Epidemics and the SIR model

To understand how a disease moves through a group of people, you can sort everyone into three buckets. There are the people who can still catch it, the people who have it right now and can pass it on, and the people who have already had it and recovered, who cannot catch it again. Doctors call these three groups susceptible, infected, and recovered, which is where the name SIR comes from.

An outbreak is just people moving from the first bucket to the second, and then from the second to the third. Every day, some healthy people catch it from the sick ones, and some sick people get better. Whether the outbreak grows or shrinks depends on a tug of war between those two flows.

There is a single number that sums it up. It is called R nought, and it means: on average, how many other people does one sick person infect before they recover? If that number is above one, each case makes more than one new case, and the outbreak grows. If it is below one, the disease is fading with every step, and it dies out on its own. Getting R nought below one is the whole goal of fighting an epidemic.

Here is the part that surprises people. You do not have to make everyone immune to stop a disease. Once enough people are immune, a sick person keeps bumping into people who cannot catch it, and the chain of spread breaks even though plenty of people could still, in theory, get sick. That shield is called herd immunity, and it is why vaccinating most of a town protects the few who cannot be vaccinated.

The SIR model turns those three buckets into numbers that change over time: \(S\) susceptible, \(I\) infected, \(R\) recovered, adding up to the whole population \(N\). Two rules move people between them. Susceptible people become infected at a rate set by how often they meet infected people, and infected people recover at a steady rate. Writing \(\beta\) for the transmission rate and \(\gamma\) for the recovery rate, the flows are

\[ \frac{dS}{dt} = -\beta\,\frac{S I}{N}, \qquad \frac{dI}{dt} = \beta\,\frac{S I}{N} - \gamma I, \qquad \frac{dR}{dt} = \gamma I. \]

The basic reproduction number is the ratio of those two rates, \(R_0 = \beta / \gamma\). It is the average number of people one case infects when almost everyone is still susceptible, at the very start. But as the disease spreads, the susceptible pool shrinks, and what actually matters is the effective reproduction number \(R_{\text{eff}} = R_0 \cdot S/N\). The epidemic keeps growing only while \(R_{\text{eff}} > 1\).

That single condition explains the shape of every outbreak. Early on \(S \approx N\), so \(R_{\text{eff}} \approx R_0\) and, if that is above one, cases climb roughly exponentially. But each new infection removes someone from \(S\), so \(R_{\text{eff}}\) slides downward. The epidemic peaks at the exact moment \(R_{\text{eff}} = 1\), when the susceptible fraction has fallen to \(1/R_0\). After that the curve turns over and cases decline, not because the disease got weaker, but because it is running out of people to infect.

The same maths gives the herd immunity threshold. If a fraction \(1 - 1/R_0\) of the population is already immune, whether from past infection or vaccination, then \(R_{\text{eff}}\) starts below one and a new outbreak cannot even get going. For a disease with \(R_0 = 4\) that threshold is three quarters of the population; for measles, with \(R_0\) around 15, it is roughly 95 percent, which is why measles needs such high vaccination rates to hold back.

An outbreak also does not usually burn through everyone. It tends to overshoot the threshold and then stop, leaving a slice of the population untouched, and slowing transmission (flattening the curve) lowers and delays the peak so that hospitals are not overwhelmed all at once, buying time even when the eventual total is similar.

Where the model comes from. The equations are the Kermack and McKendrick model of 1927, the founding result of compartmental epidemiology. The mass-action term \(\beta S I / N\) assumes homogeneous mixing, that any infected person is equally likely to contact any susceptible one. It is a mean-field approximation, deliberately crude, and its power is that it turns an epidemic into two parameters with clear meaning: a contact-and-transmission rate \(\beta\) and a recovery rate \(\gamma\), whose reciprocal \(1/\gamma\) is the mean infectious period.

The threshold theorem. Kermack and McKendrick's central result is a threshold: an introduced infection grows if and only if \(R_0 = \beta/\gamma > 1\). Linearising \(dI/dt\) near the disease-free state gives an early growth rate \((\beta - \gamma)\), so cases rise exponentially when \(R_0 > 1\) and decay when \(R_0 < 1\). Because \(S\) only ever falls, the incidence \(dI/dt\) changes sign exactly once, at \(S/N = 1/R_0\); that is the unique peak.

The final size. Dividing the equations and integrating removes time and yields the final size relation for the fraction \(z\) ever infected, \(1 - z = e^{-R_0 z}\). It has a positive solution only when \(R_0 > 1\), and even for large \(R_0\) some susceptibles always escape, because transmission collapses once \(S\) drops below the threshold. The epidemic overshoots herd immunity: it does not stop when \(R_{\text{eff}}\) first hits one, it coasts past, so the attack rate exceeds \(1 - 1/R_0\).

Herd immunity and control. Immunising a fraction \(p > 1 - 1/R_0\) drives the initial effective reproduction number below one and prevents a major outbreak; this is the quantitative basis for vaccination targets. Note that the classic threshold assumes random mixing and a homogeneous population, so real thresholds shift with contact structure, age patterns and heterogeneity in susceptibility, often downward.

Beyond basic SIR. The framework extends readily. Adding an exposed-but-not-yet-infectious compartment gives SEIR; letting recovered individuals lose immunity gives SIRS, which can sustain endemic cycles; adding births and deaths lets a disease persist and recur. Stochastic versions (the Reed and Frost chain-binomial, branching-process approximations) capture what the deterministic model cannot, that an outbreak with \(R_0 > 1\) can still go extinct by chance while it is small, with extinction probability near \(1/R_0\). And on real contact networks the well-mixed assumption breaks, so network and metapopulation models refine, without overturning, the core insight: an epidemic is a threshold phenomenon governed by how many further cases each case produces. That logic connects directly to how immunity is built and remembered, the subject of the immune system, and to how pathogens evolve under pressure, as in antibiotic resistance.