Mathematical analysis of autonomous and non-autonomous age-structured reaction-diffusion-advection population model

In this paper, we study an age-structured reaction-diffusion-advection population model. First, we use a non-densely defined operator to the linear age-structured reaction-diffusion-advection population model in a patchy environment. By spectral analysis, we obtain the asynchronous exponential growth of the population model. Then we consider nonlinear death rate and birth rate, which all depend on the function related to the generalized total population, and we prove the existence of a steady state of the system. Finally, we study the age-structured reaction-diffusion-advection population model in non-autonomous situations. We give the comparison principle and prove the eventual compactness of semiflow by using integrated semigroup. We also prove the existence of compact attractors under the periodic situation.     

Theory and approximation of solutions to a harvested Hierarchical age-structured population model

This article is concerned with theoretic analysis and numerical approximation of solutions to a hierarchical age-structured population model, in which the vital rates of an individual depend more on the number of older individuals. The well-posedness of the model is rigorously treated by means of fixed point principle, and an algorithm and convergence analysis are presented. An example is used to show the effectiveness of the numerical method.     

Dynamical analysis of age-structured pertussis model with covert infection

An age-structured pertussis model with covert infection is proposed to understand the effect of covert infection on the recurrence of pertussis. It is found that vaccination only for young children does not have a decisive effect on whooping cough control. It is shown that although the vaccine coverage rate is relatively high, the model has a backward bifurcation for a larger covert infection rate. In addition, sufficient conditions for the disease-free steady state to be globally asymptotically stable are obtained.     

Hopf bifurcation of an age-structured compartmental pest-pathogen model

This paper investigates an age-structured compartmental pest-pathogen model by using the theory of integrated semigroup. We study the stability of the steady state of the model by analyzing the associated characteristic transcendental equation. It is shown that Hopf bifurcation occurs at a positive steady state as bifurcating parameter passes a sequence of critical values.     

Optimal harvesting for periodic age-dependent population dynamics with logistic term

We prove an asymptotic behavior result for an age-dependent population dynamics with logistic term and periodic vital rates. We investigate next an optimal harvesting problem related to a periodic age-structured model with logistic term. Existence of an optimal control and necessary optimality conditions are established. A conceptual algorithm to approximate the optimal pair is derived and some numerical experiments are presented.     

Numerical approximation of a stochastic age-structured population model in a polluted environment with Markovian switching

In this paper, a stochastic age-structured population model with Markovian switching is investigated in a polluted environment. Both the stochastic disturbance of environment and the Markovian switching are incorporated into the model. By Itô formula and several assumptions, the boundedness in the qth moment of exact solutions of model are proved. Furthermore, making use of truncated Euler–Maruyama (EM) method, the strong convergence criterion of numerical approximation in the qth moment is established, and the rate of convergence is estimated. Numerical simulations are carried out to illustrate the theoretical results. Our results indicate that the truncated EM method can be used for stochastic age-structured population system in a polluted environment.     

Existence-uniqueness and monotone approximation for an erythropoiesis age-structured model

We develop a monotone approximation to the solution of an age-structured model which describes the regulation of erythropoiesis, the process in which red blood cells are developed. The convergence of this approximation to the unique solution of the model is also established.     

Numerical solution of the nonlinear age-structured population models by using the operational matrices of Bernstein polynomials

In this paper a numerical method for solving the nonlinear age-structured population models is presented which is based on Bernstein polynomials approximation. Operational matrices of integration, differentiation, dual and product are introduced and are utilized to reduce the age-structured population problem to the solution of algebraic equations. The method in general is easy to implement, and yields good results. Illustrative examples are included to demonstrate the validity and applicability of the new technique.     

