Abstract

In this paper, a novel coronavirus SIDARTHE epidemic model system is constructed using a Caputo-type fuzzy fractional differential equation. Applying Caputo derivatives to our model is motivated by the need to more thoroughly examine the dynamics of the model. Here, the fuzzy concept is applied to the SIDARTHE epidemic model for finding the transmission of the coronavirus in an easier way. The existence of a unique solution is examined using fixed point theory for the given fractional SIDARTHE epidemic model. The dynamic behaviour of COVID-19 is understood by applying the numerical results along with a combination of fuzzy Laplace and Adomian decomposition transform. Hence, an efficient method to solve a fuzzy fractional differential equation using Laplace transforms and their inverses using the Caputo sense derivative is developed, which can make the problem easier to solve numerically. Numerical calculations are performed by considering different parameter values.

1. Introduction

The official report of the outbreak of COVID-19 from Wuhan, China, due to coronavirus was first released on December 31, 2019, by World Health Organization (WHO). Despite the fact that Chinese cities instituted lockdowns to limit infection transmission, the efforts were inadequate, and the sickness has now spread over the world. Because of this, the WHO classified COVID-19 as a global pandemic in March 2020. Initially, this unanticipated global pandemic wreaked havoc on nearly every aspect of human life. COVID-19 has been sweeping the globe in waves since the first case was recorded. On November 24, 2021, WHO reported a novel SARS-CoV-2 variant. The B.1.1.529 variant was named after the fifteenth Greek alphabet Omicron. In less than a week, this highly altered strain spread across six continents, raising worldwide health concerns. Omicron has been discovered in over 50 nations across six continents since its discovery. Compared to variants alpha, beta, gamma, and delta of SARS-CoV-2, omicron emerges as the most distinct and unique variant. It possesses six mutations compared to the Wuhan variety, resulting in greater transmissibility and vaccination resistance. Omicron’s rapid spread has made it the most common coronavirus strain, overtaking the previous globally ubiquitous Delta variant. However, it is still unsure whether it has a higher transmission level than the delta variant. The host viral kinetics model of SARS-CoV-2 under the Caputo fractional-order operator are discussed in [1]. Furthermore, a new study reveals that its spread is compared to wildfire in the context of the delta variant’s continuous proliferation and innate immunity. Omicron is predicted to make the delta a more prevalent variety if current trends continue. Mathematical models are useful for determining how an infection behaves when it enters a population and determining whether it can be eradicated or will continue in different settings [2]. The transmission of this disease is caused by tiny particles or droplets called aerosols that carry the virus into the atmosphere. The aerosols are released by a contaminated person while sneezing, coughing, or exhaling. Many researchers and scientists are continuously working to reduce the transmission of this vicious disease throughout the world [3]. A fractional order pandemic model is developed to examine the spread of COVID-19 with and without the omicron variant and its relationship with heart attack using real data from the United Kingdom. Infectious diseases is the discipline that focuses on illustrating various factors starting from the appearance to the evolution responsible for the spread of the infection among the human population so that suitable methods can be adopted to prevent and fight against the diseases.

One of the most common tools for describing the transmission of the disease is mathematical modelling, which has played a significant role in epidemiology and has resulted in framing necessary measures to protect people from infectious diseases. The most influential work in the field of mathematical epidemiology was first investigated by Kermack and McKendrick as the SIR model in the year 1927 [4]. A modified model of the SIR (Susceptible, Infected, Recovered) epidemic was introduced in order to detect the confirmed number of infected cases and consecutive burdens on isolation wards and ICUs [5]. Also, Nelstruck developed the variables used in the proposed model by introducing a SIR epidemic model and explaining how to dominate the spread of the disease. To restore the pandemic with the involvement of social distancing and lockdown, Gerberry and Milner presented the SEIQR model in [6]. Using the ordinary differential equation (DE), the authors in [7] investigate how the vaccination rate and the fraction of avoided contacts affect the population dynamics in the SIR model. The nonlinear biological SIR models using Feed-Forward Artificial Neural Networks (FFANN) optimized with a global search of genetic algorithm aided with rapid local search interior-point IP algorithms, i.e., FEANN-GAIP. is discussed in [8]. The authors in [9] discussed the stochastic first order Runge–Kutta method is taken into account for model simulation. The numerical solutions of the multispace fractal-fractional Kuramoto–Sivashinsky equation (MSFFKS) and the multispace fractal-fractional Korteweg de Vries equation (MSFFKDV) are discussed in [10]. To execute this, the entire population is partitioned into five units: suspected, exposed, infected, isolated, and recovered from the disease [1115]. From the publication of Anwar Zeb et al. epidemiological’s SEIQR model with isolation class in 2020, mathematical epidemiology has expanded in numerous directions, involving biology and computer science [1619]. Some recent studies have focused on this area of research [2028].

