Modeling Disease Transmission Near Eradication: An Equation Free Approach

October 2, 2014

Abstract: 

Although disease transmission in the near eradication regime is inherently stochastic, deterministic quantities such as the probability of eradication are of interest to policy makers and researchers. Rather than running large ensembles of discrete stochastic simulations over long intervals in time to compute these deterministic quantities, we create a data-driven and deterministic ‘‘coarse’’ model for them using the Equation Free (EF) framework. In lieu of deriving an explicit coarse model, the EF framework approximates any needed information, such as coarse time derivatives, by running short computational experiments.However, the choice of the coarse variables (i.e., the state of the coarse system) is critical if the resulting model is to be accurate. In this manuscript, we propose a set of coarse variables that result in an accurate model in the endemic and near eradication regimes, and demonstrate this on a compartmental model representing the spread of Poliomyelitis.

Diagram of the equation-free modeling process. In lieu of a single, lengthy, detailed simulation, a sequence of lifting, simulation, restriction, and projection operators are used instead.

Conclusion: Our ambition in this manuscript was to demonstrate that the EF method can be used to reduce the computational cost associated with determining useful quantities such as the probability of eradication, and to outline the types of coarse variables and lifting operators that are required to carry out these computations. In the relatively small problem studied here, the EF framework enables a factor of two speedup with a modest loss of accuracy. For larger problems, such as the more detailed model presented in, additional techniques such as the gap–tooth scheme are promising approaches to obtain more dramatic decreases in the computational cost. Ultimately, the purpose of EF is to act as a computational framework that allows systems level tasks to be performed without the need to derive explicit coarse governing equations or pay the full computational cost associated with running many detailed simulations