Dynamical Analysis And Optimal Control Of An Epidemic Model

Cholera is an ancient disease,capable of causing periodic epidemic disease,still remains an important cause of morbidity and mortality in some parts of the global.It's necessary to establish a mathematical model to study the transmission law and formulate reasonable optimal control strategy to effectively control the epidemic of infectious diseases.In this paper,we study a SIQRB model for cholera through quarantine,using the Pontryagin maximum principle to search for an optimal strategy to control the spread of epidemics,and take reasonable control measures to reduce the harm caused by the epidemic.This paper take the isolation measures in the process of controlling the spread of infectious diseases,and assume that the people affected in treatment until recovery during the period of separation,consider the existence of both ”environment-human” and”human-human” transmission pathways in the process of infectious disease transmission,and the disease-related mortality of the infected and the quarantined during the epidemic period,also ignore the possible loss of immunity of the convalesces in the short term,study a SIQRB cholera epidemic model with quarantine measures.Firstly,prove the well-posedness of the solution to the infectious disease model,discuss existence and uniqueness conditions for the disease-free equilibrium and the endemic equilibrium of the infectious disease model,prove the equilibrium point is globally asymptotically stable.Adopt dynamic quarantine strategy in the process of controlling the spread of disease,use the Pontryagin maximum principle to find the optimal control strategy.Combining with the Haiti's cholera outbreak in 2010,through the numerical simulation to verify the rationality of model in this paper.For the optimal control problem studied,clearly know the intensity of the isolation measures adopted to control the spread of infectious diseases and the appropriate time which the infectious individuals may leave quarantine,in order to minimize the number of infectious individuals and bacteria concentration,as well as the costs associated with the quarantine,and illustrate the effectiveness of the optimal quarantine measures for the patients.