The multistep Laplace optimized decomposition method for solving fractional-order coronavirus disease model (COVID-19) via the Caputo fractional approach

1 Introduction

In 2019, a new strain of coronavirus, COVID-19, was discovered in the Wuhan, China. The pandemic may have been started by snakes or bats, although this has been ruled out by other specialists. Fever is one of the first indications of infection, followed by coughing and breathing difficulty. Pneumonia, acute respiratory syndrome of extreme severity, kidney failure, as well as death can occur as a result of infection’s successive phases [1]. The virus has spread quickly and with a high level of severity. As a result, it has become a public health threat on a never-before-seen scale. Researchers and scientists are trying to figure out how this virus works and how to get rid of it, so they are looking for treatments and vaccines. Transmission occurs primarily through droplets and person-to-person contact. Getting the COVID-19 vaccine, wearing a mask, staying at home, social distancing, and washing hands frequently are all highly recommended ways to reduce the spread of infection.

Fundamentally, many biologists and mathematicians are interested in the dynamics of diseases, for example, the model of HIV-1 infection of CD4+ T-cells is studied in [2], the rubella disease model is utilized in [3], the susceptible, infected, and recovered model is discussed in [4], and the model of childhood disease is given in [5]. Mathematicians always aim to model real-world behaviors that they observed in their daily activities. In a new glimpse, several important biological problems have been modeled successfully the last time using the concept of the Caputo fractional derivative (CFD), such as the fractional hybrid thermostat model [6], fractional human liver model [7], fractional measles epidemic model [8], and others [9,10,11,12,13,14,15,16,17]. However, many researchers have exercised fractional mathematical models to simulate coronavirus transmission as utilized in [18,19,20].

Fractional calculus is a branch of mathematics that aims to comprehend real-world phenomena that are represented by noninteger-order derivatives. In this context, fractional calculus theory and its illustrated applications are gaining popularity around the world. Because of its effective properties, fractional calculus has found widespread use in modeling dynamic processes in a variety of well-known fields of science, engineering, biology, medicine, and many others [21,22,23,24,25,26,27,28,29,30,31]. In general, no method exists that produces an exact solution to the fractional differential problems (FDPs), but only approximate solutions are possible. So, several methods for solving FDPs have been developed with several descriptions and mathematical formulations as included in [32,33,34,35,36,37].

In this article, we present a novel approach named the multistep Laplace optimized decomposition method (MLODM) to obtain analytical approximation solutions for the COVID-19 model using the CFD. The MLODM possesses the combined behavior of Laplace transformation and optimized decomposition method, where the optimized decomposition method is a powerful technique that uses an approach of linearization of nonlinear equations for obtaining a better series solution, which provides efficient algorithms for analytic approximate solutions, and when compared to the traditional Adomian decomposition approach, it gives effective precision and convergence [38]. Besides the solutions of the Laplace optimized decomposition method (LODM) converge in a relatively narrow region, and it had sluggish convergence. For this reason, we modify the LODM to give a novel technique called MLODM that gives good results in a longer time.

Anyhow, let us consider the following COVID-19 model, where the total populations of humans here are divided into four classes [18]: S ( ζ ) : immunocompromised individuals, E ( ζ ) : exposed or infected individuals who do not transmit the infection, I ( ζ ) : people who have been confirmed to have a disease and are infectious, and R ( ζ ) : recovering people. So, the COVID-19 model in CFD concerning 0 < α ≤ 1 with ζ ≥ 0 is given by [18]

subjected to the corresponding initial conditions:

Hither, in (1) and (2), δ is the rate of recruitment, λ 1 and λ 2 are the incidence rates, respectively, ω is the relapse rate, ν is the natural death rate, η is the rate at which COVID-19-exposed people join the infectious class, υ is the percentage of infected people who recover, and ψ is the death rate of infected class due to the SARS-CoV-2 virus [18]. Indeed, the total population N ( ζ ) is given, at each instant of time, by

Fundamentally and analytically, the MLODM when handling the COVID-19 model has many practical features and positive advantages. First, the approximate solution is very close when compared with the fourth-order Runge-Kutta method (RKM4), and so it can be observed by the obtained outcomes that the MLODM is very sufficient and accurate. Second, high accuracy can be achieved that is valid for a longer period by using a small number of Adomian polynomials, subdomain [ ζ k − 1 , ζ k ] , and time steps ζ . Third, it is a simple method to apply and does not need advanced mathematical tools or a skilled programmer. Fourth, it is universal as it can be used to solve other types of linear and nonlinear fractional problems and systems. Fifth, the proposed multistep approach can be applied to other transforms like Fourier or Sumudu, which is its the main characteristic.

