Background Information
1.1 Theorical Background
1.1.2 Fractional-Order Derivative
In this subsection, we present definitions and results that show the advantages of using fractional-order derivatives in epidemiological models. The main bibliography used to
present the results is [29].
Definition 1.1.1. (Fractional integral of Riemann-Liouville, [116,29] ) Let𝛼 ∈ ℝ+,𝑏 > 0 and𝑓 ∈ 𝕃𝑝([0, 𝑏] → ℝ𝑚), with1 ≤ 𝑝 ≤ ∞. The fractional integral of Riemann-Liouville, for 𝑡 ∈ [0, 𝑏], of order𝛼, is given by
𝕀𝛼𝑡𝑓 (𝑡) = 1 Γ(𝛼) ∫
𝑡 0
(𝑡 − 𝑠)𝛼−1𝑓 (𝑠)𝑑𝑠.
whereΓ(⋅)is the Gamma function, when𝛼 ∈ ℕ, we haveΓ(𝛼) = 𝛼!We define𝐴𝐶𝑛[0, 𝑏]as the set of functions with order derivative𝑛 − 1absolutely continuous in[0, 𝑏][47].
Definition 1.1.2. (Fractional derivative of Riemann-Liouville, [116,29]) Let𝛼 ∈ ℝ+,𝑏 > 0, 𝑓 ∈ 𝐴𝐶𝑛[0, 𝑏], and𝑛 = [𝛼](part entire of𝛼). The fractional derivative of Riemann-Liouville of order𝛼, is given by
𝐃𝛼𝑡𝑓 (𝑡) = 𝐃𝑛𝑡𝕀𝑛−𝛼𝑡 𝑓 (𝑡) = 𝑑𝑛 𝑑𝑡𝑛(
1 Γ(𝑛 − 𝛼) ∫
𝑡 0
(𝑡 − 𝑠)𝑛−𝛼−1𝑓 (𝑠)𝑑𝑠
). (1.7)
Definition 1.1.3. (Fractional derivative in the Caputo sense, [116,29]) Let𝛼 ∈ ℝ+,𝑏 > 0 and𝑓 ∈ 𝐴𝐶𝑛[0, 𝑏]. For𝑡 ∈ [0, 𝑏], the fractional derivative in the Caputo sense of order𝛼 is given by
𝑐𝔻𝛼𝑡 𝑓 (𝑡) = 𝐃𝛼𝑡(𝑓 (𝑡) − 𝑓 (0)). (1.8) For𝛼 ∈ (0, 1), we have that
𝑐𝔻𝛼𝑡 𝑓 (𝑡) = 𝕀1−𝛼𝑡 ̇𝑓(𝑡), (1.9) where ̇𝑓(𝑡)represent of the first derivative of𝑓.
Other definition of the fractional derivative in the Caputo sense is:
Definition 1.1.4. (See [99,54]) For𝛼 > 0, with𝑛 − 1 < 𝛼 < 𝑛,𝑛 ∈ ℕ, the fractional derivative in the sense of Caputo is defined as
𝑐𝔻𝛼𝑡 𝑓 (𝑡) = 𝑑𝛼𝑓 (𝑡)
𝑑𝑡𝛼 ∶= 1
Γ(𝑛 − 𝛼) ∫
𝑡 0
(𝑡 − 𝑠)𝑛−𝛼−1𝑓𝑛(𝑠)𝑑𝑠.
Now, we present definitions and statistical results that we will use.
Definition 1.1.5. (Beta distribution (Beta), [99,29]) A random variable3,𝕏, follows the Beta distribution if its probability density function is given by
𝑓𝕏(𝑥) = 𝑓𝕏(𝑥; 𝑝, 𝑞) = 1
𝐵(𝑝, 𝑞)𝑥𝑝−1(1 − 𝑥)𝑞−1𝐈(0,1)(𝑥), (1.10) where𝑝, 𝑞 > 0,𝐈(0,1)is the indicator function of interval(0, 1)and𝐵(⋅)is the beta function.
