On existence and semi-analytical results to fractional order mathematical model of COVID-19

Abstract Corona virus disease 2019 (Covid-19) is an illness caused by a natural corona virus called severe acute respiratory syndromes corona virus 2 (SARS-COV-2 formally called Covid-19) which is respiratory illness and has been declared the Covid-19 outbreak a global pandemic by World Health Organization (WHO). Presently, Covid-19 becomes a health concern around the globe. In present article, we investigated the dynamics of Covid-19 infectious disease under the fractional order derivative from both theoretical as well as analytical aspects. Using fixed-point theory results, we developed the indispensable conditions for the existence of the solution of the proposed model. Further we have used the techniques of Laplace transform coupled with the Adomian’s decomposition method to obtain the semi analytical solution for the model under consideration. Finally we have provided some graphical presentation corresponding to different fractional order derivatives via Matlab for the desired solutions.


Introduction
The world community, despite of having modern technologies in every walk of life, is striving to fight an unknown enemy termed as corona virus, which affected the developed world more intensively as compared to the developing countries. At the closing of the year 2019, the world was strike by a virus named as novel corona virus in short . The Corona virus was first identified in 2019, in the city of Wuhan, China (Jasper, Kin, & Zheng, 2020). Despite the fact that there is no scientific authentication, the scientists and the researchers are still believe that the virus has been transferred from animals to beings (Ji, Wang, Zhao, Zai, & Li, 2020). Instantaneously, it is also perceptible that the transfer of the virus occurs from person to person (Chen & Guo, 2016). The World Health Organization (WHO) on October 10th, 2020, reported that the virus has hit 210 countries around the globe. The total number of deaths reported as 1,072,711 out of 37,110,933 people have been affected. The death ratio in countries like Italy, Spain and the USA were higher as compared to the other countries. The symptoms of the virus include difficulties in respiratory procedure, likewise breath spar, mild temperature, cold and other indications alike. However, other symptoms such as gastroenteritis and acoustic illness of conflicting extremity have also been reported (Wang et al., 2006).
The main cause of the eruption of the virus is, whenever the affected individuals cough or sneeze which produced small droplets and people who comes in contact with the droplets are attacked by the virus. To stop the escalation of virus, several authorities of the world community adopted the policy of a full-fledged lock down and completely closed all the activities to ensure safety for their respective citizens. At this crucial time, the world health communities and law enforcement authorities put forward their level best to prevent the escalation of the virus in the world community. At the first instance, the scientists and researchers found that the main causes of the Covid-19 are bats where these viruses are identical to the virus spread in China in 2003 termed as Severe Acute Respiratory Syndromes (SARS) (Ge et al., 2013;Lu, 2020). Later on, the newly born virus  has been compared with the virus SARS and MERS (Middle East Respiratory Syndromes) to classify the virus and presented a reliable strategy against Covid-19. For the said purpose, the researchers used to the previous studies dealing with SARS and MERS. Lu (Zhou et al., 2020) claimed that Covid-19 relates SARS-Cov and MERS-Cov. For further study, the readers are referred to (Benvenuto et al., 2020;Kumar, Chauhan, Momani, & Hadid, 2020;Rambaut, 2020;Shen et al., 2021;Tian et al., 2020).
Initially, it was believed that bats are the primary sources of the COvid-19. Therefore the disease was compared with Severe Acute Respiratory Syndromes (SARS) which was originated from China and then covered almost the rest of the world in 2003 (Ge et al., 2013;Lu, 2020). Later on, this virus was also compared to Middle East Respiratory Syndromes (MERS) to examine the suitable class for the virus. Beside this several other researchers argued that this Covid-19 virus belongs to the Beta-Corona genus (Benvenuto et al., 2020;Kumar, Chauhan, et al., 2020;Rambaut, 2020;Shen et al., 2021;Tian et al., 2020). It is important to note that the main source of spreading this infectious disease is due to the droplets of infected individuals excreted through nose or mouth. For instance, if infected individual is in contact with any susceptible vector through his droplet (sneeze or cough), then the susceptible has a great risk to be attacked by the virus. For precautionary measures, various authorities around the globe adopted the policies including complete lock down and smart lock down.
In the present paper, we presented a mathematical approach to Covid-19 infectious disease. We investigate the existence perspective as well as develop the numerical scheme for the underlying Covid-19 model. For qualitative analysis, we used the tools of fixed-point theory such as Layer-Schauder and Arzila-Ascoli theorem, in order to obtain the desired results. To investigate the approximate solution of the underlying model, we utilize the techniques of Laplace transform coupled with Adomian decomposition (1). The concerned techniques work on the combination of Laplace and Adomian. With the help of Adomian, the problem is decomposed into polynomial and Laplace convert that differential equation into simple algebraic equations, which then solved and converted back to subsidiary equations. At the end of the work, we acquired the approximate solution as infinite series for different fractional value of l and provided it plots. For the aforementioned computational work, we utilized the tools of Matlab.
There are various numerical techniques to obtain the concerned solutions for different biological models, such as Homotopy Perturbation Method (HPM), Variation Iteration Method (VIM), Adomian Decomposition Method (ADM), Differential Transform (DTM) and Generalization Differential Transform (GDTM), for details (Odibat & Momani, 2008;Hu, Luo, & Lu, 2008;Dehghan, Yousefi, & Lotfi, 2011;Kumar, Singh, & Kumar, 2015;Anjum, He, & He, 2021;Ain, He, Anjum, & Ali, 2020;Anjum & He, 2019;Anjum, Suleman, Lu, He, & Ramzan, 2020;Adomian, 1994). Probably, Laplace Adomain Decomposition Method (LADM) is one the reliable and efficient techniques to obtained numerical approximation of different problems. Therefore, the concerned technique has been adopted for the numerical computation of our proposed problem. George Adomian ð1923À1996Þ introduced a method to solve nonlinear differential equation, known as Adomian Decomposition Method (Koonin & Cetron, 2009), which we utilized by replacing the terms in the series form and then each term is calculated through a polynomial generated by power series. The series is considered in the manner adopted in (16) and (17). The method adopted in (17) is the representation of nonlinear terms involved in the system, while (16) has been used for the representation of linear terms that are involved in the system under consideration.
Note that modeling is a powerful tool to describe a real word situation in mathematical concepts. In recent decade, the concerned technique has attained considerable attention of researchers, due to verity of applications in all discipline of sciences. Mathematical modeling plays a key role in investigation of dynamical behavior of infected diseases and its control at the earlier stages. The area of research involving the study of infectious diseases is an area of interest for the researchers nowadays. In the connection, we predict and investigate the dynamical behavior of fractional order Covid-19 model (1), through Caputo fractional operator. We develop the mechanism through which an infectious disease is spread in the community. The proposed model is specially considered for the ongoing pandemic in the world started from China and covered almost all the countries of the world. The capture fractional order Covid-19 model is given by with initial conditions where 0<l 1 and the analogous communication is granted as follows (Gao, Veeresha, Prakasha, & Baskonus, 2020;. The parameter involved in the model (1) is described in Table 1. We also imposed the assumption the all parameter are non-negatives. The compartments and parameters contained in the model (1) are illustrated in Tables 1 and 2, respectively while the numerical values of the parameters are stated in (Table 3).
This article is organized as follows: We investigate semi-analytical solution for the proposed model (1). In the first section of this work, the authors presented existence of the solutions for the concerned model. We utilize the tools of fixed point to obtain the conditions for existence of the solutions. The deserted results are obtained via Layer-Schauder and Arzila a-Ascoli theorems. Since, for the problem involving non-linearity the study and development of the criteria for the existence of the solution of the problem is very tough, therefore the researchers paid more attention to obtain approximate solutions for such a problem. The second part of this work, is committed to the general produced for obtain analytical solutions of the aforementioned Covid-19 viral disease model. In order to obtain the analytical solution, we used the techniques based on the combination of Adomian decomposition and Laplace transform known as LADM. With help of aforesaid techniques, we obtained the solution in the form of infinite series. In Section 3, the authors have provided the numerical solution and discussion of the model under consideration. Using Matlab, we obtained the deserted results in the form of infinite series up to three terms for different fractional order derivatives and presented through plots.
Theorem 2.5 (Godefroy, 1901). Consider T B7 !B be a compact and continuous mapping corresponding to the bounded set then it assures one fixed point of the operator T :