This article is organized as follows. Section 2 presents basic definitions of fractional calculus and the Laplace transform that we shall use. In Section 3, the essential steps of the MLODM to solve nonlinear FDP will be detailed. We apply the MLODM to the COVID-19 model to show its efficiency and compare it with the RKM4 in Section 4. Numerically simulated results and central processing unit (CPU) computational time cost are shown in Section 5. Finally, we give a conclusion in Section 6 with some future planning.

2 Preliminaries

Fractional calculus has been introduced and developed as an excellent mathematical tool to describe the memory and hereditary characteristics of many materials and processes in a variety of fields of pure and applied science. In recent literature, it has been used to formulate many nonlinear differential problems and exploited to provide a comprehensive and clear explanation of dynamics, dispersion, wave propagation, and evolutionary models because of space time change [21,22,23,24,25,26,27,28,29,30,31].

In this section, we present some fundamental definitions and properties of the fractional calculus theory and the Laplace transform approach. For more results, interesting properties, and highlights, the readers can refer to the works in [21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37].

3 The MLODM: formulation and steps

This section is divided into a couple of portions as follows: first, the LODM for the FDPs. Second, the MLODM for the FDPs. Anyhow, with the help of the Laplace transform and decomposition method, a full formulation will be utilized in our discussion.

Let us consider the following FDP insight into the CFD scheme:

subjected to the corresponding initial condition:

where ζ > 0 , 0 < α ≤ 1 , M : R → R is a nonlinear term, and H : [ ( 0 , ∞ ) → R is a given analytic function.

First, let us try the LODM for the FDPs: the optimized decomposition method is used to provide an analytical approximate solution for a wide class of nonlinear problems. This utilized approach is mainly based on the new construction of the solution series that depends on the Taylor approximation of the nonlinear equation [38].

Now, we will propose a new technique called LODM to find the analytical approximate solution for (10) and (11). Fundamentally, we apply the Laplace transform to (10) as follows:

By using Definition 5, we have

By using the constraint condition z ( 0 ) = d 0 in (11) and applying the inverse Laplace transform to both sides of (14), we obtain

Let z ( ζ ) be the solution of (15) in which the optimized decomposition method assumes that z ( ζ ) can be decomposed by the infinite series of the formation:

Hither, M is a nonlinear term that appears in (10), and it can be denoted by

where R j ( ζ ) takes the formation:

Recalling that R j ( ζ ) is called the Adomian polynomials as described in [34]. Anyhow, after substituting (16) and (17) into (15), one obtained

where

The components of { w j ( ζ ) } j = 0 ∞ are determined as follows

such that the constant b is expressed as follows:

and

The Taylor series expansion of g of two variables about ζ = 0 is

Then, let us solve g ( D α [ z ( 0 ) ] , z ( 0 ) ) = 0 for D α [ z ( 0 ) ] , say, D α [ z ( 0 ) ] = c 0 and z ( 0 ) = d 0 . Following this by substituting the results in (23), we have

Second, let us complete the formation by applying the MLODM for FDPs. The solutions given by the LODM are valid for a short period of time, and hence, we next introduce a novel technique of MLODM, which gives good results in a longer time. To do this, we construct the general form of the MLODM.

Let [ 0 , T ] is the time period in which we want to find a solution to (10) and (11). Anyhow, [ 0 , T ] is divided into m subintervals [ ζ k − 1 , ζ k ] with k = 1 , 2 , ⋯ , m of equal length Δ ζ . So, (10) will become a system that takes the following formation:

Further, by applying the same procedure, (11) can be re-written as follows:

where a 1 = d 0 , 0 < α ≤ 1 , ζ > 0 , and k = 1 , 2 , ⋯ , m .

Finally, the basic steps of algorithm of the MLODM that is used to compute the approximate solutions of (1) and (2) can be given in short as follows:

• Apply the LODM to find z 1 ( ζ ) on [ ζ 0 , ζ 1 ] , where ζ = 0 by using z 1 ( 0 ) = d 0 .

