In this paper, the existence of periodic solutions for a time dependent age-structured population model is studied. The averaged net reproductive number is introduced as the main parameter to determine the dynamical b...In this paper, the existence of periodic solutions for a time dependent age-structured population model is studied. The averaged net reproductive number is introduced as the main parameter to determine the dynamical behaviors of the model. The existence of a global parameterized branch of periodic solutions of the model is obtained by using the contracting mapping theorem in a periodic and continuous function space. The global stability of the trivial equilibrium is studied and a very practical stability criteria for the model is obtained. The dynamics of the linear time-periodic model is similar to that of the linear model.展开更多
The importance of epidemiology in our life has stimulated researchers to extend the classic Susceptibles-Infectives-Removed (SIR) model to sophisticated models by including more factors in order to give detailed trans...The importance of epidemiology in our life has stimulated researchers to extend the classic Susceptibles-Infectives-Removed (SIR) model to sophisticated models by including more factors in order to give detailed transmission dynamics of epidemic diseases. However, the integration of the quarantine policy and age-structure is less addressed. In this work we propose an age-structured MSIQR (temporarily immune-susceptibles-infectives-quarantined-removed) model to study the impact of quarantine policies on the spread of epidemic diseases. Specifically, we investigate the existence of steady state solutions and stability property of the proposed model. The derived explicit expression of the basic reproductive number shows that the disease-free equilibrium is globally asymptotically stable if, and that the unique endemic equilibrium exists if. In addition, the stability conditions of the endemic equilibrium are derived.展开更多
This paper studied a structured model by age of tuberculosis. A population divided into two parts was considered for the study. Each subpopulation is submitted to a program of vaccination. It was allowed the migration...This paper studied a structured model by age of tuberculosis. A population divided into two parts was considered for the study. Each subpopulation is submitted to a program of vaccination. It was allowed the migration of vaccinated people only between the two patches. After the determination of and , the local and global stability of the disease-free equilibrium was studied. It showed the existence of three endemic equilibrium points. The theoretical results were illustrated by a numeric simulation.展开更多
This paper establishes an age-structured Tuberculosis(TB)model to investigate the joint impacts of information and immigration of population on the spread of TB disease.Mathematically,we show that the model is point d...This paper establishes an age-structured Tuberculosis(TB)model to investigate the joint impacts of information and immigration of population on the spread of TB disease.Mathematically,we show that the model is point dissipative,and the semi-flow generated by the model has the property of asymptotic smoothness,and then study the existence and global stability of positive steady state by the direct Lyapunov functional.Numerically,by using Matlab software,we verify the theoretical results,and further explore the influence of information(including information coverage and disease-related memory delay)and immigration on the final size of TB disease.The simulation results show that both information coverage and immigration are positive correlated with the final size of disease,and disease-related memory delay can affect the arrival time of positive steady state,which implies us that improving information coverage,enlarging disease-related memory,and reducing the immigration of population(especially latent and infected individuals)can effectively control the progression of TB disease.展开更多
A nonlinear optimal control problem in the Lotka-McKendrick population model is studied.It describes rational management of age-structured farmed populations in aquaculture and indoor farms.Employing generalized funct...A nonlinear optimal control problem in the Lotka-McKendrick population model is studied.It describes rational management of age-structured farmed populations in aquaculture and indoor farms.Employing generalized functions,we prove the impulse nature of optimal harvesting.Exact analytic solutions for sustainable harvesting strategies are obtained and used to analyze the optimal dynamics of harvesting age and rotation under technological innovations.展开更多
Smoking is a serious global public health problem.The ability to quit smoking is closely related to age;in addition,personal determination and education usually play an important role in quitting smoking.In order to c...Smoking is a serious global public health problem.The ability to quit smoking is closely related to age;in addition,personal determination and education usually play an important role in quitting smoking.In order to capture such characteristics,we developed a novel age-structured smoking dynamical model.By defining the smoking generation number Ro,the local stability,global stability of the boundary equilibrium and endemic equilibrium are obtained using Lyapunov functions.The uniform persistence and the well-posedness and asymptotic smoothness of the solutions are also studied.Sensitivity analyzes show that the lower the age of onset of smoking and the higher the determination to stop,the greater the likelihood of quitting smoking and numerical studies support the theoretical results.展开更多
