Dynamic Analysis Of Age-structured Infectious Disease Model

Age is an important consideration in the transmission of infectious diseases,which leads to the fact that the research on age-structured models for infectious diseases and their dynamics has attracted much attention.Several types of epidemic models with immune age-structure are established in this thesis due to the possibility of immune fading.The dynamic behaviors of the models are analyzed in terms of the following aspects.(1)A single-group age-structured model with general incidence is investigated.Based on this single-group model,a multi-group model is further studied in consideration of heterogeneity in population.The basic reproductive numbers R₀ for the two models are defined respectively,and the global stabilities of the equilibria are proved by constructing Lyapunov functions correspondingly.The results show that when R₀<1,the disease-free equilibrium is globally asymptotically stable.Otherwise,when R₀>1,the endemic equilibrium is globally asymptotically stable.In these two models,the immune age cannot change the stability of the equilibrium.(2)An age-structured model of SIRS type with general incidence is presented and analyzed.By reformulating the model as an abstract Cauthy problem and applying theorem related with Hille-Yosida operator,the dynamic properties are investigated,including stability of equilibria and the condition for periodic solution bifurcated by the destabilization of endemic equilibrium.According to the distribution of the roots of the characteristic equation,the existence of local Hopf bifurcation due to the perturbation of immune age is discussed.Meanwhile,numerical simulations are conducted to illustrate the influence of immune age on the dynamical behaviors of the model.(3)An age-structured model of SIRS type with general incidence and Logistic growth is proposed.The dynamics of the model are investigated using Hille-Yosida operator.By linearizing model system at endemic equilibrium,the existence of pure virtual roots is studied when the basic reproductive number R₀>1 and the condition for Hopf bifurcation is discussed.At the same time,numerical simulations are presented to show the influence of critical immune age,which is chosen as bifurcation parameter,and to show the effect of Logistic growth on the solutions of the model.