On age-structured SIS epidemic model for time dependent population

1.IntroductionandPreliminaryAgestructureinepidemicmodelshasbeenconsideredbymanyauthors,becauseoftherecoghtionthattransmissiondynamicsofcertaindiseasescouldnotbecorrectlydescribedbythetraditionalepidemicmodelswithnoagedependence.Especially,Busenbergetal.II'2]giveacompleteanalysisofafairlygeneralSISmodelwithagestructureandasteady-statetotalpopulation,showingtheekistenceofathresholdforendemicstates.In[tis],theyassumethatthepopulationhasreacheditssteadystate.Althoughitisnotilladeqilatetoassumet…     

The exact solution of nonlinear age-structured population model

In this paper, the representation of the exact solution of age-structured population model is obtained. Based on this, an effective numerical algorithm for solving the approximate solution of population model is given. The final numerical experiment shows that our method is effective.     

PERSISTENCE IN AN AGE-STRUCTURED POPULATION FOR A PATCH-TYPE ENVIRONMENT

A discrete-time model for an age-structured population in a patch-type environment is presented and analyzed. Comparison techniques for difference equations are used to find sufficient conditions for population persistence or extinction. The persistence and extinction theorem is used to define the critical patch number, the threshold for population persistence. Several examples are presented which illustrate the results of the theorems. The model is applied to a watersnake population.     

Mathematical Analysis for an Age-Structured Heroin Epidemic Model

In this paper, an age-structured heroin epidemic model, where the susceptibility of individuals and the relapse of heroin users in treatment are described by two age-dependent variables, is formulated and analyzed. The basic reproduction ratio of the model is derived and proved to be a threshold condition, which completely determines the global behaviors of the model. The asymptotic smoothness of the semiflow generated by the family of solutions, uniform persistence and existence of an interior global attractor have been presented for establishing and defining a Lyapunov functional on this attractor. Some control strategies of heroin and two special cases of the model formulation are addressed.

     

Global dynamics of an age-structured cholera model with both human-to-human and environment-to-human transmissions and saturation incidence

In this paper, an age-structured cholera model with both human-to-human and environment-to-human transmissions and saturation incidence is proposed. In the model, we consider the infection age of infectious individuals and the biological age of pathogen in the environment. It is verified that the global dynamics of the model is completely determined by the basic reproduction number. Asymptotic smoothness is verified as a necessary argument. By analyzing corresponding characteristic equations, we discuss the local stability of each of feasible steady states. Uniform persistence is shown by using the persistence theory for infinite dimensional dynamical system. The global stability of each of feasible steady states is established by using suitable Lyapunov functionals and LaSalle’s invariance principle. Numerical simulations are carried out to illustrate the theoretical results.     

The Continuous Solutions of an Age-structured Host-Vector Epidemic Model

1.IntroductionAgehasbeenrecognizedasanimportantfactorinthedynamicsofepidemicprocessforalongperiod.Age-dependentmodelshavebeenanalyzedextensivelyandgreatattentionhasbeenpaidinconnectionwiththeanalysisofrealepidemics(Capasso[1J,El-DeMa[2j,Webb[3]).Inthispaperwestudyahost-vectorSISmodelwithagestructure.LetusconsiderthecasethatavectorisresPOnsibleforthespreadofthediseaseamongthehostpopulation.Thevectorpopulationsplayanimportantroleinsomeepidemicdiseases.Ca-passohaspresentedsomehost-vectorepi…     

Mathematical model of hematopoiesis dynamics with growth factor-dependent apoptosis and proliferation regulations

We consider a nonlinear mathematical model of hematopoietic stem cell dynamics, in which proliferation and apoptosis are controlled by growth factor concentrations. Cell proliferation is positively regulated, while apoptosis is negatively regulated. The resulting age-structured model is reduced to a system of three differential equations, with three independent delays, and existence of steady states is investigated. The stability of the trivial steady state, describing cells dying out with a saturation of growth factor concentrations is proven to be asymptotically stable when it is the only equilibrium. The stability analysis of the unique positive steady state allows the determination of a stability area, and shows that instability may occur through a Hopf bifurcation, mainly as a destabilization of the proliferative capacity control, when cell cycle durations are very short. Numerical simulations are carried out and result in a stability diagram that stresses the lead role of the introduction rate compared to the apoptosis rate in the system stability.     

