Interpreting Breakthrough Infections Given Assortative Mixing of Partially Vaccinated Populations

Abstract

Declining vaccine coverage across the United States has increased the risk of outbreaks of vaccine-preventable diseases. Even when vaccines have low primary failure rates, conventional epidemic theory predicts a strongly nonlinear, positive relationship between vaccine coverage and the fraction of breakthrough infections in vaccinated individuals. These breakthrough infections may generate misconceptions that vaccines are not working and accelerate declines in confidence and coverage. Here, we set out to test predictions of conventional epidemic theory that assumes random mixing between individuals irrespective of vaccine status. In contrast to expectations from random mixing models, we find a far lower fraction of breakthrough infections in measles outbreak data from seven states in the United States. To explore this discrepancy, we evaluate an alternative, compartmental disease model that accounts for preferential mixing (‘assortativity’) between people with the same vaccination status. The model predicts significantly lower fractions of breakthrough infections, consistent with observations from measlesoutbreak data. Next, we leverage the deviation between statewide and school-level vaccine MMR coverage across kindergartens in sixteen states, finding substantial assortativity in all cases. Our model accounting for preferential mixing predicts the total number of breakthrough infections is nonlinear, peaking at intermediate coverage below vaccine-derived herd immunity. Nationally, 94% of counties that report MMR coverage are above the model-predicted breakthrough-maximizing coverage, suggesting that they are at risk for increasing breakthrough infections if coverage declines. Vaccination outreach and monitoring campaigns should develop proactive strategies to contextualize breakthrough infections before low levels of primary failure contributes to population-scale increases in preventable disease.

1. Introduction

Vaccine coverage is declining in the United States, contributing to outbreaks of vaccine-preventable diseases (Kiang et al., 2025; Dong et al., 2025b). The ongoing measles outbreak in the U.S. exceeded 2,200 confirmed cases in 2025, spreading largely but not entirely within unvaccinated individuals. Large-scale measles outbreaks are also unfolding in Canada and Mexico, with >5,000 and >6,000 confirmed cases in 2025 alone, respectively. The Pan American Health Organization declared the loss of measles-free status and the reestablishment of endemic measles transmission in the Americas on November 10, 2025 (Pan American Health Organization, 2025). Decreasing vaccine coverage increases the outbreak risk for other vaccine-preventable illnesses, including polio and diptheria (Kiang et al., 2025). Changes in vaccine coverage may also shift the relative and total prevalence of infections in vaccinated individuals (i.e., breakthrough infections associated with vaccine failure (Heininger et al., 2012)).

Understanding the impact of reduced vaccine coverage on breakthrough infections has important implications for strategic communication, epidemiological analysis, and public health preparedness. Part of what fuels vaccine skepticism among individuals is the notion that vaccines are at best ineffective and at worst dangerous (including the idea that vaccination can increase the risk of infection) and therefore not worth the perceived cost (Kata, 2010). Recent viral social media posts have cited the apparently high proportion of mumps and measles infections in vaccinated individuals to suggest that vaccines are ineffective (e.g., https://archive.ph/Hxd6V, https://archive.ph/bCaaj, accessed and archived December 22, 2025). While the rate of vaccine failure is often quite low, the prevalence of infections in vaccinated versus unvaccinated individuals may be quite substantial when coverage is high (Arima and Oishi, 2018; Orenstein et al., 1985; Poland, 1994). For example, at least 15% of measles cases in New Mexico in 2025 were breakthrough infections (New Mexico Department of Health, 2025).

