What SIR represents
The model divides a closed population into susceptible (S), infected and infectious (I), and recovered with permanent immunity (R). The flow is S → I → R, preserving S + I + R = N. Initial conditions are I₀ infected individuals, S₀ = N − I₀, and R(0) = 0.
Initial N and I₀ are integer counts; compartment dynamics are a continuous approximation and may have fractional values. I₀ = 0 produces a scenario without infection: the model introduces no spontaneous cases.
This family of models originates in Kermack and McKendrick (1927). It describes infection dynamics under defined assumptions; parameters for a specific population require estimation from data.
Transmission and recovery
β and γ have units day⁻¹. β combines contacts and transmission probability; it is a rate, not a daily probability, and can exceed 1 day⁻¹. γ is the recovery rate; for γ > 0, the mean infectious period is 1/γ.
A constant γ implies an exponential distribution of time spent in I: 1/γ is a mean, not an identical recovery time for every individual. With γ = 0, the model becomes the SI limit and recovery does not occur. With β = γ = 0, all compartments stay constant and β/γ is undefined.
Basic and effective reproduction numbers
With γ > 0, R₀ is the average number of secondary infections per infected individual in a fully susceptible population. R₀ > 1 means more than one infection per infected individual on average under that condition; it does not specify how many infections each individual causes. With I > 0, growth in the current scenario depends on Re(t): I grows when Re > 1 and declines when Re < 1. R₀ > 1 does not guarantee growth if few susceptible individuals remain.
If Re crosses 1 from above, the maximum of I occurs at S* = N/R₀. The dotted chart line marks that susceptible count, not an immune population. If the scenario starts with Re(0) ≤ 1, the maximum is at t = 0.
Re = 1 means an instantaneous zero slope of I; S and R can continue to change, so this does not imply an endemic equilibrium. These relationships are described by Morris and colleagues (2021).
Herd immunity threshold (HIT)
The herd immunity threshold is the theoretical immune percentage that prevents initial infection growth under homogeneous mixing and perfect protective immunity. For R₀ > 1, that coverage reduces Re to 1; for R₀ ≤ 1, the threshold is zero. It does not predict how many people will recover or the coverage required with an imperfect vaccine.
Incidence does not vanish when Re crosses 1 during an outbreak: infected individuals still transmit. The cumulative infection percentage can exceed HIT while I declines.
During an outbreak, N − S includes infected and recovered individuals. The cumulative infection percentage therefore differs from the immune percentage. The scenario starts without initial immunity or vaccination; HIT is a theoretical reference. R₀ = ∞ and HIT = 100 % with β > 0 and γ = 0 are formal limits, not a recovery scenario.
Prevalence, incidence and cumulative infections
I(t) is the number of infected individuals at an instant; prevalence as a proportion is I(t)/N. Incidence c(t) = βS(t)I(t)/N is a rate of new infections per day. Its maximum may precede the maximum of I because recovery also affects the infected population.
The incidence chart represents that instantaneous rate. New cases between two times are obtained by integrating c(t) over that interval. Cumulative infections up to the horizon are N − S(tMax) = I(tMax) + R(tMax), including I₀.
Continuous time, horizon and decline
SIR time is continuous: peaks and threshold crossings may occur between whole days. For example, 166.8 days equals 166 days, 19 hours and 12 minutes. The incidence unit “per day” does not turn time into discrete daily intervals.
The horizon bounds the observed interval. If a curve is still growing at the end, the observed maximum refers to that interval and the peak has not yet been reached. A maximum may also occur at the start or be constant throughout the interval.
The first decline below I = 1 is reported as a crossing time even when the outbreak decreases from the start. In continuous SIR, I generally approaches zero without reaching it in finite time: fewer than one expected individual is a convention, not evidence of real stochastic extinction.
Decimal results do not describe uncertainty in the rates, differences between populations or individual variability.
Examples and model limitations
The examples use illustrative constant rates. The references provide context for R₀; they do not establish a calibration of the displayed β and γ pairs. The COVID-19 example refers to early estimates from 2020. R₀ estimates depend on population, variant, setting and method; they are not universal constants for each disease.
- Homogeneous mixing: no age structure, geography or contact networks.
- Immediate infectiousness: no latent stage before I. SEIR includes an exposed compartment before the infectious one; the latent period and clinical incubation are distinct concepts.
- Permanent immunity: no reinfection or waning immunity.
- Constant rates and population: no varying interventions, vaccination, births, migration or mortality.
- Deterministic dynamics: no random fluctuations, parameter uncertainty or stochastic extinction, effects that matter when few infected individuals remain.