# Jacob Curran-Sebastian (University of Manchester and University of Copenhagen)

**Approximating First Passage and Peak Timing Distributions for Epidemics**

Understanding the timing of the peak of a disease outbreak is an important part of epidemic forecasting. The time taken for an outbreak to become large is inherently stochastic, however, the disease dynamics can be well approximated by a deterministic model once a sufficient number of cases is reached. We present analytic and numerical methods for approximating the distribution of times at which a given number of cases is reached using a branching process model, known as the First Passage Time (FPT) distribution. Once a threshold number of cases, which we denote Z^*, has been reached, we project the FPT distribution forward in time using a deterministic model in order to obtain the peak timing distribution. Importantly, our results require a fraction of the computational cost of running full Monte Carlo Simulations. We begin with a simple SIR model and extend the results to include more general multitype models.