Conventional epidemic theory posits that the fraction of breakthrough infections in an outbreak may be leveraged to infer vaccine efficacy or vaccine coverage (Orenstein et al., 1985; Althaus and Salathé, 2015; Bhatia et al., 2025). However, such inference generally assumes random mixing within the population, whereas unvaccinated individuals may cluster within relatively isolated communities, to which outbreaks may be confined (Munday et al., 2024; Wilson et al., 2021; Newcomer et al., 2024). Realistic assortativity within populations (i.e., limited transmission-relevant interactions between vaccinated and unvaccinated individuals) may protect vaccinated individuals from infection and reduce the number of breakthrough infections. Likewise, efforts to estimate key epidemiological parameters based on breakthrough infections may be systematically biased if they fail to account for assortativity. Changes in overall vaccine coverage and spatial clustering of unvaccinated people also have important implications for the incidence of breakthrough infections. The proliferation of anti-vaccine rumors, including across mainstream and governmental sources, is simultaneously driving an overall increase in vaccine refusal (and nonmedical vaccine exemptions from vaccine requirements), and could reduce assortativity as vaccine hesitancy expands (Dong et al., 2025a; Carpiano et al., 2023; Fattah et al., 2026). Both trends could increase the burden of breakthrough infections as larger epidemics extend into (previously) highly vaccinated communities.

Here, we assess the link between vaccination coverage, primary failure, assortativity, and breakthrough infections. To begin, we compare data on the documented fraction of breakthrough measles infections across seven states to expectations from a compartmental model assuming random mixing. The gap between theory and state-level data suggests random mixing models are limited in their predictive power for breakthrough incidence. As an alternative, we develop and analyze an expanded epidemic model with preferential mixing. Accounting for assortativity can reconcile observations of low incidence of breakthrough infections with high coverage rates – reinforced by inference of moderate, realized assortativity from school-level MMR coverage for kindergarteners across sixteen states. While assortativity reduces the number of breakthrough infections, dwe find that decreases in coverage typically increases breakthrough infections, potential reinforcing anti-vaccine perspectives and catalyzing a synergistic feedback loop. Our findings support the need for principled approaches to anticipate and contextualize breakthrough infections to avoid mis-perceptions that can worsen outbreaks.

2. Results

2.1. Model-predicted vaccine coverage exceeds surveyed coverage

To understand the expected relationship between breakthrough infections and vaccine coverage, we assess a conventional epidemic model of disease transmission and vaccination for a vaccine with primary failure rate $ϵ$ and population-wide vaccine coverage $p$ (see Equation S1 for equations and Figure S1 for a compartmental diagram). We consider the relative and total number of breakthrough infections at equilibrium, focusing on parameters consistent with measles and the MMR vaccine (Table S1, with initial conditions in Table S2).

In instances where vaccine coverage is insufficient to eliminate a disease, it is possible to link the fraction of breakthrough infections (relative to total infections), $f_V$, to vaccine coverage, $p$. Assuming no assortativity, the fraction of breakthrough infections increases with vaccine coverage, potentially sharply, such that $f_V=\frac{p\epsilon }{1-p(1-\epsilon )}$ (Figure 1A, subsection S1.2). For example, if coverage drops to $p=0.9$ assuming that the MMR vaccine has failure rate $\epsilon =0.03$ (Centers for Disease Control and Prevention, 2025a) then $f_V\approx 0.21$ of total infections should be breakthrough infections. The positive relationship between coverage and the breakthrough fraction is maintained when two doses of a vaccine are administered (consistent with MMR), and the fraction of breakthrough cases that only received one dose is consistently greater than the fraction of two-dose breakthroughs (section S2).

Figure 1: The model-predicted vaccine coverage is consistently below the surveyed kindergarten vaccine coverage.

(A) The predicted positive relationship between vaccine coverage ($p$) and the breakthrough fraction ($f_V$). For each of the seven states included in the analysis, the point indicates the reported breakthrough fraction for measles in 2025 and the vertical line corresponds to the model-predicted vaccine coverage. Model predictions are for parameters consistent with measles and the MMR vaccine: $m=\frac{1}{365\times 80}{\mathrm{d}\mathrm{a}\mathrm{y}}^{-1},\epsilon =0.03,\gamma =0.1{\mathrm{d}\mathrm{a}\mathrm{y}}^{-1}$ , $\beta =15\times (\gamma +m){\mathrm{d}\mathrm{a}\mathrm{y}}^{-1}$ (i.e., ${ℛ}_0=15$). The population has size 10M and there is no assortativity ($\varphi =0$). See Figure S3 for a supplemental analysis using different values of ${ℛ}_0$ and $ϵ$. (B) Comparison of model-predicted vaccine coverage to surveyed kindergarten MMR coverage for school year 2024–2025. The point indicates the breakthrough fraction across all cases with known vaccination status and the horizontal bars correspond to uncertainty in the breakthrough fraction given that some cases have unknown vaccination status (see section 4 for further explanation). The dashed line indicates perfect correspondence.

