The Impact of Neglecting Vaccine Unwillingness in Epidemiology Models

This study demonstrates that neglecting vaccine unwillingness in epidemiological models introduces significant errors in both long-term equilibrium predictions and short-term epidemic wave simulations, with the magnitude of error depending on the disease's infectivity and the speed of vaccination programs.

The Gist

Imagine you are trying to predict how a fire will spread through a forest. You have a hose (the vaccine) that can put out the fire, but you know that some trees are "stubborn" and won't let you spray them, while others are eager to be saved.

This paper asks a simple but crucial question: Does it matter if we pretend every tree is willing to get sprayed, or do we need to build a model that specifically accounts for the "stubborn" trees?

The author, Glenn Ledder, says the answer depends entirely on how much time you have to watch the fire.

The Two Time Scales: The "Long Haul" vs. The "Big Wave"

The paper splits the problem into two different scenarios, like looking at a marathon runner versus a sprinter.

1. The Endemic Scenario (The Long Haul / The Marathon)

The Metaphor: Imagine a slow-burning fire that has been going on for years. The forest is constantly regrowing new trees (births) and old trees are dying (deaths). The fire is always there, just simmering.

• The Mistake: Many models assume that because we have a hose, we can just spray everyone eventually. They might say, "Okay, we can't spray 30% of the trees, so let's just turn down the water pressure on the hose for everyone."
• The Reality: In this long-term scenario, the "stubborn" trees (the vaccine-refusers) act like a permanent shield for the fire. Because the fire burns so slowly over decades, the "willing" trees get sprayed and become immune very quickly. The fire then only has the "stubborn" trees to feed on.
• The Verdict: If you pretend the stubborn trees don't exist (or just turn down the water pressure), your model is completely wrong. You will think the fire will die out, but it won't. The "stubborn" trees create a safe haven where the fire can hide forever.
  • Simple takeaway: For long-term planning, you must draw a separate box for the people who won't get vaccinated. You can't just pretend they are there but less important.

2. The Epidemic Scenario (The Big Wave / The Sprint)

The Metaphor: Now imagine a massive, sudden wildfire that sweeps through the forest in a few weeks. It's a sprint. The fire moves so fast that it burns trees before the firefighters can even get their hoses ready.

• The Mistake: Again, some models just turn down the water pressure to account for the stubborn trees.
• The Reality: Here, the answer is a bit more nuanced.
  • If the fire is super fast (Highly Infectious) and the hose is slow: The fire burns everything before the water hits. In this case, it doesn't matter much if you model the stubborn trees separately or just turn down the water. The fire wins either way.
  • If the fire is slower and the hose is fast: This is where the model matters. If the fire moves slowly enough that the firefighters can actually save the "willing" trees, then the "stubborn" trees become the only fuel left. If you don't model them correctly, you might think the fire will be put out, but it will actually rage on through the stubborn trees.
  • Simple takeaway: For short-term outbreaks, turning down the water pressure (adjusting the rate) sometimes works as a shortcut, but it's risky. If the vaccine is good and the outbreak isn't too fast, you really need to model the stubborn trees separately to get an accurate prediction.

The "Turn Down the Water" Trick

The paper tests a specific idea: Instead of making the model complicated by adding a "stubborn tree" category, can we just say, "Okay, our hose is 30% less effective because 30% of trees won't take it"?

• In the Long Haul (Endemic): No. This trick fails miserably. It gives you the wrong answer about whether the disease will disappear or stay forever.
• In the Sprint (Epidemic): Maybe, but be careful. If the disease is slow and the vaccine is fast, this trick creates a big error. If the disease is super fast, the error is smaller, but it's still better to be precise.

Why This Matters in the Real World

The author uses data suggesting that in many places (like the US), about 30% of people might refuse a vaccine for a new disease.

• If you are a policy maker planning for next year (Endemic): You cannot ignore the 30%. If you do, you might think you've reached "herd immunity" and stop worrying, only to find the disease is still circulating because it's hiding in the unvaccinated group.
• If you are predicting the next big wave (Epidemic): You need to know how fast the vaccine can be delivered. If you can vaccinate quickly, the 30% refusal rate becomes a huge problem that changes the outcome of the wave. If you can't vaccinate fast, the refusal rate matters less because the disease spreads too fast for the vaccine to help anyway.

The Bottom Line

The paper concludes that while it's tempting to keep math models simple by just "turning down the volume" on vaccination rates, this is a dangerous shortcut.

To get the right answer, especially for long-term disease control, you need to build the "stubborn" people into the map of the model from the start. You can't just pretend they are part of the crowd; you have to acknowledge they are in a different line.

Technical Summary

Here is a detailed technical summary of the paper "The Impact of Neglecting Vaccine Unwillingness in Epidemiology Models" by Glenn Ledder.

1. Problem Statement

The paper addresses a critical flaw in standard compartmental epidemiological models (such as SIR and SVIR): the assumption that vaccination rates apply uniformly to the entire susceptible population ($S$). In reality, a significant fraction of the population is unwilling or unable to receive vaccines due to behavioral or logistical reasons.

The study investigates two primary questions:

1. Magnitude of Error: How much error is introduced into key model outcomes (e.g., basic reproduction number, equilibrium infection rates, epidemic wave size) by neglecting vaccine unwillingness?
2. Mitigation Strategy: Can this error be sufficiently reduced by simply adjusting the vaccination rate constant (scaling it down by the fraction of willing individuals) rather than explicitly modeling a separate "unwilling" susceptible compartment?

The analysis distinguishes between two time scales:

• Endemic Time Scale: Long-term behavior focusing on equilibrium states (Disease-Free Equilibrium vs. Endemic Equilibrium).
• Epidemic Time Scale: Short-term behavior focusing on the initial outbreak wave and numerical simulations.

