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Infinite horizon optimal control of a SIR epidemic under an ICU constraint

Lorenzo Freddi, Dan Goreac

TL;DR

A $\Gamma-convergence argument is developed to reduce the problem to a finite horizon allowing to use a state constrained version of Pontryagin's theorem to characterize the structure of the optimal controls of a SIR epidemic on an infinite horizon.

Abstract

The aim of this paper is to provide a rigorous mathematical analysis of an optimal control problem of a SIR epidemic on an infinite horizon. A state constraint related to intensive care units (ICU) capacity is imposed and the objective functional linearly depends on the state and the control. After preliminary asymptotic and viability analyses, a $Γ$-convergence argument is developed to reduce the problem to a finite horizon allowing to use a state constrained version of Pontryagin's theorem to characterize the structure of the optimal controls. Illustrating examples and numeric simulations are given according to the available data on Covid-19 epidemic in Italy.

Infinite horizon optimal control of a SIR epidemic under an ICU constraint

TL;DR

A $\Gamma-convergence argument is developed to reduce the problem to a finite horizon allowing to use a state constrained version of Pontryagin's theorem to characterize the structure of the optimal controls of a SIR epidemic on an infinite horizon.

Abstract

The aim of this paper is to provide a rigorous mathematical analysis of an optimal control problem of a SIR epidemic on an infinite horizon. A state constraint related to intensive care units (ICU) capacity is imposed and the objective functional linearly depends on the state and the control. After preliminary asymptotic and viability analyses, a -convergence argument is developed to reduce the problem to a finite horizon allowing to use a state constrained version of Pontryagin's theorem to characterize the structure of the optimal controls. Illustrating examples and numeric simulations are given according to the available data on Covid-19 epidemic in Italy.
Paper Structure (16 sections, 25 theorems, 100 equations, 10 figures)

This paper contains 16 sections, 25 theorems, 100 equations, 10 figures.

Table of Contents

  1. Introduction
  2. Analysis of the controlled system
  3. An elementary viability analysis
  4. Greedy control strategy and examples
  5. The optimal control problem for a general cost
  6. The case of a linear cost
  7. Reduction to finite horizon problems
  8. The case of a cost depending only on the control ($\lambda_2=0$)
  9. The general case
  10. The finite horizon problem: optimality conditions
  11. Pontryagin approach. General considerations
  12. Optimality conditions for problem ${\mathcal{P}}^T_{\lambda_1,\lambda_2}$ with $T<\infty$
  13. Remarks and consequences
  14. Structure of the optimal controls
  15. Bocop simulations
  16. ...and 1 more sections

Key Result

Theorem 2.1

For every control $u\in U$ and every initial condition $i_0\in(0,1)$ and $s_0\in(0,1-i_0]$ the controlled system stateeq admits a unique solution $(s,i)\in Y$. Moreover, $(s(t),i(t))\in\mathbb{T}:=\{(s,i)\in(0,1]^2\ :\ s+i\le1\}$ for every $t\in I$.

Figures (10)

  • Figure 1: The epidemic trajectory
  • Figure 2: Epidemic trajectories with $u=0$ (in green) and $u=u_{\max}$ (in red)
  • Figure 3: The curves $\Phi_{0}$ and $\Phi_{\max}$
  • Figure 4: The greedy strategy (in black) starting from $(s_0,i_0)\in\mathcal{B}\setminus\mathcal{A}$
  • Figure 5: The curve $\Psi_0$
  • ...and 5 more figures

Theorems & Definitions (45)

  • Theorem 2.1
  • Remark 2.2: Notation
  • Lemma 2.3
  • Proposition 2.4
  • Remark 2.5
  • Theorem 2.6
  • Lemma 2.7
  • Remark 2.8
  • Definition 3.1
  • Theorem 3.2
  • ...and 35 more