Analysis Of Dynamics Of Several Tuberculosis Models With Imperfect Vaccination And Early And Late Latency

Tuberculosis(TB)is a chronic infectious disease of the lungs caused by mycobacterium tuberculosis.Highly pathogenic TB has the characteristics of high mortality,and it not only harms human health but also affects national economic and social stability.Therefore,it is of great theoretical and practical significance to study the TB transmission rules and control measures of highly pathogenic TB.In this thesis,according to the TB infection stages of early and late latency and considering two interventions of incomplete vaccination and treatment,three TB models are established and their dynamic behaviors are analyzed by using the modeling idea of infectious disease dynamics.The details are as follows:The first chapter briefly introduces the background of TB,the current research status at home and abroad and the main works of this paper.The second chapter gives some basic concepts of epidemic dynamics,definitions,and lemmas related to this paper.In chapter three,based on the transmission mechanism of TB,an ordinary differential equation model with bilinear incidence is established,and the qualitative analysis and optimal control strategy are given,taking into account the characteristics of early and late latency and the control measures of neonatal vaccination and treatment of active TB patients.First,the control reproduction numberR₀ is calculated by using the next generation matrix method.By constructing appropriate Lyapunov functions,it is proved that the disease-free equilibrium is globally asymptotically stable if R₀?1 while the endemic equilibrium is globally asymptotically stable if R₀?1.Secondly,four control variables are introduced into the model,and the optimal control strategy is analyzed by using the Hamilitonian function and the Pontryagin maximum principle.Finally,some suggestions for controlling the spread of TB are given by numerical simulation.Chapter four builds a corresponding stochastic model based on the ordinary model in Chapter three and considers the interference of environmental white noise and analyzes the long-term behavior of the model.The existence and uniqueness of the model solution,the existence of the ergodic stationary distribution and the extinction of the disease are discussed by constructing suitable Lyapunov function and applying It(?)'s formula and some inequality techniques.Finally,several numerical examples are shown to verify the feasibility of the theoretical results.Chapter five establishes a corresponding stochastic Markov switching model based on the stochastic model in Chapter four further considering the perturbation of color noise,and its asymptotic properties are discussed.At last,the feasibility of the obtained theoretical results is verified by numerical examples.