When Outbreaks Breed Outbreaks: The Hawkes Point Process for Self-Exciting Epidemics
Modeling the Domino Effect of Disease Transmission
🎲 Introduction: The Chain Reaction of Infection
Imagine dropping a single domino in a vast arrangement—watching as it topples others, which topple still more, creating cascading waves of falling pieces. This is the essence of self-excitation in epidemic dynamics: each infection doesn’t just add to the count—it actively generates more infections, which generate even more.
Enter the Hawkes point process, a powerful statistical model originally developed to describe earthquake aftershocks [1] but now revolutionizing how we understand infectious disease transmission. Unlike traditional models that treat cases as independent events, the Hawkes process explicitly captures how each infection event increases the probability of future infections—exactly what happens in real epidemics.
Developed by Alan Hawkes in 1971 [2], this framework treats disease cases as a temporal point process where the “intensity” (instantaneous rate) of new cases depends on the entire history of past cases. It’s particularly well-suited for diseases with clear transmission chains, superspreading events, or clustered outbreak patterns.
From tracking norovirus in cruise ships to modeling SARS-CoV-2 superspreading events, the Hawkes process provides a mathematically elegant way to quantify how much each case contributes to future transmission—and when the epidemic might finally fizzle out [3-4].
🧮 Model Description: The Mathematics of Contagion Cascades
The Hawkes process models the occurrence of infection events as points in time {t₁, t₂, …, tₙ} on the interval [0, T], where the intensity (rate) of new events depends on past events through a triggering function.
Core Intensity Equation
λ(t) = μ + ∑₍tᵢ < t₎ α · e⁻ᵠ⁽ᵗ⁻ᵗⁱ⁾
This deceptively simple equation captures the full dynamics of self-exciting processes:
- λ(t): Intensity (expected rate) of new infections at time t
- μ (mu): Background rate—spontaneous infections not caused by previous cases
- α (alpha): Jump size—how much each past infection increases the intensity
- ω (omega): Decay rate—how quickly the infectious influence of past cases fades
- tᵢ: Time of the i-th infection event
- T: Total observation horizon (days)
The Branching Process Interpretation
The Hawkes process has an elegant cluster representation that makes its epidemiological meaning crystal clear:
- Immigrant events: Occur as a Poisson process with rate μ—these are “index cases” not caused by local transmission
- Offspring events: Each immigrant (and its descendants) generates secondary cases according to an exponential delay distribution with rate ω
- Mean offspring number: n = α / ω—this is the branching ratio, equivalent to the basic reproduction number R₀ in simple cases
🔑 Critical insight: When n < 1, the process is subcritical—outbreaks eventually die out. When n ≥ 1, the process becomes supercritical—leading to sustained transmission or explosive growth.
Probability Distribution of Events
Given the intensity λ(t), the probability of observing events at times {t₁, …, tₙ} is:
P(t₁, …, tₙ) = [∏₍i=1₎ⁿ λ(tᵢ)] · exp(−∫₀ᵀ λ(s) ds)
This likelihood function enables parameter estimation from real outbreak data using maximum likelihood or Bayesian methods.
📊 Key Parameter Definitions and Typical Values
Understanding these parameters is essential for interpreting Hawkes model results in epidemiological contexts.
| μ | Background rate | 0.01 – 1.0/day | Higher μ = more imported/spontaneous cases |
| α | Triggering amplitude | 0.1 – 5.0 | Higher α = stronger transmission per case |
| ω | Decay rate | 0.2 – 2.0/day | Higher ω = shorter infectious period |
| n = α/ω | Branching ratio | 0.1 – 2.0 | n ≈ R₀ for simple transmission scenarios |
| T | Observation window | 7 – 365 days | Longer T = better parameter estimation |
Epidemiological Interpretation Examples
- Influenza in a school: μ = 0.1/day (imported cases), α = 1.2, ω = 0.6/day → n = 2.0 (high transmission)
- Foodborne outbreak: μ = 0.5/day (common source), α = 0.3, ω = 1.0/day → n = 0.3 (limited secondary transmission)
- Endemic disease: μ = 0.8/day, α = 0.4, ω = 0.8/day → n = 0.5 (stable low-level transmission)
The mean infectious period is 1/ω, so ω = 0.5/day corresponds to a 2-day average infectious period.
⚠️ Assumptions and Applicability: When Hawkes Shines
The Hawkes point process is powerful but works best under specific epidemiological conditions.
✅ Ideal Applications
- Clustered outbreak data: Clear transmission chains or superspreading events
- Individual-level timing: Exact or approximate infection times available
- Limited population size: Where depletion of susceptibles isn’t dominant (early outbreak phase)
- Homogeneous mixing: Within the observed population or setting
- Single strain: No significant antigenic variation during observation period
❌ Limitations and Challenges
- Large populations: Susceptible depletion becomes important, violating the linear Hawkes assumption
- Spatial heterogeneity: Requires spatial extensions (see below)
- Multiple introductions: High μ can mask transmission dynamics
- Reporting delays: Observed times may not reflect true infection times
💡 Pro Tip: The Hawkes process works best for outbreak investigations (hospitals, cruise ships, communities) rather than large-scale population epidemics where susceptible depletion dominates [5].
🚀 Model Extensions and Variants: Beyond Basic Self-Excitation
The basic Hawkes framework has inspired numerous sophisticated extensions for real-world epidemiological challenges.
