In this paper,based on the SVIQR model we develop a stochastic epidemic model with multiple vaccinations and time delay.Firstly,we prove the existence and uniqueness of the global positive solution of the model,and co...In this paper,based on the SVIQR model we develop a stochastic epidemic model with multiple vaccinations and time delay.Firstly,we prove the existence and uniqueness of the global positive solution of the model,and construct suitable functions to obtain sufficient conditions for disease extinction.Secondly,in order to effectively control the spread of the disease,appropriate control strategies are formulated by using optimal control theory.Finally,the results are verified by numerical simulation.展开更多
In order to study the influence of stochastic disturbance and environment switching on the HPV infection and provide a theoretical basis for the development of effective HPV disease prevention measures,in this paper w...In order to study the influence of stochastic disturbance and environment switching on the HPV infection and provide a theoretical basis for the development of effective HPV disease prevention measures,in this paper we establish a kind of two-sex stochastic HPV epidemic model with white noise and Markov switching.We show that the model has a unique local positive solution and a unique global positive solution.Then we identify the threshold conditions for the persistence of the HPV epidemic,and verify the persistence of the disease using the Lyapunov method and the Ito^formula.At last,the numerical simulation is carried out to illustrate the rationality of the theoretical results.展开更多
<span style="font-family:Verdana;">In this paper we build and analyze two stochastic epidemic models with death. The model assume</span><span style="font-family:Verdana;"><span...<span style="font-family:Verdana;">In this paper we build and analyze two stochastic epidemic models with death. The model assume</span><span style="font-family:Verdana;"><span style="font-family:Verdana;"><span style="font-family:Verdana;">s</span></span></span><span style="font-family:Verdana;"><span style="font-family:Verdana;"><span style="font-family:Verdana;"> that only susceptible individuals (S) can get infected (I) and may die from this disease or a recovered individual becomes susceptible again (SIS model) or completely immune (SIR Model) for the remainder of the study period. Moreover, it is assumed there are no births, deaths, immigration or emigration during the study period;the community is said to be closed. In these infection disease models, there are two central questions: first it is the disease extinction or not and the second studies the time elapsed for such extinction, this paper will deal with this second question because the first answer corresponds to the basic reproduction number defined in the bibliography. More concretely, we study the mean-extinction of the diseases and the technique used here first builds the backward Kolmogorov differential equation and then solves it numerically using finite element method with FreeFem++. Our contribution and novelty </span></span></span><span style="font-family:Verdana;"><span style="font-family:Verdana;"><span style="font-family:Verdana;">are</span></span></span><span style="font-family:Verdana;"><span style="font-family:Verdana;"><span style="font-family:Verdana;"> the following: however the reproduction number effectively concludes the extinction or not of the disease, it does not help to know its extinction times because example with the same reproduction numbers has very different time. Moreover, the SIS model is slower, a result that is not surprising, but this difference seems to increase in the stochastic models with respect to the deterministic ones, it is reasonable to assume some uncertainly.</span></span></span>展开更多
Nonlinearity and randomness are both the essential attributes for the real world,and the case is the same for the models of infectious diseases,for which the deterministic models can not give a complete picture of the...Nonlinearity and randomness are both the essential attributes for the real world,and the case is the same for the models of infectious diseases,for which the deterministic models can not give a complete picture of the evolution.However,although there has been a lot of work on stochastic epidemic models,most of them focus mainly on qualitative properties,which makes us somewhat ignore the original meaning of the parameter value.In this paper we extend the classic susceptible-infectious-removed(SIR)epidemic model by adding a white noise excitation and then we utilize the large deviation theory to quantitatively study the long-term coexistence exit problem with epidemic.Finally,in order to extend the meaning of parameters in the corresponding deterministic system,we tentatively introduce two new thresholds which then prove rational.展开更多
In this work,we consider a stochastic epidemic model with vaccination,healing and relapse.We prove the existence and the uniqueness of the positive solution.We establish sufficient conditions for the extinction and th...In this work,we consider a stochastic epidemic model with vaccination,healing and relapse.We prove the existence and the uniqueness of the positive solution.We establish sufficient conditions for the extinction and the persistence in mean of the stochastic system.Moreover,we also establish sufficient conditions for the existence of ergodic stationary distribution to the model,which reveals that the infectious disease will persist.The graphical illustrations of the approximate solutions of the stochastic epidemic model have been performed.展开更多
In this paper,a stochastic SiS epidemic infectious diseases model with double stochastic perturbations is proposed.First,the existence and uniqueness of the positive global solution of the model are proved.Second,the ...In this paper,a stochastic SiS epidemic infectious diseases model with double stochastic perturbations is proposed.First,the existence and uniqueness of the positive global solution of the model are proved.Second,the controlling conditions for the extinction and persistence of the disease are obtained.Besides,the effects of the intensity of volatility Si and the speed of reversion 1 on the dynamical behaviors of the model are discussed.Finally,some numerical examples are given to support the theoretical results.The results show that if the basic reproduction number R_(0)^(8)<1,the disease will be extinct,that is to say that we can control the threshold R_(0)^(8)to suppress the disease outbreak.展开更多
