The SIR Diffusion Model

🦠 The SIR Diffusion Model: A Reaction–Diffusion Approach to Epidemic Spread

📖 Introduction

Epidemic modeling has long relied on compartmental models to understand how diseases spread in populations. One of the pioneering frameworks is the SIR model (Susceptible–Infectious–Recovered), introduced by Kermack and McKendrick in 1927 [1].

The classic SIR model uses ordinary differential equations (ODEs) to describe how the number of susceptible (S), infectious (I), and recovered (R) individuals change over time. It helped establish fundamental epidemiological concepts such as epidemic thresholds and herd immunity.

However, the traditional SIR model assumes that populations are well-mixed, neglecting spatial movement and local spread. To address this limitation, researchers extended the model by incorporating diffusion terms representing random movement of individuals in space.

The result is the SIR Diffusion Model, a type of reaction–diffusion system, which captures how infections propagate through both time and geography. It has been used to model diseases like plague, rabies, and COVID-19, where spatial spread is significant.

📐 Model Description

The SIR Diffusion Model expands the classical SIR system by introducing spatial diffusion.
Let S(x, t), I(x, t), and R(x, t) represent the densities of susceptible, infectious, and recovered individuals at location x and time t.

The governing equations are:

  ∂S/∂t = Dₛ ∇²S – β S I / N
  ∂I/∂t = Dᵢ ∇²I + β S I / N – γ I
  ∂R/∂t = Dᵣ ∇²R + γ I

The Laplacian operator (∇²) describes spatial diffusion in two dimensions:

  ∇²f = ∂²f/∂x² + ∂²f/∂y²

Here:

  • Dₛ, Dᵢ, Dᵣ are diffusion coefficients for susceptible, infectious, and recovered groups.
  • β (beta) is the transmission rate.
  • γ (gamma) is the recovery rate.
  • N is the total population (assumed constant).

These equations couple local infection dynamics with spatial spread, producing traveling infection waves observable in real epidemics.

🔧 Parameter Definitions and Typical Values

SymbolMeaningTypical RangeNotes
βTransmission rate0.1 – 0.5 day⁻¹Higher β → faster spread (e.g., COVID-19 early outbreaks ≈ 0.3)
γRecovery rate0.07 – 0.3 day⁻¹Reciprocal of infectious period (≈ 3–14 days)
Dₛ, Dᵢ, DᵣDiffusion coefficients1 – 50 km²/dayReflect human or animal mobility
NPopulation densitycontext-dependentConstant in closed systems

📊 Example:
If β = 0.3 day⁻¹, γ = 0.1 day⁻¹, and Dᵢ = 10 km²/day, the infection front travels at approximately
  c ≈ 2√(Dᵢ × β / γ) ≈ 7 km/day —
a realistic value observed in regional outbreaks.

🤔 Assumptions and Applicability

The model relies on several assumptions:

  • The population is closed (no births or migration).
  • Local homogeneous mixing occurs within small regions.
  • Continuous space is assumed; movement is diffusive.
  • Permanent immunity follows recovery.

✅ Best suited for: locally transmitted diseases such as measles, rabies, or plague.
⚠️ Less suitable for: diseases spread by long-range travel (e.g., influenza, COVID-19 during global transmission phases).

🔄 Model Extensions and Variants

The SIR Diffusion Model serves as a foundation for several important extensions:

  1. SIR with Vital Dynamics – Includes birth/death rates (μ) for long-term endemic behavior.
  2. SIRS (Waning Immunity) – Allows recovered individuals to become susceptible again at rate ν.
  3. SEIR (Latent Period) – Adds an Exposed (E) compartment with incubation rate σ to represent delay before infectiousness.
  4. SIRD (Mortality) – Incorporates a Deceased (D) group with mortality rate α, suitable for high-fatality infections.
  5. Cross-Diffusion or Behavioral Models – Introduce adaptive movement (e.g., susceptibles avoiding infected areas), modeling behaviors like social distancing.

🏁 Conclusion

The SIR Diffusion Model beautifully merges infection dynamics and spatial movement, revealing how epidemics spread as waves through populations.
It provides key insights into both the rate and pattern of infection propagation.

As a cornerstone of spatial epidemiology, this model helps researchers evaluate:

  • Epidemic wave speeds,
  • Effects of population density and mobility,
  • And the impact of interventions like lockdowns or travel restrictions.

Within an Epidemiological PDE Toolbox, the SIR Diffusion Model and its variants (SIRS, SEIR, SIRD) form a versatile framework for analyzing, forecasting, and controlling disease spread across landscapes. 🌍

📚 References

[1] Kermack, W. O., & McKendrick, A. G. (1927). A Contribution to the Mathematical Theory of Epidemics. Proceedings of the Royal Society A, 115(772), 700–721.
https://doi.org/10.1098/rspa.1927.0118

[2] Bailey, N. T. J. (1975). The Mathematical Theory of Infectious Diseases (2nd ed.). Griffin, London.
https://openlibrary.org/books/OL4910522M/The_mathematical_theory_of_infectious_diseases

[3] Anderson, R. M., & May, R. M. (1982). Directly Transmitted Infectious Diseases: Control by Vaccination. Science, 215(4536), 1053–1060.
https://doi.org/10.1126/science.7058346

[4] Diekmann, O., & Heesterbeek, J. A. P. (2000). Mathematical Epidemiology of Infectious Diseases: Model Building, Analysis and Interpretation. Wiley.
https://www.wiley.com/en-us/Mathematical+Epidemiology+of+Infectious+Diseases%3A+Model+Building%2C+Analysis+and+Interpretation-p-9780471492412

[5] Hethcote, H. W. (2000). The Mathematics of Infectious Diseases. SIAM Review, 42(4), 599–653.
https://doi.org/10.1137/S0036144500371907

[6] Murray, J. D. (2003). Mathematical Biology II: Spatial Models and Biomedical Applications (3rd ed.). Springer.
https://doi.org/10.1007/b98869

[7] Brauer, F., & Castillo-Chavez, C. (2012). Mathematical Models in Population Biology and Epidemiology (2nd ed.). Springer.
https://doi.org/10.1007/978-1-4614-1686-9

[9] Noble, J. V. (1974). Geographic and Temporal Development of Plagues. Nature, 250, 726–729.
https://doi.org/10.1038/250726a0

[11] Davydovych, V., Dutka, V., & Cherniha, R. (2023). Reaction–Diffusion Equations in Mathematical Models Arising in Epidemiology. Symmetry, 15(11), 2025.
https://doi.org/10.3390/sym15112025