# Stochastic Optimal Control Analysis of a Mathematical Model: Theory and Application to Non-Singular Kernels

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction to the Problem and Model Formulation

## 2. Qualitative Analysis of the Global Non-Negative Solution

**Theorem**

**1.**

**Proof.**

## 3. Extinction of the Proposed Model

**Lemma**

**1.**

**Theorem**

**2.**

**Proof.**

## 4. The Stationary Distribution of the Disease

**Lemma**

**2**

- 1.
- In the set U and its neighbor thereof, the eigenvalue of the diffusion matrix $A\left(t\right)$ that has the smallest magnitude is bounded away from the origin;
- 2.
- If x is in ${R}^{d}\backslash U$, the average time τ in which a path starts from x reaching U is $<\infty $, and ${sup}_{x\in K}{E}^{x}\tau <\infty \phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\forall \phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}K\subset {R}^{n}$ where K is compact. Besides, for an integrable function $f(\xb7)$ w.r.t measure Π, we have:

**Theorem**

**3.**

**Proof.**

**Case**

**1.**

**Case**

**2.**

## 5. Stochastic Optimal Control

- The control measure ${u}_{1}\left(t\right)$ shows physically the discussion about non-singular and singular kernels on social research forums such as Google scholar and Researchgate for example "https://pubpeer.com/";
- The variable ${u}_{2}\left(t\right)$ describes the size of papers published/accepted, the books, etc., about non-singular and singular kernels;
- The variable ${u}_{3}$ stands for the qualitative aspects such as the fairness of publishers and the editorial board;
- The measure ${u}_{4}\left(t\right)$ denotes a conference presenting good talks on the subject of fractional derivatives and integrals.

## 6. Numerical Simulations

#### 6.1. Numerical Simulations for the Stationary Distribution and Extinction

**Example**

**1.**

**Example**

**2.**

#### 6.2. Numerical Simulations for Stochastic Optimality

## 7. Conclusions and Prediction

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Simulations of $({F}^{c}\left(t\right),I\left(t\right),{I}^{P}\left(t\right),{I}^{N}\left(t\right),R\left(t\right),D\left(t\right))$ for the stochastic model (1) with its corresponding deterministic version. (

**a**) ${F}^{C}\left(t\right)$: researchers using fractional differential operators with non-singular kernels. (

**b**) $I\left(t\right)$: researchers that have been affected by the harmful papers. (

**c**) ${I}^{P}\left(t\right)$: researchers that have been affected, but still have positive opinions about non-singular kernel derivatives. (

**d**) ${I}^{N}\left(t\right)$: researchers that have been affected and have negative opinions about non-singular kernel derivatives. (

**e**) $R\left(t\right)$: researchers that have overcome divisive criticism. (

**f**) $D\left(t\right)$: researchers that die, or retire.

**Figure 2.**Simulations of $({F}^{c}\left(t\right),I\left(t\right),{I}^{P}\left(t\right),{I}^{N}\left(t\right),R\left(t\right),D\left(t\right))$ for the stochastic model (1) with its corresponding deterministic version. (

**a**) ${F}^{C}\left(t\right)$: researchers using fractional differential operators with non-singular kernels. (

**b**) $I\left(t\right)$: researchers that have been affected by the harmful papers. (

**c**) ${I}^{P}\left(t\right)$: researchers that have been affected, but still have positive opinions about non-singular kernel derivatives. (

**d**) ${I}^{N}\left(t\right)$: researchers that have been affected and have negative opinions about non-singular kernel derivatives. (

**e**) $R\left(t\right)$: researchers that have overcome divisive criticism. (

**f**) $D\left(t\right)$: researchers that die, or retire.

**Figure 3.**The probability distribution histogram of $({F}^{c}\left(t\right),I\left(t\right),{I}^{P}\left(t\right),{I}^{N}\left(t\right),R\left(t\right),D\left(t\right))$ for the stochastic model (1). (

**a**) ${F}^{C}\left(t\right)$: researchers using fractional differential operators with non-singular kernels. (

**b**) $I\left(t\right)$: researchers that have been affected by the harmful papers. (

**c**) ${I}^{P}\left(t\right)$: researchers that have been affected, but still have positive opinions about non-singular kernel derivatives. (

**d**) ${I}^{N}\left(t\right)$: researchers that have been affected and have negative opinions about non-singular kernel derivatives. (

**e**) $R\left(t\right)$: researchers that have overcome divisive criticism. (

**f**) $D\left(t\right)$: researchers that die, or retire.

**Figure 4.**Simulations of $({F}^{c}\left(t\right),I\left(t\right),{I}^{P}\left(t\right))$ for the stochastic model (1) with its corresponding deterministic version, both with and without control. (

