Epidemiological models can be used to study the impact and spread of disease through a population. The continuous-time version of these models takes on the form of a system of ordinary differential equations (ODEs) in which the state variables are the population or population fraction of members belonging to different states relative to the disease, such as susceptible or infected. Often these models are coarse and the precise number of individuals in a given class, or the numerical parameters explaining the interactions and transitions among the different classes, are not known with certainty. These quantities may be bounded with intervals or probability distributions.

In this presentation, we demonstrate a method for the verified solution of nonlinear epidemiological models, thus computing rigorous bounds on the predicted population of classes over a given time frame, based on the specified ranges of the uncertain parameters and/or initial conditions. The method is based on the general approach described by Lin and Stadtherr [1], which uses an interval Taylor series to represent dependence on time, and uses Taylor models to represent dependence on uncertain values. We also demonstrate an approach for the propagation of uncertain probability distributions in one or more model parameter and/or initial condition. Assuming an uncertain probability distribution for each parameter and/or initial condition of interest in the epidemiological model, we use a method, based on Taylor models and probability boxes (p-boxes) and recently described by Enszer et al. [2], that propagates these distributions through the dynamic model. As a result, we obtain a p-box describing the probability distribution for each state variable at any given time of interest. The traditional Monte Carlo simulation approach may not bound all possible population trajectories because it is impossible to sample the complete parameter space with a finite number of trials, but this Taylor model method provides completely rigorous results that fully capture all possible dynamics of the system under uncertain conditions.

The use of this method in epidemiology is demonstrated using the SIRS model, and other variations of Kermack-McKendrick models, including a case with time-dependent transmission [3]. The results of Monte Carlo simulation are compared to those of this method.

[1] Lin, Y., Stadtherr, M.A. Validated Solutions of Initial Value Problems for Parametric ODEs. Applied Numerical Mathematics, 57: pp. 1145--1162, 2007.

[2] Enszer, J.A., Lin, Y., Ferson, S., Corliss, G.F., Stadtherr, M.A. Propagating Uncertainties in Modeling Nonlinear Dynamic Systems. In Proceedings of the 3rd International Workshop on Reliable Engineering Computing, Georgia Institute of Technology, Savannah, GA: pp. 89--105, 2008.

[3] J. Dushoff et. al. Dynamical Resonance can Account for Seasonality of Influenza Epidemics. Proceedings of the National Academy of Sciences, 101: pp. 16915--16916, 2004.