Computers, Materials & Continua DOI:10.32604/cmc.2022.013906 | |

Article |

Structure Preserving Algorithm for Fractional Order Mathematical Model of COVID-19

1Department of Mathematics, University of Management and Technology, Lahore, Pakistan

2Department of Mathematics and Statistics, The University of Lahore, Lahore, Pakistan

3Stochastic Analysis & Optimization Research Group, Department of Mathematics, Air University, Islamabad, 44000, Pakistan

4Department of Mathematics, National College of Business Administration and Economics, Lahore, Pakistan

5Department of Mathematics, Faculty of Sciences, University of Central Punjab, Lahore, 54500, Pakistan

6Faculty of Mathematics and Statistics, Ton Duc Thang University, Ho Chi Minh City, 72915, Vietnam

7Department of Mathematics, College of Arts and Science at Wadi Aldawaser, Prince Sattam Bin Abdulaziz University, Alkharj, 11991, Kingdom of Saudi Arabia

*Corresponding Author: Ilyas Khan. Email: ilyaskhan@tdtu.edu.vn

Received: 30 August 2021; Accepted: 10 November 2021

Abstract: In this article, a brief biological structure and some basic properties of COVID-19 are described. A classical integer order model is modified and converted into a fractional order model with

Keywords: Coronavirus pandemic model; deterministic ordinary differential equations; numerical methods; convergence analysis

Novel coronavirus is a spherical or pleomorphic shaped, particle having single stranded (positive sense) RNA (Ribonucleic Acid) linked to a nucleoprotein surrounded by a special type of protein. The outer surface of the coronavirus contains the projections of the club-shaped structure. The Classification of the coronaviruses depends upon the appearance of the outer surface (whether it is crown like or halo like), the replication mechanism and the distinct features related to the chemistry of the virus. In general, these viruses belong to OC43-like or 229E-like serotypes. Avian and mammalian species serve as hosts for them. Both types are similar with respect to morphology and chemical structure. Corona viruses present in human beings and animals are antigenically similar. These are capable of attacking on different types of tissues in animals. But, in human beings this family of viruses generally cause only the upper respiratory tract infection. This virus belons to the subclass Orthocoronavirinae, class Coronaviridae, order Nidovirales, realm Riboviria, kingdom Orthcornavirae and phylum Pisuviricota. The dimension of this virus varies from 26 to 32 kilobases, which is largest in the class of RNA viruses. They have distinct protruded club or clove shaped studs or spikes [1]. Like other corona viruses, COVID-19 also contains protein in the form of spikes ejecting outside from the surface. These spikes cling with the host (human) cells then its genome bears a structural change and the viral membrane fuse with the host cell cytoplasm. After this step, the viral genes of the COVID-19 enter into the host cell for replication and multiplication of the viruses. Depending upon the protease of the host cell, cleavage reaction permits it to reach into the host cell by endocytosis or fusion. After entering into the host cell, the virus becomes uncovered and their genome attacks on the cell cytoplasm. The genome of the coronavirus works as a messenger and it is translated by the ribosomes of the host cells. These viruses are divided into four categories as alpha coronavirus, beta coronavirus, gamma coronavirus and delta coronavirus. The first two viruses infect the mammals while the last two viruses initially attack the birds. The genera and species of these viruses are described as follows: the species Alphacoronavirus 1, Human coronavirus 229E, Human coronavirus NL63, Miniopterus bat coronavirus 1, Miniopterus bat cor onavirus HKU8, Porcine epidemic diarrhea virus, Rhinolophus bat coronavirus HKU2 and Scotophilus bat coronavirus 512 belong to the Alpha coronavirus. While the species, Betacoronavirus 1(Bovine Coronavirus, Human coronavirus OC43), Hedgehog coronavirus 1, Human coronavirus HKU1, Middle East respiratory syndrome-related coronavirus, Murine coronavirus, Pipistrellus bat coronavirus HKU5, Rousetlus bat coronavirus HKU9, Severe acute respiratory syndrome- related coronavirus (SARS-Cov, SARS-Cov-2) and Tylonycteris bat coronavirus HKU4 belong to Beta coronavirus. Furthermore, the species Avian coronavirus and Beluga whale coronavirus SW1 are the members of the Gamma coronavirus. Lastly, the Bulbul coronavirus HKU11 and Porcine coronavirus HKU15 species are the family members of the Delta coronavirus. Coronaviruses are deleterious to health with high risk factor. Some of them have more than 30% mortality rate, for instance MERS-COV. But other are not so harmful like as common cold. All types of the coronaviruses can be the causative agent of cold with prime symptoms including fever, sore throat and swollen adenoids. Moreover, they can cause primary viral pneumonia or secondary bacterial pneumonia or bronchitis in the same way as that of pneumonia [2]. The SARS-COV appeared in 2003, resulted in severe acute respiratory syndrome (SARS). It effected both the upper and lower respiratory tract due to an unmatched pathogenesis. There are six classes of human coronaviruses that are known so far, each specie is categorized into two types. There are seven types of human coronaviruses. Four coronaviruses which show mild symptoms are: Human coronavirus OC43 (HCOV-OC43),

In this section, we will present some fundamental definitions of non-integer order derivatives, their key properties and notations used in this article.

