Dynamics Analysis Of A Class Of Infectious Disease Model With Optimal Control

Epidemic is one of the most harmful and terrible diseases to human health.In history, infectious diseases have brought great disaster to human survival and the people’s livelihood.Mathematical models have become one of the most important methods in analysing the spread of infectious diseases and suggesting optimal control strategy.However,in the past studies,the control of infectious diseases were mostly focused on fitting the epidemic data,predicting disease trends and using qualitative analysis methods to study the effect of disease prevention by controlling the basic reproductive number,but did not consider the time-variable disease control policy actually.In this paper,by reading a lot of literature,we mainly consider the following research questions:Firstly,because of improvement of medical facilities and the medical level of the doctors,vaccinatives and quarantives have becoming more and more,due to this,this paper establish the SVEIQR model which contains the vaccinative and quarantive and find the basic reproductive numberR0.By choosing different parameters,the model is simulated,the results show that the disease-free equilibrium is globally asymptotical stability when R0is no more than one,at the same time,disease will be eliminated;when R0is more than one,the endemic equilibrium is globally asymptotical stability and it will become endemic disease. Secondly, using the Pontryagin maximum principle in a simple SEIR model,by introducing the Hamilton function,we find the designing method of optimal control and prove the existence and uniqueness of the quadratic optimal control and give the characterization of quadratic optimal control and linear optimal control.What’s more,by using the dual problem and cost function,we give another method of proving the existence of optimal control.Further,we consider a SVEIQR model and give the characterization of the optimal control when the objective function contains three control variables(vaccinative cost,quarantive cost,treating cost).Finally,by simulating,we tests the effectiveness of optimal vaccinating and optimal quaranting.