An Age-Structured Model For The Transmission Dynamics Of Measles

Measles is a common acute respiratory infection,the disease has been studied for hundreds of years and its transmission is understood in greater detail.Generally speaking,the transmission of measles goes through incubation period,prodromal period,eruption period and recovery,so there are four groups involved:suscepti-ble,latent,sick and recovered.Before measles vaccination,infection occurred every 2 to 3 years,vaccination has greatly reduced the incidence of disease.In addi-tion,recognized that the age-structure of the population affects the transmission trend of infectious diseases,therefore,this paper is based on Earn et al.(Science 287:667-670,2000)and Huang et al.(Theory in Biosciences 137:185-195,2018),which are articles about the ODE models of measles,we modeled infectious diseases with age-structure,explored the impact of vaccination on transmission trends and the age distribution of infected populations.We constructed a Banach Space,bringing in a linear operator and an nonlinear operator to transform the age-structure model into an abstract Cauchy problem,and obtained the existence and uniqueness of solutions;Secondly,in the process of analyzing the stability of disease-free equilibrium,we defined the basic reproduction number R0,and proved that when R0<1,the disease-free equilibrium is globally asymptotically stable;Next,we proved the existence,uniqueness and local asymp-totic stability of disease-endemic equilibrium when R0>1;Lastly,the numerical simulation not only verified the theory and showed the importance of improving vaccination rates for preventing and controlling measles,but also founded that most of those infected were young people.