Theoretical and semi-analytical results to a biological model under Atangana–Baleanu–Caputo fractional derivative

This manuscript is related to finding a solution of the SIR model under Mittag-Leffler type derivative. For the required results, we use Laplace transform together with Adomian decomposition method (LADM). The mentioned method is a powerful tool to deal with various linear and nonlinear problems of “fractional order differential equations (FODEs)”. Also, we study some results devoted to qualitative theory for the concerned model. Computational results show the verification of the established analysis. Briefly, we state that qualitative theory for the existence of solution is important to ensure whether the considered problem has a solution or not. Further ensuring the existence of solution, we investigate approximate solution which is computed in the form of infinite series. The results are graphically displayed to analyze the adopted procedure for solving nonlinear FODEs under ABC derivative.

where α is the birth rate, N = x(t) + y(t) + z(t), a represents the unrelated death rate, b is a disease-related death rate, δ is infectious rate, and β is the removal rate. Further, x stands for the density of susceptible, y for infected, and z for recovered individuals, respectively. Onward of the said model has been investigated very well, see [5][6][7][8].
Riemann, Liouville, Euler, and Fourier have made a significant contribution in the eighteenth century in the area mentioned above. Various aspects of mathematical modeling may not be well described via ordinary calculus since derivatives of noninteger order are in fact definite integrals that provide accumulation. The concerned accumulation includes the corresponding integer counterpart as a special case. Further such operators permit greater freedom in degree as compared to integer order (for details, see [9][10][11][12][13][14][15][16][17][18][19][20][21]). In the said area, by considering different aspects, great work has been done in [12][13][14]. Differential operator with noninteger order has not been uniquely defined. There are several definitions in the literature. On the basis of kernels, there are two concepts. One definition involving a singular kernel is often called power law, while the second one contains nonsingular kernel of exponential and Mittag-Leffler type. The differential operators involving Mittag-Leffler and exponential type kernel have been recently introduced by Atangana, Baleanu, Caputo, and Fabrizo (see [19,[22][23][24][25]). This derivative exhibits the singular kernel by a nonsingular kernel [20,21,[26][27][28][29]. Since the differential and integral operators of ABC type are nonlocal and nonsingular, such operators reduce the complication in numerical analysis of many problems. Further in some problems, the mentioned operators play excellent roles in description of many hereditary and memory terms. Therefore, the mentioned operators have been considered in the recent time in an increasing way for investigating physical and biological problems. In this regard, a number of methods available in literature have been applied to compute solutions under these derivatives. To compute the approximate and analytical solution, a famous decomposition method was used as the best tool for many problems. Therefore, in this article, we utilize Laplace Adomian decomposition method (LADM) for the series solution of SIR model (1) under ABC derivative. We consider the biological model (1) and use the ABC derivative for the model with order μ such that μ ∈ (0, 1] as given by under the condition Then, we get the results in the form of an analytical solution of the SIR model. Moreover, we exhibit the approximate solution for distinct fractional order μ ∈ (0, 1]. In addition, we study some results about the qualitative analysis and stability analysis for the concerned model. Further, the right-hand sides of model (2) vanish at zero as for the general problem in [20], Theorem 3.1. Via fixed point theory and nonlinear analysis, we establish some results regarding the existence and stability of solution. Then, we compute the required series solution via the proposed method for model (2).

Auxiliary results
We recall some fundamental results here.

Definition 3 "The Laplace transform of the ABC derivative of a function φ(t)" is defined by
Lemma 1 For 0 < μ < 1, the solution of the problem is provided by Definition 4 The operator ϕ k : Y → Y for k = 1, 2, 3 defined as is Hyers-Ulam (HU) stable if, for any positive number c l (l = 1, 2, 3, . . . 9), l (l = 1, 2, 3) and for every solution ( x, y, z) ∈ Y obeying the relation with (x, y, z) ∈ Y of (7), the following hold: Definition 5 If δ l for l = 1, 2, 3, . . . n are eigenvalues of the matrix N , then the spectral radius is denoted as (N ) and is defined as Moreover, if (N ) < 1, this implies that N tends to zero.
and the matrix tends to zero, then (7) is Hyers-Ulam stable.

Qualitative results for the proposed model (2)
In this part of the manuscript, we study qualitative results for problem (2). Here, we express right-hand sides of (2) as follows: We select such that The concerned norm may be defined as Then the Picard operator is given as We present the following theorem. (15), there exists at most one solution to the considered model (2).

Theorem 2 In view of the Banach contraction theorem under the Picard operator as defined in
Proof In this regard, applying AB I μ on model (2), we obtain Using Lemma 1 and writing (16) in a simple form, one has where and x, y, z), Using (17) and (18), the operator in (15) is defined as follows: Thus, the model under our study satisfies the result such that d <d M .
On further simplification, one has To compute (22), we proceed as follows: As is a contraction, so we have kd < 1, thus T is a contraction. Therefore, our concerned problem (18) has the required solution.

Stability results
Theorem 3 If d < 1 holds, then the matrix N also converging to zero is Hyers-Ulam stable.
Proof Taking any two solutions (x, y, z), ( x, y, z), we have In the same fashion, one has where c i = ϑ(μ) +θ(μ)t μ k for each i = 1, 2, . . . , 9. Now, the matrix N given by converges to zero. Hence, the system is Hyers-Ulam stable.

Analytical results for the proposed model
Here, we are going to apply LADM to obtain general results for the considered model (2).
Now, we are going to consider x(t), y(t), z(t) in terms of infinite series as follows: We resolve nonlinear terms as follows: where A q (x, y) can be defined as .

Computational results
In this section of the paper, our computational results about a series solution of the concerned model are represented. To obtain the main goal, we apply LADM for the solution corresponding to the values given in Table 1. In view of Table 1, we exhibit the results,

Concluding remarks
We have discussed LADM for a biological model of the SIR model using the ABC operator. Also, we developed some results about the qualitative theory and Hyers-Ulam stability analysis. The methodology utilized here for dynamical problems under ABC operator of derivative is very rarely applied in the literature. Further on providing some graphs of approximate results, we have illustrated the procedure. The mentioned tool may be used in the future to handle more complicated problems under the aforementioned operator.