The present paper deals with the periodic behaviours of an age-structured populationmodel.The period-similarity is proposed,which reveals a certain similar structure between thosepopulation models with distinct age st...The present paper deals with the periodic behaviours of an age-structured populationmodel.The period-similarity is proposed,which reveals a certain similar structure between thosepopulation models with distinct age structure.In addition,other results show that the fluctuations ofan age-structured population are closely related with the age structure.展开更多
In this paper,we formulate an age-structured HIV model,in which the influence of humoral immunity and the infection age of the infected cells are considered.The model is governed by three ordinary differential equatio...In this paper,we formulate an age-structured HIV model,in which the influence of humoral immunity and the infection age of the infected cells are considered.The model is governed by three ordinary differential equations and two first-ordered partial differential equations and admits three equilibria:disease-free,immune-inactivated and immune-activated equilibria.We introduce two important thresholds:the basic reproduction number R〇and immune-activated reproduction number R\and further show the global stability of above three equilibria in terms of R〇and Ri,respectively.The numerical simulations are presented to illustrate our results.展开更多
This paper deals with the global dynamics of a tuberculosis(TB)model with agestructure and delay.We perform some rigorous analyses for the model,including presenting an explicit formula for the basic reproduction numb...This paper deals with the global dynamics of a tuberculosis(TB)model with agestructure and delay.We perform some rigorous analyses for the model,including presenting an explicit formula for the basic reproduction number of the model,addressing the persistence of the solution semi-flow and the existence of the global attractor.Based on these analyses,we establish some results on stability and instability of equilibrium of the system.Finally,some numerical examples are provided to illustrate our obtained results.展开更多
In this paper,we propose detailed and reasonable viral dynamics by using a multi-compartment model that incorporating the age since the infection of multiple infectedcells,multiple target cells(Langerhans-cells and CD...In this paper,we propose detailed and reasonable viral dynamics by using a multi-compartment model that incorporating the age since the infection of multiple infectedcells,multiple target cells(Langerhans-cells and CD4^(+)T-cells),multiple viral strains(CCR5 and CXR4 HIV)and multiple infection routes(cell-to-cell and cell-to-virus).Thebasic reproduction number,R_(0),of the whole model is derived from two transmissionmechanisms:one is the potential trigger from the infection routes for a single target celland other is the joint effect of multiple viral infections for multi-target cells.Accordingly,we study the global stability of the steady states for the single target model.For thewhole model,we prove that the infection-free steady state is globally asymptoticallystable if R_(0)<1,whereas viruses persist uniformly if R_(0)>1.Numerical simulations arecarried out to illustrate the theoretical results.Sensitive analyses expound the effect ofmodel parameters on the comprehensive reproduction number.It is remarkable to findthat simultaneous control of HlV infection for two target cells can effectively reducethe viral loads within-host.Finally,our work suggests that the synergetic mechanism ofmulti-target cells and multi-strain cannot be ignored during treatment.展开更多
The existence and uniqueness of positive steady states for the age-structured MSEIR epidemic model with age-dependent transmission coefficient is considered. Threshold results for the existence of endemic states are e...The existence and uniqueness of positive steady states for the age-structured MSEIR epidemic model with age-dependent transmission coefficient is considered. Threshold results for the existence of endemic states are established; under certain conditions, uniqueness is also shown.展开更多
This paper constructed and studied a nonresident computer virus model with age structure and two delays effects. The non-negativity and boundedness of the solution of the model have been discussed, and then gave the b...This paper constructed and studied a nonresident computer virus model with age structure and two delays effects. The non-negativity and boundedness of the solution of the model have been discussed, and then gave the basic regeneration number, and obtained the conditions for the existence and the stability of the virus-free equilibrium and the computer virus equilibrium. Theoretical analysis shows the conditions under which the model undergoes Hopf bifurcation in three different cases. The numerical examples are provided to demonstrate the theoretical results.展开更多
This paper is concerned with optimal harvesting problems for a system consisting oftwo populations with age-structure and interaction of predator-prey. Existence and uniquenessof non-negative solutions to the system a...This paper is concerned with optimal harvesting problems for a system consisting oftwo populations with age-structure and interaction of predator-prey. Existence and uniquenessof non-negative solutions to the system and the continuous dependence of solutions on controlvariables are investigated. Existence of optimal policy is discussed, optimality conditions arederived by means of normal cone and adjoint system techniques.展开更多
