Asymptotic Behaviors Of Several Structured Population And HIV Models

Structured population models and epidemic models are important research topics in the field of differential equations and biological mathematics.In this dissertation,within a semigroup framework and by using methods of Hille-Yosida operators,spectral analysis,the Perron-Frobenius theory as well as bifurcation theory,this thesis studies the long time behavior of solutions to several kinds of structured population models and epidemic models with finite delay,includ-ing well-posedness of the solutions,regularities,local and global stability,asyn-chronous exponential growth,uniform persistence,as well as existence of the periodic solutions.The results obtained in this paper generalize some existing conclusions in the relevant literatures to some degree.The dissertation contains six chapters.In Chapter 1 we first introduce the background and research status of struc-tured population dynamics and epidemic dynamics.We then present the main work and the main results of this dissertation.Chapter 2 is devoted to the study of a spatially and size-structured popula-tion dynamics model with delay in the birth process.By using C0-semigroup the-ory,spectral analysis arguments,Perron-Frobenius and bifurcation theory some dynamical properties of the system are investigated,including the asymptotical stability,asynchronous exponential growth at the null equilibrium as well as Hopf bifurcation occurring at the positive steady state of the system.Some examples are presented and simulated to illustrate the obtained results.The asymptotic behavior of a size-structured population model with infinite states-at-birth and distributed delay in birth process is investigated in Chapter 3.Particularly,we discuss the effect of time lag on the long-term dynamics.First,we formally linearize the system around a steady state and study the lin-earized system by using C0-semigroup framework and spectral analysis methods.In this manner we achieve some results on linearized stability,instability and asynchronous exponential growth for the system under some conditions.The obtained results are also illustrated by some examples and simulations.In Chapter 4,we analyze an age-structured HIV infection model which in-corporates a logistic growth term for the target cells and antiretroviral therapy.By presenting an explicit formula for the reproductive number of the model,ad-dressing the persistence of the solution semi-flow and the existence of a global attractor,some results on stability and instability for the system are established.The existence of Hopf bifurcation is also obtained around the positive equilib-rium.Finally,some numerical examples are provided to illustrate our obtained results.Chapter 5 deals with an age-structured HIV infection model with logistic growth for target cells and both virus-to-cell and cell-to-cell infection routes.Based on the existence of the infection free and infection equilibria and some rigorous analyses for the considered model,the asymptotic stability of these e-quilibria is discussed via determining the distribution of eigenvalues.We also address the persistence of the solution semi-flow by proving the existence of a global attractor.Furthermore,Hopf bifurcations occurring at the positive steady state is exploited here.At last,some numerical examples are provided to illustrate the obtained results.In Chapter 6,we propose and analyze an age-structured model for within-host HIV virus dynamics which is incorporated with both virus-to-cell and cell-to-cell infection routes,and proliferations of both uninfected and infected cells in the form of logistic growth.The model turns out to be a hybrid system with two differential-integral equations and one first order partial differential equation.We perform some rigorous analyses for the considered model.Among the interesting dynamical behaviours of the model is the occurrence of backward bifurcation in terms of the basic reproduction number R0 at R0 =1,which raises new challenges for effective infection control.Based on our analytical results,we also discuss the cause of such a backward bifurcation.