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Epidemiology: SIR Spread (R0) Calculator

Calculate the Basic Reproduction Number (R0) of a pathogen using the SIR model. Determine whether a disease will spread as an epidemic or die out.

Calculate the foundational Basic Reproduction Number (R₀) of a pathogen to mathematically diagnose whether a disease will explosively cascade or extinguish.

Infections / Day

The raw number of new bodies one patient infects over a 24-hour cycle.

Recovery Kinetics (Linked)

Fraction / Day
Days

These fields automatically synchronize via γ = 1 / D

Spread Trajectory Bounds

Basic Reproduction Number (R₀)

2.50
Expected Secondary Infections Per Patient
Calculated Population StatusEPIDEMIC SPREADING
The Sub-Zero Goal
0.0 (Safe)1.02.0+ (Danger)
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What is The SIR Model & Epidemic Thresholds?

The SIR (Susceptible-Infected-Recovered) model partitions a population into three compartments and tracks flow between them using differential equations. The single most important output is R0 ('R-naught'): if R0 exceeds 1.0, each infected person infects more than one other person on average, and the disease spreads exponentially. If R0 falls below 1.0, the chain of transmission breaks and the outbreak dies out. Public health interventions — masks, vaccines, quarantine — work by reducing beta (fewer contacts) or increasing gamma (faster recovery or isolation).

Mathematical Foundation

Basic Reproduction Number
R0=βγwhereγ=1DR_0 = \frac{\beta}{\gamma} \quad \text{where} \quad \gamma = \frac{1}{D}
R0R_0= Basic Reproduction Number — average number of secondary infections caused by one infected individual in a fully susceptible population.
β\beta= Effective contact rate — new infections produced per infected person per day.
γ\gamma= Recovery rate — fraction of the infected pool that recovers (or is removed) each day.
DD= Duration of infectiousness in days. If contagious for 5 days, gamma = 1/5 = 0.20/day.

Laws & Principles

  • The R0 = 1.0 Threshold: This is the critical dividing line. Above 1.0, case counts grow exponentially. Below 1.0, the outbreak decays. Every public health intervention aims to push the effective reproduction number below this boundary.
  • Herd Immunity Threshold: The fraction of the population that must be immune to stop transmission is 1 - (1/R0). For measles (R0 ~15), this means ~93% must be immune. For influenza (R0 ~1.5), only ~33% is needed.
  • Generation Time: R0 measures total secondary infections, but the generation time (how quickly those infections happen) determines outbreak speed. Two diseases with R0 = 3 can behave very differently if one has a 2-day generation time and the other has a 14-day generation time.

Step-by-Step Example Walkthrough

" A novel respiratory virus emerges. Patients are contagious for 10 days (D=10). Each infected person transmits to 0.3 new people per day (beta = 0.3). "

  1. 1. Calculate gamma from duration: 1/10 = 0.10 per day.
  2. 2. Calculate R0: beta / gamma = 0.3 / 0.1 = 3.0.
  3. 3. Since R0 = 3.0 > 1.0, the disease will spread exponentially.
  4. 4. Herd immunity threshold: 1 - (1/3.0) = 66.7% of the population must be immune to halt transmission.
Final Result: R0 = 3.0. Each patient infects 3 others on average. Without intervention, cases will triple each generation. Achieving 67% population immunity through vaccination would bring the effective R below 1.0 and stop the epidemic.

Quick Answer: What does R0 tell you?

R0 is the average number of people one infected person will infect in a fully susceptible population. If R0 is above 1.0, the disease spreads. Below 1.0, it dies out. Enter the contact rate and recovery rate above to calculate R0 instantly — or use the preset buttons for measles, COVID-19, and influenza.

The R0 Formula

R0 = β ÷ γ = β × D

Where β is the effective contact rate (infections/person/day), γ is the recovery rate (1/D), and D is the duration of infectiousness in days.

Historical Outbreak Scenarios

Measles in an Unvaccinated School

  1. Parameters: β = 1.8/day, D = 10 days.
  2. R0: 1.8 × 10 = 18. Each child infects 18 others.
  3. Herd immunity: 1 - 1/18 = 94.4%. Nearly everyone must be vaccinated to prevent outbreaks.
  4. Outcome: Without vaccination, measles will infect essentially 100% of susceptible contacts.

Seasonal Influenza

  1. Parameters: β = 0.6/day, D = 3 days.
  2. R0: 0.6 × 3 = 1.8. Each case generates ~2 secondary cases.
  3. Herd immunity: 1 - 1/1.8 = 44%. Achievable with moderate vaccination coverage.
  4. Outcome: Flu spreads but is controllable. Hand hygiene and vaccination together can push R below 1.0.

Known Pathogen R0 Values

Pathogen R0 Range Transmission Herd Immunity Threshold
Measles12 – 18Airborne92 – 95%
Chickenpox10 – 12Airborne + contact90 – 92%
SARS-CoV-2 (Delta)5 – 8Respiratory droplets80 – 88%
Influenza1.5 – 2.0Respiratory droplets33 – 50%
Ebola1.5 – 2.5Bodily fluids33 – 60%

Epidemiological Best Practices

Do This

  • ✓Distinguish R0 from Rt. R0 assumes a fully susceptible population. The effective reproduction number Rt accounts for current immunity levels. Use Rt for real-time outbreak tracking; use R0 for baseline characterization of a novel pathogen.
  • ✓Use serial interval data when available. The serial interval (time between successive cases) gives a more precise estimate of generation time than the full infectious period, especially for diseases with pre-symptomatic transmission.

Avoid This

  • ✗Don't treat R0 as a fixed constant. R0 varies by population density, behavior, climate, and healthcare infrastructure. The same pathogen can have R0 = 2 in a rural area and R0 = 6 in a crowded city.
  • ✗Don't confuse R0 with case fatality rate. A disease with R0 = 15 (measles) is not necessarily deadlier than one with R0 = 2 (Ebola). R0 measures transmissibility, not severity. Ebola is far more lethal despite spreading much less efficiently.

Frequently Asked Questions

What does R0 = 1.0 mean exactly?

R0 = 1.0 means each infected person infects exactly one other person on average. The number of active cases stays constant — it neither grows nor shrinks. This is the epidemic threshold: anything above 1.0 grows exponentially, anything below decays toward zero.

How do vaccines reduce R0?

Vaccines remove people from the susceptible pool. If 70% of a population is vaccinated against a disease with R0 = 3, the effective reproduction number drops to Rt = 3 × (1 - 0.70) = 0.9. Since 0.9 is below 1.0, the outbreak cannot sustain itself — this is herd immunity.

Why is measles R0 so much higher than flu?

Measles virus particles can remain suspended in air for up to 2 hours after an infected person leaves a room. It has both an extremely high contact rate (airborne transmission to anyone sharing the space) and a long infectious period (~10 days). Flu requires closer contact and has a shorter infectious window (~3 days).

What are the limitations of the SIR model?

The basic SIR model assumes a well-mixed population (everyone contacts everyone equally), permanent immunity after recovery, and no births or deaths during the epidemic. Real populations have heterogeneous contact networks, waning immunity, age structure, and geographic clustering — all of which require extended models (SEIR, SIRS, agent-based) for accurate forecasting.

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