Disease elimination and re-emergence in differential-equation models

• We developed a novel framework to analyze disease elimination and re-emergence.
• We provide a simulation free method to determine if disease elimination is possible.
• We model measles eliminations and re-emergences in Iceland from 1924 to 1938.
• Iceland was likely to experience a measles re-emergence shortly after October 1927.
• Undocumented measles re-emergences in Iceland were unlikely from 1930 to 1936.

Traditional differential equation models of disease transmission are often used to predict disease trajectories and evaluate the effectiveness of alternative intervention strategies. However, such models cannot account explicitly for probabilistic events, such as those that dominate dynamics when disease prevalence is low during the elimination and re-emergence phases of an outbreak. To account for the dynamics at low prevalence, i.e. the elimination and risk of disease re-emergence, without the added analytical and computational complexity of a stochastic model, we develop a novel application of control theory. We apply our approach to analyze historical data of measles elimination and re-emergence in Iceland from 1923 to 1938, predicting the temporal trajectory of local measles elimination and re-emerge as a result of disease migration from Copenhagen, Denmark.