We revisit the interplay between breakthrough infections and changes in coverage in the context of the measles outbreaks in the United States in 2025. Prior work assuming random mixing has leveraged realized measurements of the fraction of breakthrough infections to infer failure rates (Orenstein et al., 1985) and vaccine coverage (Althaus and Salathé, 2015). The reported fraction of breakthrough infections across seven states ranges from 0 – 0.21. As a result, the theoretically inferred vaccine coverage rate, $p$, assuming random mixing falls below 80% in five states and below 60% in three states (Figure 1A). In contrast, the surveyed kindergarten vaccine coverage, $p_{\text{school}}$, exceeded 85% in all states (Figure 1B). Theoretical predictions for vaccine coverage are substantially below surveyed measures for all states, with the exception of New Mexico. In North Dakota, no cases were observed in vaccinated individuals among 36 total cases, consistent with a theoretically predicted vaccine coverage near 0 in contrast to the actual state-wide coverage of 90%.

2.2. State-level breakthrough fraction and vaccine coverage are consistent with moderate assortativity

We hypothesize that non-random mixing based on vaccination status may help resolve the discrepancy between model-predicted and surveyed vaccine coverage. We set out to test this hypothesis by incorporating assortativity into an expanded model of disease dynamics in a partially vaccinated population. In practice, we subdivide the population into vaccinated and unvaccinated groups and incorporate a preferential mixing rate $\varphi$, which corresponds to the share of an individual’s contacts that will be reserved for people with concordant vaccination status (versus contacts with the entire population; Fig. S1). Using the same disease parameters in Fig. 1 for measles, we find that assortative mixing reduces the expected fraction of breakthrough infections for all values of coverage (Fig. 2). In sensitivity analysis, these results were robust to alternative assumptions on ${ℛ}_0$ and primary failure rate (vaccine efficacy) (Figure S3). Across states, the relationship between surveyed vaccine coverage and the breakthrough fraction is positive. As hypothesized, observed low breakthrough fractions (below 0.2) at high vaccine coverage rates (above 85%) are consistent with moderate to high levels of assortativity in mixing relevant to measles transmission ($\varphi >0.6$) (Figure 2).

Figure 2: Assortativity reduces the breakthrough fraction.

Lines illustrate the relationship between vaccine coverage ($p$) and the breakthrough fraction ($f_V$) for different levels of assortativity ($\varphi$), ranging from 0 (no assortativity) to 0.98 (high assortativity). Parameters are consistent with measles and the MMR vaccine (see Table S1). See Figure S3 for a supplemental analysis using different values of ${ℛ}_0$ and $ϵ$. Black points indicate surveyed kindergarten vaccine coverage for school year 2024–2025 against the reported breakthrough fraction for measles in 2025 and vertical bars correspond to uncertainty in the breakthrough fraction given that some cases have unknown vaccination status (see Methods and Figure 1 for more information).

