The SIR model of an epidemic

Abstract

The SIR model is a three-compartment model of the time development of an epidemic. After normalizing the dependent variables, the model is a system of two non-linear differential equations for the susceptible proportion $S$ and the infected proportion $I$. After normalizing the time variable there is only one remaining parameter. This largely expository article is mainly about aspects of this model that can be understood with calculus. It also discusses an alternative exactly solvable model that appeared in early work of Kermack and McKendrick. This model may be obtained by replacing $SI$ factors by $\sqrt{2S-1}I$ factors. For a mild epidemic, where $S$ is decreasing from 1 but remains fairly close to 1, this is a reasonable approximation.

Extracted equations

• dS/dτ = -r0*S*I
• dI/dτ = (r0*S - 1)*I
• dR/dτ = I

Paper claims vs. our run

• Peak infection occurs when effective reproduction number r0*S = 1
  not-testable
  no fidelity score recorded
• S decreases exponentially as S = S0*exp(-r0*R)
  not-testable
  no fidelity score recorded
• Final removed R∞ satisfies 1 - R∞ = S0*exp(-r0*R∞)
  not-testable
  no fidelity score recorded
• For r0=2, final R∞ ≈ 0.80
  not-testable
  no fidelity score recorded
• For r0=3, final R∞ ≈ 0.94
  not-testable
  no fidelity score recorded
• For r0=6, final R∞ ≈ 0.997
  not-testable
  no fidelity score recorded
• Infection curve is asymmetric about peak, especially for large r0
  not-testable
  no fidelity score recorded

Parameters

Run notes

equations tried: ['dS/dτ = -r0*S*I', 'dI/dτ = (r0*S - 1)*I', 'dR/dτ = I']

Cross-field hypotheses involving this paper

• building energy systems and control → epidemiology
  Physics-informed machine learning (PIML) priors embedded in compartmental epidemic models

  If we embed domain knowledge about disease transmission mechanisms (e.g., contact rates, incubation periods, recovery kinetics) as physics priors into data-driven SIR variants, we can improve epidemic forecasting accuracy and physical consistency when extrapolating beyond training data. This mirrors how BESTOpt embeds thermodynamic and HVAC physics into neural network modules to improve generalization.

  Experiment: Baseline: standard SIR fit to synthetic or historical epidemic data; Variant: PIML-SIR with neural network correction terms constrained by transmission-rate bounds and conservation laws. Compare out-of-sample forecasting error and physical plausibility (e.g., do predicted contact rates stay within epidemiological bounds?) on held-out epidemic curves.

  novelty 0.55testability 0.62business 0.35sim: ode
• epidemiology → epidemiology, network disease dynamics
  none — both papers are within epidemiology and use identical SIR-ODE frameworks

  Paper B is not a transfer of mechanism from Paper A to a different field; both papers study SIR epidemic models in epidemiology using ODEs. Paper B simply extends Paper A's classical mass-action SIR to network-structured populations and proves equivalence under certain conditions.

  Experiment: Not applicable — the prompt requires two papers from DIFFERENT fields. Both papers are epidemiology/network disease dynamics.

  novelty 0.00testability 0.00business 0.00sim: none