R0 and herd immunity

The reproduction number in R: why an epidemic needs R0 above one, and why immunising a fraction one minus one over R0 of the population stops it spreading.

The SIR model said an epidemic needs the basic reproduction number above one. Turn that around and it becomes the most useful idea in disease management: if you can drag the effective reproduction number below one, by removing enough susceptibles before the pathogen arrives, the outbreak cannot sustain itself. That is herd immunity, and the fraction you have to protect follows directly from the reproduction number.

The effective reproduction number

The basic reproduction number counts secondary cases when everyone is susceptible. But a pathogen invading a population where some hosts are already immune meets fewer susceptibles, so each case infects proportionally fewer others. The effective reproduction number is the basic one scaled by the susceptible fraction; immunise a fraction of the population and you cut it by that fraction.

R0 <- 4
Reff <- function(v) R0*(1 - v)      # v = fraction immunised (or already immune)
c(none = Reff(0), quarter = Reff(0.25), half = Reff(0.5), three_quarters = Reff(0.75))

          none        quarter           half three_quarters 
             4              3              2              1 

With a basic reproduction number of 4, an untouched population has each case seeding four more. Immunise half and each seeds two; the epidemic still grows, just slower. Immunise three quarters and each case seeds exactly one: the pathogen treads water, neither growing nor shrinking. Push past that and every case seeds fewer than one, the chain of transmission breaks, and the outbreak dies even though most individuals were never touched. That tipping fraction is the herd immunity threshold.

v_crit <- function(R0) 1 - 1/R0      # herd immunity threshold
c(R0_2 = v_crit(2), R0_4 = v_crit(4), R0_10 = v_crit(10))

 R0_2  R0_4 R0_10 
 0.50  0.75  0.90 

The threshold is one minus one over the reproduction number. For a pathogen with a reproduction number of 4 it is 0.75: protect three quarters of the population and the remaining quarter is shielded too, because the pathogen can no longer find enough hosts to keep going. The more transmissible the pathogen, the higher the bar: a reproduction number of ten demands ninety per cent coverage.

R0g <- seq(1, 12, 0.1)
dg <- data.frame(R0 = R0g, v = v_crit(R0g))
ggplot(dg, aes(R0, v)) +
  geom_line(colour = col_a, linewidth = 1) +
  geom_point(data = data.frame(R0 = 4, v = herd), aes(R0, v), colour = col_b, size = 2.6) +
  labs(x = "Basic reproduction number", y = "Fraction that must be immune",
       title = "More transmissible pathogens need more immunity")

Figure 1: The herd immunity threshold against the basic reproduction number. The fraction that must be immune to stop spread rises steeply for low reproduction numbers and approaches the whole population as transmissibility climbs.

Where the threshold comes from

The threshold is exactly the point where the effective reproduction number crosses one. Immunising a fraction leaves the reproduction number at the basic value times the susceptible fraction, and setting that equal to one and solving gives the herd immunity threshold. Plot the effective reproduction number against coverage and it is a straight line from the basic value down to zero, crossing one at the threshold.

vg <- seq(0, 1, 0.01)
dr <- do.call(rbind, lapply(c(2, 4, 10), function(r)
  data.frame(v = vg, Reff = r*(1 - vg), R0 = paste0("R0 = ", r))))
ggplot(dr, aes(v, Reff, colour = R0)) +
  geom_hline(yintercept = 1, colour = col_ref, linetype = "dotted") +
  geom_line(linewidth = 1) +
  coord_cartesian(ylim = c(0, 10)) +
  scale_colour_manual(values = c("R0 = 2" = col_a, "R0 = 4" = col_b, "R0 = 10" = col_ref)) +
  labs(x = "Fraction immunised", y = "Effective reproduction number", colour = NULL,
       title = "Immunity below the threshold only slows the epidemic")

Figure 2: The effective reproduction number against the fraction immunised, for three pathogens. Each line falls from its basic value to zero; where it crosses one (dotted) is the herd immunity threshold, further right for more transmissible pathogens.

Why this matters beyond vaccination

Herd immunity is usually discussed for vaccines, but the same threshold governs any disease in any population, which is why it belongs to ecology and not just medicine. In wildlife the immune fraction can come from prior infection sweeping through, and the threshold predicts when a pathogen can re-invade a recovering population: as births and deaths refill the susceptible pool, the effective reproduction number climbs back toward the basic value and the next outbreak becomes possible. That refilling is what turns a one-off epidemic into recurrent or endemic disease. The threshold also underpins culling and control decisions, though only when the reproduction number is measured correctly, which is harder than it looks and is the checking post. And the whole logic, an invader that needs each unit to more than replace itself, is the same threshold that governs whether a species persists in a patchy landscape.

Related tutorials

• The SIR epidemic model
• Host density and endemic disease
• Checking an epidemic model
• Extinction risk and population viability

References

• Anderson RM, May RM 1985. Nature 318(6044):323-329 (10.1038/318323a0).
• Fine PEM 1993. Epidemiologic Reviews 15(2):265-302 (10.1093/oxfordjournals.epirev.a036121).