2.3. Differences between state- and school-level vaccine coverage are consistent with moderate assortativity

We estimated realized levels of assortativity relevant to measles transmission in the U.S., by analyzing vaccine coverage data for kindergarteners across sixteen states (including four of the seven shown in Figure 1 and Figure 2) over a multiyear period, (see Sec. 4). We estimated assortativity for each state annually by comparing school-level and statewide kindergarten MMR coverage and taking a population-weighted mean across school-level values (Equation 2). School-based assortativity estimates ($\tilde{\varphi }$) exceeded 0.2 in all cases, indicating moderate assortativity (Fig. 3). The average, estimated assortativity since the 2019 – 2020 school year ranged from 0.27 in Michigan to 0.60 in Maryland (median: 0.37; mean: 0.39) (Figure 3, Table S3). States with greater estimated assortativity also had a lower breakthrough fraction, as expected (compare Figs. 2 and 3; ordering of states by ascending estimated assortativity or descending breakthrough fraction is: Michigan, Utah, South Carolina, and North Dakota). Estimated assortativity values are significantly positive in all cases. However, estimated assortativity was consistently below model-predicted values as inferred from the fraction of breakthrough infections (0.6 or greater, see Fig. 2) – an issue we return to in the Discussion. We did not find evidence in any state for significant changes in estimated assortativity over time (Table S3, Figure S5).

Figure 3: Estimated assortativity across sixteen states is moderate.

Distribution of boostrapped estimated assortativity values ($\tilde{\varphi }$) across all years available for each state, arranged vertically by descending estimated assortativity. Values of 0 would correspond to no assortativity (i.e., all schools have the same vaccine coverage) while values of 1 would correspond to complete separation between vaccinated and unvaccinated populations. The four states in black (ND, MI, SC, and UT) also had measles epidemics in 2025 and were included in the analysis of breakthrough fractions (see Figure 1 and Figure 2). *New York excludes New York City. See Table S5 for information on data sources and limitations.

2.4. Breakthrough infections are maximized at intermediate vaccine coverage and low assortativity

We leverage both empirical and model findings to explore scenarios of decreasing coverage and assortativity on the total number of breakthrough infections. As a baseline assuming random mixing, total infections (regardless of vaccination status) and infections in the unvaccinated population increase monotonically with vaccine coverage below the herd immunity threshold $p_c=\frac{1}{1-\epsilon }\left(1-\frac{1}{{ℛ}_0}\right)$ (Anderson and May (1991), subsection S1.3). As a result, the total number of breakthrough infections should follow a unimodal function of coverage and peak at a level $\hat{p}$ below the vaccine-induced herd immunity threshold, $\hat{p}=\frac{1}{1-\epsilon }\left(1-\frac{1}{\sqrt{{ℛ}_0}}\right)$. (Fig. 4 and subsection S1.3). This relationship remains unimodal in the presence of assortativity. As shown in Fig. 4, assortativity typically decreases the total number of breakthrough infections, although it can also make elimination more difficult when vaccine coverage is high (see Figure S6). Assortativity also generally decreases the vaccine coverage level $\hat{p}$ at which breakthrough infections are maximized and the peak number of breakthrough infections (although $\hat{p}$ is nearly constant between 0.2 and 0.6, the range of assortativity values estimated across school-level data). However, $\hat{p}$ may be greater when there is some assortativity compared to no assortativity. The unimodal relationship between the number of breakthrough infections and vaccine coverage holds across different ${ℛ}_0$ and vaccine failure rates (Figure S4).

Figure 4: Breakthrough infections peak at intermediate vaccine coverage.

Lines indicate annual new cases per 100K people depending on vaccine coverage ($p$) and assortativity ($\varphi$, colors). The left panel shows log-scaled cases in unvaccinated people (dashed lines) and vaccinated people (solid lines, breakthrough infections). The right panel shows linear-scaled breakthrough infections, with points corresponding to the peak number of breakthrough infections for different levels of assortativity ($\hat{p}$). Parameters are consistent with measles and the MMR vaccine (${ℛ}_0=15$ and failure rate $ϵ=0.03$, see Table S1). See Figure S4 for a supplemental analysis using different values of ${ℛ}_0$ and $ϵ$.

We predict that decreasing vaccine coverage could lead to increases in the total number of breakthrough infections even as the relative fraction of breakthrough infections decreases. In the case of measles, assuming ${ℛ}_0=15$ and $\epsilon =0.03$ and no assortativity, $p_c\approx 0.96$ and $\hat{p}\approx 0.76$, i.e., breakthrough infections should increase rapidly as coverage decreases moderately (i.e., by 5%–15%). Moreover, if coverage drops such that $p<\hat{p}$ then public health efforts to increase coverage are predicted to lead to more breakthrough infections given the unimodal relationship between coverage and the total number of breakthrough infections. This response arises as a result of the balance between the protective benefits of increasing vaccinations that decrease the total burden of illness while increasing the size of the vaccinated population.

