MPC without Terminal Ingredients Tailored to the SEIR Compartmental Epidemic Model

TL;DR

This paper tackles constrained SEIR epidemic control using model predictive control (MPC) without terminal ingredients. It proves recursive feasibility and convergence to the continuum of disease-free equilibria under a designed quadratic running cost and a sufficiently long forecast horizon, leveraging the admissible set $\mathcal{A}$ and the safe invariant box $\mathbb{X}_{\mathcal{M}}$. The results show that a subset $\mathcal{A}'$ acts as a domain of attraction for $\mathcal{E}_{\mathrm{nom}}$, enabling stable eradication without terminal constraints. Numerical simulations demonstrate that shorter horizons than those required by terminal-ingredient MPC can still achieve convergence, with higher $\lambda$ values reducing infections but potentially prolonging the epidemic, suggesting practical computational savings and design insights for intervention strategies.

Abstract

We consider the SEIR compartmental epidemic model subject to state and input constraints (a cap on the proportion of infectious individuals and limits on the allowed social distancing and quarantining measures, respectively). We present a tailored model predictive control (MPC) scheme without terminal conditions. We rigorously show recursive feasibility and asymptotic convergence of the MPC closed loop to the continuum of disease-free equilibrium points for suitably designed quadratic running cost and a sufficiently long prediction horizon (forecast window). Moreover, we establish the viability kernel (a.k.a. the admissible set) as a domain of attraction of the continuum of equilibria.

Figures (7)

• Figure 1: Admissible set, $\mathcal{A}$, and invariant box, $\mathbb{X}_{\mathcal{M}}$. From $\mathbf{x}^0\in\mathcal{A}$, with an admissible input, it is possible for the state to always satisfy the infection cap and to asymptotically settle at a point where $x_1\leq \frac{\gamma_{\max}}{\beta_{\min}}$. From $\mathbf{x}^0\in\mathbb{X}_{\mathcal{M}}$ the state will satisfy the infection cap for any admissible input and settle at a point where $x_1\leq \frac{\gamma_{\mathrm{nom}}}{\beta_{\mathrm{nom}}}$.
• Figure 2: Sketch of the proof of Proposition \ref{['lem:V_inf_uni_bounded_on_A']}.
• Figure 3: Closed-loop state trajectories of the constrained SEIR system, $\mathcal{S}$, produced by the MPC algorithm without terminal ingredients, Algorithm \ref{['alg:mpc']}, with varying values for $\lambda$ (0.99 cyan, 0.7 magenta, 0.5 green, 0.2 dark blue, 0.01 red). The blue points indicate the boundary of the admissible set, $[\partial \mathcal{A}]_- := \partial \mathcal{A} \cap \mathrm{int}(\mathbb{X})$; the grey box indicates the invariant box, $\mathbb{X}_{\mathcal{M}}$; and the blue plane indicates the set $\{\mathbf{x}\in\mathbb{R}^3: x_1 + x_2 + x_3 = 1\}$.
• Figure 4: Closed-loop behaviour of the exposed compartment under the MPC feedback with varying $\lambda$ (first 100 days).
• Figure 5: Closed-loop behaviour of the infectuous compartment under the MPC feedback with varying $\lambda$ (first 100 days).

Theorems & Definitions (12)

• Lemma 1
• proof
• Lemma 2
• proof
• Proposition 1
• proof
• remark 1
• Proposition 2
• proof
• Theorem 1