Dynamic Study Of Ebola And Pertussis Epedemic Models

Mathematical models have been widely used in describing infectious diseases transmission.The aim of the Center for Disease Control and Prevention（CDC）is to reduce or eliminate the spread of infectious diseases in the population through the formulation of disease prevention and control measures.The mathematical models can provide theoretical basis more accurately for the establishment of measures,which is an important significance of infectious disease dynamics.In this paper,we construct the Ebola model following the Gamma distribution and the pertussis model with age structure.We also compare the difference between the evaluations of the mathematical models that the process of the disease transmission follows different assumptions.In addition,we analysis that age structure is an important aspect when modeling pertussis models.（1）In the chapter 2,we estabilish and analyze several Ebola models with various stage distributions,including exponential,Gamma and arbitrary distributions.These models are used to evaluate the effectiveness of the control strategies such as timely isolation（or hospitalization）and burial and to identify potential discrepancies between the results from models with exponential and Gamma distributions.Through numerical simulations,the differences how these models affect the final size,peak size and peak time are revealed.Analysis of different models for intervention evaluation results can provide CDC with theoretical basis to select effective interventions according to specific circumstance.（2）In section 2.4,we establish a class of SEITR model with treatment,derive the potential assumptions of model and obtain the integral differential equations corresponding to the general distribution.By introducing exponential distribution or Gamma distribution in the special stage of disease transmission,the complex Integro-differential equation is reduced to ODE equation.For the disease-free equilibrium of the model that follow the exponential distribution,we use the Vieta theorem method to prove its local stability and apply global attractor approach to cetify the global asymptotical stability of the model.Sensitivity analysis of parameters that influence the control reproduct number?_c is studied,and we found that the change of effective transimation rateβis the most effective one.At last,we deduce the final scale relation and simulate it with Matlab,and then compare the difference of final size under different distribution.（3）In chapter 3,we first set up a partial differential equation（PDE）with continuous age-structured that individuals can be infected twice in the lifetime,and derive the formulas of the effective reproductive number?and the basic reproductive number?₀,furthermore,we give their biological explanations in the form of integral.It is proved that if?<1,the disease-free equilibrium is globally asymptotically stable,and if?>1,there exists endemic equilibrium.Secondly,we construct a three-infection pertussis transimation model,and obtain the expression of the infection probability F（a）at the age a by three methods,including the mathematical model,biological meaning interpretation and probability statistics.We conclude that if we assume that a specific infection stage follows exponential distribution,the expressions of F（a）obtained by the three methods are the same,which shows that the mathematical model is reasonable in explaining the spread of disease.