3A random variable is a variable whose value is unknown or a function that assigns values to each of an experiment’s outcomes [52].
1.1 | THEORICAL BACKGROUND
Remark 1.1.6. When𝑝 = 𝑞 = 1, the Beta distribution becomes the uniform distribution over the range(0, 1).
Definition 1.1.6. (See [99,29]) Let𝕏be a continuous random variable with the probability density function𝑓𝕏(⋅). The expectation or expected value of𝕏is given by
𝐸[𝕏] = ∫
∞
−∞
𝑥𝑓𝕏(𝑥)𝑑𝑥. (1.11)
Proposition 1.1.7. (See [99,29]) Let𝑔 ∶ ℝ → ℝand𝕏be a continuous random variable with the probability density function𝑓𝕏(⋅). The expectation or expected value of𝑔(𝕏)is given by
𝐸[𝑔(𝕏)] = ∫
∞
−∞
𝑔(𝑥)𝑓𝕏(𝑥)𝑑𝑥. (1.12)
Let’s mention important properties of fractional derivatives in the Caputo sense.
Lemma 1.1.8. Let𝑛 − 1 < 𝛼 < 𝑛,𝑛 ∈ ℕ,𝛼 ∈ ℝand𝑓 (𝑡)be such that 𝑐𝔻𝛼𝑡 𝑓 (𝑡)exists. Then,
• lim𝛼→𝑛 𝑐𝔻𝛼𝑡 𝑓 (𝑡) = 𝑓(𝑛)(𝑡),
• lim𝛼→𝑛−1 𝑐𝔻𝛼𝑡 𝑓 (𝑡) = 𝑓(𝑛−1)(𝑡) − 𝑓(𝑛−1)(0).
where𝑓(𝑛)and𝑓(𝑛−1)are the classical integer derivatives of𝑓 of order𝑛and𝑛 − 1 respec-tively.
Lemma 1.1.9. (Linearly) Let 𝑛 − 1 < 𝛼 < 𝑛,𝑛 ∈ ℕ,𝜆 ∈ ℂand the function𝑓 (𝑡)and g(t) be such that both 𝑐𝔻𝛼𝑡 𝑓 (𝑡)and 𝑐𝔻𝛼𝑡 𝑔(𝑡)exist. The Caputo fractional derivative is a linear operator, i.e:
𝑐𝔻𝛼𝑡(𝜆𝑓 (𝑡) + 𝑔(𝑡)) = 𝜆𝑐𝔻𝛼𝑡 𝑓 (𝑡) + 𝑐𝔻𝛼𝑡 𝑔(𝑡).
Lemma 1.1.10. (Non-commutative) Let𝑛 − 1 < 𝛼 < 𝑛and𝑛, 𝑚 ∈ ℕand𝑓 (𝑡)be such that
𝑐𝔻𝛼𝑡 𝑓 (𝑡)exists. Then,
𝑐𝔻𝛼𝑡 𝐃𝑚𝑓 (𝑡) = 𝑐𝔻𝛼+𝑚𝑡 𝑓 (𝑡) ≠ 𝐃𝑚(𝑐𝔻𝛼𝑡 𝑓 (𝑡)).
Memory Effect
The fractional calculus is a great tool that can be employed to describe real-life phe-nomena with so-called memory effect. One way to introduce the memory effect into a mathematical model is to change the order of the derivative of a classical model so that it is non-integer [47,29].
Let𝑓 be a real function defined in[0, 𝑡], 𝑡1, 𝑡2 ∈ [0, 𝑡] are such that0 < 𝑡1 < 𝑡2, and 𝐹 = (𝕀𝛼𝑓 )(𝑡2) − (𝕀𝛼𝑓 )(𝑡1), for𝛼 ∈ ℝ+. If𝛼 ≠ 1it can be seen that the value of𝐹 depends on the entire range of𝑓 over[0, 𝑡2], whereas𝐹 depends only on the range of𝑓 over[𝑡1, 𝑡2]if
𝛼 = 1[29]: Note that if𝛼 = 1, the second integral is eliminated and we obtain:
𝐹 = ∫
𝑡2
𝑡1
𝑓 (𝑠)𝑑𝑠.