Global boundedness of the solutions to a Gurtin-MacCamy system

This paper is concerned with the analysis of a generalized Gurtin-MacCamy model describing the evolution of an age-structured population. The problem of global boundedness is studied. Namely we ask whether there are simple general assumptions that one can make on the vital rates in order to have boundedness of the solution. Next a fully implicit finite difference scheme along the characteristic is considered to approximate the solution of the system. Global boundedness of the numerical solutions is investigated. The optimal rate of convergence of the scheme is obtained in the maximum norm. Numerical examples are presented.     

Second-order characteristic schemes in time and age for a nonlinear age-structured population model

In this paper, we analyze two new second-order characteristic schemes in time and age for an age-structured population model with nonlinear diffusion and reaction. By using the characteristic difference to approximate the transport term and the average along the characteristics to treat the nonlinear spatial diffusion and reaction terms, an implicit second-order characteristic scheme is proposed. To compute the nonlinear approximation system, an explicit second-order characteristic scheme in time and age is further proposed by using the extrapolation technique. The global existence and uniqueness of the solution of the nonlinear approximation scheme are established by using the theory of variation methods, Schauder’s fixed point theorem, and the technique of prior estimates. The optimal error estimates of second order in time and age are strictly proved for both the implicit and the explicit characteristic schemes. Numerical examples are given to illustrate the performance of the methods.     

Global asymptotic stability for an age-structured model of hematopoietic stem cell dynamics

We investigate a system of two nonlinear age-structured partial differential equations describing the dynamics of proliferating and quiescent hematopoietic stem cell (HSC) populations. The method of characteristics reduces the age-structured model to a system of coupled delay differential and renewal difference equations with continuous time and distributed delay. By constructing a Lyapunov–Krasovskii functional, we give a necessary and sufficient condition for the global asymptotic stability of the trivial steady state, which describes the population dying out. We also give sufficient conditions for the existence of unbounded solutions, which describe the uncontrolled proliferation of HSC population. This study may be helpful in understanding the behavior of hematopoietic cells in some hematological disorders.     

A population dynamics model with strong maternal care

A two-sex age-structured nondispersing population dynamics deterministic model is presented taking into account strong maternal and weak paternal care of offspring. The model includes a weighted harmonic-mean type pair formation function and neglects the spatial dispersal and separation of pairs. It is assumed that each sex has pre-reproductive and reproductive age intervals. All adult individuals are divided into single males, single females, permanent pairs, and female-widows taking care of their offsprings after the death of their partners. All pairs are of two types: pairs without offspring under parental care at the given time and pairs taking child care. All individuals of pre-reproductive age are divided into young and juvenile groups. The young offspring are assumed to be under parental or maternal (after the death of their father) care. Juveniles can live without parental or maternal care but they cannot reproduce offsprings. It is assumed that births can only occur from couples. The model consists of nine integro-PDEs subject to the conditions of integral type. A class of separable solutions is studied, and a system for macro-moments evolving in time is derived in the case of age-independent vital ones. __________ Translated from Lietuvos Matematikos Rinkinys, Vol. 46, No. 2, pp. 215–255, April–June, 2006.     

Hopf bifurcation in a size-structured population dynamic model with random growth

This paper is devoted to the study of a size-structured model with Ricker type birth function as well as random fluctuation in the growth process. The complete model takes the form of a reaction-diffusion equation with a nonlinear and nonlocal boundary condition. We study some dynamical properties of the model by using the theory of integrated semigroups. It is shown that Hopf bifurcation occurs at a positive steady state of the model. This problem is new and is related to the center manifold theory developed recently in [P. Magal, S. Ruan, Center manifold theorem for semilinear equations with non-dense domain and applications to Hopf bifurcation in age-structured models, Mem. Amer. Math. Soc., in press] for semilinear equation with non-densely defined operators.