At assortativity consistent with the mean estimated value across states (Figure 3, $\tilde{\varphi }=0.39$), the critical coverage level $p_c$ exceeds 0.99 ($p_c=0.9993$) and the breakthrough infection-maximizing coverage $\hat{p}$ is approximately 0.77. Of the 2,141 counties in the United States that reported MMR coverage in kindergarteners for school year 2022–2023, fewer than 3% of counties (55 counties) had coverage exceeding assortativity-adjusted $p_c$, while nearly 92% of counties (1,964 counties) have coverage between assortativity-adjusted $p_c$ and $\hat{p}$ (Figure 5). Together this means that over 94% of counties have coverage exceeding $\hat{p}$ (2,019 counties for the value of $\hat{p}$ derived using the national mean estimated $\varphi$; with a range between 2,009 to 2,041 counties for values when accounting for variation in assortativity across states, see Table S3). These counties are at risk of increased breakthrough infections if vaccine coverage declines further (Figure 5).

Figure 5: Most counties have vaccine coverage above $\hat{p}$, meaning that efforts to increase coverage could lead to more breakthrough infections.

In the map, counties are shaded depending on how their kindergarten MMR coverage for school year 2022–2023 (Dong et al., 2025b) to the critical coverage $\hat{p}$ and the vaccine-induced herd immunity threshold $p_c$ calculated for the mean assortativity estimated across states ($\tilde{\varphi }=0.39$) and realistic parameters for measles and the MMR vaccine ($\hat{p}=0.77$ and $p_c>0.99$). Counties without available data for this time period (across 13 states) are white. The bottom figure is a histogram of vaccine coverage across the 2,141 counties that reported vaccine coverage and the vertical lines indicate $\hat{p}$ and $p_c$. The dotted lines indicate the range of values for $\hat{p}$ for when $\varphi$ is between 0.27 and 0.60 (as estimated from school-level data, see Figure 3). Across the range of estimated $\varphi$ values, $p_c$ is nearly invariant.

3. Discussion

Declining vaccine coverage and expanded vaccine hesitancy could lead to more breakthrough infections and increase the risk of a positive feedback loop driving vaccine hesitancy and increases in vaccine-preventable disease. Analysis of an epidemic transmission model with preferential mixing within groups of vaccinated and unvaccinated individuals shows that once coverage has dropped below vaccine-induced herd immunity thresholds, decreasing vaccination rates can increase breakthrough infections, undermining public health objectives. We also find evidence of preferential mixing (i.e., ‘assortativity’) based on vaccination status – supported by comparing data on recent outbreaks to transmission model predictions and by analyzing differences between state- and school-level vaccination coverage. Vaccine assortativity makes outbreaks like the current U.S. measles outbreak more likely, but also reduces the proporton of breakthrough infections. Moving forward, reduced vaccine coverage and decreases in assortativity could lead to substantial increases in the total number of breakthrough infections.

The present analysis comes with limitations. We model primary failure as an all-or-nothing process (where people are either fully protected or completely unprotected) rather than a leaky mechanism (wherein all vaccinated people are partially protected), but disease dynamics are less sensitive to the mode of failure when vaccine effectiveness is relatively high, as is the case for MMR (Lee et al., 2025). We also do not consider secondary failure (i.e., waning immunity), which may be particularly important in highly vaccinated populations where incidence has been relatively low and cases occur predominantly in adults (Yang et al., 2020; Leung et al., 2025). Nonetheless, the results are generally robust to the primary failure rate and ${ℛ}_0$, suggesting the relationships between vaccine coverage, assortativity, and breakthrough infections are relevant across varied contexts (Figure S3, Figure S4). Efforts to characterize the precise mechanism of vaccine failure and implications for breakthrough infections are especially important in the context of discussions about next-generation vaccine development and updated measles vaccination strategies in elimination contexts (Yang et al., 2020; Poland and Jacobson, 2012).

