On age-structured SIS epidemic model for time dependent population

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

Asymptotic behavior of an age-structured population model with diffusion

A reaction-diffusion system with stage-structure is studied. We provide well-posedness of the model and prove that time-dependent solutions evolve either towards a positive equilibrium or to the trivial one. Under suitable conditions, a branch of positive equilibrium is shown to exist.     

A model of an age-structured population in a multipatch environment

We propose a model of an age-structured population divided into N geographical patches. We distinguish two time scales, at the fast time scale we have the migration dynamics and at the slow time scale the demographic dynamics. The demographic process is described using the classical McKendrick-von Foerster model for each patch, and a simple matrix model including the transfer rates between patches depicts the migration process.Assuming that 0 is a simple strictly dominant eigenvalue for the migration matrix, we transform the model (an e.d.p. problem with N state variables) into a classical McKendrick-von Foerster model (scalar e.d.p. problem) for the global variable: total population density. We prove, under certain assumptions, that the semigroup associated to our problem has the property of positive asynchronous exponential growth and so we compare its asymptotic behaviour to that of the transformed scalar model. This type of study can be included in the so-called aggregation methods, where a large scale dynamical system is approximately described by a reduced system. Aggregation methods have been already developed for systems of ordinary differential equations and for discrete time models.An application of the results to the study of the dynamics of the Sole larvae is also provided.     

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.     

An improved model of age-structured population dynamics

The classical model of age-dependent population dynamics is improved. Instead of the traditional renewal equation, a new approach is developed to describe the reproduction process of the population. The composition of a population is redefined to contain the pre-birth individuals, and the disadvantages of the classical model avoided. Moreover, the improved model turns out to be an initial value problem, which is mathematically more convenient to deal with. Existence and uniqueness results for the nonlinear nonautonomous system of model equations are obtained. It is shown that the classical model and its time delay generalization are two degenerate cases of the improved model.     

Nonlinear stability of traveling wavefronts in an age-structured population model with nonlocal dispersal and delay

This paper is concerned with the nonlinear stability of traveling wavefronts for a single species population model with nonlocal dispersal and age structure. By using the weighted energy method together with the comparison principle, we prove that the traveling wavefront is exponentially stable, when the initial perturbation around the wavefronts decays exponentially at –∞, but it can be arbitrarily large in other locations. In particular, our result implies that the time delay is harmless for stability of traveling wavefronts of the model.     

Global analysis of an age-structured SEIR model with immigration of population and nonlinear incidence rate

Epidemic models with infection age of infectious individuals have been extensively studied, however, most of the existing works ignore the combined effects of immigration and nonlinear incidence. In this paper, we incorporate both the effects of immigration and nonlinear incidence, based on which we formulate an SEIR epidemic model. We give a rigorous mathematical analysis on some necessary technical materials. Then, by constructing a Lyapunov functional, we show that the endemic equilibrium is globally asymptotically stable. Numerical simulations of an application are given to support our theoretical results.     

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.     

A nonautonomous epidemic model on time scales

We obtain conditions for permanence and extinction of the infection for a nonautonomous SIQR model defined on an arbitrary time scale. The threshold conditions are given by some numbers that play the role of the basic reproduction number in this setting. As a particular case of our result, we recover several threshold conditions obtained in the literature, on discrete or continuous time, for autonomous, periodic models and general nonautonomous models and we also discuss some new situations, including an aperiodic time scale.     

Extinction and periodic oscillations in an age-structured population model in a patchy environment

We consider an age-structured single-species population model in a patch environment consisting of infinitely many patches. Previous work shows that if the nonlinear birth rate is sufficiently large and the maturation time is small, then the model exhibits the usual transition from the trivial equilibrium to the positive (spatially homogeneous) equilibrium represented by a traveling wavefront. Here we show that (i) if the birth rate is so small that a patch alone cannot sustain a positive equilibrium then the whole population in the patchy environment will become extinct, and (ii) if the birth rate is large enough that each patch can sustain a positive equilibrium and if the maturation time is moderate then the model exhibits nonlinear oscillations characterized by the occurrence of multiple periodic traveling waves.     

A numerical method for nonlinear age-structured population models with finite maximum age

We propose a new numerical method for the approximation of solutions to a non-autonomous form of the classical Gurtin-MacCamy population model with a mortality rate that is the sum of an intrinsic age-dependent rate that becomes unbounded as the age approaches its maximum value, plus a non-local, non-autonomous, bounded rate that depends on some weighted population size. We prove that our new quadrature based method converges to second-order and we show the results of several numerical simulations.     

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.     

The dynamics of an age-structured TB transmission model with relapse

This paper deals with the global dynamics for a tuberculosis transmission model with age-structure and relapse. The time delay in the progression from the latent individuals to becoming the infectious individuals is also considered in our model. 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 semiflow and the existence of a global attractor. Based on these analyses, we establish some results about stability and instability of the solutions for our model. At end, the model is applied to describe tuberculosis transmission in China. The number of the total population and the number of the annual newly reported TB cases both match the statistical data well. The number of the total population, the latent individuals, the infectious individuals, the Purified Protein Derivative (PPD) positive rate, and the prevalence rate from 2020 to 2035 all are presented.     

Slow and fast scales for superprocess limits of age-structured populations

A superprocess limit for an interacting birth-death particle system modeling a population with trait and physical age-structures is established. Traits of newborn offspring are inherited from the parents except when mutations occur, while ages are set to zero. Because of interactions between individuals, standard approaches based on the Laplace transform do not hold. We use a martingale problem approach and a separation of the slow (trait) and fast (age) scales. While the trait marginals converge in a pathwise sense to a superprocess, the age distributions, on another time scale, average to equilibria that depend on traits. The convergence of the whole process depending on trait and age, only holds for finite-dimensional time-marginals. We apply our results to the study of examples illustrating different cases of trade-off between competition and senescence.     

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.