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

Chowell, Gerardo

14. The effective reproduction number, R t , with quantified uncertainty While the basic reproduction number, commonly denoted by R 0 , gauges the transmission potential of an infectious disease epidemic in a fully susceptible population during the early epidemic take off (Anderson & May 1982) , the effective reproduction number R t captures changes in transmission potential over time (Chowell et al., 2016c; Nishiura et al., 2009) . We can characterize the effective reproduction number and its uncertainty during the early epidemic exponential growth phase (Wallinga & Lipsitch, 2007) . When the early dynamics follow sub-exponential growth, another method relies on the generalized-growth model (GGM) to characterize the profile of growth from early incidence data (Chowell et al., 2016c) . In particular, the GGM can reproduce a range of growth dynamics from constant incidence (p ¼ 0) to exponential growth (p ¼ 1) . We can generate the uncertainty associated with the effective reproduction number during the study period directly from the uncertainty associated with the parameter estimates ðb r i ; b p i Þ where i ¼ 1; 2; …; S:. That is, R tj ðb r i ; b p i Þ provides a curve of the effective reproduction number for each value of the parameters b r i ; b p i where i ¼ 1; 2; …; S: Then, we can compute the curves R tj ðb r i ; b p i Þ based on the incidence at calendar time t j denoted by Iðt j ; b r i ; b p i Þ, and the discretized probability Fig. 19 . Top panels display the empirical distributions of the growth rate r, the deceleration of growth parameter p and the effective reproduction number R eff based on fitting the GGM to the first 20 days of the 1918 influenza pandemic in San Francisco. We assumed an exponential distribution for the generation interval of influenza with a mean of 4 days and variance of 16. The bottom panel shows the fit of the GGM to the first 20 days of the 1918 influenza pandemic in San Francisco. Circles correspond to the data while the solid red line corresponds to the best fit obtained using the generalized-growth model (GGM). The blue lines correspond to the uncertainty around the model fit. We estimated the deceleration of growth parameter at 0.95 (95%CI: 0.95, 1.0), an epidemic growth profile with uncertainty bounds that includes exponential growth dynamics (i.e., p ¼ 1) during the early growth trajectory of the pandemic in Madrid. distribution of the generation interval denoted by r tj . The effective reproduction number R tj ðb r i ; b p i Þcan be estimated using the renewal equation (Chowell et al., 2016c; Nishiura et al., 2009 ):