📄 infectious diseases

Modeling epidemiological patterns of smallpox in Copenhagen in the 19th century after the introduction of the vaccine

This paper utilizes a mechanistic SEIR model and historical hospital data to demonstrate that the return of smallpox in Copenhagen after a 16-year hiatus was primarily driven by the waning of vaccine-induced immunity after approximately 20 years, which shifted the disease's burden from children to young adults.

Original authors: Eilersen, A., Poder, S. K., Grenfell, B. T., Sneppen, K., Simonsen, L.

Published 2026-06-21

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CC BY 4.0

Original authors: Eilersen, A., Poder, S. K., Grenfell, B. T., Sneppen, K., Simonsen, L.

Original paper licensed under CC BY 4.0 (https://creativecommons.org/licenses/by/4.0/). ⚕️ This is an AI-generated explanation of a preprint that has not been peer-reviewed. It is not medical advice. Do not make health decisions based on this content. Read full disclaimer

Imagine Copenhagen in the early 1800s as a crowded city where a terrifying monster called Smallpox used to roam freely. This monster was a childhood killer; almost every child got sick with it, and many didn't survive.

Then, in 1801, the city introduced a "shield" called the vaccine. By 1810, the government made it mandatory for children to get this shield to prove they were ready for church confirmation. The result was immediate and magical: the monster vanished. For 14 years, the city enjoyed a peaceful "honeymoon." No smallpox. No fear.

But then, in 1824, the monster returned. However, it looked different. It wasn't attacking little kids anymore; it was sneaking up on young adults.

The Mystery
Scientists wanted to know: Why did the monster come back, and why was it targeting adults instead of children?

To solve this, the authors of this paper acted like detectives. They dug up old hospital records from Copenhagen (specifically from a quarantine hospital called "Søkvæsthuset") that listed thousands of patients, their ages, and whether they had been vaccinated. They then built a giant digital "simulator" (a computer model) to replay history and test different theories.

The Suspects
The researchers had four main suspects for why the monster returned:

The "Lost Generation": A group of people born right after the vaccine started but before it became mandatory. They were too old to get the vaccine as babies, but the monster had already disappeared before they grew up, so they never caught it naturally either. They were sitting ducks.
The "Broken Shield" (Vaccine Failure): Maybe the vaccine didn't work for some people at all.
The "Leaky Shield" (Leaky Vaccine): Maybe the vaccine was like a raincoat with holes—it slowed the rain down but didn't stop it completely.
The "Fading Shield" (Waning Immunity): Maybe the shield worked perfectly at first, but over time, it just wore out and disappeared.

The Investigation
The researchers ran their computer simulation over and over, tweaking the rules to see which theory matched the real historical data.

The "Lost Generation" Theory: The model confirmed that this group was indeed very sick during the first few outbreaks. They were the "low-hanging fruit" for the virus. However, this group was temporary. Once they got sick (or died), they were gone. This didn't explain why the monster kept coming back for decades.
The "Broken" or "Leaky" Shield: The data showed that very few young vaccinated children got sick. If the vaccine was broken or leaky, we would have seen sick toddlers. We didn't. So, the shield was working great for kids.
The "Fading Shield" Theory: This was the winner. The data showed that people who got vaccinated as children in 1805 started getting sick again around 1830. That's about 25 years later.

The Verdict
The paper concludes that the vaccine was a hero, but it had an expiration date.

Think of the vaccine like a fresh coat of paint on a house. For the first 20 to 25 years, the paint is thick, shiny, and keeps the rain (the virus) out completely. But after about 25 years, the paint starts to peel and fade. The house is still standing, but it's no longer fully protected.

Because the city vaccinated almost everyone as children, the virus had nowhere to go for 14 years (the honeymoon). But as the "paint" on the first vaccinated generation started to fade (around 25 years later), the virus found a new target: those now-adults whose shields had worn thin. The disease didn't disappear; it just shifted its playground from the nursery to the living room.

Why This Matters
The study shows that when you vaccinate a whole generation of children, you can accidentally create a "gap" in the future. If the protection fades after 25 years, the disease can return as an adult illness. The authors note that we see similar patterns today with other diseases like Rotavirus and Pertussis (whooping cough), proving that looking at history helps us understand how vaccines behave over the long haul.

In short: The vaccine saved Copenhagen, but because the protection faded after about 25 years, the virus returned to attack the very people it had protected as children, turning a childhood disease into an adult one.

