Qualitative Analysis And Data Simulation Of Tuberculosis Transmission Models

As an ancient infectious disease, Tuberculosis (TB) remains to be a serious harm to human life health for thousands of years. China is one of the22TB high burden countries in the world. In recent years, many countries initially with low TB incidence and a great number of developing countries suffer a resurgence of TB due to population growth, increasing mobility of the population, the spread of HIV infection, multi-drug resistant TB increases and other reasons. Facing the severe situation of TB epidemic, The World Health Organization has taken effective mea-sures to control the prevalence of TB and many countries have taken the appropriate measures accordingly. In this thesis, four tuberculosis dynamics transmission mod-els were built according to the modeling thought of epidemic dynamics method, the mechanism of the spread of the tuberculosis, and other factors.This thesis consists of six chapters. The first chapter introduces the background of tuberculosis, the epidemic of tuberculosis in China, the current research status at home and abroad of the TB models, the fundamental concepts of epidemic dynamics, and the general method to calculate the basic reproductive number.In chapter two, we set up and simulate a mathematical model by using the tuberculosis data reported monthly by China CDC from January2005to Decem-ber2012. The optimal parameter values of the model are obtained by fixing some parameter values and initial values, and using MATLAB tool fminsearch, which is part of optimization toolbox. Based upon these values, we get the basic reproduc-tive number of the disease for each year. By virtue of the Chi-square test of the statistical inference, the optimal parameter values of the model are proved to be reasonable. The discussion of the basic reproductive number reveals that the mere dependence on the treatment of latent and infectious individuals (chemoprophylaxis and therapeutics) cannot eliminate tuberculosis in China. The reason is that al-most47percent infectious individuals in China do not go to hospital for treatment. Therefore, the most critical factor in eliminating tuberculosis in China is to increase the treatment rate of infectious individuals. We also give some qualitative analysis of the model. By constructing Lyapunov function, we get the result that the disease-free equilibrium is globally asymptotically stable when the basic reproductive is less than unity. We also get that the cinditions of two, one or zero endemic equilibria. At last, we get the local stability of the endemic equilibrium under certain conditions.In chapter three, a TB dynamic model with isolation and incomplete treatment is proposed. By constructing Lyapunov function, we get the result that the disease-free equilibrium is globally asymptotically stable when the basic reproductive is less than unity. When the basic reproductive number is greater than unity, the disease-free equilibrium is unstable and a unique endemic equilibrium exists. The local asymptotical stability of the endemic equilibrium is proved by Routh-Hurwitz criterion. The global stability of the endemic equilibrium is proved by Generalized Dulac-Benxison criterion when the disease induced death rate d=0. Numerical simulations support our analytical results. We get the result that the number of infectious individuals is increasing as the value of the incomplete treatment rate increase through numerical simulations. We also discuss the optimal strategy for tuberculosis control according to the theoretical results.In chapter four, we study a TB dynamic transmission model with non-monotonic incidence rate, which describes the psychological effect of certain serious diseases on the community when the number of infectious individuals is getting larger. The psychological effect in TB transmission is mainly reflected in a particular social environment, such as schools, hospitals, prisons, etc. By constructing Lyapunov function, we get the result that the disease-free equilibrium is globally asymptoti-cally 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 a unique endemic equilibrium exists. The local asymptotical stability of the endemic equilibrium is proved by Routh-Hurwitz criterion. The global stability of the endemic equilibrium is proved by constructing the second compound system. Though the basic reproduction number R0does not depend on α (describes the psychological effect of the general public toward the infectious individuals) explicit-ly, numerical simulations indicate that when the disease is endemic, the steady state value of the infectious individuals decreases as α increases.In chapter five, we study a TB dynamic transmission model with strong Allee effect. By using qualitative theory of ordinary differential equation, the stability of the disease-free equilibria were obtained. The model has three disease-free equilibria (0,0,0,0),(m,0,0,0),(M,0,0,0). The equilibrium (0,0,0,0) is always stable and the equilibrium (m,0,0,0) is always unstable. The equilibrium (M,0,0,0) is stable if AB-kβM>0and unstable if AB-kβM<0. We also get the basic reproductive number R0. If R0>1, there exists a unique endemic equilibrium. Through Routh-Hurwitz criterion and some algebraic methods, we get the conditions of the stabilities of the endemic equilibrium. The endemic equilibrium is stable if Ro>1and S1<S*<M while unstable if R0>1and m<S*<S1. If S*=S1, the model undergos a Hopf bifurcation at the endemic equilibrium. Using the center manifold theory and the normal form theory, the explicit formulae which determine the stability and the direction of the bifurcating period solutions are derived. Our results show that the strong Allee effect impact the qualitative features of TB epidemics.In chapter six, we arrive at a conclusion by summarizing the current work and setting the research targets afterwards.