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

Authors: P. Chellamani; K. Julietraja; Ammar Alsinai; Hanan Ahmed
Journal: Complexity
 (2022)

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## 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.


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## Body

## 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 [11–15]. 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 [16–19]. Some recent studies have focused on this area of research [20–28].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 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.Figure1 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.Figure 1
Daily new confirmed COVID-19 cases per million people.Figure 2
Current COVID-19 cases worldwide.


## 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.(1)S′ℏ=−SℏαIℏ+βDℏ+γAℏ+δRℏ,I′ℏ=SℏαIℏ+βDℏ+γAℏ+δRℏ−ε+ζ+λIℏ,D′ℏ=εIℏ−η+νDℏ,A′ℏ=ζIℏ−θ+ρ+κAℏ,R′ℏ=ηDℏ+θAℏ−ν+ξRℏ,T′ℏ=ρAℏ+νRℏ−σ+τTℏ,H′ℏ=λIℏ+νDℏ+κAℏ+ξRℏ+σTℏ,E′ℏ=τTℏ.Figure 3
Graphical representation of the SIDARTHE compartmental.Table 1
SIDARTHE epidemic model transmission rates.
ParametersDescriptionValuesαInfection rate as a result of contact between a susceptible person and an infected case0.57βInfection rate as a result of contact between a susceptible person and a diagnosed case0.0114γInfection rate as a result of contact between a susceptible and an ailing case0.456δInfection rate as a result of contact between a susceptible case and a recognised case0.0114εRate of symptomless infection patients being found0.171ζThe probability that a person with the disease is ignorant of their infection0.1254θRate of detection of infected cases with symptoms0.3705ηThe probability that an affected person knows they have the disease0.1254ρUnidentified infected case causes life-threatening symptoms0.0171υThe confirmed infected case causes life-threatening symptoms0.0274τRate of death0.01λ, κ, σ, ν, ξHealing rates for the five stages of infected cases0.0342 (λ and ν)0.0342 (λ and ν)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 [32–36]). After that, fuzzy FODEs were expanded from FODEs [37–39].In order to establish the uniqueness and existence theory of solutions, several academics have explored FODEs and fuzzy integral equations [40–45]. 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 [46–51]. 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 Dℏγfℏ=dγ/dxγfℏ in (1). That is,(2)DℏγAsℏ=−AsℏαBsℏ+βCsℏ+γDsℏ+δEsℏ,DℏγBsℏ=Asℏ+αBsℏ+βCsℏ+γDsℏ+δEsℏ−ε+ζ+λBsℏ,DℏγCsℏ=εBsℏ−η+νCsℏ,DℏγDsℏ=ζBsℏ−θ+ρ+κDsℏ,DℏγEsℏ=ηCsℏ+θDsℏ−γ+ξEsℏ,Dℏγℱsℏ=ρDsℏ+νEsℏ+σ+τFsℏ,DℏγGsℏ=λBsℏ+ρCsℏ+κDsℏ+ξEsℏ+σℱsℏ,Dℏγℋsℏ=τℱsℏ,where Ssℏ-Susceptible, Isℏ-Infection, Dsℏ-Diagnosed, Asℏ-Ailing, Rsℏ-Recognized, Tsℏ-Threatened, Hsℏ-Healed, and Esℏ-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:(3)DℏγAsℏ=−Asℏαs¯ℏℬsℏ+βs¯ℏCsℏ+γs¯ℏDsℏ+δs¯ℏEsℏ,Dℏγℬsℏ=Asℏ+αs¯ℏℬsℏ+βs¯ℏCsℏ+γs¯ℏDsℏ+δs¯ℏEsℏ−εs¯ℏ+ζs¯ℏ+λs¯ℏℬsℏ,DℏγCsℏ=εs¯ℏℬsℏ−ηs¯ℏ+ρs¯ℏCsℏ,DℏγDsℏ=ζs¯ℏℬsℏ−ϑs¯ℏ+ρs¯ℏ+κs¯ℏDsℏ,DℏγEsℏ=ηs¯ℏCsℏ+ϑs¯ℏDsℏ−γs¯ℏ+ξs¯ℏEsℏ,Dℏγℱsℏ=ρs¯ℏDsℏ+γs¯ℏsEℏ+σs¯ℏ+τ¯ℏℱsℏ,DℏγGsℏ=λs¯ℏBsℏ+ρs¯ℏCsℏ+κs¯ℏDsℏ+ξs¯ℏEsℏ+σs¯ℏℱsℏ,Dℏγℋsℏ=τ¯ℏℱsℏ.For0<γ≤1 and μ∈0,1, the fuzzy initial condition is given by the following equation:(4)A˜0,μ=A−0,μ,A¯0,μ,ℬ˜0,μ=ℬ−0,μ,ℬ¯0,μ,C˜0,μ=C−0,μ,C¯0,μ,D˜0,μ=D−0,μ,D¯0,μ,E˜0,μ=E−0,μ,E¯0,μ,ℱ˜0,μ=ℱ−0,μ,ℱ¯0,μ,G˜0,μ=G−0,μ,G¯0,μ,ℋ˜0μ=ℋ−0,μ,ℋ¯0,μ.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]).
Letρ−ϑ,ρ¯ϑ be the parametric form of a fuzzy number (FN) ρ, where 0≤ϑ≤1, and meets the following criteria:(1)
ρ¯ϑ is the left continuous, decreasing over (0, 1] and bounded, whereas the right continuous is at 0(2)
ρ¯ϑ is right continuous, decreasing over [0,1] and bounded, whereas the left continuous is at 0(3)
ρ¯ϑ≤ρ¯ϑ
Also, ifρϑ¯=ρϑ¯=0, then ϑ is called a crisp number.Definition 2 (see [46, 49]).
LetΦ:0,b⟶ℝ be a continuous fuzzy function. Then, the fuzzy fractional integral (FFI) according to t is given by the following equation:(5)IκΦℏ=1Γℏ∫0tt−ζκ−1Φζdζ,κ,ζ∈0,∞.
Furthermore, ifΦ∈CF0,b∩LF0,b, then FFI is defined as follows:(6)IκΦℏp=IκΦ¯pℏ,IκΦ¯pℏ,0≤p≤1,where CF0,b and LF0,b are the spaces of fuzzy Lebesgue integrable functions and fuzzy continuous correspondingly. Here,(7)IκΦ¯pℏ=1Γℏ∫0tt−ζκ−1Φ¯pℏdζ,κ,ζ∈0,∞,IκΦ¯pℏ=1Γℏ∫0tt−ζκ−1Φ¯pℏdζ,κ,ζ∈0,∞.Definition 3 (see [55, 56]).
IfΦ∈CF0,b∩LF0,b be a fuzzy function, Φ=Φ¯pℏ,Φ¯pℏ and t∈0,b, then the FF Caputo’s derivative is defined as follows:(8)DβΦt0p=DβΦ¯pt0,DβΦ¯pt0,0≤β,p≤1,where(9)DβΦ¯pt0=1Γn−β∫0tt−ζn−β−1dndζnΦ¯pζdζt=t0,DβΦ¯pt0=1Γn−β∫0tt−ζn−β−1dndζnΦ¯pζdζt=t0,whenever the right-side integrals converge and n=β.Definition 4.
