The Hidden Flaw in Vaccine Models That Makes Shots Look Less Effective Than They Are

Somewhere between a clinical trial and a public health model, something goes wrong. A vaccine gets tested, its effectiveness is measured, a number is produced — say, 30% protection against infection — and that number gets entered into an epidemic simulation. Seems straightforward. But for a large class of vaccines, that number means something subtly different in the real world than it does inside a mathematical model. The result, according to a new paper from the University of Melbourne, is that models are systematically underestimating how well vaccines actually work — and overestimating how bad an outbreak will be.

In one of the paper's illustrative scenarios, the stakes are concrete: with a pathogen whose basic reproduction number ($R_0 = 2.6$, meaning each case generates roughly 2.6 new ones without immunity), 60% vaccine coverage, and a measured vaccine effectiveness of 30%, the standard modeling approach predicts that 82% of the population will be infected over the course of the outbreak. Apply the corrected approach the researchers propose, and that figure drops to 68% — a difference of 14 percentage points, representing millions of people in any large country (Middleton et al., 2026).

That gap doesn't come from bad data or wrong assumptions about biology. It comes from a mismatch between two different ways of counting infections — and from decades of modelers not fully accounting for the difference.

The Science

The paper, by Casey Middleton, Oliver Eales, James McCaw, and Freya Shearer of the University of Melbourne's Infectious Disease Dynamics Unit, extends earlier theoretical work to produce practical, actionable guidance for epidemic modelers. Their focus is on the parameterization problem: the process of translating empirical vaccine effectiveness estimates into the numbers that mathematical models actually use.

Epidemic models — the kind used to plan vaccination campaigns, set coverage targets, and project future case burdens — need to represent what vaccines do to transmission. To do that, they need a numerical value for the degree of protection a vaccine confers. The natural place to get that value is from vaccine effectiveness (VE) studies: clinical trials, observational studies, or test-negative designs that compare infection rates in vaccinated and unvaccinated people.

The trouble is that not all VE statistics measure the same thing, and the thing a model needs is not always the thing a study reports.

The researchers work through the problem using the classic SIR framework (Susceptible–Infectious–Recovered), the workhorse of epidemiological modeling, extended to include vaccinated compartments. Their simulation study generates synthetic epidemic data under controlled conditions, then tests whether different parameterization strategies recover the true population-level vaccine effect.

What They Found

The key conceptual divide is between two types of vaccines — and two types of statistics.

A leaky vaccine gives all recipients partial protection. Think of it as turning down the volume on infection risk for everyone: if you're vaccinated, every exposure is somewhat less likely to result in infection, but you're never fully immune. Most real-world vaccines behave somewhere in this direction. An all-or-nothing vaccine, by contrast, gives some recipients complete, sterilizing immunity and does nothing for others. The mathematical behavior of these two types is meaningfully different.

The same split applies to how VE is measured. The most common approach is the cumulative attack rate ratio (ARR) — you follow a group of vaccinated and unvaccinated people, count who gets infected by the end of the study, and compute the relative risk. This is what most clinical trials and test-negative studies report. Less common are hazard ratio (HR) estimators, which capture the instantaneous risk of infection per unit time or per exposure event, and require finer-grained data on transmission chains or infection timing.

Here is the crux of the problem, first identified theoretically decades ago and now given practical modeling guidance: for a leaky vaccine, the ARR-based estimate will systematically understate the true per-exposure protection over time. Why? Because as an epidemic progresses, the most susceptible unvaccinated people get infected and are removed from the pool — a phenomenon called differential depletion of susceptibles. The remaining unvaccinated people are, in a sense, a hardier group. That makes the vaccinated cohort look relatively less protected than they actually are, because the comparison group has changed. The measured ${\text{VE}}_{\text{measured,R}}$ drifts downward even if the vaccine's biological protection hasn't changed at all.

Now here is where models go wrong. The vaccine effect parameter in a leaky SIR model — call it $\alpha$ — is a hazard ratio: it tells the model how much to scale down the infection risk per unit time for a vaccinated person. But modelers routinely plug in the ARR-based VE estimate directly as $\alpha$. This sets a false equivalence between two different statistical quantities. The model then replicates the depletion-of-susceptibles bias a second time, compounding the underestimate. The modeled population-level vaccine effect, ${\text{VE}}_{\text{modeled,R}}$, ends up lower than the real-world measured value — leading to pessimistic predictions about outbreak size and herd immunity (Middleton et al., 2026).

Standard vs. Adjusted Parameterization: Predicted Total Infections

Label	Value
Standard parameterization (α = VE_measured)	82 %
Adjusted parameterization (α_adjusted = 0.51)	68 %

Source: Middleton et al. (2026), Section 4.1 illustrative scenario.

The fix the researchers propose is elegant. Rather than using ${\text{VE}}_{\text{measured,R}}$ directly as $\alpha$, they solve backwards: given the measured VE, the population's vaccine coverage, and the basic reproduction number, what value of ${\alpha }_{\text{adjusted}}$ would cause the model's simulated population-level vaccine effect to match the measured VE? The answer is always higher than the naïve value — ${\alpha }_{\text{adjusted}}>{\text{VE}}_{\text{measured,R}}$ — because you need to give the model a stronger per-person protection to counteract the bias it would otherwise re-introduce.

In the scenario above, a measured ${\text{VE}}_{\text{measured,R}}=0.30$ maps to an adjusted parameter of ${\alpha }_{\text{adjusted}}=0.51$ — a value 70% higher than what a modeler would use under standard practice. The adjusted model runs a slower, lower epidemic, and produces a final infection count 14 percentage points lower.

