Anticipating resurgence

Leading indicators for vaccine-preventable disease

John M. Drake

Regents’ Professor, Odum School of Ecology & Center for the Ecology of Infectious Diseases, University of Georgia

Department of Epidemiology

Columbia University Mailman School of Public Health

October 1, 2026

Collaborators

Pej Rohani

Suzanne O’Regan

Tobias Brett

Eamon O’Dea

Amy Winter

Andrew Tredennick

Mallory Harris

Bogdan Epureanu

Amin Ghadami

Shiyang Chen

Paige Miller

Matt Ferrari

Andrew Park

Fabian Dablander

Measles has resurged three times in eleven years

Weekly US measles cases, 2006–2025, show three major outbreaks: 2014, 2018–19 and 2025. Drake & Rohani, Future Microbiol 2026.

Resurgence looks sudden because the threshold is crossed before the cases arrive

In a simulation matched to California’s 2006 pertussis resurgence, the tipping point (R₀ = 1) comes roughly a year before cases take off. O’Regan & Drake, Theor Ecol 2013.

Early warning belongs in the watch and warning phases of epidemic intelligence

Epidemic intelligence has three phases (watch, warning and emergency), and each calls for different data, questions and tools. After Han & Drake, EMBO Rep 2016.

The plan for today

Why the approach to R = 1 should be detectable
Whether the signal holds up in messy surveillance data
Where it has worked
US measles, where it didn’t
What that failure teaches us

Why the threshold is detectable

A system losing resilience recovers from shocks more slowly

A ball in a shallow basin returns slowly after a push, so its trajectory becomes more variable and more autocorrelated. Scheffer et al., Science 2012.

For measles, herd immunity leaves almost no margin

$R_e = R_0 \, \frac{S}{N} \qquad\qquad p_c = \frac{1 - 1/R_0}{E}$

$R_0$	basic reproduction number: secondary cases from one infection in a wholly susceptible population
$R_e$	effective reproduction number in the current population
$S/N$	fraction of the population susceptible
$p_c$	critical vaccination coverage
$E$	vaccine efficacy against infection

With $R_0$ ≈ 15 and vaccine efficacy E ≈ 0.95, the critical coverage is about 93%.

Declining vaccination is a slow driver acting on fast transmission

$\dot S = \mu(1-p) - \beta S I - \eta S - \mu S \qquad \dot I = \beta S I + \eta S - (\gamma + \mu) I$

$\dot p = \epsilon f(S, I, R), \qquad 0 < \epsilon \ll 1$

Stochastic transmission is described exactly by a master equation

$\frac{dP(\sigma, t)}{dt} = \sum_{\tilde\sigma \ne \sigma} T(\sigma \mid \tilde\sigma)\, P(\tilde\sigma, t) - \sum_{\tilde\sigma \ne \sigma} T(\tilde\sigma \mid \sigma)\, P(\sigma, t)$

Transitions from $\sigma = (S, I)$ go to $(S{+}1, I{-}1)$ , $(S{-}1, I)$ , $(S{+}1, I)$ , $(S, I{-}1)$ or $(S, I{+}1)$ .

O’Regan & Drake, Theor Ecol 2013.

The van Kampen expansion turns the fluctuations into a linear Fokker–Planck equation

$S = N\phi_S + \sqrt{N}\,\xi_S, \qquad I = N\phi_I + \sqrt{N}\,\xi_I$

$\frac{\partial \Pi(\xi, t)}{\partial t} = -\sum_{i,j} A_{ij} \frac{\partial (\xi_j \Pi)}{\partial \xi_i} + \frac{1}{2}\sum_{i,j} B_{ij} \frac{\partial^2 \Pi}{\partial \xi_i \partial \xi_j}$

The mean field gives the deterministic model; the fluctuations are Gaussian, with covariance set by the Jacobian $A$ .

System-size expansion from statistical physics: van Kampen, Adv Chem Phys 1976.

