Fractional study of the Covid-19 model with different types of transmissions

We investigate a mathematical system of the recent COVID-19 disease focusing particularly on the transmissibility of individuals with different types of signs under the Caputo fractional derivative. To get the approximate solutions of the fractional order system we employ the fractional-order Alpert multiwavelet(FAM). The fractional operational integration matrix of Riemann-Liouville (RLFOMI) employing the FAM functions is considered. The origin system will be transformed into a system of algebraic equations. Also, an error estimation of the supposed scheme is considered. Satisfactory results are gained under various values of fractional order with the chosen initial conditions (ICs).


Introduction
Mathematical analysis and modeling are necessary ingredients for understanding the dynamics of disease epidemiology. The usage of such models in epidemiology gives us an ability to study the dynamics of diseases and supports controlling strategies. COVID-19 pandemic has attracted many researcher to examine disease regarding transmission and control of the pandemic. Many of researcher have tried to do this aim using different types of fractional and integer-order operators. For example, a real-time evaluation of the epidemiological specifications of new coronavirus infections has been done. Details of this investigation can be read in [1]. A review of the 2019-nCoV's transition risk and its implications for public health measures were described in [2] in another paper. Additionally, citation 3 provides a statistical estimate of the impact of non-pharmaceutical therapies in truncating the Covid-19 [3]. Performs an approximation of the Covid-19's exponential pickup measure. Additionally, the COVID-19 epidemic was mathematically modelled in [4] with a trace of awareness campaigns. Some other works with integer-order operators can be seen in [5][6][7][8][9][10]. Recently, researchers have paid notable attention to use the of fractional-order differential and integral operators in modeling different types of systems since such operators are efficient tools for modeling real-world phenoms carrying complex dynamics.
Investigations on the non-integer-order systems showed that the fractional-order operators can display complex dynamical behaviors better than their integer-order counterparts. Regarding fractional operators, different types of studies have been done on the Covid-19 system. For instance [11], reported on approximative solutions and stability analyses for the non-integer order dynamics of Covid-19. A collocation method to simulate the non-integer order Covid-19 model under the Caputo operator can be seen in [12]. In [13] computational modeling of novel Covid-19 under Mittag-Leffler Power Law was studied. A numerical study of the Covid-19 model with reinfection and the significance of quarantine was reported in [14]. Also, in [15] a numerical study and stability analysis of a novel COVID-19 model were investigated. A new SITR mathematical model for predicting the COVID-19 of India with lockdown effect can be found in [16]. A SEIHDR model for COVID-19 with inter-city networked coupling effects has been done in [17]. A mathematical model of the SIR epidemic model(COVID-19) with non-integer derivatives including stability and numerical analysis can be seen in [18]. A case study of the Covid-19 epidemic in India via new generalized Caputo type fractional derivatives was done in [2]. Some other works using fractional operators can be found in [19][20][21][22][23][24][25][26][27]. In this work we consider a new deterministic system to design its fractional model. The integer order of this system is as follows [25]. (1.1) Where S(t) denotes the number of susceptible individuals and E(t) shows those who have been exposed yet are not yet contagious. I s (t) defines the folks with severe symptoms who are contagious, I m (t) implies individuals with minimal symptoms who are contagious and I a (t) shows the number of asymptomatic individuals. The number of hospitalized individuals in H(t) and individuals in intensive cure unit in I cu . Also, R(t) and D(t) define recovered by immunityindividuals and deceased persons, respectively, at time t. To design the fractional order model of the considered system we need some elementary definitions of the fractional calculus.
To solve the Eq. (1.5) we use the collocation method related to the fractional-order Alpert multiwavelts (FAMWs). First, consider the following problem The key goal of this study is to use a collocation scheme related to FAMWs to solve the fractional Covid-19 system (1.5) employing the fractional integration operational matrix (OMFI). Thus, we attain the OMFI for the FAMWs. This is done by expanding FAMWs into the piecewise Taylor functions' fractional order. At last, the solution of FPDEs is reduced using the matrix the process of solving algebraic problems.
Organization of the work is as follows: In Section 2 we provide few essential definitions of fractional calculus and the fractional model of the considered system. Section 3 is devoted to create the fractional-order multiwavelets of Alpert on [0, 1] and constructing the Riemann-Liouville FOMI for the main problem solution. The approximation solution of Eqs. (1.6) and (1.7) employing FAMWs basis together with collocation scheme is presented in section 4. The error analysis can be seen in Section 5. In section 6, results and discussion can be read and conclusion can be read in the last section.
Next section will be showing the fractional-order Alpertmultiwavelets.

