# Epidemic models with uncertainty in the reproduction number

## Abstract

One of the first quantities to be estimated at the start of an epidemic is the basic reproduction number, \({\mathcal{R}_0}\). The progress of an epidemic is sensitive to the value of \({\mathcal{R}_0}\), hence we need methods for exploring the consequences of uncertainty in the estimate. We begin with an analysis of the *SIR* model, with \({\mathcal{R}_0}\) specified by a probability distribution instead of a single value. We derive probability distributions for the prevalence and incidence of infection during the initial exponential phase, the peaks in prevalence and incidence and their timing, and the final size of the epidemic. Then, by expanding the state variables in orthogonal polynomials in uncertainty space, we construct a set of deterministic equations for the distribution of the solution throughout the time-course of the epidemic. The resulting dynamical system need only be solved once to produce a *deterministic stochastic* solution. The method is illustrated with \({\mathcal{R}_0}\) specified by uniform, beta and normal distributions. We then apply the method to data from the New Zealand epidemic of H1N1 influenza in 2009. We apply the polynomial expansion method to a Kermack–McKendrick model, to simulate a forecasting system that could be used in real time. The results demonstrate the level of uncertainty when making parameter estimates and projections based on a limited amount of data, as would be the case during the initial stages of an epidemic. In solving both problems we demonstrate how the dynamical system is derived automatically via recurrence relationships, then solved numerically.

## Keywords

*SIR*model Kermack–McKendrick model Basic reproduction number Uncertainty

## Mathematics Subject Classification

92D30## Preview

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