The SIR model

Concept

The SIR model

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Statement

Compartmental modelling of an epidemic.

Why it matters

basic-reproduction-number gives a single threshold number for whether an outbreak grows; the SIR model is the underlying dynamical system that number is extracted from, and it is the standard quantitative tool for turning qualitative statements about disease spread ("more contagious," "more people are susceptible") into an actual predicted epidemic curve — how fast cases rise, how high they peak, and how many people are ultimately infected before the outbreak dies out on its own.

Everything downstream in this unit, from herd-immunity thresholds to vaccination-strategy targets, is derived by analysing this same system of equations under different parameter choices or added compartments; the SIR model is the minimal version from which those extensions (transmission-dynamics, SEIR-type models) all build outward.

Hypotheses

The population is divided into three mutually exclusive compartments — Susceptible (\(S\)), Infected (\(I\)), Recovered/removed (\(R\)) — and every individual occupies exactly one compartment at any time.This deliberately collapses individual variation (age, contact pattern, immune status) into three aggregate counts; the model tracks compartment sizes only, not which specific individuals occupy them, trading individual-level realism for a small, analytically tractable system. The population is well-mixed: any infected individual is equally likely to contact any susceptible individual, and the contact rate \(\beta\) and recovery rate \(\gamma\) are constant parameters, the same for every individual in their compartment.Without well-mixing, the risk of infection would depend on each individual's specific contact network rather than only on the aggregate fractions \(S/N\) and \(I/N\), and the simple mass-action term \(\beta SI\) used in Step 1 below would not correctly describe the true contact-dependent transmission rate (transmission-dynamics relaxes this assumption explicitly). Recovery confers permanent immunity, and there are no births, deaths, or migration over the timescale considered (a closed population).This is why \(R\) is often read as "removed" rather than strictly "recovered" — it can equally represent immune recovery or disease-induced death, since both permanently exit the pool of transmission-relevant individuals in the same way. Diseases that do not confer lasting immunity, or epidemics slow enough that background births and deaths matter, require the SIS or SEIR-with-demography extensions rather than this closed-population SIR form.

Proof

\frac{dS}{dt} = -\beta\frac{SI}{N}

New infections occur at a rate proportional to the product of susceptible and infected counts (mass-action mixing, Hypotheses), scaled by the transmission rate \(\beta\) and normalised by total population \(N\) so the rate depends on the fraction infected, not the absolute count alone; susceptibles are depleted exactly as fast as new infections occur. A

\frac{dI}{dt} = \beta\frac{SI}{N} - \gamma I

The infected compartment gains exactly the new infections lost from \(S\) in Step 1, and loses individuals at rate \(\gamma I\) as they recover or are removed; \(\gamma\) is the reciprocal of the mean infectious period, so \(1/\gamma\) individuals remain infectious, on average, for \(1/\gamma\) units of time. A

\frac{dR}{dt} = \gamma I

Removed individuals accumulate at exactly the rate infected individuals leave the infectious compartment (Step 2); since \(S+I+R=N\) is conserved throughout (Hypotheses' closed population), this equation is not independent — it is fixed once Steps 1 and 2 are specified. A

\frac{dI}{dt} > 0 \iff \beta\frac{S}{N} > \gamma \iff \frac{S}{N} > \frac{1}{R_0}, \qquad R_0 \equiv \frac{\beta}{\gamma}

Dividing Step 2's condition for growth (\(\beta SI/N > \gamma I\)) by \(I\) and rearranging isolates the basic reproduction number \(R_0=\beta/\gamma\) (developed fully in basic-reproduction-number): the epidemic grows exactly when the susceptible fraction exceeds the threshold \(1/R_0\), and shrinks once it falls below it. B

\text{Since } S(t) \text{ is non-increasing (Step 1) and bounded below by } 0\text{, } I(t) \text{ must eventually decline to zero: every closed-population SIR epidemic terminates.}

\(S\) can only fall over time (new infections never create new susceptibles), so \(S/N\) crosses below the threshold \(1/R_0\) at latest when \(S\) reaches its floor; once it does, Step 4 guarantees \(dI/dt<0\), and \(I\) decays monotonically toward zero thereafter — the epidemic ends not because the disease "runs out" of virulence, but purely because it exhausts a sufficient supply of susceptibles. B

