Bifurcation analysis and optimal control of COVID-19 with exogenous reinfection and media coverages

In this paper,a SEIR epidemic model related to media coverage and exogenous reinfections is established to explore the transmission dynamics of COVID-19.The basic reproduction number is calculated using the next generation matrix method.First,the existence of equilibrium points is investigated,and different kinds of equilibrium points indicate that the disease may disappear,or exist that result in different quantity of susceptible individuals,pre-symptomatic infected individuals and symptomatic infected individuals.The stability of the equilibria is discussed by a geometric approach,and it is found that controlling reproduction number to be lower than 1 is not suficient for eradication of COVID-19.Second,transcritical bifurcation is explored,and it is found that improving the ratio of exogenous reinfection may lead to backward bifurcation under poor medical conditions,which indicates that two endemic equilibrium points appear.Third,to investigate the infuence of parameters on the basic reproduction,sensitivity analysis is done to choose relatively sensitive parameters,and the parameters for treatment and media coverage are selected.An optimal control model is established to balance the treatment and media awareness.By exploring the existence and the uniqueness of the optimal control solution,the optimal control strategies are given.Finally,we run numerical simulations to verify the theoretical analysis on actual data of China,and the data from the four different states of India is used for forecasting the situation of infected individuals in a short period.It is found by the simulation that the co-function of treatment and media coverage results in the reduced number of infectious individuals.