Modeling and analysis of COVID-19 epidemics with treatment in fractional derivatives using real data from Pakistan

Abstract

Coronaviruses are a large family of viruses that cause different symptoms, from mild cold to severe respiratory distress, and they can be seen in different types of animals such as camels, cattle, cats and bats. Novel coronavirus called COVID-19 is a newly emerged virus that appeared in many countries of the world, but the actual source of the virus is not yet known. The outbreak has caused pandemic with 26,622,706 confirmed infections and 874,708 reported deaths worldwide till August 31, 2020, with 17,717,911 recovered cases. Currently, there exist no vaccines officially approved for the prevention or management of the disease, but alternative drugs meant for HIV, HBV, malaria and some other flus are used to treat this virus. In the present paper, a fractional-order epidemic model with two different operators called the classical Caputo operator and the Atangana-Baleanu-Caputo operator for the transmission of COVID-19 epidemic is proposed and analyzed. The reproduction number $R_0$ is obtained for the prediction and persistence of the disease. The dynamic behavior of the equilibria is studied by using fractional Routh-Hurwitz stability criterion and fractional La Salle invariant principle. Special attention is given to the global dynamics of the equilibria. Moreover, the fitting of parameters through least squares curve fitting technique is performed, and the average absolute relative error between COVID-19 actual cases and the model's solution for the infectious class is tried to be reduced and the best fitted values of the relevant parameters are achieved. The numerical solution of the proposed COVID-19 fractional-order model under the Caputo operator is obtained by using generalized Adams-Bashforth-Moulton method, whereas for the Atangana-Baleanu-Caputo operator, we have used a new numerical scheme. Also, the treatment compartment is included in the population which determines the impact of alternative drugs applied for treating the infected individuals. Furthermore, numerical simulations of the model and their graphical presentations are performed to visualize the effectiveness of our theoretical results and to monitor the effect of arbitrary-order derivative.

Fig. 1

The daily COVID-19 cases time series in Pakistan from March 24 to April 20, 2020, and the best fitted curve from the proposed model

Fig. 2

Long-term prediction for infectious population in Pakistan using the proposed COVID-19 model under the a Caputo and b ABC operators with $\vartheta =0.973$

Fig. 3

Profile for the susceptible population using the proposed COVID-19 model under the a Caputo and b ABC operators with different values of $\vartheta$

Fig. 4

Profile for the exposed population using the proposed COVID-19 model under the a Caputo and b ABC operators with different values of $\vartheta$

Fig. 5

Profile for the quarantined population using the proposed COVID-19 model under the a Caputo and b ABC operators with different values of $\vartheta$

Fig. 6

Profile for the asymptomatic population using the proposed COVID-19 model under the a Caputo and b ABC operators with different values of $\vartheta$

Fig. 7

Profile for the symptomatic population using the proposed COVID-19 model under the a Caputo and b ABC operators with different values of $\vartheta$

Fig. 8

Profile for the isolated population using the proposed COVID-19 model under the a Caputo and b ABC operators with different values of $\vartheta$

Fig. 9

Profile for the treated population using the proposed COVID-19 model under the a Caputo and b ABC operators with different values of $\vartheta$

Fig. 10

Profile for the recovered population using the proposed COVID-19 model under the a Caputo and b ABC operators with different values of $\vartheta$

Fig. 11

Profile for the symptomatic I(t) population using the proposed COVID-19 model under the a Caputo and b ABC operators with different values of $\delta$ (transmission rate) while taking $\vartheta =0.853$

Fig. 12

Profile for the symptomatic I(t) population using the proposed COVID-19 model under the a Caputo and b ABC operators with different values of $\rho$ (rate of joining treatment class) while taking $\vartheta =0.926$

Fig. 13

Dynamical behavior of the basic reproduction number $R_0$ for varying values of $\delta$ (transmission rate) and ${\beta }_3$ (recovery rate from symptomatic class)

Fig. 14

Dynamical behavior of the basic reproduction number $R_0$ for varying values of $r_1$ (rate at which exposed become infected) and $r_2$ (rate at which quarantined are isolated)

Fig. 15

Dynamical behavior of the basic reproduction number $R_0$ for varying values of ${\beta }_1$ (recovery rate from quarantined class) and ${\beta }_2$ (recovery rate from asymptomatic class)