2.1 Non Integer Order Derivatives

Fractional order derivatives have been defined by many researchers in a number of ways according to the nature of the kernel used therein. Some basic fractional order operators are defined in this section. Firstly, the Riemann–Liouville non integer order derivative of order

where

The importance of this operator, when applied to solve a system of fractional differential equations is that it can be associated with initial conditions of classical order, which results in an initial value problem in the desired form as,

A very useful definition relating to this article by using the classical finite differences on a uniform mesh partionised in

where h is defined as the difference of

where

This expression is derived from the famous Euler method. Consider the fractional differential equation

Now, by applying the G-L scheme on a uniform mesh, we obtain the following expression

where e and

where

Furthermore, ei and

Lemma 1 [17]: Let

Also the following two relations are satisfied

and

In this section, we present the GL-NSFD hybrid scheme is formulated by combining the GL scheme for numerical approximation of the fractional order derivatives and NSFD scheme constructed by using the standard rules designed by Mickens. More details can be seen in [18]. The system of equations for COVID-19 is described as follows:

In this portion, we will construct the proposed scheme. The discretization of fractional derivative

The above formula is used on the left hand side of Eq. (6) to get the following expression

After some simplifications, we have the final form as,

Similar procedure is adopted for the remaining compartments and we have the final forms as,

2.3 Positivity of the Solution

In this portion, positivity of the solution will be investigated. Positivity is an important feature of the compartmental models. Since, the state variables in these type of models describe the size of the population that cannot be negative. So, positivity is the basic requirement of the solutions at every moment of time. Following result is helpful in this regard.

Theorem: Assume that all the unknowns and parameters arose in the model are non-negative i.e.,

Proof: Taking in to account the Eqs. (11) to (15) for n = 0, we have

From the restrictions imposed on the state variables and parameters, it is evident that

Since the state variables in the model represent the subpopulation of a certain compartment. So the sum of values of all the state variables must be less than or equal to the total population or equivalently the sum of solutions at any time must be bounded. The following result is helpful in this regard.

Theorem: Let

Proof: Considering the Eqs. (11) to (15), we have

By applying the principle of mathematical induction,

for n = 0, we have

In the same way

Now, we calculate the expression (16) for n = 1, and obtain the following relations,

The above inequalities help us to reach at

Now, let

For some

where,

Now for

In the same fashion a adopted before, we conclude that

So, the given expression is true for all positive values of n.

Hence, the solutions are bounded

In this portion, we will investigate the stability of the model at both the points of equilibria i.e., at a corona free equilibrium point and a corona existing equilibrium point.

The corona free equilibrium state of the model is given as

The corona virus existing equilibrium state is calculated as

where,

Theorem: The corona free equilibrium

Proof: The corona-free equilibrium

Notice that the two Eigen values are repeated as

where,

By using the Routh–Hurwitz Criterion of 2nd order polynomial as,

Hence, all Eigenvalues are negative and by Routh-Hurwitz criteria the given equilibrium point C1 is locally asymptotically stable.

Theorem: The corona existence equilibrium

Proof: The corona existence equilibrium

For the eigen values, the Jacobean matrix at

Notice that, the Eigen values are

where,

By using the Routh–Hurwitz Criterion of 3rd order polynomial, we get the following expression:

and

Thus, we have concluded that all Eigenvalues are negative and by Routh Hurwitz criteria, the given equilibrium point C2 is locally asymptotically stable. Here, we will present a suitable numerical example and graphical solutions of the state variables involved in the model. This whole stuff is presented with the aid of computer simulations.

3 Numerical Example and Simulations

In this portion, an example of the fractional order COVID-19 model is provided. The parametric values are mentioned in Tab. 1. Also, non-negative initial conditions are considered.

Computer aided graphs are submitted to support our assertions. These sketches support the fact that proposed numerical device is a structure preserving tool for solving the nonlinear fractional systems. The device encounters the positivity, stability and boundedness of the solutions. All the graphs in Fig. 1 reveals that all the subpopulations converge at the virus free equilibrium point (with different values of

with a certain rate of convergence according to the value of

In this study, some biological and physical features of the novel corona virus-2019 are described. A classical

Funding Statement: The authors received no specific funding for this study.

Conflicts of Interest: The authors declare that they have no conflicts of interest to report regarding the present study.

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