Now, we study the memory effect in derivatives and integrals of fractional-order based on the expected values of a random variable. The following proposition characterizes three fractional operators that depend on expected values of a random variable (𝐸[𝕏]).
Proposition 1.1.11. Let𝛼 ∈ ℝ+and𝑓 ∈ 𝐴𝐶[0, 𝑏]. Under these conditions, we have
The previous proposition is presented and proved in [29]. Let’s present the proof of (1.15).
1.1 | THEORICAL BACKGROUND
= 𝑡1−𝛼
Γ(2 − 𝛼)𝐸[ ̇𝑓(𝑡)(𝑡𝑊 )], where𝑊 ∼Beta(1, 1 − 𝛼). Thus, we obtain (1.15).
Remark 1.1.12. We can rewrite Caputo’s derivative as follows [29]:
𝑐𝔻𝛼𝑡 𝑓 (𝑡) = 𝑡−𝛼
Γ(1 − 𝛼)𝐸[𝑓 (𝑡𝑊 ) − 𝑓 (0)] + 𝑡1−𝛼
Γ(3 − 𝛼)𝐸[ ̇𝑓(𝑡)(𝑡𝑉 )]. (1.16) The Hysteresis Phenomenon
When the current state of a system is influenced by its historical past, that system is said to be influenced by the phenomenon of hysteresis. This is a typical kernel used to define integral and fractional differential operators, such as the Caputo derivative. Fractional operators can be interpreted from the statistical approach, through the mathematical expectation, where the past history of the system follows aBetadistribution. As the𝛼 parameter is between 0 and 1, recent times are more influential than past times in this distribution [19,29].
We can use fractional operator (1.13)-(1.15) to interpret the hysteresis effect. More precisely, the rate of variation of the Caputo derivative is given explicitly by the weighted average of all past derivatives as we can see in Formula (1.13). The other operators have similar explanations [29].
As𝑈 has distribution𝐵(1, 𝛼), the random variable𝑆 = 𝑡𝑈 has the values in the interval (0, 𝑡). Thus, from (1.13), 𝕀𝛼𝑡𝑓 (𝑡)coincides with 𝑡𝛼
Γ(𝛼 + 1)𝐸[𝑓 (𝑆)], 0 ≤ 𝑠 ≤ 𝑡. The 𝕀𝛼𝑡𝑓 (𝑡)is affected by all previous values in𝑡 because the expected value𝐸[𝑓 (𝑆)]is affected as well.
Given that𝐸[𝑓 (𝑆)]is in (1.14) and (1.15), we may deduce that𝐃𝛼𝑡𝑓 (𝑡)and 𝑐𝔻𝛼𝑡 𝑓 (𝑡)also have memory effect. With the formulas (1.13)-(1.15), we arrive at the following interpretations [29]:
• The fractional integral𝕀𝛼𝑡𝑓 (𝑡)is proportional to the weighted average of𝑓 (𝑠), conside-ring all prior values𝑠of𝑡, distributed by aBetadistribution;
• 𝐃𝛼𝑡𝑓 (𝑡)is the sum of two amounts proportional to the weighted average of𝑓 (𝑠);
• 𝑐𝔻𝛼𝑡 𝑓 (𝑡) is proportional to the weighted average of the classic derivative ̇𝑓(𝑡)(𝑠), considering all prior values𝑠 < 𝑡 , distributed by aBetadistribution;
• If𝛼 = 1, then𝑈 ∼ 𝐵(1, 1),𝑈 has uniform distribution such that 𝕀𝛼𝑡 = 𝑡𝐸[𝑓 (𝑡𝑈 )] =
∫0𝑡𝑓 (𝑠)𝑑𝑠. Therefore, 𝐃𝛼𝑡𝑓 (𝑡) = 𝑑
𝑑𝑡𝕀0𝑡𝑓 (𝑡) = 𝑑
𝑑𝑡𝑓 (𝑡) = ̇𝑓(𝑡)(𝑡) and 𝑐𝔻𝛼𝑡[𝑓 (𝑡) − 𝑓 (0)] = 𝑑
𝑑𝑡𝕀0𝑡[𝑓 (𝑡) − 𝑓 (0)] = 𝑑
𝑑𝑡[𝑓 (𝑡) − 𝑓 (0)] = ̇𝑓(𝑡)(𝑡).Consequently, when𝛼 = 1the fractional and classical calculus coincide.
