A Fuzzy Fractional Order Approach to SIDARTHE Epidemic Model for COVID-19

. In this paper, a novel coronavirus SIDARTHE epidemic model system is constructed using a Caputo-type fuzzy fractional diferential 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 fnding the transmission of the coronavirus in an easier way. Te existence of a unique solution is examined using fxed point theory for the given fractional SIDARTHE epidemic model. Te 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 efcient method to solve a fuzzy fractional diferential 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 diferent parameter values.


Introduction
Te ofcial report of the outbreak of COVID-19 from Wuhan, China, due to coronavirus was frst released on December 31, 2019, by World Health Organization (WHO). Despite the fact that Chinese cities instituted lockdowns to limit infection transmission, the eforts were inadequate, and the sickness has now spread over the world. Because of this, the WHO classifed 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 frst case was recorded. On November 24, 2021, WHO reported a novel SARS-CoV-2 variant. Te B.1.1.529 variant was named after the ffteenth 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. Te 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 wildfre 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 diferent settings [2]. Te transmission of this disease is caused by tiny particles or droplets called aerosols that carry the virus into the atmosphere. Te 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 fght against the diseases.
One of the most common tools for describing the transmission of the disease is mathematical modelling, which has played a signifcant role in epidemiology and has resulted in framing necessary measures to protect people from infectious diseases. Te most infuential work in the feld of mathematical epidemiology was frst investigated by Kermack and McKendrick as the SIR model in the year 1927 [4]. A modifed model of the SIR (Susceptible, Infected, Recovered) epidemic was introduced in order to detect the confrmed 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 diferential equation (DE), the authors in [7] investigate how the vaccination rate and the fraction of avoided contacts afect the population dynamics in the SIR model. Te nonlinear biological SIR models using Feed-Forward Artifcial 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]. Te authors in [9] discussed the stochastic frst order Runge-Kutta method is taken into account for model simulation. Te numerical solutions of the multispace fractalfractional 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 fve units: suspected, exposed, infected, isolated, and recovered from the disease [11][12][13][14][15]. 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 [16][17][18][19]. Some recent studies have focused on this area of research [20][21][22][23][24][25][26][27][28].
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 diferential operators are nonlocal operators, they can better represent some dynamic system processes and natural physics processes when compared to integer order diferential equation. Te Caputo fractional operator is more fexible for analysis and handles the initial and boundary value problems. It is also widely used to defne the time-fractional derivatives in fractional partial diferential equations. Tis motivates us to solve the fuzzy fractional diferential equations in the Caputo sense. Te fractional diferential equations with fuzzy solutions, as well as fuzzy boundary and initial value problems can be solved using the fuzzy Laplace transform technique. Another signifcant beneft is that it ofers direct problem-solving without frst generating nonhomogeneous diferential equations and then fguring out a general solution. Te uniqueness and existence of the solution to the following fuzzy fractional model are explained using fxed 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. Te trend emphasises the need to fnding suitable recovery measures in controlling the spread. Te individual country-based report on the confrmed cases per million is illustrated in Figure 2. Te 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.

SIDARTHE Fractional Mathematical Model for the COVID-19 Outbreak
Trough 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 confrmed and suspected cases of infection as well as diferent stages of sickness. Te 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-Treatened (found to be infected with life-threatening symptoms) H-Healed (recovered) E-Extinct (dead) Te 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 insignifcant [31]. Te graphical representation of the SIDARTHE compartmental model is shown in Figure 3. Te SIDARTHE COVID-19 model transmission rates are given in Table 1.   2 Complexity Te 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.
In order to establish the uniqueness and existence theory of solutions, several academics have explored FODEs and fuzzy integral equations [40][41][42][43][44][45]. It is challenging to evaluate exact solutions for each fuzzy FODE when working with them. Numerous eforts have been undertaken by mathematicians to solve fuzzy FODEs using a variety of strategies, including the spectral techniques, integral transform methods, and perturbation method [46][47][48][49][50][51]. A stability investigation of fuzzy DEs was conducted by some researchers [52].
Firstly, the (1) is transformed into a fractional order diferential scheme by denoting D Next, we will examine the model (2) with a fuzzy fractional-order derivative. Te fuzzy fractional order derivative of (2) is given as follows: For 0 < c ≤ 1 and μ ∈ [0, 1], 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.

