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 ${\mathrm{SEI}}_1{I}_{2}R$ model, using the Caputo fractional derivative. Here, ${\mathrm{SEI}}_1{I}_{2}R$ 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 $\left(R_0\right)$ 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.