Power-series solution of the L-fractional logistic equation

Applied Mathematics Letters 154 (2024) 109085 Available online 30 March 2024 0893-9659/© 2024 The Author(s). Published by Elsevier Ltd. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Contents lists available at ScienceDirect Applied Mathematics Letters journal homepage: www.elsevier.com/locate/aml Regular article Power-series solution of the L-fractional logistic equation Marc Jornet a, Juan J. Nietob,∗ aDepartament de Matemàtiques, Universitat de València, 46100 Burjassot, Spain bCITMAga, Departamento de Estatística, Análise Matemática e Optimización, Universidade de Santiago de Compostela, 15782 Santiago de Compostela, Spain ARTICLE INFO Keywords: Logistic model Fractional calculus Non-integer differential equation Leibniz and Caputo fractional derivative Analytic solution Euler numbers ABSTRACT We consider the L-fractional derivative, which has been proposed in the literature to study fractional differentials in geometry and processes in mechanics. Our context is population growth and epidemiology, for which the use of L-derivatives is motivated by transitions. Using power series, we solve the logistic differential equation model under this fractional derivative. Several conclusions on the method, the derivative, and the singularity of the associated kernel are reached. Fractional Euler numbers, related to the logistic map and the famous Riemann zeta function, are also introduced. 1. Introduction Fractional differential equations are studied to extend classical models based on integer-order derivatives. For these equations, derivatives are of arbitrary order, with any real or even complex number [1]. There are many notions of fractional derivative in the literature, with several applications [2,3] (modeling of COVID-19, Cholera, etc.). The (Leibniz) L-fractional derivative of an absolutely continuous function 𝑥∶ [0, 𝑇 ]→Ris [4,5] 𝐿 𝐷𝛼𝑥(𝑡) = 𝐶𝐷𝛼𝑥(𝑡) 𝐶𝐷𝛼𝑡,(1.1) where 𝑡∈ [0, 𝑇 ]is time, 𝛼∈ (0,1) is the fractional order of differentiation, and 𝐶𝐷𝛼𝑥(𝑡) = 1 𝛤(1 − 𝛼)∫𝑡 0 𝑥′(𝜏) (𝑡−𝜏)𝛼d𝜏(1.2) is the Caputo fractional derivative of 𝑥with first-order derivative 𝑥′, being 𝛤the gamma function. This fractional derivative (1.1) was envisioned to deal with fractional differentials in geometry, d𝛼𝑥(𝑡) = 𝐿 𝐷𝛼𝑥(𝑡) d𝛼𝑡. (1.3) Variations of it have been suggested, such as the 𝛬-fractional derivative based on the Riemann–Liouville fractional operator [6]. Most of the applications of these new fractional derivatives appear in mechanics [4,6]. An L-fractional differential equation is an equation of the form 𝐿 𝐷𝛼𝑥(𝑡) = 𝑓(𝑡, 𝑥(𝑡)).