Fractional-order Alpert multiwavelet system
Now, we provide few definitions related to the interpolating scaling functions(ISFs), Alpert multiwaveletsas well as the functions for fractional-order Alpert multiwavelets (AMFs). Also, the OMFI of the fractional order Alpert multi-wavelets will be constructed. Suppose P m as the mth-degree Legendre polynomial which has roots τ e when e ¼ 0, 1, …, m À 1. We define the ISFs on [0, 1]: φ e ðtÞ ¼ where We extend the following function which has the degree less than m with the function φ 0 , …, φ mÀ1 as hðtÞ ¼  (2.4) FAMs hold the two-scale relation as [30]. where w(t) ¼ t μÀ1 and δ ij called the Kronecmer symbol. (2.8)
We can approximate q(t) as follows and we can calculate A j , B j , j ¼ J 0 þ 1, …, J by employing the refinement equations in (2.2) and (2.5) successively [30]. By employing such a representation we can avoid the integral computations appeared in Eqs. (2.13) and (2.16).

Operational matrix (OMFI)
In this section, we set the OMFI for FAMs. We define set of FPTfs on [0, 1] by [31].
We define the fractional-order scaling functions using FPTFs as follows where TY(t) is a vector with dimension 2 J m Â 1. Employing (2.21) along with properties of I μ t , we get (2.23) Γðϑ 0 þ 1Þ Employing G μ , OMFI of wavelets of fractional-order Alpert can be gained: A N Â N system of equations is gained to solve for F e J;m unknowns. This system can be solved using Newton iteration method. After solving the system and plugging its solution in Eq. (3.1), the approximate solution of (1.1)-(1.2) can be obtained.To get the numerical solutions Eqs. (1.6) and (1.7) using FAMWs, it suffices to replace G À1 μ B μ J0;J ðtÞ instead of the vector Φ ! μ J ðtÞ.

Error estimation
Now, we get few estimations regarding AMFs truncated error q À P μ;K J by the Sobolev norms concept. Definition of the Sobolev norm ς ! 0 in (a, b) is [32]. which tells that if y is infinitely smooth and the rate of convergence of P μ;m J to q is faster than 1 2 J to the power of m À z and any power of 1 mÀ1 , which is better than the classic spectral approaches. [33].
Notice, owing to Eqs.

Results and discussion
Now, we investigate the performance of the supposed method for the fractional system (1.5) under different values of fractional orders. For case 1, we consider the fractional orders of α ¼ 0.8, 0.9 and 0.95 which the corresponding solutions of this case can be seen in Figs. 1-3. A decreasing trend in the number of susceptible individuals can be seen in Fig. 1. On the other hand, the those who have been exposed yet are not yet contagious, asymptomatic individuals, and the infectious individuals with severe symptoms experience an increasing pattern in their numbers. But, after a sharp increasing trend, the number of hospitalized individuals and individuals in intensive cure units in see an increasing pattern after 40 days. For the second case, we have selected α ¼ 0.85, 0.91 and 1. Similar trends in the number of individuals belonging to each group can be observed under the selected fractional orders. Approximate solutions of the considered fractional system regarding the second case are plotted in Figs. 4-6.

Conclusion
In this work, we designed a fractional model of the Covid-19 system employing the Caputo fractional order derivative. An efficient numerical scheme based on the FAMWs was used to get the approximate solutions of the new designed system. The suggested scheme was established by developing OMFI. The computational outlay of the suggested technique was lessened due to the matrix's sparsity nature. Using the collocation technique we derived the approximate solutions of the new fractional system under different values of fractional orders. Figures of the solutions for each dependent variable were presented, clearly.