The epidemic transmission model may also underestimate the benefits of current vaccination practices and the risks of assortativity based on vaccine status. Although the epidemic model assumes a single dose vaccination for simplicity, MMR, and most other childhood vaccines, require multiple doses. In an expanded two-dose model, we show that a larger share of breakthrough infections occur in people who have only received a single dose, underscoring the importance of counting breakthrough infections based on dose number in order to avoid underestimating the effectiveness of complete vaccination (Figure S2). We assume that breakthrough infections are identical to infections in unvaccinated people. In practice, breakthrough infections may be less infectious and associated with fewer symptoms (Cherry and Zahn, 2018; Leung et al., 2025; Sundell et al., 2019; Evans et al., 2024), while increased activity of vaccinated people who may have more mild symptoms or assume that they are unable to become infected could lead to higher effective transmission rates (Park et al., 2020; Pedroza-Meza et al., 2026). Also, children below vaccination age constitute a sizable proportion of the unvaccinated population and may be at greater risk of severe illness, meaning that cases in unvaccinated individuals are especially concerning (Leung et al., 2025).

Considering a variety of epidemiological outcomes may provide additional insight into the impacts of vaccination campaigns depending on various goals, which may depend on the disease or vaccine (e.g., reducing clinical severity or preventing infections in certain groups). Although we focus on the potential for diminished assortativity to shift disease burden toward vaccinated people in the context of sustained transmission, our model does not consider epidemic risk following re-introduction, which may be considerably greater in spatially clustered populations (Truelove et al., 2019). This work underscores the importance of considering the combined impacts of vaccine coverage and assortative mixing on disease risk both across the population at large and among vaccinated people.

While we argue that assortativity may partially explain the tendency for the model-predicted fraction of breakthrough infections to exceed observed values based on state-level coverage, limitations in data collection may also contribute to the gap between model-predicted and estimated assortativity (see Fig. 3). Although measles cases in the United States in 2025 exceeded cases over the past two decades and these outbreaks remain concerning from a public health standpoint, the small number of cases reported in any given state further complicate attempts to draw inference based on the breakthrough fraction. State-level reports of vaccine coverage may exceed coverage within local communities where outbreaks occurred (Fitzpatrick et al., 2025). For example, Mohave County, where the outbreak in Arizona has been centered, had county-level MMR coverage in kindergarteners of 76% compared to 89% coverage at the state level (Dong et al., 2025b; Centers for Disease Control and Prevention, 2025b). Although vaccine coverage is generally included in outbreak reports (Rahimi et al., 2025) and school-level coverage for kindergarteners is reported by several states, we recommend improvement of assessment and reporting of local vaccine coverage (including in school-aged children of different grades) and standardization of reporting practices across regions (Dolan et al., 2019). Efforts to measure vaccine coverage in preschool and adult populations and describe coverage and contact patterns outside of school settings (including contact rates dependent on vaccination status) are necessary to improve understanding of transmission dynamics, and underlying risk (Gastañaduy et al., 2020; Salmon et al., 2006; Bednarczyk and Sundaram, 2025; Zhou et al., 2026). Of particular relevance, the vaccine series commences at twelve months of age and is completed between ages four and five (Centers for Disease Control and Prevention, 2025a), meaning that contact rates by age, binned by vaccine eligibility, are especially important to understand the potential for transmission from contact with unvaccinated individuals (Taube et al., 2025; Nelson et al., 2021; Andrejko et al., 2022). Epidemiological analyses at granular geographic scales (i.e., sub-state and even sub-county) will help guide response, given that fine-scale heterogeneity in vaccine coverage exists and substantially shapes epidemic dynamics (Fitzpatrick et al., 2025; Masters et al., 2020; Truelove et al., 2019; Fattah et al., 2026; Dong et al., 2025b; Zhou et al., 2026).

