key: cord-0688503-608w4o6o
authors: Kumar, Sunil; Kumar, Ranbir; Momani, Shaher; Hadid, Samir
title: A study on fractional COVID‐19 disease model by using Hermite wavelets
date: 2021-02-07
journal: Math Methods Appl Sci
DOI: 10.1002/mma.7065
sha: 20f68cfdaaef32b46bfd6bb21284273bf26963c1
doc_id: 688503
cord_uid: 608w4o6o

The preeminent target of present study is to reveal the speed characteristic of ongoing outbreak COVID‐19 due to novel coronavirus. On January 2020, the novel coronavirus infection (COVID‐19) detected in India, and the total statistic of cases continuously increased to 7 128 268 cases including 109 285 deceases to October 2020, where 860 601 cases are active in India. In this study, we use the Hermite wavelets basis in order to solve the COVID‐19 model with time‐ arbitrary Caputo derivative. The discussed framework is based upon Hermite wavelets. The operational matrix incorporated with the collocation scheme is used in order to transform arbitrary‐order problem into algebraic equations. The corrector scheme is also used for solving the COVID‐19 model for distinct value of arbitrary order. Also, authors have investigated the various behaviors of the arbitrary‐order COVID‐19 system and procured developments are matched with exiting developments by various techniques. The various illustrations of susceptible, exposed, infected, and recovered individuals are given for its behaviors at the various value of fractional order. In addition, the proposed model has been also supported by some numerical simulations and wavelet‐based results.

The coronavirus infection is the most deadly and dangerous infectious infection in the whole globe. The COVID-19 infection, the short form of corona virus infection (2019), is sourced by serious intense respiratory syndrome corona virus 2 (SARS-CoV-2). 1 Most individual infected red to the COVID-19 infection will experience mild to mitigate respiratory sickness and instaurate without requiring special treatment. The outbreak of a harmful and extremely infected virus of the current period is a corona infection, and it is recognized first time in the Wuhan place (China) on December 2019. 2 In January 2020, the World Health Organization (WHO) divulged the COVID-19 infection to be a social fitness difficulty and recognized it as a outbreak on 11 March 2020. Since then, it killed over 1 61 402 on April 19 over the infected case of 23 47 875 individuals in more than 180 countries. From the starting up to the present day, there is no drug or dose available to completely recovered infected individual. In the initial stages of this infectious infection outbreak, patients feel tiredness, dry cough, and fever sometimes lead to breathing problems. Recently, Lin et al. discussed an accurate system for the COVID-19 infection, which accurately captures the time line of the COVID-19 infection in Lin et al. 3 In 2020, Chen et al. 4 discussed a mathematical system for reproducing the stage-based transmission of the COVID-19 infection. Recently, Khan et al. 5 formulated mathematical system to study on the COVID-19 infection. In maximum cases, the infected individuals have few same symptoms which combine dry cough (68%) and fever (88%). Few of the cases have symptoms that combine tissue and joint pain, fatigue, respiratory sputum manufacture, sore throat, and headache. The COVID-19 infection disperses at a broad intensity between individual in the close touch with infected individual. According to WHO, the most common incubation period areas to 1-14 days. 6 In 2020, many researchers devoted themselves to study about COVID-19 in previous studies. [7] [8] [9] [10] 10 The recorded cases of SARS-Cov-2, to 1 April 2020, till 18 April 2020, in India has been shown in Figure 1 , and comparison between the number of recovered individual and number of deaths is displayed in Figure 2 . Fractional calculus (FC) is dealing with the calculus of derivatives and integral of arbitrary-order real or complex. [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] There has been a powerful development in fractional differential equations (FDEs) in last decades due to its popularity in distinct research areas of science and technology. FDEs have an advantage in modelling real-life phenomena because it will reduce the errors arising from the ignored parameters. There are numerous examples of the mathematical model, which are consists of FDEs. To mention some, FDEs are used in breast cancer, 28 hepatitis B virus, 29 and Nipah Virus. 30 The dengue model is discussed through FDEs in Kilicman and Hamdan, 31 and in Dubey et al., 32 Baba and Ghanbari, 33 and Gao et al. 34 the food chain model, tuberculosis, and rubella, respectively.