Result

\frac{dS}{dt}=-\beta\frac{SI}{N},\quad \frac{dI}{dt}=\beta\frac{SI}{N}-\gamma I,\quad \frac{dR}{dt}=\gamma I \ ;\quad \text{epidemic grows} \iff \frac{S}{N}>\frac1{R_0}

Reading. Three coupled equations, driven by only two parameters (\(\beta\), the transmission rate, and \(\gamma\), the recovery rate), reproduce the entire qualitative shape of an epidemic curve — initial exponential-like growth while susceptibles are abundant, a peak once the threshold in Step 4 is crossed, and an eventual decline as the susceptible pool is depleted.

Scope. Valid for a well-mixed, closed population with permanent immunity and constant \(\beta,\gamma\) (Hypotheses); real outbreaks with heterogeneous contact structure, waning immunity, or significant births/deaths during the epidemic require the extensions transmission-dynamics and herd-immunity build on this base.

Corollaries & converses

herd-immunity's threshold, \(S/N<1/R_0\), is exactly Step 4's growth condition read in reverse: immunising (moving individuals directly from \(S\) to \(R\)) enough of the population to push \(S/N\) below \(1/R_0\) prevents an epidemic from growing at all, without requiring every individual to be immune.
vaccination-strategy's population-level targets are a direct, practical corollary: since vaccination acts on the SIR system exactly like Step 3's removal mechanism (moving individuals to \(R\) without them passing through \(I\)), the same threshold condition determines the minimum vaccinated fraction needed to stop transmission.
Converse: observing that an outbreak has stopped growing, even while \(S\) remains well above zero, is direct evidence (via Step 4) that \(S/N\) has already crossed below \(1/R_0\) — herd immunity has been reached, even though a substantial susceptible population still remains.

Fails without

Drop the well-mixed assumption (Hypotheses): if contact is instead structured (households, workplaces, spatial clustering), the mass-action term \(\beta SI/N\) in Steps 1–2 systematically overstates transmission once local susceptible pools within a cluster are depleted even while the population-wide \(S/N\) remains high; real epidemics on structured contact networks generally grow more slowly and peak lower than the well-mixed SIR model predicts for the same \(\beta\) and \(\gamma\) (transmission-dynamics addresses this directly).
Drop permanent immunity (Hypotheses): if recovered individuals can become susceptible again (an SIS or SIRS system rather than SIR), Step 3's one-way flow into \(R\) no longer holds, \(S\) is no longer monotonically non-increasing, and Step 5's guarantee that every epidemic terminates on its own no longer applies — the disease can instead settle into a sustained endemic equilibrium rather than dying out.

Common errors

Assuming an epidemic only stops once the entire population has been infected or immunised; Step 4–5 show it stops as soon as \(S/N\) falls below \(1/R_0\), which for a typical \(R_0\) leaves a substantial fraction of the population never infected in that wave (Worked examples).
Confusing \(R_0=\beta/\gamma\) (Step 4), a constant property of the disease and setting at the outset, with the effective reproduction number at a later time, which depends on the shrinking susceptible fraction \(S/N\) and falls below \(R_0\) as the epidemic progresses.
Treating \(\beta\) as simply "how contagious the disease is" in isolation; \(\beta\) folds together both the pathogen's intrinsic transmissibility and the population's contact rate, so identical pathogens can have different effective \(\beta\) values in populations with different mixing patterns (transmission-dynamics).
Assuming the epidemic curve must be symmetric (decline mirrors rise); the SIR equations generally produce an asymmetric curve, with the decline phase shaped by both the shrinking \(S\) and the ongoing constant-rate removal \(\gamma I\), not by an inherent symmetry.

Discussion

The compartmental structure formalised in the Proof traces to the work of Ronald Ross (whose earlier mosquito-malaria models introduced compartmental thinking to epidemiology) and, more directly, to William Kermack and Anderson McKendrick's 1927 paper, which derived essentially the SIR system given here and proved the threshold behaviour of Step 4 — now often called the Kermack–McKendrick threshold theorem.