(1.4) In this paper, we investigate the use of power-series expansions to solve (1.4) and thus extend the analysis conducted for linear equations in [7]. We deal with an important model from epidemiology and population growth: the logistic equation. Power series ∗Corresponding author. E-mail addresses: [email protected] (M. Jornet), [email protected] (J.J. Nieto). https://doi.org/10.1016/j.aml.2024.109085 Received 1 February 2024; Received in revised form 17 March 2024; Accepted 27 March 2024 Applied Mathematics Letters 154 (2024) 109085 2 M. Jornet and J.J. Nieto have been applied to this model and some of its variations: for Caputo logistic equation [8], Caputo SIS equations [9], Caputo generalized proportional logistic equation [10], and Caputo compartmental models [11]. Alternative methods to power series have been used for other fractional derivatives [12] (implicit solutions). Notice that 𝐶𝐷𝛼𝑡𝛽=𝛤(𝛽+ 1) 𝛤(𝛽−𝛼+ 1) 𝑡𝛽−𝛼(1.5) for 𝛽 > 0. In particular, 𝐶𝐷𝛼𝑡=𝑡1−𝛼∕𝛤(2 − 𝛼), hence the L-fractional derivative can be rewritten as 𝐿 𝐷𝛼𝑥(𝑡) = 𝛤(2 − 𝛼) 𝑡1−𝛼 𝐶𝐷𝛼𝑥(𝑡).(1.6) For power 𝛽= 0,𝐶𝐷𝛼1=0, so that 𝐿 𝐷𝛼1=0. That is, the fractional derivative of any constant is zero. This property is very important when dealing with initial states in (1.4) and power series. 2. Motivation of L-fractional models The most basic population model is given by the exponential, or Malthusian, equation: 𝑥′=𝑟𝑥. (2.1) When justifying the validity of (2.1), one employs differentials: d𝑥=𝑟𝑥d𝑡. (2.2) This equation means that, when an infinitesimal time element d𝑡passes, a proportion 𝑟d𝑡of the population enters/exits the population. If one wants to generalize (2.2) via a fractional index, in order to capture a wider range of dynamics, then fractional infinitesimal time elements d𝛼𝑡may be considered [13,14]. If d𝛼𝑡passes, then a proportion 𝑟d𝛼𝑡of the population flows: d𝛼𝑥=𝑟𝑥d𝛼𝑡. (2.3) Hence 𝐿 𝐷𝛼𝑥=𝑟𝑥, (2.4) by (1.3). The interpretation of 𝑟≡𝑟𝛼in (2.4) is different to (2.1), because it is a fractional rate of transition. When 𝛼= 1, then 𝑟 is the usual rate and 1∕𝑟is the mean time of exponential transition. In both cases (2.1) and (2.4), the units of 𝑟are time−1(it is a rate), and 𝐿 𝐷𝛼𝑥represents a true velocity. Since (2.3) and (2.4) define the basic population movements, the fractional version of subsequent models such as logistic or SIR follows. 3. Fractional logistic differential equation 3.1. Ordinary and Caputo equations The ordinary logistic differential equation 𝑥′=𝑥(1 − 𝑥) = 𝑥−𝑥2(3.1) can be explicit solved. One has the constant solutions 𝑥= 0 and 𝑥= 1. For an initial condition 𝑥(0) = 𝑥0∉ {0,1}, one obtains the expression 𝑥(𝑡) = 𝑥0 𝑥0+ (1 − 𝑥0)e−𝑡. The classical logistic function appears when 𝑥0= 1∕2: 𝑥(𝑡) = 1 1+e−𝑡.