Infectious Disease Dynamics

The reproduction number was being used to make decisions before anyone had written it down carefully. George Macdonald did for malaria in 1952 what Ross had begun, with field data in hand, and extracted a conclusion that reorganised a global programme. The reproduction number for a vector-borne infection depends on the biting rate squared — the mosquito must bite once to acquire the parasite and once to transmit it — and on mosquito survival raised to a high power, because the parasite needs about ten days of development inside the insect before it can be passed on. An intervention that shortens adult mosquito life therefore does far more than one that reduces larval numbers. Indoor residual spraying follows directly from the exponent.

The smallpox campaign supplied the other great practical result. The WHO's stated plan was to vaccinate 80% of the population of every endemic country, and in much of west Africa that was not achievable. William Foege, short of vaccine in eastern Nigeria, instead found each case and vaccinated the people around it, and the people around them. Surveillance and containment worked because smallpox has a visible rash, no symptomless carriers, and relatively slow spread — so cases could be found faster than they transmitted. The last natural case was in 1977, and the disease is the only human infection eradicated. The method does not transfer: polio, with mostly symptomless infection, has resisted the same approach for three decades.

Roy Anderson and Robert May, arriving from ecology rather than medicine, completed the unification in 1979. Hosts are a resource that infection depletes and birth replenishes; immunity is depletion; virulence is a trait under selection. The framework explained the two-year cycles of measles in pre-vaccination cities as the time taken for births to restock the susceptible pool, and it yielded a warning that has repeatedly proved right: a vaccination programme that reduces transmission without reaching the threshold raises the average age at infection, and for rubella, chickenpox and polio, infection at an older age is more dangerous.

Write $S$, $I$ and $R$ for the fractions susceptible, infectious and recovered, $\beta$ for the transmission rate and $\gamma$ for the recovery rate, so that $R_0 = \beta/\gamma$. The equations are

$$\frac{dS}{dt} = -\beta S I, \qquad \frac{dI}{dt} = \beta S I - \gamma I, \qquad \frac{dR}{dt} = \gamma I.$$

Infections grow while $dI/dt > 0$, which requires $\beta S > \gamma$, that is

$$S > \frac{\gamma}{\beta} = \frac{1}{R_0}.$$

So prevalence peaks exactly when the susceptible fraction falls to $1/R_0$, and the immune fraction at that moment is the herd-immunity threshold

$$H = 1 - \frac{1}{R_0}.$$

At the peak, however, a large number of people are currently infectious, and each of them still infects someone — just fewer than one person each on average. Those infections continue, so the epidemic overshoots. Dividing the first equation by the third and integrating gives the final size relation for the fraction $z$ ever infected:

$$1 - z = e^{-R_0 z}.$$

Solving numerically:

$R_0$	Herd-immunity threshold $1 - 1/R_0$	Final size $z$
1.5	33%	58%
2.5	60%	89%
5	80%	99.3%
15 (measles)	93%	>99.99%

For an infection with $R_0 = 2.5$, letting the epidemic run infects 89% of the population, while reaching the same immunity by vaccination requires 60%. The gap — 29 percentage points of a population — is the overshoot, and it is the entire quantitative case for not relying on infection to produce herd immunity.

Two cautions belong with the table. First, $R_0$ is an average, and the variance matters enormously. If a minority of cases cause most transmission — measured by the dispersion parameter $k$, estimated near 0.1 for SARS-CoV-2 and for SARS before it — then most introductions fizzle out while occasional events infect dozens. With $R_0 = 2.5$ and $k = 0.1$, roughly 10 to 20% of cases account for 80% of transmission. The same mean produces a different epidemic: more failed chains, faster explosions when a chain catches, and interventions aimed at crowded indoor gatherings doing far more than their share.

Second, the final-size formula assumes a homogeneously mixing population with no behaviour change. Real populations are neither, which lowers the effective overshoot and is why observed attack rates in a first wave are usually well below the table's numbers. The structure of the argument survives: the threshold is where transmission turns over, not where the epidemic stops.

The last turning point in this thread is not a result but a change in role. In 2001, during Britain's foot-and-mouth epidemic, models fitted to the first weeks' data were used to argue for pre-emptive slaughter on farms adjoining infected ones, and about six million animals were killed. In 2020, projections of hospital demand were central to decisions that closed schools and borders across dozens of countries.

Both episodes exposed the same mismatch. The models produce conditional statements — if contact rates stay at this level, the peak falls here — and they are received as predictions. Their parameters are weakly identified from the available data, as the open problem above describes, so the range of defensible outputs is wide, and which end of that range reaches a minister depends on presentation rather than on epidemiology. The models were not wrong to be used; nothing else answered the question being asked. What was missing, and is still largely missing, is any settled practice for auditing a model that is about to justify a decision of that size — the kind of scrutiny that statistical inference applies to a published estimate, applied instead to a projection under pressure of days.