The Applications Of Infectious Disease Models With Impulsive Effects Differential Equations

As we all know, in infectious disease dynamics in a long time, the main used of the mathematical model is the so-called "store room" (compartment) model, its basic idea from Kermack and Mckendrick was founded in 1927, has still to be widely used and continuous developed. Most of the early epidemic models for the total populations were constant assumptions. These assumptions were correct only when a shorter time, the closed environment, natural fertility and mortality can balance each other, and the mortality rate due to illness and less time to set up can be neglected. In recent years, International research on infectious disease dynamics is rapid development, a large number of mathematical models were used to analyze a variety of infectious diseases, which, because of pulse differential equations models closer to the actual cause of the widespread concern in many scholars and get an in-depth development and a large number can be used in drug dynamics, population dynamics, infectious diseases, and other fields. This is because the population dynamics and infectious diseases in many natural phenomena and man-made factors interfere with the role of pulse to describe may more precise. In this paper, we consider the pulse models of population dynamics and infectious disease, analyze the complex nature of the system and the stability of periodic solutions.In the second chapter, we study the standard model of SIR with pulse vaccination. Zhien Ma et al.[30] obtained disease-free equilibrium conditions on SIR model. Based on this SIR model, joining impulsiv effect and by using Floquet multiplier theorem, we discuss the disease-free cycle of the local asymptotic stability. The theoretical results obtained show that the percentage of increasing vaccination can prevent the continued development of infectious diseases. In the third chapter, we study a vertical transmission and the pulse delay vaccination SIR epidemic model, as the descendants of those infected at birth may be infected by the mothers of their infection, such as hepatitis, tuberculosis, and so on. This is called vertical transmission. Recently, a vertical model of infectious diseases was investigated by Michael [27]. The author[38] obtained some initial works of pulse immunization under vertical transmission conditions and obtained some conditions of disease which disappeared or tend to balance point. Since the disease in the incubation period of deaths related to delay the phenomenon, the author [38] did not consider the delay which played important role in infectious diseases model. Therefore, in accordance with the biological point of view, in this chapter, we will study a impulsive delay infectious diseases model. Because the stability of time-delay impulsive differential equations are more difficult to discuss, and on the model for infectious diseases, people often only concerned about the disease, susceptible to the existence and extinction of the disease and continued to mean the demise of the disease and epidemic. We study the attractive behavior and persistence of periodic solutions of the model of disease-free cycle with impulses and delays. The results obtained improve and extend the corresponding ones in the literature[38].In addition, when establishing the infectious disease model with the disease spread, the spread of the disease, the incubation period and immunization period often can not be ignored. If compared to the model without delays, the epidemic models with delays are better to describe the spread of the disease situation. In the fourth chapter, we establish and study delay SEIRS infectious disease model with impulses. Cooke studied the behavior of balance point of non-pulse model of disease-free. The literature[35] presented the threshold and stability conditions of the model. Due to two time-delays, it is difficult to discuss the balance point. This article has more details on the two special circumstances SEIS and SIRS. Wen-Di Wang [36] obtained some conditions of local stability and global stability at the balance state of the endemic system. In this chapter, we consider the pulse of the susceptibility to carry out vaccination. On the basis of this, we increase the pulse effects. The delay and pulse make the model more complex. Through analysis , the conditons of global stability of periodic solutions are obtained. The results obtained imply that a shorter or a longer cycle of pulse vaccination period can eliminate the disease under other unchanged factors.