*4.4. Type-1 Diabetes Mellitus: Parameter a*<sup>15</sup> *as a Function of Fractional-Order q*

T1DM is a common autoimmune disease that originates when the pancreatic *β*-cells cannot produce insulin at normal levels, and patients will require hormone dosage for their entire life. [84]. The fractional-order system (5) exhibits this condition when the density of *β*-cells distinguished by *a*<sup>15</sup> reduces and, therefore, the pancreas may not secrete sufficient insulin to stabilize the glucose concentration.

The bifurcation diagram of Equation (5) is shown in Figure 9a when considering *a*<sup>15</sup> as critical parameter with *q* = 0.95. As can be seen, the system (5) exhibits different types of steady behaviors for specific values of *a*15. However, whether this parameter decreases, the system behaves chaotically, as is demonstrated in Proposition 3.

**Figure 9.** (**a**) Bifurcation diagram varying the T1DM parameter *a*<sup>15</sup> and setting *q* = 0.95, and (**b**) Two-dimensional bifurcation diagram for *a*<sup>15</sup> and fractional-order *q*. The unbounded behavior is represented by the green regions; chaotic behavior is denoted by red regions, and the periodic behavior (healthy behavior) is given by the blue regions.

**Proposition 3.** *When q*<sup>1</sup> = *q*<sup>2</sup> = *q*<sup>3</sup> ≡ *q* = 0.95 *and a*<sup>1</sup> = 2.04*, a*<sup>7</sup> = 2.01*, a*<sup>8</sup> = 0.22*, a*<sup>15</sup> = 0.26*, the system* (5) *exhibits a chaotic attractor.*

**Proof.** By selecting *q* = 0.95, *a*<sup>15</sup> = 0.26, and *w* = 100, we attain the characteristic polynomial at *E*<sup>1</sup> = (0.823, 1.881, 1.366) as

$$
\lambda^{285} + 2.159\lambda^{190} + 61.954\lambda^{95} - 106.369,\tag{27}
$$

being *λ* = 1.0048 the unstable root. At the equilibrium point *E*<sup>2</sup> = (0.618, 0.883, 0.867), we obtain

$$
\lambda^{285} + 1.765 \lambda^{190} + 1.535 \lambda^{95} + 17.503,\tag{28}
$$

with unstable roots *λ*1,2 = 1.0091 ± 0.0137*i*. Therefore, *ρ* = (*π*/2*w*) − 0.0135 > 0. In this manner, the proposed system (1) fulfills Theorem 3 for generating a chaotic attractor.

The Lyapunov exponents when *q* = 0.95 and *a*<sup>1</sup> = 2.04, *a*<sup>7</sup> = 2.01, *a*<sup>8</sup> = 0.22, *a*<sup>15</sup> = 0.26 are *λ*<sup>1</sup> = 1.7733, *λ*<sup>2</sup> = 0, and *λ*<sup>3</sup> = −24.3966. Similarly previous case, Figure 9b presents the two-dimensional bifurcation diagram relating *a*<sup>15</sup> and the fractional-order *q*. The green, red, and blue colors denote unbounded, chaotic, and steady-state behaviors, respectively. A healthy behavior, free of T1DM, is found for fractional-orders lowers than *q* < 0.925. Those results may imply that lower fractional-orders mitigate the effect of the reduction of population density of *β*-cells for T1DM.
