Limitations and applications in a fractional Barbalat’s Lemma

Abstract

Barbalat’s Lemma is a mathematical result that can lead to the solution of many asymptotic stability problems. On the other hand, Fractional Calculus has been widely used in mathematical modeling, mainly due to its potential to make explicit the dependence of previous stages through nonlocal operators. In this work, we present a fractional Barbalat’s Lemma and its proof, as proposed in [31]. The proof is analyzed in order to show an imprecision. In fact, for orders $0<\alpha <1$, we are not able to get the supreme limit of the integrand. Then, a counterexample and a corrected version of the lemma are presented, according to [9]. The objective of this work is to draw attention to the potential and limitations of a fractional Barbalat’s Lemma, given its wide use in recent articles. In a fractional SIR model, we exhibit the constraint of the result by introducing a non-periodic relapse. So, the supreme limit could not be verified. Also in this context, we provide a general discussion of the classical Calculus’ properties that are not inherited if we change the integer orders to fractional ones.

Introduction

Abel is remembered as the first mathematician that applied the Arbitrary-Order Calculus, more known as Fractional Calculus, to a real-world problem. In fact, Abel’s solution for the tautochrone problem, based upon mathematical analysis, leads to a fractional integral. After that, some pioneers of the applications of Fractional Calculus are recalled in [27]. However, it was only after the first International Conference on Fractional Calculus and Applications, in 1974, that the Fractional Calculus has been broadly used in several areas. One can cite applications in epidemiology [2], quantum transport [7], fibrillation [26], diffusion [24], and so on. In addition, behaviors of long memory type can also be studied via delay differential equations [3], which allows a cross between the Fractional Calculus and the theory of DDE’s, expanding the range of applications and studies. Thus, it is natural that the theories of the classical Calculus, in particular the integer-order differential equations, are revisited, in view of their possible adaptation to Fractional Calculus. This adaptation, however, is not always immediate. For instance, the Mean Value theorem is not local in Fractional Calculus, which implies that the sign of the derivative is not sufficient to indicate the monotonicity.

In this context, we aim to verify the validity of a fractional Barbalat’s Lemma. In ODE’s theory, Barbalat’s Lemma is a mathematical result concerning the asymptotic properties of functions and their derivatives. When used correctly for dynamical systems, it can lead to the solution of many asymptotic stability problems, among which we emphasize compartmental epidemiological models ([4, 10]). In general terms, it deals with the convergence to zero of a sufficiently well-behaved function whose integral is bounded.

An intuitive idea for the generalization of this lemma would be to consider the fractional integral of order $\alpha$ in the statement. In 2017, this generalization was publicized in [31]. However, in 2015, [9] had already demonstrated that the lemma is not valid for $0<\alpha <1$. Even though the same authors are cited in [31], the proved fallibility of the lemma is not considered.

The Barbalat’s Lemma as proposed in [31] is used in many works from last years, some of which also cite the authors of [9]. Two examples from 2020 are [12] and [30]. Already in 2022, we found in [13] the same lemma, used to investigate the global stability of a fractional-order HBV infection. Also in 2022, the result is used for the Caputo-Fabrizio fractional derivative, in an asymptotic analysis of novel coronavirus disease via fractional-order epidemiological model [14].

Thus, given the importance of the lemma for recent biomathematical models, it is necessary to provide a dialogue between the proof of [31] and the counterexample of [9]. For this, we introduce in Sections 2 and 3 some preliminary discussion and the classic Barbalat’s Lemma. So, in Section 4, we discuss the generalization problems. Hoping to attract the attention of researchers to the care with the use of the fractional Barbalat’s Lemma, we also provide some numerical results. In Section 5, we discuss conditions for the validity of fractional Barbalat’s Lemma and, in Section 6, we end with a brief discussion about Lyapunov’s fractional theory and an example of application in the Caputo fractional SIR model with and without relapse.

The Fractional Calculus and some of its uncommon behaviors

Fractional Calculus is over 320 years old, but only in the last few decades has it shown substantial growth as an area of research. It is a fertile area, including allowing to deal explicitly with the “memory effect” of many phenomena, considering the dependence of previous stages on materials or processes and, in this context, optimizing the modeling of phenomena and systems. However, Fractional Calculus has some unusual behaviors and, in this sense, many classical Calculus’ theories need to be revised before they can continue to be used. In [21], we draw attention to properties that are not inherited from the integer-order Calculus, beginning with the issue of different definitions, then moving on to Mean Value theorems and topics of “persistence” of properties.

As the focus of this work is to discuss different approaches about a fractional Barbalat’s Lemma and parallel questions, we begin with some basic considerations. Not only the Barbalat’s Lemma cannot be directly translated to fractional integrals, but even the sign of the derivative is no longer valid as an indication of the monotone behavior of the function, as we see in Subsection 2.2.

Preliminaries

The Fractional Calculus probably was born in 1695, when, as the legend says, l’Hôpital asked Leibniz about the meaning of a derivative of order 1/2. Over next centuries, important advances were made by Liouville, Riemann, Grünwald, Caputo, and many others. However, it was only after the first International Conference on Fractional Calculus and Applications, in 1974, that the number of researchers in Fractional Calculus showed great growth. Currently, congresses and symposia take place more frequently, and the reader may refer to the reference [5] for a chronology of publications in Fractional Calculus until 2019, as well as for general results.

Below we consider [a, b] a finite real interval, and $\alpha$ a real number such that $0\le n-1<\alpha <n$, with n integer. The extension of the idea of the iterated integral leads to the following definition:

Definition 1

(Riemann-Liouville integral in finite intervals) The Riemann-Liouville integral of an arbitrary order $\alpha$ is set to $t\in [a,b]$ by

$$\begin{array}{c}\hfill {\displaystyle I_{a+}^{\alpha }f(t)=\frac{1}{\mathrm{\Gamma }(\alpha )}{\int }_a^t{(t-\theta )}^{\alpha -1}f(\theta )d\theta .}\end{array}$$ 2.1

After introducing the arbitrary-order integral, it is natural to search for the definition of the corresponding derivative:

Definition 2

[Riemann-Liouville derivative in finite intervals] The Riemann-Liouville derivative of an arbitrary order $\alpha$ is set to $t\in [a,b]$ by

$$\begin{array}{c}\hfill {\displaystyle D_{a+}^{\alpha }f(t)=D^n[I_{a+}^{n-\alpha }f(t)]=\frac{1}{\mathrm{\Gamma }(n-\alpha )}\left(\frac{d^n}{d{t}^n}\right){\int }_a^t{(t-\theta )}^{n-\alpha -1}f(\theta )d\theta ,}\end{array}$$ 2.2

with $D^n$ representing the integer-order derivative.

