# Long-Term Bifurcation and Stochastic Optimal Control of a Triple-Delayed Ebola Virus Model with Vaccination and Quarantine Strategies

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## Abstract

**:**

## 1. Introduction

## 2. Stochastic Long-Run Dynamics of Ebola Model

**Lemma**

**1.**

**Theorem**

**1.**

- 1.
- The stationary case (${\mathcal{S}}_{\circ}^{\u2606}>1$)—that is, ecosystem (3)—admits a single ergodic limiting distribution ${\pi}_{\u2606}^{E}(\xb7)$. In other words, Ebola epidemic persists.
- 2.
- The eradication case (${\mathcal{S}}_{\circ}^{\u2606}<1$)—that is, the Ebola epidemic—will disappear with full probability.

**Proof.**

## 3. Stochastic Optimal Control Strategies

- A1:
- The isolation of the vulnerable population as well as of infectives is represented by the control measure ${u}_{1}$. This control strategy aims to minimize the value of the transmission coefficient $\beta $.
- A2:
- The rate of quarantine of the infected population is controlled via the function ${u}_{2}$. This strategy aims to reduce the number of infectives by shifting them into the quarantine class.

## 4. Parameters Estimation and Curve Fitting

## 5. Numerical Verification

#### 5.1. Stochastic Long-Run Behavior of Ebola Virus Model

#### 5.2. Stochastic Optimal Control Strategies

## 6. Conclusions

- In Theorem 1, we have proved the main result related to the stationarity and extinction of the Ebola virus. It is worthy to mention that the analysis of these long-time properties is very significant for the underlying perturbed systems. Especially, in the case of epidemiological models, the ergodicity offers a general idea of the infection permanence.
- For controlling the rapid spread of the disease, we assumed two control parameters, and these were incorporated into the stochastic model. The stochastic model was analyzed with the help of Pontryagin principle, and the required optimality conditions were derived therein. To check the validity of the theoretical results and effectiveness of the control parameters, we plotted the models by simulating the same in MATLAB. Through simulations, the obtained results for persistence/extinction of the Ebola infection are verified. It was concluded that by using both the control variables, one can easily eliminate the Ebola from the community with minimum cost.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 2.**Fitting of the deterministic model with the Ebola virus statistical data by using the values of parameter from Table 2.

**Figure 5.**The first row represents the paths of the affected category for the following different time lags: ${\varsigma}_{1}=30$, ${\varsigma}_{1}=100$ and ${\varsigma}_{1}=200$. The second row shows the associated histograms and the probability density functions for ${\mathcal{I}}_{\mathbf{s}}$.

**Figure 6.**The first row represents the paths of the affected category for the following different time lags: ${\varsigma}_{2}=30$, ${\varsigma}_{2}=100$ and ${\varsigma}_{2}=200$. The second row shows the associated histograms and the probability density functions for ${\mathcal{I}}_{\mathbf{s}}$.

**Figure 7.**The first row represents the paths of the affected category for the following different time lags ${\varsigma}_{1}=50$, ${\varsigma}_{1}=150$ and ${\varsigma}_{1}=400$. The second row shows the associated histograms and the probability density functions for ${\mathcal{I}}_{\mathbf{s}}$.

**Figure 8.**The long-run behavior of the Ebola without and with control strategies: (first column) curves achieved from the solution of the deterministic system and (second column) trajectories of the random system.

**Figure 9.**The curves exhibit the time-evolution of the control measures obtained from simulating the models: deterministic and stochastic.

**Table 1.**Definition of some subsets used in the demonstration of Theorem 1, where ${x}_{\u2606}>{\rm Y}>0$ are two constants to be selected later.

Subset | Definition |
---|---|

${\mathbb{W}}_{a}$ | $\left(\right)$ |

${\mathbb{W}}_{b}$ | $\left(\right)$ |

${\mathbb{W}}_{c}$ | $\left(\right)$ |

${\mathbb{W}}_{\mathrm{d}}$ | $\{(t,\omega )\in [-\varsigma ,\infty [\times \Omega |\phantom{\rule{2.84544pt}{0ex}}{\mathcal{I}}_{\mathbf{s}}(t,\omega )\ge {x}_{\u2606},\phantom{\rule{2.84544pt}{0ex}}\mathrm{or},\phantom{\rule{2.84544pt}{0ex}}{\mathcal{I}}_{\mathbf{e}}(t,\omega )\ge {x}_{\u2606}\}$ |

${\mathbb{W}}_{g}$ | $\{(t,\omega )\in [-\varsigma ,\infty [\times \Omega |\phantom{\rule{2.84544pt}{0ex}}{\rm Y}\le {\mathcal{I}}_{\mathbf{s}}(t,\omega )\le {x}_{\u2606},\phantom{\rule{2.84544pt}{0ex}}\mathrm{and},\phantom{\rule{2.84544pt}{0ex}}{\rm Y}\le {\mathcal{I}}_{\mathbf{e}}(t,\omega )\le {x}_{\u2606}\}$ |

**Table 2.**The estimated and fitted values of the parameters obtained from fitting the model against the real Ebola cases reported in Western Guinea from the first forty weeks of the 2015 Ebola epidemic [18].

Parameter | Epidemiological Meaning | Value | Source |
---|---|---|---|

$\mathcal{A}$ | Recruitment rate into susceptible class | 10 | Estimated |

h | The Ebola transmission rate | $0.0004$ | Fitted |

u | The normal death rate of each class | $1.2531\times {10}^{-3}$ | Fitted |

v | The vaccination rate of the susceptible individuals | $0.00027$ | Estimated |

z | The recovery rate of quarantine class | $0.13531$ | Fitted |

q | The quarantine rate of infected class | $0.802529$ | Fitted |

w | The recovery rate of infectious individuals | $0.0209$ | Fitted |

${\alpha}_{1}$ | The disease-related death rate of infected class | $1.0135\times {10}^{-6}$ | Fitted |

${\alpha}_{2}$ | The disease-related death rate of isolated class | $3.1969\times {10}^{-2}$ | Fitted |

${\varsigma}_{1}$ | Time delay associated with ${\mathcal{I}}_{\mathbf{s}}$ | 21 | Fitted |

${\varsigma}_{2}$ | Time delay associated with ${\mathcal{I}}_{\mathbf{e}}$ | 21 | Estimated |

${\varsigma}_{3}$ | Time delay associated with ${\mathcal{I}}_{\mathbf{q}}$ | 21 | Estimated |

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**MDPI and ACS Style**

Din, A.; Khan, A.; Sabbar, Y.
Long-Term Bifurcation and Stochastic Optimal Control of a Triple-Delayed Ebola Virus Model with Vaccination and Quarantine Strategies. *Fractal Fract.* **2022**, *6*, 578.
https://doi.org/10.3390/fractalfract6100578

**AMA Style**

Din A, Khan A, Sabbar Y.
Long-Term Bifurcation and Stochastic Optimal Control of a Triple-Delayed Ebola Virus Model with Vaccination and Quarantine Strategies. *Fractal and Fractional*. 2022; 6(10):578.
https://doi.org/10.3390/fractalfract6100578

**Chicago/Turabian Style**

Din, Anwarud, Asad Khan, and Yassine Sabbar.
2022. "Long-Term Bifurcation and Stochastic Optimal Control of a Triple-Delayed Ebola Virus Model with Vaccination and Quarantine Strategies" *Fractal and Fractional* 6, no. 10: 578.
https://doi.org/10.3390/fractalfract6100578