Forecasted Size of Measles Outbreaks Associated With Vaccination Exemptions for Schoolchildren

Key Points Question What is the expected size of measles outbreaks in Texas at current (ie, 2018) and decreased vaccination rates? Findings In this decision analytical model study of metropolitan statistical areas in Texas, 1 in 20 measles introductions was forecasted to be associated with outbreaks of more than 400 cases in the Austin and Dallas metropolitan areas, according to 1000 simulated outbreaks in each metropolitan area; 64% of cases were forecasted to occur in children for whom a vaccine has been refused and 36% in others (ie, bystanders). Increased vaccine exemptions were forecasted to be associated with increases in the expected size of measles outbreaks. Meaning The 2018 vaccination rates in multiple metropolitan areas may permit large measles outbreaks, which could infect not only vaccine refusers but also other members of the population.


eAppendix 1. Measles Model
The basic reproduction number, R0, is the expected number of new infections caused by a single infectious person in a completely susceptible population. It is reported as 12-18 for measles [5,6]. FRED simulations are calibrated to produce a specified R0 by adjusting the probability that an interaction between an infectious agent and a susceptible agent results in the susceptible agent becoming infected.
Iterating over different values of this transmissibility parameter gave a best fit of 13.3 (12.0-14.7), from 100 simulations of a completely susceptible Austin-Round Rock MSA population. In each calibration simulation, 100 school students were randomly selected to seed an outbreak, and the average number of secondary infections caused by these 100 agents used to calculate the R0 for the transmissibility parameter value. 100 initial cases were used to ensure a representative distribution in the number of secondary infections, however this may have resulted in a slight decrease in the measured value of R0 if agents were competing for the same secondary infections. Public school districts more reliably reported their vaccination rates than individual private schools; as each public school district represents multiple schools, the overall percentage of unmatched schools was only 5.1%. Additional factors include that a minority of schools providing their address as a Post Office (PO) box, rather than their street address, when supplying vaccination data. Furthermore, some schools may have opened, closed or moved address from when the school data used in the synthetic population was collected and when the vaccination data was collected.
In some instances, this made it unfeasible to match vaccination data to a synthetic school.
It was necessary to infer the vaccination rates of schools without reported vaccination data. The probability density function of the vaccination rates of the schools in each county was estimated using kernel density estimation with machine learning on the reported vaccination rates. A grid search leave-one-out cross-validation was used to optimize the bandwidth of each kernel density estimation. This was performed with scikit-learn's GridSearchCV algorithm [3]. A vaccination rate for each unmatched school was then drawn from the probability distribution of that school's county and type (i.e. public or private). With the combined matched and inferred data, the simulations used an overall student vaccination rate of 98.9%; this is consistent with the overall reported vaccination rate of seventh graders in Texas in 2017-18 [1].
It is recommended that at least four data samples be used when performing a kernel density estimation [4]. If at least four vaccination rates in a county for a public or private school were not available, reported vaccination rates from both types of schools were used for kernel density estimation. If the number of reported vaccination rates available was still fewer than four, a kernel density estimation was not performed. Instead a reported vaccination rate for a school in the county was randomly drawn and used instead. eFigure 1 shows a sample distribution of school vaccination rates in Lubbock County, Texas. The reported vaccination rate of the schools in the county are shown, alongside the estimated vaccination rates of four schools. The corresponding probability density function was generated using a kernel density estimation on the reported data.

eFigure 1. Probability Density Function (Blue Line) of Vaccination Rates of Schools in Lubbock County, Texas
Gray bars indicate the reported vaccination rate of schools in the county which are matched to schools in the FRED simulation. The red bars indicate the estimated vaccination rate of four schools, which were unmatched to reported vaccination rates, drawn from the probability density function.
Measles was modeled with several disease stages. An agent exposed to measles took 11.5 ± 1.23 days (median and dispersion of a lognormal distribution) to enter the 'latent' stage. After one day in the latent state, agents spent three days in the 'fever' stage, before entering the 'rash' stage for four days. A recovery period of 10 days followed, before the agent acquired immunity [7]. Agents were infectious during the latent, fever and rash stages. Agents had a 50% chance of household confinement during the fever stage and 95% during the rash stage.
The measles outbreaks were simulated for 270 simulation days. This approximates the length of the school year. The mixing behavior of children may differ during the school summer vacation.