Although the SIR system has no general closed-form solution for \(S(t)\), \(I(t)\), \(R(t)\) as explicit functions of time, it is fully tractable numerically, and several exact implicit relationships can be derived, including a formula for the total epidemic size (the final fraction ever infected) purely in terms of \(R_0\), obtained by eliminating time between the \(dS\) and \(dR\) equations — a genuinely useful, non-obvious result given how simple the original three equations look.

Common misconception: that a disease with a higher \(R_0\) always causes more total deaths or a larger total epidemic. Total impact depends jointly on \(R_0\), the case-fatality rate, and the population's response (vaccination-strategy); a high-\(R_0\), low-severity disease and a low-\(R_0\), high-severity disease can pose very different public-health challenges despite the SIR framework's growth condition (Step 4) depending on \(R_0\) alone.

Worked examples

\text{Take } \beta = 0.5\ \text{day}^{-1},\ \gamma = 0.1\ \text{day}^{-1}, \text{ so } R_0 = \beta/\gamma = 5.

By Step 4, the epidemic grows as long as \(S/N > 1/R_0 = 0.2\); starting near \(S/N\approx1\) (almost everyone susceptible at the outset), the epidemic grows until the susceptible fraction is driven down to \(20\%\), at which point growth stops and \(I\) peaks. A

\text{Peak infection occurs exactly when } S/N = 1/R_0 = 0.2, \text{ i.e. when } 80\% \text{ of the population has already been infected or is currently infected.}

This follows directly from Step 4's threshold condition applied at the turning point \(dI/dt=0\); note that the epidemic does not stop growing only once everyone is infected — it stops once the remaining susceptible fraction alone falls to \(20\%\), well before exhausting the population. A

R_0=5 \ \Rightarrow\ \text{peak at } S/N=0.20 \ \Rightarrow\ \text{herd-immunity threshold} = 1-\tfrac1{R_0}=80\%

Reading. A disease with \(R_0=5\) requires roughly \(80\%\) of the population to be immune (through infection or vaccination) before transmission stops growing — the same \(1-1/R_0\) formula developed fully in herd-immunity, derived here directly from Step 4's threshold condition.

Scope. The identical calculation applies for any \(R_0\); larger \(R_0\) values require a correspondingly higher immune fraction before the threshold condition is satisfied.

Problems

A disease has \(\beta = 0.3\ \text{day}^{-1}\) and a mean infectious period of \(5\) days. Compute \(R_0\) and the herd-immunity threshold.
Solution
Mean infectious period \(=1/\gamma=5\) days, so \(\gamma=0.2\ \text{day}^{-1}\). \(R_0=\beta/\gamma=0.3/0.2=1.5\). Herd-immunity threshold \(=1-1/R_0=1-1/1.5\approx0.33\), so roughly \(33\%\) of the population needs to be immune before the epidemic stops growing.
Explain, using Step 1, why \(S(t)\) can never increase over the course of an SIR epidemic, and what this implies (via Step 5) about the long-run fate of \(I(t)\).
Solution
Step 1 gives \(dS/dt=-\beta SI/N\), which is always \(\le0\) since \(\beta,S,I,N\ge0\); \(S\) is therefore non-increasing throughout. By Step 5, since \(S\) can only fall (never regenerating new susceptibles, per the Hypotheses' permanent-immunity assumption), \(S/N\) must eventually cross below the threshold \(1/R_0\) of Step 4 and stay below it, forcing \(dI/dt<0\) from that point onward, so \(I(t)\to0\): every closed-population SIR epidemic dies out on its own, without needing external intervention, purely by exhausting susceptibles.
Two populations face the same pathogen (\(R_0=4\) in both), but Population B has already vaccinated \(60\%\) of its members before the outbreak begins (vaccinated individuals move directly into \(R\)). Determine whether the epidemic can grow in Population B, using Step 4.
Solution
With \(60\%\) already in \(R\), the initial susceptible fraction is \(S/N=0.4\). The herd-immunity threshold is \(1/R_0=1/4=0.25\), so growth requires \(S/N>0.25\). Since \(0.4>0.25\), Step 4's condition for growth is still satisfied, and the epidemic can still grow in Population B — \(60\%\) vaccination coverage is short of the \(1-1/R_0=75\%\) needed to prevent growth entirely for this \(R_0\).