Technical Summary: Modeling Epidemiological Patterns of Smallpox in 19th Century Copenhagen

Problem Statement
Following the introduction of Edward Jenner's smallpox vaccine in 1798, Denmark implemented a mandatory, free childhood vaccination program by 1810. This initiative successfully eliminated smallpox from Copenhagen for a 14-year "honeymoon period." However, the disease returned in 1824 with a distinct epidemiological profile: it became a milder illness predominantly affecting young adults rather than children. The authors seek to explain the mechanisms behind this resurgence and the shift in the age distribution of cases. Specifically, the study investigates four potential factors: (1) waning vaccine-induced immunity, (2) the existence of a "lost generation" (individuals born too late for natural immunity but too early for vaccination), (3) primary vaccine failure (vaccination not inducing immunity), and (4) vaccine "leakiness" (reduced but not eliminated risk of infection).

Methodology
The study integrates historical archival data with a mechanistic mathematical model.

Data Sources: The authors utilized records from the Copenhagen smallpox quarantine hospital, Søkvæsthuset, covering the period 1824–1835, which captured approximately 3,000 medically attended cases (both mild and severe) due to strict quarantine policies. Data on vaccination status and age were derived from contemporary medical reports (Møhl, Hoppe, Wendt) and official public health records. For later outbreaks (1860s–1870s), where quarantine policies were relaxed and mild cases were underreported, the analysis relied on mortality time series.
Mathematical Model: A deterministic, age-structured SEIR (Susceptible-Exposed-Infected-Recovered) model was constructed using ordinary differential equations (ODEs).
- Structure: The population was divided into 70 one-year age cohorts. The model included compartments for natural immunity and vaccine-induced immunity. To simulate waning immunity, the vaccinated immune compartment ( $V$ ) was subdivided into four sub-compartments to generate a Gamma-distributed duration of immunity.
- Vaccination Dynamics: The model simulated the rollout of the vaccine from 1801 to 1810 (voluntary) and the subsequent mandatory phase (1810 onwards), reaching an effective coverage of ~90% of surviving children. The maximum age for vaccination ( $m$ ) was modeled as declining linearly from 8 years to 1 year by 1824.
- Parameters: The basic reproduction number ( $R_0$ ) was set to 7 to reflect Copenhagen's high density. The latent and infectious periods were set to 12 days. The duration of vaccine-induced immunity ( $\tau_V$ ) was the primary parameter fitted to the data.
- Scenarios: The model was run to compare the "waning immunity" hypothesis against "leaky vaccine" and "low coverage/failure" scenarios. Stochastic extinction and reintroduction of the virus via immigration were also incorporated.

Key Results

Waning Immunity as the Primary Driver: The model demonstrated that a vaccine-induced immunity duration of approximately 20–30 years (with a mean fit of ~25 years) best reproduced the observed epidemiological patterns. This duration accurately predicted the 14-year honeymoon period, the ~6-year interval between subsequent epidemic peaks, and the gradual increase in the mean age of infected individuals.
Rejection of Alternative Hypotheses:
- Vaccine Failure/Low Coverage: Scenarios assuming high rates of primary vaccine failure or low coverage (e.g., 60%) failed to fit the data. These models predicted a resurgence of smallpox in young children, which contradicted the historical record showing very few cases in recently vaccinated cohorts.
- Leaky Vaccine: While a highly effective leaky vaccine (reducing transmission by 90%) could mathematically approximate the mean age data, it produced poor fits for the age distributions of vaccinated cases and failed to explain the absence of cases in recently vaccinated children.
The "Lost Generation": The model confirmed the existence of a "lost generation" born between the early 1790s and 1810. These individuals were too old to be vaccinated in childhood but too young to have acquired natural immunity before the disease was eliminated. They were overrepresented in the unvaccinated case counts during the initial post-honeymoon outbreaks (1824, 1828–30).
Model Fit: The simulation successfully matched the observed shift in the mean age of cases (coefficient of determination $R^2 = 0.76$ ) and the timing of epidemic peaks. The model predicted outbreaks occurring roughly every six years, consistent with the observed 5.67-year interval.

Significance and Claims
The authors claim that their study provides a quantitative explanation for the historical shift of smallpox from a childhood disease to an adult disease in the early vaccine era. The primary contribution is the identification of waning vaccine immunity (averaging 25 years) as the dominant factor driving the resurgence and changing epidemiology, rather than vaccine failure or leakage.

The paper highlights that the "honeymoon period" was a result of temporary herd immunity created by the vaccination campaign, which protected both vaccinated and unvaccinated populations until immunity waned. The study also underscores the transient nature of the "lost generation" vulnerability, which dissipated after the first few post-honeymoon outbreaks.

The authors conclude that these historical dynamics are relevant to modern epidemiology, noting parallels with the shifting age demographics of Rotavirus and Pertussis following their respective vaccine introductions. They assert that historical data, when combined with modern mechanistic modeling, offers valuable insights into the long-term behavior of infectious diseases under vaccination programs, particularly regarding the duration of protection and the potential for epidemic recurrence.

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