Fitting dynamic models to epidemic outbreaks with quantified uncertainty: A primer for parameter uncertainty, identifiability, and forecasts

Chowell, Gerardo

In previous Examples #2 and #3, we measured the uncertainty of model parameters estimated from data by constructing confidence intervals using the empirical distribution of the parameters. However, it is possible to use the empirical distributions of the parameters to assess the uncertainty associated with composite parameters whose values depend on several existing model parameters and are often useful to gauge the behavior of the modeled system. For instance, a key epidemiological parameter to measure the transmissibility of a pathogen is the basic reproduction number, R 0 (Anderson & May 1991; Diekmann, Heesterbeek, & Metz, 1990; van den Driessche & Watmough, 2002) . This parameter is a function of several parameters of the epidemic model including transmission rates and infectious periods of the epidemiological classes that contribute to new infections. This is an important parameter as it often serves as a threshold parameter for SEIR-type compartmental models. If R 0 > 1 then an epidemic is expected to occur whereas values of R 0 < 1cannot sustain disease transmission. For instance, for the simple SEIR model, the basic reproduction number is given by: 17 . Long-term forecasts derived using the GRM by estimating parametersr, p and K with quantified uncertainty when the model is fitted to an increasing length of the growth phase (40, 60, …, 140 days) of a synthetic daily incidence curve simulated using the same GRM model with parameters r ¼ 0:2; p ¼ 0:8; a ¼ 1; and K ¼ 1000. Using only data of the early epidemic growth phase (before the inflection point occurring around day 50), the model is underdetermined and significantly underestimates the incidence curve. Forecasts are gradually improved particularly when the model is calibrated using data past the epidemic's inflection point. The cyan curves correspond to the uncertainty during the model calibration period while the gray curves correspond to the uncertainty in the forecast. The mean (solid red line) and 95% CIs (dashed red lines) of the model fit are also shown. The vertical line separates the calibration and forecasting periods.