(3.2) The solution to (3.1) can also be expanded in the form of a power series as 𝑥(𝑡) = ∑∞ 𝑛=0 𝑥𝑛𝑡𝑛, so that 𝑥′(𝑡) = ∞ ∑ 𝑛=1 𝑥𝑛𝑛𝑡𝑛−1 = ∞ ∑ 𝑛=0 𝑥𝑛+1(𝑛+ 1)𝑡𝑛(3.3) 𝑥2(𝑡) = ∞ ∑ 𝑛=0 (𝑛 ∑ 𝑚=0 𝑥𝑚𝑥𝑛−𝑚)𝑡𝑛(3.4) (Cauchy product for series). Then, from 3and imposing equality between the coefficients, 𝑥𝑛+1 =1 𝑛+ 1 [𝑥𝑛− 𝑛 ∑ 𝑚=0 𝑥𝑚𝑥𝑛−𝑚]. Applied Mathematics Letters 154 (2024) 109085 3 M. Jornet and J.J. Nieto For example, for 𝑥0= 1∕2, 𝑥(𝑡) = 1 2+1 4𝑡−1 48 𝑡3+1 480 𝑡5+⋯.(3.5) Under convergence of the series (positive radius of convergence around 𝑡= 0), these formal operations are valid. In [8], the Caputo fractional logistic equation 𝐶𝐷𝛼𝑥=𝑥(1 − 𝑥) = 𝑥−𝑥2(3.6) was studied. The authors found the solution to (3.6) in the form of a power-series expansion in terms of 𝑡𝛼, as 𝑥(𝑡) = ∞ ∑ 𝑛=0 𝑥𝑛(𝑡𝛼)𝑛= ∞ ∑ 𝑛=0 𝑥𝑛𝑡𝛼𝑛.(3.7) Formally, 𝐶𝐷𝛼𝑥(𝑡) = ∞ ∑ 𝑛=0 𝑥𝑛 𝐶𝐷𝛼(𝑡𝛼𝑛) = ∞ ∑ 𝑛=1 𝑥𝑛 𝛤(𝑛𝛼 + 1) 𝛤((𝑛− 1)𝛼+ 1) 𝑡𝛼(𝑛−1) = ∞ ∑ 𝑛=0 𝑥𝑛+1 𝛤((𝑛+ 1)𝛼+ 1) 𝛤(𝑛𝛼 + 1) 𝑡𝛼𝑛.(3.8) Then, from (3.6), 𝑥𝑛+1 =𝛤(𝑛𝛼 + 1) 𝛤((𝑛+ 1)𝛼+ 1) [𝑥𝑛− 𝑛 ∑ 𝑚=0 𝑥𝑚𝑥𝑛−𝑚](3.9) is the recursive relation for the coefficients, where 𝑥0=𝑥(0) is the initial state. 3.2. L-fractional equation In this paper, we consider the L-fractional logistic equation 𝐿 𝐷𝛼𝑥=𝑥(1 − 𝑥) = 𝑥−𝑥2,(3.10) which is equivalent to the non-autonomous Caputo equation 𝐶𝐷𝛼𝑥=𝑡1−𝛼 𝛤(2 − 𝛼)𝑥(1 − 𝑥). We work with the formal power-series expansion 𝑥(𝑡) = ∞ ∑ 𝑛=0 𝑥𝑛𝑡𝑛.(3.11) In contrast to (3.6), the power series is classical and not written in terms of 𝑡𝛼. The L-fractional derivative of (3.11) is 𝐿 𝐷𝛼𝑥(𝑡) = ∞ ∑ 𝑛=0 𝑥𝑛 𝐿 𝐷𝛼(𝑡𝑛) = ∞ ∑ 𝑛=1 𝑥𝑛 𝛤(𝑛+ 1)𝛤(2 − 𝛼) 𝛤(𝑛+1−𝛼)𝑡𝑛−1 = ∞ ∑ 𝑛=0 𝑥𝑛+1 𝛤(𝑛+ 2)𝛤(2 − 𝛼) 𝛤(𝑛+2−𝛼)𝑡𝑛.(3.12) Notice that (1.5) and (1.6) have been used to compute 𝐿 𝐷𝛼(𝑡𝑛). Then, placing the power series into (3.10), we derive ∞ ∑ 𝑛=0 𝑥𝑛+1 𝛤(𝑛+ 2)𝛤(2 − 𝛼) 𝛤(𝑛+2−𝛼)𝑡𝑛= ∞ ∑ 𝑛=0 𝑥𝑛𝑡𝑛− ∞ ∑ 𝑛=0 (𝑛 ∑ 𝑚=0 𝑥𝑚𝑥𝑛−𝑚)𝑡𝑛. By equating the coefficients, 𝑥𝑛+1 𝛤(𝑛+ 2)𝛤(2 − 𝛼) 𝛤(𝑛+2−𝛼)=𝑥𝑛− 𝑛 ∑ 𝑚=0 𝑥𝑚𝑥𝑛−𝑚. This yields the recursive relation 𝑥𝑛+1 =𝛤(𝑛+2−𝛼) 𝛤(𝑛+ 2)𝛤(2 − 𝛼)[𝑥𝑛− 𝑛 ∑ 𝑚=0 𝑥𝑚𝑥𝑛−𝑚],(3.13) for 𝑛≥0. The initial condition is 𝑥(0) = 𝑥0. The first values of 𝑥𝑛, from (3.13), are 𝑥1= − (𝑥0− 1)𝑥0, 𝑥2=1 2(2 − 𝛼)(𝑥0− 1)𝑥0(2𝑥0− 1), 𝑥3=1 12 (3 − 𝛼)(2 − 𝛼)(𝑥0− 1)𝑥0(2(2𝛼− 5) (𝑥0− 1)𝑥0+𝛼− 2), 𝑥4=1 288 (4 − 𝛼)(3 − 𝛼)(2 − 𝛼)2(𝑥0− 1)𝑥0(2𝑥0− 1)(2(𝛼(2𝛼− 11) + 21) (𝑥0− 1)𝑥0+ (3 − 𝛼)(2 − 𝛼)). Applied Mathematics Letters 154 (2024) 109085 4 M. Jornet and J.J. Nieto The well-known property 𝛤(𝑧+ 1) = 𝑧𝛤 (𝑧)of the gamma function has been applied. For 𝑥0= 1∕2 (L-fractional logistic function), we have 𝑥1=1 4, 𝑥2= 0, 𝑥3= − 1 96 (3 − 𝛼)(2 − 𝛼), 𝑥4= 0, 𝑥5=(5 − 𝛼)(4 − 𝛼)(3 − 𝛼)2(2 − 𝛼)2 23040 , 𝑥6= 0, 𝑥7= − ((𝛼− 9)𝛼+ 25)(7 − 𝛼)(6 − 𝛼)(5 − 𝛼)(4 − 𝛼)(3 − 𝛼)3(2 − 𝛼)3 232243200 , 𝑥8= 0. When 𝛼→1−, these coefficients converge to the classical ones that solve (3.1). 