1. Spatio-Temporal Hawkes Process
For outbreaks with spatial components (e.g., disease spread across neighborhoods):
λ(s, t) = μ(s) + ∑₍tᵢ < t₎ α · e⁻ᵠ⁽ᵗ⁻ᵗⁱ⁾ · K(s − sᵢ)
Where K(s − sᵢ) is a spatial kernel (e.g., Gaussian) describing how transmission probability decreases with distance [6].
2. Multivariate Hawkes Process
For multiple interacting diseases or strains:
λₖ(t) = μₖ + ∑₍l=1₎ᴸ ∑₍tᵢˡ < t₎ αₖₗ · e⁻ᵠₖₗ⁽ᵗ⁻ᵗⁱˡ⁾
Where αₖₗ represents cross-excitation from strain l to strain k [7].
3. Non-Parametric Hawkes Process
When the triggering function shape is unknown:
λ(t) = μ + ∑₍tᵢ < t₎ g(t − tᵢ)
Where g(·) is estimated non-parametrically using kernel methods or splines [8].
4. Marked Hawkes Process
When cases have additional attributes (severity, age, location):
λ(t) = μ + ∑₍tᵢ < t₎ α(mᵢ) · e⁻ᵠ⁽ᵗ⁻ᵗⁱ⁾
Where mᵢ is the mark (attribute) of the i-th event, and α(mᵢ) allows transmission intensity to depend on case characteristics [9].
5. Epidemic-Type Aftershock Sequence (ETAS) Model
Originally developed for earthquakes, now used for disease superspreading:
λ(t) = μ + ∑₍tᵢ < t₎ κ · (mᵢ − m₀) · (t − tᵢ + c)⁻ᵖ
Where mᵢ represents “magnitude” (e.g., number of secondary cases), capturing how larger events trigger more offspring [10].
6. Time-Varying Hawkes Process
For interventions that change transmission dynamics:
λ(t) = μ(t) + ∑₍tᵢ < t₎ α(t) · e⁻ᵠ⁽ᵗ⁻ᵗⁱ⁾
Where μ(t) and α(t) can change at intervention times or follow smooth trends.
🎯 Conclusion: Capturing the Ripple Effects of Infection
The Hawkes point process represents a paradigm shift in epidemic modeling—from viewing cases as independent observations to recognizing them as interconnected events in a causal chain. By explicitly modeling how each infection increases the probability of future infections, it captures the fundamental self-exciting nature of disease transmission.
What makes this approach particularly valuable is its dual interpretation: as both a point process with time-varying intensity and as a branching process with clear epidemiological parameters. The branching ratio n = α/ω provides an intuitive measure of transmission potential, while the decay rate ω reveals the timescale of infectiousness.
In an era of precision public health, where outbreak investigations increasingly rely on detailed case timing and contact tracing data, the Hawkes process offers a statistically rigorous framework for quantifying transmission dynamics in real time. Whether you’re investigating a hospital outbreak, analyzing superspreading events, or modeling the early phase of an emerging pathogen, this model provides your Statistical Epidemics Toolbox with a powerful lens for understanding how outbreaks truly propagate—one infection at a time, each creating ripples that extend far into the future.
📚 References
[1] Ogata, Y. (1988). Statistical models for earthquake occurrences and residual analysis for point processes. Journal of the American Statistical Association, 83(401), 9–27. https://doi.org/10.1080/01621459.1988.10478560
[2] Hawkes, A. G. (1971). Spectra of some self-exciting and mutually exciting point processes. Biometrika, 58(1), 83–90. https://doi.org/10.1093/biomet/58.1.83
[3] Meyer, S., Elias, J., & Höhle, M. (2012). A space-time conditional intensity model for invasive meningococcal disease occurrence. Biometrics, 68(2), 607–616. https://doi.org/10.1111/j.1541-0420.2011.01684.x
[4] Chiang, Y. Y., & Chen, Y. C. (2021). Modeling COVID-19 superspreading events using Hawkes processes. PLoS ONE, 16(8), e0256015. https://doi.org/10.1371/journal.pone.0256015
[5] Browning, E., Danon, L., & House, T. (2021). Superspreading dynamics in a structured population. Journal of The Royal Society Interface, 18(184), 20210621.
[6] Mohler, G. O., Short, M. B., Brantingham, P. J., Schoenberg, F. P., & Tita, G. E. (2011). Self-exciting point process modeling of crime. Journal of the American Statistical Association, 106(493), 100–108. https://doi.org/10.1198/jasa.2011.ap09546
[7] Bacry, E., Mastromatteo, I., & Muzy, J. F. (2015). Hawkes processes in finance. Market Microstructure and Liquidity, 1(01), 1550005.
[8] Lewis, P. A., & Mohler, G. (2011). A nonparametric EM algorithm for multiscale Hawkes processes. Journal of Nonparametric Statistics, 23(4), 1011–1024.
[9] Reynaud-Bouret, P., & Schbath, S. (2010). Adaptive estimation for Hawkes processes; application to genome analysis. Annals of Statistics, 38(5), 2781–2822. https://doi.org/10.1214/10-AOS806
[10] Zhuang, J., Ogata, Y., & Vere-Jones, D. (2002). Stochastic declustering of space-time earthquake occurrences. Journal of the American Statistical Association, 97(458), 369–380. https://doi.org/10.1198/016214502760046925