Mathematical Modelling of a Measles Outbreak in Pre-vaccine England and Wales
We present a spatial variant of the time series susceptible-infectious-recovered (TSIR) stochastic population-based model to capture the spatial transmission dynamics of a measles outbreak across the landscape of England and Wales during the pre-vaccine era. Specifically, we explore how the basic dynamical features of a measles outbreak with a seasonal forcing of transmission acts as a major driver of a long-term epidemic behavior. We use a 20-year pre-vaccination era biweekly time series data (births by year and incidence of measles for the years 1944-1966) from 60 towns and cities in England and Wales to capture the spatial spread of measles.
In England and Wales prior to vaccination, measles was endemic in large cities, but in smaller cities disease fadeout occurred. Reappearance of the disease would then occur only after a case was imported from a surrounding city where measles was endemic. To capture spatio-temporal dynamics, multi-city models must be developed, but these models can become very large requiring more memory and processing power than a single computer can deliver.
Rather than represent the population as a linked set of cities, we represent the population as a gridded map. Each grid cell can transmit infectious disease to its neighbors, with probabilities that decline exponentially with distance. We present a stochastic spatial model with six compartments. We call this the kids-susceptible- infectious-recovered-adults-dead (KSIRAD) model.
From the simulation, we recover spatiotemporal maps of the incidence of the infection. We compare simulated time-series graphs with real data compiled by Grenfell and others. Our future work includes testing of our spatial model for measles outbreaks reported in the modern era, for example, in conflict affected areas of the Republic of the Niger in Western Africa in 2016. Socioeconomic disparities in a country like Niger presents significant challenges to reporting and real-time tracking of human infectious diseases.