The authors of [29] show the influence ofBetadistribution on fractional operators, in particular on the Riemann-Liouville and Caputo fractional operators.
Numerical Method
The algorithm used in this work to numerically solve nonlinear differential equations of fractional-order can be found in [48,49,50]. The algorithm has the structure of a PECE (Predict-Evaluate-Correct-Evaluate) method and combines a fractional-order algorithm with a classical method. The approach chosen is Adams-Bashforth-Moulton for both integrators. The key to deriving the method in the fractional variant is to use the trape-zoidal quadrature product formula. This algorithm is independent of the𝛼−parameter and behaves very similar to the classical Adams-Bashforth-Moulton method. The stability prop-erties do not change in the fractional version compared to the classical algorithm.
Optimal Control Problem with FDE
The following definitions are used to formulate and study the fractional-order optimal control problem.
We assume that𝛼 ∈ ℝ+,𝑏 > 0,𝑓 ∈ 𝐴𝐶𝑛[𝑎, 𝑏], and𝑛 = [𝛼]. We define the left-sided and right-sided fractional integral Riemann-Louville for𝑓 ∶ ℝ+⟶ ℝ,𝛼 > 0are:
𝑎𝕀𝛼𝑡 𝑓 (𝑡) ∶= 1 Γ(𝛼) ∫
𝑡 𝑎
𝑓 (𝑠)𝑑𝑠
(𝑡 − 𝑠)1−𝛼, (Left) (1.17)
𝑡𝕀𝛼𝑏𝑓 (𝑡) ∶= 1 Γ(𝛼) ∫
𝑏 𝑡
𝑓 (𝑠)𝑑𝑠
(𝑠 − 𝑡)1−𝛼. (Right) (1.18)
Note:Let’s define 𝕀𝛼𝑡 𝑓 (𝑡) = 0𝕀𝛼𝑡 𝑓 (𝑡).
The left-sided and right-sided Riemann–Liouville fractional derivatives are define as [33,66]:
𝑎𝐃𝛼𝑡 𝑓 (𝑡) = 𝑑𝑛 𝑑𝑡𝑛(
1 Γ(𝑛 − 𝛼) ∫
𝑡 𝑎
(𝑡 − 𝑠)𝑛−𝛼−1𝑓 (𝑠)𝑑𝑠
), (Left) (1.19)
𝑡𝐃𝛼𝑏𝑓 (𝑡) = 𝑑𝑛 𝑑𝑡𝑛(
(−1)𝑛 Γ(𝑛 − 𝛼) ∫
𝑏 𝑡
(𝑠 − 𝑡)𝑛−𝛼−1𝑓 (𝑠)𝑑𝑠
). (Right) (1.20)
Note:Let’s denote𝐃𝛼𝑡𝑓 (𝑡) = 0𝐃𝛼𝑡 𝑓 (𝑡).
The left-sided and right-sided fractional derivatives proposed by Caputo are given by [33,66]:
𝑐
𝑎𝔻𝛼𝑡 𝑓 (𝑡) = 1 Γ(𝑛 − 𝛼) ∫
𝑡 𝑎
(𝑡 − 𝑠)𝑛−1−𝛼𝑓𝑛(𝑠)𝑑𝑠, (Left) (1.21)
𝑐
𝑡𝔻𝛼𝑏𝑓 (𝑡) = (−1)𝑛 Γ(𝑛 − 𝛼) ∫
𝑏 𝑡
(𝑠 − 𝑡)𝑛−1−𝛼𝑓𝑛(𝑠)𝑑𝑠. (Right) (1.22) Note:Let’s define 𝑐𝔻𝛼𝑡 𝑓 (𝑡) = 𝑐0𝔻𝛼𝑡 𝑓 (𝑡).
