The Complexity Control Of Singular Epidemic Models

Natural resources are important for the survival and development of the human.However,over-exploitation of renewable resources has lead to a lot of society problems,such as the scarcity of water,the extinction of species and outbreak of infectious disease.Thus,how to use and manage the renewable resources in a sustainable manner is the hot topic in society.Nowadays,the research on the control of population dynamical systems and epidemic dynamical systems has got some results.But there is less work on the control of singular eco-epidemic model.In this paper,based on population dynamics and epidemic dynamics,three singularity models are proposed.Using the theory of singular system,the stability of the systems are disscussed,the existense conditions of the bifurcation and chaos are given.In chapter 2,a Logistic singular SIRS model is proposed,using linearization theory of system and pole placement,the controller which makes the model stable at the disease-free is obtained.Simulations is done to check the rationality of the results.In chapter 3,the singular SIRS model with vertical infection is set up.For the case of the disease transmission rate with seasonality,the threshold of the singular SIRS model with vertical infection is given.Using pole placement,the controller which makes the system stable at disease-free equilibrium.Numerical simulation is done to illustrate the rationality of results.For the case of the disease transmission rate with non-seasonality,the chaotic phenomena is confirmed by numerical simulations.Using singular system theory,tracking control which makes the disease disappear is designed..In chapter 4,based on the theory of eco-epidemiology and the principle of economics,a singular eco-epidemiological model with infected predator is built up.Then,using bifurcation theory,some sufficient conditions of singularity induced bifurcation for the model are obtained.Further,the controllers which make the model stable around disease-free equilibrium are obtained.Finally,numerical simulations are carried out to prove our results.In all,some controller are designed to make the disease disappear and to offer a basis for controlling disease.