In the present article, we are discussing the dynamics of the COVID-19 infection system recommended by Khan et al. 5 with time fractional Caputo derivative.

with primary conditions (0) = 1 , (0) = 2 , (0) = 3 , (0) = 4 , (0) = 5 and (0) = 6 , where  is the overall population of individual. Again,  divided into five subpart in the manner that susceptible individual (t), exposed individual (t), infected (symptomatic) individual (t), asymptotically infected (t), and the recovered or the removed individual (t). The individual of the market or reservoir is stand for (t). During the 1980s, wavelet analysis because of their outstanding application in image and signal processing became popular tools in many branches of science and engineering. More than that, speciality such as orthogonality, good localization of wavelets have attracted many researchers. Wavelets sanction the precise representation of variety of operator and functions. 35, 36 Therefore, wavelets have significant role in many fields like image processing, time frequency analysis, and signal processing. 37 Basis of wavelet is relatively incipient and has received attention for solving various types of FDEs. Legendre, Hermite, and Laguerre wavelets are used to solve FDEs in previous studies. [38] [39] [40] [41] The Bernoulli wavelet is used to solve coupled models of nonlinear arbitrary-order integro-DEs in Wang et al. 42 Recently, Kumar et al. have discussed Bernstein wavelets method for solving SIR epidemic. 43 There are some research articles on Hermite wavelets for solving FDEs. 44, 45 Best of our knowledge, there is not any article based on Hermite wavelets for solving Biological models in [0, t l ]. The work is set up as pursue; in part 2, we provide some basic essential explanations of FC. In Part 3, we construct the Hermite wavelets for arbitrary interval and also discussed the convergence analysis. We develop the operational for Hermite wavelets with the help of block pulse functions in Part 4. In Part 5, we use Hermite wavelets and Adams-Bashforth-Moulton (ABM) to solve the COVID-19 infection model. Numerical simulation and discussion are given in Part 6. Finally, achieving remarks are given in Part 7.

There are some essential explanations of arbitrary derivative and integral in previous studies. [46] [47] [48] Definition 2.1. The Riemann-Liouville (RL) integral operator of order is described as

where t − 1 *Υ(t) is the convolution product of t − 1 and Υ(t).

Arbitrary-order derivative in the sense of Caputo's is described as

Let k, M be positive integer. The Hermite wavelets nm (t) for n = 1, 2, 3, … , 2 k−1 and m = 0, 1, 2, … , M − 1 are described on the interval [0, t l ) as

where  m (t) is the well-known Hermite polynomial of degree m with respect to the weight function w(t) = √ 1 − t 2 on the real line R and fulfill the following recurrence relation.

Let Ω k, M be the space spanned by Hermite wavelets for nm , that is,

Since Υ 0 ∈ Ω k, M is the particular perfect approximation, then there exists particular coefficients

where Λ and G vectors are defined as

and

Choosing k = 2, M = 4, and the collocation points as t i = 2i − 1∕2m, i = 1, 2, … ,m = 2 k−1 M, we obtained the Hermite wavelets matrix as 

Theorem 3.1. Let Υ(t) ∈ L 2 [0, t l ] be the function and Υ 0 (t) ∈ Ω k, M is the approximation of Υ(t) then

Proof. Let Υ (i) (t) be the continuous function, where i = 0, 1, 2, … M. Then there exist  ∈ N such that

Then by Taylor ′ s formula

Since, { nm (t)} is the family of piecewise function. As Ω k,M = span{ nm (t)}, therefore

where t l ∈ N is the fixed number and when M is sufficiently large then || Υ ||→0. Hence, Hermite wavelets approximation is convergent.

In the following section, the Hermite wavelets operational matrix for integer-order integration and fractional-order integration, respectively, are obtained. These operational matrices play vital role in the proposed method for solving the problem.

The block pulse functions (BPFs) defined over the interval [0, t l ) as

where = 0, 1, 2, … ,m and m = [b 1 , b 2 , b 3 , … , bm] . The useful properties of BPFs are listed in previous studies. [48] [49] [50] [51] Here, we will use BPFs to construct the Hermite wavelets operational matrix of arbitrary-order integration.

and l = (l + 1) + 1 − 2l + 1 + (l − 1) + 1 , for l = 1, 2, 3, … ,m − 1. Now, we develop the Genocchi wavelets operational matrix for arbitrary-order integration, P . Let

Then,

Using the above fact the operational matrix, P for = 0.5, M = 4, k = 2, and t l = 1 is given as 

This is the operational matrix of BPFs of 8 × 8 order.

We have formulated proposed schemes for the study of arbitrary-order COVID-19 infection system.

