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 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.

Assessing the role of quarantine and isolation as control strategies for COVID-19 outbreak: A case study

“…Finally, various simulations (time series and contour plots) are carried out to assess the transmission dynamics of the virus based on different values of the parameters with conclusion that the contacts among individuals and elongation in the quarantine period are the most effective strategies to combat with the pandemic. This study will be revisited in future with introduction of tools concerning optimal control theory in order to identify best controls among the available management strategies.Other possible directions of interest are to construct stochastic [25] or fractional-order [26] , [27] , [28] , [29] versions of the model proposed in the present study.…”

Modeling the Dynamics of Drug Spreading in China

“…Since we used the historical data of 2015−2018 to ascertain the parameter values and the data of 2019 to verify the projected trend, the result showed an overall well goodness of fit, and we have reasons to believe that this value of is an authentic manifestation of the drug situation in China for the modeling period. Given the illicit nature of drug-taking behaviors in China and the low reported number of existing drug users in official data, the scale of drug epidemic is by no means comparable to those of infectious diseases, and it seems reasonable that the of drug-using behaviors is smaller than those of infectious diseases by an order of magnitude [ 11 ]. Based on this and the already-decreasing number of drug users observed since 2018, it would not be difficult to understand the larger impact of Intervention 3 or 4 compared with Intervention 1 or 2 on the already-shrinking drug epidemic.…”

“…Compartmental dynamic modeling has undergone extensive development in the past decades and has proved itself as an effective tool in the research of infectious diseases, including influenza, smallpox, coronavirus, HIV, etc. [ 9 , 10 , 11 , 12 , 13 ]. In recent years, the application of compartmental dynamic models has been extended to other research fields, for instance, dynamics of the spread of alcoholism, cigarette smoking, internet virus, rumors, or drug-using behavior [ 14 , 15 , 16 , 17 ].…”

Mathematical model to assess the imposition of lockdown during COVID-19 pandemic

“…Many recent researches in literature studied mathematical modeling of COVID-19 [23] , [24] . Some of these researches were concerned with fractional order models [25] , [26] , [27] , [28] , [29] , [30] , [31] .…”