Figure 1.
Critical Community Size of four immunizing childhood infections.
The x axis shows the size of unvaccinated proportion of the populations of each country (log scale), the y axis shows the proportion of years where incidence reported to the WHO is greater than zero (years range between 1998 and 2011 for rubella and mumps and 1980 and 2011 for pertussis and measles). Colours indicate island states (red); the size of points indicates the number of years for which there was data. Vertical lines show previously reported CCS values; higher values for measles refers to Niger [10], lower to America and the UK [3], [24]; for pertussis values refer to England and Wales [12]. The grey lines show a fitted linear regression, weighted to reflect sample size for each country (Table 1). The extreme positive outlier for measles (reflecting no years with more than one case at a relatively large population size) is the Democratic People's Republic of Korea. For all infections, islands with no years with no cases reported tend to be islands like the United Kingdom and New Zealand, likely to have highly effective surveillance systems.
Figure 2.
Estimates of the Critical Community Size.
a) Estimates of the distribution of the population size at which no years with no cases are expected based on linear regressions described in Table 1 and shown in Fig. 1 for the total unvaccinated population and for unvaccinated births; here encompassing parameter uncertainty; b) Violin plots showing the distribution of sizes of the unvaccinated populations for which no years with no cases were recorded for each of the infections (p0 = 0 indicates no zeros in the time-series, and therefore points along the y = 0 line in Fig. 1); vertical dotted lines indicated previous estimates of the CCS for each of the infections, see Fig. 1 for a description.
Table 1.
Weighted linear regression linking log size of the unvaccinated population and proportion of years for which no cases were reported; followed by the same but taking the size of the number of unvaccinated births as the covariate; corresponding estimates of the CCS, obtained as the point at which the fitted line intersects with zero are shown in Fig. 2.
Table 2.
Main effects for the model of the probability of extinction for each of the 4 infections identified using the ‘step’ function in R; and then eliminating variables not significant at the 0.05 level; standard errors shown in brackets; stars indicate significance with ° for p-values <0.1, * for p<0.05, ** for p<0.01, *** for p<0.001.
Figure 3.
Predicted probability of extinction.
The x axis is population size for four childhood infections and the y axis is probability of extinction for island nations (red) and mainland nations (black) and showing upper and lower standard errors, taken at the median log proportion of resident migrants (an index of connectivity of –3.52) and median human development index (0.67). Parameters underlying these predictions are shown in Table 1.
Figure 4.
Relative risks of extinction on an island.
The probability of extinction for island countries divided by the probability of extinction for non-island countries for 4 childhood infections relative to extinction on the mainland; showing only a fraction of the distribution for mumps for clarity. The proportion of migrants and HDI are set to the median across all countries (–3.52 and 0.67 respectively), and the population size of unvaccinated children is set to 1e5. Median values are 1.92 (1.04–3.37) for rubella, 1.51 (1.07–2.16) for measles, 10.26 (4.06–25.29) for mumps, and 2.27 (1.57,3.32) for pertussis; brackets indicate 2.5% and 97.5% quantiles from prediction made across 2000 samples from the estimated multivariate normal distribution of the parameters.