Compartment Structure and ODEs
The SIR model assumes a closed population of constant size $N = S + I + R$ with homogeneous mixing (every individual equally likely to contact every other). The governing ODEs are:
\[\dot{S} = -\frac{\beta S I}{N}\] \[\dot{I} = \frac{\beta S I}{N} - \gamma I\] \[\dot{R} = \gamma I\]The term $\beta SI/N$ is the mass action incidence: $\beta$ is the transmission rate (contacts per unit time times probability of transmission per contact), and $I/N$ is the probability that a random contact is infectious. An alternative standard incidence formulation uses $\beta SI$ (density-dependent transmission, appropriate for non-mixing populations).
The parameter $\gamma$ is the recovery rate; the mean infectious period is $1/\gamma$. For COVID-19, $1/\gamma \approx 5$–10 days; for measles, $\approx 8$ days.
Since $\dot{S} + \dot{I} + \dot{R} = 0$, the total population is conserved. The system reduces to two independent equations (the equation for $R$ is redundant).
Basic Reproduction Number $R_0$
The basic reproduction number $R_0$ is the expected number of secondary infections caused by a single infectious individual introduced into a fully susceptible population:
\[R_0 = \frac{\beta}{\gamma}\]Epidemic growth or decline is governed by whether $R_0$ exceeds 1:
- $R_0 < 1$: $I(t)$ decreases monotonically; no epidemic.
- $R_0 = 1$: threshold case; epidemic stalls.
- $R_0 > 1$: epidemic grows initially (while $S/N \approx 1$).
The early exponential growth rate (Malthusian parameter) is:
\[r_0 = \beta - \gamma = \gamma(R_0 - 1)\]so $I(t) \approx I_0 e^{r_0 t}$ initially, with doubling time $T_d = \ln 2 / r_0$.
| Disease | $R_0$ | $1/\gamma$ (days) |
|---|---|---|
| Measles | 12–18 | 8 |
| Smallpox | 5–7 | 14 |
| Influenza (1918) | 2–3 | 4 |
| COVID-19 (original) | 2–3 | 5–10 |
| COVID-19 (Omicron) | 8–15 | 3–5 |
Herd Immunity and Final Size
An epidemic subsides not because the pathogen disappears, but because susceptibles become depleted. As $S$ declines, the effective reproduction number falls:
\[R_\text{eff}(t) = R_0 \cdot \frac{S(t)}{N}\]The epidemic peaks when $R_\text{eff} = 1$, i.e., when $S = N/R_0$. The fraction of the population that must be immune (either through infection or vaccination) to prevent epidemic spread is the herd immunity threshold:
\[p_c = 1 - \frac{1}{R_0}\]For measles with $R_0 = 15$, $p_c \approx 0.93$; vaccination coverage must exceed 93% to prevent outbreaks.
The final size of the epidemic (total fraction ever infected, $R_\infty/N$) satisfies the implicit equation obtained by integrating $\dot{R}/\dot{S} = -\gamma/\beta = -1/R_0$:
\[\ln\!\frac{S(0)}{S(\infty)} = R_0\!\left(1 - \frac{S(\infty)}{N}\right)\]For a fully susceptible initial population ($S(0) = N$), with $z = 1 - S(\infty)/N$ (final attack rate):
\[z = 1 - e^{-R_0 z}\]This transcendental equation has a non-trivial solution for $R_0 > 1$. For $R_0 = 2$, numerically $z \approx 0.797$, meaning ~80% of the population is eventually infected.
SEIR Extension: Latency Period
Many infections have a latent period $1/\sigma$ during which individuals are infected but not yet infectious. The SEIR model adds an Exposed compartment:
\[\dot{S} = -\frac{\beta SI}{N}\] \[\dot{E} = \frac{\beta SI}{N} - \sigma E\] \[\dot{I} = \sigma E - \gamma I\] \[\dot{R} = \gamma I\]The basic reproduction number remains $R_0 = \beta/\gamma$ (the latent period delays but does not alter ultimate spread). The initial growth rate satisfies the quadratic:
\[r_0^2 + (\sigma + \gamma)r_0 - \sigma\gamma(R_0 - 1) = 0\]giving $r_0 = \frac{1}{2}\left[-(\sigma+\gamma) + \sqrt{(\sigma+\gamma)^2 + 4\sigma\gamma(R_0-1)}\right]$, which is smaller than the SIR growth rate for the same $R_0$, consistent with the flattening effect of latency.
For COVID-19, $1/\sigma \approx 4$–5 days (incubation) and $1/\gamma \approx 5$–8 days (infectious period).
Stochastic SIR and Branching Process Approximation
The deterministic SIR ignores demographic randomness. In the stochastic SIR (continuous-time Markov chain), events are:
- Infection: $S \to S-1, I \to I+1$ at rate $\beta SI/N$
- Recovery: $I \to I-1, R \to R+1$ at rate $\gamma I$
For large $N$, the stochastic model approximates the ODE. But for small $I$ (near outbreak start or end), stochasticity matters crucially: the probability of extinction before takeoff is significant even when $R_0 > 1$.
When $I \ll N$ (early outbreak), $S \approx N$ and infections approximately follow a Galton-Watson branching process. Each infectious individual gives rise to a Poisson($R_0$) new cases. The probability of extinction $q$ satisfies:
\[q = G(q) = e^{R_0(q-1)}\]For $R_0 \leq 1$, $q = 1$ (certain extinction). For $R_0 > 1$, extinction probability is $q < 1$, the smaller root of $q = e^{R_0(q-1)}$. For $R_0 = 2$, $q \approx 0.203$: about 20% of introductions self-extinguish stochastically.
The critical slowing down near $R_0 = 1$ makes the effective $R_0$ hard to estimate from early case counts. Confidence intervals around $R_0$ estimates are typically estimated via maximum likelihood from the exponential growth phase or via Bayesian inference using generation interval distributions.
COVID-19 Application
During the COVID-19 pandemic, the SEIR framework was extended to include:
- Asymptomatic infectious compartments ($I_A$) with reduced $\beta$
- Hospitalization and ICU compartments for healthcare capacity planning
- Age-stratified contact matrices ${c_{ij}}$ from POLYMOD surveys
- Vaccination compartments with waning immunity
The effective reproduction number was estimated in near-real-time using the renewal equation:
\[R_t = \frac{I_t}{\sum_s I_{t-s} w_s}\]where $w_s$ is the generation interval distribution (discretized). Methods such as EpiEstim (Cori et al. 2013) use a Bayesian gamma prior on $R_t$ and update with Poisson-distributed incidence data.
SEIR fits to UK data (March–April 2020) estimated $R_0 \approx 2.4$–3.1, falling to $R_t < 1$ within 2 weeks of the national lockdown (Ferguson et al. 2020, Imperial Report 9).