In our work, the fractional-order (Caputo’s sense) SIDARTHE epidemic model are investigated for the COVID-19 infection system mathematically. Since the fractional order differential operators are nonlocal operators, they can better represent some dynamic system processes and natural physics processes when compared to integer order differential equation. The Caputo fractional operator is more flexible for analysis and handles the initial and boundary value problems. It is also widely used to define the time-fractional derivatives in fractional partial differential equations. This motivates us to solve the fuzzy fractional differential equations in the Caputo sense. The fractional differential equations with fuzzy solutions, as well as fuzzy boundary and initial value problems can be solved using the fuzzy Laplace transform technique. Another significant benefit is that it offers direct problem-solving without first generating nonhomogeneous differential equations and then figuring out a general solution. The uniqueness and existence of the solution to the following fuzzy fractional model are explained using fixed point theory. In addition, the numerical results from the fuzzy Laplace transform based on the Adomian decomposition are helpful in understanding the physical behaviour of COVID-19 with dynamical structures.

Figure 1 is presented to understand the trend in the spread of COVID-19 among the human populations in countries with a high rate of infection. The trend emphasises the need to finding suitable recovery measures in controlling the spread. The individual country-based report on the confirmed cases per million is illustrated in Figure 2. The data report is obtained from the website “our world in data.” So, in order to overcome the increase in the impact of the infection due to corona virus, this article aims to study and perform mathematical analysis for a better understanding of the pandemic.

2. SIDARTHE Fractional Mathematical Model for the COVID-19 Outbreak

Through the use of SIDARTHE, Giordano et al. [29] created a model of the COVID-19 outbreak and contrasted its reaction with the actual data in Italy. SIDARTHE makes a distinction between confirmed and suspected cases of infection as well as different stages of sickness.

The entire population is divided into eight disease stages:S-Susceptible (uninfected)I-Infected (undetected, asymptomatic infection)D-Diagnosed (asymptomatic infection detected)R-Recognised (detected)A-Ailing (undetected, symptomatic infected)T-Threatened (found to be infected with life-threatening symptoms)H-Healed (recovered)E-Extinct (dead)

The likelihood of re-exposure to the virus after recovery is not included. Although there are anecdotal occurrences in the literature [30], the reinfection rate value seems to be insignificant [31]. The graphical representation of the SIDARTHE compartmental model is shown in Figure 3. The SIDARTHE COVID-19 model transmission rates are given in Table 1.

The model, SIDARTHE, distinguishes between infections that have been discovered and those that have not, as well as between illnesses that are potentially life-threatening (severe and major) and those that are not (moderate and minor), both of which call for admission to an intensive care unit.

2.1. Fuzzy Fractional Order of SIDARTHE Epidemic Model

The concept of fuzzy calculus and fractional order differential equations (FODEs) have been developed over the past few years as extensions of current calculus and DEs (see [3236]). After that, fuzzy FODEs were expanded from FODEs [3739].

In order to establish the uniqueness and existence theory of solutions, several academics have explored FODEs and fuzzy integral equations [4045]. It is challenging to evaluate exact solutions for each fuzzy FODE when working with them. Numerous efforts have been undertaken by mathematicians to solve fuzzy FODEs using a variety of strategies, including the spectral techniques, integral transform methods, and perturbation method [4651]. A stability investigation of fuzzy DEs was conducted by some researchers [52].

Firstly, the (1) is transformed into a fractional order differential scheme by denoting in (1). That is,where -Susceptible, -Infection, -Diagnosed, -Ailing, -Recognized, -Threatened, -Healed, and -Extinct.

Next, we will examine the model (2) with a fuzzy fractional-order derivative. The fuzzy fractional order derivative of (2) is given as follows:

For and , the fuzzy initial condition is given by the following equation:

We were motivated to suggest a unique fractional calculus-based coronavirus infection system in order to brief on the current status. With the common RNA properties present in COVID-19, the physical behaviour of such an infection system is improved by the suggested model, which is close to the actual behaviour of such a system.

