Questions: Mathematical Models of Disease Transmission

Question 1

In an SIR model, the epidemic peak (maximum number of infectious individuals) occurs at which condition?

• A. When the entire susceptible population has been infected
• B. When the exposed (E) compartment reaches its maximum
• C. When the fraction of susceptibles S/N equals 1/R₀, so that each case infects exactly one other
• D. When the recovery rate γ equals the transmission rate β

Answer: (C)

The infectious compartment grows when dI/dt = βSI/N − γI > 0, i.e., when βS/N > γ, or S/N > γ/β = 1/R₀. The peak occurs when dI/dt = 0, which is exactly when S/N = 1/R₀. At that point, each infectious individual infects exactly one other on average — the turning point of the epidemic. After this, S has been depleted below the threshold and the epidemic declines. Option D confuses the condition γ = β (which would mean R₀ = 1, the epidemic threshold) with the peak condition.

Question 2

An SIR model has R₀ = 3. A student argues: 'Since R₀ > 1, the epidemic will keep growing until everyone is infected.' What does the model actually predict?

• A. The student is correct — R₀ > 1 guarantees the entire population will eventually be infected
• B. The epidemic grows initially but peaks and declines once S/N falls to 1/R₀ ≈ 33%, leaving roughly 67% susceptible uninfected
• C. The epidemic peaks when S/N = 33% but then oscillates indefinitely without ending
• D. The epidemic declines immediately because the herd immunity threshold has already been reached

Answer: (B)

R₀ > 1 means the epidemic grows initially, but not indefinitely. As infections spread, the susceptible pool S is depleted. When S/N falls to 1/R₀ = 1/3, each case generates exactly one new case — the peak. After the peak, S continues to fall below 1/R₀ and the epidemic declines. The final attack rate (fraction ever infected) is less than 1 — for R₀ = 3, roughly 94% will be infected before the epidemic ends, but this comes from solving the final size equation, not from 'everyone gets it.' The student's intuition confuses 'epidemic grows when R₀ > 1' with 'everyone gets infected.'

Question 3

In an SIR model, an epidemic can peak and decline without any external intervention, even when R₀ > 1.

• T. True
• F. False

Answer: True

The epidemic's own dynamics create the decline: as infections accumulate, susceptibles are converted to recovered individuals, depleting the pool available for new transmission. Once S/N < 1/R₀, the effective reproduction number Rₜ = R₀ × (S/N) falls below 1, and the infectious class begins to shrink. No intervention is needed — the epidemic is self-limiting. This is why all historical epidemics eventually ended even without vaccines or effective treatments.

Question 4

Adding an exposed (E) compartment to create an SEIR model increases the epidemic peak size compared to the equivalent SIR model, because more individuals are 'loaded' in the pre-infectious stage before the peak.

• T. True
• F. False

Answer: False

The E compartment (latent period) actually delays and flattens the epidemic curve — it reduces the peak height and shifts it later in time. Individuals in E are infected but not yet infectious, so they slow the rate at which the infectious class I grows. The total attack rate (final epidemic size) is similar, but the peak is lower and later. This is why diseases with longer incubation periods (like COVID-19) tend to produce slower-building, more prolonged outbreaks than diseases with very short latent periods.

Question 5

Explain why the transmission term βSI/N in the SIR model causes epidemics to be self-limiting, even without any interventions.

Model answer: The term βSI/N is the rate of new infections. It is proportional to S (the number of susceptibles). As the epidemic spreads, individuals move from S to I to R, permanently depleting S. As S shrinks, the product βSI/N decreases — fewer susceptibles means fewer new infections per unit time, even if the number infectious I is still large. Eventually S/N falls below 1/R₀, meaning each infectious case generates fewer than one new case, and the epidemic declines. The epidemic consumes the very fuel (susceptibles) that sustains it.

This self-limiting property is one of the most important insights from compartmental models. It explains why epidemic curves are bell-shaped rather than growing without bound. It also gives rise to the herd immunity threshold: if enough of the population is immune before an outbreak, S/N is already below 1/R₀ at the start, and the epidemic cannot grow at all.