SIR Model — Recovered
Rate at which infected individuals recover and gain immunity.
Formula
Variables
Example
γ = 0.1, I = 0.1 → dR/dt = 0.01
How it works
This equation says that the recovered group grows in direct proportion to how many people are currently infected and how quickly each infected person leaves infection. In the standard SIR model, recovery is modeled as a constant per-capita rate γ, so each infected individual has the same average chance per unit time of recovering, making the total recovery flow equal to γI. The result increases the recovered fraction R while simultaneously reducing the infected fraction, which is why this term is the bookkeeping link between infection dynamics and immunity.
When to use it
Use this term when modeling infectious diseases in which recovery confers immunity, such as in basic epidemic models for measles-like or COVID-style compartmental analysis. It is especially useful when you want to estimate how quickly the immune/recovered pool accumulates from a known infected fraction and recovery rate.
Worked example, step by step
Given: γ = 0.25 per day, I = 0.16 Step 1: Substitute into dR/dt = γI Step 2: dR/dt = 0.25 × 0.16 Step 3: dR/dt = 0.04 per day Answer: The recovered fraction is increasing at 0.04 of the population per day.
Common mistakes to avoid
- Using the total population instead of the infected fraction I
- Treating γ as a percentage without converting it to a rate per unit time
- Forgetting that this gives a rate of change of R, not the amount recovered over an entire outbreak
Did You Know?
In the SIR model, S + I + R = 1 always — every person is in exactly one compartment at any moment.
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More in Epidemiology & Public Health
View allIncidence Rate
BasicRate at which new cases of a disease appear in a population over a given period.
Prevalence
BasicProportion of a population that has a particular condition at a point in time.
Basic Reproduction Number
IntermediateAverage number of secondary infections produced by one infected individual in a fully susceptible population.
Herd-Immunity Threshold
IntermediateMinimum fraction of the population that must be immune to prevent sustained spread.
Effective Reproduction Number
IntermediateActual average infections per case when part of the population is already immune.
SIR Model — Susceptible
AdvancedRate of change of the susceptible population in the SIR compartmental model.