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Analysis of the Adomian decomposition method to estimate the COVID-19 pandemic

Several techniques, including mathematical models, have been explored since the onset of COVID-19 transmission to evaluate the end outcome and implement drastic measures for this illness. Using the currently infected, noninfected, exposed, susceptible, and recovered cases in the Indian community, we created a mathematical model to describe the transmission of COVID-19. In particular, we used the semianalytical Adomian decomposition method without considering any discretization to perform the first-order differential equations related to COVID-19 cases. According to our early findings, rigorous initial isolation for 22–25 days would reduce the number of exposed and newly infected people. As a result of the downstream effect, the number of suspected and recovered persons would remain stable, assuming that social distance is properly recognized. In a larger sense, the parameters established by our mathematical model may aid in the refinement of future pandemic tactics.

Importantly, preventive measures such as social distancing and lockdown in populated areas have proven to be highly effective in this pandemic. As there is no potential therapy (drugs, vaccines), several scientific approaches have either been tested or are in development. Several mathematical models using the epidemiological data were also tested to estimate the transmission rate of the virus and to monitor the predominant factors contributing to this pandemic [9].
Herein we have collected the epidemiological, pathophysiological, and relevant data related to COVID-19 and employed a rational approach to forecast the time required to effectively control COVID-19 in India considering the ideal situation of social distancing and lockdown being followed adequately and appropriately in different location in India [10]. We combined a stochastic transmission model with data on cases of coronavirus disease 2019  in India to estimate how transmission had varied over time from March 14, 2020 to March 26, 2020. To estimate the early dynamics of transmission in India, we fitted a stochastic transmission dynamic model to multiple publicly available datasets on cases in India. The four datasets we fitted were: (1) infected patients, (2) exposed individuals, (3) asymptomatic patients, and (4) patients recovered. Based on these estimates, we calculated the probability that newly introduced cases (suspected people) might generate outbreaks in other areas. Hence, we used an additional dataset for comparison with model outputs, i.e., (5) suspected individuals and (6) reservoir individuals.
In the current study, we have evaluated the epidemiological, pathophysiological, and other publicly available COVID-19 associated data from the Indian population. Besides infected patients, exposed persons, and convalesced patients, we have exclusively analyzed the suspected persons to accurately assess the early dynamics of transmission in India.

Integral transform
It is a mathematical operator which produces a new function f(t) by integrating the multiple of a function F(t) and another function K(ζ, t), which is called the Kernel between the suitable limits. This process of transform of that function into another is called integral transform, which is written as There are so many integral transforms generally named after the mathematicians who developed them such as Laplace Transform, Mellin Transform, Hankel Transform, Fourier Transform, etc.
The main aim of integral transform is that it gives powerful working methods for solving initial value problems and the initial-boundary value problems for the linear differential and integral equations. It has many mathematical and physical applications.
The choice of taking a suitable integral transform for a particular differential equation is very important because it decides to convert not only the derivative into an appropriate differential equation but also the terms of boundary values into algebraic structures.
Also, Harinder Singh noted the advanced numerical methods for differential equations [11] and fractional differential equations [12], which may also be helpful in analysis for this.
For the complete solution of the problem, it is also necessary to convert the equation into its original form by applying inverse integral transform.
Hence, inverse integral transform is the contrapositive part of the integral transform so that the solution may be obtained in a more simple way.
In this subsection of this chapter, we have studied Laplace integral transform, Adomian Laplace transform, iterative Laplace transform.

Laplace transform
The Laplace Transform of the function F(t), t > 0 is given by And the inverse Laplace transform is given by There are some functions with their Laplace transforms that are given in the list.

Initial approach to design the input equation
Recently, a few independent studies by Chen [20] and Zhang [21] successfully applied mathematical models on the coronavirus disease (COVID-19). Herein we have used and adapted the mathematical equation proposed by Khan and colleagues [22], which is presented as In this equation, N represents the total population which is segregated into five subclasses, namely, susceptible people S(t), exposed people E(t), infected people I(t), asymptotically infected people A(t), and recovered people R(t). The detailed parameters relevant to this equation are defined in Table 1.