At $R_0 = 1$ , return time, variance and autocorrelation all diverge

$\Lambda = \beta - (\gamma + \mu) \;\to\; 0 \quad \text{as} \quad R_0 \to 1$

Characteristic return time $\tau \approx -1/\Lambda$ diverges (a linear approximation, valid only near the critical point)
Variance diverges
Lag-one autocorrelation goes to one

O’Regan & Drake 2013; Drake et al., PLOS Comput Biol 2019.

Near the critical immunization level, the potential well goes flat

The potential function of the SIR model with vaccination at immunization levels from 90% to 100%; the well is flattest at the critical level (about 94%), where R_e = 1. Drake et al., PLOS Comput Biol 2019.

This is the ball in the basin again, but now the basin is computed from the SIR model with vaccination rather than drawn. The mathematical object is called a potential function. Think of it as a landscape. The state of the system, here prevalence, is the ball. The ball rolls downhill toward the bottom of the well, which is the equilibrium, and the steepness of the walls tells you how hard the system is pulled back after a disturbance.

Each white curve is the potential at a different immunization level, from 90 percent at the front to 100 percent at the back. The yellow line traces the bottom of the well, the equilibrium prevalence, which slides toward zero as immunization rises. At 90 percent, in pink, the ball overshoots and rolls back and forth with a period of about a year. Those are damped oscillations, the familiar epidemic cycles of a childhood infection. At high immunization, in blue, the walls are steep, and a disturbance dies away within weeks without oscillating. In between, at the critical immunization level of about 94 percent, where R-effective equals one, the well is almost flat. You can see it in the inset at the top right. A flat well means almost no restoring force, so a disturbance takes a very long time to disappear.

That’s critical slowing down. It’s why variance and autocorrelation rise, and it’s what we’ll be looking for in the data.

[~2 min]

Near the threshold, the indicators have simple closed forms

Indicator	Formula
Mean	$\mu_1 = \dfrac{\zeta/\gamma}{1 - R_0}$
Variance	$\sigma^2 = \dfrac{\zeta/\gamma}{(1 - R_0)^2}$
Index of dispersion	$\sigma^2/\mu_1 = \dfrac{1}{1 - R_0}$
Coefficient of variation	$\sigma/\mu_1 = (\zeta/\gamma)^{-1/2}$
Autocorrelation	$AC(\tau) = \exp\!\big(-(1 - R_0)\gamma\tau\big)$

Birth–death–immigration process with sparking rate ζ. Brett, Drake & Rohani, J R Soc Interface 2017.

In simulation, variance and autocorrelation rise before herd immunity is lost

In a city of 8.5 million that starts 92% protected, vaccination declines (blue) and R_e rises (red); rolling autocorrelation and variance climb before the outbreak. Drake, Rohani & Winter, Future Microbiol 2026.

Does the signal hold up in surveillance data?

Case reports are aggregated and noisy versions of the true dynamics

True infections (black) differ from perfect weekly reports (blue) and from reports with sampling error (red). Brett et al., PLOS Comput Biol 2018.

Everything so far comes from a simple compartmental model, and that’s the model the theory was derived from. Real epidemics are messier: case reports are incomplete, transmission is seasonal, and populations are structured. So the next question is still a theoretical one. Should we expect critical slowing down to show up under those more realistic conditions? We answer it by simulating the complications and checking whether the indicators still work.

The first complication is observation. Nobody observes the number infected. Health departments see cases, aggregated by week or month and thinned by under-reporting. The black line is the truth, blue is perfect weekly reporting, and red is weekly reporting with realistic sampling error. Postdoctoral associate Toby Brett asked whether we can still see critical slowing down despite this corruption of information.

[~1.5 min]

Most indicators still work under realistic reporting

Seven of ten indicators keep high AUC unless reporting error is highly overdispersed. Brett et al. 2018.

Seasonal forcing doesn’t erase critical slowing down

Simulated emergence with no, low and high seasonality, with wavelet power below. Miller et al. 2017.

