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An update to the critical vaccination threshold for measles herd immunity

This paper presents a mathematical model incorporating waning vaccine-induced immunity to derive updated measles herd immunity thresholds, demonstrating that even minimal immunity loss can fundamentally alter transmission dynamics and render traditional critical vaccination targets insufficient for achieving herd immunity in highly vaccinated populations.

Original authors: Mahmuda Ruma, Elena Aruffo, Alexandra Teslya, Jane Heffernan

Published 2026-06-30

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Mahmuda Ruma, Elena Aruffo, Alexandra Teslya, Jane Heffernan

Original paper licensed under CC BY 4.0 (https://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 the measles virus as a very persistent, mischievous ghost that loves to jump from person to person. For decades, public health officials have believed that if we vaccinate enough people, we can build an invisible "force field" around our communities. This force field is called herd immunity. The old rule of thumb was simple: if you vaccinate 95% of the population, the ghost can't find anyone to jump to, and the outbreak stops. It was like having a wall so high that the ghost simply couldn't climb over it.

However, this new research suggests that the wall might have some tiny, invisible cracks that we didn't notice before.

The "Leaky Bucket" Analogy

The researchers, led by a team from York University and Utrecht University, propose a new way of thinking about our immunity.

In the past, we assumed that once you got the measles vaccine, your immunity was like a solid steel vault door—it stayed locked forever. But recent evidence suggests that for some people, that door might slowly rust over time. It doesn't fall off immediately, but it gets a little weaker, year after year.

The authors built a mathematical model to simulate this. They imagined the population not just as "Vaccinated" or "Unvaccinated," but as a group of people whose vaccine protection is slowly leaking.

The Vault (V): People with full, fresh protection.
The Rusty Door (W): People whose protection is starting to fade. They aren't fully open yet, but they are a little more vulnerable than before.
The Open Door (S): People who have lost their protection entirely and are fully susceptible again.

The Big Discovery: The "Impossible" Threshold

The most surprising finding in the paper is that even if the "rust" (waning immunity) happens incredibly slowly—so slowly that it takes a lifetime to become a problem—it changes the math completely.

Think of it like trying to fill a bathtub with a leaky drain.

Old Math: If you turn the faucet on (vaccination) to 95%, the tub fills up, and the water stays there.
New Math: The researchers found that if the drain is even slightly open, no matter how slowly, you might never be able to fill the tub to the top, even if you keep the faucet running at 100%.

In technical terms, they found that in certain realistic scenarios, it is mathematically impossible to achieve herd immunity just by vaccinating people, because the protection is slowly fading. The "critical vaccination threshold" (the magic number of people you need to vaccinate) might actually be higher than 100%—which is impossible to reach.

Does This Mean We Are Doomed?

No. This is the most important part of the story.

Just because the "force field" (herd immunity) might be mathematically impossible to maintain perfectly doesn't mean the ghost (measles) will run wild. The paper explains that even if we can't reach that perfect 100% shield, the risk of a massive outbreak can still be extremely low.

Imagine the rusty door again. Even if the door isn't perfectly sealed, it's still much harder for the ghost to get through than if the door were wide open. The study shows that even with these "leaky" vaccines, the virus still struggles to spread, especially if the people with rusty doors are still somewhat protected.

The Real-World Test: Southwestern Ontario

To prove their model works, the team applied it to a real measles outbreak that happened in Southwestern Ontario in 2024-2025.

They looked at the data: 769 confirmed cases.
They used their "leaky bucket" math to work backward.
The Result: Their model estimated that about 86% of the population was vaccinated (which is high, but not the mythical 95%+).
They calculated that the virus was spreading, but the "rusty" immunity in the population helped slow it down.
They found that about 77% of the infections happened in people who were fully susceptible (never vaccinated or lost immunity long ago), while the rest happened in people whose immunity was fading.

The Takeaway

This paper is a wake-up call for the math we use to plan public health.

The Old Rules Need Updating: The simple formula "Vaccinate 95% to stop measles" might be too simple if vaccine protection fades over time.
It's Not All Bad News: Even if we can't mathematically guarantee a perfect "herd immunity" shield, high vaccination rates still keep outbreaks small and manageable.
The "Leak" is Real: We need to accept that vaccine protection might not last a lifetime for everyone, and our models need to account for that slow fade.

In short: The wall might have a few cracks, but as long as we keep building it high and wide, the ghost still can't get in to cause a disaster. We just need to be smarter about how we measure the wall's strength.

Technical Summary: An Update to the Critical Vaccination Threshold for Measles Herd Immunity

Problem Statement
The classical assumption in measles epidemiology is that vaccine-induced immunity is lifelong, leading to a critical vaccination threshold ( $p_c$ ) of $1 - 1/R_0$ to achieve herd immunity. However, recent evidence suggests that measles vaccine-induced immunity may wane over time, potentially rendering individuals susceptible decades after vaccination. This waning challenges existing public health metrics, specifically the control reproduction number ( $R_c$ ) and the critical vaccination threshold ( $p_c$ ). The paper addresses the need to re-evaluate these metrics in populations where vaccine immunity is not permanent, using the 2024-2025 measles outbreak in Southwestern Ontario, Canada, as a case study.

