A closure approximation technique for epidemic models

Abstract

Stochastic modelling causes an infinite set of ordinary differential equations for the moments. Closure models are useful as they recast this infinite set into a finite set of ordinary differential equations. A general closure principle is developed, which we believe all closure models should fulfil. In the Liouville model, if all covariances are zero initially, then they remain at zero for all times. Our closure principle assumes that this logical implication also should apply for the closed Liouville model. A specific covariance closure (CC) model is developed based on the Dirac distribution. It states that all covariances up to order n-1 differ from zero, whereas all covariances of order n and higher equal zero. The CC approximation is compared with Keeling's [Journal of Animal Ecology 69 (2000), pp. 725–736, Journal of Theoretical Biology 205 (2000), pp. 269–281] approximation based on the lognormal distribution and with the central moment closure approximation based on the multivariate normal distribution. Closure is applied to the classic Kermack and McKendrick [Proceedings of the Royal Statistical Sociecty, Series A 115 (1927), pp. 700–721] equations for epidemic growth. For the main examples studied the CC approximation of orders 7–12 outperforms Keeling's approximation.