Another problem is that many vaccine-preventable diseases are strongly seasonal. For childhood infections, school-term forcing is exactly the kind of periodic signal that could swamp a slow trend. Paige Miller simulated emergence with no, low and high seasonal amplitude, left to right. Here’s how to read each column. The top row is R-naught: the black line includes the seasonal forcing, and the blue dashed line is its average, which is flat for ten years and then climbs to one at year fifteen. The dashed gray line marks where that climb begins. The middle row is weekly incidence. The bottom row is the wavelet power spectrum: time runs left to right, period runs down the vertical axis from a few weeks at the top to four years at the bottom, and red means high power. Ignore everything outside the white curve, where edge effects make the spectrum unreliable.

The thing to look for is spectral reddening. As the system slows down, its fluctuations shift toward longer periods, so power moves down the vertical axis. In the non-seasonal panel on the left, the lower right corner turns red after year ten. With strong seasonality, on the right, the annual cycle puts a band of power right at one year, and that band is what makes the problem hard.

[~2 min]

Wavelet filtering stays reliable as seasonality grows

AUC of the top indicators against seasonal amplitude β₁/β₀, under three kinds of preprocessing. Miller et al., Theor Biol Med Model 2017, Fig. 3.

Early warning signals appear even in models too complex to analyze

Lag-one autocorrelation rises before R_eff = 1 in six models, from a birth–death process to the FRED agent-based simulation. Brett et al., PLOS Comput Biol 2020.

Where it has worked

Malaria in Kenya’s western highlands resurged after years of control

Monthly malaria cases at a tea estate in Kericho, Kenya, 1965–2002. Harris, Hay & Drake, Biol Lett 2020.

Four indicators warned of Kericho’s resurgence two years ahead

Four indicators became significant about 24 months before the transition. Harris, Hay & Drake 2020.

Variance rose before the 1983 monkeypox epidemic in Central and West Africa

Statistically significant increases in variance appear in the bifurcation window. Unpublished analysis.

A second resurgence, this time of a pathogen held in check by a vaccine against a different virus. Smallpox vaccination protects against monkeypox, and when smallpox vaccination stopped after eradication, that cross-protection began to wane. Monkeypox cases in Central and West Africa rose through the early 1980s to an epidemic peak in December 1983. The gray bars are monthly cases and the points are rolling variance, with red and blue marking statistically significant increases. Variance rose through the window before the peak.

One caution before you believe this too much. Surveillance for monkeypox was intensified in the 1980s to verify smallpox eradication, with more than 20,000 specimens tested, so some of that rise may be reporting rather than transmission. This analysis is unpublished.

[~1.5 min]

Signals have also appeared before mumps and pertussis resurgence

Mumps, England: a combined decision function rose about four years before the 2004–05 outbreak
Pertussis, US states: resurgent states were distinguishable from non-resurgent ones by the early 1990s
Measles, Niger: variance and autocorrelation rose before re-emergence and elimination in model-constrained simulations

Brett & Rohani, PLOS Biol 2020; Tredennick et al., J R Soc Interface 2022.

For COVID-19, overlapping timescales scrambled the signal

Indicator trends before Europe’s second COVID-19 wave point both ways across countries. Dablander et al. 2022.

US measles, 2006–2025

Three outbreaks stand out against years of near-zero incidence

Weekly US measles cases, 2006–2025. Colored areas are the 11 outbreaks; the dashed line is the 6-cases-per-week threshold used to define them. Drake & Rohani 2026.

We tested whether variance and autocorrelation rose before three major outbreaks

Weekly cases, 2006–2025, from a curated domestic dataset plus CDC NNDSS
Rolling-window variance and lag-one autocorrelation (our open-source R package)
Trend tests with Spearman’s ρ adjusted for autocorrelation
Sensitivity grid: 3 detrending × 3 window bandwidths

Drake & Rohani, Future Microbiol 2026.

Neither indicator rose consistently before any of the three outbreaks

Autocorrelation showed a weak positive trend over the full series, significant only under some detrending choices. Variance showed none.

In the windows before 2014, 2018–19 and 2025, neither increased consistently.

Apparent signals appeared only with very large bandwidths and weren’t robust. Drake & Rohani, Future Microbiol 2026.