This paper is to investigate positive periodic solutions of a biological system composed of two competing species. The existence and uniqueness of nonnegative solutions to the model for a set of given vital rates and ...This paper is to investigate positive periodic solutions of a biological system composed of two competing species. The existence and uniqueness of nonnegative solutions to the model for a set of given vital rates and initial distribution are treated and the contractive property of the solutions explored. Based on these results, some simple conditions for the global existence of positive periodic orbits are established by means of Horn's asymptotic fixed point theorem.展开更多
Three analytic algorithms based on Adomian decomposition, homotopy perturbation and homotopy analysis methods are proposed to solve some models of nonlinear age-structured population dynamics and epidemiology. Truncat...Three analytic algorithms based on Adomian decomposition, homotopy perturbation and homotopy analysis methods are proposed to solve some models of nonlinear age-structured population dynamics and epidemiology. Truncating the resulting convergent infinite series, we obtain numerical solutions of high accuracy for these models. Three numerical examples are given to illustrate the simplicity and accuracy of the methods.展开更多
This article focuses on the study of an age structured SEIRS epidemic model with a vaccination program when the total population size is not kept at constant. We first give the explicit expression of the reproduction ...This article focuses on the study of an age structured SEIRS epidemic model with a vaccination program when the total population size is not kept at constant. We first give the explicit expression of the reproduction number in the presence of vaccine ( is the exponent of growth of total population), and show that the infection-free steady state is linearly stable if and unstable if , then we apply the theoretical results to vaccination policies to determine the optimal age or ages at which an individual should be vaccinated. It is shown that the optimal strategy can be either one- or two-age strategies.展开更多
For some infectious diseases such as mumps,HBV,there is evidence showing that vaccinated individuals always lose their immunity at different rates depending on the inoculation time.In this paper,we propose an age-stru...For some infectious diseases such as mumps,HBV,there is evidence showing that vaccinated individuals always lose their immunity at different rates depending on the inoculation time.In this paper,we propose an age-structured epidemic model using a step function to describe the rate at which vaccinated individuals lose immunity and reduce the age-structured epidemic model to the delay differential model.For the age-structured model,we consider the positivity,boundedness,and compactness of the semiflow and study global stability of equilibria by constructing appropriate Lyapunov functionals.Moreover,for the reduced delay differential equation model,we study the existence of the endemic equilibrium and prove the global stability of equilibria.Finally,some numerical simulations are provided to support our theoretical results and a brief discussion is given.展开更多
文摘In this paper, the existence of periodic solutions for a time dependent age-structured population model is studied. The averaged net reproductive number is introduced as the main parameter to determine the dynamical behaviors of the model. The existence of a global parameterized branch of periodic solutions of the model is obtained by using the contracting mapping theorem in a periodic and continuous function space. The global stability of the trivial equilibrium is studied and a very practical stability criteria for the model is obtained. The dynamics of the linear time-periodic model is similar to that of the linear model.
文摘The importance of epidemiology in our life has stimulated researchers to extend the classic Susceptibles-Infectives-Removed (SIR) model to sophisticated models by including more factors in order to give detailed transmission dynamics of epidemic diseases. However, the integration of the quarantine policy and age-structure is less addressed. In this work we propose an age-structured MSIQR (temporarily immune-susceptibles-infectives-quarantined-removed) model to study the impact of quarantine policies on the spread of epidemic diseases. Specifically, we investigate the existence of steady state solutions and stability property of the proposed model. The derived explicit expression of the basic reproductive number shows that the disease-free equilibrium is globally asymptotically stable if, and that the unique endemic equilibrium exists if. In addition, the stability conditions of the endemic equilibrium are derived.
文摘This paper studied a structured model by age of tuberculosis. A population divided into two parts was considered for the study. Each subpopulation is submitted to a program of vaccination. It was allowed the migration of vaccinated people only between the two patches. After the determination of and , the local and global stability of the disease-free equilibrium was studied. It showed the existence of three endemic equilibrium points. The theoretical results were illustrated by a numeric simulation.
基金the anonymous referees for their careful reading and helpful comments which led to an improvement of our paper.This work was supported by National Natural Science Foundation of China(Nos.12126349,11601293,11901326,11901477)the Natural Science Foundation of Shanxi Province(No.201901D211160)+1 种基金the Natural Science Foundation of Guizhou Province(No.Qian Ke He Jichu-ZK[2021]Yiban002)the Project funded by China Postdoctoral Science Foundation(2019M653816XB).
