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Compartmental models (epidemiology) - mklabgr - 09-01-2026
Compartmental Models in Epidemiology — Summary
Compartmental models are mathematical tools used to describe how an infectious disease spreads through a population by dividing individuals into different states, or compartments. The most famous example is the SIR model, where $S(t)$ represents susceptible individuals, $I(t)$ infectious individuals, and $R(t)$ individuals who have been removed from transmission through recovery, immunity, or death. The basic transition is
$S \rightarrow I \rightarrow R$.
The classical SIR model is described by the system of differential equations
$\frac{dS}{dt}=-\frac{\beta SI}{N}$,
$\frac{dI}{dt}=\frac{\beta SI}{N}-\gamma I$,
$\frac{dR}{dt}=\gamma I$,
where $\beta$ represents the transmission rate, $\gamma$ represents the recovery or removal rate, and the total population is
$N=S+I+R$.
One of the most important quantities in epidemic modelling is the basic reproduction number
$R_0=\frac{\beta}{\gamma}$.
The quantity $R_0$ represents the expected number of secondary infections caused by a single infected individual in a completely susceptible population. In general, if $R_0>1$, an epidemic can grow, while if $R_0<1$, the number of infections tends to decline.
The basic SIR model can be extended to represent more realistic diseases. The SEIR model, for example, introduces an additional exposed compartment $E$, producing the transition
$S \rightarrow E \rightarrow I \rightarrow R$.
The compartment $E$ represents individuals who have been infected but are not yet infectious. This is useful for diseases that have a significant latent or incubation period.
Another important model is the SIS model
$S \rightarrow I \rightarrow S$,
which is appropriate for diseases where recovery does not produce lasting immunity. Other extensions include the SIRD model, where deaths are represented explicitly, as well as models incorporating vaccination, temporary immunity, births, deaths, age structure, spatial movement, social networks, and changing contact rates.
Compartmental models may be either deterministic or stochastic. Deterministic models are usually expressed using systems of ordinary differential equations, while stochastic models treat infection, recovery, and other transitions as probabilistic events.
A central quantity during an ongoing epidemic is the effective reproduction number, usually written as $R_t$ or $R_e$. Unlike $R_0$, which assumes a completely susceptible population, $R_t$ changes over time as immunity develops, vaccination increases, or human behaviour changes. When $R_t>1$, infections tend to increase, whereas when $R_t<1$, infections tend to decrease.
Compartmental models are widely used to estimate epidemic peaks, study the possible final size of an outbreak, calculate vaccination thresholds, and investigate the effects of interventions such as isolation, vaccination, social distancing, or reductions in contact rates. They therefore provide an important connection between differential equations, probability, dynamical systems, statistics, and public-health decision making.
Key Takeaways
- The main idea is to divide a population into epidemiological compartments.
- The classical model is $S \rightarrow I \rightarrow R$.
- The fundamental reproduction number is $R_0=\frac{\beta}{\gamma}$.
- If $R_0>1$, an epidemic can initially grow; if $R_0<1$, transmission tends to decline.
- The SEIR model uses $S \rightarrow E \rightarrow I \rightarrow R$ to include a latent period.
- The SIS model uses $S \rightarrow I \rightarrow S$ when immunity is temporary or absent.
- More advanced models can incorporate vaccination, mortality, age structure, networks, mobility, and changing behaviour.
- Mathematically, compartmental epidemiology relies heavily on differential equations, dynamical systems, probability, numerical methods, and parameter estimation.
ARTICLE