Outbreaks didn’t get larger or longer on the way to 2025

Outbreak duration (r = 0.44, p = 0.18) and size (r = 0.47, p = 0.16) show no trend with start date. Drake & Rohani, Future Microbiol 2026.

Fade-outs stayed as frequent as ever

The number of weeks with zero, fewer than two, or fewer than six cases in a trailing 52-week window shows no secular decline. Drake & Rohani, Future Microbiol 2026.

Four explanations for the missing signal

Direction. Approach from below is intrinsically harder to detect.
Hovering, not drifting. The US may sit in a weakly subcritical regime.
Response. Outbreak control truncates the dynamics the theory describes.
Aggregation. National totals average over asynchronous local dynamics.

Why? I see four explanations, and they aren’t exclusive. First, direction. Suzanne O’Regan and I showed that critical slowing down is intrinsically harder to detect when transmission approaches the threshold from below, as in resurgence, than from above, as in elimination. Second, the US may not be drifting toward the threshold at all. It may be hovering just below it, where importations plus the episodic buildup of susceptibles produce outbreaks like these without any bifurcation. Third, response. Every outbreak triggers case investigation, exclusion and vaccination campaigns, so outbreak size and duration reflect us as much as the virus. Fourth, the national series averages over communities whose immunity gaps are very different and whose dynamics aren’t synchronized.

My own bet is on the second and third.

[~1.5 min]

What it will take

Outbreak response has to be part of the model

Theory treats outbreak size as intrinsic dynamics, but for measles every outbreak is truncated by case finding, exclusion and vaccination campaigns.

We need to tell drifting from hovering

The same outbreak record can come from a system sliding toward R = 1 or one idling just below it.

Signals will be local before they are national

Immunity gaps are concentrated in schools, counties and communities, so indicators should be computed where transmission happens.

Evidence is still thin, and a warning isn’t a decision

Only 16 of 37 studies of resilience indicators for outbreaks used real data
Few show an unambiguous rise in routine surveillance before an outbreak
Action thresholds must weigh the asymmetric costs of false alarms and misses

Delecroix et al., PLOS Glob Public Health 2023; Drake, Rohani & Winter, Future Microbiol 2026.

The approach to resurgence should be detectable; for US measles, so far, it hasn’t been

The work now is closing the gap between the theory and the way public health surveillance and response actually operate.

Acknowledgments

Pej Rohani, Andrew W. Park, Bogdan Epureanu, Matt Ferrari, Eamon O’Dea, Shiyang Chen, Eric Marty, Paige Miller, Tobias S. Brett, Amin Ghadami, Mallory Harris, Suzanne O’Regan, Fabian Dablander, Hans Heesterbeek, Denny Borsboom, Andrew Tredennick, Amy Winter, Shaun Truelove

Supported by NSF award 2426740; NSF Predictive Intelligence for Pandemic Prevention award 2200158; and NIGMS/NIH award U01GM110744 (MIDAS). The content is solely the responsibility of the authors.

References (1 of 3)

Brett TS, Drake JM, Rohani P. 2017. Anticipating the emergence of infectious diseases. J R Soc Interface 14:20170115.
Brett TS, O’Dea EB, Marty É, Miller PB, Park AW, Drake JM, Rohani P. 2018. Anticipating epidemic transitions with imperfect data. PLOS Comput Biol 14:e1006204.
Brett TS, Ajelli M, Liu Q-H, Krauland MG, Grefenstette JJ, van Panhuis WG, Vespignani A, Drake JM, Rohani P. 2020. Detecting critical slowing down in high-dimensional epidemiological systems. PLOS Comput Biol 16:e1007679.
Brett TS, Rohani P. 2020. Dynamical footprints enable detection of disease emergence. PLOS Biol 18:e3000697.
Chen S, O’Dea EB, Drake JM, Epureanu BI. 2019. Eigenvalues of the covariance matrix as early warning signals for critical transitions in ecological systems. Sci Rep 9:2572.
Dablander F, Heesterbeek H, Borsboom D, Drake JM. 2022. Overlapping timescales obscure early warning signals of the second COVID-19 wave. Proc R Soc B 289:20211809.
Delecroix C, van Nes EH, van de Leemput IA, Rotbarth R, Scheffer M, ten Bosch Q. 2023. The potential of resilience indicators to anticipate infectious disease outbreaks, a systematic review and guide. PLOS Glob Public Health 3:e0002253.

