Path Integral Approach to Uncertainties in SIR-type Systems

Abstract

In this paper I show how path integral techniques can be used to put measures on histories in "susceptible-infectious-recovered" (SIR)-type systems. The standard SIR solution emerges as the classical saddle point of the action describing the measure. One can then expand perturbatively around the background solution, and this paper goes on to work out the covariance of fluctuations around the background solution. Using a Green's function type approach, one simply needs to solve additional ordinary differential equations; an explicit matrix inversion is not required. The computed covariance matrix should be useful in the construction of fast likelihoods for fitting the parameters of SIR-type models to data. A comparison of the predictions of the approach to an ensemble of simulations is presented.