Adomian decomposition method (ADM)
George Adomian [23] established the Adomian decomposition method (ADM) in the 1980s. The ADM has received much attention in recent years in applied mathematics and in the field of infinite series solution. It is an effective method to solve many types of linear, nonlinear, ordinary, or partial differential equations and integral transforms (such as the Volterra and Fredholm integral transforms). After that, he extended the modified Adomian decomposition method for solving the nonlinear Emden-Fowler system of differential Eqs. Alderremy, Elzaki, and Chamekh [24] also wrote a research article on the solution of the Emden-Fowler system using MADM (modified Adomian decomposition method). It may be relevant to point out that by ADM we can solve the Mathematical Model of coronavirus for people.
Using the Adomian decomposition method, the differential term is first written as an In general form, L is used for the highest differential operator used in the equation and similarly L À1 for that times the integral operator.
Using L, the given equation can be written as Now taking the inverse operator L À1 on both sides, we get Now, the Adomian decomposition suggests that f(x) can be written as the summation of continue infinite series Comparing both sides, we get f 0 ¼ 1 4 4! and so on. Using these in the above equation

Analysis with the Adomian decomposition method
The Adomian Decomposition method (ADM) is a semianalytical method for solving ordinary and partial nonlinear differential equations. The method was developed from the 1970s to the 1990s by George Adomian. Considering the Adomian decomposition method (ADM) [23] can effectively solve several types of linear, nonlinear, ordinary, or partial differential equations and integral transforms (such as the Volterra and Fredholm integral transforms) and the method has lately been modified in several studies [24]. We used Eq.
(1) as input for the ADM to perform numerical simulation, data fitting, and the estimation of different values. Hence, using the L operator for the first-order differential equation, we wrote Eq. (1) as Using L 2 1 on both sides, we obtained System (3) is the solution for the given model.
Solving in the recurrence manner and writing the (n + 1) th term, the nonlinearity terms SI, SA, and SQ can also be written as W n . U n , V n , W n are further decomposed as follows: Chapter 10 • Analysis of COVID-19 mathematical model 179 Using these notations, the last equation can be written as Similarly, solving in the same manner all the other five equations of the system of differential Eq. (1) gives the solutions as Eqs. (4) and (5) are the recurrence solution for the given mathematical model of the system.
We also considered finding a numerical solution (approximation) of the iterative values of using the parametric values (particularly estimated or fitted) from Table 1. Notably, as described by the World Health Organization (WHO) [1], 35% of the Indian population live in cities or urban areas of India [10]; we therefore considered a parameter for the total population (N ¼ 481747192) that corresponds to 35% of India's population.
Hence, the formulated equation can be written as: Based on the parameters given in Table 1, the initial conditions are given below. By using these inputs to system (3) and solving, we obtained the following equation in the end: The complete solution is given by the following series form: These systems of solution are obtained by solving these separate partitions simultaneously.

Results and discussion
On the basis of the results obtained, we found that under the ideal situation of an appropriate and efficient implementation of social distancing, it was possible to curb COVID-19 within 22-25 days of strict lockdown (Fig. 1A). As a downstream effect, the number of exposed individuals would be expected to increase very slowly by up to 4 months from onset of disease (Fig. 1B). It is noteworthy that this graph (Fig. 1C) also depends on the number of newly infected persons. As the numbers of exposed individuals were insignificant up to the first 50 days, i.e., only after 150 days (approximately 5 months), the drastic increase in the number of exposed persons would be accounted for. Even if the number of suspects at the outbreak of the disease would be 50 crore within 4-5 months, the numbers could still be reduced to zero (Fig. 1D). Furthermore, the number of suspected recovered individuals followed the same trend (Fig. 1E), thus indicating that if social distance had been adequate and effective, stable disease control would have been achieved.
Overall, the parameters defined by our mathematical model may help to refine the future strategies to control similar pandemic situations. In the context of limitations, consideration of the additional factors (gender stratification, age-related death risk, epidemiological factor, smoking behavior, etc.) can further improve this model.

Conclusion
The main aim of this study was to determine an eventual numerical reality (month-wise), such as when under an ideal situation of social distancing in the lockdown period we could expect a significant reduction in the number of COVID-19 cases, and when we could consider the situation to be under control. The time duration of clinical progression, such as pathogen entry and its life cycle in the human body, and recovering duration is also discussed, which can help in epidemic analysis and its control (Fig. 2). During the early stages of cell attachment, the pathogen attaches to the cell surface in 5 min postinfection and hijacks the cell's function within 30 min [27]. The virus infects the lung cells and reaches a peak stage within 10-13 days [28]. By this time, the immune cells of the body get activated and produce a cytokine microenvironment to destruct the virus [29]. The virus has a mean incubation period of 5.1 days and the patient recovers within 2-3 weeks postinfection [30,31]. Therefore, such model projections are highly beneficial to forecast health-care facility (including the number of critical/intensive care units, ventilators, safety kits, medicines, etc.) and would be needed for successful treatment and management of COVID-19. By applying the Adomian decomposition method on the Indian population impacted by COVID-19, we could show that the strict initial isolation for 22-25 days would have a significant positive impact on the whole pandemic situation. Finally, the presented mathematical model would also consider other necessary interventions and initiatives that our government may need to take in order to curb the disease burden.

Funding
No funding was received for this work.

Conflicts of interests
The authors declare no conflict of interests.

Early stages of infection following transmission Various stages of lung infection
a.