(Laplace transform (LT)). LetF is fuzzy real number, α∈0,1 and G>0. Then, the FF differential is(10)Dℏαxℏ=ft,xℏ,ℏ∈0,G,x0∈F,where f0,G×R⟶R and x0∈R. If α=1, then it reduces to a fractional differential equation.Theorem 1 (see [49]).
IfΦ∈CF0,b∩LF0,b, then the fuzzy LT of fractional derivative in Caputo’s sense is(11)LDβΦℏp=sβLΦℏ−sβ−1Φ0,where 0≤p≤1, and 0<β≤1.Theorem 2 (Arzela-ascoli theorem).
The sequencefnn∈ℕ of continuous function defined on an interval I=a,b is said to be an uniformly bounded sequence, if ∃ a number M∋(12)fn℘≤M,for each fn in the sequence and for every x∈a,b (Here, M must not depend on n or x).The sequencefnn∈ℕ is an uniformly equicontinuous if for every ϵ>0∃δ>0 such that(13)fn℘−fny<ϵ,whenever ℘−y<δ∀ functions fn in the sequence (Here, δ may be depend on ϵ, but not ℘, y or n).Theorem 3 (Schauder Fixed Point Theorem).
AssumingΛ:M⟶M is compact and let M be a convex, closed subset of a bounded Banach space X. Then, Λ has atleast one fixed point in M.


## 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 [32–36]). After that, fuzzy FODEs were expanded from FODEs [37–39].In order to establish the uniqueness and existence theory of solutions, several academics have explored FODEs and fuzzy integral equations [40–45]. 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 [46–51]. 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 Dℏγfℏ=dγ/dxγfℏ in (1). That is,(2)DℏγAsℏ=−AsℏαBsℏ+βCsℏ+γDsℏ+δEsℏ,DℏγBsℏ=Asℏ+αBsℏ+βCsℏ+γDsℏ+δEsℏ−ε+ζ+λBsℏ,DℏγCsℏ=εBsℏ−η+νCsℏ,DℏγDsℏ=ζBsℏ−θ+ρ+κDsℏ,DℏγEsℏ=ηCsℏ+θDsℏ−γ+ξEsℏ,Dℏγℱsℏ=ρDsℏ+νEsℏ+σ+τFsℏ,DℏγGsℏ=λBsℏ+ρCsℏ+κDsℏ+ξEsℏ+σℱsℏ,Dℏγℋsℏ=τℱsℏ,where Ssℏ-Susceptible, Isℏ-Infection, Dsℏ-Diagnosed, Asℏ-Ailing, Rsℏ-Recognized, Tsℏ-Threatened, Hsℏ-Healed, and Esℏ-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:(3)DℏγAsℏ=−Asℏαs¯ℏℬsℏ+βs¯ℏCsℏ+γs¯ℏDsℏ+δs¯ℏEsℏ,Dℏγℬsℏ=Asℏ+αs¯ℏℬsℏ+βs¯ℏCsℏ+γs¯ℏDsℏ+δs¯ℏEsℏ−εs¯ℏ+ζs¯ℏ+λs¯ℏℬsℏ,DℏγCsℏ=εs¯ℏℬsℏ−ηs¯ℏ+ρs¯ℏCsℏ,DℏγDsℏ=ζs¯ℏℬsℏ−ϑs¯ℏ+ρs¯ℏ+κs¯ℏDsℏ,DℏγEsℏ=ηs¯ℏCsℏ+ϑs¯ℏDsℏ−γs¯ℏ+ξs¯ℏEsℏ,Dℏγℱsℏ=ρs¯ℏDsℏ+γs¯ℏsEℏ+σs¯ℏ+τ¯ℏℱsℏ,DℏγGsℏ=λs¯ℏBsℏ+ρs¯ℏCsℏ+κs¯ℏDsℏ+ξs¯ℏEsℏ+σs¯ℏℱsℏ,Dℏγℋsℏ=τ¯ℏℱsℏ.For0<γ≤1 and μ∈0,1, the fuzzy initial condition is given by the following equation:(4)A˜0,μ=A−0,μ,A¯0,μ,ℬ˜0,μ=ℬ−0,μ,ℬ¯0,μ,C˜0,μ=C−0,μ,C¯0,μ,D˜0,μ=D−0,μ,D¯0,μ,E˜0,μ=E−0,μ,E¯0,μ,ℱ˜0,μ=ℱ−0,μ,ℱ¯0,μ,G˜0,μ=G−0,μ,G¯0,μ,ℋ˜0μ=ℋ−0,μ,ℋ¯0,μ.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]).
Letρ−ϑ,ρ¯ϑ be the parametric form of a fuzzy number (FN) ρ, where 0≤ϑ≤1, and meets the following criteria:(1)
ρ¯ϑ is the left continuous, decreasing over (0, 1] and bounded, whereas the right continuous is at 0(2)
ρ¯ϑ is right continuous, decreasing over [0,1] and bounded, whereas the left continuous is at 0(3)
ρ¯ϑ≤ρ¯ϑ
Also, ifρϑ¯=ρϑ¯=0, then ϑ is called a crisp number.Definition 2 (see [46, 49]).
LetΦ:0,b⟶ℝ be a continuous fuzzy function. Then, the fuzzy fractional integral (FFI) according to t is given by the following equation:(5)IκΦℏ=1Γℏ∫0tt−ζκ−1Φζdζ,κ,ζ∈0,∞.
Furthermore, ifΦ∈CF0,b∩LF0,b, then FFI is defined as follows:(6)IκΦℏp=IκΦ¯pℏ,IκΦ¯pℏ,0≤p≤1,where CF0,b and LF0,b are the spaces of fuzzy Lebesgue integrable functions and fuzzy continuous correspondingly. Here,(7)IκΦ¯pℏ=1Γℏ∫0tt−ζκ−1Φ¯pℏdζ,κ,ζ∈0,∞,IκΦ¯pℏ=1Γℏ∫0tt−ζκ−1Φ¯pℏdζ,κ,ζ∈0,∞.Definition 3 (see [55, 56]).
IfΦ∈CF0,b∩LF0,b be a fuzzy function, Φ=Φ¯pℏ,Φ¯pℏ and t∈0,b, then the FF Caputo’s derivative is defined as follows:(8)DβΦt0p=DβΦ¯pt0,DβΦ¯pt0,0≤β,p≤1,where(9)DβΦ¯pt0=1Γn−β∫0tt−ζn−β−1dndζnΦ¯pζdζt=t0,DβΦ¯pt0=1Γn−β∫0tt−ζn−β−1dndζnΦ¯pζdζt=t0,whenever the right-side integrals converge and n=β.Definition 4.
(Laplace transform (LT)). LetF is fuzzy real number, α∈0,1 and G>0. Then, the FF differential is(10)Dℏαxℏ=ft,xℏ,ℏ∈0,G,x0∈F,where f0,G×R⟶R and x0∈R. If α=1, then it reduces to a fractional differential equation.Theorem 1 (see [49]).
