Fuzzy fractional-order model of the novel coronavirus

In this paper, a novel coronavirus infection system with a fuzzy fractional differential equation defined in Caputo’s sense is developed. By using the fuzzy Laplace method coupled with Adomian decomposition transform, numerical results are obtained for better understanding of the dynamical structures of the physical behavior of COVID-19. Such behavior on the general properties of RNA in COVID-19 is also investigated for the governing model. The results demonstrate the efficiency of the proposed approach to address the uncertainty condition in the pandemic situation.


Introduction
Recently, the whole globe has been suffering from a novel coronavirus pandemic, which was named "2019 novel coronavirus", abbreviated by "2019-nCoV", and claimed to outbreak for the first time in Wuhan city, central China [1]. It has been observed that 2019-nCoV is transmitted from animal to human; as many infected claimed that they had been infected due to a local fish and wild animal market in Wuhan as early as 28 November [2]. Soon after, some researchers confirmed that the transmission also happens from a person to a person [3]. According to the data reported by WHO (World Health Organization), on March 21, 2020, the reported laboratory confirmed human infections in 187 countries, territories, or areas around the world have reached more than 292,142, including 12,784 death cases [4]. Even in some countries, like Italy and Spain, the death rate was as high as almost 0.066. This verifies the severity and high infectivity of 2019-nCoV. It is confirmed that most people infected with 2019-nCoV will experience mild to moderate respiratory illness, such as breath difficulty, low fever, sick, cough, and other symptoms. However, other symptoms such as gastroenteritis and neurological diseases of varying severity have also been reported [5]. The 2019-nCoV is transmited mainly through droplets from the nose when an infected person coughs or sneezes. Once a person breaths the droplets from infected people in the air, he/she will be exposed to the danger of getting the infection. As a result, the best way to prevent the virus is to avoid meetings and touching other people. For this purpose, the Chinese government decided to lock down Wuhan city and cut or limit the transportation system of the country, including airplanes, trains, buses, and pri-for the understanding of the dynamical structures of the physical behavior of 2019nCoV. We define the system of six equations illustrating the outbreak of the coronavirus in the form of nonlinear fractional order differential equations (FODEs), involving the susceptible people S k (t), the exposed population E k (t), total infected strength I k (t), asymptotically infected population A k (t), the total number of humans recovered R k (t), reservoir M k (t), and corresponding interaction, which are presented as follows [13]: where n k represents the rate of birth, m k represents the death of infected population, b k represents the transmission coefficient, b l represents disease transmission coefficient, κ is transmissibility multiple, ω k and ω k denote signified incubation period, γ k and γ k represent the recovery rate of I k and A k , respectively, ξ and η denote the influence of the virus from I k and A k to M k , and ν represents the rate of eliminating the virus from M k . The parameters are explained in Table 1.
In the last few years, modern calculus and DEs have been extended to fuzzy calculus and FODEs [14][15][16][17][18], respectively. Then FODEs were extended to fuzzy FODEs [19][20][21]. FODEs and fuzzy integral equations have been studied by many researchers to establish the existence and uniqueness theory of solutions [22][23][24][25][26][27]. When dealing with fuzzy FODEs, it is really tedious to compute more precise solutions to every fuzzy FODE. A lot of efforts have been made by mathematicians in solving fuzzy FODEs by using various methods like perturbation method, integral transform methods, as well as spectral techniques [28][29][30][31][32][33]. Some researchers performed stability analysis of fuzzy DEs [34]. Here, we are going to investigate model (1) with a fuzzy fractional-order derivative where the uncertainty lies in the initial data. For 0 < γ ≤ 1, associated to fuzzy initial condition, for α ∈ [0, 1], Regarding the above explanations and to address the current uncertain situation, we were motivated to propose a novel coronavirus infection system under fuzzy fractional calculus. In fact, considering the proposed model which also enhances the physical behavior of such an infection system, we ensure that the model is closer to the real behavior of a system evolving the general properties of RNA in COVID-19.

Definition 2 ([35]
) On a fuzzy number μ, the p-level set is defined by where p ∈ (0, 1] and x ∈ R. , v(ϑ)) and w = (w(ϑ), w(ϑ)) be two fuzzy numbers in their parametric form. The Hausdorff distance between v and w is defined by In E, the metric ρ has the following properties: (iv) (E, ρ) is a complete metric space.
Definition 6 ([31]) Let Θ : R → E be a fuzzy mapping. Then Θ is called continuous if for any > 0 ∃δ > 0 and a fixed value of λ 0 ∈ [ζ 1 , ζ 2 ], we have Definition 7 ( [28,31]) Let Φ be a continuous fuzzy function on [0, b] ⊆ R, a fuzzy fractional integral in Riemann-Liouville sense corresponding to t is defined by are the spaces of fuzzy continuous functions and fuzzy Lebesgue integrable functions, respectively, then fuzzy fractional integral is defined as

