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Our recent works discuss the meaning of an arbitrary-order SIR model. We claim that arbitrary-order derivatives can be obtained through special power-laws in the infectivity and removal functions. This work intends to summarize previous ideas and show new results on a meaningful model constructed with Mittag-Leffler functions. We emphasize the tric...

## Contexts in source publication

**Context 1**

... Figure 4.1 illustrates the change of the solution, where the dashed line corresponds to the solution from the time t = 90. In Figure 4.2, we have the equivalent trajectories for a maximum time T = 3000. ...

**Context 2**

... Figure 4.1 illustrates the change of the solution, where the dashed line corresponds to the solution from the time t = 90. In Figure 4.2, we have the equivalent trajectories for a maximum time T = 3000. ...

**Context 3**

... consider the example below, in Figure 4.3, where the value of ℜ(t ′ ) given by Eq. (4.3) for the peak point t ′ is ℜ peak ≈ 0.9078 < 1. ...

**Context 4**

... will be discussed later. Also, is also possible to have no more peaks if we start in a previous point, as we can see in Figure 4.4. In both examples, we consider N = 10 6 and the initial condition is (N − 1, 1, 0). ...

**Context 5**

... also can observe ℜ peak > 1. In some of these cases, we can take an earlier day, where ℜ(t) > ℜ peak > 1, and the disease starts decreasing even faster, as illustrated in Figure 4.5. In other words, "the point where the epidemic must start to decrease" is really not the peak point anymore! ...

**Context 6**

... In this same Figure 4.5, we consider different starting points along the originally constructed I curve with initial condition (N − 1, 1, 0). So, it illustrates the evolution of the I compartment if we start modeling at different starting points along the same curve. ...

**Context 7**

... the Figure table 4.6, we display the ℜ 0 S(0)/N and the ℜ S 0 S(0)/N for each curve. ...

**Context 8**

... we exhibit an unusual feature observed in the initial instants of the model simulations. One can easily see in the Figure 4.5 that the blue and purple curves have a small depression before a rise. This type of behavior does not appear in the integer-order model. ...

**Context 9**

... in the model (3.10)-(3.12), if β > α, we have dI/dt < 0 for t small enough, as illustrated in the Figures 4.8-4.9. When α = β, this behavior is not observed. ...

## Similar publications

This chapter summarizes and concludes the contribution of this thesis in Section 6.1 and Section 6.2, respectively. Section 6.3 provides an overview of future work projects.

## Citations

A presentation of the latest results obtained in studies of Fractional Models by the team formed by Sandro R. Mazorche, Noemi Z. Monteiro and Matheus T. Mendonça. A brief historical presentation of Fractional Calculus with an emphasis on the turn of the 19th to the 20th century, where the figure of Magnüs G. Mittag-Leffler is the center of the initial discussion. Then a presentation of the Harmonic Oscillator, where a comparison is made: Fractional model x Integer model. In the third part, the SIR model is presented, and two fractional formulations are discussed: first, the model described changing the integer derivative by the Caputo derivative, where we will see solution behavior, lost of monotonicity and a correction in Barbalat's lemma for the fractional case. Then, we will see the description of the SIR model through a construction involving the derivative of Riemann-Liouville, where the central idea of the construction is presented, as well as the calculation of the asymptotic equilibrium points and stability results. Ending with a story involving Mittag-Leffler, Volterra and Fredholm.
The presentation is in the link: https://www.youtube.com/watch?v=i7w4kreAG_Q

https://link.springer.com/article/10.1007/s13540-022-00111-6
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<α<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.