A fractional order mathematical model for COVID-19 dynamics with quarantine, isolation, and environmental viral load

Abstract

COVID-19 or coronavirus is a newly emerged infectious disease that started in Wuhan, China, in December 2019 and spread worldwide very quickly. Although the recovery rate is greater than the death rate, the COVID-19 infection is becoming very harmful for the human community and causing financial loses to their economy. No proper vaccine for this infection has been introduced in the market in order to treat the infected people. Various approaches have been implemented recently to study the dynamics of this novel infection. Mathematical models are one of the effective tools in this regard to understand the transmission patterns of COVID-19. In the present paper, we formulate a fractional epidemic model in the Caputo sense with the consideration of quarantine, isolation, and environmental impacts to examine the dynamics of the COVID-19 outbreak. The fractional models are quite useful for understanding better the disease epidemics as well as capture the memory and nonlocality effects. First, we construct the model in ordinary differential equations and further consider the Caputo operator to formulate its fractional derivative. We present some of the necessary mathematical analysis for the fractional model. Furthermore, the model is fitted to the reported cases in Pakistan, one of the epicenters of COVID-19 in Asia. The estimated value of the important threshold parameter of the model, known as the basic reproduction number, is evaluated theoretically and numerically. Based on the real fitted parameters, we obtained $R_0\approx 1.50$ . Finally, an efficient numerical scheme of Adams-Moulton type is used in order to simulate the fractional model. The impact of some of the key model parameters on the disease dynamics and its elimination are shown graphically for various values of noninteger order of the Caputo derivative. We conclude that the use of fractional epidemic model provides a better understanding and biologically more insights about the disease dynamics.

Keywords: COVID-19; Caputo fractional model; Environmental impact; Parameter estimations; Quarantine and isolation; Real data; Simulation; Stability analysis.

Figure 1

Curve fitting (solid red line) to the confirmed infected cases using model (6), while the data are consider from 1 March to 30 June 2020

Figure 2

The impact of contact rate ${\zeta }_1$ on cumulative symptomatic, asymptomatic, and hospitalized COVID-19 individuals for $\alpha =1,\alpha =0.95,\alpha =0.90,\alpha =0.85$

Figure 3

The impact of contact rate ${\zeta }_2$ on cumulative symptomatic, asymptomatic, and hospitalized COVID-19 individuals for $\alpha =1,\alpha =0.95,\alpha =0.90,\alpha =0.85$

Figure 4

The impact of ψ on cumulative symptomatic, A, and H COVID-19 individuals for $\alpha =1,\alpha =0.95,\alpha =0.90,\alpha =0.85$

Figure 5

The influence of quarantine rate of exposed individuals ${\epsilon }_1$ on I, A, and H individuals for $\alpha =1,\alpha =0.95,\alpha =0.90,\alpha =0.85$

Figure 6

The impact of hospitalization (self-isolation) rate on $I,A$, and H individuals for $\alpha =1,\alpha =0.95,\alpha =0.90,\alpha =0.85$

Figure 7

The impact of viral influence of A individuals to the environment on $I,A$, and H individuals for $\alpha =1,\alpha =0.95,\alpha =0.90,\alpha =0.85$

Figure 8

The impact of $m_2$ to the environment on symptomatic, asymptomatic, and hospitalized COVID-19 individuals for $\alpha =1,\alpha =0.95,\alpha =0.90,\alpha =0.85$

Figure 9

The impact of removal rate $m_3$ on symptomatic, A, and H COVID-19 individuals for $\alpha =1,\alpha =0.95,\alpha =0.90,\alpha =0.85$