Theoretical Analysis Of A Class Of Infectious Disease Models With Age Structure

Emerging infectious diseases have been seriously endangering human health and social development,attracting widespread attention around the world to their prevention and control.In this paper,we not only consider age as a factor that cannot be ignored in the transmission of infectious diseases,but also how a series of social problems(lack of medical resources,triaging pressure on health care workers,etc.)that arise when emerging infectious diseases occur will affect the dynamical behavior of infectious diseases.Therefore,we develop a class of mathematical models of infectious diseases with an age-structured and analyze the global dynamical behavior of the models,mainly including the following research contents.Firstly,an age-structured infectious disease model with medical resource constraints is established.The existence of the equilibria are proved by defining the basic reproduction number R0 of this model;and then the local stability of the equilibriua are determined by analyzing its characteristic equations using linearization methods.Subsequently,by using Laplace transform,convolution theorem,Lebesgue-Fatou lemma,etc.,we prove the uniform persistence of the system under the threshold condition.Furthermore,the global stability of the equilibria are studied by constructing appropriate Lyapunov function and using LaSalle invariance principle.Finally,we analyze the impact of the key factors of the system on the spread of infectious diseases through numerical simulation.In the presence of limited medical resources,the results suggest that equitable distribution for the limited medical resources is significant when treating low-risk and high-risk diseases at the same time and that keeping a resource sharing coefficient at a moderate level helps to eliminate the disease.Secondly,considering that the number of patients with low risk infectious diseases vary over time,based on the above research model,the saturated incidence rate of low risk infectious diseases is introduced,and an infectious disease model with two age structures is established.First,we analyze the well-posedness of the solution of this model,including boundedness and asymptotic smoothness.Then,two basic reproduction numbers R01 and R02 of this model are defined respectively according to the biological significance,the existence of the equilibria is proved,and the threshold conditions of disease outbreak are also obtained.Subsequently,the linearization method is used to deal with the model,and the local stability of the equilibria is discussed.Furthermore,the uniform persistence of the system is investigated,and the global stability of the equilibria is proved by constructing an appropriate Lyapunov function.Finally,numerical simulations are used to verify the correctness of the theoretical results.