We also present the definition of Caputo derivative of arbitrary order, for which, among other characteristics, the derivative of a constant is zero:

Definition 3

(Caputo derivative in finite intervals) The Caputo derivative with arbitrary order $\alpha$ is defined for $t\in [a,b]$ by

$$\begin{array}{c}\hfill {\displaystyle {}^C{D}_{a+}^{\alpha }f(t)=I_{a+}^{n-\alpha }[D^{n}f(t)]={\displaystyle \frac{1}{\mathrm{\Gamma }(n-\alpha )}}{\int }_a^t{(t-\theta )}^{n-\alpha -1}{\displaystyle \frac{d^n}{d{\theta }^n}}f(\theta )d\theta .}\end{array}$$ 2.3

In the next section, we see that the sign of the fractional derivative does not imply monotonicity, which, in particular, is a challenge to fractional Lyapunov theories.

The sign of the fractional derivative does not imply monotonicity

In classical Calculus, the sign of the derivative indicates when the function is decreasing or increasing. However, in Arbitrary-Order Calculus, this feature does not have a simple correspondent: the sign of the arbitrary-order derivative is not indicative of monotonicity. This is a great challenge in the recent plenty of publications in Fractional Calculus. In fact, based on an understandable mistake about the fractional Mean Value theorem, many authors (e.g. [11, 13]) state the following result, especially in the case of the arbitrary-order Caputo derivative:

Let $\alpha \in (0,1)$ and suppose $f(t),{}^C{D}_{a+}^{\alpha }f(t)\in C[a,b]$. It follows from the Mean Value theorem that, if ${}^C{D}_{a+}^{\alpha }f(t)\ge 0$ in the interval [a, b], then f is nondecreasing in [a, b]. Similarly, if ${}^C{D}_{a+}^{\alpha }f(t)\le 0$ in the interval [a, b], then f is nonincreasing in that interval.

The problem is that, without other hypotheses, this result is not valid. The example that we discuss is based on [6] and, in [15], we also can see that this assertion fails in the SIR model.

Consider the function $y(t)=t^3-1.5t^2+0.5t$. We have $y^{\prime }(t)=3t^2-3t+0.5$. Given any $\alpha \in (0,1)$, the arbitrary-order derivative in the Caputo sense of y(t) is given by

$$\begin{array}{c}\hfill {}^C{D}_{0+}^{\alpha }y(t)={\displaystyle \frac{t^{1-\alpha }}{\mathrm{\Gamma }(4-\alpha )}}(6t^2-3(3-\alpha )t+0.5(3-\alpha )(2-\alpha )).\end{array}$$ 2.4

Thus, ${}^C{D}_{0+}^{\alpha }y(t)$ has no sign variation in the interval

$$\begin{array}{c}\hfill T_{\alpha }=\left[0,,,{\displaystyle \frac{3(3-\alpha )-\sqrt{3(1+\alpha )(3-\alpha )}}{12}}\right].\end{array}$$ 2.5

Now, we notice that

$$\begin{array}{c}\hfill {\displaystyle \frac{d}{d\alpha }}[{\displaystyle \frac{3(3-\alpha )-\sqrt{3(1+\alpha )(3-\alpha )}}{12}}]={\displaystyle \frac{1}{12}}[-{\displaystyle \frac{\sqrt{3}(1-\alpha )}{\sqrt{(1+\alpha )(3-\alpha )}}}-3]<0.\end{array}$$ 2.6

So,

$$\begin{array}{cc}\hfill {\displaystyle \frac{3(3-\alpha )-\sqrt{3(1+\alpha )(3-\alpha )}}{12}}& >{\displaystyle \frac{3(3-1)-\sqrt{3(1+1)(3-1)}}{12}}\hfill \\ \hfill & ={\displaystyle \frac{3-\sqrt{3}}{6}}.\hfill \end{array}$$ 2.7

This implies that, for any $\alpha \in (0,1)$, ${}^C{D}_{0+}^{\alpha }y(t)$ has no sign variation in the interval $T_{\alpha }=\left[0,,,(3-\sqrt{3}),/,6,+,{ϵ}_{\alpha }\right]$ for some ${ϵ}_{\alpha }>0$, as illustrated in Figure 1.

Fig. 1.

Integer and fractional derivatives

However, y(t) has a local maximum at $(3-\sqrt{3})/6$, as we can see in Figure 2. So, despite the fact that we have ${}^C{D}_{0+}^{\alpha }y(t)\ge 0$ for all $t\in T_{\alpha }$, the function y(t) is not monotonous in $T_{\alpha }$.

Fig. 2.

Behavior of y(t)

In [16], we also discuss the difficulty of defining initial conditions for IVP, since, as the arbitrary-order derivatives are nonlocal, all the past must be taken into account. Moreover, we highlight the difference in the analysis of extreme points and equilibrium, in comparison with classical Calculus, between some other aspects that deserve attention.

The classic Barbalat’s Lemma

Now, we focus on the classic Calculus’ result that is supposed to be extended to Fractional Calculus. But, before discussing Barbalat’s Lemma, we recall two points about asymptotic properties of functions and their derivatives. Given a function of t, the following facts are important:

Remark 1

[25] The limit $df/dt\to 0$ does not imply convergence of f. Consider, for example, the function $f(t)=sin(logt)$. While

$$\begin{array}{c}\hfill {\displaystyle \frac{\mathit{df}}{\mathit{dt}}}={\displaystyle \frac{cos(logt)}{t}}\to 0,\end{array}$$ 3.1

the f function continues to oscillate, more and more slowly, as illustrated in Fig. 3

Fig. 3.

Remark 1: $f(t)=sin(log(t))$

Remark 2

[25] Convergence of f does not imply $df/dt\to 0$.

For example, the nonnegative function $f(t)=e^{-t}{sin}^2(e^{2t})$ tends to zero, but its derivative,

$$\begin{array}{c}\hfill {\displaystyle \frac{\mathit{df}}{\mathit{dt}}}=4e^{t}cos(e^{2t})sin(e^{2t})-e^{-t}{sin}^2(e^{2t}),\end{array}$$ 3.2

is unlimited, as illustrated in Fig. 4 .

Fig. 4.

