A fractional-order mathematical model for COVID-19 outbreak with the effect of symptomatic and asymptomatic transmissions
Zeeshan Ali, Faranak Rabiei, Mohammad Mehdi Rashidi, Touraj Khodadadi
Abstract
Abstract The purpose of this paper is to investigate the transmission dynamics of a fractional-order mathematical model of COVID-19 including susceptible ( $$\textsc {S}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>S</mml:mi> </mml:math> ), exposed ( $$\textsc {E}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>E</mml:mi> </mml:math> ), asymptomatic infected ( $$\textsc {I}_1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>I</mml:mi> <mml:mn>1</mml:mn> </mml:msub> </mml:math> ), symptomatic infected ( $$\textsc {I}_2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>I</mml:mi> <mml:mn>2</mml:mn> </mml:msub> </mml:math> ), and recovered ( $$\textsc {R}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>R</mml:mi> </mml:math> ) classes named $$\mathrm {SEI_{1}I_{2}R}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>SEI</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:msub> <mml:mi>I</mml:mi> <mml:mn>2</mml:mn> </mml:msub> <mml:mi>R</mml:mi> </mml:mrow> </mml:math> model, using the Caputo fractional derivative. Here, $$\mathrm {SEI_{1}I_{2}R}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>SEI</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:msub> <mml:mi>I</mml:mi> <mml:mn>2</mml:mn> </mml:msub> <mml:mi>R</mml:mi> </mml:mrow> </mml:math> 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 $$(R_{0})$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>(</mml:mo> <mml:msub> <mml:mi>R</mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> 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 $$R_{0} < 1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>R</mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:mo><</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> whereas the endemic equilibrium becomes stable for $$R_{0} > 1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>R</mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:mo>></mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> 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.