Measles Age-Structured Model

This vignette demonstrates how to build and run an age-structured model of transmission within a U.S. county. The example is an outbreak of measles in Washington County, Utah.

WARNING: The data and model used in this vignette are for demonstration purposes only and do not reflect real-world conditions accurately. The results are not intended to forecast or predict a measles outbreak size in this location.

First we define the age groups that we want use in our model. Here we choose groupings for which we might have reason to assume different levels of immunization against measles. We define the age_limits vector using the lower limit (minimum age) of each group, starting with 0:

# under 1, 1-4, 5-11, 12-17, 18-24, 25-44, 45-69, 70 plus
age_limits <- c(0, 1, 5, 12, 18, 25, 45, 70)

Next we collect census data from Washington County, Utah, using our custom age group choice:

# Get data for Washington County, Utah
washington_data <- getCensusData(
  state_fips = getStateFIPS("Utah"),
  county_name = "Washington County",
  year = 2024,
  age_groups = age_limits,
  csv_path = getCensusDataPath()
)

Now we specify immunity levels of our population age groups due to vaccination. The first age group (under 1 year) has no immunity because the first measles vaccine is not given until age one. Then we assume rising immunity with age.

NOTE: These vaccination and immunization assumptions are not validated by data and should not be taken as the actual levels of immunization against measles in Washington County.

age_immunity <- c(0, 0.77, 0.83, 0.85, 0.87, 0.90, 0.92, 1)

Now we can begin defining the input arguments to our finalsize function so that we can estimate the measles outbreak size in this county after an introduction.

popsize <- washington_data$age_pops

initV <- round(age_immunity * popsize)  # initially immune cases 

initI <- rep(0, length(popsize))  # initial infectious cases
initI[6] <- 1                     # assume 1 initial case in the 6th age group (25-44)

initR <- rep(0, length(popsize))  # no recent prior measles outbreaks

Transmission matrix ingredients: contact matrix, relative susceptibility and transmissibility, and the basic reproduction number (\(R_0\)). The contact matrix uses Polymod contact survey data, adjusted for local population distribution:

contactmatrix <- contactMatrixPolymod(age_limits, popsize)
relsusc <- rep(1, length(popsize))      # Assume no age differences in susceptibility
reltransm <- rep(1, length(popsize))    # or transmissibility per contact
R0 <- 10

Now we can estimate the final size of an outbreak using the default, deterministic solution produced by the finalsize() function:

fs <- finalsize(popsize, R0, contactmatrix, relsusc, reltransm, initR, initI, initV)
round(sum(fs))
#> [1] 7573

Under the model assumptions and with \(R_0\) = 10, there could be an outbreak of more than 7,500 measles cases in this county.

names(fs) <- washington_data$age_labels
round(fs)
#> under1   1to4  5to11 12to17 18to24 25to44 45to69 70plus 
#>    372    648   1673   1748    876   1414    843      0

Here’s an example using the “hybrid” method option in the finalsize() function, which runs stochastic simulations from the initial conditions until the outbreak dies out or reaches an “escape” threshold, after which the stochastic simulation is suspended and replaced with the deterministic solution above.

fs_hybrid <- finalsize(popsize, R0, contactmatrix, relsusc, reltransm, initR, initI, initV, method = "hybrid", nsims = 30)
colnames(fs_hybrid) <- washington_data$age_labels
fs_hybrid
#>       under1 1to4 5to11 12to17 18to24 25to44 45to69 70plus
#>  [1,]      0    0     3      1      0      4      4      0
#>  [2,]    372  648  1673   1748    876   1414    843      0
#>  [3,]      0    0     0      0      0      1      0      0
#>  [4,]    372  648  1673   1748    876   1414    843      0
#>  [5,]      0    2     0      1      0      7      2      0
#>  [6,]      0    0     0      0      0      1      0      0
#>  [7,]      0    0     0      0      0      1      0      0
#>  [8,]      0    0     0      0      0      1      0      0
#>  [9,]      0    0     0      0      0      1      0      0
#> [10,]      0    0     0      0      0      2      1      0
#> [11,]      0    0     0      0      0      1      0      0
#> [12,]      0    0     0      0      0      1      2      0
#> [13,]      0    0     0      0      0      1      0      0
#> [14,]      0    0     1      1      0      2      0      0
#> [15,]      0    0     0      0      0      2      0      0
#> [16,]      0    0     0      0      0      1      0      0
#> [17,]      0    0     0      0      0      2      0      0
#> [18,]      0    0     0      0      0      1      0      0
#> [19,]    372  648  1673   1748    876   1414    843      0
#> [20,]      0    0     0      0      0      1      0      0
#> [21,]    372  648  1673   1748    876   1414    843      0
#> [22,]      0    0     0      0      0      1      0      0
#> [23,]      0    0     0      0      0      1      0      0
#> [24,]      0    0     0      0      0      1      0      0
#> [25,]      0    0     1      0      1      3      0      0
#> [26,]      0    0     0      0      0      1      0      0
#> [27,]      0    0     0      0      0      1      0      0
#> [28,]      0    1     0      0      0      4      0      0
#> [29,]      0    0     0      0      0      1      0      0
#> [30,]    372  648  1673   1748    876   1414    843      0

The results show that not every introduction of one infectious individual (in the 25 to 44 age group) will lead to a large outbreak.

NOTE: All above results should be taken with caution, as this simulation is a simplification of reality that does not consider, for example, individuals’ changing behavior, e.g. dramatically reducing contact rates or taking other precautionary actions upon observing a measles outbreak occurring in the community. The model also makes the highly unrealistic assumption of spatially uniform mixing across a large county with geographically dispersed communities, which could make the results unreliable.