Optimal control of the COVID-19 pandemic: controlled sanitary deconfinement in Portugal

Abstract

The COVID-19 pandemic has forced policy makers to decree urgent confinements to stop a rapid and massive contagion. However, after that stage, societies are being forced to find an equilibrium between the need to reduce contagion rates and the need to reopen their economies. The experience hitherto lived has provided data on the evolution of the pandemic, in particular the population dynamics as a result of the public health measures enacted. This allows the formulation of forecasting mathematical models to anticipate the consequences of political decisions. Here we propose a model to do so and apply it to the case of Portugal. With a mathematical deterministic model, described by a system of ordinary differential equations, we fit the real evolution of COVID-19 in this country. After identification of the population readiness to follow social restrictions, by analyzing the social media, we incorporate this effect in a version of the model that allow us to check different scenarios. This is realized by considering a Monte Carlo discrete version of the previous model coupled via a complex network. Then, we apply optimal control theory to maximize the number of people returning to "normal life" and minimizing the number of active infected individuals with minimal economical costs while warranting a low level of hospitalizations. This work allows testing various scenarios of pandemic management (closure of sectors of the economy, partial/total compliance with protection measures by citizens, number of beds in intensive care units, etc.), ensuring the responsiveness of the health system, thus being a public health decision support tool.

Figure 1

Fraction of confirmed active cases per day in Portugal. Red line: from March 2 to May 17, 2020. Yellow line: from May 17 to June 9, 2020. Green line: from June 9 to July 29, 2020. The drastic jump down in the real data (black points) corresponds to the day when the Portuguese authorities announced 9844 recovered individuals on May 24.

Figure 2

Probability distribution ($\mathbf{P}(\mathbf{u})$) for each opinion ($\mathbf{u}$). The opinion ranges from zero to one, zero meaning no intention to follow the government policies while one means complete adhesion to this policy. The blue values correspond to the Portuguese situation in April 2020 while the yellow ones are for the situation in July 2020.

Figure 3

Evolution of the number of infected individuals (normalized by the total population) with time. (a) Red crosses correspond to the experimental recordings while the blue line is the fit of the SAIRP model with opinion. The bluish shadow marks the uncertainty of the model. (b) Blue line is the fit of the SAIRP model coupled with the opinion distribution, corresponding to April 2020, and the yellow line is the evolution of the model coupled with the state of social opinion as in July 2020.

Figure 4

Active infected individuals: comparison of the solution of the SAIRP model with the optimal control problem. Linear and quadratic fit for the time where there is no transfer from P to S, in terms of $u_{max}\in [0.05;0.95]$ under the constraints $I⩽0.60\times I_{max}$ and $I⩽2/3\times I_{max}$. (a) Fraction of active infected individuals. (b) Control u satisfying the constraint $I(t)⩽0.60\times I_{max}$. (c) Linear fit for the time with no transfer from P to S for $0<u_{max}⩽0.5$. (d) Quadratic fit for the time with no transfer from P to S for $0<u_{max}⩽0.95$.

Figure 5

Number of hospital beds occupation for the optimal control solutions. (a) Number of hospital beds for $u_{max}\in \{0.05,0.10,0.15,0.20\}$ subject to $I(t)⩽0.60\times I_{max}$ varying between 5% and 15% of the number of infected individuals. (b) Number of hospital beds for $u_{max}\in \{0.05,0.10,0.15,0.20,0.25\}$ under the state constraint $I(t)⩽0.60\times I_{max}$, representing between 5% and 15% of the number of active infected individuals. (c) ICU hospital bed occupancy for $u_{max}\in \{0.05,0.10,0.15,0.20,0.25\}$ under the state constraint $I(t)⩽0.60\times I_{max}$. The ICU beds occupation represents between 1.5% and 3% of the number of active infected individuals.