A fractional-order differential equation model of COVID-19 infection of epithelial cells

Abstract

A novel coronavirus disease (COVID-19) appeared in Wuhan, China in December 2019 and spread around the world at a rapid pace, taking the form of pandemic. There was an urgent need to look for the remedy and control this deadly disease. A new strain of coronavirus called Severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) was considered to be responsible for COVID-19. Novel coronavirus (SARS-CoV-2) belongs to the family of coronaviruses crowned with homotrimeric class 1 fusion spike protein (or S protein) on their surfaces. COVID-19 attacks primarily at our throat and lungs epithelial cells. In COVID-19, a stronger adaptive immune response against SARS-CoV-2 can lead to longer recovery time and leads to several complications. In this paper, we propose a mathematical model for examining the consequence of adaptive immune responses to the viral mutation to control disease transmission. We consider three populations, namely, the uninfected epithelial cells, infected cells, and the SARS-CoV-2 virus. We also take into account combination drug therapy on the dynamics of COVID-19 and its effect. We present a fractional-order model representing COVID-19/SARS-CoV-2 infection of epithelial cells. The main aim of our study is to explore the effect of adaptive immune response using fractional order operator to monitor the influence of memory on the cell-biological aspects. Also, we have studied the outcome of an antiviral drug on the system to obstruct the contact between epithelial cells and SARS-CoV-2 to restrict the COVID-19 disease. Numerical simulations have been done to illustrate our analytical findings.

Keywords: Adaptive immune response; COVID-19; Epithelial cell; Fractional-order; SARS-CoV-2.

Fig. 1

Best fits of model (3) to the viral load data in patients from Germany .

Fig. 2

Trajectories of 3 model variables without drug therapy with $\alpha =0.7,0.8,0.9,1.0$ along the time t (days), $p=2.5<{\delta }_v=2.6$.

Fig. 3

Trajectories of 3 model variables without drug therapy with $\alpha =0.7,0.8,0.9,1.0$ along the time t (days), $p=2.7>{\delta }_v=2.6$.

Fig. 4

Numerical solutions of fractional order SARS-CoV-2 model with the drug therapy. Comparison of the efficacy in different time intervals for ${ϵ}_1={ϵ}_2=0.02$ (small) for $t\le 10$ days and ${ϵ}_1={ϵ}_2=0.9$ (high) for $t>10$ days when $R_0>1$ keeping $\alpha =1$ and $T=20$ days.