3.3. Remarks In contrast to Caputo equations, the L-fractional solution is 𝐶∞and real analytic, expanded in terms of 𝑡𝑛instead of 𝑡𝛼𝑛. This property is intuitive, because the units of the L-fractional derivative (1.6) are population∕time, instead of population∕time𝛼. Since the solution 𝑥is smooth and not merely absolutely continuous, we can apply integration by parts on (1.2) and (1.1), so that 𝐶𝐷𝛼𝑥(𝑡) = 1 𝛤(1 − 𝛼)(𝑡1−𝛼 1 − 𝛼𝑥′(0) + 1 1 − 𝛼∫𝑡 0 (𝑡−𝜏)1−𝛼𝑥′′(𝜏)d𝜏)=1 𝛤(2 − 𝛼)(𝑡1−𝛼𝑥′(0) + ∫𝑡 0 (𝑡−𝜏)1−𝛼𝑥′′(𝜏)d𝜏), 𝐿 𝐷𝛼𝑥(𝑡) = 𝑥′(0) + 1 𝑡1−𝛼∫𝑡 0 (𝑡−𝜏)1−𝛼𝑥′′(𝜏)d𝜏. Then, these fractional derivatives have a non-singular kernel, (𝑡−𝜏)=(𝑡−𝜏)1−𝛼, that is continuous on [0, 𝑇 ]. However, the L- fractional derivative has the denominator 𝑡1−𝛼, so that 𝐶𝐷𝛼𝑥(0) = 0 ≠𝐿 𝐷𝛼𝑥(0) in general. This is an important property, considering the well-known deficiencies of fractional operators with non-singular kernels [15]. For example, the Caputo–Fabrizio derivative presents the restriction 𝐶𝐹𝐷𝛼𝑥(0) = 0, so for applications one needs to work with the Losada-Nieto integral problem [16]. On the other hand, since 𝐿 𝐷𝛼1=0, the initial condition can be set as for ordinary and Caputo fractional differential equations, 𝑥(0) = 𝑥0. This is not the case for the Riemann–Liouville or the 𝛬-fractional derivatives [6]. 3.4. Convergence Convergence of (3.11), with (3.13), is proved in what follows. There is a constant 𝐶 > 0, independent of 𝛼∈ (0,1) and of 𝑛, such that 𝛤(𝑛+2−𝛼) 𝛤(𝑛+ 2)𝛤(2 − 𝛼)≤𝐶. (3.14) Let 𝑦0=|𝑥0|, 𝑦𝑛+1 =𝐶(𝑦𝑛+ 𝑛 ∑ 𝑚=0 𝑦𝑚𝑦𝑛−𝑚).(3.15) By the triangular inequality and induction of 𝑛, it holds 𝑦𝑛≥|𝑥𝑛|. Then, it suffices to prove that ∑∞ 𝑛=0 𝑦𝑛𝑡𝑛<∞on a certain interval [0, 𝛿). Formally, if we consider 𝜓(𝑡) = ∑∞ 𝑛=0 𝑦𝑛𝑡𝑛, then 𝜓(𝑡) = 𝑦0+ ∞ ∑ 𝑛=1 𝑦𝑛𝑡𝑛=𝑦0+𝑡 ∞ ∑ 𝑛=0 𝑦𝑛+1𝑡𝑛=𝑦0+𝐶𝑡 ∞ ∑ 𝑛=0 𝑦𝑛𝑡𝑛+𝐶𝑡 ∞ ∑ 𝑛=0 (𝑛 ∑ 𝑚=0 𝑦𝑚𝑦𝑛−𝑚)𝑡𝑛=𝑦0+𝐶𝑡 (𝜓(𝑡) + 𝜓(𝑡)2). This can be viewed as an algebraic equation with a functional root. Indeed, let 𝜙∶R2→R,𝜙(𝑡, 𝑤) = 𝑤−𝑦0−𝐶𝑡(𝑤+𝑤2). This map is analytic, 𝜙(0, 𝑦0)=0and 𝜕𝑤𝜙(0, 𝑦0)=1≠0. By the implicit-function theorem, 𝜙(𝑡, 𝑤(𝑡)) = 0 for a unique analytic function 𝑤∶ (−𝛿, 𝛿)→Rsuch that 𝑤(0) = 𝑦0. These conditions imply that 𝑤=𝜓is analytic. 3.5. Euler numbers Like in [10,17], a class of fractional Euler numbers can be introduced from (3.13). Euler numbers are very important because they can be related to the logistic function and the Riemann zeta function [8], 𝜁(𝑠) = ∑∞ 𝑛=1 𝑛−𝑠. Let us revisit [8]. The sum defining 𝜁(𝑠)is convergent when Re(𝑠)>1, so 𝜁is defined on the complex plane Cby analytic continuation. The logistic function (3.2), corresponding to the initial condition 𝑥0= 1∕2, satisfies 𝑥(𝑡) = 1 1+e−𝑡= ∞ ∑ 𝑛=0 1 2𝐸𝑛 𝑡𝑛 𝑛!, where {𝐸𝑛∶𝑛= 0,1,2,…} are the Euler numbers. Then 𝑥𝑛=𝐸𝑛 𝑛!