2. Methodology

The author employs a generic SVIR model (Susceptible, Vaccinated, Infectious, Removed) with demographics (births and deaths).

Model Structure

• Compartmentalization: The susceptible population is split into:
  • $S_W$: Willing susceptibles.
  • $S_U$: Unwilling/Unable susceptibles.
  • $V$: Vaccinated individuals (derived from $S_W$).
  • $I$ and $I_V$: Infectious individuals (unvaccinated and vaccinated, respectively).
• Vaccination Benefits: The model assumes imperfect vaccination with three potential benefits relative to unvaccinated individuals:
  1. Reduced infection rate (factor $\sigma \le 1$).
  2. Reduced infectiousness (factor $\eta \le 1$).
  3. Faster recovery (factor $\alpha \ge 1$).
     These are combined into a single parameter representing relative disease burden: $\xi = \frac{\sigma \eta}{\alpha}$.
• Comparative Models:
  • Full Model: Explicitly tracks $S_W$ and $S_U$ with a vaccination rate $\phi$ applied only to $S_W$.
  • Naive Model: Assumes all susceptibles are willing but scales the vaccination rate constant by the willingness fraction $W$ (i.e., using rate $W\phi$ instead of $\phi$).

Scaling and Analysis

The study utilizes non-dimensional scaling to analyze the system on two distinct time scales:

1. Epidemic Scaling ($\tau$): Time is scaled by the infectious duration ($1/(\gamma+\mu)$). Demographic terms (birth/death) are treated as small perturbations ($O(\epsilon)$). The focus is on the total fraction of the population escaping infection during the first wave.
2. Endemic Scaling ($T$): Time is scaled by the mean lifespan ($1/\mu$). The focus is on the stability of equilibria and long-term infection rates.

The author derives analytical expressions for the Basic Reproduction Number ($R_v$) and the Endemic Equilibrium (EDE) using the Next Generation Matrix method and Routh-Hurwitz stability criteria. Numerical simulations are used for the epidemic time scale to compare the "Full Model" against the "Naive Model."

3. Key Contributions

• Analytical Proof of Rate Constant Inadequacy (Endemic): The paper mathematically proves that for endemic scenarios, simply reducing the vaccination rate constant to account for unwillingness yields no improvement in accuracy compared to neglecting unwillingness entirely. The leading-order terms for the equilibrium infection rate and stability conditions are identical in both the naive and the fully corrected models.
• Differentiation of Time Scales: The study establishes that the importance of modeling unwillingness depends entirely on the time scale of interest.
  • Endemic: Unwillingness must be modeled structurally (separate compartments).
  • Epidemic: The error depends on the interplay between disease infectiousness ($R_0$) and vaccination speed ($\theta$).
• Quantification of Error: The paper provides specific bounds and graphical evidence showing that neglecting unwillingness leads to significant overestimation of vaccination efficacy, particularly for highly effective vaccines and fast vaccination programs.

4. Key Results

A. Endemic Time Scale (Long-term)

• Basic Reproduction Number ($R_v$): The effective reproduction number is given by $R_v \approx R_0(U + \xi W)$, where $U$ is the unwilling fraction and $W$ is the willing fraction.
• Stability: The condition for a Disease-Free Equilibrium (DFE) is $R_v < 1$.
• Critical Finding: The vaccination rate constant ($\theta$) does not appear in the leading-order formulas for $R_v$ or the equilibrium infection rate.
  • Implication: In the long term, willing susceptibles are vaccinated so quickly relative to the disease dynamics that the rate of vaccination is irrelevant; only the fraction of the population that can be vaccinated matters.
  • Conclusion: Adjusting the rate constant in a naive model fails to correct the error. The structural separation of willing/unwilling populations is mandatory for accurate endemic modeling.

B. Epidemic Time Scale (Short-term)

• Error Magnitude: The error introduced by neglecting unwillingness is significant when:
  1. The disease is less infectious (lower $R_0$).
  2. The vaccination program is relatively fast (high $\theta$).
  3. The vaccine is highly effective (low $\sigma$).
• Mitigation via Rate Constant:
  • For slow vaccination rates, scaling the rate constant ($W\theta$) provides a "noticeable but not large" correction.
  • For fast vaccination rates, scaling the rate constant results in significant errors in predicting the peak infection size and the final fraction of the population infected.
• Simulation Results: Graphs show that for fast vaccination and high efficacy, the naive model (even with scaled rates) significantly underestimates the number of infections compared to the full compartmental model.

5. Significance and Implications

• Public Health Policy: Ignoring vaccine unwillingness leads to overly optimistic projections of disease eradication and herd immunity thresholds. Policymakers relying on naive models may underestimate the resources needed or the final size of an epidemic.
• Modeling Best Practices:
  • Endemic Models: It is absolutely vital to incorporate vaccine unwillingness as a structural compartment (separating $S_W$ and $S_U$). Simply adjusting parameters is mathematically insufficient.
  • Epidemic Models: While adjusting the rate constant is better than nothing, it is not a substitute for structural complexity, especially for fast-moving vaccination campaigns against less infectious diseases.
• Generalizability: Although the study uses a generic model, the authors argue that the conclusions hold for more complex disease profiles. The fundamental biological reality—that unwilling individuals cannot be vaccinated regardless of the rate constant—remains the dominant factor.

Conclusion: The paper concludes that for accurate epidemiological modeling, vaccine unwillingness must be explicitly built into the rate diagram (state variables) rather than merely adjusting rate constants. This is non-negotiable for endemic analysis and highly recommended for epidemic analysis to avoid significant predictive errors.