Questions: SIR Compartmental Models for Infectious Disease

Question 1

An epidemic with R₀ = 4 is spreading. After the peak, daily cases are declining — but 40% of the population is still susceptible. A student claims: 'The epidemic is declining because the virus is running out of people to infect.' What does the SIR model actually say?

• A. The student is correct — case declines always indicate that susceptibles are nearly exhausted
• B. Cases are declining because the susceptible fraction has dropped below 1/R₀ = 25%, so each case now generates fewer than one new case on average — not because susceptibles are exhausted
• C. The epidemic is declining because the virus has mutated to a less transmissible variant
• D. The model predicts cases cannot decline while 40% of the population remains susceptible

Answer: (B)

Post-peak decline is driven by the epidemic threshold, not by exhaustion of susceptibles. When S/N falls below 1/R₀, the effective reproduction number Rₑ = R₀ × S/N drops below 1, so each infected person generates fewer than one new case and incidence falls. With R₀ = 4, the threshold is S/N = 0.25 — the epidemic peaks and begins declining when 75% are immune, even though 25% (not 0%) remain susceptible. A substantial susceptible fraction always survives uninfected.

Question 2

In the SIR model, what does the herd immunity threshold represent?

• A. The fraction of the population that must be vaccinated to achieve zero new infections
• B. The minimum immune fraction (1 − 1/R₀) at which each infectious case generates on average fewer than one new case, causing incidence to decline
• C. The fraction of the population that will ultimately be infected before the epidemic ends
• D. The susceptible fraction below which the pathogen cannot survive at all

Answer: (B)

The herd immunity threshold is 1 − 1/R₀ — the immune fraction at which the effective reproduction number Rₑ falls below 1. It does not require zero susceptibles; it requires enough immune individuals that transmission chains shrink on average. Note that it is not the same as the final attack rate (how many are ultimately infected), which is determined by the final size equation and is always larger than the herd immunity threshold for R₀ > 1.

Question 3

In the SIR model, the epidemic reaches its peak number of infectious individuals at exactly the moment the susceptible fraction crosses the herd immunity threshold (S/N = 1/R₀).

• T. True
• F. False

Answer: True

The peak of I occurs when dI/dt = 0, which requires β(I/N)S − γI = 0, simplifying to S/N = γ/β = 1/R₀. At this exact moment — when the susceptible fraction first equals 1/R₀ — new infections precisely balance recoveries. After this point, susceptibles continue to be depleted, Rₑ stays below 1, and I declines. This is also the instant the herd immunity threshold is first crossed.

Question 4

In an SIR epidemic, the epidemic ends mainly when essentially most susceptibles have been infected — the final uninfected individuals are those who happened to avoid contact with any infectious person by chance.

• T. True
• F. False

Answer: False

The SIR model shows that a predictable fraction of susceptibles always escapes infection, determined by the final size equation: ln(S∞/S₀) = −R₀(1 − S∞/N). The epidemic self-extinguishes before reaching all susceptibles because the susceptible pool is depleted enough that Rₑ < 1, and the epidemic shrinks to zero before infecting everyone. The survivors are 'saved' by the depletion dynamics of the epidemic, not by chance avoidance — and the exact fraction is a deterministic function of R₀ alone.

Question 5

Why does an SIR epidemic continue to decline in cases even when a substantial fraction of the population remains susceptible? Explain the mechanism.

Model answer: The epidemic declines whenever the effective reproduction number Rₑ = R₀ × S/N falls below 1 — meaning each infectious case generates fewer than one new case on average. This happens when the susceptible fraction S/N drops below 1/R₀ (the herd immunity threshold), even if many susceptibles remain. The mechanism is depletion: as infected individuals recover and acquire immunity, the density of susceptibles in the population decreases, reducing the force of infection β(I/N). New infections occur more slowly than recoveries, so I shrinks. The epidemic ends not from exhaustion of all susceptibles, but because the susceptible pool has been depleted enough to make sustained transmission impossible.

Students often confuse 'epidemic ending' with 'all susceptibles infected.' The SIR model clarifies that the epidemic is self-limiting through a threshold effect — depletion drives Rₑ below 1 before susceptibles are exhausted.