Remark 2: $f(t)=e^{-t}{sin}^2(e^{2t})$

So, given that a function tends to a finite limit, what additional requirement can guarantee that its derivative converges to zero? Barbalat’s Lemma indicates that the derivative itself must have some smoothness. More precisely,

Lemma 1

[Barbalat] If the differentiable function f(t) has a finite limit when $t\to \infty$, and if df/dt is uniformly continuous, then $df/dt\to 0$ when $t\to \infty$.

The proof of the lemma can be consulted at [25]. To apply Barbalat’s Lemma to the analysis of dynamical systems, we usually use the following immediate corollary:

Lemma 2

(Lemma “Lyapunov-type”) [25] If a scalar function V(x, t) satisfies the following conditions:

• V(x, t) is bounded from below,

• $\dot{V}(x,t)$ is negative semidefinite,

• $\dot{V}(x,t)$ is uniformly continuous in time,

then $\dot{V}(x,t)\to 0$ when $t\to \infty$.

In fact, V approaches a finite limit $V_{\infty }$, such that $V_{\infty }\le V(x(0),0)$ (this does not require uniform continuity). The above lemma then follows from Barbalat’s Lemma.

In this work, we use the following version of the lemma, which can be obtained directly from the first one:

Lemma 3

[Barbalat] If the uniformly continuous nonnegative function f(t) in $[0,\infty ]$ is such that ${\int }_0^{t}f(\tau )d\tau <C$, for some constant C and all $t>0$, then $f(t)\to 0$ when $t\to \infty$.

This version is broadly used in mathematical modeling. For instance, it was used to prove the boundedness of solutions of a viral infection in a pest control model, in [10]. In 2017, was published a result for the convergence of a function based on its fractional derivative and integration, given by

Lemma 4

[28] If ${\int }_0^{t}f(\tau )d\tau$ has a finite limit as $t\to +\infty$, and if ${}^C{D}_{0+}^{\alpha }f(t)$ is bounded, where $f:[0,+\infty )\to \mathbb{R}$, then $f(t)\to 0$ as $t\to +\infty$, where $0<\alpha <1$.

However, this is not always applicable, so an intuitive idea for the generalization of Lemma 3 is to consider the fractional integral of order $\alpha$ in its statement. Also in 2017, this generalization was publicized in [31], as we discuss in next section.

Fractional Barbalat’s Lemma: the generalization problem

As mentioned in the Introduction, in Theorem 3.1 of [31] and related references it is found the following fractional Barbalat’s Lemma:

Let $\varphi :\mathbb{R}\to \mathbb{R}$ be a uniformly continuous function on $[t_0,\infty ]$, and, with $\alpha \in (0,1)$, p and M two positive constants, $I_{t_0+}^{\alpha }\mid \varphi (t){\mid }^p\le M$ for all $t>0$. Then,

$$\begin{array}{c}\hfill \underset{t\to \infty }{lim}\varphi (t)=0.\end{array}$$

The case $\alpha =1$ is the traditional case. Let us discuss the presented demonstration. We assume for the sake of contradiction that there exists a positive scalar $\epsilon$ and a sequence ${(t_k)}_{k\in \mathbb{N}}$ with $t_k\to \infty$ such that $\mid \varphi (t_k)\mid >\epsilon$. Let us assume without loss of generality that $\mid t_{k+1}-t_k\mid >{\delta }_0$ for all k. This implies that the intervals $[t_k-{\delta }_0/2,t_k+{\delta }_0/2]$ do not overlap. By the uniform continuity of $\varphi$, there is a $\delta$, which we will assume without loss of generality less than ${\delta }_0/2$, such that

$$\begin{array}{c}\hfill \mid \varphi (t^{\prime })-\varphi (t^{\prime \prime })\mid <{\displaystyle \frac{\epsilon }{2}},\end{array}$$ 4.1

for any $t^{\prime },t^{\prime \prime }$ such that $\mid t^{\prime }-t^{\prime \prime }\mid <\delta .$ Then, for every t in $[t_k-\delta ,t_k+\delta ]$, we have

$$\begin{array}{c}\hfill \mid \varphi (t)\mid \ge \mid \varphi (t_k)\mid -\mid \varphi (t_k)-\varphi (t)\mid >{\displaystyle \frac{\epsilon }{2}},\end{array}$$ 4.2

whence

$$\begin{array}{c}\hfill \mid \varphi (t){\mid }^p>{\left({\displaystyle \frac{\epsilon }{2}}\right)}^p.\end{array}$$ 4.3

If $0<\alpha <1$, and $n>k$, we have $-2\delta /(t_n-t_k+\delta )>-1$. So, by Bernoulli’s inequality,

$$\begin{array}{c}\hfill {\left(1,-,{\displaystyle \frac{2\delta }{t_n-t_k+\delta }}\right)}^{\alpha }\le 1-{\displaystyle \frac{2\delta \alpha }{t_n-t_k+\delta }},\end{array}$$ 4.4

that is,

$$\begin{array}{c}\hfill 1-{\left(1,-,{\displaystyle \frac{2\delta }{t_n-t_k+\delta }}\right)}^{\alpha }\ge {\displaystyle \frac{2\delta \alpha }{t_n-t_k+\delta }}.\end{array}$$ 4.5

Then, for $n>k$, we have

$$\begin{array}{cc}& {(t_n-t_k+\delta )}^{\alpha }-{(t_n-t_k-\delta )}^{\alpha }\hfill \\ \hfill & ={(t_n-t_k+\delta )}^{\alpha }\left[1,-,{\left(1,-,{\displaystyle \frac{2\delta }{t_n-t_k+\delta }}\right)}^{\alpha }\right]\hfill \\ \hfill & \ge {(t_n-t_k+\delta )}^{\alpha }\left({\displaystyle \frac{2\delta \alpha }{t_n-t_k+\delta }}\right)\ge 2\delta \alpha {(t_n-t_1+\delta )}^{\alpha -1}.\hfill \end{array}$$ 4.6