Let us consider a fractional model of arbitrary-order COVID-19 infection system as

where l Λ T m = [ l Λ 1 , l Λ 2 , l Λ 3 … , l Λm], l = 1, 2, 3, 4, 5, 6 are unknown coefficients. By using the primary conditions and definition of arbitrary-order integral operator, we obtain

Further, using the properties of BPFs in (9), we obtain

Then, we have 

and (t)(t) = ( 1 Λ T m Pm ×m Φm ×m m + (0))( 6 Λ T m Pm ×m Φm ×m m + (0))

Now, we substitute Equations (8), (10), (11) , and (12) into the COVID-19 model (1) of fractional order; we obtain the system of algebraic equations. By adopting, Newton iteration method to solve nonlinear algebraic equations; we can find the anonymous coefficients. On substituting anonymous coefficients into the (10), we can achieve the required solutions.

Here, mainly we will describe that famous numerical scheme ABM method as a pair to construct a predictor-corrector method to solve arbitrary-order COVID-19 infection system. On applying ABM method 52,53 on Equation (1), we obtained the predictor values and the corresponding corrector values as follows. To change it into discrete form, let h = t l − 0∕m, t n = nh, n = 0, 1, 2, … ,m − 1, and ∈ (0, 1]

where a ,n+1 = ⎧ ⎪ ⎨ ⎪ ⎩ n +1 − (n − )(n + 1) , i = 0, (n − + 2) +1 + (n − ) +1 − 2(n − + 1) +1 , i 0 ≤ ≤ n,

This is the formulation of arbitrary-order COVID-19 infection system by using ABM scheme.

The primary objective of proposed section is to present numerical simulation and graphical illustration of susceptible, exposed, infected, and recovered individuals of the arbitrary-order COVID-19 infection system. Further, we observed the many behaviors susceptible, exposed, infected, and recovered peoples in arbitrary-order COVID-19 infection system by using Hermite wavelet and ABM methods. A comparative study of the susceptible, exposed, infected, and recovered individuals is represented through portrayed 03-20.

Here, we consider the arbitrary-order COVID- 5 The numerical solutions obtained by Hermite wavelets method and ABM are displayed in Figures 3-8 . Further, we noticed that the arbitrary-order derivative has a remarkable effect on the dynamics of COVID-19. Susceptible individual (t), exposed individual (t), infected individual (t), asymptotic infected individual (t), recovered individual (t), and individual in reservoir (t) against time t in days for large amounts of are displayed in Figures 9-20 . Moreover, we observe that the fractional operator provides more flexibility than the integer operator. It is observed that the COVID-19 infection mathematical model depends on time fractional derivative; the current study may guidance to figure out the harmful virus.

Illustration 3 displays behaviors of susceptible peoples by using Hermite wavelet method, and a comparison study through Illustration 1 represented between susceptible peoples in arbitrary-order COVID-19 infection system by using Hermite wavelet and ABM methods at M = 4, k = 6, and = 1. The number of susceptible peoples is decreasing in arbitrary-order COVID-19 infection system. We observed that nature of the susceptible peoples is identical by both schemes. Illustrations 4-7 display the nature of exposed, infected, asymptotically infected, and recovered peoples in arbitrary-order COVID-19 infection system by using Hermite wavelet and ABM methods at M = 4, k = 6, and = 1. Further, we observed through Illustrations 4-7 that the number of exposed, infected, asymptotically infected, and recovered peoples is continuously increasing in arbitrary-order COVID-19 infection system, but it is also notice from figure 7 that the total number of recovered peoples is increasing very fast according time. It is also observed through monthly serological survey on COVID-19 that the total number recovery peoples has been jumped. The recovery rate stands at 64.53% among COVID-19 patients. Further, various nature of susceptible, exposed, infected, and recovered individuals in arbitrary-order COVID-19 infection system represented by various Figures 8-20. 

Novel coronavirus is a extremely infectious infection worldwide. The thousand certified cases and thousands of infection have been reported in various countries. The infection has turn into outbreak, disturbing almost all countries of the world, and has caused enormous budgetary, social, and psychological load on countries. In this article, the COVID-19 infection mathematical model has been determined numerically via Hermite wavelets scheme. Moreover, solutions obtained by Hermite wavelets are compared with solutions obtained by ABM predictor corrector scheme. Firstly, Hermite wavelets function approximation and convergence investigation have been recommended. In other way, an arbitrary-order integral operator for [0, t l ] has been derived by which the COVID-19 infection can be reconstructed into simply determined system of algebraic equations. Finally, we have analyzed the results through graphically.

It was concluded that susceptible peoples contract the infection through direct contact with infected individuals, as well as indirectly through the presence of coronavirus in the environment. Further, we have examined through study that there is no best option other than social distancing, isolation, and palliative measures in present time.

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This research is supported by Ajman University grant 2020-COVID-19-07.

https://orcid.org/0000-0003-0620-1068