[Paper Review] A Mean-field Approach for Controlling Singularly Perturbed Multi-population SIS Epidemics

P T Akhil, Eitan Altman+1 more|arXiv (Cornell University)|Feb 15, 2019

Mathematical and Theoretical Epidemiology and Ecology ModelsMedicine21 references3 citations

TL;DR

This paper proposes a mean-field singular perturbation approach to control multi-population SIS epidemics with one mobile community and multiple isolated communities. By modeling cross-community interactions as controllable rates, it derives a reduced-order ODE system via singular perturbation theory, enabling optimal control via Pontryagin’s minimum principle. The key result is that infection can be sustained in all communities—despite dying out in isolation—when mobile nodes facilitate transmission, and the control policy is provably near-optimal for large populations.

ABSTRACT

We consider a multi-population epidemic model with one or more (almost) isolated communities and one mobile community. Each of the isolated communities has contact within itself and, in addition, contact with the outside world but only through the mobile community. The contact rate between the mobile community and the other communities is assumed to be controlled. We first derive a multidimensional ordinary differential equation (ODE) as a mean-field fluid approximation to the process of the number of infected nodes, after appropriate scaling. We show that the approximation becomes tight as the sizes of the communities grow. We then use a singular perturbation approach to reduce the dimension of the ODE and identify an optimal control policy on this system over a fixed time horizon via Pontryagin's minimum principle. We then show that this policy is close to optimal, within a certain class, on the original problem for large enough communities. From a phenomenological perspective, we show that the epidemic may sustain in time in all communities (and thus the system has a nontrivial metastable regime) even though in the absence of the mobile nodes the epidemic would die out quickly within each of the isolated communities.

Motivation & Objective

• To analyze whether cross-community interactions via a mobile population can sustain epidemics in isolated communities where infection would otherwise die out.
• To develop a scalable control strategy that maximizes information spread (or infection) over a finite time horizon while minimizing control costs on mobile community interactions.
• To establish a rigorous link between the original stochastic epidemic process and a reduced deterministic fluid limit via mean-field and singular perturbation approximations.
• To prove that the optimal control policy derived from the reduced system is close to optimal for the original system in large-population regimes.

Proposed method

• Derive a multidimensional ordinary differential equation (ODE) as a mean-field fluid approximation of the infected population dynamics after scaling community sizes to infinity.
• Apply singular perturbation theory to separate fast dynamics (within isolated communities) from slow dynamics (in the mobile community), reducing the system to a one-dimensional ODE.
• Use Pontryagin’s minimum principle to derive an optimal control policy on the reduced ODE system for a fixed time horizon.
• Establish tightness of the fluid approximation by proving that the empirical distribution of infected nodes in the original process converges to the reduced system’s trajectory as population size increases.
• Bound the approximation error using Doob’s and Markov’s inequalities, showing convergence in probability with rate $ O(1/ ext{log } n) $ under specific scaling.
• Validate the near-optimality of the control policy by proving that the value function of the reduced system converges to that of the original system as $ n \to \infty $.

Experimental results

Research questions

• RQ1 Can a mobile community sustain infection in isolated communities where the epidemic would otherwise die out quickly?
• RQ2 What is the optimal control policy for maximizing infection spread over a finite time horizon while minimizing control costs?
• RQ3 How accurate is the reduced-order ODE model in approximating the behavior of the full stochastic epidemic process in large populations?
• RQ4 Can the fluid limit and singular perturbation reduction be rigorously justified in terms of convergence and near-optimality?
• RQ5 What is the rate of convergence of the fluid approximation and control policy to the original system as community sizes grow?

Key findings

• Infection can be sustained in all isolated communities due to cross-community transmission via the mobile population, even when each isolated community would otherwise experience a rapid die-out of infection.
• The reduced one-dimensional ODE system accurately captures the long-term behavior of the full system, with the fluid approximation error bounded by $ O(1/\log n) $ under appropriate scaling.
• The optimal control policy derived from the reduced system is provably close to optimal for the original system, with the difference in value functions vanishing as $ n \to \infty $.
• The convergence rate of the approximation error is $ O(1/\log n) $, and the probability of large deviation from the fluid limit decays as $ O(1/\log n) $.
• The Lipschitz constants in the analysis scale as $ O(1/\varepsilon) $, where $ \varepsilon $ controls the timescale separation, and the error bounds are shown to vanish under suitable scaling of $ \varepsilon \to 0 $ with $ \varepsilon \geq C/\log n $.
• The value function of the original problem converges to that of the reduced problem, with the difference bounded by a constant times $ O(1/\log n) $, confirming the near-optimality of the control policy.