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.