文摘This paper establishes an age-structured Tuberculosis(TB)model to investigate the joint impacts of information and immigration of population on the spread of TB disease.Mathematically,we show that the model is point dissipative,and the semi-flow generated by the model has the property of asymptotic smoothness,and then study the existence and global stability of positive steady state by the direct Lyapunov functional.Numerically,by using Matlab software,we verify the theoretical results,and further explore the influence of information(including information coverage and disease-related memory delay)and immigration on the final size of TB disease.The simulation results show that both information coverage and immigration are positive correlated with the final size of disease,and disease-related memory delay can affect the arrival time of positive steady state,which implies us that improving information coverage,enlarging disease-related memory,and reducing the immigration of population(especially latent and infected individuals)can effectively control the progression of TB disease.
文摘A nonlinear optimal control problem in the Lotka-McKendrick population model is studied.It describes rational management of age-structured farmed populations in aquaculture and indoor farms.Employing generalized functions,we prove the impulse nature of optimal harvesting.Exact analytic solutions for sustainable harvesting strategies are obtained and used to analyze the optimal dynamics of harvesting age and rotation under technological innovations.
基金supported by the National Natural Science Foundation of China(NSFC:11961024,11801047)the Natural Science Foundation of Chongqing under Grant(cstc2019jcyj-msxmX0755,cstc2018jcyjAX0606)+1 种基金the Team Building Project for Graduate Tutors in Chongqing(JDDSTD201802)Group Building Scientific Innovation Project for universities in Chongqing(CXQT21021).
文摘Smoking is a serious global public health problem.The ability to quit smoking is closely related to age;in addition,personal determination and education usually play an important role in quitting smoking.In order to capture such characteristics,we developed a novel age-structured smoking dynamical model.By defining the smoking generation number Ro,the local stability,global stability of the boundary equilibrium and endemic equilibrium are obtained using Lyapunov functions.The uniform persistence and the well-posedness and asymptotic smoothness of the solutions are also studied.Sensitivity analyzes show that the lower the age of onset of smoking and the higher the determination to stop,the greater the likelihood of quitting smoking and numerical studies support the theoretical results.
文摘The present paper deals with the periodic behaviours of an age-structured populationmodel.The period-similarity is proposed,which reveals a certain similar structure between thosepopulation models with distinct age structure.In addition,other results show that the fluctuations ofan age-structured population are closely related with the age structure.
基金supported in part by Guangdong Natural Science Foundation(Nos.2016A030313426,2020A1515010445).
文摘In this paper,we formulate an age-structured HIV model,in which the influence of humoral immunity and the infection age of the infected cells are considered.The model is governed by three ordinary differential equations and two first-ordered partial differential equations and admits three equilibria:disease-free,immune-inactivated and immune-activated equilibria.We introduce two important thresholds:the basic reproduction number R〇and immune-activated reproduction number R\and further show the global stability of above three equilibria in terms of R〇and Ri,respectively.The numerical simulations are presented to illustrate our results.
基金supported by the National Infect ious Disease Science and Technology Major Project Grant 2017ZX10201302-007by the National Natural Science Foun-dation of China(Grant Nos.11926328,11926329,11971281 and 11501443)+2 种基金by the Natural Science Basic Research Plan in Shaanxi Province of China Grant 2020JQ-693by the Natural Science Foundation of Shaanxi Provincial Department of Education Grant 18JK0092supported by the NUPTSF(Grant No.NY220093).
文摘This paper deals with the global dynamics of a tuberculosis(TB)model with agestructure and delay.We perform some rigorous analyses for the model,including presenting an explicit formula for the basic reproduction number of the model,addressing the persistence of the solution semi-flow and the existence of the global attractor.Based on these analyses,we establish some results on stability and instability of equilibrium of the system.Finally,some numerical examples are provided to illustrate our obtained results.
基金This research was supported by the National Natural Science Foundation of China(11971013,11571170).