IfΦ∈CF0,b∩LF0,b, then the fuzzy LT of fractional derivative in Caputo’s sense is(11)LDβΦℏp=sβLΦℏ−sβ−1Φ0,where 0≤p≤1, and 0<β≤1.Theorem 2 (Arzela-ascoli theorem).
The sequencefnn∈ℕ of continuous function defined on an interval I=a,b is said to be an uniformly bounded sequence, if ∃ a number M∋(12)fn℘≤M,for each fn in the sequence and for every x∈a,b (Here, M must not depend on n or x).The sequencefnn∈ℕ is an uniformly equicontinuous if for every ϵ>0∃δ>0 such that(13)fn℘−fny<ϵ,whenever ℘−y<δ∀ functions fn in the sequence (Here, δ may be depend on ϵ, but not ℘, y or n).Theorem 3 (Schauder Fixed Point Theorem).
AssumingΛ:M⟶M is compact and let M be a convex, closed subset of a bounded Banach space X. Then, Λ has atleast one fixed point in M.


## 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:(14)ZMℏ=−Asℏαs¯ℏℬsℏ+βs¯ℏCsℏ+γs¯ℏDsℏ+δs¯ℏΕsℏ,YMℏ=Asℏ+αs¯ℏℬsℏ+βs¯ℏCsℏ+γs¯ℏDsℏ+δs¯ℏΕsℏ−εs¯ℏ+ζs¯ℏ+λs¯ℏℬsℏ,XMℏ=εs¯ℏℬsℏ−ηs¯ℏ+ρs¯ℏCsℏ,WMℏ=ζs¯ℏℬsℏ−ϑs¯ℏ+ρs¯ℏ+κs¯ℏDsℏ,VMℏ=ηs¯ℏCsℏ+ϑs¯ℏDsℏ−γs¯ℏ+ξs¯ℏΕsℏ,UMℏ=ρs¯ℏDsℏ+γs¯ℏsΕℏ+σs¯ℏ+τ¯ℏℱsℏ,QMℏ=λs¯ℏBsℏ+ρs¯ℏCsℏ+κs¯ℏDsℏ+ξs¯ℏΕsℏ+σs¯ℏℱsℏ,QMℏ=τ¯ℏℱsℏ.LetMℏ=ℏ,JsℏJ=A,ℬ,C,D,E,ℱ,G,ℋ and 0<γ≤1, and then substituting (4) in (3), the given model (3) can be written as follows:(15)DtγAsℏ=ZMℏ,Dtγℬsℏ=YMℏ,DtγCsℏ=XMℏ,DtγDsℏ=WMℏ,DtγΕsℏ=VMℏ,Dtγℱsℏ=UMℏ,DtγGsℏ=QMℏ,Dtγℋsℏ=QMℏ.Using the initial conditions along with FFIIr as defined in Definition 2, we get the first term as follows:(16)Asℏ=A¯0,μ+1Γγ∫0ℏℏ−℘γ−1Z℘,As℘d℘,Asℏ=A¯0,μ+1Γγ∫0ℏℏ−℘γ−1Z℘,As℘d℘.Which is the same as follows:(17)Asℏ=A¯0,μ+1Γγ∫0ℏℏ−℘γ−1Z℘,As℘d℘.Next, the second term will be(18)ℬsℏ=ℬ¯0,μ+1Γγ∫0ℏℏ−℘γ−1Y℘,As℘d℘,ℬsℏ=ℬ¯0,μ+1Γγ∫0ℏℏ−℘γ−1Y℘,As℘d℘,which implies(19)ℬsℏ=ℬ¯0,μ+1Γγ∫0ℏℏ−℘γ−1Y℘,As℘d℘.The third term will be(20)Csℏ=C¯0,μ+1Γγ∫0ℏℏ−℘γ−1X℘,As℘d℘,Csℏ=C¯0,μ+1Γγ∫0ℏℏ−℘γ−1X℘,As℘d℘,which yields(21)Csℏ=C¯0,μ+1Γγ∫0ℏℏ−℘γ−1X℘,As℘d℘.Similarly, the fourth term is(22)Dsℏ=D¯0,μ+1Γγ∫0ℏℏ−℘γ−1W℘,As℘d℘,Dsℏ=D¯0,μ+1Γγ∫0ℏℏ−℘γ−1W℘,As℘d℘,which is the same as follows:(23)Dsℏ=D̃0,μ+1Γγ∫0ℏℏ−℘γ−1W℘,As℘d℘.Hence the other terms will be(24)Εsℏ=Ε̃0,μ+1Γγ∫0ℏℏ−℘γ−1V℘,As℘d℘,(25)ℱsℏ=ℱ̃0,μ+1Γγ∫0ℏℏ−℘γ−1U℘,As℘d℘,(26)Gsℏ=G̃0,μ+1Γγ∫0ℏℏ−℘γ−1Q℘,As℘d℘,(27)ℋsℏ=ℋ̃0,μ+1Γγ∫0ℏℏ−℘γ−1Q℘,As℘d℘,where As℘=JsℏJ=A,ℬ,C,D,E,ℱ,G,ℋ.LetB=B1×B2×B3×B4×B5×B6×B7×B8 be a Banach space under the fuzzy norm. Then,(28)JsℏJ=A,ℬ,C,D,Ε,ℱ,G,ℋ=maxℏ∈0,TAsℏ+ℬsℏ+Csℏ+Dsℏ+Εsℏ+ℱsℏ+Gsℏ+ℋsℏ|.Therefore, (26) and (27) can be written as follows:(29)Ls̃ℏ=L̃0,μ+1Γγ∫0ℏℏ−Δγ−1⊖Δ,L̃sΔdΔ,where(30)Ls¯ℏ=Asℏ,ℬsℏ,Csℏ,Dsℏ,Esℏ,ℱsℏ,Gsℏ,ℋsℏ,L̃0,μ=Ã0,μ,ℬ̃0,μ,C̃0,μ,D̃0,μ,Ẽ0,μ,ℱ̃0,μ,G̃0,μ,ℋ̃0,μ,⊖t,L̃κℏ=Zt,As℘,Yt,As℘,Xt,As℘,Vt,As℘,Ut,As℘,Qt,As℘,Qt,As℘.Several assumptions have been made on the nonlinear function⊖:B⟶B as follows:(1)
(C-1)∃ constant ℏL>0∋ for each L̃s1ℏ, L̃s2ℏ∈B(31)⊖t,L̃s1ℏ−⊖t,P̃s2k≤ℏLL̃s1ℏ−L̃s2ℏ.(2)
(C-2)∃ constant, ML>0 and NL>0∋(32)⊖t,L̃sℏ≤MLL̃sℏ+NL.Theorem 4.