We present an epidemic model which can incorporate essential biological detail as well as the intrinsic demographic stochastieity of the epidemic process, yet is very simple, enabling rapid generation of a large numbe...We present an epidemic model which can incorporate essential biological detail as well as the intrinsic demographic stochastieity of the epidemic process, yet is very simple, enabling rapid generation of a large number of simulations, A deterministic version of the model is also derived, in the limit of infinitely large populations, and a final-size formula for the deterministic model is proved. A key advantage of the model proposed is that it is possible to write down an explicit likelihood functions for it, which enables a systematic procedure for fitting parameters to real incidence data, using maximum likelihood.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
We present a mathematical analysis of the transmission of certain diseases using a stochastic susceptible-exposed-infectious-treated-recovered(SEITR)model with multiple stages of infection and treatment and explore th...We present a mathematical analysis of the transmission of certain diseases using a stochastic susceptible-exposed-infectious-treated-recovered(SEITR)model with multiple stages of infection and treatment and explore the effects of treatments and external fluctuations in the transmission,treatment and recovery rates.We assume external fluctuations are caused by variability in the number of contacts between infected and susceptible individuals.It is shown that the expected number of secondary infections produced(in the absence of noise)reduces as treatment is introduced into the population.By defining RT,n and R T,n as the basic deterministic and stochastic reproduction numbers,respectively,in stage n of infection and treatment,we show mathematically that as the intensity of the noise in the transmission,treatment and recovery rates increases,the number of secondary cases of infection increases.The global stability of the disease-free and endemic equilibrium for the deterministic and stochastic SEITR models is also presented.The work presented is demonstrated using parameter values relevant to the transmission dynamics of Influenza in the United States from October 1,2018 through May 4,2019 influenza seasons.展开更多
In this paper, we introduce stochasticity into an SIR epidemic model with vaccina- tion. The stochasticity in the model is a standard technique in stochastic population modeling. When the perturbations are small, by t...In this paper, we introduce stochasticity into an SIR epidemic model with vaccina- tion. The stochasticity in the model is a standard technique in stochastic population modeling. When the perturbations are small, by the method of stochastic Lyapunov functions, we carry out a detailed analysis on the dynamical behavior of the stochastic model regarding of the basic reproduction number R0. If R0 ≤ 1, the solution of the model is oscillating around a steady state, which is the disease-free equilibrium of the corresponding deterministic model. If R0 〉 1, there is a stationary distribution and the solution has the ergodic property, which means that the disease will prevail.展开更多
In this paper,we propose a new mathematical model based on the association between susceptible and recovered individual.Then,we study the stability of this model with the deterministic case and obtain the conditions f...In this paper,we propose a new mathematical model based on the association between susceptible and recovered individual.Then,we study the stability of this model with the deterministic case and obtain the conditions for the extinction of diseases.Moreover,in view of the association between susceptible and recovered individual perturbed by white noise,we also give sufficient conditions for the extinction and the permanence in mean of disease with the white noise.Finally,we have numerical simulations to demonstrate the correctness of obtained theoretical results.展开更多
This paper presents a law of large numbers result,as the size of the population tends to infinity,of SIR stochastic epidemic models,for a population distributed over distinct patches(with migrations between them)and d...This paper presents a law of large numbers result,as the size of the population tends to infinity,of SIR stochastic epidemic models,for a population distributed over distinct patches(with migrations between them)and distinct groups(possibly age groups).The limit is a set of Volterra-type integral equations,and the result shows the effects of both spatial and population heterogeneity.The novelty of the model is that the infectivity of an infected individual is infection age dependent.More precisely,to each infected individual is attached a random infection-age dependent infectivity function,such that the various random functions attached to distinct individuals are i.i.d.The proof involves a novel construction of a sequence of i.i.d.processes to invoke the law of large numbers for processes in,by using the solution of a MacKean-Vlasov type Poisson-driven stochastic equation(as in the propagation of chaos theory).We also establish an identity using the Feynman-Kac formula for an adjoint backward ODE.The advantage of this approach is that it assumes much weaker conditions on the random infectivity functions than our earlier work for the homogeneous model in[20],where standard tightness criteria for convergence of stochastic processes were employed.To illustrate this new approach,we first explain the new proof under the weak assumptions for the homogeneous model,and then describe the multipatch-multigroup model and prove the law of large numbers for that model.展开更多
基金supported by the Fundamental Research Funds for the Central Universities(No.3122025090)。
文摘In this paper,based on the SVIQR model we develop a stochastic epidemic model with multiple vaccinations and time delay.Firstly,we prove the existence and uniqueness of the global positive solution of the model,and construct suitable functions to obtain sufficient conditions for disease extinction.Secondly,in order to effectively control the spread of the disease,appropriate control strategies are formulated by using optimal control theory.Finally,the results are verified by numerical simulation.