The combination of theory and data-driven analyses suggest that projected increases in the number of breakthrough infections could necessitate shifts in surveillance and quarantine strategies that currently assume infections will be rare in vaccinated people (Cherry and Zahn, 2018; Cassini et al., 2024; Poland and Jacobson, 2012; Leung et al., 2025). Research to characterize how information about breakthrough infection affects vaccine decision-making, coupled with models that incorporate the behavior-disease feedback loop between breakthrough infections, vaccine hesitancy, and declining coverage, may help inform interventions (Pedroza-Meza et al., 2026). Efforts to increase public trust in vaccination and address the specific concerns of different communities are especially urgent to counter the diffusion of vaccine hesitancy (which could also decrease assortativity) (Carpiano et al., 2023; Hijano et al., 2025). We recommend that communication and monitoring efforts anticipate the link between changes in coverage and increases in breakthrough incidence. Improving baseline expectations and managing interpretations of breakthrough infections may help avoid avoidable errors in vaccination initiatives.

4. Methods

Epidemic transmission model with assortativity

We propose an epidemic transmission model that partitions individuals into the following compartments: susceptible and unvaccinated $\left(S_U\right)$, infected and unvaccinated $\left(I_U\right)$, susceptible and vaccinated $\left(S_V\right)$, infected and vaccinated ($I_V$), recovered from an infection without vaccination ($R_U$), and vaccinated or recovered from a breakthrough infection $\left(R_V\right)$. We assume that infection confers sterilizing immunity. We track the size of the population of individuals who are unvaccinated ($N_U=S_U+I_U+R_U$) or vaccinated ($N_V=S_V+I_V+R_V$) and assume a stable population size $\left(N_T=N_U+N_V\right)$. Births occur at a constant rate $b$, to balance population-level deaths (independent of infection status) happening at a constant per capita rate (i.e., $b=m{N}_T$). Unvaccinated people are vaccinated with probability $p$ (the vaccine coverage).

We assume that the vaccine has no effect (primary failure) with a probability $\epsilon$, and provides sterilizing immunity otherwise. Contacts may be assortative, meaning that people are more likely to contact others who share their vaccination status. Assortativity is parameterized by $\varphi$, corresponding to the proportion of contacts reserved for people with concordant vaccination status versus the entire population (Busenberg and CastilloChavez, 1991). We allow $\varphi$ to vary from 0 (no asssortativity; random mixing) to 1 (full assortativity; no contact between people with discordant vaccination status). Accounting for assortativity, infected individuals contribute equally to the force of infection with transmission coefficient $\beta$ irrespective of their vaccine status. Finally, infected individuals recover at the same per capita rate $\gamma$. The complete model is:

$$\begin{array}{l}{\dot{S}}_U=b(1-p)-\beta S_U\left(\varphi \frac{I_U}{N_U}+(1-\varphi )\frac{I_U+I_V}{N_U+N_V}\right)-m{S}_U\\ {\dot{S}}_V=bp\epsilon -\beta S_V\left(\varphi \frac{I_V}{N_V}+(1-\varphi )\frac{I_U+I_V}{N_U+N_V}\right)-m{S}_V\\ {\dot{I}}_U=\beta S_U\left(\varphi \frac{I_U}{N_U}+(1-\varphi )\frac{I_U+I_V}{N_U+N_V}\right)-\gamma I_U-m{I}_U\\ {\dot{I}}_V=\beta S_V\left(\varphi \frac{I_V}{N_V}+(1-\varphi )\frac{I_U+I_V}{N_U+N_V}\right)-\gamma I_V-m{I}_V\\ {\dot{R}}_U=\gamma I_U-m{R}_U\\ {\dot{R}}_V=bp\left(1-\epsilon \right)+\gamma I_V-m{R}_V.\end{array}$$ (1)

The equations were solved numerically using the ode23t solver in MATLAB (The MathWorks, Inc., 2025) for the range of parameter values and initial conditions provided in Table S1 and Table S2. The time step-size was set adaptively by the solver, under the constraint that the relative error tolerance for each integration step was set to 10⁻⁸ for each state variable and all state variables had to be non-negative at all times. To obtain the equilibrium system state for each set of parameter values, the differential equations were integrated up to T = 5 × 10⁵ days, which was adequate time for the system to reach equilibrium. The state of the system at time T was taken to be the equilibrium state. The MATLAB parallel computation toolbox was used to solve for steady states across different parameter values. Additional analyses and figures were conducted in RStudio using R version 4.2.3 (Posit team, 2025; R Core Team, 2021).

