A fractional-order mathematical model for COVID-19 outbreak with the effect of symptomatic and asymptomatic transmissions

The purpose of this paper is to investigate the transmission dynamics of a fractional-order mathematical model of COVID-19 including susceptible ( S ), exposed ( E ), asymptomatic infected ( I_1 ), symptomatic infected ( I_2 ), and recovered ( R ) classes named SEI_1I_2R model, using the Caputo fractional derivative. Here, SEI_1I_2R model describes the effect of asymptomatic and symptomatic transmissions on coronavirus disease outbreak. The existence and uniqueness of the solution are studied with the help of Schaefer- and Banach-type fixed point theorems. Sensitivity analysis of the model in terms of the variance of each parameter is examined, and the basic reproduction number (R_0) to discuss the local stability at two equilibrium points is proposed. Using the Routh–Hurwitz criterion of stability, it is found that the disease-free equilibrium will be stable for R_0 < 1 whereas the endemic equilibrium becomes stable for R_0 > 1 and unstable otherwise. Moreover, the numerical simulations for various values of fractional-order are carried out with the help of the fractional Euler method. The numerical results show that asymptomatic transmission has a lower impact on the disease outbreak rather than symptomatic transmission. Finally, the simulated graph of total infected population by proposed model here is compared with the real data of second-wave infected population of COVID-19 outbreak in India.