基金supported by the Scientific Research Project of Tianjin Municipal Educational Commission(No.2021KJ058)。
文摘In order to study the influence of stochastic disturbance and environment switching on the HPV infection and provide a theoretical basis for the development of effective HPV disease prevention measures,in this paper we establish a kind of two-sex stochastic HPV epidemic model with white noise and Markov switching.We show that the model has a unique local positive solution and a unique global positive solution.Then we identify the threshold conditions for the persistence of the HPV epidemic,and verify the persistence of the disease using the Lyapunov method and the Ito^formula.At last,the numerical simulation is carried out to illustrate the rationality of the theoretical results.
文摘<span style="font-family:Verdana;">In this paper we build and analyze two stochastic epidemic models with death. The model assume</span><span style="font-family:Verdana;"><span style="font-family:Verdana;"><span style="font-family:Verdana;">s</span></span></span><span style="font-family:Verdana;"><span style="font-family:Verdana;"><span style="font-family:Verdana;"> that only susceptible individuals (S) can get infected (I) and may die from this disease or a recovered individual becomes susceptible again (SIS model) or completely immune (SIR Model) for the remainder of the study period. Moreover, it is assumed there are no births, deaths, immigration or emigration during the study period;the community is said to be closed. In these infection disease models, there are two central questions: first it is the disease extinction or not and the second studies the time elapsed for such extinction, this paper will deal with this second question because the first answer corresponds to the basic reproduction number defined in the bibliography. More concretely, we study the mean-extinction of the diseases and the technique used here first builds the backward Kolmogorov differential equation and then solves it numerically using finite element method with FreeFem++. Our contribution and novelty </span></span></span><span style="font-family:Verdana;"><span style="font-family:Verdana;"><span style="font-family:Verdana;">are</span></span></span><span style="font-family:Verdana;"><span style="font-family:Verdana;"><span style="font-family:Verdana;"> the following: however the reproduction number effectively concludes the extinction or not of the disease, it does not help to know its extinction times because example with the same reproduction numbers has very different time. Moreover, the SIS model is slower, a result that is not surprising, but this difference seems to increase in the stochastic models with respect to the deterministic ones, it is reasonable to assume some uncertainly.</span></span></span>
基金supported by the National Natural Science Foundation of China(No.12172167)。
文摘Nonlinearity and randomness are both the essential attributes for the real world,and the case is the same for the models of infectious diseases,for which the deterministic models can not give a complete picture of the evolution.However,although there has been a lot of work on stochastic epidemic models,most of them focus mainly on qualitative properties,which makes us somewhat ignore the original meaning of the parameter value.In this paper we extend the classic susceptible-infectious-removed(SIR)epidemic model by adding a white noise excitation and then we utilize the large deviation theory to quantitatively study the long-term coexistence exit problem with epidemic.Finally,in order to extend the meaning of parameters in the corresponding deterministic system,we tentatively introduce two new thresholds which then prove rational.
文摘In this work,we consider a stochastic epidemic model with vaccination,healing and relapse.We prove the existence and the uniqueness of the positive solution.We establish sufficient conditions for the extinction and the persistence in mean of the stochastic system.Moreover,we also establish sufficient conditions for the existence of ergodic stationary distribution to the model,which reveals that the infectious disease will persist.The graphical illustrations of the approximate solutions of the stochastic epidemic model have been performed.
