Frequently, real-world problems can be mathematically modeled using differential equations equipped with some initial conditions. Thus, Caputo’s definition of fractional-order derivative is appropriate to replace the classical derivative used in the original models. In addition, the Laplace transform of Caputo’s derivative can be written in terms of the given initial conditions of the problems and Caputo’s derivative of a constant is equal to zero. Hence, we will reasonably use this fractional-order derivative for our delayed fractional-order glucose–insulin interaction model.
A function \(f(t)\) (\(t>0\)) is said to be in the space \(C_{q}\) (\(q \in \mathbb{R}\)) if it can be expressed as \(f (t) = t^{p}g(t)\) for some \(p>q\), where \(g(t)\) is continuous in \([0,\infty )\). The function is also said to be in the space \(C_{q}^{m}\) if \(f^{(m)} \in C_{q}\), \(m \in \mathbb{N}\) (see [17, 38] and the references cited therein for further details).
Definition 2.1
([17])
The Riemann–Liouville fractional integral operator of order \(q > 0\) of a function \(f \in C_{q}\) with \(a \geq 0\) is defined as
$$ {}_{\mathrm{RL}}J_{a}^{q}f(t)=\frac{1}{\varGamma (q)} \int _{a}^{t} (t-\tau )^{q-1}f( \tau ) \,d\tau , \quad t>a, $$
(4)
where \(\varGamma (z)=\int _{0}^{\infty } e^{-u} u^{z-1} \,du\) is the gamma function.
Definition 2.2
([17])
For a positive real number q, the Caputo fractional derivative of order q with \(a \geq 0\) is defined in terms of the Riemann–Liouville fractional integral, i.e., \({}_{C}D_{a}^{q}f(t)={}_{\mathrm{RL}}J_{a}^{m-q}f^{(m)}(t)\), or it can be expressed as
$$ {}_{C}D_{a}^{q}f(t)= \textstyle\begin{cases} \frac{1}{\varGamma (m-q )} \int _{a}^{t} \frac{f^{ (m )} (\tau )}{ (t-\tau )^{q-m+1}}\,d\tau , & m-1< q < m, \\ \frac{d^{m}}{dt^{m}}f(t),& q=m, \end{cases} $$
(5)
where \(t \geq a\), \(f \in C_{-1}^{m}\), and \(m\in \mathbb{N}\). In particular, when \(0< q\leq 1\), we have
$$ {}_{C}D_{a}^{q}f(t)= \frac{1}{\varGamma (1-q )} \int _{a} ^{t} \frac{f' (\tau )}{ (t-\tau )^{q}}\,d\tau . $$
(6)
Corollary 2.1
([39])
Let
\(0< q \leq 1\). Suppose
\(f \in C[a,b]\)
and
\({}_{C}D_{a}^{q} f\in (a,b]\). If
\({}_{C}D_{a}^{q} f(t) \geq 0\)
for all
\(t \in (a,b)\), then the function
f
is non-decreasing, and if
\({}_{C}D_{a}^{q} f(t) \leq 0\)
for all
\(t \in (a,b)\), then the function
f
is non-increasing.
Lemma 2.1
(Fractional comparison principle in [40])
Let
\(x(0)=y(0)\)
and
\({}_{C}D_{a}^{q} x(t) \leq {}_{C}D_{a}^{q} y(t)\), where
\(0< q \leq 1\). Then
\(x(t) \leqslant y(t)\).
The Laplace transforms of the Caputo fractional derivative and some types of the Mittag-Leffler functions are as follows.
Lemma 2.2
([17])
The Laplace transform of the Caputo fractional derivative of order
\(m-1< q< m\)
is
$$ \mathscr{L}\bigl\{ {}_{C}D_{a}^{q}f(t) \bigr\} =s^{q}F(s)-\sum^{m-1}_{k=0}s^{q-k-1}f ^{(k)}(a), $$
(7)
where
\(F(s)= \mathscr{L} \{f(t)\}\).
Definition 2.3
([17])
The single parameter Mittag-Leffler function is defined by
$$ E_{q} (t) = \sum_{k=0}^{\infty } \frac{t^{k}}{ \varGamma (q k+1)},\quad q >0, $$
(8)
and the two parameter Mittag-Leffler function can be defined by
$$ E_{q, \beta } (t) = \sum_{k=0}^{\infty } \frac{t^{k}}{\varGamma (q k+ \beta )}, \quad q, \beta >0. $$
(9)
It is not difficult to see that \(E_{q,1} (t)=E_{q} (t)\).
