Bias in the estimated reporting fraction due to vaccination in the time-series SIR model

The time-series Susceptible-Infectious-Recovered (TSIR) model has been a standard tool for studying the non-linear dynamics of acute, immunizing infectious diseases. The standard assumption of the TSIR model, that vaccination is equivalent to a reduction in the recruitment of susceptible individuals, or the birth rate, can lead to a bias in the estimate of the reporting fraction and of the total incidence. We show that this bias increases with the level of vaccination due to a double counting of individuals who are infected prior to the age of vaccination. We present a simple correction for this bias by discounting the observed number of cases by the product of the number that occur prior to the average age of vaccination and the vaccination coverage during the initial susceptible reconstruction step of the TSIR model fitting. We generate a time series of measles cases using an age-structured SIR transmission model with vaccination after birth (at 9 months of age) and illustrate the bias with the standard TSIR fitting method. We then illustrate that our proposed correction eliminates the bias in the estimated reporting fraction and total incidence. We note further that this bias does not impact the estimates of the seasonality of transmission.