SIR Epidemic Model Calculator
Overview
The SIR model is a classic, deterministic compartment model for infectious disease dynamics. It divides a closed population into three groups:
- S: susceptible (can become infected)
- I: infectious (currently infected and able to transmit)
- R: removed/recovered (no longer infectious; typically assumed immune)
Given initial conditions and two rate parameters—β (infection/transmission rate) and γ (recovery/removal rate)—the model produces an “epidemic curve”: infections rise, peak, and then fall as susceptibility is depleted. This calculator simulates that curve numerically over time using a discrete time step.
What the inputs mean
- Total population (N): the total number of people in the modeled population. In a basic SIR model, N stays constant.
- Initial infected (I₀): number of infectious people at time 0.
- Initial recovered (R₀): number of removed/recovered people at time 0.
- Initial susceptible (S₀): implied by the others: S₀ = N − I₀ − R₀. For the model to make sense you should have N ≥ I₀ + R₀.
- Infection rate (β) (per day): controls how quickly susceptible people become infected through contact with infectious people. Larger β makes outbreaks grow faster and peak higher (all else equal).
- Recovery rate (γ) (per day): controls how quickly infectious people leave the I compartment. The average infectious period is approximately 1/γ days.
- Time step (Δt) (days): the simulation increment. Smaller values generally improve numerical stability/accuracy.
- Steps: number of iterations. Total simulated duration is approximately Steps × Δt days.
Model equations (continuous time)
The standard SIR differential equations (Kermack–McKendrick) are:
These equations assume that transmission is proportional to the product S·I (homogeneous mixing) and scaled by 1/N.
Key derived quantities
- Basic reproduction number: R₀ = β/γ. If R₀ > 1 and most people are susceptible, infections can initially grow; if R₀ < 1, infections tend to die out.
- Effective reproduction number: Re(t) = (β/γ) · (S(t)/N). Even with R₀ > 1, Re(t) declines as S decreases; the epidemic starts shrinking once Re(t) drops below 1.
- Average infectious duration: about 1/γ days.
Discrete-time simulation (Euler method)
This calculator advances the model using a forward Euler discretization with time step Δt:
- Sn+1 = Sn − Δt · β · Sn In / N
- In+1 = In + Δt · β · Sn In / N − Δt · γ · In
- Rn+1 = Rn + Δt · γ · In
In general, smaller Δt reduces numerical error. If Δt is too large, Euler updates can overshoot and produce unrealistic values (e.g., negative S or I), especially when β is high or when the epidemic changes rapidly.
How to interpret results
- Peak infected: the maximum of I(t). This is a proxy for maximum concurrent infectious burden.
- Time to peak: when I(t) reaches its maximum; depends strongly on β and γ.
- Final size: how many people have ever been infected by the end of the simulation, often approximated by R(T) − R₀ (assuming recovered represent cumulative infections in the basic model).
- Early growth vs decline: if your initial state yields Re(0) > 1, I(t) will initially increase; if Re(0) < 1, I(t) typically decreases immediately.
To explore interventions conceptually, lowering β (reduced contact/transmission) or raising γ (faster recovery/isolation) reduces R₀ and tends to reduce the peak and total infections.
Worked example
Suppose:
- N = 1000
- I₀ = 1
- R₀ = 0 → S₀ = 999
- β = 0.3 per day
- γ = 0.1 per day
Then:
- R₀ = β/γ = 0.3/0.1 = 3
- Re(0) = 3 · (999/1000) ≈ 2.997 > 1, so infections initially grow.
- Average infectious period ≈ 1/γ = 10 days.
If you run the simulation long enough, I(t) will rise from 1, reach a peak, and eventually fall as S(t) declines and Re(t) drops below 1. Try decreasing β to 0.15 (keeping γ = 0.1): R₀ becomes 1.5 and the peak should be smaller and later; decreasing β below 0.1 makes R₀ < 1 and I(t) typically shrinks from the start.
Parameter comparison (what changes when you adjust inputs)
| Change | Typical effect on the epidemic curve | Why |
|---|---|---|
| Increase β | Faster growth, higher and earlier peak; larger total infected | More transmission per S–I contact increases incidence |
| Decrease β | Slower growth, lower peak; may prevent a major outbreak | Reduces R₀ and Re(t) |
| Increase γ | Lower peak and shorter outbreak duration | People leave I faster (shorter infectious period) |
| Increase I₀ | Earlier visible outbreak; peak may occur sooner | More initial infectious seed accelerates early dynamics |
| Decrease Δt | More stable/accurate numerical trajectory (slower to compute) | Reduces discretization error from Euler steps |
Assumptions & limitations
- Closed population: N is constant (no births, migration). Deaths are not modeled separately unless you interpret “removed” as including deaths.
- Homogeneous mixing: everyone is equally likely to contact everyone else; no households, networks, age structure, or geography.
- No incubation/latent period: infections become infectious immediately (contrast with SEIR models).
- Constant rates: β and γ are treated as fixed over time; real epidemics have changing behavior, seasonality, interventions, and waning immunity.
- Deterministic dynamics: randomness is ignored; small outbreaks in small populations can differ substantially from deterministic predictions.
- Numerical approximation: Euler stepping can introduce error; very large Δt may produce unstable or unrealistic values.
- Not medical advice: this tool is for education and scenario exploration, not for clinical or public-health decision making.
Reference: Kermack, W.O. & McKendrick, A.G. (1927). “A Contribution to the Mathematical Theory of Epidemics.”