Adjusted Vaccine Effect Parameter by Mechanism (Target VE = 0.30)

Label	Value
All-or-nothing (c → 0)	0.3 ᾱ adjusted
Heterogeneous (c = 10)	0.46 ᾱ adjusted
Leaky (c → ∞)	0.48 ᾱ adjusted

Source: Middleton et al. (2026), Figure 3 and Section 4.2. R₀ = 2.6, 60% vaccine coverage.

The researchers extend this correction to heterogeneous vaccines — a more general class where vaccine-derived protection varies across individuals, described by a beta distribution with mean $\bar{\alpha }$ and a shape parameter $c$. All-or-nothing and leaky vaccines are the extreme ends of this spectrum ($c \to 0$ and $c\to \infty$, respectively). For a scenario where ${\text{VE}}_{\text{modeled,R}}=0.30$ is the target, an all-or-nothing vaccine needs no adjustment ($\bar{\alpha} = 0.30$), but a heterogeneous vaccine with $c=10$ requires ${\bar{\alpha }}_{\text{adjusted}}=0.46$, and a fully leaky vaccine requires ${\bar{\alpha }}_{\text{adjusted}}=0.48$ (Middleton et al., 2026).

The paper also derives the implications for herd immunity thresholds — the fraction of a population that needs to be immune before an epidemic can no longer sustain itself. This threshold, often expressed as $1-1/R_0$ for a fully effective vaccine, shifts when vaccines are imperfect and when you account for the parameterization correction. Across several $R_0$ and VE scenarios, the adjusted parameterization predicts lower herd immunity thresholds than standard practice — meaning the population coverage needed to stop an epidemic may be more within reach than existing models suggest.

One important caveat the researchers flag: the accuracy of the adjusted parameterization depends on how closely the model's assumptions about population context match the context in which the original VE study was conducted. If a VE study was done in a population with 56% vaccine coverage and you apply the resulting adjusted $\alpha$ to a model with 80% coverage, some residual error will remain. Their simulation study shows this clearly — when the study and model settings match, the adjusted approach is highly accurate; as they diverge, accuracy degrades. The standard approach, however, is consistently worse across this entire range (Middleton et al., 2026). The adjusted method still outperforms naïve incorporation even when contexts differ.

Why This Changes Things

The implications ripple outward in several directions.

First, there is the direct question of model accuracy in pandemic response. During COVID-19, and in ongoing planning for influenza, RSV, and other respiratory pathogens, governments relied heavily on mathematical models to set vaccination targets, allocate doses, and decide when outbreak-control measures could be relaxed. If those models systematically underestimated vaccine impact — treating vaccines as less protective than they are — then their recommendations may have been systematically more cautious than necessary. This isn't a flaw in any single model or team; it reflects a gap in the field's parameterization standards.

Second, herd immunity thresholds matter enormously for public health messaging and policy. A threshold of 80% coverage sounds vastly more achievable than 90%. If standard parameterization has been nudging modeled thresholds upward — making the bar for population immunity look harder to clear — that has real consequences for how policymakers and the public think about vaccination campaigns.

Third, this work highlights a subtle but important distinction that is easy to lose sight of: a statistic from an observational study and a parameter in a mechanistic model are not the same kind of object, even when they carry similar-sounding names. ${\text{VE}}_{\text{measured,R}}$ is a population-level ratio summarizing what happened in a specific cohort over a specific period. $\alpha$ in a leaky model is a per-unit-time scaling of individual infection hazard. Conflating them isn't laziness; it's a conceptual error that has persisted partly because the difference is small in many practical scenarios and partly because the field lacked clear guidance on when and how to correct for it.

The University of Melbourne team provides that guidance in concrete, practical form. Their Table 1 — a decision matrix matching vaccine mechanisms to study statistics to parameterization strategy — is the kind of tool that modelers can use directly. Their root-finding algorithm for calculating ${\alpha }_{\text{adjusted}}$ is implementable in any standard scientific computing environment.

What's Next

The paper is explicit about what it doesn't solve. The adjusted parameterization works best when modelers have a reliable estimate of the epidemiological context — particularly the vaccine coverage and $R_0$ — in which the original VE study was conducted. In practice, that information isn't always reported alongside VE estimates, and even when it is, the relevant population may have changed.

There is also the harder question of the shape parameter $c$ for heterogeneous vaccines. The distribution of individual-level protection across a vaccinated population is genuinely difficult to measure; most empirical studies report only a population average. The paper shows that the adjusted mean ${\bar{\alpha }}_{\text{adjusted}}$ varies with $c$, meaning that getting the heterogeneous correction right requires knowing something about how protection is distributed — information that is rarely available. Getting better empirical handles on this distribution is an open research priority.

The paper focuses on protection against infection acquisition, which is the VE outcome most directly tied to transmission dynamics. But real-world vaccine programs also care about protection against severe disease, hospitalization, and death — outcomes that involve different biological mechanisms and different study designs. Extending the parameterization framework to these outcomes, and to multi-dose vaccine schedules, waning immunity, and variant-specific protection, remains work to be done.

What Middleton and colleagues have delivered is something undervalued in methods-focused research: a clear, practically actionable correction to a systematic error that has likely been biasing public health models for years. The math is not exotic. The fix is computable. And the stakes — better epidemic forecasts, more accurate vaccine policy decisions, more honest herd immunity estimates — are as high as anything in applied epidemiology.

Vaccines are more powerful than our models have been giving them credit for. Closing that gap starts here.