2.2. Basic Definitions

In this section, basic preliminary concepts relating to fuzzy fractional used in the following sections are discussed.

Definition 1 (see [53, 54]). Let be the parametric form of a fuzzy number (FN) , where , and meets the following criteria:(1) is the left continuous, decreasing over (0, 1] and bounded, whereas the right continuous is at 0(2) is right continuous, decreasing over [0,1] and bounded, whereas the left continuous is at 0(3)Also, if , then is called a crisp number.

Definition 2 (see [46, 49]). Let be a continuous fuzzy function. Then, the fuzzy fractional integral (FFI) according to is given by the following equation:Furthermore, if , then FFI is defined as follows:where and are the spaces of fuzzy Lebesgue integrable functions and fuzzy continuous correspondingly. Here,

Definition 3 (see [55, 56]). If be a fuzzy function, and , then the FF Caputo’s derivative is defined as follows:wherewhenever the right-side integrals converge and .

Definition 4. (Laplace transform (LT)). Let is fuzzy real number, and . Then, the FF differential iswhere and . If , then it reduces to a fractional differential equation.

Theorem 1 (see [49]). If , then the fuzzy LT of fractional derivative in Caputo’s sense iswhere , and .

Theorem 2 (Arzela-ascoli theorem). The sequence of continuous function defined on an interval is said to be an uniformly bounded sequence, if a number for each in the sequence and for every (Here, must not depend on or ).

The sequence is an uniformly equicontinuous if for every such thatwhenever functions in the sequence (Here, may be depend on , but not , y or n).

Theorem 3 (Schauder Fixed Point Theorem). Assuming is compact and let M be a convex, closed subset of a bounded Banach space X. Then, has atleast one fixed point in .

3. Major Contribution

The uniqueness and existence of the solution to the succeeding FF model are described.

3.1. Uniqueness and Existence

In this section, using fixed point theory the uniqueness and existence of the ensuing FF model are examined. Considering the right side of the model, we have the following equation:

Let and , and then substituting (4) in (3), the given model (3) can be written as follows:

Using the initial conditions along with FFI as defined in Definition 2, we get the first term as follows:

Which is the same as follows:

Next, the second term will bewhich implies

The third term will bewhich yields

Similarly, the fourth term iswhich is the same as follows:

Hence the other terms will bewhere .

Let be a Banach space under the fuzzy norm. Then,

Therefore, (26) and (27) can be written as follows:where

Several assumptions have been made on the nonlinear function as follows:(1)(C-1) constant for each , (2)(C-2) constant, and

Theorem 4. Model (3) has at least one solution under Assumption (C-1).

Let be a subset of which is a convex and closed fuzzy set. The mapping is defined as follows:

For any , we have the following equation:

Since the operator is implied to be bounded by the last inequality, we obtain (M) ⊆ M. Next, we demonstrate the complete continuous of the operator . Furthermore, if and , then

According to the previous inequality, the right side goes to zero . Hence,

As a result, the operator is equicontinuous. The operator is entirely continuous according to Theorem 4, and as proved earlier, is bounded. According to Theorem 3, system (3) has at least one solution.

Theorem 5. The system (5) has a unique solution, if (C-1) holds and .

Let , then

Hence, is a contraction. Therefore, system [5] has a unique solution according to the Banach contraction theorem.

3.2. Procedure

A usual method is given to solve the equation using the fuzzy Laplace Transform.

Inserting the Laplace Transform in an equation and by initial conditions, we arrive

Then, we set the following equation:

After making some modifications, we set the following equation:

The infinite series solution is given by the following equation:where are the Adomian polynomials, representing nonlinear terms. Applying the infinite series in equation [24], the fuzzy LT equation will become the following equation:

Now, applying the inverse LT, the above equation becomes the following equation:

When comparing the terms on both sides, the series’ initial (first) terms are taken into account.and the second terms in the series are

Similarly, the remaining terms are , , , , , and .

Hence, the general series solution will be

That is,

Similarly, the other terms are

The above-given equations are written as follows:

Using the above-given equations, the spread of the virus can be studied by applying the real data.

4. Results and Discussion

In this section, we determine the terms of the series solution for a particular triangular FN.

Considering the model (2) with initial conditions,

Using the above-given initial conditions, the first term of the series solution of the equation is