Epidemics and the SIR Model: Threshold, Peak and Extinction

1. Introduction

In January 1918, an unusual influenza begins circulating in American military barracks. Within months, it spreads to every continent and kills between 50 and 100 million people — more than World War I. In 2020, a new coronavirus paralyzes the global economy within weeks. These events raise a fundamental question: can we predict the trajectory of an epidemic?

The partial answer is yes — provided one understands the assumptions and limitations of the models. The SIR model, proposed by Kermack and McKendrick in 1927, is the starting point of modern mathematical epidemiology. Simple in its formulation, it captures the essential mechanisms of infectious disease spread and allows key quantities such as the basic reproduction number R₀ to be defined.

2. The S, I, R Compartments

The SIR model divides the population into three mutually exclusive compartments. S denotes susceptible individuals: they have not yet been infected and can be. I denotes infectious individuals: they carry the pathogen and can transmit it. R denotes removed individuals: they have been infected and are now immune (or deceased, in simplified versions). The total population N = S + I + R is assumed constant in the basic version.

These compartments are abstractions. In reality, the boundary between susceptible and infectious is not sharp: there is an incubation period during which the individual is infected but not yet contagious. The boundary between infectious and recovered depends on the duration of the infectious period, which varies between individuals. These simplifications are deliberate and must be made explicit.

3. The SIR Model Equations

The SIR model is described by three coupled ordinary differential equations. The first describes the decrease in susceptibles: dS/dt = −βSI/N. The second describes the dynamics of infectious individuals: dI/dt = βSI/N − γI. The third describes the accumulation of recovered individuals: dR/dt = γI.

The parameter β is the transmission rate: it represents the average number of effective contacts per unit time between an infectious individual and susceptibles. The parameter γ is the removal rate: its inverse 1/γ is the average duration of the infectious period. These two parameters summarize all the disease biology in the basic model.

4. R₀, Epidemic Threshold and Herd Immunity

The basic reproduction number R₀ is defined as the average number of secondary cases produced by one infectious individual in a fully susceptible population. In the SIR model, R₀ = β/γ. If R₀ < 1, each infectious individual produces on average less than one secondary case: the epidemic dies out. If R₀ > 1, the epidemic can grow. This threshold is the central quantity of mathematical epidemiology.

The effective reproduction number Rₜ is R₀ multiplied by the fraction of the population still susceptible: Rₜ = R₀ × S/N. At the start of the epidemic, S ≈ N and Rₜ ≈ R₀. As the epidemic progresses, S decreases and Rₜ declines. When Rₜ = 1, the number of infectious individuals is at its maximum. When Rₜ < 1, the epidemic declines.

Herd immunity in the SIR model framework is reached when the fraction of the immunized population exceeds 1 − 1/R₀. For a disease with R₀ = 4, the threshold is 1 − 1/4 = 75%. For measles itself (R₀ between 12 and 18 in unvaccinated populations), the threshold is about 92–95%. These values assume homogeneous immunity in the population, which is rarely the case.

5. Time Curves and Phase Space

The time curves of the SIR model show three characteristic phases. In the growth phase, I increases exponentially as long as S is close to N. In the peak phase, I reaches its maximum when Rₜ = 1, that is, when S = N/R₀. In the decline phase, I decreases because the fraction of susceptibles is too low to sustain transmission. The epidemic dies out before all susceptibles are infected: a fraction of the population escapes infection even without prior immunity.

The phase space of the SIR model is two-dimensional if we use S and I (R being determined by R = N − S − I). The trajectory starts near (N, 0) — almost the entire population is susceptible, very few infectious — and evolves toward a point on the I = 0 axis — the epidemic is over. This trajectory is a monotone curve in the (S, I) plane: I increases then decreases while S decreases continuously.

6. Spiral in Phase Space: When and Why

The trajectory of the classic SIR model in the (S, I) plane is not a spiral. It is a curve that rises, reaches a maximum and falls, without ever returning toward its starting point. There is no oscillation around an interior equilibrium. The final equilibrium is on the I = 0 axis, which is not an interior fixed point but a line of equilibria.

Spiral trajectories can appear in extensions of the SIR model that introduce additional mechanisms. The SIRS model (with waning immunity) can produce damped oscillations around an endemic equilibrium if parameters are in a certain range. The model with demographic turnover (births and deaths) can also produce oscillations. Seasonality of the transmission rate can force annual oscillations.

Models with delays — for example, a delay between infection and contagiousness — can produce sustained oscillations or chaotic behavior. Contact-network-structured models, where individuals do not mix homogeneously, can produce complex dynamics with successive waves. In all these cases, the spiral in phase space is a property of the extended model, not of the basic SIR model.

7. Extensions: SEIR, SIRS, Structured Models

The SEIR model adds an E compartment for exposed individuals: infected but not yet contagious. This compartment captures the incubation period, which is important for diseases like COVID-19 (incubation 2–14 days) or Ebola (2–21 days). Adding E slows the initial growth of the epidemic and changes the shape of the time curves.

The SIRS model replaces permanent immunity with temporary immunity: recovered individuals return to the S compartment after some time. This model can produce recurrent epidemics and, under certain conditions, endemic oscillations. It is relevant for diseases like seasonal influenza, where immunity wanes and new variants emerge.

Age-structured models divide the population into age groups with different contact rates. Contact-network-structured models explicitly represent interactions between individuals. These approaches are more realistic but also more data- and computation-intensive. They are used for major pandemics where contact heterogeneities are important.

8. Real Data and Model Limitations

Fitting the SIR model to real data is difficult for several reasons. Reported cases underestimate actual infections: under-reporting is systematic and varies across surveillance systems. Delays between infection, symptoms, testing and reporting blur the time curves. Mortality data are more reliable than case data, but they reflect infection with an additional delay.

Behaviors change in response to the epidemic. When people become aware of the risk, they reduce contacts, decreasing β. When epidemic pressure decreases, behaviors return to normal. This behavioral feedback is not captured by the basic SIR model and can produce successive waves that the model does not predict.

The homogeneous population assumption is particularly problematic. Human contacts are highly heterogeneous: some individuals have far more contacts than others (super-spreaders). Family, occupational and geographic structures create transmission clusters. These heterogeneities can reduce the effective herd immunity threshold below the prediction of the homogeneous model.

9. Vaccination and Control

Vaccination reduces the number of susceptibles. In the SIR model, vaccinating a fraction p of the population is equivalent to moving these individuals directly into the R compartment. The effective reproduction number becomes Rₜ = R₀(1 − p). For the epidemic to be unable to start, we need Rₜ < 1, i.e., p > 1 − 1/R₀. This is the herd immunity condition.

Non-pharmaceutical interventions — physical distancing, mask wearing, school closures, quarantine — act primarily by reducing β. Their effectiveness depends on implementation and population adherence. More complex models can estimate the impact of each measure separately, but interactions between measures and behavioral substitution effects complicate interpretation.

The SIR model illustrates a general principle: epidemics are collective phenomena governed by thresholds. Below the threshold R₀ = 1, an epidemic cannot establish itself. Above it, it can grow. Vaccination and control measures act by lowering Rₜ below this threshold. This principle is robust even if the quantitative details depend on the chosen model.