230 of Japan Society for the Promotion of Science to the third author
文摘In this article, we establish the global asymptotic stability of a disease-free equilibrium and an endemic equilibrium of an SIRS epidemic model with a class of nonlin- ear incidence rates and distributed delays. By using strict monotonicity of the incidence function and constructing a Lyapunov functional, we obtain sufficient conditions under which the endemic equilibrium is globally asymptotically stable. When the nonlinear inci- dence rate is a saturated incidence rate, our result provides a new global stability condition for a small rate of immunity loss.
文摘This paper considers two differential infectivity（DI） epidemic models with a nonlinear incidence rate and constant or varying population size. The models exhibits two equilibria, namely., a disease-free equilibrium O and a unique endemic equilibrium. If the basic reproductive number σ is below unity,O is globally stable and the disease always dies out. If σ〉1, O is unstable and the sufficient conditions for global stability of endemic equilibrium are derived. Moreover,when σ〈 1 ,the local or global asymptotical stability of endemic equilibrium for DI model with constant population size in n-dimensional or two-dimensional space is obtained.
文摘In a polluted environment, considering the biological population infected with a kind of disease and hunted by human beings, we formulate a nonautonomous SIR population-epidemic model with time-varying impulsive release and general nonlinear incidence rate and investigate dynamical behaviors of the model. Under the reasonable assumptions, the sufficient conditions which guarantee the globally attractive of the disease-free periodic solution and the permanence of the infected fish are established, that is, the infected fish dies out if , whereas the disease persists if . To substantiate our theoretical results, extensive numerical simulations are performed for a hypothetical set of parameter values.
基金Supported by the National Natural Science Foundation of China(11101323)Supported by the Natural Science Basic Research Plan in Shaanxi Province of China(2014JQ1038)Supported by the Xi’an Polytechnic University Innovation Fund for Graduate Students(CX201608)
文摘In this paper,an SEIR model with nonlinear incidence rates are studied.The basic reproduction number R_0 characterizes the disease transmission dynamics: if R_0≤ 1,the disease-free equilibrium is globally asymptotically stable and the disease always dies out,if R_0> 1 then there is a unique endemic equilibrium which is globally asymptotically stable and the disease persists.
文摘In this paper, a susceptible-vazcinated-exposed-infectious-recovered epidemic model with waning immunity and continuous age structures in vaccinated, exposed and infectious classes has been formulated. By using the Fluctuation lemma and the approach of Lyapunov functionals, we establish a threshold dynamics completely determined by the basic reproduction number. When the basic reproduction number is less than one, the disease-free steady state is globally asymptotically stable, and otherwise the endemic steady state is globally asymptotically stable.
文摘This paper presents a study on a new rumor propagation model with nonlinear propagation rate and secondary propagation rate. We divide the total population into three groups, the ignorant, the spreader and the aware. The nonlinear incidence rate describes the psychological impact of certain serious rumors on social groups when the number of individuals spreading rumors becomes larger. The main contributions of this work are the development of a new rumor propagation model and some results of deterministic and stochastic analysis of the rumor propagation model. The results show the influence of nonlinear propagation rate and stochastic fluctuation on the dynamic behavior of the rumor propagation model by using Lyapunov function method and stochastic related knowledge. Numerical examples and simulation results are given to illustrate the results obtained.
基金supported by the National Natural Science Foundation of China(Grant No.11871475)the Hunan Provincial Innovation Foundation for Postgraduate(Grant No.CX20200096)the Fundamental Research Funds for the Central Universities of Central South University(Grant No.2020zzts024).
