Pareto rules for malaria super-spreaders and super-spreading

Abstract

Heterogeneity in transmission is a challenge for infectious disease dynamics and control. An 80-20 "Pareto" rule has been proposed to describe this heterogeneity whereby 80% of transmission is accounted for by 20% of individuals, herein called super-spreaders. It is unclear, however, whether super-spreading can be attributed to certain individuals or whether it is an unpredictable and unavoidable feature of epidemics. Here, we investigate heterogeneous malaria transmission at three sites in Uganda and find that super-spreading is negatively correlated with overall malaria transmission intensity. Mosquito biting among humans is 90-10 at the lowest transmission intensities declining to less than 70-30 at the highest intensities. For super-spreaders, biting ranges from 70-30 down to 60-40. The difference, approximately half the total variance, is due to environmental stochasticity. Super-spreading is thus partly due to super-spreaders, but modest gains are expected from targeting super-spreaders.

Fig. 1

Mosquito counts and the modeled daily expectation. In all plots, the y-axis shows the square root of the observations. The total anopheline mosquito counts and daily expectation S_d for a Tororo, b Kanungu, and c Jinja. The number of sporozoite-positive mosquitoes for d Tororo, e Kanungu, and f Jinja

Fig. 2

Biting weights and results of the variance components analysis. a Distribution of biting weights (i.e., the points are the fitted values ω_h,n from the fitting procedure) and a Gamma distribution fitted to describe the points using MLE (solid lines, plotted to the 99th quantile) for Jinja, Gamma(1.03, 1.03); Kanungu, Gamma(0.71, 0.71); Tororo, Gamma(2.41, 2.41). b Proportion of variance explained by biting weights (ω), seasonality in the HBR (S), the estimated sampling variance (the smallest sliver), and environmental stochasticity and measurement errors (ε)

Fig. 3

The Pareto analysis and fractions by site for super-spreaders or super-spreading for all (HBR) or sporozoite-positive (EIR) anopheline mosquitoes. The data are sorted naturally (for super-spreading) or by the biting weight (for super-spreaders). The point where the CDF crosses the line 1−X is the Pareto fraction. The gray lines in the background show the analysis for each month. The colored line shows the CDF for all the data for each site. For emphasis, the Pareto fraction for each month was also plotted in color. Note that for some months, super-spreaders have a Pareto fraction that is less than 50-50; the houses that tend to get the most bites need not account for half the bites in any particular month. The analyses are shown for Tororo (a–d); Kanungu (e–h); and Jinja (i–l) for super-spreading (a, c, e, g, i, k) and for super-spreaders (b, d, f, h, j, l); for all anophelines (a, b, e, f, i, j), and for sporozoite-positive anophelines (c, d, g, h, k, l)

Fig. 4

The counts by household and summary of the Pareto analysis for anopheline mosquito counts. Top row shows analysis for all anophelines, and the bottom for sporozoite-positive anophelines. a Anopheline mosquito catch counts by month and household (each household is on one line), sorted within each site by the median counts for each household. The study ended after 42 months for Jinja and Kanungu; 69 months of data from Tororo are presented here. Darker colors indicate higher counts. b Monthly Pareto fractions (e.g., 0.9 is 90-10) for super-spreading (circles) and super-spreader (x’s) by mean monthly HBR. Linear fit lines for super-spreading (solid) and super-spreaders (dashed) are shown with confidence intervals in gray. c The Pareto index plotted vs. the logged mean monthly HBR for both super-spreading (circles) and super-spreaders (x’s). The dashed lines are values of the Pareto index that give the 80-20, 70-30, and 55-45 distributions. The range is restricted to 1–5 (outliers are identifiable in panel b). d Sporozoite-positive anopheline mosquito catch counts by month and household (each household is on one line), sorted within each site by the median counts for each household. Darker colors indicate higher counts. e Monthly Pareto fractions for super-spreading (circles) and super-spreaders (x’s) by mean monthly EIR. Linear fit line for super-spreading (line) and super-spreaders (dashed) are shown with confidence intervals in gray. f The Pareto index plotted vs the logged mean monthly EIR for both super-spreading (circles) and super-spreaders (x’s). As in (c), the range is restricted to 1–5 (outliers are identifiable in panel e)