State-level outbreak and vaccination data

We compiled publicly available data from twelve state departments of health that CDC lists as providing realtime updates on measles case reports (https://www.cdc.gov/measles/data-research/index.html, last accessed 15 December 2025) (see Table S4 for data and sources). Six of the states (Michigan, New Mexico, North Dakota, South Carolina, Texas, and Utah) report cases at the state level categorized by vaccination status (where vaccinated indicates at least one dose of MMR) and one state (Arizona) reports the total number of cases and the percentage of cases in unvaccinated people. All states except for Texas disaggregate unvaccinated cases from cases with unknown vaccination status. For our calculation, we assumed at least half of the unknown/unvaccinated cases were unvaccinated and classified the remainder as unvaccinated. We calculated the fraction of breakthrough infections ($f_V$) as the fraction of all known cases occurring in people with at least one dose of MMR. Across all states, we estimated a lower bound on $f_V$ by assuming that people with unknown status are vaccinated at one third the rate of cases with known status; we estimated an upper bound by assuming they were three times more likely to be vaccinated.

School-level vaccination data

We examined school-level vaccine coverage to approximate assortativity in school attendance based on vaccine status. We collected data by first examining state health department websites across all fifty states and then requesting data directly if they were not available on public dashboards. Annual MMR coverage (two doses completed) and number of students at the school level for kindergartens across the state were available for twelve states. Michigan and Iowa report coverage for all required vaccines rather than just MMR, Missouri data are at the zip code level, South Carolina reports data for a sample of schools, and several states suppressed data from schools with few students enrolled for privacy (see Table S5 for detailed information about school-level data). In total, we included sixteen states in the analysis. Preferential attendance based on vaccination status (${\varphi }_i$) was approximated by comparing school-level vaccine coverage $\left(p_i\right)$ to state-level coverage ($P$, estimated from each year’s school-level data):

$${\varphi }_i=\left\{\begin{array}{ll}\frac{p_i-P}{1-P},& p_i\ge P\\ \frac{P-p_i}{P},& p_i\le P\end{array}\right.$$ (2)

We then took a state-wide, annual average of this metric ($\tilde{\varphi }$), weighting by the share of the state’s children attending each school. We bootstrapped calculations by resampling 1,000 times from schools in a given state and year and taking the 95% confidence interval across this distribution of values. In a supplemental analysis, we tested whether estimated assortativity has significantly changed in any state since the 2019 – 2020 school year (with Bonferroni correction, α = 0.05/16 = 0.003).

Significance Statement:

Infections among vaccinated people may exacerbate vaccine hesitancy. Despite reports of ‘breakthrough infections’ within ongoing measles outbreaks, we show that the realized fraction of breakthrough infections is far lower than predicted by conventional epidemic theory that assumes random mixing. Combining epidemic models and school-level vaccine coverage data, we show that breakthrough infections may be partially limited via preferential mixing (‘assortativity’) based on vaccination status. Given current MMR coverage, most of the country is at risk for increasing breakthrough infections if vaccine coverage declines further. We conclude that enhanced vaccine monitoring and outreach campaigns are needed to confront a potential positive feedback loop between increasing breakthrough infections and declining vaccine coverage that could substantially increase the burden of vaccine-preventable disease.

Data, Materials, and Software Availability

All associated code has been uploaded to Github at https://github.com/WeitzGroup/Breakthrough-Infections/ and is archived on Zenodo at doi.org/10.5281/zenodo.18332220. Publicly available outbreak and vaccination data across six states are included in the repository, while data obtained via correspondence with public health departments in ten states are withheld.