The Birth of Epidemic Modeling: Daniel Bernoulli and the Smallpox Inoculation Controversy
1 Introduction
Mathematical modeling of infectious diseases is now a cornerstone of public health decision-making. During COVID-19, we all became familiar with concepts like “flattening the curve” and reproduction numbers. But did you know that the first mathematical model for infectious disease control was developed nearly 300 years ago? In 1760, a Swiss mathematician named Daniel Bernoulli created a revolutionary approach to evaluate a controversial medical procedure: smallpox inoculation (Bernoulli 1766). His work represents the birth of what we now call mathematical epidemiology (Dietz and Heesterbeek 2002).
2 Who Was Daniel Bernoulli?
Daniel Bernoulli (1700-1782) came from a family of brilliant mathematicians. While his relatives focused on pure mathematics, Daniel pursued medicine alongside his mathematical studies (Hald 2003). This unique combination placed him perfectly to bridge these two worlds when a major public health controversy erupted in 18th century Europe.
3 The Smallpox Crisis
Today, we celebrate smallpox as the first human disease to be completely eradicated through vaccination. But in Bernoulli’s time, smallpox was a devastating illness that:
- Killed approximately 400,000 Europeans annually
- Caused death in 1 out of every 7-10 infected individuals
- Left survivors permanently scarred and sometimes blind
- Affected nearly everyone, with most contracting it during childhood
4 The Controversial Solution: Variolation
In the early 18th century, a precursor to vaccination called “variolation” (or inoculation) was introduced to Europe (Blower and Bernoulli 2004). This procedure involved:
- Taking a small amount of pus from a person with a mild case of smallpox
- Inserting it under the skin of a healthy person
- Inducing a usually milder form of the disease
- Conferring lifelong immunity if the person survived
This technique reduced the mortality rate from smallpox from about 15% to approximately 1-2%. However, it was extremely controversial because:
- It deliberately infected healthy people with a deadly disease
- Some people died from the procedure itself
- Inoculated individuals could spread smallpox to others
- Religious and ethical objections were raised against “interfering with God’s will”
5 Bernoulli’s Breakthrough: The First Disease Model
Bernoulli recognized that this controversy needed more than opinions—it needed data and mathematical analysis. In 1760, he presented his work “Essai d’une nouvelle analyse de la mortalité causée par la petite vérole” (Essay on a new analysis of the mortality caused by smallpox) to the Royal Academy of Sciences in Paris (Bernoulli 1766).
His approach was revolutionary because:
- It was the first mathematical model specifically designed to evaluate a public health intervention
- It separated the risk of dying from smallpox from other causes of death
- It calculated potential years of life gained through inoculation
- It used real demographic data from the city of London
6 The Model Explained Simply
Bernoulli’s approach can be understood through a simple analogy:
Imagine 1,000 children born in the same year. Without inoculation, these children face two risks as they age: - The risk of contracting and possibly dying from smallpox - The risk of dying from any other cause
Bernoulli used available data to estimate: - How many people would get smallpox each year - How many would die from it - How many would die from other causes
He then compared this natural scenario with one where everyone was inoculated against smallpox. Although inoculation carried its own small risk of death, Bernoulli’s calculations showed that widespread inoculation would:
- Increase life expectancy by about 3 years (significant for that time period)
- Save thousands of lives across a population
7 Mathematical Innovation: The Life Table Approach
Bernoulli’s work was mathematically innovative because he:
- Created what we now call a “dynamic life table” - tracking population changes over time
- Separated specific causes of death (what epidemiologists now call “competing risks”)
- Introduced differential equations to model population changes due to disease (Dietz and Heesterbeek 2002)
In modern mathematical notation, his core equation looked something like:
Where: -
While this may look intimidating, the concept is straightforward: Bernoulli tracked how many people would die from different causes under different scenarios (J. A. P. Heesterbeek and Dietz 1996).
8 Impact and Legacy
Bernoulli’s mathematical analysis had profound effects:
Quantified benefits: He showed that inoculation would increase life expectancy by approximately 3 years - remarkable in an era when life expectancy was around 30 years (Blower and Bernoulli 2004).
Risk comparison: He demonstrated that the 1-2% risk of death from inoculation was far smaller than the lifetime risk of dying from natural smallpox infection.
Policy influence: His work helped shift public opinion and policy toward accepting inoculation as a worthwhile public health measure.
Methodological innovation: He established mathematical modeling as a valid approach to public health questions (Hethcote 2000).
9 The Birth of Mathematical Epidemiology
Bernoulli’s approach contained the seeds of modern infectious disease modeling:
- It used mathematics to predict disease outcomes
- It quantified the impact of interventions
- It separated competing causes of death
- It estimated population-level effects from individual risks
The field that Bernoulli pioneered would later be expanded by other mathematical innovators like Ronald Ross (Ross 1908, 1911) and the team of Kermack and McKendrick (Kermack and McKendrick 1927), eventually leading to the sophisticated models we use today (Anderson and May 1991).
10 Modern Applications of Bernoulli’s Approach
The principles Bernoulli established are still used to answer questions like:
- How many lives will a new vaccine save? (Ferguson et al. 2006)
- Is a screening program worth its costs?
- Which age groups should receive priority for limited medical resources?
- What is the optimal strategy for controlling an epidemic? (Riley et al. 2003)
Modern epidemic models have incorporated additional mathematical sophistication, particularly around the concept of the basic reproduction number (R₀) and the construction of next-generation matrices (Diekmann, Heesterbeek, and Roberts 2010; Wallinga and Lipsitch 2007), but they still follow Bernoulli’s fundamental approach of using mathematics to guide public health decisions.
11 Conclusion
Daniel Bernoulli’s 1760 smallpox model represents a pivotal moment in the history of public health (Bernoulli 1766). By applying mathematical reasoning to a contentious medical issue, he demonstrated that data and analysis could guide decisions more effectively than opinions alone.
As we continue to face infectious disease challenges in the modern world, from emerging pathogens to vaccine hesitancy, Bernoulli’s approach remains relevant (Keeling and Rohani 2008). His work reminds us that mathematical modeling is not just an academic exercise but a powerful tool for saving lives and informing policy.
The next time you hear about disease models predicting the course of an epidemic or evaluating the impact of interventions, remember that this approach began with a mathematician applying his skills to a pressing public health problem nearly three centuries ago (H. Heesterbeek and Roberts 2015).