Definition 8 ([31]) If a fuzzy function
, 0 ≤ p ≤ 1 and t 1 ∈ (0, b), then the fuzzy fractional Caputo's derivative is defined as whenever the integrals on the right-hand sides converge and n = β .
, then for 0 ≤ p ≤ 1, and 0 < β ≤ 1, the Laplace transform of fuzzy fractional derivative in Caputo's sense is given by

Main results
In the following section, the existence and uniqueness of solution to the subsequent fuzzy fractional model are discussed; and we provide the procedure for finding a semianalytic solution of model (2) by using fuzzy Laplace transform.

Existence and uniqueness
In this section, by the use of fixed point theory, the existence and uniqueness of the subsequent fuzzy fractional model is discussed. Consider the right-hand sides of model (2): where Ψ , Ξ , f , g, h, and y are fuzzy functions. Thus, for 0 < γ ≤ 1, the given model (2) can be written as: with fuzzy initial conditions Now applying fuzzy fractional integral I r and using initial conditions, we get Let us define a Banach space as B = B 1 × B 2 under the fuzzy norm: One can write equation (4) as wherẽ , We make several assumptions on the nonlinear function Θ : B → B as follows: (C-1) There exists constant K ℵ > 0 such that for each‫א‬ k 1 (t),‫א‬ k 2 (t) ∈ B, (C-2) There exist constants M ℵ > 0 and N ℵ > 0 such that Theorem 2 Under Assumption (C-2), the considered model (3) has at least one solution.
Proof Let A = ‫א{‬ k (t) ∈ B : ‫א‬ k (t) ≤ r} ⊂ B be a closed and convex fuzzy set, and ψ : A → A be a mapping defined as For any‫א‬ k (t) ∈ A, we have From the last inequality, we have ψ(A) ⊂ A, which implies that the operator ψ is bounded. Next we show that the operator ψ is completely continuous. For this, let φ 1 , φ 2 ∈ [0, T] be such that φ 1 < φ 2 , then From the last inequality, we see that the right-hand side goes to zero as φ 2 → φ 1 . Hence, Thus, the operator ψ is equicontinuous. By Arzela-Ascoli theorem, the operator ψ is completely continuous, also ψ is bounded as proved earlier. Therefore, system (3) has at least one solution by Schauder's fixed point theorem.
Hence ψ is a contraction. Hence, by Banach contraction theorem, system (3) has a unique solution.

Procedure for solution
Here a general method is provided in order to find the solution of the considered system by the fuzzy Laplace transform. Taking fuzzy Laplace transform of (3) and using initial conditions, we get The infinite series solution is given by: where Z 1 n , Z 2 n , and Z 3 n are Adomian polynomials, representing nonlinear terms. So the last equation becomes Taking the inverse Laplace transform, we have Comparing the terms on both sides, we consider the first two terms of the series V(0, α), I(0, α), Similarly, we can find the other terms. Hence, the series solution of the considered system is given by

Numerical results and discussion
We consider a table corresponding to the parameters involved in the model. Consider the proposed model (2) with initial conditions as given in Table 1:  Applying the proposed procedure to (2) and using initial conditions, we have The second term of the series solution is For the sake of simplicity, assume that Now the third term of the series is In Figs. 1-6 we presented comparisons of approximate fuzzy and approximate normal solutions for the considered model at the given uncertainty against various fractional order. We see that as the susceptible class value is decreasing, the exposed papulation increases and hence infection spreads with different rate due to various fractional order. Similarly, the death cases are increasing so the recovered class also grows and the asymptotically infected class also increases, and hence the population of virus in the reservoir is growing. From the figures we observe that fuzzyness along with fractional calculus pro-     Remark 1 Regarding the provided results, it is clear that the lower bound is an increasing set-valued function and an upper bound is a decreasing one, which proves that the solutions are fuzzy numbers. Also, it is worthy to mention that for general cases, similar results can be obtained under fuzzy differentiability.
Remark 2 Considering the fact that stochastic and random parameters are more complex to address, the uncertainty can lead to an increase in the computation cost, so employing fuzzy concepts for modeling such real-world systems can be the most suitable choice.

Conclusions
In this paper, we have demonstrated the existence and uniqueness of the solution to the fuzzy fractional order model of COVID-19 infection using the Banach fixed point theorem. We also established a proper procedure for the fuzzy Laplace transform coupled with Adomian decomposition method to obtain an approximate solution for the proposed model. We have presented comparisons between fuzzy and normal results up to three terms to depict the efficiency of this approach. We observed that fuzzyness coupled with fractional calculus approach excellently produced global dynamics of those problems where uncertainty lies in the data.