Existence results for the proposed model
We are interested to investigate that weather the solution of dynamical problem/model exists or not.
To answer this query we impose the mechanisms of fixed-point theory to inspect the existence of the solution of the considered problem. In addition, we disclose the right hand side of the problem (1)  (2) Assume that B ¼ Cð½0, T, R 6 Þ be the Banach spaces, with (3) In-view of (2) and (3), the considered system (1) can be disclose in the form of Solution of Equation (4) is written as, We assume that the following suppositions hold for the existence of considered problemðH 1 Þ Assume that 9 constants C ! ÃÃ and M ! ÃÃ , ʯ ! ÃÃ ðt, WðtÞÞ C ! ÃÃ jWðtÞj þ M ! ÃÃ , 8W 2 B: ðH 2 Þ Assume that 9 a constant L ! ÃÃ >0, for every W, W Ã 2 B ʯ j! ÃÃ ðt, WÞÀ! ÃÃ ðt, W Ã Þj L ! ÃÃ jwÀw Ã j: Further, we make use of Leray-Schauder fixedpoint theorem to verify that the solution of the problem we studied does exist.
Theorem 1.1. In the light of supposition ðH 1 Þ and continuity of ! ÃÃ : ½0, T Â R 6 7 !R, the integral equation (5) has at least one solution and hence the considered model (1) with mC ! ÃÃ <1, must have at-least one solution, where v ¼ t l Cðlþ1Þ : Proof. Assume, that ðH 1 Þ holds and defined D ¼ fWðtÞ B : jjWjj B n, t 2 0, T ½ g, be a convex closed subset of B and n ! v 0 þvM ! ÃÃ 1ÀvC ! ÃÃ : Also defined an operator T D7 !D 8 W 2 D and jW 0 j ¼ v 0 : ! ÃÃ ðs, WðsÞ ds, Hence T ðDÞ D and hence the operator T is continuous.
w Now assume that t 1 <t 2 2 ½0, T, we need to prove that the operator T is completely continuous. For this let, If t 2 ! t 1 , then right side of (6) approaches to Zero, ) jjT Wðt 2 ÞÀT Wðt 1 Þjj ! 0 as t 2 ! t 1 : Hence the operator T is equi-continuous and bounded. Therefore by Arzila-Ascoli theorem T is relatively compact which implies that T is completely continuous. By making use of the Schauder fixed-point theorem we can claim that the discussed system (1) must have at least one solution.
Theorem 1.2. If the supposition (H 2 Þ holds true and t l L ! ÃÃ <Cðl þ 1Þ, then the considered problem has unique solution. Proof. If W , W Ã 2 B, and the operator T B ! B defined above. Now we consider the following Thus, T is continuous. Therefore the system (1) has unique solution.