**a**) ${F}^{c}\left(t\right)$ in the case of the deterministic model both with and without control. (

**b**) ${F}^{c}\left(t\right)$ in the case of the stochastic model both with and without control. (

**c**) $I\left(t\right)$ in the case of the deterministic model both with and without control. (

**d**) $I\left(t\right)$ in the case of the stochastic model both with and without control. (

**e**) ${I}^{P}\left(t\right)$ in the case of the deterministic model both with and without control. (

**f**) ${I}^{P}\left(t\right)$ in the case of the stochastic model both with and without control.

**Figure 5.**Simulations of $({I}^{N}\left(t\right),R\left(t\right),D\left(t\right))$ for the stochastic model (1) with its corresponding deterministic version, both with and without controls. (

**a**) ${I}^{N}\left(t\right)$ in the case of the deterministic model both with and without control. (

**b**) ${I}^{N}\left(t\right)$ in the case of the stochastic model both with and without control. (

**c**) $R\left(t\right)$ in the case of the deterministic model both with and without control. (

**d**) $R\left(t\right)$ in the case of the stochastic model both with and without control. (

**e**) $D\left(t\right)$ in the case of the stochastic model both with and without control. (

**f**) $D\left(t\right)$ in the case of the stochastic model both with and without control.

**Figure 6.**Simulations of $({u}_{1}\left(t\right),{u}_{2}\left(t\right),{u}_{3}\left(t\right),{u}_{4}\left(t\right))$ for the stochastic model (1) with its corresponding deterministic version. (

**a**) Trajectories of the optimal control of the deterministic system with and without control. (

**b**) Trajectories of the optimal control of the stochastic system with and without control.

**Table 1.**Values of the parameters for simulating Model (1).

Parameters | Unite | ${\mathit{V}}_{1}$ | ${\mathit{V}}_{2}$ | ${\mathit{V}}_{3}$ |
---|---|---|---|---|

$\Lambda $ | Per week | 0.05 | 2.5 | 2.8 |

d | Per week | 1/170.365 | 1/70.365 | 1/70.365 |

$\beta $ | Per week | 0.08 | 0.9 | 0.8 |

${\gamma}_{1}$ | Per week | 0.01 | 0.08 | 0.8 |

${\gamma}_{2}$ | Per week | 0.04 | 0.07 | 0.5 |

${\gamma}_{3}$ | Per week | 0.004 | 0.07 | 0.004 |

${\varphi}_{2}$ | Per week | 0.02 | 0.04 | 0.04 |

$\tau $ | Per week | 0.004 | 0.004 | 0.005 |

$\delta $ | Per week | 0.01 | 0.03 | 0.03 |

$\sigma $ | Per week | 0.03 | 0.1 | 0.1 |

$\psi $ v | Per week | 0.01 | 0.01 | 0.01 |

${\varphi}_{1}$ | Per week | 0.02 | 0.001 | 0.01 |

${\eta}_{1}$ | Noise intensity | 0.4 | 0.4 | 2 |

${\eta}_{2}$ | Noise intensity | 0.3 | 0.2 | 0.3 |

${\eta}_{3}$ | Noise intensity | 0.4 | 0.5 | 0.4 |

${\eta}_{4}$ | Noise intensity | 0.5 | 0.5 | 0.5 |

${\eta}_{5}$ | Noise intensity | 0.6 | 0.6 | 0.1 |

${\eta}_{6}$ | Noise intensity | 0.2 | 0.1 | 0.2 |

${F}^{c}\left(0\right)$ | Initial value | 100 | 60 | 430 |

$I\left(0\right)$ | Initial value | 10 | 50 | 10 |

${I}^{P}\left(0\right)$ | Initial value | 06 | 40 | 30 |

${I}^{N}\left(0\right)$ | Initial value | 03 | 35 | 20 |

$R\left(0\right)$ | Initial value | 00 | 15 | 10 |

$D\left(0\right)$ | Initial value | 02 | 15 | 10 |

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**MDPI and ACS Style**

Din, A.; Ain, Q.T.
Stochastic Optimal Control Analysis of a Mathematical Model: Theory and Application to Non-Singular Kernels. *Fractal Fract.* **2022**, *6*, 279.
https://doi.org/10.3390/fractalfract6050279

**AMA Style**

Din A, Ain QT.
Stochastic Optimal Control Analysis of a Mathematical Model: Theory and Application to Non-Singular Kernels. *Fractal and Fractional*. 2022; 6(5):279.
https://doi.org/10.3390/fractalfract6050279

**Chicago/Turabian Style**

Din, Anwarud, and Qura Tul Ain.
2022. "Stochastic Optimal Control Analysis of a Mathematical Model: Theory and Application to Non-Singular Kernels" *Fractal and Fractional* 6, no. 5: 279.
https://doi.org/10.3390/fractalfract6050279