Optimal Control of an SIR Model with Noncompliance as a Social Contagion

TL;DR

The paper develops a mechanistic SIR framework that explicitly separates compliant and noncompliant populations and models the spread of noncompliance as a social contagion. It derives the disease-free equilibria and the reproduction number $\mathscr R_0(s,s^*)$ using a next-generation matrix, then formulates a four-dimensional optimal-control problem with controls $(\alpha,\eta,\mu,\nu)$ that balance infection burden, noncompliance, and control costs via a Pontryagin framework. Existence of an optimal control is established, and explicit PMP-based control laws are computed with a forward-backward sweep algorithm. Numerical results across three scenarios show how synergies between disease-controls and noncompliance-controls can dramatically reduce total cost, while identifying regimes where high baseline noncompliance undermines intervention efficacy. The work suggests directions for future extensions to stochastic settings, multiplex networks, and richer control portfolios to enhance real-world applicability.

Abstract

We propose and study a compartmental model for epidemiology with human behavioral effects. Specifically, our model incorporates governmental prevention measures aimed at lowering the disease infection rate, but we split the population into those who comply with the measures and those who do not comply and therefore do not receive the reduction in infectivity. We then allow the attitude of noncompliance to spread as a social contagion parallel to the disease. We derive the reproductive ratio for our model and provide stability analysis for the disease-free equilibria. We then propose a control scenario wherein a policy-maker with access to control variables representing disease prevention mandates, treatment efforts, and educational campaigns aimed at encouraging compliance minimizes a cost functional incorporating several cost concerns. We characterize optimal controls via the Pontryagin optimality principle and present simulations which demonstrate the behavior of the control maps in several different parameter regimes.

Figures (7)

• Figure 1: Table of parameters for each scenario. Recall in particular that $\overline \eta, \overline \mu, \overline \nu$ are upper bounds for the control variables $\eta,\mu,\nu$, which represent increase in recovery rate due to seeking treatment, decrease in spread rate of noncompliance due to public health initiatives, and recovery rate for noncompliance due to educational campaigns, respectively. Not mentioned is the control variable $\alpha$, representing decrease in infection rate among the compliant susceptible population due to nonpharmaceutical intervention measures, since $\alpha$ will take values in $[0,1].$
• Figure 2: Simulation of Scenario 1 in the absence of controls.
• Figure 3: Simulation of Scenario 1 with a "balanced" policy-maker: cost weights $c_1 = 1, c_2 = 0.1$. The strategy is to quell noncompliance using $\mu,\nu$ so as to keep the infection rate low. Since the majority of the population is compliant, $\alpha$ and $\nu$ can be deployed to effectively lower the reproductive ratio. However, it is not lowered so much that $\mathscr R_0(\tfrac{b}{\delta},0) < 1.$ A small outbreak of the disease is allowed to occur.
• Figure 4: Simulation of Scenario 1 with a public health oriented policy-maker: cost weights $c_1 = 5, c_2 = 0$. In this case, even though there is no explicit cost associated with noncompliance, because the policy-maker desires to keep infections low, the optimal strategy employs $\mu$ and $\nu$ even more strongly than in the balanced case. The disease controls $\alpha$ and $\eta$ are also increased so as to drive $\mathscr R_0(\tfrac{b}{\delta},0) < 1$ and take advantage of the stability of the DFE $(\tfrac{b}{\delta},0)$ until the disease has been effectively eradicated.
• Figure 5: Simulation of Scenario 1 with a health oriented policy-maker: cost weights $c_1 = 5, c_2 = 0$, with either treatment $\eta$ unavailable (top), or control of noncompliance $\mu,\nu$ unavailable (bottom). Note the $y$-axis change in the top left.

Theorems & Definitions (7)

• Proposition 2.1
• proof
• Proposition 2.2
• proof
• Theorem 2.3
• Theorem 3.1
• proof