Model (3) has at least one solution under Assumption (C-1).LetM=L̃sℏ∈B:L̃sℏ≤r be a subset of B which is a convex and closed fuzzy set. The mapping ϵ:M⟶M is defined as follows:(33)εL̃sℏ=L̃0,μ+1Γγ∫0ℏℏ−Δγ−1⊖Δ,L̃sΔdΔ.For anyL̃sℏ∈M, we have the following equation:(34)εLS¯ℏ=maxℏ∈0,TL˜0,μ+1Γγ∫0ℏℏ−Δ−1HΔ,L̃sΔdΔ,≤L˜0,μ+1Γγ∫0ℏℏ−Δ−1⊖Δ,L̃sΔdΔ,≤L˜0,μ+1Γγ∫0ℏℏ−Δ−1ILL̃sℏ+JLdΔ,≤L˜0,μ+1ΓγℏγΓγ+1ILL̃sℏ+JL.Since the operatorε 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 ϕ1,ϕ2∈0,T and ϕ1<ϕ2, then(35)εLS¯ℏϕ2−εLS¯ℏϕ1=1Γγ∫0ϕ2ϕ2−Δγ−1HΔ,L̃sdΔ−1Γγ∫0ϕ1ϕ1−Δγ−1⊖Δ,L̃sΔdΔ,≤ϕ2γ−ϕ1γILP̃sℏ+JLΓγ+1.According to the previous inequality, the right side goes to zeroϕ2⟶ϕ1. Hence,(36)εL̃sℏϕ2−εL̃sℏϕ1⟶0asϕ2⟶ϕ1.As a result, the operatorϵ is equicontinuous. The operator ϵ is entirely continuous according to Theorem 4, and as proved earlier, ϵ 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 κγsL<Γγ+1.LetL̃s1ℏ,L̃s2ℏ∈B, then(37)εL̃s1ℏ−εL̃s2ℏ=maxκ∈0,T1Γγ∫0ℏℏ−Δ−1⊖Δ,L̃s1ΔdΔ−1Γγ∫0ℏℏ−Δ−1⊖Δ,L̃s2ΔdΔ,≤κγΓγ+1sLL̃s1ℏ−L̃s2ℏ.Hence,ϵ 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(38)LDℏγAsℏ=LZMℏ,LDℏγℬsℏ=LYMℏ,LDℏγCsℏ=LXMℏ,LDℏγDsℏ=LWMℏ,LDℏγEsℏ=LVMℏ,LDℏγℱsℏ=LUMℏ,LDℏγGsℏ=LQMℏ,LDℏγℋsℏ=LQMℏ.Then, we set the following equation:(39)xγLAsℏ=xγ−1A˜0,μ+LZMℏ,xγLℬsℏ=xγ−1ℬ˜0,μ+LYMℏ,xγLCsℏ=xγ−1C˜0,μ+LXMℏ,xγLDsℏ=xγ−1D˜0,μ+LWMℏ,xγLEsℏ=xγ−1E˜0,μ+LVMℏ,xγLℱsℏ=xγ−1ℱ˜0,μ+LUMℏ,xγLGsℏ=xγ−1G˜0,μ+LQMℏ,xγLℋsℏ=xγ−1ℋ˜0,μ+LQMℏ.After making some modifications, we set the following equation:(40)xγLAsℏ=1xA˜0,μ+1xγLZMℏ,xγLBsℏ=1xxγ−1ℬ˜0,μ+1xγLYMℏ,xγLCsℏ=1xxγ−1C˜0,μ+1xγLXMℏ,xγLDsℏ=1xxγ−1D˜0,μ+1xγLWMℏ,xγLEsℏ=1xxγ−1Ε˜0,μ+1xγLVMℏ,xγLℱsℏ=1xxγ−1ℱ˜0,μ+1xγLUMℏ,xγLGsℏ=1xxγ−1G˜0,μ+1xγLQMℏ,xγLℋsℏ=1xxγ−1ℋ˜0,μ+1xγLQMℏ.The infinite series solution is given by the following equation:(41)Asℏ=∑ℓ=0∞Asℓℏ,ℬsℏ=∑ℓ=0∞ℬsℓℏ,Csℏ=∑ℓ=0∞Csℓℏ,Dsℏ=∑ℓ=0∞Dsℓℏ,Esℏ=∑ℓ=0∞Esℓℏ,ℱsℏ=∑ℓ=0∞ℱsℓℏ,Gsℏ=∑ℓ=0∞Gsℓℏ,ℋsℏ=∑ℓ=0∞ℋsℓℏ,Asℏℬsℏ=∑n=0∞Z1,ℓ,ℬsℏAsℏ=∑n=0∞Z2,ℓ,where Z1,ℓ,Z2,ℓ are the Adomian polynomials, representing nonlinear terms. Applying the infinite series in equation [24], the fuzzy LT equation will become the following equation:(42)L∑ℓ=0∞Asℓℏ=1xÃ0,μ+1xγLZMℏ,L∑ℓ=0∞ℬsℓℏ=1xℬ̃0,μ+1xγLYMℏ,L∑ℓ=0∞Csℓℏ=1xC̃0,μ+1xγLXMℏ,L∑ℓ=0∞Dsℓℏ=1xD̃0,μ+1xγLWMℏ,L∑ℓ=0∞Esℓℏ=1xẼ0,μ+1xγLVMℏ,L∑ℓ=0∞ℱsℓℏ=1xℱ̃0,μ+1xγLUMℏ,L∑ℓ=0∞Gsℓℏ=1xG̃0,μ+1xγLQMℏ,L∑ℓ=0∞ℋsℓℏ=1xℋ̃0,μ+1xγLQMℏ.Now, applying the inverse LT, the above equation becomes the following equation:(43)∑ℓ=0∞Asℓℏ=A˜0,μ+L−11xγLZMℏ,∑ℓ=0∞ℬsℓℏ=ℬ˜0,μ+L−11xγLYMℏ,∑ℓ=0∞Csℓℏ=C˜0,μ+L−11xγLXMℏ,∑ℓ=0∞Dsℓℏ=D˜0,μ+L−11xγLWMℏ,∑ℓ=0∞Esℓℏ=Ε˜0,μ+L−11xγLVMℏ,∑ℓ=0∞ℱsℓℏ=ℱ˜0,μ+L−11xγLUMℏ,∑ℓ=0∞Gsℓℏ=G˜0,μ+L−11xγLQMℏ,∑ℓ=0∞ℋsℓℏ=ℋ˜0,μ+L−11xγLQMℏ.When comparing the terms on both sides, the series’ initial (first) terms are taken into account.