• For k ≥ 2 , the LODM is used with z k ( ζ k − 1 ) over [ ζ k − 1 , ζ k ] .

• The same process is repeated m times to obtain a sequence of approximate solutions as follows:

It should be mentioned that those steps could be applied using the MATHEMATICA 11 program to obtain the required results.

4 Application: COVID-19

Temporal fractional evolution models are excellent tools for the figuration of nonlinear phenomena of dynamic diseases and for understanding the basic physics, phase properties, and evolutionary dynamics that covered these models.

In this portion, the MLODM in view of the CFD is profitably applied to solve the COVID-19 model in its fractional version. Anyhow, let us reconsider the COVID-19 model utilized in (1)–(3) again in which 0 < α ≤ 1 and ζ ≥ 0 .

is subjected to the corresponding initial conditions:

Hither, we will apply the proposed MLODM to the time domain [ 0 , 200 ] , where we will split it into subdomains with a step ∆ ζ = 1 . First, let S k ( ζ ) , E k ( ζ ) , I k ( ζ ) , and R k ( ζ ) with k = 1 , 2 , … , 200 are the approximate solutions of (29) in each subdomain [ ζ k − 1 , ζ k ] with k = 1 , 2 , … , 200 . Then, (29) can be re-written as follows:

is subjected to the corresponding initial conditions:

By applying the Laplace transform to both sides of (31), we obtain the following:

In view of (13)–(15) and by using constraint conditions in (32), we obtain

The optimized decomposition method solutions of (34) are of the following form:

Finally, by using (21) with k = 1 , 2 , … , 200 , we obtain

Hither, the Adomian polynomial components are given by

Further, the constants b i with i = 1 , 2 , 3 , 4 are given as follows:

5 Numerical simulations

Numerical simulations are used to demonstrate the theoretical results reported in earlier sections. For this, various simulation outcomes of the COVID-19 model are discussed and studied as well in this portion. Some graphical representative results are presented with physical interpretations for several fractional parameters to support the theoretical framework and to give a clear visualization of the behavior of the proposed model. Further, numerical comparisons with the RKM4 are made to illustrate the effectiveness and simplicity of the presented MLODM. All calculations and representative results are performed by using the mathematica computing system.

To perform numerical simulation, we use the initial conditions as follows [20]: S ( 0 ) = 220 , E ( 0 ) = 100 , I ( 0 ) = 3 , and R ( 0 ) = 0 with time steps ζ = 200 . Anyhow, the parameter values with their mathematical descriptions are tabulated in Table 1 beside their numerical values for the simulations of the utilized fractional-order COVID-19 model.

First, the information and needed requirements shown in Figures 1 and 2 are tabulated in Table 2. However, in Figure 1(a), we plot the susceptible population of (29) for α = 1 , where the dotted lines denote the MLODM solution of susceptible class S ( ζ ) and the solid lines denote the RKM4 solution of susceptible class S ( ζ ) . Figure 1(b) represents the exposed population of (29) for α = 1 , where the dotted lines and solid lines denote the MLODM solution and the RKM4 solution of the exposed class E ( ζ ) , respectively. Figure 1(c) and (d) illustrate the infectious population I ( ζ ) and the recovered population R ( ζ ) , respectively, of (29) for α = 1 , where the MLODM solution represents the dotted lines and the RKM4 solution represents the solid lines. We can observe from the numerical approximation results drawn in Figure 1(a)–(d) that the approximate solutions exhibited using MLODM have a high level of accuracy.

Figure 1

The MLODM solution versus the RKM4 solution at α = 1 and 0 ≤ ζ ≤ 200 of the COVID-19 model as (a) susceptible class S ( ζ ) , (b) exposed class E ( ζ ) , (c) infected class I ( ζ ) , and (d) recovered class R ( ζ ) .

Figure 2

The MLODM solution at α ∈ { 1,0.9,0.85,0.75,0.65 } and 0 ≤ ζ ≤ 200 of the COVID-19 model as (a) susceptible class S ( ζ ) , (b) exposed class E ( ζ ) , (c) infected class I ( ζ ) , and (d) recovered class R ( ζ ) .