Methodology
The authors developed a deterministic mathematical model extending the standard Susceptible-Infected-Recovered (SIR) framework to an SIRVWS structure. The population is divided into:

S: Fully susceptible individuals.
I: Infected individuals originating from the susceptible class.
R: Recovered individuals with lifelong immunity.
V: Vaccinated individuals with full protection.
W: Vaccinated individuals in a waning immunity stage (partially immune).
$I_W$ : Infected individuals originating from the waning immunity class.

The model incorporates demographic dynamics (births and deaths) and specific transitions for waning immunity ( $V \to W \to S$ ) and immune boosting ( $W \to V$ upon exposure). Key parameters include the waning rates ( $k_1, k_2$ ), the reduction in infectiousness for partially immune individuals ( $G$ ), the boosting proportionality constant ( $\nu$ ), and the probability of infection versus boosting upon exposure ( $q$ ).

Using the Next Generation Matrix method, the authors derived analytical expressions for:

The Control Reproduction Number ( $R_c$ ).
The Critical Vaccination Threshold ( $p_c$ ) required for herd immunity.
The Probability of an outbreak and the Final Outbreak Size.

These theoretical results were applied to the Southwestern Ontario Public Health Region (SOPHR) outbreak. The authors utilized Latin Hypercube Sampling (LHS) to generate 100,000 parameter sets within plausible ranges to simulate epidemic dynamics, fitting the model to reported case data (769 confirmed cases as of August 2025) to estimate parameters such as $R_0$ , waning rates, and the effective population size involved in transmission.

Key Contributions and Results

Revised Metrics: The study derives updated equations for $R_c$ and $p_c$ that explicitly account for waning immunity. The authors demonstrate that the classical formulas ( $R_c = R_0(1-p)$ and $p_c = 1 - 1/R_0$ ) serve as lower bounds; the true thresholds are higher when waning is considered.
Impossibility of Herd Immunity: A primary finding is that even extremely slow waning rates (corresponding to immunity half-lives of 77–99 years) can fundamentally alter herd immunity dynamics. The analysis identifies realistic parameter regimes where vaccine-induced herd immunity cannot be achieved, regardless of high vaccine coverage, because the threshold $p_c$ exceeds 1.
Outbreak Probability vs. Herd Immunity: Despite the potential inability to achieve herd immunity under certain waning conditions, the model indicates that the probability of an outbreak may remain very low if the type reproduction numbers are sufficiently reduced (e.g., through low susceptibility of the waning class or reduced infectiousness).
Application to SOPHR Outbreak:
- The model estimated a basic reproduction number ( $R_0$ ) of approximately 12.3 and a control reproduction number ( $R_c$ ) of 2.7 for the SOPHR outbreak.
- The estimated vaccination uptake was approximately 86%.
- Waning rates were estimated at $k_1 \approx 0.009$ year $^{-1}$ (V to W) and $k_2 \approx 0.007$ year $^{-1}$ (W to S), corresponding to immunity half-lives of roughly 77 and 99 years, respectively. These values align with recent serological studies suggesting susceptibility can return ~51 years post-vaccination.
- Approximately 77% of infections occurred in the fully susceptible class ( $S$ ), consistent with reports that the majority of cases were in unvaccinated individuals.
- The effective population involved in transmission ( $N_0$ ) was estimated to range between 4,400 and 6,500 individuals, a small fraction of the total regional population.
Sensitivity to Coverage: Simulations comparing 78% and 92% vaccine uptake showed that higher coverage leads to a higher $R_0$ estimate (due to model fitting constraints) but a smaller total outbreak size and earlier peak time. Conversely, lower coverage (78%) resulted in a larger final outbreak size and a peak occurring approximately 6 months later.

Significance and Claims
The paper claims that widely used measles herd immunity thresholds may require re-evaluation in highly vaccinated populations where vaccine-induced immunity wanes. The authors assert that their updated mathematical definitions provide a more accurate basis for public health decision-making than classical models that assume lifelong immunity.

The study highlights a counter-intuitive dynamic: even if the theoretical threshold for herd immunity cannot be met due to waning, the actual risk of a large outbreak may remain low due to the reduced susceptibility and infectiousness of partially immune individuals. The authors conclude that while their model offers necessary updates to risk assessment tools, it relies on assumptions of homogeneous mixing and ignores age structure and stochastic effects, suggesting these as limitations for future research. The work does not propose new vaccination strategies but rather provides a refined mathematical framework for interpreting outbreak data and assessing the long-term sustainability of herd immunity in the context of waning vaccine protection.