(44)A¯s0ℏ=A¯0,μ,ℬ¯s0ℏ=ℬ¯0,μ,C¯s0ℏ=C¯0,μ,D¯s0ℏ=D¯0,μ,Ε¯s0ℏ=Ε¯0,μ,ℱ¯s0ℏ=ℱ¯0,μ,G¯s0ℏ=G¯0,μ,ℋ¯s0ℏ=ℋ¯0,μ,A¯s0ℏ=A¯0,μ,ℬ¯s0ℏ=ℬ¯0,μ,C¯s0ℏ=C¯0,μ,D¯s0ℏ=D¯0,μ,Ε¯s0ℏ=Ε¯0,μ,ℱ¯s0ℏ=ℱ¯0,μ,G¯s0ℏ=G¯0,μ,ℋ¯s0ℏ=ℋ¯0,μ,and the second terms in the series are(45)A¯s1ℏ=L−11xνL−A¯s0α¯ℬ¯s0ℏ+β¯C¯s0ℏ+γ¯D¯s0ℏ+δ¯E¯s0ℏ,A¯s1ℏ=L−11xνL−A¯s0α¯ℬ¯s0ℏ+β¯C¯s0ℏ+γ¯D¯s0ℏ+δ¯E¯s0ℏ,ℬ¯s1ℏ=L−11xνL−A¯s0α¯ℬ¯s0ℏ+β¯C¯s0ℏ+γ¯D¯s0ℏ+δ¯E¯s0ℏ,ℬ¯s1ℏ=L−11xνL−A¯s0α¯ℬ¯s0ℏ+β¯C¯s0ℏ+γ¯D¯s0ℏ+δ¯E¯s0ℏ.Similarly, the remaining terms areC¯s1ℏC¯s1ℏ, D¯s1ℏD¯s1ℏ, E¯s1ℏE¯s1ℏ, ℱ¯s1ℏℱ¯s1ℏ, G¯s1ℏG¯s1ℏ, and ℋ¯s1ℏℋ¯s1ℏ.Hence, the general series solution will be(46)A¯sℏ=∑r=0∞A¯srℏ,A¯sℏ=∑r=0∞A¯srℏ.That is,(47)Ãsℏ=∑r=0∞Ãsrℏ.Similarly, the other terms are(48)ℬ¯sℏ=∑r=0∞ℬ¯srℏ,ℬ¯sℏ=∑r=0∞ℬ¯srℏ,C¯sℏ=∑r=0∞C¯srℏ,C¯sℏ=∑r=0∞C¯srℏ,D¯sℏ=∑r=0∞D¯srℏ,D¯sℏ=∑r=0∞D¯srℏ,E¯sℏ=∑r=0∞E¯srℏ,E¯sℏ=∑r=0∞E¯srℏ,ℱ¯sℏ=∑r=0∞ℱ¯srℏ,ℱ¯sℏ=∑r=0∞ℱ¯srℏ,G¯sℏ=∑r=0∞G¯srℏ,G¯sℏ=∑r=0∞G¯srℏ,ℋ¯sℏ=∑r=0∞ℋ¯srℏ,ℋ¯sℏ=∑r=0∞ℋ¯srℏ.The above-given equations are written as follows:(49)A˜sℏ=∑r=0∞A˜srℏ,ℬ˜sℏ=∑r=0∞ℬ˜srℏ,C˜sℏ=∑r=0∞C˜srℏ,D˜sℏ=∑r=0∞D˜srℏ,E˜sℏ=∑r=0∞E˜srℏ,ℱ̃sℏ=∑r=0∞ℱ̃srℏ,G̃sℏ=∑r=0∞G̃srℏ,ℋ̃sℏ=∑r=0∞ℋ̃srℏ.Using the above-given equations, the spread of the virus can be studied by applying the real data.


## 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:(14)ZMℏ=−Asℏαs¯ℏℬsℏ+βs¯ℏCsℏ+γs¯ℏDsℏ+δs¯ℏΕsℏ,YMℏ=Asℏ+αs¯ℏℬsℏ+βs¯ℏCsℏ+γs¯ℏDsℏ+δs¯ℏΕsℏ−εs¯ℏ+ζs¯ℏ+λs¯ℏℬsℏ,XMℏ=εs¯ℏℬsℏ−ηs¯ℏ+ρs¯ℏCsℏ,WMℏ=ζs¯ℏℬsℏ−ϑs¯ℏ+ρs¯ℏ+κs¯ℏDsℏ,VMℏ=ηs¯ℏCsℏ+ϑs¯ℏDsℏ−γs¯ℏ+ξs¯ℏΕsℏ,UMℏ=ρs¯ℏDsℏ+γs¯ℏsΕℏ+σs¯ℏ+τ¯ℏℱsℏ,QMℏ=λs¯ℏBsℏ+ρs¯ℏCsℏ+κs¯ℏDsℏ+ξs¯ℏΕsℏ+σs¯ℏℱsℏ,QMℏ=τ¯ℏℱsℏ.LetMℏ=ℏ,JsℏJ=A,ℬ,C,D,E,ℱ,G,ℋ and 0<γ≤1, and then substituting (4) in (3), the given model (3) can be written as follows:(15)DtγAsℏ=ZMℏ,Dtγℬsℏ=YMℏ,DtγCsℏ=XMℏ,DtγDsℏ=WMℏ,DtγΕsℏ=VMℏ,Dtγℱsℏ=UMℏ,DtγGsℏ=QMℏ,Dtγℋsℏ=QMℏ.Using the initial conditions along with FFIIr as defined in Definition 2, we get the first term as follows:(16)Asℏ=A¯0,μ+1Γγ∫0ℏℏ−℘γ−1Z℘,As℘d℘,Asℏ=A¯0,μ+1Γγ∫0ℏℏ−℘γ−1Z℘,As℘d℘.Which is the same as follows:(17)Asℏ=A¯0,μ+1Γγ∫0ℏℏ−℘γ−1Z℘,As℘d℘.Next, the second term will be(18)ℬsℏ=ℬ¯0,μ+1Γγ∫0ℏℏ−℘γ−1Y℘,As℘d℘,ℬsℏ=ℬ¯0,μ+1Γγ∫0ℏℏ−℘γ−1Y℘,As℘d℘,which implies(19)ℬsℏ=ℬ¯0,μ+1Γγ∫0ℏℏ−℘γ−1Y℘,As℘d℘.The third term will be(20)Csℏ=C¯0,μ+1Γγ∫0ℏℏ−℘γ−1X℘,As℘d℘,Csℏ=C¯0,μ+1Γγ∫0ℏℏ−℘γ−1X℘,As℘d℘,which yields(21)Csℏ=C¯0,μ+1Γγ∫0ℏℏ−℘γ−1X℘,As℘d℘.Similarly, the fourth term is(22)Dsℏ=D¯0,μ+1Γγ∫0ℏℏ−℘γ−1W℘,As℘d℘,Dsℏ=D¯0,μ+1Γγ∫0ℏℏ−℘γ−1W℘,As℘d℘,which is the same as follows:(23)Dsℏ=D̃0,μ+1Γγ∫0ℏℏ−℘γ−1W℘,As℘d℘.Hence the other terms will be(24)Εsℏ=Ε̃0,μ+1Γγ∫0ℏℏ−℘γ−1V℘,As℘d℘,(25)ℱsℏ=ℱ̃0,μ+1Γγ∫0ℏℏ−℘γ−1U℘,As℘d℘,(26)Gsℏ=G̃0,μ+1Γγ∫0ℏℏ−℘γ−1Q℘,As℘d℘,(27)ℋsℏ=ℋ̃0,μ+1Γγ∫0ℏℏ−℘γ−1Q℘,As℘d℘,where As℘=JsℏJ=A,ℬ,C,D,E,ℱ,G,ℋ.LetB=B1×B2×B3×B4×B5×B6×B7×B8 be a Banach space under the fuzzy norm. Then,(28)JsℏJ=A,ℬ,C,D,Ε,ℱ,G,ℋ=maxℏ∈0,TAsℏ+ℬsℏ+Csℏ+Dsℏ+Εsℏ+ℱsℏ+Gsℏ+ℋsℏ|.Therefore, (26) and (27) can be written as follows:(29)Ls̃ℏ=L̃0,μ+1Γγ∫0ℏℏ−Δγ−1⊖Δ,L̃sΔdΔ,where(30)Ls¯ℏ=Asℏ,ℬsℏ,Csℏ,Dsℏ,Esℏ,ℱsℏ,Gsℏ,ℋsℏ,L̃0,μ=Ã0,μ,ℬ̃0,μ,C̃0,μ,D̃0,μ,Ẽ0,μ,ℱ̃0,μ,G̃0,μ,ℋ̃0,μ,⊖t,L̃κℏ=Zt,As℘,Yt,As℘,Xt,As℘,Vt,As℘,Ut,As℘,Qt,As℘,Qt,As℘.Several assumptions have been made on the nonlinear function⊖:B⟶B as follows:(1)
(C-1)∃ constant ℏL>0∋ for each L̃s1ℏ, L̃s2ℏ∈B(31)⊖t,L̃s1ℏ−⊖t,P̃s2k≤ℏLL̃s1ℏ−L̃s2ℏ.(2)
(C-2)∃ constant, ML>0 and NL>0∋(32)⊖t,L̃sℏ≤MLL̃sℏ+NL.Theorem 4.