References (2 of 3)

Drake JM, Brett TS, Chen S, Epureanu BI, Ferrari MJ, Marty É, Miller PB, O’Dea EB, O’Regan SM, Park AW, Rohani P. 2019. The statistics of epidemic transitions. PLOS Comput Biol 15:e1006917.
Drake JM, Rohani P. 2026. No evidence for critical slowing down before measles outbreaks in the US, 2006–2025. Future Microbiol 21:135–140.
Drake JM, Rohani P, Winter A. 2026. Can vaccine-preventable disease resurgence be anticipated? Leading indicators and tipping points. Future Microbiol 21:321–327.
Ghadami A, O’Dea EB, Drake JM, Rohani P, Epureanu BI. 2023. Anticipating epidemic transitions in metapopulations with multivariate spectral similarity. Nonlinear Dyn 111:17605–17615.
Han BA, Drake JM. 2016. Future directions in analytics for infectious disease intelligence: toward an integrated warning system for emerging pathogens. EMBO Rep 17:785–789.
Harris MJ, Hay SI, Drake JM. 2020. Early warning signals of malaria resurgence in Kericho, Kenya. Biol Lett 16:20190713.
Kaul RB, Evans MV, Murdock CC, Drake JM. 2018. Spatio-temporal spillover risk of yellow fever in Brazil. Parasit Vectors 11:488.

References (3 of 3)

Miller PB, O’Dea EB, Rohani P, Drake JM. 2017. Forecasting infectious disease emergence subject to seasonal forcing. Theor Biol Med Model 14:17.
O’Dea EB, Drake JM. 2019. Disentangling reporting and disease transmission. Theor Ecol 12:89–98.
O’Dea EB, Drake JM. 2022. A semi-parametric, state-space compartmental model with time-dependent parameters for forecasting COVID-19 cases, hospitalizations and deaths. J R Soc Interface 19:20210702.
O’Regan SM, Drake JM. 2013. Theory of early warning signals of disease emergence and leading indicators of elimination. Theor Ecol 6:333–357.
O’Regan SM, O’Dea EB, Rohani P, Drake JM. 2020. Transient indicators of tipping points in infectious diseases. J R Soc Interface 17:20200094.
Scheffer M, Carpenter SR, Lenton TM, et al. 2012. Anticipating critical transitions. Science 338:344–348.
Tredennick AT, O’Dea EB, Ferrari MJ, Park AW, Rohani P, Drake JM. 2022. Anticipating infectious disease re-emergence and elimination: a test of early warning signals using empirically based models. J R Soc Interface 19:20220123.
van Kampen NG. 1976. The expansion of the master equation. Adv Chem Phys 34:245–309.

Backup

The second factorial moment separates rising transmission from rising reporting

The mean rises whether reporting or transmission increases; the second factorial moment rises only with transmission. O’Dea & Drake, Theor Ecol 2019.

Resurgence is harder to see coming than elimination

Elimination: coverage rising

Emergence: coverage falling

Theoretical indicators from the van Kampen expansion for SIS and SIR models. O’Regan & Drake, Theor Ecol 2013.

Each phase needs its own analytical tool

Watch: spillover risk mapping

Warning: early warning signals

Emergency: forecasting

Kaul et al. 2018; Drake et al. 2019; O’Dea & Drake 2022.

Wavelet filtering ranks with variance; wavelet reddening degrades with seasonality

AUC of each indicator across seasonal amplitude under three kinds of preprocessing. Miller et al., Theor Biol Med Model 2017.

Pertussis spectra across US states converged as vaccination pushed toward the threshold

Multivariate spectral similarity of state-level pertussis series, 1938–1980. Ghadami et al., Nonlinear Dyn 2023.

Trajectories spiral more slowly toward equilibrium near the critical immunization level

Deterministic SIR solutions in S–I phase space for six immunization levels and two initial conditions. Drake et al., PLOS Comput Biol 2019.

