Seasonality and the persistence and invasion of measles

Abstract

The critical community size (CCS) for measles, which separates persistent from extinction-prone populations, is arguably the best understood stochastic threshold in ecology. Using simple models, we explore a relatively neglected relationship of how the CCS scales with birth rate. A predominantly positive relationship of persistence with birth rate is complicated by the accompanying dynamical transitions of the underlying deterministic process. We show that these transitions imply a lower CCS for high birth rate less developed countries and contrary to the experience in lower birth rate, industrial countries, the CCS may increase after vaccination. We also consider the evolutionary implications of the CCS for the origin of measles; this analysis explores how the deterministic and stochastic thresholds for invasion and persistence set limits on the mechanism by which this highly infectious pathogen could have successfully colonized its human host.

Figure 1

Changing patterns of persistence with birth rate. (a) Annual fade-outs against population size: calculated from 50 replicates of a 24-year time-series of the stochastic SEIR model for three representative birth rates. Black, red and green curves correspond to a birth rate of 20, 30 and 40 per thousand, respectively. Error bars indicate standard errors (calculated using the variance between the replicates). (b) Mean annual fade-outs for population size against birth rate (50 replicates, 24-year time-series). The England and Wales CCS (500 000) is marked with a cross, with bands of uncertainty due to clumping of the population sizes in the data. Black shades represent no fade-outs over 50 replicates. Grey shades represent up to 0.3 annual fade-outs in 50 replicates (below which fade-out would be unlikely in a short time-series). The expected CCS for England and Wales after vaccination is indicated by a circle. (c) The average power of the largest population size (3 million) around the annual (black), biennial (red) and triennial (green) modes of the Fourier spectrum against birth rate. (d) Trough and peak size estimated from case distribution, plotted against birth rate. The trough size at the England and Wales fit (cross) and after vaccination (circle) at 70% are indicated.

Figure 2

Annual, biennial and triennial dynamics and the definition of the trough and peak size. (a–c) Ten-year time-series of case reports from one replicate of the stochastic model illustrating the triennial, biennial and annual epidemics experienced as the birth rate is increased (N=3 million, B=8, 20 and 42 per thousand). (d–f) The corresponding case distributions for triennial, biennial and annual epidemics. Averaged over 50 replicates of the stochastic model, this distribution gives the probability of a given value of x(t)=log(1+cases(t)) occurring in a given week. The tails of the distribution used to define the peak and trough size are indicated.

Figure 3

Expected dynamical transitions and implications for the impact of vaccination on persistence from high birth rate countries. (a) Trough depth (tail of case distribution) for a large population N=3 million at a fixed birth rate (20 per thousand) for a range of transmission rates (β≈R₀/5.0 contacts per day) and seasonal amplitudes (α). The equivalent range of birth rates for a fixed R₀=20 is given on the second right-hand axis (per thousand population). (b) Complementing (a), we plot the dominant mode of the Fourier spectrum against transmission rate (β) and seasonal amplitude of variation (α). Blue, predominantly annual epidemics; red, biennial; green, triennial. The three topologically different paths following vaccination from a community experiencing annual dynamics at high birth rate (b=40 per thousand) are indicated. (i) R₀≈12, (ii) R₀≈6 and (iii) R₀≈20. The path corresponding to the predicted impact of vaccination on England and Wales (b=20 per thousand, R₀=20) is also marked as cross to circle.

Figure 4

Persistence during invasion: interplay of transmission. and birth rates. (a) The first invasion of a well-mixed population results in inevitable extinction due to exhaustion of susceptibles after a time T_E and first possible re-invasion a time T_I later. (b) The dependence of T_E, the time to the first extinction, on transmission and birth rates, defined as the first time when (I≤1, E≤1). (c) The dependence of T_I, the time to first possible re-invasion, on transmission and birth rates, defined as the first time after T_E that (dI/dt)>0