Sensitivity analysis of epidemic forecasting and spreading on networks with probability generating functions

Epidemic forecasting tools embrace the stochasticity and heterogeneity of disease spread to predict the growth and size of outbreaks. Conceptually, stochasticity and heterogeneity are often modelled as branching processes or as percolation on contact networks. Mathematically, probability generating functions (PGFs) provide a flexible and efficient tool to describe these models and quickly produce forecasts. While their predictions are probabilistic-i.e. distributions of outcome-they depend deterministically on the input distribution of transmission statistics and/or contact structure. Since these inputs can be noisy data or models of high dimension, traditional sensitivity analyses are computationally prohibitive and are therefore rarely used. Here, we use statistical condition estimation to measure the sensitivity of stochastic polynomials representing noisy generating functions. In doing so, we can separate the stochasticity of their forecasts from potential noise in their input. For standard epidemic models, we find that predictions are most sensitive at the critical epidemic threshold (basic reproduction number R0 = 1) only if the transmission is sufficiently homogeneous (dispersion parameter k > 0.3). Surprisingly, in heterogeneous systems (k ≤ 0.3), sensitivity is highest for values of R0 > 1. We expect our methods will improve the transparency and applicability of PGFs as epidemic forecasting tools.