基金supported by National Science and Technology Innovation 2030 of China Next-Generation Artificial Intelligence Major Project(Grant No.2018AAA0101800)Key Project of Technological Innovation and Application Development Plan of Chongqing(Grant No.cstc2020jscx-dxwtBX0044)+1 种基金Key projects of Mathematics and Finance Research Center of Sichuan University of Arts and Science(No.SCMF202201)National College Students Innovation and Entrepreneurship Training Program(No.S202110619028,202210619035).
文摘In this paper,a stochastic SiS epidemic infectious diseases model with double stochastic perturbations is proposed.First,the existence and uniqueness of the positive global solution of the model are proved.Second,the controlling conditions for the extinction and persistence of the disease are obtained.Besides,the effects of the intensity of volatility Si and the speed of reversion 1 on the dynamical behaviors of the model are discussed.Finally,some numerical examples are given to support the theoretical results.The results show that if the basic reproduction number R_(0)^(8)<1,the disease will be extinct,that is to say that we can control the threshold R_(0)^(8)to suppress the disease outbreak.
文摘We present an epidemic model which can incorporate essential biological detail as well as the intrinsic demographic stochastieity of the epidemic process, yet is very simple, enabling rapid generation of a large number of simulations, A deterministic version of the model is also derived, in the limit of infinitely large populations, and a final-size formula for the deterministic model is proved. A key advantage of the model proposed is that it is possible to write down an explicit likelihood functions for it, which enables a systematic procedure for fitting parameters to real incidence data, using maximum likelihood.
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We present a mathematical analysis of the transmission of certain diseases using a stochastic susceptible-exposed-infectious-treated-recovered(SEITR)model with multiple stages of infection and treatment and explore the effects of treatments and external fluctuations in the transmission,treatment and recovery rates.We assume external fluctuations are caused by variability in the number of contacts between infected and susceptible individuals.It is shown that the expected number of secondary infections produced(in the absence of noise)reduces as treatment is introduced into the population.By defining RT,n and R T,n as the basic deterministic and stochastic reproduction numbers,respectively,in stage n of infection and treatment,we show mathematically that as the intensity of the noise in the transmission,treatment and recovery rates increases,the number of secondary cases of infection increases.The global stability of the disease-free and endemic equilibrium for the deterministic and stochastic SEITR models is also presented.The work presented is demonstrated using parameter values relevant to the transmission dynamics of Influenza in the United States from October 1,2018 through May 4,2019 influenza seasons.
文摘In this paper, we introduce stochasticity into an SIR epidemic model with vaccina- tion. The stochasticity in the model is a standard technique in stochastic population modeling. When the perturbations are small, by the method of stochastic Lyapunov functions, we carry out a detailed analysis on the dynamical behavior of the stochastic model regarding of the basic reproduction number R0. If R0 ≤ 1, the solution of the model is oscillating around a steady state, which is the disease-free equilibrium of the corresponding deterministic model. If R0 〉 1, there is a stationary distribution and the solution has the ergodic property, which means that the disease will prevail.
基金This work was supported financially by the Natural Science Foundation of Shandong Province under Grant No.ZR2017MA045.
文摘In this paper,we propose a new mathematical model based on the association between susceptible and recovered individual.Then,we study the stability of this model with the deterministic case and obtain the conditions for the extinction of diseases.Moreover,in view of the association between susceptible and recovered individual perturbed by white noise,we also give sufficient conditions for the extinction and the permanence in mean of disease with the white noise.Finally,we have numerical simulations to demonstrate the correctness of obtained theoretical results.
文摘This paper presents a law of large numbers result,as the size of the population tends to infinity,of SIR stochastic epidemic models,for a population distributed over distinct patches(with migrations between them)and distinct groups(possibly age groups).The limit is a set of Volterra-type integral equations,and the result shows the effects of both spatial and population heterogeneity.The novelty of the model is that the infectivity of an infected individual is infection age dependent.More precisely,to each infected individual is attached a random infection-age dependent infectivity function,such that the various random functions attached to distinct individuals are i.i.d.The proof involves a novel construction of a sequence of i.i.d.processes to invoke the law of large numbers for processes in,by using the solution of a MacKean-Vlasov type Poisson-driven stochastic equation(as in the propagation of chaos theory).We also establish an identity using the Feynman-Kac formula for an adjoint backward ODE.The advantage of this approach is that it assumes much weaker conditions on the random infectivity functions than our earlier work for the homogeneous model in[20],where standard tightness criteria for convergence of stochastic processes were employed.To illustrate this new approach,we first explain the new proof under the weak assumptions for the homogeneous model,and then describe the multipatch-multigroup model and prove the law of large numbers for that model.
