Stability Analysis Of Two Classes Of SIR Models With Discontinuous Treatment

The theory of differential equations with discontinuous right-hand sides has developedrelatively sufficiently in the sixties or seventies of the20th centuries. It is generally known thatthese equations can be widely used in many practical cases, such as the dynamical analysis of theepidemic models. In the20th centuries, since many infectious diseases have emerged andquickly began to sicken patients, the study of infectious disease models is always the focus of us.In order to strengthen the prevention and cure of infectious disease, one of the crucial issues isthe stability analysis of infectious disease models with treatment strategy. People are eager toanalyze and predict the transmission process quantitatively, and determine how to reduce thetreatment cost to the minimum through the analysis of the established discontinuous model. Itplays a vital role in the prevention and control of the diseases, the reducing of the social cost, andthe keep of the sustainable development of resources. In this paper, the dynamical analysis oftwo classes of SIR model with discontinuous treatment is proposed. The structure of this paper isarranged as following:In Chapter1, we introduce the outline of the background and history of differentialequations with discontinuous right-hand sides, the history and focus of the infectious modelsdynamical analysis, and the basic knowledge of this paper.In Chapter2, we introduce a discontinuous treatment into a class of infectious model, andanalyze the global asymptotic behavior of two types of equilibrium point under some hypothesis.Two examples satisfying our hypothesis conditions of this section are proposed to confirm ourconclusions. In the end, conclusions that guarantee the finite time convergence of the root of ourmodel are discussed.In Chapter3, the dynamical behavior analysis of an infectious model with distributed delayand discontinuous treatment is given, conditions making the equilibriums to be globally stableare proposed and Lyapunov functions are structured to consider the global asymptotic stability ofequilibriums. At the same time, the influence of the hypothesis to the model’s global behavior isdiscussed.Finally, we summarize our main results in chapter4.