Model (3) has at least one solution under Assumption (C-1).LetM=L̃sℏ∈B:L̃sℏ≤r be a subset of B which is a convex and closed fuzzy set. The mapping ϵ:M⟶M is defined as follows:(33)εL̃sℏ=L̃0,μ+1Γγ∫0ℏℏ−Δγ−1⊖Δ,L̃sΔdΔ.For anyL̃sℏ∈M, we have the following equation:(34)εLS¯ℏ=maxℏ∈0,TL˜0,μ+1Γγ∫0ℏℏ−Δ−1HΔ,L̃sΔdΔ,≤L˜0,μ+1Γγ∫0ℏℏ−Δ−1⊖Δ,L̃sΔdΔ,≤L˜0,μ+1Γγ∫0ℏℏ−Δ−1ILL̃sℏ+JLdΔ,≤L˜0,μ+1ΓγℏγΓγ+1ILL̃sℏ+JL.Since the operatorε 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 ϕ1,ϕ2∈0,T and ϕ1<ϕ2, then(35)εLS¯ℏϕ2−εLS¯ℏϕ1=1Γγ∫0ϕ2ϕ2−Δγ−1HΔ,L̃sdΔ−1Γγ∫0ϕ1ϕ1−Δγ−1⊖Δ,L̃sΔdΔ,≤ϕ2γ−ϕ1γILP̃sℏ+JLΓγ+1.According to the previous inequality, the right side goes to zeroϕ2⟶ϕ1. Hence,(36)εL̃sℏϕ2−εL̃sℏϕ1⟶0asϕ2⟶ϕ1.As a result, the operatorϵ is equicontinuous. The operator ϵ is entirely continuous according to Theorem 4, and as proved earlier, ϵ 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 κγsL<Γγ+1.LetL̃s1ℏ,L̃s2ℏ∈B, then(37)εL̃s1ℏ−εL̃s2ℏ=maxκ∈0,T1Γγ∫0ℏℏ−Δ−1⊖Δ,L̃s1ΔdΔ−1Γγ∫0ℏℏ−Δ−1⊖Δ,L̃s2ΔdΔ,≤κγΓγ+1sLL̃s1ℏ−L̃s2ℏ.Hence,ϵ 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(38)LDℏγAsℏ=LZMℏ,LDℏγℬsℏ=LYMℏ,LDℏγCsℏ=LXMℏ,LDℏγDsℏ=LWMℏ,LDℏγEsℏ=LVMℏ,LDℏγℱsℏ=LUMℏ,LDℏγGsℏ=LQMℏ,LDℏγℋsℏ=LQMℏ.Then, we set the following equation:(39)xγLAsℏ=xγ−1A˜0,μ+LZMℏ,xγLℬsℏ=xγ−1ℬ˜0,μ+LYMℏ,xγLCsℏ=xγ−1C˜0,μ+LXMℏ,xγLDsℏ=xγ−1D˜0,μ+LWMℏ,xγLEsℏ=xγ−1E˜0,μ+LVMℏ,xγLℱsℏ=xγ−1ℱ˜0,μ+LUMℏ,xγLGsℏ=xγ−1G˜0,μ+LQMℏ,xγLℋsℏ=xγ−1ℋ˜0,μ+LQMℏ.After making some modifications, we set the following equation:(40)xγLAsℏ=1xA˜0,μ+1xγLZMℏ,xγLBsℏ=1xxγ−1ℬ˜0,μ+1xγLYMℏ,xγLCsℏ=1xxγ−1C˜0,μ+1xγLXMℏ,xγLDsℏ=1xxγ−1D˜0,μ+1xγLWMℏ,xγLEsℏ=1xxγ−1Ε˜0,μ+1xγLVMℏ,xγLℱsℏ=1xxγ−1ℱ˜0,μ+1xγLUMℏ,xγLGsℏ=1xxγ−1G˜0,μ+1xγLQMℏ,xγLℋsℏ=1xxγ−1ℋ˜0,μ+1xγLQMℏ.The infinite series solution is given by the following equation:(41)Asℏ=∑ℓ=0∞Asℓℏ,ℬsℏ=∑ℓ=0∞ℬsℓℏ,Csℏ=∑ℓ=0∞Csℓℏ,Dsℏ=∑ℓ=0∞Dsℓℏ,Esℏ=∑ℓ=0∞Esℓℏ,ℱsℏ=∑ℓ=0∞ℱsℓℏ,Gsℏ=∑ℓ=0∞Gsℓℏ,ℋsℏ=∑ℓ=0∞ℋsℓℏ,Asℏℬsℏ=∑n=0∞Z1,ℓ,ℬsℏAsℏ=∑n=0∞Z2,ℓ,where Z1,ℓ,Z2,ℓ are the Adomian polynomials, representing nonlinear terms. Applying the infinite series in equation [24], the fuzzy LT equation will become the following equation:(42)L∑ℓ=0∞Asℓℏ=1xÃ0,μ+1xγLZMℏ,L∑ℓ=0∞ℬsℓℏ=1xℬ̃0,μ+1xγLYMℏ,L∑ℓ=0∞Csℓℏ=1xC̃0,μ+1xγLXMℏ,L∑ℓ=0∞Dsℓℏ=1xD̃0,μ+1xγLWMℏ,L∑ℓ=0∞Esℓℏ=1xẼ0,μ+1xγLVMℏ,L∑ℓ=0∞ℱsℓℏ=1xℱ̃0,μ+1xγLUMℏ,L∑ℓ=0∞Gsℓℏ=1xG̃0,μ+1xγLQMℏ,L∑ℓ=0∞ℋsℓℏ=1xℋ̃0,μ+1xγLQMℏ.Now, applying the inverse LT, the above equation becomes the following equation:(43)∑ℓ=0∞Asℓℏ=A˜0,μ+L−11xγLZMℏ,∑ℓ=0∞ℬsℓℏ=ℬ˜0,μ+L−11xγLYMℏ,∑ℓ=0∞Csℓℏ=C˜0,μ+L−11xγLXMℏ,∑ℓ=0∞Dsℓℏ=D˜0,μ+L−11xγLWMℏ,∑ℓ=0∞Esℓℏ=Ε˜0,μ+L−11xγLVMℏ,∑ℓ=0∞ℱsℓℏ=ℱ˜0,μ+L−11xγLUMℏ,∑ℓ=0∞Gsℓℏ=G˜0,μ+L−11xγLQMℏ,∑ℓ=0∞ℋsℓℏ=ℋ˜0,μ+L−11xγLQMℏ.When comparing the terms on both sides, the series’ initial (first) terms are taken into account.