⋅2, 𝑥(𝑛)(0) = 𝐸𝑛 2(3.16) Applied Mathematics Letters 154 (2024) 109085 5 M. Jornet and J.J. Nieto Fig. 1. Graph of 𝑥𝑁(𝑡). First panel: L-equation. Second panel: Caputo equation. Table 1 Some L-fractional Euler numbers. 𝐸𝛼 𝑛𝑛= 1 𝑛= 2 𝑛= 3 𝑛= 4 𝑛= 5 𝑛= 6 𝑛= 7 𝑛= 8 𝑛= 9 𝑛= 10 𝛼= 0.10.5 0 −0.131579 0 0.073477 0 −0.0537188 0 0.0456643 0 𝛼= 0.70.5 0 −0.192308 0 0.233880 0 −0.506359 0 1.6341 0 𝛼= 0.99 0.5 0 −0.247525 0 0.486089 0 −2.01594 0 14.2791 0 𝛼= 1 0.5 0 −0.25 0 0.5 0 −2.125 0 15.5 0 are the terms of the power series of 𝑥and the 𝑛th derivatives, respectively. See (3.5). On the other hand, the Euler zeta function, 𝜁𝐸(𝑠)=2∑∞ 𝑛=1(−1)𝑛𝑛−𝑠, satisfies 𝜁𝐸(𝑠) = −2(1 − 21−𝑠)𝜁(𝑠).(3.17) It is known that, at negative integers, 𝜁𝐸(−𝑘)=−𝐸𝑘, 𝑘 = 1,2,…,(3.18) so that, by (3.16)–(3.18), 𝑥(𝑛)(0) = −𝜁𝐸(−𝑛) 2= (1 − 21+𝑛)𝜁(−𝑛), 𝑛 = 1,2,…. We can define L-fractional Euler numbers 𝐸𝛼 𝑛, based on the solution of (3.10) with 𝑥0= 1∕2. Notice that, above, 𝐸𝑛is the 𝑛th derivative of 2𝑥(𝑡) = 2∕(1 + e−𝑡)at 𝑡= 0, where 𝑥solves the ordinary logistic equation for 𝑥0= 1∕2. Consequently, we define 𝐸𝛼 𝑛as the 𝑛th 𝛼-derivative (𝐿 𝐷𝛼◦ ⋯ ◦𝐿 𝐷𝛼) of 2𝑥𝛼(𝑡)at 𝑡= 0, where 𝑥𝛼(𝑡) = ∑∞ 𝑛=0 𝑥𝛼 𝑛𝑡𝑛solves the L-fractional logistic equation with 𝑥0= 1∕2. By successive differentiation in (3.12), one gets 2𝑥𝛼(𝑡) = ∞ ∑ 𝑛=0 2𝑥𝛼 𝑛𝑡𝑛= ∞ ∑ 𝑛=0 𝐸𝛼 𝑛 𝛤(2 − 𝛼)𝑛∏𝑛 𝑗=1 𝛤(𝑗+1) 𝛤(𝑗+1−𝛼) 𝑡𝑛= ∞ ∑ 𝑛=0 𝐸𝛼 𝑛 𝛤(2 − 𝛼)𝑛𝛤(1 + 𝛼)𝑛∏𝑛 𝑗=1 (𝑗 𝑗−𝛼)𝑡𝑛. Thus, 𝐸𝛼 𝑛= 2𝑥𝛼 𝑛𝛤(2 − 𝛼)𝑛 𝑛 ∏ 𝑗=1 𝛤(𝑗+ 1) 𝛤(𝑗+1−𝛼)= 2𝑥𝛼 𝑛𝛤(2 − 𝛼)𝑛𝛤(1 + 𝛼)𝑛 𝑛 ∏ 𝑗=1 (𝑗 𝑗−𝛼).(3.19) The coefficients 𝑥𝛼 𝑛are defined recursively by (3.13). When 𝛼→1−, the behavior is consistent: 𝐸𝛼 𝑛→2𝑥𝑛⋅(𝑛!) = 𝐸𝑛. For Caputo fractional equations, the Euler numbers 𝐶𝐸𝛼 𝑛are 𝐶𝐸𝛼 𝑛= 2𝑥𝛼 𝑛𝛤(𝑛𝛼 + 1), where 𝑥𝛼 𝑛solves (3.9). 3.6. Numerical example For 𝑥0= 0.5and 𝛼= 0.1, we plot in Fig. 1 the truncated solution 𝑥𝑁(𝑡)for several odd degrees 𝑁. We work with the L-solution (first panel) and the Caputo solution (second panel). The curves are not the same and may capture different dynamics. A clear difference is the value of 𝑥′(0): for the first panel, it is finite, but for the second panel, it is ∞[15]. When 𝛼becomes smaller, differences between the fractional derivatives are more notable. The rate of convergence of the sums (visual overlap) is slower in the L-fractional case, likely due to the inclusion of 𝑡1−𝛼for 𝐿 𝐷𝛼which increases the nonlinearity. The profile of the L-fractional solution is more similar to the case 𝛼= 1, probably due to the differential nature of the derivative, (1.3). Essentially, the evolution with 𝛼∈ (0,1] is smoother. These facts agree with the plots presented in [7] for linear equations