Dynamics Of Tuberculosis Epidemic Models With Age Structure

Tuberculosis(TB)is a chronic infectious disease caused by mycobacterium tu-berculosis infection which can be fatal.It is one of the three infectious diseases with high mortality rates in our country.TB remains to be a serious harm to hu-man life for thousands of years.In recent years,more developing countries suffer a resurgence of TB due to population growth,increasing mobility of the population,multi-drug resistant TB,the spread of HIV infection and so on.Facing the severe situation of TB,The World Health Organization and many countries have taken effective measures to control the prevalence of TB.In this thesis,three TB dynamics transmission models have been built according to the epidemic perspectives,the mechanism of the spread of TB and other factors.The thesis consists of five chapters.In the first chapter,we first introduce the background of TB,the basic models of infectious diseases and age-structure models,the current research status of age-structure models.Second we introduce the basic concepts and the basic knowledge of epidemic dynamics such as Lyapunov-Lasalle invariant principle,Routh-Hurwitz criterion,the basic reproductive number.In the second chapter,we establish and study a SEIR TB dynamic model with latency age and relapse age and consider the factors of incomplete treatment,recurrence.First,we obtain the existence of the disease-free equilibrium and the expression of the basic reproductive number.We prove the uniform persistence of the model.Second,by constructing a Lyapunov function,we get the results that the disease free equilibrium is globally asymptotically stable when the basic reproductive number is less than unity.When the basic reproductive number is greater than unity,the disease free equilibrium is unstable and there is a unique endemic equilibrium which is globally asymptotically stable.Final we give the numerical simulations which further validate our theoretical results.In the third chapter,we construct a SVEIR tuberculosis model with age struc-ture and consider the factors of incomplete treatment,recurrence,vaccination fail-ure.First,we obtain the existence of the disease-free equilibrium and the expression of the basic reproductive number.We prove the uniform persistence of the mod-el.Second,by constructing a Lyapunov function,we show that the disease free equilibrium is globally asymptotically stable when the basic reproductive number is less than unity.When the basic reproductive number is greater than unity,the disease free equilibrium is unstable and there is a unique endemic equilibrium which is globally asymptotically stable.In the fourth chapter,we examine an age-structure TB dynamic model for limited medical conditions.Because TB has a longer incubation period,we consider the fast-low process factor.In the western regions,especially in remote rural areas the medical conditions axe limited,therefore,in our model,we introduce a saturated treatment function.Similar threshold dynamics axe established.In the fifth chapter,we summarize the current work and address the future research goals.