# Real-Time Optimization of Social Distancing to Mitigate COVID-19 Pandemic Using Quantized Extremum Seeking

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## Abstract

**:**

## 1. Introduction

## 2. COVID-19 Outbreak Modeling

#### 2.1. SIR Modeling

#### 2.2. Bifurcation Analysis

#### 2.3. Constrained Objective

## 3. Social Distancing Real-Time Optimization

#### 3.1. Classical Discrete-Time Extremum Seeking

#### 3.2. Discrete-Time Quantized Extremum Seeking

## 4. Quantized ESC Application to the SEAIR Model

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Evolution of objective function (6) with respect to the input ${\alpha}_{a}$, describing the cost of pandemic mitigation with respect to the social distancing level. This figure highlights a unique optimum represented by the black star. Continuous line: steady-state values. Dashed line: transient values after 200 days.

**Figure 2.**Discrete perturbation-based extremum seeking [26]. The input u is modulated with the dither signal d perturbing the measured objective function $h=J$. The latter signal is then demodulated in two steps: first by removing the continuous component and low frequencies through a high-pass filter with cut-off frequency ${f}_{HP}$, then by multiplying the filtered signal ${h}_{HP}$ by the dither signal to isolate the information on the gradient $\widehat{\xi}$ at $\omega $. The integration of the gradient estimate provides the input estimate $\widehat{u}$.

**Figure 4.**Application of discrete QESC to system (1)—time evolution of the states. In blue: QESC with bias compensation—In dashed red: QESC without bias compensation. Even if the state variables present almost identical transient trajectories, the latter diverge after 200 days and end up in different steady-states. Non-intuitively, converging to a closer neighborhood of the cost objective optimum (i.e., optimizing social distancing) unfortunately leads to slightly higher casualties while still limiting the number of infections.

**Figure 5.**Application of discrete QESC to system (1): time evolution of the input $u={\alpha}_{a}$ and output $y=J$. Blue line: QESC with bias compensation. Dashed red line: QESC without bias compensation. The impact of bias compensation is highlighted by the faster decrease of the discrete social distancing level (as ${\alpha}_{a}$ increases) after 200 days.

**Figure 6.**Application of discrete QESC to system (1): time evolution of the gradient $\widehat{\xi}$, the dither magnitude a and the bias $\delta $. Blue line: QESC with bias compensation. Dashed red line: QESC without bias compensation. This figure shows the small but important impact of the bias estimation during the transient period, which allows gaining one quantized level on ${\alpha}_{a}$ as illustrated in Figure 5.

**Figure 7.**Application of discrete QESC to system (1)—evolution of the output $y=J$ with respect to the input $u={\alpha}_{a}$. Black arrows indicate the convergence direction. Continuous line: QESC with bias compensation. Dashed line: QESC without bias compensation. The blue lines indicate the transient cost functions in 50, 100 and 360 days, while the red line indicates the cost function steady-state after 1000 days. Blue stars show the ending state of the QESC algorithms while the red star indicates the numerical steady-state optimum ${y}^{*}={J}^{*}=-2.1665$ and ${u}^{*}={\alpha}_{a}^{*}=0.28$. This diagram confirms that, thanks to the bias estimation, the QESC is able to drive the system at the closest quantized level of the optimum ${J}^{*}\left({\alpha}_{a}^{*}\right)$.

N | ${\mathit{\alpha}}_{\mathit{i}}$$\left[{\mathit{d}}^{-1}\right]$ | $\mathit{\kappa}$$\left[{\mathit{d}}^{-1}\right]$ | $\mathit{\beta}$$\left[{\mathit{d}}^{-1}\right]$ | l$\left[{\mathit{d}}^{-1}\right]$ | $\mathit{\rho}$$\left[\mathit{d}\right]$ | $\mathit{\gamma}$$\left[{\mathit{d}}^{-1}\right]$ |
---|---|---|---|---|---|---|

$1.1\phantom{\rule{4pt}{0ex}}{10}^{7}$ | $0.01$ | $0.3$ | $0.025$ | $0.5$ | $0.1$ | $0.1$ |

**Table 2.**Parameter values of the objective function (6).

${\mathit{I}}_{\mathbf{ref}}$ | ${\mathit{\alpha}}_{\mathit{a},\mathbf{ref}}$$\left[{\mathit{d}}^{-1}\right]$ | ${\mathit{\eta}}_{\mathit{\psi}}$ | ${\mathit{\eta}}_{\mathit{\varphi}}$ | ${\mathit{\eta}}_{\mathit{P}}$ | $\mathit{\u03f5}$ |
---|---|---|---|---|---|

2 | $0.5$ | 1 | 1 | 200 | $0.3$ |

h$\left[{\mathit{d}}^{-1}\right]$ | $\mathit{\omega}$$\left[{\mathit{d}}^{-1}\right]$ | ${\mathit{a}}_{-}$ | ${\mathit{a}}_{0}$ | ${\mathit{\sigma}}_{\mathit{a}}$ | ${\mathit{\gamma}}_{\mathit{a}}$ | ${\mathit{u}}_{+}-{\mathit{u}}_{-}$ | ${\mathit{k}}_{\mathit{I}}=\frac{1}{{\mathit{\tau}}_{\mathit{I}}}$ |
---|---|---|---|---|---|---|---|

$0.99$ | $\frac{2\phantom{\rule{4pt}{0ex}}\pi}{205}$ | $0.005$ | $0.05$ | $0.15$ | $0.7$ | $0.025$ | $\frac{1}{7}$ |

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**MDPI and ACS Style**

Dewasme, L.; Vande Wouwer, A. Real-Time Optimization of Social Distancing to Mitigate COVID-19 Pandemic Using Quantized Extremum Seeking. *COVID* **2022**, *2*, 1077-1088.
https://doi.org/10.3390/covid2080079

**AMA Style**

Dewasme L, Vande Wouwer A. Real-Time Optimization of Social Distancing to Mitigate COVID-19 Pandemic Using Quantized Extremum Seeking. *COVID*. 2022; 2(8):1077-1088.
https://doi.org/10.3390/covid2080079

**Chicago/Turabian Style**

Dewasme, Laurent, and Alain Vande Wouwer. 2022. "Real-Time Optimization of Social Distancing to Mitigate COVID-19 Pandemic Using Quantized Extremum Seeking" *COVID* 2, no. 8: 1077-1088.
https://doi.org/10.3390/covid2080079