The Research Of Distributed Parameter Systems With Impulsive Effects And Applications

Distributed parameter systems are infinite dimensional systems. For the past 46 years distributed parameter systems, those described by partial differential equations, or partial integrodifferential equations, or coupled equations of partial differential equations and ordinary differential equations, have undergone a rapid development and have gotten fairly rich results in modelling, stability and control of mechanical systems, evironmental systems, air-and space-crafts. The theory of distributed parameter systems has been applied in population systems, the study of population process and the control of epidemics, etc.Infectious diseases are caused by parasites, and easily transmitted from person to person. In general, the procedure of infectious diseases transmission is as the following: when there is an adequate contact of a suceptible with an infective so that transmission occurs, then the susceptible enters the exposed class of those in the latent period, who are infected but not yet infections. After the latent period ends, the individual enters the class of infectives, who are infectious in the sense that they are capable of transmitting the infection. When the infectious period ends, the individual enters the recovered class, or go back into suceptible class. Many infectious diseases transmission depend on the chrological age and infection-age of host, such as age-dependent disease transmission and age-dependent severity of infection. In addition, some diseases can be transmitted by disease vectors.Vaccination is a commonly used method for controlling infectious diseases. Generally, there are two vaccination strategies: constant vaccination and pulse vaccination. In order to avoid that disease vectors yield antibodies to insecticides, we usually apply pulse strategy in eliminating pests.Establishing and studying epidemic model can provide theory in the prevention, control and forest of infectious diseases, and reveals the transmission characteristics of infectious diseases. Starting in 1927 Kermack and Mckendrick published initial papers on epidemic models, a tremendous varity of models have been formulated. We find that most authors ignored the factor of chrological age and infection-age of host, and used ordinary differential equations to describe epidemic models. Realistic infectious disease models include both time and age as independent variables, because the recovered fraction usually increases with age, risks from an infection may be related to age. In this case partial differential equations can rigorously describe diseases transmission other than ordinary differential equations. We all know that the optimization and control of some biological phenomena(vaccinating susceptibles, or poisoning for disease vectors in a short time)are impulsive. So, impulsive differential equations provide a description of model of discrete perturbations. Furthermore, pulse vaccination proportion and interpulse time which are in initial conditions, are two controlled parameters, the systems are reparable systems if the total population is constant. Therefore, it is of value in theory and of significant in practice to investigate the application of distributed parameter systems with impulse effects on epidemic models, and has received a great attention.In this dissertation, we establish and study three epidemic models: epidemic model with impulsive vaccination which are modelled by coupled equations of partial differential equations and ordinary differential equations; host-vector epidemic model with impulsive effects; epidemic model with impulsive vaccination on age interval. The main innovation and results obtained in this dissertation may be summarized as the following:We study the application of a pulse vaccination strategy to eradicate hepatitis B and C modelled by SIV epidemic model. Since infection-age is an important factor of hepatitis progression, we incorporate the infection-age into the model. We derive the condition in which eradication solution is a global attractor, this condition depends on pulse vaccination proportion p and interpulse time T. By using Floquet theory we obtain the condition of the global stability of the solution, which shows that large enough pulse vaccination proportion and relatively small interpulse time lead to the eradication of hepatitis B and C. Further we use the bifurcation theory and Lyapunov- Schmidt series expansion to show the existence of the positive periodic solution.Establishing a tuberculosis(TB) model with impulsive vaccination. Since TB is a slowly progressing disease, and has a latent period, we incorporate the infection-age(the time lapsed since infection in latent) into the model, and use ordinary defferential equations and partial defferential equations to describe the model. We demonstrate the global asymptotic stability of the eradication solution if pulse vaccination proportion is large enough and interpulse period is relatively small.We develop a mathematical model for the disease(malaria and Chagas disease)which can be transmitted via vector and through blood transfusion in host population. The host population is structured by the chronological age. We assume that the instantaneous death and infection rates depend on the age, and obtain the existence of the nonnegative solution of the system. Applying semigroup theory, spectrum theory, etc, we investigate the existence and stability of the infection-free and the endemic steady state.We obtain an age-structured epidemic model for malaria with impulsive effect, and consider the effect of blood transfusion and infected vector transmission. An effective way to prevent the malaria is to control the size of mosquito population. So, we use a proportional periodic impulsive poisoning for mosquito population. We derive the condition in which eradication solution is locally asymptotically stable. The condition shows that large enough pulse reducing proportion and relatively small interpulse time lead to the eradication of the diseases.We present the epidemic model with impulsive vaccination on age interval concerning there is age-dependent severity of infection for some diseases, such as TB and measles. By means of the semigroup theory and integral equation theory, we establish the existence and uniqueness of nonnegative solution to ensure that the model equations are well-posed. Further we get an infection-free periodic solution of the system.The main meaning of this paper is to establish epidemic model with impulsive vaccination and host-vector epidemic with impulsive effects in order to describe the transmission dynamics of some diseases, and to provide mathematical proof for the existence and asymptotic property of the solution of the system.