Measles transmission during a large outbreak in California

Abstract

A large measles outbreak in 2014-2015, linked to Disneyland theme parks, attracted international attention, and led to changes in California vaccine policy. We use dates of symptom onset and known epidemic links for California cases in this outbreak to estimate time-varying transmission in the outbreak, and to estimate generation membership of cases probabilistically. We find that transmission declined significantly during the course of the outbreak (p = 0.012), despite also finding that estimates of transmission rate by day or by generation can overestimate temporal decline. We additionally find that the outbreak size and duration alone are sufficient in this case to distinguish temporal decline from time-invariant transmission (p = 0.014). As use of a single large outbreak can lead to underestimates of immunity, however, we urge caution in interpretation of quantities estimated from this outbreak alone. Further research is needed to distinguish causes of temporal decline in transmission rates.

Keywords: Immunization; MMR; Subexponential transmission; Vaccination; Wallinga-Teunis estimation.

Fig. 1.

Serial intervals of reported Disney transmission links (gray bars) overlaid with estimated serial interval distribution (black curve).

Fig. 2.

Transmission network of 2014–2015 outbreak indicating onset date, age, and vaccination status for each case and known transmission links.

Fig. 3.

Estimated reproduction number R for each case by date and aver-aged over two-week periods, taking into account reported index cases and transmission links. Cases

Fig. 4.

Estimated number R for each case, by age and vaccination status. Estimates of R by case are shown in black, and box-and-whisker plot indicates median and quartiles.

Fig. 5.

Estimated generation membership of cases by day. Generation membership is estimated using the Wallinga-Teunis method, including information about known transmission links (see text for details). At each day, the estimated number of cases belonging to each generation is shown as a colored bar.

Fig. 6.

Estimated reproduction number R by generation and estimated probability of generation membership per case. Mean estimated R by generation is plotted as orange bars, and probability of membership of each case in each generation is plotted as dots shaded from light blue to black. Points of probability zero are omitted. Note that there are relatively few points of probability near ½. Note also the similarity between estimated R by generation and estimated R by two-week period (Fig. 3). Because of this similarity, this paper focuses on estimation of R by generation, while estimation of R by date is a potential subject of future research. (Fig. 7).

Fig. 7.

Distribution of estimates of rate of decline of R by generation in simulations with constant R, against true R (top to bottom) and k (left to right). Rates of quenching of transmission $\widehat{\tau }$ obtained by fitting function R = R_ie^−τm to Wallinga-Teunis estimated R by generation m in simulated outbreaks, conditioned on outbreak size of 131 cases. Horizontal gray line shows the true value τ = 0, and horizontal colored line shows quenching rate $\widehat{\tau }$ estimated for Disney outbreak, illustrating the significance test reported in the text. Boxes mark median and interquartile range of estimates. The k = 0.1 case includes outliers $\widehat{\tau }$ not shown.

Fig. 8.

Size and duration in branching processes with constant R and with quenched R, compared to Disney outbreak. (Top left) Likelihood of outbreak size within 5 cases and duration within 5 days of Disney measurements in simulation with constant R; (Top right) Distribution of size and duration given maximum likelihood constant R, constant R fit to Disney outbreak duration only, and constant R fit to Disney outbreak size only (black dot is size and duration of Disney outbreak); (Middle left) Distribution of outbreak duration given three constant R estimates (vertical line is duration of Disney outbreak); (Middle right) Distribution of outbreak size given three constant R estimates (vertical line is size of Disney outbreak); (Bottom left) Likelihood surface for R_i and τ parameters describing quenched and unquenched R; (Bottom right) Distribution of size and duration given maximum likelihood quenched R and quenched R sequence estimated by Wallinga-Teunis method (black dot is size and duration of Disney outbreak). Simulated with k = 0.40.