This study employs a semi-analytical approach, called Optimal Homotopy Asymptotic Method (OHAM), to analyze a coronavirus (COVID-19) transmission model of fractional order. The proposed method employs Caputo's fractional derivatives and Reimann-Liouville fractional integral sense to solve the underlying model. To the best of our knowledge, this work presents the first application of an optimal homotopy asymptotic scheme for better estimation of the future dynamics of the COVID-19 pandemic. Our proposed fractional-order scheme for the parameterized model is based on the available number of infected cases from January 21 to January 28, 2020, in Wuhan City of China. For the considered real-time data, the basic reproduction number is R_{0} ≈ 2.48293 that is quite high. The proposed fractional-order scheme for solving the COVID-19 fractional-order model possesses some salient features like producing closed-form semi-analytical solutions, fast convergence and non-dependence on the discretization of the domain. Several graphical presentations have demonstrated the dynamical behaviors of subpopulations involved in the underlying fractional COVID-19 model. The successful application of the scheme presented in this work reveals new horizons of its application to several other fractional-order epidemiological models.

Humans have invented many scientific methods so far and have set in motion several steps to avoid, even to cure, some of the lethal diseases. Although they believed they had conquered nature, corona-virus appeared killing thousands of people in China. Coronavirus has also been spread in many countries from Africa to Europe. Coronavirus 2019 (COVID-19) is a contagious virus causing infection to the respiratory system and is widely spread from humans to humans. The first infected case of this new COVID-19 disease was identified on December 31, 2019 in the city of Wuhan, China, the capital of Hubei province [

Mathematical models play a vital role not only in understanding the dynamics of infection but also in investigating the recommendable conditions under which the disease will persist or wiped out. Presently, governments and researchers have shown great concerns to COVID-19 because of high transmission rate and noteworthy disease induced death rate. The COVID-19 virus is generally transferred when an infected person releases droplets generated by sneezing, coughing or exhaling. The confirmed cases of COVID-19 have reached nearly fifty four million all around the globe, and more than 1.3 million deaths have been caused by this virus. As of November 13, 2020, according to Worldometers [

Keenly tracking the corona virus transmission, researchers have organized to speed up the diagnostic processes, and several types of vaccines are under investigation against COVID-19. For example, Cao et al. [

Over the last thirty years, fractional derivatives have captivated the numerous researchers after recognition of the fact that in comparison to the classical derivatives, fractional derivatives are more reliable operators to model the real world physical phenomenons. In dynamical problems, fractional calculus (FC) based modeling is receiving a rapid popularity nowadays. The mathematical modeling of many physical and engineering models based on the idea of FC exhibits highly precise and accurate experimental results as compared to the models based on conventional calculus. The non-integer differential operators such as Caputo, Caputo-Fabrizio and ABC are fractional operators that transform the ordinary model to generalized model. In this article, we present a novel research on a fractional order dynamical model that underpins the propagation of coronavirus infectious disease and provide some forecasting with real world data. We extend an integer-order model formulation to a fractional order model by adding the Caputo sense of fractional derivative. The reason of using the Caputo fractional derivative is that it possesses several basic characteristics of fractional calculus. Moreover, the transmission behavior described in the model can be better defined by using Caputo operator. Research based on derivatives in Caputo sense and its applications to different models emerging in various disciplines of engineering and other sciences can be observed in several past studies [

Presuming that the transmission occurs primarily within the population of bats and afterwards the transmission occurs to wild animals usually termed as hosts. Hunting of these carriers and then their transportation to the supply markets of seafood are considered as virus reservoirs. By exposing to the market, people get the risk of infection. In the following subsections, we revisit the evolution of the COVID-19 evolutionary tracks from bats to humans in the form of three mathematical models.

From the mathematical modeling point of view, let us denote the size of entire population of bats by

The susceptible bat population is hired via birth rate _{b} after having completed their incubation period and therefore, get included in the infected subpopulation _{h}), whereas

Let

The governing

This model presents changing aspects of transmission of COVID-19 among human that are due to the close contacts of human population with the contagious environment without direct contact to virus hosts. Peoples' birth and natural death rates are denoted by the parameters _{p} is coefficient of disease transmission. Transmission amongst asymptomatically infected healthy individuals may occur in the form of

To develop fractional order COVID-19 model, we describe some basic definitions from fractional calculus, which play vital role in fractional calculus for solving fractional order system of differential equations. These definitions consist of fractional integral operator of a function

The integral operator of fractional order in Riemann–Liouville sense with order α ≥ 0 of

Initial conditions obeyed by the model are:

Integrating over

It implies that the total population has an upper bound of

The solution of the polynomial equation

Clearly

The essential reproduction quantity (

The spectral radius

Parameter | Description | Value |
---|---|---|

Rate of removing virus from |
0.01 | |

Contribution rate by |
0.000398 | |

Rate of recovery or removal from |
0.854302 | |

Rate of recovery or removal from |
0.09871 | |

Period of incubation of |
0.00047876 | |

Period of incubation |
0.005 | |

The proportion of asymptomatic infection | 0.1234 | |

Disease transmission from |
0.000001231 | |

Transmissibility multiple | 0.02 | |

Rate of contact of |
0.05 | |

Rate of natural mortality | ||

Total initial population | 8266000 | |

Birth rate |

Now we develop OHAM scheme for solving underlying fractional order COVID-19 model. OHAM technique is known for its rapid convergence as compared to other techniques. It produces an approximate closed form of the desired solutions and, hence, is known as semi-analytical approach. Procedure of our approximate OHAM scheme has been derived by following the relevant principles as described in literature [

We construct the homotopy equations by defining the real valued functions

In the above relations

For

And so on.

Adopting the same procedure presented in Steps 1–4, we find the following approximate solutions for the state variables

This section is dedicated for presentation of the closed form semi-analytical solutions for all of the state variables and demonstration of their dynamics through graphical exhibition of related simulation results. Model 4 has total six equations. That means COVID-19 fractional model is to analyzed by observing the behaviors of six subpopulations

The considered values for model parameters are presented by

For the fractional analysis of human susceptible population we calculate the auxiliary constants (

The auxiliary constants (

0.6 | ||

0.7 | ||

0.8 | ||

0.9 | ||

1 |

1 |

Considering various orders of the fractional derivative, above necessary conditions provide the relevant optimum values of auxiliary constants for the exposed population and are presented in

0.6 | ||

0.7 | ||

0.8 | ||

0.9 | ||

1 |

The OHAM scheme based second order solution for the exposed population

The values of auxiliary constant

Solving the following equations for

0.6 | ||

0.7 | ||

0.8 | ||

0.9 | ||

1 |

The values of auxiliary constant

The values of

0.6 | ||

0.7 | ||

0.8 | ||

0.9 | ||

1 |

The following total error function corresponding to recovered population is obtained from

Solving the following necessary conditions we obtain the values of

0.6 | ||

0.7 | ||

0.8 | ||

0.9 | ||

1 |

Applying the least square approach for the auxiliary constant of

The solutions of following equations give the values of

0.6 | ||

0.7 | ||

0.8 | ||

0.9 | ||

1 |

In this study, closed form semi-analytical approximate solution has been presented for fractional order COVID-19 model in Caputo sense of derivative operator. The underlying model involves parameters that were extracted by fitting real data into the dynamical model. The fitted parameters are responsible for a high reproduction number

The authors are thankful to their respective departments and Universities.