Hence, it follows that, for $t>t_n$,

$$\begin{array}{cc}\hfill I_{t_0+}^{\alpha }\mid \varphi (t){\mid }^p& \ge {\displaystyle \frac{1}{\mathrm{\Gamma }(\alpha )}}{\int }_{t_0}^{t_n}\mid \varphi (\tau ){\mid }^p{(t_n-\tau )}^{\alpha -1}d\tau \hfill \\ \hfill & \ge {\displaystyle \frac{1}{\mathrm{\Gamma }(\alpha )}}\sum _{k=1}^{n-1}{\int }_{t_k-\delta }^{t_k+\delta }\mid \varphi (\tau ){\mid }^p{(t_n-\tau )}^{\alpha -1}d\tau \hfill \\ \hfill & +{\displaystyle \frac{1}{\mathrm{\Gamma }(\alpha )}}{\int }_{t_n-\delta }^{t_n}\mid \varphi (\tau ){\mid }^p{(t_n-\tau )}^{\alpha -1}d\tau \hfill \\ \hfill & \ge {\displaystyle \frac{{\epsilon }^p}{2^p\alpha \mathrm{\Gamma }(\alpha )}}\sum _{k=1}^{n-1}[{(t_n-t_k+\delta )}^{\alpha }-{(t_n-t_k-\delta )}^{\alpha }]+{\displaystyle \frac{{\epsilon }^p{\delta }^{\alpha }}{2^p\alpha \mathrm{\Gamma }(\alpha )}}\hfill \\ \hfill & \ge {\displaystyle \frac{{\epsilon }^p(n-1)\delta {(t_n-t_1+\delta )}^{\alpha -1}}{2^{p-1}\mathrm{\Gamma }(\alpha )}}+{\displaystyle \frac{{\epsilon }^p{\delta }^{\alpha }}{2^p\alpha \mathrm{\Gamma }(\alpha )}}.\hfill \end{array}$$ 4.7

A touchy situation occurs here, since it is considered $\gamma =max\{t_{k+1}-t_k\},$ for $k=1,2,\cdots ,n-1.$ With this assumption, we have

$$\begin{array}{cc}\hfill I_{t_0+}^{\alpha }\mid \varphi (t){\mid }^p& \ge {\displaystyle \frac{{\epsilon }^p(n-1)\delta {[(n-1)\gamma +\delta ]}^{\alpha -1}}{2^{p-1}\mathrm{\Gamma }(\alpha )}}+{\displaystyle \frac{{\epsilon }^p{\delta }^{\alpha }}{2^p\alpha \mathrm{\Gamma }(\alpha )}}\hfill \\ \hfill & ={\displaystyle \frac{{\epsilon }^p{(n-1)}^{\alpha }\delta }{2^{p-1}\mathrm{\Gamma }(\alpha ){[\gamma +\delta /(n-1)]}^{1-\alpha }}}+{\displaystyle \frac{{\epsilon }^p{\delta }^{\alpha }}{2^p\alpha \mathrm{\Gamma }(\alpha )}}.\hfill \end{array}$$ 4.8

So, letting $n\to \infty$, we get $I_{t_0+}^{\alpha }\mid \varphi (t){\mid }^p\to \infty$, contradiction. This implies that $\varphi (t)\to 0$.

Remark 3

The proof above considers $\gamma =max\{t_{k+1}-t_k\},k=1,2,\cdots ,n-1.$ Thus, $\gamma$ depends on n. If we write ${\gamma }_n$ to remember this dependency, what we get is

$$\begin{array}{c}\hfill I_{t_0+}^{\alpha }\mid \varphi (t){\mid }^p\ge {\displaystyle \frac{{\epsilon }^p{(n-1)}^{\alpha }\delta }{2^{p-1}\mathrm{\Gamma }(\alpha ){[{\gamma }_n+\delta /(n-1)]}^{1-\alpha }}}+{\displaystyle \frac{{\epsilon }^p{\delta }^{\alpha }}{2^p\alpha \mathrm{\Gamma }(\alpha )}}.\end{array}$$ 4.9

When doing $n\to \infty$, we should be able to study ${lim}_{n\to \infty }{\gamma }_n$. If we have $lim{sup}_{n\to \infty }{\gamma }_n=L$ for finite L, then the proof is valid. However, the proof completion can be invalid if we have $lim{sup}_{n\to \infty }{\gamma }_n=\infty$. As in the case of the function exhibited in Proposition 1, we can have $I_{t_0+}^{\alpha }\mid \varphi (t){\mid }^p$ limited, even though $\varphi (t)\nrightarrow 0$. The idea is that the influence of the past is decreasing, with has relation with the Short Memory principle of the arbitrary-order operators [23].

In fact, for $0<\alpha <1$, a direct generalization of Barbalat’s Lemma is not true, as shown in next proposition, published in 2015:

Proposition 1

[9] If $0<\alpha <1$, there exists a nonnegative uniformly continuous function f such that $I_{0+}^{\alpha }f(t)<M$ for all t, with M a positive constant, but f does not converge to zero when t goes to infinity.

The proof uses the following lemma:

Lemma 5

[9] If f is a bounded function that vanishes for all $t>T$, then $I_{0+}^{\alpha }f\to 0$. Also, $I_{0+}^{\alpha }f$ will be a uniformly continuous function.

Proof

Proceeding with the proof of Proposition 1, let p(t) be a null function at all points except the intervals $[t_i,t_i+\delta ]$, where it takes the value 1 (nonperiodic pulse), with ${(t_i)}_{i\in \mathbb{N}}$ an increasing divergent sequence to be specified and $\delta$ a fixed positive real to be specified.

Note that, for every t and every $\tau <t$, the function $p(\tau )$ can be written as $p(\tau )={\sum }_{i=1}^n{p}_i(\tau ),$ where $n=max\{i:t\ge t_i\}$ and $p_i(t)$ is a function that vanishes outside the interval $[t_i,t_i+\delta ]$, in which it takes the value 1.

For every i, we have $I_{0+}^{\alpha }{p}_i(t)<C_1$, where $C_1$ is a positive constant. Indeed,

• If $t<t_i$, then $I_{0+}^{\alpha }{p}_i(t)=0$;