(44)A¯s0ℏ=A¯0,μ,ℬ¯s0ℏ=ℬ¯0,μ,C¯s0ℏ=C¯0,μ,D¯s0ℏ=D¯0,μ,Ε¯s0ℏ=Ε¯0,μ,ℱ¯s0ℏ=ℱ¯0,μ,G¯s0ℏ=G¯0,μ,ℋ¯s0ℏ=ℋ¯0,μ,A¯s0ℏ=A¯0,μ,ℬ¯s0ℏ=ℬ¯0,μ,C¯s0ℏ=C¯0,μ,D¯s0ℏ=D¯0,μ,Ε¯s0ℏ=Ε¯0,μ,ℱ¯s0ℏ=ℱ¯0,μ,G¯s0ℏ=G¯0,μ,ℋ¯s0ℏ=ℋ¯0,μ,and the second terms in the series are(45)A¯s1ℏ=L−11xνL−A¯s0α¯ℬ¯s0ℏ+β¯C¯s0ℏ+γ¯D¯s0ℏ+δ¯E¯s0ℏ,A¯s1ℏ=L−11xνL−A¯s0α¯ℬ¯s0ℏ+β¯C¯s0ℏ+γ¯D¯s0ℏ+δ¯E¯s0ℏ,ℬ¯s1ℏ=L−11xνL−A¯s0α¯ℬ¯s0ℏ+β¯C¯s0ℏ+γ¯D¯s0ℏ+δ¯E¯s0ℏ,ℬ¯s1ℏ=L−11xνL−A¯s0α¯ℬ¯s0ℏ+β¯C¯s0ℏ+γ¯D¯s0ℏ+δ¯E¯s0ℏ.Similarly, the remaining terms areC¯s1ℏC¯s1ℏ, D¯s1ℏD¯s1ℏ, E¯s1ℏE¯s1ℏ, ℱ¯s1ℏℱ¯s1ℏ, G¯s1ℏG¯s1ℏ, and ℋ¯s1ℏℋ¯s1ℏ.Hence, the general series solution will be(46)A¯sℏ=∑r=0∞A¯srℏ,A¯sℏ=∑r=0∞A¯srℏ.That is,(47)Ãsℏ=∑r=0∞Ãsrℏ.Similarly, the other terms are(48)ℬ¯sℏ=∑r=0∞ℬ¯srℏ,ℬ¯sℏ=∑r=0∞ℬ¯srℏ,C¯sℏ=∑r=0∞C¯srℏ,C¯sℏ=∑r=0∞C¯srℏ,D¯sℏ=∑r=0∞D¯srℏ,D¯sℏ=∑r=0∞D¯srℏ,E¯sℏ=∑r=0∞E¯srℏ,E¯sℏ=∑r=0∞E¯srℏ,ℱ¯sℏ=∑r=0∞ℱ¯srℏ,ℱ¯sℏ=∑r=0∞ℱ¯srℏ,G¯sℏ=∑r=0∞G¯srℏ,G¯sℏ=∑r=0∞G¯srℏ,ℋ¯sℏ=∑r=0∞ℋ¯srℏ,ℋ¯sℏ=∑r=0∞ℋ¯srℏ.The above-given equations are written as follows:(49)A˜sℏ=∑r=0∞A˜srℏ,ℬ˜sℏ=∑r=0∞ℬ˜srℏ,C˜sℏ=∑r=0∞C˜srℏ,D˜sℏ=∑r=0∞D˜srℏ,E˜sℏ=∑r=0∞E˜srℏ,ℱ̃sℏ=∑r=0∞ℱ̃srℏ,G̃sℏ=∑r=0∞G̃srℏ,ℋ̃sℏ=∑r=0∞ℋ̃srℏ.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,(50)A¯0,μ=5μ−1,1−5μ,ℬ¯0,μ=5μ−1,1−5μ,C¯0,μ=5μ−1,1−5μ,D¯0,μ=5μ−1,1−5μ,E¯0,μ=5μ−1,1−5μ,ℱ¯0,μ=5μ−1,1−5μ,G¯0,μ=5μ−1,1−5μ,ℋ¯0,μ=5μ−1,1−5μ.Using the above-given initial conditions, the first term of the series solution of the equation is(51)A¯s0ℏ,μ=5μ−1,A¯s0ℏ,μ=1−5μ,ℬ¯s0ℏ,μ=5μ−1,ℬ¯s0ℏ,μ=1−5μ,C¯s0ℏ,μ=5μ−1,C¯s0ℏ,μ=1−5μ,D¯s0ℏ,μ=5μ−1,D¯s0ℏ,μ=1−5μ,E¯s0ℏ,μ=5μ−1,E¯s0ℏ,μ=1−5μ,ℱ¯s0ℏ,μ=5μ−1,ℱ¯s0ℏ,μ=1−5μ,G¯s0ℏ,μ=5μ−1,G¯s0ℏ,μ=1−5μ,ℋ¯s0ℏ,μ=5μ−1,ℋ¯s0ℏ,μ=1−5μ.Similarly, the next term of the series solution is given as follows:(52)A¯s1ℏ,μ=−5μ−1δ5μ−1ℏ+β5μ−1+γ5μ−1+δ5μ−1ℏηΓη+1,A¯s1ℏ,μ=−1−5μδ1−5μℏ+β1−5μ+γ1−5μ+δ1−5μℏηΓη+1,ℬ¯s1ℏ,μ=5μ−1+α5μ−1ℏ+β5μ−1+γ5μ−1+δ5μ−1−ε+ζ+λ5μ−1ℏηΓη+1,ℬ¯s1ℏ,μ=1−5μ+α1−5μℏ+β1−5μ+γ1−5μ+δ1−5μ−ε+ζ+λ1−5μℏηΓη+1,C¯s1ℏ,μ=ε5μ−1+η+ρ5μ−1ℏηΓη+1,C¯s1ℏ,μ=ε1−5μ+η+ρ1−5μℏηΓη+1,D¯s1ℏ,μ=ζ5μ−1+θ+ρ+κ5μ−1ℏηΓη+1,D¯s1ℏ,μ=ζ1−5μ+θ+ρ+κ1−5μℏηΓη+1,E¯s1ℏ,μ=η5μ−1+θ5μ−1−γ+ξ5μ−1ℏηΓη+1,E¯s1ℏ,μ=η1−5μ+θ1−5μ−γ+ξ1−5μℏηΓη+1,ℱ¯S1ℏ,μ=ρ5μ−1+γ5μ−1−σ+τ5μ−1ℏηΓη+1,ℱ¯S1ℏ,μ=ρ1−5μ+γ1−5μ−σ+τ1−5μℏηΓη+1,G¯S1ℏ,μ=λ5μ−1+ρ5μ−1+K5μ−1+ξ5μ−1ℏηΓη+1,G¯S1ℏ,μ=λ1−5μ+ρ1−5μK1−5μ+ξ5μ−1ℏηΓη+1,ℋ¯S1ℏ,μ=τ¯5μ−1ℏηΓη+1,ℋ¯S1ℏ,μ=τ¯5μ−1ℏηΓη+1.Proceeding the steps in the similar manner, we can easily find the third, fourth, fifth, and the remaining terms of the series solution. Hence, we obtain the following