General procedure of LADM for Caputo derivative
To derive the general procedure of LADM for the Caputo's fractional derivative, we consider the following differential equation where c stands for Caputo fractional derivative, N represents the nonlinear term, R represent the linear term involved in the given equation and f(t) is a source term. Taking Laplace on both sides of (7) and rearranging the terms yields the following.
By applying the definition of Laplace transform for Caputo's fractional derivative we get the following Let us the required solution may be expressed in the form of infinity series as uðtÞ ¼ X 1 n¼0 u n ðtÞ: Further the nonlinear term is decomposed as NuðtÞ ¼ X 1 n¼0 P n ðtÞ, where, By plugging (9) and (10) in (8), we get the following By comparison the terms on both sides of (11), we have After evaluating the inverse Laplace transforms, we get the required solution as uðtÞ ¼ u 0 þ u 1 ðtÞ þ u 2 ðtÞ þ u 3 ðtÞ þ ::::::::

General procedure for approximate solution
In the segment, we developed the scheme for approximate solution of our proposed problem (1). Taking Laplace transform on (1), we have Applying the Laplace transform in the sense of Caputo fractional derivative on (13), we get Applying inverse Laplace and plugging the initial conditions on (14), we have an À mSðtÞ À bSðtÞIðtÞ À bkSðtÞAðtÞ À cSðtÞMðtÞ Assuming the solutions SðtÞ, EðtÞ, IðtÞ, AðtÞ, RðtÞ and MðtÞ in the form of infinite series give by The non-linear terms are represented as where Y n ðtÞ, Z n ðtÞ and X n ðtÞ are Adomian's polynomials defined as Y n ðtÞ ¼ Using (16) and (17) in (15), we gets Now, applying inverse Laplace on (18), we obtained Putting the above values of parameters and l ¼ 0:99 in (20), we have   By assigning the values to the terms defined in system (1) from (Gao et al., 2020;, we get the following graphs.
From Figures 2-7, it can be observe that the rise of the population of infected individuals is related to the rise in the population of exposed individuals which is clearly related to the up rise of the class of the susceptible population. Therefore infection is going up with asymptotic infection that goes up. Therefore, the rate of recovery is increasing as infection and so more death cases will occur. Hence the denseness of the class consist of removed population is raising up. The rate of increase is expressed by taking different values of the arbitrary order of the fractional derivative. It is evident that having smaller order of the fractional derivative the rate is quickly moving up and by making the order greater the rate is becoming slow.

Conclusion
At present, Covid-19 becomes the pandemic and a health concern around the globe. The world social and economic sectors have been smashed by this infectious disease. Several authorities like medical professionals, scientists and other researchers are trying their level best to put forward a reliable strategy for stopping the escalation of the virus in the community. The main aim of this article is to present and appropriate mathematical approach to Covid 19 infectious disease. For instance Using fixed-point theory and nonlinear analysis, we have proved the existence of the proposed Covid-19 model. By Laplace Adomian decomposition method, we have derived semi-analytical results. Graphical presentations are given to illustrate the dynamics of the model.