and the Mittag-Leffler function. In Table 1, some Euler numbers (3.19) are reported and compared with 𝛼= 1. The computations of this paper were conducted with the software Mathematica®(Wolfram Research, Inc., version 12.1, Champaign, 2020). Applied Mathematics Letters 154 (2024) 109085 6 M. Jornet and J.J. Nieto 4. Conclusions Since the L-fractional derivative defines a differential, it has a geometrical meaning. Then, usual models of population transitions can be extended from integer to non-integer order. The method of power series is a powerful tool to derive local analytic solutions to L-fractional differential equations, as shown with the logistic model. In contrast to the Caputo derivative, the series are not fractional, but classical analytic. The convergence of the series is proved by means of the implicit-function theorem. Euler numbers can be generalized to a fractional index. Since the solutions to these L-fractional differential equations are smooth and not merely absolutely continuous, the fractional operator can actually be written as an integral with respect to a non-singular kernel (by integration by parts), but with no controversies at the initial instant. It seems that the factor 1∕𝑡1−𝛼, or 1∕𝐷𝛼𝑡, avoids issues associated to non-singular kernels. Further, this factor makes dimensionality consistent, so that the vector field is a true velocity. Some graphs are plotted to illustrate the results. One observes that the Caputo solution starts with infinite slope at 𝑡= 0 and that the L-solution is more similar to the ordinary solution. Some Euler numbers are tabulated. Funding The research of J.J. Nieto was supported by the Agencia Estatal de Investigación (AEI) of Spain Grant PID2020-113275GB-I00 funded by MCIN/AEI/10.13039/501100011033 and by ‘‘ERDF A way of making Europe’’, by the ‘‘European Union’’ and Xunta de Galicia, grant ED431C 2023/12 for Competitive Reference Research Groups (2023–2026). Data availability No data was used for the research described in the article. References [1] A.A. Kilbas, H.M. Srivastava, J.J. Trujillo, Theory and Applications of the Fractional Differential Equations, North-Holland Mathematics Studies, 2006. [2] F. Ndaïrou, I. Area, J.J. Nieto, C.J. Silva, D.F. Torres, Fractional model of COVID-19 applied to Galicia, Spain and Portugal, Chaos, Solitons Fract. 144 (2021) 110652. [3] D. Baleanu, F.A. Ghassabzade, J.J. Nieto, A. Jajarmi, On a new and generalized fractional model for a real cholera outbreak, Alex. Eng. J. 61 (2022) 9175–9186. [4] K.A. Lazopoulos, A.K. Lazopoulos, Fractional vector calculus and fractional continuum mechanics, Prog. 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