Critical slowing down precedes emergence in the hinge model

Variance in a moving window rises through a period of critical slowing down before the critical point. Unpublished.

Autocorrelation slows as vaccination drives transmission toward elimination

Oscillations in the autocorrelation function lengthen as vaccine uptake approaches the critical level. Drake et al., PLOS Comput Biol 2019.

Emergence rate can be estimated by maximum likelihood

The likelihood surface from the first six years of a simulated emergence recovers the true rate of increase in R₀. Brett, Drake & Rohani, J R Soc Interface 2017.

Indicator performance depends on window size and transition speed

Indicators under 10- to 40-year approaches; red is stationary BDI theory. Brett, Drake & Rohani 2017.

The largest eigenvalue of the spatial covariance matrix is an early warning signal

In a spatial SI model, the largest covariance eigenvalue and its share of variation rise toward the transition. Chen et al., Sci Rep 2019.

Reactivity is a transient early warning signal

Reactivity rises before emergence with importation. O’Regan et al., J R Soc Interface 2020.

Vaccination reshaped the synchrony of US measles

Regional measles series and mean pairwise correlation over time, 1952–1993. O’Dea, Miller & Drake.

Anticipating resurgence

Collaborators

Measles has resurged three times in eleven years

Resurgence looks sudden because the threshold is crossed before the cases arrive

Early warning belongs in the watch and warning phases of epidemic intelligence

The plan for today

Why the threshold is detectable

A system losing resilience recovers from shocks more slowly

For measles, herd immunity leaves almost no margin

Declining vaccination is a slow driver acting on fast transmission

Stochastic transmission is described exactly by a master equation

The van Kampen expansion turns the fluctuations into a linear Fokker–Planck equation

At R0=1R_0 = 1R0​=1, return time, variance and autocorrelation all diverge

Near the critical immunization level, the potential well goes flat

Near the threshold, the indicators have simple closed forms

In simulation, variance and autocorrelation rise before herd immunity is lost

Does the signal hold up in surveillance data?

Case reports are aggregated and noisy versions of the true dynamics

Most indicators still work under realistic reporting

Seasonal forcing doesn’t erase critical slowing down

Wavelet filtering stays reliable as seasonality grows

Early warning signals appear even in models too complex to analyze

Where it has worked

Malaria in Kenya’s western highlands resurged after years of control

Four indicators warned of Kericho’s resurgence two years ahead

Variance rose before the 1983 monkeypox epidemic in Central and West Africa

Signals have also appeared before mumps and pertussis resurgence

For COVID-19, overlapping timescales scrambled the signal

US measles, 2006–2025

Three outbreaks stand out against years of near-zero incidence

We tested whether variance and autocorrelation rose before three major outbreaks

Neither indicator rose consistently before any of the three outbreaks

Outbreaks didn’t get larger or longer on the way to 2025

Fade-outs stayed as frequent as ever

Four explanations for the missing signal

What it will take

Outbreak response has to be part of the model

We need to tell drifting from hovering

Signals will be local before they are national

Evidence is still thin, and a warning isn’t a decision

The approach to resurgence should be detectable; for US measles, so far, it hasn’t been

Acknowledgments

References (1 of 3)

References (2 of 3)

References (3 of 3)

Backup

The second factorial moment separates rising transmission from rising reporting

Resurgence is harder to see coming than elimination

Each phase needs its own analytical tool

Wavelet filtering ranks with variance; wavelet reddening degrades with seasonality

Pertussis spectra across US states converged as vaccination pushed toward the threshold

Trajectories spiral more slowly toward equilibrium near the critical immunization level

Critical slowing down precedes emergence in the hinge model

Autocorrelation slows as vaccination drives transmission toward elimination

Emergence rate can be estimated by maximum likelihood

Indicator performance depends on window size and transition speed

The largest eigenvalue of the spatial covariance matrix is an early warning signal

Reactivity is a transient early warning signal

Vaccination reshaped the synchrony of US measles

At $R_0 = 1$ , return time, variance and autocorrelation all diverge