Establishment, Analysis And Research On Control For Some Epidemic Models

In combination of the research of epidemic and mathematics, the study of the trend of an infectious disease becoming an epidemic in one area by establishing mathematical infectious model on dynamics has become a project of both theoretical and practical value. It is beneficial not only in predicting development but also in the prevention and control of infectious disease.This dissertation focuses on the research of infectious patterns and relative problems on the basis of the combination of epidemic and mathematics.The first chapter is introduction, summarizing the most recent development of foreign mathematical infectious model research, bringing out the research project of this dissertation as well as its results.In the second chapter, SIR epidemic model elaborated by ordinary and partial differential equations is established on the theory of compartment model, which results in the proving of the existence, uniqueness and stability of equilibrium solution. This chapter explains also the parameter identification of system (2-1-1) and the establishment of a control zone of epidemic.In the third chapter, on the theory of compartment model, SEIR epidemic model elaborated by ordinary differential equations is established. It proves the existence and stability of equilibrium solution through the analysis of characteristic equation and eigenvalue. In explaining the local stability of equilibrium solution to such non-vanishing model disease, Routh-Hurwits solution has been applied. Through the study of SEIR epidemic model elaborated by partial differential equation and reproductive function, the existence and stability of equilibrium solution is again proved.In regard of the long period of illness, and the infectious pattern, ability and curing effect correspondent to this period, the discussion of epidemic model (P) relevant to the period of illness has been carded out. On the one hand, the uniqueness of the regular solution to the system (P) is proved through priori estimation. On the other hand, the stability of the solution to this model is supported by the theory of integro-partial differential equation.An assumption is first made in chapter five that people in one area are disturbed by two diseases, which are exclusive to each other. People have been divided into four groups: susceptible group, the first kind of infected group, the second infected group and the immune group. Firstly, on the theory of compartment model, a mathematical model (5-2-1) is established assuming that two diseases are both epidemic. Then the discussion concerning the existence and stability of the equilibrium solution to the vanishing of system disease has been carried out through the application of a small perturbation. Finally, the existence and locally asymptotical stability of the equilibrium solution to non-vanishing of the epidemic is proved through integro-partial differential equation.The main innovation of this paper is as follows:1, It discusses two optimal vaccination problems of system (2-4-1): (1) the pursuit of minimum cost on condition of some specific result; (2) the pursuit of optimal result on condition of some specific cost. Through functional analysis, the existence of the solution to these two optimal vaccination problems has been proved.2, According to the specific objective function(?)(ψ) of system (3-4-1), the necessary condition for a vaccination to be optimal is obtained.3, Improving the force of infectious. Most authors discussed mainly SIS, SIR, SEIR and MSEIR. However, they all assumed that the force of infectious is a monotonic operator of density of infected individuals I(a,t), the relation can be described byIn fact,λ(t) should be increasing as a function of the fraction of latent and infectious population at time t. In this paper, we study some epidemic models under the assumption that where E(a, t), I(a, t) denote the age-specific density of latent and infective individuals,4, Establishment and analysis for the epidemic model relevant to the period of illness and the epidemic model with two diseases. The existence and stability of equilibrium solution is proved.By means of nonlinear functional analysis, differential equations, integral equations, and modern control theory, we obtain a number of theoretical results, which are theoretically valuable and provide a solid ground for the practical use of our models.