equation:(53a)As¯ℏ=5μ−12α+β+γ+δℏηΓη+1+5μ−13ℏηΓη+12α+β+γ+δα5μ−1ε+β+γ+δ−ε+ζ+λ+βη+ν+γζ−ϑ+ρ+κ+δη+θ−ν+ξℏηΓη+1+⋯As¯ℏ=1−5μ−1−5μ2α+β+γ+δℏηΓη+1+1−5μ3ℏηΓη+12α+β+γ+δα1−5μα+β+γ+δ−ε+ζ+λ+βε−η+ν+γζ−ϑ+ρ+κ+δη+θν+ξℏηΓη+1+⋯,ℬs¯ℏ=5μ−1−5μ−1α+β+γ+δ−ε+ζ+λ5μ−1ℏηΓη+1−α5μ−1ε+β+γ+δ−ε+ζ+λ+βε−η+ν+γζ−ϑ+ρ+κ+δη+θ+ν+ξ5μ−13α+β+γ+δℏηΓη+12−ε+ζ+λ5μ−1ℏηΓη+1α+β+γ+δ5μ−1−ε+ζ+λℏηΓη+1+⋯ℬs¯ℏ=1−5μ−1−5μα+β+γ+δ−ε+ζ+λ1−5μℏηΓη+1−α1−5μα+β+γ+δ−ε+ζ+λ+βε−η+ν+γζ−ϑ+ρ+κ+δη+θ−ν+ξ1−5μ3α+β+γ+δℏηΓη+12−ε+ζ+λ1−5μℏηΓη+1α+β+γ+δ5μ−1−ε+ζ+λℏηΓη+1+⋯C¯sℏ=5μ−1+5μ−1ε−η+νℏηΓη+1+ε5μ−1α+β+γ+δ−ε+ζ+λ−η+νε−η+νℏηΓη+1ℏηΓη+1+…C¯sℏ=1−5μ+1−5με−η+νℏηΓη+1+ε1−5μα+β+γ+δε+ζ+λ−η+νε−η+νℏηΓη+1ℏηΓη+1+….(53b)D¯sℏ=5μ−1+5μ−1η−θ+ρ+κℏηΓη+1+5μ−1ℏηΓη+1ζα+β+γ+δ5μ−1−θ+ρ+κ5μ−1α+β+γ+δℏηΓη+1+⋯D¯sℏ=1−5μ+1−5μη−θ+ρ+κℏηΓη+1+1−5μℏηΓη+1ζα+β+γ+δ1−5μ−θ+ρ+κ1−5μα+β+γ+δℏηΓη+1+⋯⋅E¯sℏ=5μ−1+5μ−1η+θ−ν+ξℏηΓη+1+ηε−η+ν−θα+β+γ+δ5μ−1−ϑ+ξη+θ−ϑ+ξℏηΓη+1+⋯E¯sℏ=1−5μ+1−5μη+θ−ν+ξℏηΓη+1+−ϑ+ξη+θ−ϑ+ξ−ϑ+ξη+θ−ϑ+ξℏηΓη+1+….ℱ¯sℏ=5μ−1+5μ−1ρ+ϑ−σ+τℏηΓη+1+−ρ5μ−1α+β+γ+δ+ϑη+θ−ϑ+ξ−σ+τρ+ϑ−σ+τ5μ−1ℏηΓη+1ℏηΓη+1+…ℱ¯sℏ=1−5μ+1−5μρ+ϑ−σ+τℏηΓη+1+−ρ1−5μα+β+γ+δ+ϑη+θ−ϑ+ξ−σ+τρ+ϑ−σ+τ1−5μℏηΓη+1ℏηΓη+1+⋯⋅(53c)G¯sℏ=5μ−1+5μ−1λ+ρ+κ+ξ+σℏηΓη+1+λ5μ−1α+β+γ+δ−ε+ζ+λ+ρε−η+νκα+β+γ+δ+ξη+θ−ϑ+ξ+σρ+ϑ−σ+τ5μ−1ℏηΓη+1ℏηΓη+1+⋯G¯sℏ=1−5μ+1−5μλ+ρ+κ+ξ+σℏηΓη+1+λ1−5μα+β+γ+δ−ε+ζ+λ++ρε−η+ν−κα+β+γ+δ+ξη+θ−ϑ+ξ+σρ+ϑ−σ+τ1−5μℏηΓη+1ℏηΓη+1+⋯.ℋ¯sℏ=5μ−1+τ5μ−1ℏηΓη+1+τ5μ−1ρ+ϑ−σ+τℏηΓη+1ℏηΓη+1+…ℋ¯sℏ=1−5μ+τ1−5μℏηΓη+1+τ1−5μρ+ϑ−σ+τℏηΓη+1ℏηΓη+1+….By substituting the transmission rate values from Table1, the solutions for the SIDARTHE epidemic model can be found, as shown in Figures 4–11.Figure 4
For varyingℏ and η values, the suspected cases is increasing gradually (a) A¯sℏ. (b) A¯sℏ.
(a)(b)Figure 5
For varyingℏ and η values, the infected cases is increasing gradually (a) ℬ¯sℏ. (b) ℬ¯sℏ.
(a)(b)Figure 6
For varyingℏ and η values, the spreading level of diagonized cases are constantly increasing (a) C¯sℏ. (b) C¯sℏ.
(a)(b)Figure 7
For varyingℏ and η values, the spreading level of ailing cases is decreasing and then increasing slowly (a) D¯sℏ. (b) D¯sℏ.
(a)(b)Figure 8
For varyingℏ and η values, the detected cases are increasing fastly from day-to-day (a) E¯sℏ. (b) E¯sℏ.
(a)(b)Figure 9
For varyingℏ and η values, the life-threatening cases are increasing fastly (a) ℱ¯sℏ. (b) ℱ¯sℏ.
(a)(b)Figure 10
For varyingℏ and η values, the recovered cases are slightly decreasing and then increasing slowly (a) G¯sℏ. (b) G¯sℏ.
(a)(b)Figure 11
For varyingℏ and η values, the number of death cases are increases fastly (a) ℋ¯sℏ. (b) ℋ¯sℏ.
(a)(b)


## 5. Conclusion

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



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*Source: 5468696-2022-12-14.xml*