Lemma 2.3
([17])
The Laplace transforms for several Mittag-Leffler functions are given by
$$\begin{aligned}& \mathscr{L}\bigl\{ E_{q}\bigl(-\lambda t^{q}\bigr)\bigr\} = \frac{s^{q-1}}{s^{q}+\lambda }, \end{aligned}$$
(10)
$$\begin{aligned}& \mathscr{L}\bigl\{ t^{\beta -1}E_{q,\beta }\bigl(-\lambda t^{q}\bigr)\bigr\} = \frac{s^{q- \beta }}{s^{q}+\lambda }, \end{aligned}$$
(11)
provided that
\(s>|\lambda |^{1/q}\), where
λ
is a constant parameter.
Definition 2.4
Consider the following fractional-order system:
$$ {}_{C}D_{a}^{q} X(t)=F\bigl(X(t)\bigr), $$
(12)
where \(X(t)=(x_{1}(t),x_{2}(t),\ldots,x_{n}(t))^{T}\), \(F(t)=(f_{1}(t),f _{2}(t),\ldots,f_{n}(t))^{T}\), and \(q=(q_{1},q_{2},\ldots,q_{n})^{T}\) with \(q_{i}>0\), \(i=1,2,\ldots,n\). The equilibrium solution \(X^{*}=(x_{1}^{*},x _{2}^{*},\ldots,x_{n}^{*})^{T}\) of the system is defined by \(F(X^{*})=0\), i.e., \(f_{i}(X^{*})=0\), \(i=1,2,\ldots,n\).
Consider a general delayed fractional-order system
$$ {}_{C}D_{a}^{q}X(t) = F \bigl(X(t),X(t-\tau )\bigr), $$
(13)
where \(\tau >0\) is a time delay and \(X(t) = (x_{1}(t), x_{2}(t), \ldots , x_{n}(t))^{T} \in \mathbb{R}^{n} \). The equilibrium point \(X^{*}\) of system (13) is the solution of the equation \(F(X,X)=0\). The associated linearized system of system (13) at an equilibrium point \(X^{*}\) can be written as
$$ {}_{C}D_{a}^{q}U(t) = AU(t)+BU(t-\tau ), $$
(14)
where \(A, B\in \mathbb{R}^{n\times n}\). The characteristic equation of system (14) is
$$ \Delta (s)=\text{det}\bigl(s^{q} I-A-Be^{-s \tau }\bigr)=0, $$
(15)
where \(I\in \mathbb{R}^{n\times n}\) is the identity matrix. If \(\tau =0\), the linearized system (14) is reduced to
$$ {}_{C}D_{a}^{q}U(t) = MU(t), $$
(16)
where the coefficient matrix \(M=A+B\).
In [23, 41], the Hopf bifurcation conditions were investigated for the general delayed fractional-order system (13). Suppose that the following conditions hold, then system (13) encounters a Hopf bifurcation at the equilibrium \(X^{*}\) when \(\tau =\tau _{0}\).
-
(1)
All the eigenvalues \(\lambda _{i}\), \(i=1,2,3,\ldots,n\), of the coefficient matrix M of the linearized system of Eq. (13) satisfy the condition \(\vert \operatorname{arg}(\lambda _{i})\vert >\frac{q \pi }{2}\), \(i=1,2,3,\ldots,n\).
-
(2)
The characteristic equation \(\Delta (s)=0\) in Eq. (15) has a pair of purely imaginary roots \(\pm i \omega _{0}\) when \(\tau =\tau _{0}\).
-
(3)
\(\operatorname{Re} [ \frac{ds}{d \tau } ]_{ \tau =\tau _{0}} \neq 0\), where \(\operatorname{Re}(\cdot)\) denotes the real part of a complex number.
