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

Abstract

AbstractThe purpose of this paper is to investigate the transmission dynamics of a fractional-order mathematical model of COVID-19 including susceptible ($$\textsc {S}$$ S ), exposed ($$\textsc {E}$$ E ), asymptomatic infected ($$\textsc {I}_1$$ I 1 ), symptomatic infected ($$\textsc {I}_2$$ I 2 ), and recovered ($$\textsc {R}$$ R ) classes named $$\mathrm {SEI_{1}I_{2}R}$$ SEI 1 I 2 R model, using the Caputo fractional derivative. Here, $$\mathrm {SEI_{1}I_{2}R}$$ 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 $$(R_{0})$$ ( 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 $$R_{0} < 1$$ R 0 < 1 whereas the endemic equilibrium becomes stable for $$R_{0} > 1$$ 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.