• If $t\in [t_i,t_i+\delta ]$, then

  $$\begin{array}{c}\hfill I_{0+}^{\alpha }{p}_i(t)={\displaystyle \frac{1}{\mathrm{\Gamma }(\alpha )}}{\int }_{t_i}^t{(t-\tau )}^{\alpha -1}d\tau ={\displaystyle \frac{{(t-t_i)}^{\alpha }}{\alpha \mathrm{\Gamma }(\alpha )}}\le {\displaystyle \frac{{\delta }^{\alpha }}{\alpha \mathrm{\Gamma }(\alpha )}};\end{array}$$ 4.10
• If $t>t_i+\delta$, then

  $$\begin{array}{cc}\hfill I_{0+}^{\alpha }{p}_i(t)& ={\displaystyle \frac{1}{\mathrm{\Gamma }(\alpha )}}{\int }_{t_i}^{t_i+\delta }{(t-\tau )}^{\alpha -1}d\tau \hfill \\ \hfill & ={\displaystyle \frac{{(t-t_i)}^{\alpha }-{(t-t_i-\delta )}^{\alpha }}{\alpha \mathrm{\Gamma }(\alpha )}}\le I_{0+}^{\alpha }{p}_i(t_i+\delta )={\displaystyle \frac{{\delta }^{\alpha }}{\alpha \mathrm{\Gamma }(\alpha )}},\hfill \end{array}$$ 4.11

  since $I_{0+}^{\alpha }{p}_i(t)$ is monotonous decreasing for $t>t_i+\delta$, because, in this case,

  $$\begin{array}{c}\hfill {\displaystyle \frac{d}{\mathit{dt}}}[{(t-t_i)}^{\alpha }-{(t-t_i-\delta )}^{\alpha }]=\alpha [{(t-t_i)}^{\alpha -1}-{(t-t_i-\delta )}^{\alpha -1}]<0.\\ \hfill \end{array}$$ 4.12

By the property of Lemma 5, for all i, we have $I_{0+}^{\alpha }{p}_i(t)\to 0$. So, by the definition of convergence, there is $T_i$ such that, for every $t>T_i$, we have $I_{0+}^{\alpha }{p}_i(t)<1/i^2.$

Since ${(t_i)}_{i\in \mathbb{N}}$ is a divergent sequence, we can without loss of generality exclude unnecessary terms and recursively redefine the sequence in a way that $t_{i+1}>T_i$. In particular, this makes the maximum step ${\gamma }_n$ of Remark 3 tending to infinity, which characterizes the conflict between the two proofs.

A viable choice would be to consider $t_{i+1}=T_i+1$ and $\delta$ defined by the equality ${\delta }^{\alpha }{\alpha }^{-1}\mathrm{\Gamma }{(\alpha )}^{-1}=1$. So, we have $I_{0+}^{\alpha }{p}_{i+1}(t_{i+1}+\delta )=1$, and

$$\begin{array}{c}\hfill I_{0+}^{\alpha }{p}_{i+1}(T_{i+1})\le {\displaystyle \frac{1}{{(i+1)}^2}}\le 1,\end{array}$$ 4.13

which implies $T_{i+1}\ge t_{i+1}+\delta >t_{i+1}=T_i+1$. Although $I_{0+}^{\alpha }{p}_{i+1}(t)>I_{0+}^{\alpha }{p}_i(t)$ for all $t>t_{i+1}+\delta$, the idea is that, for $t>T_{i+1}>T_i$, $I_{0+}^{\alpha }{p}_i(t)$ is so small, and $I_{0+}^{\alpha }{p}_i(t)<I_{0+}^{\alpha }{p}_{i+1}(t)<1/{(i+1)}^2$. This is illustrated in Figure 7. Note that the right tail of each wave is above of the right tail of previous wave, but all of them are decreasing.

Fig. 7.

Each integral wave illustrate the fractional integral for each bump

Hence, $t_{i+1}-t_i>1$ and ${(t_i)}_{i\in \mathbb{N}}$ is an increasing divergent sequence, as expected. Also,

$$\begin{array}{c}\hfill t_{i+1}-(t_i+\delta )=T_i-(t_i+\delta )+1>0,\end{array}$$ 4.14

hence the intervals $[t_i,t_i+\delta ]$ do not overlap.

In this context, for any $t\in {\mathbb{R}}^{+}$, there is $n\in \mathbb{N}$ such that $t_{n+1}\le t\le t_{n+2}$. By the linearity of the integral operator, we can write

$$\begin{array}{c}\hfill I_{0+}^{\alpha }p(t)=I_{0+}^{\alpha }\left(\sum _{i=1}^n,p_i,(t),+,p_{n+1},(t)\right)=\sum _{i=1}^n{I}_{0+}^{\alpha }{p}_i(t)+I_{0+}^{\alpha }{p}_{n+1}(t).\end{array}$$ 4.15

By construction, it follows that

$$\begin{array}{c}\hfill I_{0+}^{\alpha }p(t)\le \sum _{i=1}^n{\displaystyle \frac{1}{i^2}}+I_{0+}^{\alpha }{p}_{n+1}(t)<2+C_1.\end{array}$$ 4.16

Thus, we have a limited, positive function p that does not vanish at infinity and whose fractional integral remains limited. Now let f(t) be a positive triangular function such that, for all $t>0$, we have $f(t)\le p(t)$. For example, let f be null at every point, except in the intervals $[t_i,t_i+\delta /2],$ where it takes the values $2{\delta }^{-1}(t-t_i)$ and in the intervals $[t_i+\delta /2,t_i+\delta ],$ where it takes the values $2{\delta }^{-1}(t_i+\delta -t)$.

Then, f is uniformly continuous (more than that, it is Lipschitz continuous with constant $2{\delta }^{-1}$), is bounded, positive, does not vanish at infinity, and is such that, if $C\ge 2+C_1,$ one has

$$\begin{array}{c}\hfill I_{0+}^{\alpha }f(t)\le I_{0+}^{\alpha }p(t)<C,\end{array}$$ 4.17

for all t. This completes the proof. $\square$

Numerical results

In this subsection, we discuss Proposition 1 numerically. For that, we constructed a MATLAB code for plotting functions f and their Riemann-Liouville integrals $I_{0+}^{\alpha }f$. We define the f function with bumps in each interval $[t_i,t_i+\delta ]$ defined in last section, taking

$$\begin{array}{c}\hfill g(t)=\left\{\begin{array}{c}exp\left({\displaystyle \frac{1}{(t-t_i)(t-t_i-\delta )}}\right),\text{if}t\in (t_i,t_i+\delta ),\hfill \\ 0,\text{otherwise,}\hfill \end{array}\right)\end{array}$$ 4.18

and normalize by convenience, writing $f(t)=g(t)/max(g(t)).$ Note that this function is $C^{\infty }$ and, as its derivative is bounded, f is also Lipschitz.

For the sequence ${(t_i)}_{i\in \mathbb{N}},$ we define $t_1=2$ and, based in Eq. (4.11), we solve recursively the inequatility

$$\begin{array}{c}\hfill x^{\alpha }-{(x-1)}^{\alpha }<1/i^2.\end{array}$$ 4.19

So, we take $t_{i+1}=\delta x+t_i+1$. This is sufficient for the proof’s purpose. Figure 5 exhibits a suitable function for $\alpha =0.5$ and $\delta ={(\mathrm{\Gamma }(\alpha +1))}^{1/\alpha }$, as proposed. In Figure 6, we can see closely the bumps’ profile.

