⚡ electrical engineering

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

This paper proposes an SIR model incorporating noncompliance as a social contagion, establishes the existence of optimal control strategies for minimizing disease and behavioral costs, and numerically demonstrates how policy-maker preferences influence optimal intervention strategies using the sequential quadratic Hamiltonian method.

Original authors: Chloe Ngo, Christian Parkinson, Weinan Wang

Published 2026-06-23

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Chloe Ngo, Christian Parkinson, Weinan Wang

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine a town trying to stop a contagious virus. Usually, scientists model this by assuming everyone follows the rules equally: if the government says "wear a mask," everyone puts one on. But in reality, some people listen, and some people don't.

This paper builds a new kind of model that treats not following the rules (noncompliance) like a virus itself. Just as a cold spreads from person to person, the attitude of "I'm not going to wear a mask" spreads through the crowd. If you hang out with people who don't follow safety rules, you are more likely to stop following them too.

Here is a breakdown of what the authors did, using simple analogies:

1. The Two Groups: The "Compliant" and the "Rebels"

The authors split the population into two teams:

The Compliant: These people wear masks, stay home, and get treated if they get sick. Because they follow the rules, the virus spreads slower among them.
The Non-Compliant: These people ignore the rules. The virus spreads faster among them.

The Twist: The authors realized that the "Rebel" attitude isn't fixed. It spreads like a social contagion. If a "Compliant" person spends time with "Rebels," they might start thinking, "Well, if they aren't wearing a mask, why should I?" This turns them into a Rebel.

2. The Policy-Maker's Toolkit

The paper imagines a "Policy-Maker" (like a mayor or health official) who wants to stop the disease but also has a limited budget. This person has four levers they can pull to control the situation:

The NPI Lever (Non-Pharmaceutical Interventions): Making masks and social distancing stricter. This lowers the virus's speed for the Compliant group.
The Treatment Lever: Getting sick Compliant people better faster.
The "Stop the Spread" Lever: Running campaigns to stop the Rebel attitude from spreading. (e.g., "Don't let your friends talk you out of safety.")
The "Rehab" Lever: Educational campaigns to convince Rebels to become Compliant again.

3. The Balancing Act (The Cost Problem)

The Policy-Maker has to solve a math puzzle. They want to:

Keep the number of sick people low.
Keep the number of Rebels low.
But they also want to save money. Every time they pull a lever (like mandating masks or running ads), it costs money.

The authors asked: What is the perfect mix of these levers to get the best result for the least cost?

4. The "Magic" Math Method

To find the answer, the authors had to invent a new way to do the math.

The Problem: Standard math tools often assume that if you want a little more control, you pay a little more money smoothly. But in real life, you might decide to go "all in" on a strict lockdown for a week and then stop completely. This is a "bang-bang" switch, which is hard for standard math to handle.
The Solution: They used a method called Sequential Quadratic Hamiltonian (SQH). Think of this as a very smart, step-by-step GPS for the Policy-Maker. Instead of guessing the whole route at once, it takes small steps, checks if the cost is going down, and adjusts the direction. It's proven to be more reliable than older methods that sometimes get stuck or give wrong answers.

5. What They Found (The Results)

The authors ran simulations with different "personalities" of Policy-Makers to see how they would act:

The "Balanced" Leader: This leader cares about both health and money. Their strategy? They use the "Stop the Spread" and "Rehab" levers early on to keep the Rebel attitude low. Once the population is mostly Compliant, they use the "NPI" and "Treatment" levers to crush the virus. They allow a tiny outbreak to happen because it's cheaper than trying to stop it 100%.
The "Health-First" Leader: This leader cares only about stopping the virus, regardless of the cost. They go hard on all levers immediately. They force the population to be Compliant and keep the virus rate so low that the disease dies out quickly.
The "Economy-First" Leader: If the virus spreads too fast or if too many people are already Rebels, this leader might decide it's too expensive to fight. They might just let the disease run its course because the cost of trying to stop it is higher than the cost of the sickness itself.

Key Insight: The paper shows that you can't just fight the virus; you have to fight the attitude of not following rules. If you ignore the fact that "Rebel behavior" spreads, your plans to stop the virus will fail.

Summary

This paper creates a model where bad behavior spreads like a virus. It uses advanced math to figure out how a leader should spend their money to stop both the disease and the bad behavior. They found that the best strategy depends entirely on how much the leader cares about money versus health, and that ignoring the spread of "rebelliousness" makes it impossible to control the disease effectively.

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

Problem Formulation
The paper addresses the challenge of incorporating human behavioral dynamics into epidemiological modeling, specifically focusing on the spread of noncompliance with non-pharmaceutical interventions (NPIs). The authors propose a compartmental Susceptible-Infected-Recovered (SIR) model that divides the population into two distinct groups: those who comply with public health measures (compliant) and those who do not (noncompliant).

Unlike traditional models where behavior is treated as a fixed parameter, this model treats noncompliance as a "social contagion" that spreads parallel to the disease via a mass-action mechanism. The compliant population ( $S, I, R$ ) experiences reduced infectivity due to NPIs, while the noncompliant population ( $S^*, I^*, R^*$ ) does not. The model includes four control variables representing policy-maker interventions:

$\alpha(t)$ : Reduction in infectivity/mixing for the compliant population (NPI enforcement).
$\eta(t)$ : Increased recovery rate for compliant infected individuals (treatment efforts).
$\mu(t)$ : Reduction in the spread rate of noncompliance (public health information initiatives).
$\nu(t)$ : Rate of "recovery" from noncompliance (educational campaigns).

