Characterizing the impact of spatial clustering of susceptibility for measles elimination

Abstract

Measles elimination efforts are primarily focused on achieving and maintaining national vaccination coverage goals, based on estimates of the critical vaccination threshold (V_c): the proportion of the population that must be immune to prevent sustained epidemics. Traditionally, V_c estimates assume evenly mixing populations, an invalid assumption. If susceptible individuals preferentially contact one another, communities may remain vulnerable to epidemics even when vaccination coverage targets are met at the national level. Here we present a simple method to estimate V_c and the effective reproductive number, R, while accounting for spatial clustering of susceptibility. For measles, assuming R₀ = 15 and 95% population immunity, adjustment for high clustering of susceptibility increases R from 0.75 to 1.29, V_c from 93% to 96%, and outbreak probability after a single introduction from <1% to 23%. The impact of clustering remains minimal until vaccination coverage nears elimination levels. We illustrate our approach using Demographic and Health Survey data from Tanzania and show how non-vaccination clustering potentially contributed to continued endemic transmission of measles virus during the last two decades. Our approach demonstrates why high national vaccination coverage sometimes fails to achieve measles elimination, and that a shift from national to subnational focus is needed as countries approach elimination.

Keywords: Clustering; Critical immunity threshold; Critical vaccination threshold; Disease transmission; Effective reproductive number; Heterogeneity; Measles; Outbreak risk; Stochastic individual-based model; Vaccination.

Fig. 1

Synthetic populations and corresponding τ(r) functions with no, low, medium, and high spatial clustering of susceptibility. Individuals are spatially distributed evenly across the space, and susceptibility increasingly clustered (grey = immune, red = susceptible).

Fig. 2

The association between $R_0$, $V_c^{\ast }$, and spatial clustering of non-vaccination ($\varphi$). As clustering increases, the required vaccination coverage to maintain $R$ = 1 increases. At low $R_0$, this increase due to clustering is much greater than at high $R_0$. (A) The association between $R_0$ and $V_c^{\ast }$ at the four defined levels of clustering, using the three $g\left(r\right)$ distributions. The relative increase in $V_c$ with increased clustering is relatively equivalent for $g_A(r)$ and $g_B(r)$, but lower for $g_C(r)$. (B) The relationship between $V_c^{\ast }$, $\varphi$, and $R_0$, where $\varphi$ represents the relative impact of spatial clustering on $R^{\ast }$. This relationship is highlighted for measles ($R_0$ = 15), mumps ($R_0$ = 7.7), and polio and rubella ($R_0$ = 6) , , , . For diseases like polio, with lower $R_0$, there is a substantially greater increase in absolute $V_c^{\ast }$ with increasing $\varphi$, as compared to diseases with higher $R_0$.

Fig. 3

Critical vaccination threshold, as defined by the overlapping densities of the $\tau \left(r\right)$ and $g\left(r\right)$ distributions. Plotted curves are $\tau \left(r\right)$ where $r$ is that which corresponds to the value of $1-G\left(r\right)$, and color corresponds to the resulting $V_c^{\ast }$ when $R_0$ = 15. Three levels of clustering ($\theta$ = 0.25, 0.50, 1.0) and 529 $g\left(r\right)$ distributions, with both $\alpha$ and $\beta$ range = 0.1–10.

Fig. 4

Probability of an outbreak and the outbreak probability ratio given adjustment for clustering ($\varphi$) compared with homogeneous ($\varphi$ = 1), for $R_0$ = 6 and $R_0$ = 15. The probability ratio compares the clustering-adjusted estimates of outbreak probability for each level of clustering to the homogeneous estimates at each level of successful vaccination coverage. The probability of an outbreak decreases with increasing successful vaccination proportion (v) and increases with increasing clustering of susceptibility ($\varphi )$. Until vaccination becomes high enough, the contribution of clustering is negligible and public health focus should be on increasing vaccination coverage, as highlighted by the areas in the red boxes. As successful vaccination coverage approaches elimination levels (estimated at 83% for $R_0$ = 6 and 93% for $R_0$ = 15 with the traditional critical immunization threshold equation) the relative effect of clustering on the outbreak probability increases exponentially. When vaccination is sufficiently high, prioritization should shift from increasing national vaccination coverage to the clustering of susceptibility (blue box).

Fig. 5

(A) Vaccination coverage among children aged 1–5 years by district and survey sample in Tanzania during the 1999 and 2010 Demographics and Health Surveys. (B) Empirical τ(r) functions and fitted exponential for Tanzania and Zambia, using DHS cluster data.