Fig. 5.

A Lipschitz aperiodic function

Fig. 6.

Bumps’ profile

In Figure 7, are illustrated the fractional integrals over time for each one of the bumps of Figure 5.

We note that the function does not converge to 0. However, $I_{0+}^{\alpha }f$, the sum of the waves of Figure 7, is illustrated in Figure 8, and is limited, because each bump only can occur when the definite integral until the time t is already lower than a threshold.

Fig. 8.

The limited Riemann-Liouville integral of f

Figures 9 and 10 exhibit the case $\alpha =1$, for which the classic Barbalat’s Lemma implies that $I_{0+}^{\alpha }f$ diverges. The same occurs with $\alpha >1$, as proved in Lemma 6. Of course, in this case we are not able to solve Eq. (4.19), because $1<1/i^2$ is an absurd. So, the example is merely illustrative.

Fig. 9.

A possible suitable function

Fig. 10.

Unlimited R-L integral, $\alpha =1$

Finally, we exemplify that, if the step of the sequence ${(t_i)}_{i\in \mathbb{N}}$ is limited, so the Barbalat’s Lemma with strong limit is valid, as proposed in [31]. This is precisely the heart of the counterexample constructed for the sake to prove Proposition 1. So, Figures 11 and 12 illustrate that, if without loss of generality the step is regular, the integral diverges.

Fig. 11.

A function with regular bumps

Fig. 12.

Unlimited R-L integral, $\alpha =0.5$

This detail is what [31] considers as ubiquitous in the discussed proof. But, as seen, it may be not true, and the supremum limit of the integrand may be not zero if the step size of the divergent sequence goes to infinity.

The fractional Barbalat’s Lemma, with conditions

From the previous discussion, it is not possible to generalize Barbalat’s Lemma to the case $0<\alpha <1$. However, the result is valid for $\alpha \ge 1$:

Lemma 6

[9] Let $\alpha \ge 1$ and f be a nonnegative uniformly continuous function such that $I_{0+}^{\alpha }f(t)<M$ for all t, with M a positive constant. So, f converges to zero.

Proof

The proof is given by contradiction. We assume that f does not converge to zero. Then, there is $\epsilon >0$ and an increasing divergent sequence ${(t_i)}_{i\in \mathbb{N}}$ such that $f(t_i)>\epsilon .$ Since f is uniformly continuous, there is $\delta >0$ such that, for all $i\in \mathbb{N}$, if $\mid t-t_i\mid <\delta ,$ then $\mid f(t)-f(t_i)\mid \le \epsilon /2$. So, if $t\in [t_i,t_i+\delta ]$, we have

$$\begin{array}{c}\hfill f(t)\ge f(t_i)-\mid f(t_i)-f(t)\mid >\epsilon /2.\end{array}$$ 5.1

Let p(t) be a null function at every point except if $t\in [t_i,t_i+\delta ]$, where it takes the value $\epsilon /2$. By definition, for $t>1$,

$$\begin{array}{c}\hfill \mathrm{\Gamma }(\alpha )I_{0+}^{\alpha }f={\int }_0^{t-1}{(t-\tau )}^{\alpha -1}f(\tau )d\tau +{\int }_{t-1}^t{(t-\tau )}^{\alpha -1}f(\tau )d\tau .\end{array}$$ 5.2

Since f is a positive function, then

$$\begin{array}{c}\hfill \mathrm{\Gamma }(\alpha )I_{0+}^{\alpha }f\ge {\int }_0^{t-1}{(t-\tau )}^{\alpha -1}f(\tau )d\tau .\end{array}$$ 5.3

Now, for $0\le \tau \le t-1$ and $\alpha \ge 1$, remembering that $f(t)\ge p(t)$ for all t, we have

$$\begin{array}{c}\hfill \mathrm{\Gamma }(\alpha )I_{0+}^{\alpha }f\ge {\int }_0^{t-1}f(\tau )d\tau \ge {\int }_0^{t-1}p(\tau )d\tau \ge n_t{\displaystyle \frac{\epsilon \delta }{2}},\end{array}$$ 5.4

where $n_t=max\{i:t_i+\delta \le t-1\}.$ Taking the limit when $t\to \infty ,$ we get

$$\begin{array}{c}\hfill \mathrm{\Gamma }(\alpha )I_{0+}^{\alpha }f(t)\ge \underset{t\to \infty }{lim}{n}_t{\displaystyle \frac{\epsilon \delta }{2}}=\infty ,\end{array}$$ 5.5

contradiction. Therefore, f converges to zero.

Remark 4

Since $I_{0+}^{\alpha }f$ can be expressed in terms of $t^{\alpha -1}\ast f$, where $\ast$ denotes the convolution operator, the lemma can be extended: if the convolution $g\ast f$ is uniformly bounded, where g is a nonnegative monotone increasing function ($g(t)=0$ for $t<0$) and f is a uniformly positive continuous function, then f converges to zero at infinity.

We note that (5.4) is not valid for $0<\alpha <1$. Therefore, we cannot extend the proof to these values of $\alpha$, which is supported by the previous section. However, one can at least assure the statement given in the next lemma:

Lemma 7

[9] Let f be a nonnegative bounded function such that $I_{0+}^{\alpha }f(t)<M$ for all t, with M a positive constant. So, ${lim\; inf}_{t\to \infty }f(t)=0$.

Proof

Within the context, the proof is even quite simple. Since f is bounded and nonnegative, ${lim\; inf}_{t\to \infty }f(t)$ exists and is nonnegative. Let us assume for the sake of contradiction that ${lim\; inf}_{t\to \infty }f(t)=l>0$. Given sufficiently small $\epsilon >0$, there exists $T_{\epsilon }>0$ such that $f(t)>l-\epsilon >0$ for $t>T_{\epsilon }.$

Define the function g such that g takes the values of f for $t\le T_{\epsilon }$ and $g(t)=l-\epsilon$ for $t>T_{\epsilon }$. So, given $t>T_{\epsilon }$, we have

$$\begin{array}{cc}\hfill I_{0+}^{\alpha }f(t)\ge I_{0+}^{\alpha }g(t)& ={\displaystyle \frac{1}{\mathrm{\Gamma }(\alpha )}}{\int }_0^t{(t-\tau )}^{\alpha -1}g(\tau )d\tau \hfill \\ \hfill & ={\displaystyle \frac{1}{\mathrm{\Gamma }(\alpha )}}{\int }_0^{T_{\epsilon }}{(t-\tau )}^{\alpha -1}g(\tau )d\tau +(l-\epsilon ){\displaystyle \frac{{(t-T_{\epsilon })}^{\alpha }}{\alpha }}\hfill \\ \hfill & \ge (l-\epsilon ){\displaystyle \frac{{(t-T_{\epsilon })}^{\alpha }}{\alpha }}\to \infty ,\hfill \end{array}$$ 5.6

contradiction.