Next the algorithm for solving a delayed fractional-order differential equation is briefly discussed as follows. Bhalekar and Daftardar-Gejji have modified the Adams–Bashforth method to solve delay differential equations of fractional order (FDDE) [42]. The method is described below. Consider the FDDE
$$\begin{aligned}& {}_{C}D_{a}^{q} y(t) = f\bigl(t,y(t),y(t- \tau )\bigr), \quad t\in [0,T], 0 < q \le 1, \end{aligned}$$
(17)
$$\begin{aligned}& y(t) =g(t), \quad t \in [-\tau ,0]. \end{aligned}$$
(18)
For convenience, we use the starting point \(a=0\). Then the Volterra integral equation of (17) can be written as
$$ y(t)=g(0)+\frac{1}{\varGamma (\alpha )} \int ^{t}_{0}(t-\xi )^{\alpha -1}f\bigl( \xi ,y(\xi ),y(\xi -\tau )\bigr)\,d\xi . $$
(19)
Similarly, we obtain a uniform grid for the delayed problem as \(\{t_{n}=nh: n=-k,-k+1,\ldots,-1,0,1,\ldots, N\}\), where k and N are integers such that \(h=T/N\) and \(h=\tau /k\). For the sake of simplicity, we let
$$ y_{h}(t_{j})=g(t_{j}), \quad j=-k,-k+1, \ldots,-1,0, $$
(20)
and note that
$$ y_{h}(t-\tau )=y_{h}(jh-kh)=y_{h}(t_{j-k}), \quad j=0,1,\ldots,N. $$
(21)
We assume that the approximations \(y_{h}(t_{j})\approx y(t_{j})\) for \(j=-k,-k+1,\ldots,-1,0,1,\ldots,n\) have been calculated, and we want to calculate \(y_{h}(t_{n+1})\) using the formula
$$ y(t_{n+1})=g(0)+\frac{1}{\varGamma (\alpha )} \int ^{t+1}_{0}(t_{n+1}- \xi )^{\alpha -1}f\bigl(\xi ,y(\xi ),y(\xi -\tau )\bigr)\,d\xi . $$
(22)
Applying the product trapezoidal quadrature formula to approximate the integral in (22), by substituting approximations \(y_{h}(t_{n})\) for \(y(t_{n})\), we obtain
$$\begin{aligned} y_{h}(t_{n+1}) =&g(0)+\frac{h^{\alpha }}{\varGamma (\alpha +2)}f \bigl(t_{n+1},y _{h}(t_{n+1}),y_{h}(t_{n+1}- \tau )\bigr) \\ &{}+\frac{h^{\alpha }}{\varGamma (\alpha +2)}\sum^{n}_{j=0}a_{j,n+1}f \bigl(t _{j},y_{h}(t_{j}),y_{h}(t_{j}- \tau )\bigr) \\ =& g(0)+\frac{h^{\alpha }}{\varGamma (\alpha +2)}f\bigl(t_{n+1},y_{h}(t_{n+1}),y _{h}(t_{n+1-k})\bigr) \\ &{}+\frac{h^{\alpha }}{\varGamma (\alpha +2)}\sum^{n}_{j=0}a_{j,n+1}f \bigl(t _{j},y_{h}(t_{j}),y_{h}(t_{j-k}) \bigr), \end{aligned}$$
(23)
where
$$ a_{j,n+1}= \textstyle\begin{cases} n^{\alpha +1}-(n-\alpha )(n+1)^{\alpha } & \text{if } j=0, \\ (n-j+2)^{\alpha +1}+(n-j)^{\alpha +1}-2(n-j+1)^{\alpha +1}, & \text{if } 1\le j \le n, \\ 1,& \text{if } j=n+1. \end{cases} $$
(24)
The product rectangle rule is then applied in (23) to evaluate the following predictor term:
$$\begin{aligned} y^{P}_{h}(t_{n+1}) &= g(0)+ \frac{1}{\varGamma (\alpha )}\sum^{n}_{j=0}b _{j,n+1}f\bigl(t_{j},y_{h}(t_{j}),y_{h}(t_{j}- \tau )\bigr) \\ &= g(0)+\frac{1}{\varGamma (\alpha )}\sum^{n}_{j=0}b_{j,n+1}f \bigl(t_{j},y _{h}(t_{j}),y_{h}(t_{j-k}) \bigr), \end{aligned}$$
(25)
where
$$ b_{j,n+1}=\frac{h^{\alpha }}{\alpha }\bigl((n+1-j)^{\alpha }-(n-j)^{\alpha } \bigr). $$
(26)