文摘In this paper,a reaction-diffusion SEI epidemic model with nonlinear incidence rate is proposed.The well-posedness of solutions is studied,including the existence of positive and unique classical solution and the existence and the ultimate boundedness of global solutions.The basic reproduction numbers are given in both heterogeneous and homogeneous environments.For spatially heterogeneous environment,by the comparison principle of the diffusion system,the infection-free steady state is proved to be globally asymptotically stable if R_(0)<1,if R_(0)>1,the system will be persistent and admit at least one positive steady state.For spatially homogenous environment,by constructing a Lyapunov function,the infect ion-free steady state is proved to be globally asymptotically stable if,R_(0)<1,and then the unique positive steady state is achieved and is proved to be globally asymptotically stable if R_(0)>1.Finally,two examples are given via numerical simulations,and then some control strategies are also presented by the sensitive analysis.
基金funded by Universitas Padjadjaran,Indonesia,via Hibah Riset Data Pustaka dan Daring Universitas Padjadjaran,No.1549/UN6.3.1/PT.00/2023.
文摘In the spread of infectious diseases,intervention levels play a crucial role in shaping interactions between healthy and infected individuals,leading to a nonlinear transmission process.Additionally,the availability of medical resources limits the recovery rate of infected patients,adding further nonlinear dynamics to the healing process.Our research introduces novelty by combining nonlinear incidence and recovery rates alongside waning immunity in an epidemic model.We present a modified SIRW-type model,examining the epidemic problem with these factors.Through analysis,we explore conditions for non-endemic and co-existing cases based on the basic reproduction ratio.The local stability of equilibria is verified using the Routh-Hurwitz criteria,while global stability is assessed using Lyapunov functions for each equilibrium.Furthermore,we investigate bifurcations around both non-endemic and co-existing equilibria.Numerically,we give some simulations to support our analytical findings.
文摘This paper develops an SIBR cholera transmission model with general incidence rate. Necessary and sufficient conditions for local and global asymptotic stability of the equilibria are established by Routh Hurwitz criterium, Lyapunov function, and the second additive composite matrix theorem. What is more, exploiting the DED is cover simulation tool, the parameter values of the model are estimated with the 1998-2021 cholera case data in China. Finally, we perform sensitivity analysis for the basic reproduction number to seek for effective interventions for cholera control. .
文摘In this paper, we treat the spread of COVID-19 using a delayed stochastic SVIRS (Susceptible, Infected, Recovered, Susceptible) epidemic model with a general incidence rate and differential susceptibility. We start with a deterministic model, then add random perturbations on the contact rate using white noise to obtain a stochastic model. We first show that the delayed stochastic differential equation that describes the model has a unique global positive solution for any positive initial value. Under the condition R<sub>0</sub> ≤ 1, we prove the almost sure asymptotic stability of the disease-free equilibrium of the model.
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 authors would like to thank the anonymous referees and the editor for very helpful suggestions and comments which led to improvements of our orig- inal paper. J. Wang was supported by National Natural Science Foundation of China （Nos. 11401182 and 11471089）, Natural Science Foundation of Heilongjiang Province （No. A201415）, Science and Technology Innovation Team in Higher Edu- cation Institutions of Heilongjiang Province （No. 2014TD005）, Project funded by China Postdoctoral Science Foundation （No. 2014M552295） and Project funded by Chongqing Postdoctoral Foundation （No. Xm2014024）. X. Wang is supported by the National Natural Science Foundation of China （No. 11301453）, Postdoctoral Science Foundation of China （No. 2014M562366）, Postdoctoral Science Foundation of Shaanxi Province （No. 2014010）, the Universities Young Teachers Program of Henan Province （No. 2014GGJS-093）.
文摘In this paper, the sharp threshold properties of a （2n ＋ 1）-dimensional delayed viral infection model are investigated. This model combines with n classes of uninfected tar- get cells, n classes of infected cells and nonlinear incidence rate h（x,v）. Two kinds of distributed time delays are incorporated into the model to describe the time needed for infection of uninfected target cells and virus replication. Under certain conditions, it is shown that the basic reproduction number is a threshold parameter for the existence of the equilibria, uniform persistence, as well as for global stability of the equilibria of the model.
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.