We recommend the reference [9] for an extensive study of results with stronger hypotheses and extra conditions.

Application

It is natural that one thinks beyond and, inspired in the Lyapunov-type Lemma 2 for the classical Barbalat’s Lemma, aims to extend other features of the Lyapunov’s theory to Fractional Calculus. The same authors of [9] recall in [8] that Lyapunov’s basic theory requires monotonicity with respect to time for the functional L(x(t), t). A fundamental tool to prove its monotonicity is the sign of its derivative. More than that, if $L(t)=L(x(t),t)$ is monotonous decreasing and bounded, then it converges. But, as we see in Subsection 2.2, the property of monotonicity is not easily determined in Fractional Calculus.

Thus, many issues remain controversial in fractional-order systems. In [8], several studies on Lyapunov’s basic theory are proposed, and one of the useful tools is Barbalat’s Lemma 7.

The classic SIR model

In [16], we apply the presented theory to a SIR model of arbitrary order with the Caputo derivative. One of the results obtained directly from Barbalat’s Lemma is Theorem 1. We note that the result is not as conclusive as we would like:

Theorem 1

In the fractional SIR model given by

$$\begin{array}{cc}\hfill {}^C{D}_{0+}^{\alpha }S(t)=& -{\displaystyle \frac{\beta S(t)I(t)}{N}},\hfill \end{array}$$ 6.1

$$\begin{array}{cc}\hfill {}^C{D}_{0+}^{\alpha }I(t)=& {\displaystyle \frac{\beta S(t)I(t)}{N}}-\gamma I(t),\hfill \end{array}$$ 6.2

$$\begin{array}{cc}\hfill {}^C{D}_{0+}^{\alpha }R(t)=& \gamma I(t),\hfill \end{array}$$ 6.3

we have

$$\begin{array}{c}\hfill {\displaystyle {lim\; inf}_{t\to \infty }I(t)=0.}\end{array}$$ 6.4

Proof

The proof consists in rewriting the last equation as

$$\begin{array}{c}\hfill I_{0+}^{\alpha }(\gamma I(t))=R(t)-R(0).\end{array}$$ 6.5

Since S, I and R are limited [16], it follows from Lemma 7 that

$$\begin{array}{c}\hfill {\displaystyle {lim\; inf}_{t\to \infty }I(t)=0.}\end{array}$$ 6.6

Remark 5

We cannot use here Lemma 4, because the integer-order integral of I(t) is unlimited, even though ${}^C{D}_{0+}^{\alpha }I(t)$ is bounded.

Proof

In fact, it is assumed that the function S(t) is continuous in any finite interval [0, T], and so, takes on a minimum value m. Then, for $t\in [0,T]$, we have

$$\begin{array}{c}\hfill {}^C{D}_{0+}^{\alpha }I(t)\ge -AI(t),\end{array}$$ 6.7

for some A. This implies the existence of a continuous positive function q(t) s.t.

$$\begin{array}{c}\hfill {}^C{D}_{0+}^{\alpha }I(t)=-AI(t)+q(t).\end{array}$$ 6.8

Using the Laplace Transform, we get

$$\begin{array}{c}\hfill s^{\alpha }\mathcal{L}\{I\}-s^{\alpha -1}I(0)=-A\mathcal{L}\{I\}+\mathcal{L}\{q\},\end{array}$$ 6.9

from which

$$\begin{array}{c}\hfill \mathcal{L}\{I\}={\displaystyle \frac{s^{\alpha -1}I(0)+\mathcal{L}\{q\}}{s^{\alpha }+A}}.\end{array}$$ 6.10

Applying the inverse Laplace Transform and the Mittag-Leffler function (see, e.g., [5]), we get

$$\begin{array}{c}\hfill I(t)=I(0)E_{\alpha }(-A{t}^{\alpha })+q(t)\star (t^{\alpha -1}{E}_{\alpha ,\alpha }(-A{t}^{\alpha })),\end{array}$$ 6.11

where $\star$ is the Laplace convolution. Note that $t^{\alpha -1}{E}_{\alpha ,\alpha }(-A{t}^{\alpha })$ is the derivative of $-\frac{1}{A}{E}_{\alpha }(-A{t}^{\alpha })$ [16], which is increasing for any A. Once q(t) is also positive,

$$\begin{array}{c}\hfill I(t)\ge I(0)E_{\alpha }(-A{t}^{\alpha })>0,\end{array}$$ 6.12

for all $t\in [0,T]$. The same arguments prove the non-negativity of the other compartments, what is nontrivial because of nonlocal effects.

Particularly, once $S(t)>0$ for all t, we can state $A=\gamma$ and it is possible to write

$$\begin{array}{c}\hfill {\int }_0^{T}I(t)dt\ge I(0){\int }_0^T{E}_{\alpha }(-\gamma t^{\alpha })dt,\end{array}$$ 6.13

for all T.

As $E_{\alpha }(-\gamma t^{\alpha })$ is not integrable, then ${\int }_0^{\infty }I(t)$ diverges.

Figure 13 illustrates the (S, I) plane, where is possible to observe several characteristics that do not persist when we change the order of the derivative, such as the monotonicity of S compartment, the stability region of the equilibrium ${(S,I,R)}_{\infty }$, in black, and the peak point condition. In the title, $\mu =0$ indicates the absence of vital dynamics. The problem of non-monotonicity is not solved by balancing the units of the parameters initially considered: since the orders are the same, these corrections only numerically transform them into other constants. We notice that the peak points, in blue, no longer follow the relationship ${}^C{D}_{0+}^{\alpha }I=0$, varying according to $\alpha$. The external trajectory is equivalent to the traditional model and the red dot indicates $(S,I)=(\gamma N/\beta ,0)$. However, as we see in [15], the equilibrium is not globally stable. It is also possible to note that, as the greater is $\alpha$, faster is the model.