eAppendix 3. Vaccination Data
Texas records the two-dose vaccination rate of MMR, at kindergarten and grade seven. The recorded kindergarten vaccination rates are typically lower than the seventh grade rates. This is due to provisional enrollments and delinquencies (data is collected early in the school year) as well as increased personal exemption rates among students in kindergarten [1]. A proportion of provisionally enrolled or delinquent kindergarten students may have received a single MMR dose (which is recommended to be administered at 12-15 months [8]) but were late in receiving their second dose (which is recommended to be administered at 4-6 years, before school entry [8]) rather than having parents who intend to forgo vaccination altogether. Delays in receiving a second MMR dose may be due to accidental tardiness, or intentionally delaying vaccination doses beyond the recommended times [9]. As a single MMR dose confers immunity to measles in 93% of recipients [8], these students are at less risk of contracting measles than students who have received no MMR doses.
For the purposes of this study it is assumed that the provisional enrollments and delinquencies are addressed in a timely manner, as Texas law requires these students to be excluded from schools until they are vaccinated or receive an exemption. We assume that when parents intentionally delay vaccination, the delay period is short compared to the period between kindergarten and seventh grade [10] (although this may overestimate the overall vaccination rate). The seventh grade vaccination rates are therefore assumed to provide a more accurate representation of each school's overall vaccination rate than the kindergarten rates.
Approximately 2.2% of students in the synthetic population are designated as home-schooled, in accordance with census data [11]. Home-schooled students were assumed to be vaccinated at the same rate as the background population, 94.8%. In some US states with stricter vaccination requirements, parents who do not wish to vaccinate their children may be required to home-school, potentially lowering the vaccination rate amongst home-schooled students. However, as Texas allows vaccination exemptions, home-school vaccination rates are assumed not to differ significantly from the mean population level. As home-schooled students can be assumed to interact with a small number of other students compared to students who attend schools, the impact of home-schooled students on the spread of an epidemic is expected to be less than that of other students. There is an apparent slight decrease in cases in Dallas-Fort Worth-Arlington for small decreases in vaccination rate (eFigure 2h). This suggests that more samples are required to more precisely determine the upper confidence limit of forecast cases in Dallas-Fort Worth-Arlington. This is because Dallas-Forth-Worth-Arlington has a population of 6.4 million (the fourth largest MSA in the US) and 1303 under-vaccinated schools; depending on the school in which an outbreak commences, there is variation in the expected overall size of an outbreak. Further simulations are beyond the scope of the current investigation as they would require significant computing resources.

eAppendix 5.
Vaccination Decrease: Fits Figure e3 shows fits to the upper confidence limit of the number of measles cases in each MSA. Exponential fits describe the increase in cases for reduced vaccination rates. Differences in the fits between MSAs are due to population sizes and the number of students attending under-vaccinated schools. As vaccination rates were decreased only in currently under-vaccinated schools, the number and size of these schools had a significant effect on the rate at which measles cases increased.

eAppendix 6. Synthetic Population
The agent population size in each metropolitan statistical area (MSA) used in the simulations is given in the eTable. Population sizes are based on the 2010 US census [12] and the Office of Management and Budget's 2015 designations of MSAs [13].
Agents selected to attend a public school were assigned to the one of the three closest public schools in their county. Students selected to attend a private school were assigned the closest private school to their household without regard for county boundaries [11,14]. Assuming students attend a school near their home may not be true for students attending religious or philosophically-orientated private schools. Hence the locations of infections transmitted from students in these schools to their family members and neighbors may be concentrated into a smaller geographical area than reality.
The model assumes that all medical-exempt children are vulnerable to measles, but some children may have received a medical exemption after previously contracting measles, which would lead to a higher immunity rate than estimated. We also assume medical-exempt students attend schools; severely immuno-compromised children may be more likely to be home-schooled to lessen their risk of infections. However, as medical-exempt students make up a small percent (0.2%) of the student population, these assumptions are unlikely to have a significant effect on the simulation results.
Neighborhoods were defined using a 1 km grid imposed on the household locations.
Agents who attend a school, attend their school on weekdays only. Most workers attend their workplaces on weekdays, however 20% are designated as weekend workers. Neighborhood contacts are increased by 50% on weekends to reflect weekend schedules of schools and workplaces.
The Tyler MSA has large potential outbreak sizes in part due to two schools with vaccination rates of 70% and 85%. It should be noted that one of these schools is a private school located in the adjacent Longview MSA. The algorithm used to assign students to private schools, which is based on distance and not county boundaries, assigned a significant number of students who live in the Tyler MSA to this school (which lies between the two MSAs' urban cores). The outbreak risk may thus be overestimated in the Tyler MSA and underestimated in the Longview MSA.