Basic Defnitions.
In this section, basic preliminary concepts relating to fuzzy fractional used in the following sections are discussed.
for each f n in the sequence and for every x ∈ [a, b] (Here, M must not depend on n or x).
Te sequence f n n∈N is an uniformly equicontinuous if for every ϵ > 0∃δ > 0 such that whenever |℘ − y| < δ∀ functions f n in the sequence (Here, δ may be depend on ϵ, but not ℘, y or n).

Major Contribution
Te uniqueness and existence of the solution to the succeeding FF model are described.

Uniqueness and Existence.
In this section, using fxed 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 M(Z) � (Z, J s (Z) J�A,B,C,D,E,F,G,H ) and 0 < c ≤ 1, and then substituting (4) in (3), the given model (3) can be written as follows: Using the initial conditions along with FFI I r as defned in Defnition 2, we get the frst term as follows: Which is the same as follows: Next, the second term will be Te third term will be Similarly, the fourth term is which is the same as follows: Hence the other terms will be where A s (℘) � J s (Z) J�A,B,C,D,E,F,G,H . Let Banach space under the fuzzy norm. Ten, Terefore, (26) and (27) can be written as follows: where Several assumptions have been made on the nonlinear function ⊖: B ⟶ B as follows:

Complexity
Let M � L̃s(Z) ∈ B: ‖L̃s(Z)‖ ≤ r be a subset of B which is a convex and closed fuzzy set. Te mapping ϵ: M ⟶ M is defned as follows: (33) For any L̃s(Z) ∈ M, we have the following equation: Since the operator ε is implied to be bounded by the last inequality, we obtain ε (M) ⊆ M. Next, we demonstrate the complete continuous of the operator ε. Furthermore, if ϕ 1 , ϕ 2 ∈ [0, T] and ϕ 1 < ϕ 2 , then According to the previous inequality, the right side goes to zero (ϕ 2 ⟶ ϕ 1 ). Hence, As a result, the operator ϵ is equicontinuous. Te operator ϵ is entirely continuous according to Teorem 4, and as proved earlier, ϵ is bounded. According to Teorem 3, system (3) has at least one solution.

Theorem . Te system (5) has a unique solution, if (C-1) holds and κ c s L
Hence, ϵ is a contraction. Terefore, system [5] has a unique solution according to the Banach contraction theorem.

Procedure. A usual method is given to
Ten, we set the following equation: After making some modifcations, we set the following equation: Te infnite series solution is given by the following equation: where Z 1,ℓ , Z 2,ℓ are the Adomian polynomials, representing nonlinear terms. Applying the infnite 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 (frst) terms are taken into account. A(0, μ), D(0, μ), F(0, μ), H(0, μ), A(0, μ), F(0, μ), 44) and the second terms in the series are Similarly, the remaining terms are Hence, the general series solution will be Complexity Tat is, Similarly, the other terms are Te 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.

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 frst term of the series solution of the equation is 12 Complexity Similarly, the next term of the series solution is given as follows: , , , , , , , .
Proceeding the steps in the similar manner, we can easily fnd the third, fourth, ffth, and the remaining terms of the series solution. Hence, we obtain the following equation: Z η Γ(η + 1)

Conclusion
In this work, a novel Caputo type fuzzy fractional diferential equation is used to build a coronavirus SIDARTHE epidemic model system. Using the Caputo sense derivative, an efective method for solving a fuzzy fractional diferential equation using Laplace transforms and their inverses is presented, which can efectively make the original problem simpler to solve numerically. Te efectiveness and applicability of the suggested method are demonstrated by the experimental (numerical) results in a unique way. In addition, fxed point theory is used to prove the existence of a unique solution, and the fuzzy Laplace transform method is used to fnd the semianalytical solution of the model.

Data Availability
All the data and material used in this research are included in the paper.

Conflicts of Interest
Te authors declare that they have no conficts of interest.