文摘In this paper,we propose detailed and reasonable viral dynamics by using a multi-compartment model that incorporating the age since the infection of multiple infectedcells,multiple target cells(Langerhans-cells and CD4^(+)T-cells),multiple viral strains(CCR5 and CXR4 HIV)and multiple infection routes(cell-to-cell and cell-to-virus).Thebasic reproduction number,R_(0),of the whole model is derived from two transmissionmechanisms:one is the potential trigger from the infection routes for a single target celland other is the joint effect of multiple viral infections for multi-target cells.Accordingly,we study the global stability of the steady states for the single target model.For thewhole model,we prove that the infection-free steady state is globally asymptoticallystable if R_(0)<1,whereas viruses persist uniformly if R_(0)>1.Numerical simulations arecarried out to illustrate the theoretical results.Sensitive analyses expound the effect ofmodel parameters on the comprehensive reproduction number.It is remarkable to findthat simultaneous control of HlV infection for two target cells can effectively reducethe viral loads within-host.Finally,our work suggests that the synergetic mechanism ofmulti-target cells and multi-strain cannot be ignored during treatment.
基金the Natural Science Foundation of Henan Province (No.994051200).
文摘The existence and uniqueness of positive steady states for the age-structured MSEIR epidemic model with age-dependent transmission coefficient is considered. Threshold results for the existence of endemic states are established; under certain conditions, uniqueness is also shown.
文摘This paper constructed and studied a nonresident computer virus model with age structure and two delays effects. The non-negativity and boundedness of the solution of the model have been discussed, and then gave the basic regeneration number, and obtained the conditions for the existence and the stability of the virus-free equilibrium and the computer virus equilibrium. Theoretical analysis shows the conditions under which the model undergoes Hopf bifurcation in three different cases. The numerical examples are provided to demonstrate the theoretical results.
基金Supported by the National Natural Science Foundation of China (10771048)the Research Project for Post-Graduates Creation of Zhejiang Province (YK2008054)
文摘This paper is concerned with optimal harvesting problems for a system consisting oftwo populations with age-structure and interaction of predator-prey. Existence and uniquenessof non-negative solutions to the system and the continuous dependence of solutions on controlvariables are investigated. Existence of optimal policy is discussed, optimality conditions arederived by means of normal cone and adjoint system techniques.
基金Supported by the National Natural Science Foundation of China (10771048, 11061017)
文摘This paper is to investigate positive periodic solutions of a biological system composed of two competing species. The existence and uniqueness of nonnegative solutions to the model for a set of given vital rates and initial distribution are treated and the contractive property of the solutions explored. Based on these results, some simple conditions for the global existence of positive periodic orbits are established by means of Horn's asymptotic fixed point theorem.
文摘Three analytic algorithms based on Adomian decomposition, homotopy perturbation and homotopy analysis methods are proposed to solve some models of nonlinear age-structured population dynamics and epidemiology. Truncating the resulting convergent infinite series, we obtain numerical solutions of high accuracy for these models. Three numerical examples are given to illustrate the simplicity and accuracy of the methods.
基金Supported by the NSFC (No.10371105) and the NSF of Henan Province (No.0312002000No.0211044800)
文摘This article focuses on the study of an age structured SEIRS epidemic model with a vaccination program when the total population size is not kept at constant. We first give the explicit expression of the reproduction number in the presence of vaccine ( is the exponent of growth of total population), and show that the infection-free steady state is linearly stable if and unstable if , then we apply the theoretical results to vaccination policies to determine the optimal age or ages at which an individual should be vaccinated. It is shown that the optimal strategy can be either one- or two-age strategies.
基金supported by The National Natural Science Foundation of China[12026236,12026222,12061079,11601293,12071418]Science and Technology Activities Priority Program for Overseas Researchers in Shanxi Province[20210049]The Natural Science Foundation of Shanxi Province[201901D211160,201901D211461,201901D111295]。
文摘For some infectious diseases such as mumps,HBV,there is evidence showing that vaccinated individuals always lose their immunity at different rates depending on the inoculation time.In this paper,we propose an age-structured epidemic model using a step function to describe the rate at which vaccinated individuals lose immunity and reduce the age-structured epidemic model to the delay differential model.For the age-structured model,we consider the positivity,boundedness,and compactness of the semiflow and study global stability of equilibria by constructing appropriate Lyapunov functionals.Moreover,for the reduced delay differential equation model,we study the existence of the endemic equilibrium and prove the global stability of equilibria.Finally,some numerical simulations are provided to support our theoretical results and a brief discussion is given.