Compartmental epidemic models divide a population into groups such as susceptible, exposed, infectious, and recovered. This cheat sheet helps students translate disease assumptions into differential equations and interpret the resulting dynamics. It is useful for analyzing outbreaks, vaccination strategies, quarantine, and public health interventions.
The goal is to connect model structure, parameters, and real-world meaning.
Key Facts
- In the basic SIR model, dS/dt = -beta S I / N, dI/dt = beta S I / N - gamma I, and dR/dt = gamma I.
- The total population is conserved in a closed SIR model because dS/dt + dI/dt + dR/dt = 0, so S + I + R = N.
- For the SIR model, the basic reproduction number is R0 = beta / gamma when nearly everyone is susceptible.
- An epidemic grows initially when R0 S(0) / N > 1 and declines when R0 S(t) / N < 1.
- The effective reproduction number is Re(t) = R0 S(t) / N in a homogeneous SIR model.
- In the SEIR model, dE/dt = beta S I / N - sigma E and dI/dt = sigma E - gamma I, where 1/sigma is the mean latent period.
- The herd immunity threshold in the ideal SIR model is p_c = 1 - 1/R0, assuming perfect vaccination and homogeneous mixing.
- At the infection peak in the SIR model, dI/dt = 0, so S = N/R0 if I is positive.
Vocabulary
- Compartment
- A compartment is a population group in a model, such as susceptible, exposed, infectious, or recovered.
- Transmission rate
- The transmission rate beta measures how efficiently infectious and susceptible individuals generate new infections.
- Recovery rate
- The recovery rate gamma is the per-person rate at which infectious individuals leave the infectious compartment.
- Basic reproduction number
- The basic reproduction number R0 is the expected number of secondary infections caused by one infectious person in a fully susceptible population.
- Effective reproduction number
- The effective reproduction number Re is the average number of secondary infections at a given time, accounting for current susceptibility and interventions.
- Equilibrium
- An equilibrium is a state where all compartment derivatives are zero, so the model does not change over time.
Common Mistakes to Avoid
- Using beta as the probability of infection per day without checking the model form is wrong because beta may combine contact rate and transmission probability.
- Forgetting to divide by N in mass-action incidence beta S I / N changes the units and makes the force of infection scale incorrectly with population size.
- Assuming R0 is always the same as Re is wrong because Re changes as susceptibility, behavior, vaccination, or interventions change.
- Interpreting the latent period as the infectious period in an SEIR model is wrong because exposed individuals are infected but not yet infectious in the basic SEIR structure.
- Treating model predictions as exact forecasts is misleading because parameter uncertainty, reporting bias, and simplifying assumptions can strongly affect outcomes.
Practice Questions
- 1 In an SIR model with beta = 0.36 per day and gamma = 0.12 per day, compute R0 and decide whether infections initially grow when S(0) is approximately N.
- 2 A disease has R0 = 2.5 in a homogeneous SIR model. Compute the ideal herd immunity threshold p_c = 1 - 1/R0.
- 3 In an SEIR model, sigma = 0.2 per day and gamma = 0.1 per day. Find the mean latent period and the mean infectious period.
- 4 Explain how reducing contacts, vaccinating susceptible individuals, and increasing isolation of infectious individuals would change beta, S, or gamma in a compartmental model.
Understanding Compartmental Models for Epidemics
A compartment is not a physical place. It is a label for a disease state. People move between labels at average rates, so the model describes expected behavior in a large population rather than the exact path of each person.
The transmission rate combines contact frequency with the chance that a contact passes on infection. Its value depends on the time unit.
A recovery rate of one fifth per day means the average infectious period is five days. These rates must be interpreted carefully because a fitted number can hide several real processes, including testing, isolation, behavior, and access to care.
The infection curve changes because the susceptible group is depleted. Early growth can look close to exponential when almost everyone can be infected. Later, each infectious person meets fewer susceptible people on average.
The peak of infectious cases is therefore a turning point, not the end of an outbreak. Cases can still be common after the peak, and the recovered total can keep rising for a long time. A useful habit is to distinguish incidence from prevalence.
Incidence means new infections over a period. Prevalence means the number currently infectious. Daily reported cases often resemble incidence, while hospital occupancy is closer to prevalence.
Adding an exposed compartment creates a delay between infection and infectiousness. This delay can shift the visible case peak well after a change in contact patterns. It does not automatically change the number of contacts needed to sustain spread, but it changes timing.
That timing matters when comparing a model with case reports, admissions, or deaths. Each data source has its own delay and its own missing cases. Parameter estimation means choosing values that make model output resemble data, but many different parameter combinations can fit one short time series.
For example, faster transmission with earlier detection may resemble slower transmission with later detection. Students should check which parameters are identifiable from the available measurements instead of treating every fitted value as certain.
Equilibria describe states where the compartment totals no longer change. In a simple closed model without births, an infection free equilibrium has no infectious people left. Whether a small introduction grows near that state depends on the effective reproduction number.
Stability is the mathematical version of this idea. Small disturbances fade in a stable state and grow in an unstable one. Intervention models work by changing a process in the equations.
Vaccination lowers the number susceptible. Isolation shortens the time a person can transmit. Masks and reduced contacts lower transmission.
Real populations are not perfectly mixed, so age, location, work patterns, and unequal access to care can alter results. A simple model is valuable for reasoning, but its assumptions should always be stated before using its predictions for policy.