Pulse Optimal Control Of Infectious Diseases

The infectious disease is our enemy. Mathematical models have become one of the mostimportant questions in analyzing the spread of infectious diseases and suggesting optimalcontrol strategy. In this paper, according to the study of infectious disease in domesticand foreign, especially, the development of the infectious disease model with pulse which isapplied broadly in reality, we mainly introduce two questions.Firstly, we propose an SIR infectious disease model for wild animals or poultry withbirth pulse and all newborns are susceptible, define the basic reproduction number R0 of thesystem. In virtue of the conclusion that the basic reproduction number R0 is the thresholdthat the disease will be extinct or not, we prove the basic reproduction number decreasetheoretically by culling and vaccination. At last, it is showed that the conclusion is right bynumerical simulation.Secondly, we modify the above model, focus on the practical situation and considerSIR model with vertical transmission. At the same time, it is assumed that birth pulseand pulse vaccination are carrying on synchronously. Using the stroboscopic map, we obtainthe exact periodic infection-free solution of the impulsive system; by applying the FloquetTheorem, we get the local stability of the disease-free periodic solution and the threshold ofthe sysytem. By using impulsive di?erential inequality, we prove the disease-free periodicsolution is globally asymptotically stable; numerical simulation show that the conclusion isright.In addition to, the paper applies the theory of optimal impulsive control to SIR in-fectious disease model with batch vaccination. We choose measurable performance index,give continuous Hamilton function and impulsive Hamilton function. At last, we obtain thenecessary condition of minimal treatment and vaccination cost.