The Riemann-Liouville and Caputo derivatives are related by the following formulas
1.1 | THEORICAL BACKGROUND
[66]:
𝑎𝐃𝛼𝑡 𝑓 (𝑡) = 𝑐𝑎𝔻𝛼𝑡 𝑓 (𝑡) +
𝑛−1
∑
𝑘=0
𝑓(𝑘)(𝑎)
Γ(𝑘 + 1 − 𝛼)(𝑡 − 𝑎)𝑘−𝛼, (1.23)
𝑡𝐃𝛼𝑏 𝑓 (𝑡) = 𝑐𝑡𝔻𝛼𝑏𝑓 (𝑡) +
𝑛−1
∑
𝑘=0
𝑓(𝑘)(𝑏)
Γ(𝑘 + 1 − 𝛼)(𝑡 − 𝑏)𝑘−𝛼 (1.24)
Now, we will present a general formulation of the fractional-order optimal control problem (FOCP) and obtain the necessary conditions for the optimality of the FOCP.
Finding the optimal control𝑢(𝑡)that minimizes the functional𝐽 is defined as:
𝐽 (𝑢) = ∫
𝑏 0
𝑓 (𝑡, 𝑥, 𝑢)𝑑𝑡, (1.25)
subject to the model with control
𝑐𝔻𝛼𝑡 𝑥(𝑡) = 𝑔(𝑡, 𝑥, 𝑢), (1.26)
with initial condition
𝑥(0) = 𝑥𝐼, (1.27)
where𝑥(𝑡)and𝑢(𝑡)are the state and control variables,𝑓 and𝑔are differential functions and0 < 𝛼 ≤ 1.
Theorem 1.1.13. If𝑓 (𝑥, 𝑢)is a minimizer of (1.25) satisfying the constraint (1.26) and the boundary condition (1.27), then there exists a function 𝜆 ∈ ℂ1[0, 𝑏]such that the triplet (𝑥, 𝑢, 𝜆)satisfies:
1. the state and co-state systems
𝑐𝔻𝛼𝑡 𝑥(𝑡) = 𝜕𝐻
𝜕𝜆(𝑡, 𝑥(𝑡), 𝑢(𝑡), 𝜆(𝑡)), (1.28)
𝑐
𝑡𝔻𝛼𝑏𝜆(𝑡) = 𝜕𝐻
𝜕𝑥(𝑡, 𝑥(𝑡), 𝑢(𝑡), 𝜆(𝑡)), (1.29) 2. the stationary condition
𝜕𝐻
𝜕𝑢(𝑡, 𝑥(𝑡), 𝑢(𝑡), 𝜆(𝑡)) = 0, (1.30) 3. and the transversality condition
𝑡𝕀1−𝛼𝑏 𝜆||𝑡=𝑏 = 𝜆(𝑏) = 0, (1.31) where the Hamiltonian𝐻 is defined by
𝐻 (𝑡, 𝑥, 𝑢, 𝜆) = 𝑓 (𝑡, 𝑥, 𝑢) + 𝜆 ⋅ 𝑔(𝑡, 𝑥, 𝑢). (1.32) The theorem and its proof are in [66].
Lemma 1.1.14(See [66]). The following equations are equivalent:
𝑐
𝑡𝔻𝛼𝑏𝜆(𝑡) = 𝜕𝐻
𝜕𝑥(𝑡, 𝑥(𝑡), 𝑢(𝑡), 𝜆(𝑡)), (1.33)
𝑐𝔻𝛼𝑡 𝜆(𝑏 − 𝑡) = 𝜕𝐻
𝜕𝑥(𝑏 − 𝑡, 𝑥(𝑏 − 𝑡), 𝑢(𝑏 − 𝑡), 𝜆(𝑏 − 𝑡)), (1.34) where𝛼 ∈ (0, 1].
The proof of Lemma (1.1.14) is found in [66] and we can find applications of this lemma in [66,14].