Fig. 13.

Trajectories for $I(0)=1,$ $\beta =1.5$, $\gamma =0.5,\alpha =0.2:0.1:1$

Figure 14 illustrates the behavior of the solution for $\alpha =0.9$ as $t\to \infty$. We notice autointersections of the trajectory, what does not occur in classical case.

Fig. 14.

Zoom as $t\to \infty$ for the trajectory with $\alpha =0.9.$

This is the simplest case where monotonicity and other classic behaviors are not maintained, but even so, its possible to observe numerically that ${lim}_{t\to \infty }I(t)=0$. However, we reassert that we cannot use the Barbalat’s Lemma for this limit, as it only gives us the minimum limit of I(t). This is also an important thought about other works we cite: the results may be correct; however, another kind of theory may be needed to prove them.

Although, in next section, we insert a relapse rate varying as the f function of Subsection 4.1, illustrating that the supremum limit really cannot be extracted from Barbalat’s Lemma.

SIR model with relapse - A critical example

For this discussion, we consider as basis a SIR model with relapse, based in a fractional tuberculosis model [29]. For simplicity, we disregard the birth rate, the mortalities, the vaccination and the loss of immunity, taking only the relapse rate $\sigma$. So, the model is the same that the (6.1)-(6.3), only plus the relapse rate:

$$\begin{array}{cc}\hfill {}^C{D}_{0+}^{\alpha }S(t)=& -{\displaystyle \frac{\beta S(t)I(t)}{N}},\hfill \end{array}$$ 6.14

$$\begin{array}{cc}\hfill {}^C{D}_{0+}^{\alpha }I(t)=& {\displaystyle \frac{\beta S(t)I(t)}{N}}-\gamma I(t)+\sigma (t)R(t),\hfill \end{array}$$ 6.15

$$\begin{array}{cc}\hfill {}^C{D}_{0+}^{\alpha }R(t)=& \gamma I(t)-\sigma (t)R(t).\hfill \end{array}$$ 6.16

Once relapses are mainly due to treatment fails, we could suppose that they are less and less frequent, because of the new treatment developments and dissemination. So, for the purpose of this example, we consider $\sigma (t)=\sigma f(t)$, where f is the function defined in Subsection 4.1. Once $0\le R(t)\le N$, in analogy with [16], we have

$$\begin{array}{c}\hfill \gamma I_{0+}^{\alpha }I(t)=R(t)-R(0)+\sigma I_{0+}^{\alpha }f(t)R(t)<N+\sigma N{I}_{0+}^{\alpha }f(t).\end{array}$$ 6.17

This implies that $I_{0+}^{\alpha }I(t)$ is limited by some constant and, from Lemma 7, we conclude that ${lim\; inf}_{t\to \infty }I(t)=0$. Hence, as can be seen in Figures 15-16, the infimum limit of I(t) is the same than that of the model (6.1)-(6.3).

Fig. 15.

Comparison between the I compartment of the model with and without relapse - the infimum limit is maintained, but not the supremum

Fig. 16.

$(S,I)-$plane

However, the supremum limit does not follow the same behavior. The oscillation of I is shown in Figures 15-16, indicating that ${lim\; sup}_{t\to \infty }I(t)\ne 0$. Its worth to note that ${lim\; sup}_{t\to \infty }I(t)\ne 0$ for any $\alpha \in (0,1]$, and this does not invalidates the integer-order analysis. In fact, in the integer case, i.e., when $\alpha =1$, Barbalat’s Lemma hypothesis fail, once ${\int }_0^{\infty }f(t)$ is not limited (see Figure 10). By other hand, if $\alpha <1$, the fractional integral $I_{0+}^{\alpha }f(t)$ is limited, even though ${lim\; sup}_{t\to \infty }I(t)\ne 0$. This is the key point of the fractional Barbalat’s Lemma. An extend discussion of the model (6.14)-(6.16) is still open.

Conclusion

Although Arbitrary-Order Calculus is almost as old as the classical Calculus, with its origins in Leibniz, l’Hôpital and Bernoulli, its greatest expansion has occurred only in recent decades. Thus, it is natural that many results need validation from other researchers, until the theory of Fractional Calculus is consolidated as that of Classical Calculus. This process can take decades, and the contribution of each researcher is important in assembling the puzzle.

In this sense, this work reviews the proof of a fractional Barbalat’s Lemma published in [31], where there is an imprecision. Unconsciously, one of the main assumptions in the proof is the existence of a limit to the step of a sequence ${(t_i)}_{i\in \mathbb{N}}$ such that $f(t_i)>ϵ$. However, this is not mandatory. So, based in [9], a counterexample and a corrected result are presented, which uses the infimum limit.

Then, we discuss in short the Lyapunov’s theory for fractional systems, showing another tricky point: the sign of a fractional derivative does not imply its monotonic behavior. In fact, there are several characteristics that do not persist when we change the order of the derivative.

Next, we show a brief application of Barbalat’s Lemma 7 in the SIR model with Caputo fractional derivative. It is important to note that we could not guarantee that the limit of the I compartment is zero, but only its infimum limit. Even thought in the simplest example the strong limit of I(t) seems to be valid, we also discuss an example with relapse in which it is not. In fact, inserting a relapse rate $\sigma (t)$, varying as the f function of Subsection 4.1, one can see that the supreme limit cannot be extracted from Barbalat’s Lemma. We remind that a broader study of this example is still open. Indeed, relapse models are useful for tuberculosis, herpes, and other major diseases.

For completeness, we recall in this last section that, in our research project, we are interested in the following question: is it possible to build fractional SIR-type models with precise mathematical and biological basis like that of the original construction did by Kermack & McKendrick? What characteristics are maintained simply exchanging orders? Does the change in the order of derivatives automatically establish consistent models, regarding the parameters’ definition, physical meaning, conservation, and units? The use of techniques to solve fractional models analytically or numerically is an interesting field in itself. However, it is important to try to verify how, where, and why the fractional derivatives interfere with the model. In this context, a fractional model can be obtained through time-since-infection dependent Mittag-Leffler based laws in the infectiousness and removal functions. Thus, in [19], we follow the footsteps of Angstmann, Henry & McGann [1], where they use the probabilistic language of Continuous Time Random Walks (CTRW). The Riemann-Liouville derivative appears throughout the construction, and following up results and applications are discussed in [17, 20, 22, 18].

Declarations

Conflict of interest

The authors declare that they have no conflict of interest.