The objective is to formulate an optimal control problem where a policy-maker minimizes a cost functional that balances the costs of infections, the costs of noncompliance, and the economic costs of implementing controls.

Methodology
The authors employ a rigorous mathematical framework combining dynamical systems theory, optimal control theory, and numerical analysis:

Model Analysis: They derive the basic reproductive ratio ( $R_0$ ) for the system and analyze the stability of disease-free equilibria (DFE). They establish conditions under which the system remains stable or becomes unstable based on the interplay between disease transmission and behavioral transmission parameters.
Existence of Optimal Controls: A significant theoretical contribution is the proof of the existence of optimal controls. The authors establish this for a generic scenario where:
- Control variables appear nonlinearly in the dynamics.
- Control costs are not required to be smooth or strongly convex (allowing for $L^1$ control costs).
- The admissible control set includes discontinuous control maps (specifically, functions of bounded variation, $BV$).
  The proof utilizes the direct method in the calculus of variations, relying on the weak closure of the control-to-state operator and the lower-semicontinuity of the cost functional.
Numerical Resolution: To solve the optimal control problem, the authors utilize the Sequential Quadratic Hamiltonian (SQH) method. They argue that SQH is superior to the standard Forward-Backward Sweep Method (FBSM) because it possesses a robust convergence theory. The algorithm iteratively updates control maps using quadratic perturbations to the Hamiltonian, ensuring sufficient decrease in the cost functional at each step. The authors prove the convergence of the SQH iterates.

Key Results
The paper presents numerical simulations across three distinct scenarios to demonstrate how optimal strategies depend on parameter regimes and policy-maker preferences (encoded in the cost weights):

Scenario 1 (Baseline): With moderate infection rates and a compliant initial population, the optimal strategy depends heavily on the policy-maker's priorities.
- A "balanced" policy-maker (weighing infection and noncompliance costs) employs a synergistic strategy: using $\mu$ and $\nu$ to suppress noncompliance early, allowing $\alpha$ and $\eta$ to effectively lower $R_0$ without driving it below 1 immediately, thus accepting a small outbreak to minimize total cost.
- A "public health-oriented" policy-maker (high infection cost, zero noncompliance cost) aggressively uses all controls to drive $R_0 < 1$ and eradicate the disease, leveraging the stability of the fully compliant DFE.
- Sensitivity: Removing treatment ( $\eta=0$ ) reduces the efficacy of control efforts by roughly one-third. Removing behavioral controls ( $\mu=\nu=0$ ) renders disease controls largely ineffective, as the population shifts to noncompliance, reducing cost reduction to ~10%.
- L1 Costs: The method successfully handles $L^1$ control costs (where $d_2=0$ ), resulting in "bang-bang" type controls for $\eta, \mu, \nu$ (taking only 0 or maximum values), while $\alpha$ remains smooth due to the quadratic nature of the dynamics in $\alpha$ .
Scenario 2 (High Spread): When the disease infection rate and the spread rate of noncompliance are high, and a significant portion of new births are noncompliant, controls become less effective. Only a policy-maker with very high infection cost weights ( $c_1 \ge 6$ ) finds it optimal to fully suppress the disease; otherwise, the optimal strategy is to let the disease run its course due to the prohibitive cost of control.
Scenario 3 (Baseline Proclivity): Increasing the baseline proportion of noncompliant new births ( $\xi$ ) reveals a critical threshold. When $\xi < 0.5$ , it is optimal to suppress noncompliance. However, when $\xi \ge 0.5$ , behavioral controls ( $\mu$ ) become ineffective (dropping to zero), and the system cannot be guided to a disease-free equilibrium.

Significance and Claims
The paper claims three primary contributions:

Sociological/Modeling: It provides a mechanistic framework where adherence to NPIs is a dynamic epidemiological quantity rather than a fixed parameter. It demonstrates that optimal control strategies are highly sensitive to policy-maker preferences and that ignoring the dynamics of noncompliance leads to suboptimal or ineffective strategies. It highlights that interventions targeting noncompliance are justified by their downstream effects on disease transmission, even if noncompliance itself carries no direct cost.
Mathematical (Existence): It provides a general existence framework for optimal controls in biological models that allows for nonlinear dynamics, non-smooth/non-convex costs (specifically $L^1$ ), and discontinuous controls. This relaxes common assumptions (like control-affine systems and $L^2$ costs) often made for analytical convenience but which are physically unrealistic for certain policy interventions.
Numerical: It advocates for the use of the SQH method in epidemiology, arguing it offers better convergence properties than the widely used FBSM while remaining implementable. The authors demonstrate that SQH can effectively resolve complex optimal control problems involving $L^1$ costs and discontinuous solutions.

The authors conclude that while the study is qualitative, it underscores the necessity of integrating behavioral dynamics into epidemiological models and suggests that future work could incorporate stochasticity, network structures, evolutionary game theory, and more granular control variables (e.g., vaccination, media control).