Growth rate and acceleration analysis of the COVID-19 pandemic reveals the effect of public health measures in real time

Yuri Tani Utsunomiya; Adam Taiti Harth Utsunomiya; Rafaela Beatriz Pintor Torrecilha; Silvana Cassia Paulan; Marco Milanesi; Jose Fernando Garcia

For simplicity, assume that the cumulative number of COVID-19 cases over time (i.e., the growth curve) in a specific country or territory follows an unknown sigmoidal function (Figure 1a) . Such assumption is common in the analysis of growth data and has been applied to a wide range of problems, from tumor (10) to bacterial (11) growth. Although empirical data from China ( Figure 1b) and South Korea (Figure 1c ) seemed to support it well, that assumption will be substantially relaxed later in our framework to accommodate complex dynamics in the evolution of COVID-19 prevalence. We define growth rate and growth acceleration as the first and second order derivatives, respectively, of the prevalence of COVID-19 in respect to time. In our framework, we selected MR to approximate these derivatives over competing models that are frequently used to describe the behavior of sigmoidal growth curves, such as the Gompertz model (12, 13) , because: (i) it is dependent on a single free parameter, the "smooth factor", which represents the number of neighboring days used in local regression; (ii) growth rate and acceleration estimates are approximated by ordinary least squares equations, which are computationally inexpensive; (iii) we performed extensive simulations of growth curves and found that it produces reasonably accurate estimates of growth rate (median R 2 = 0.99 with smooth factor of 3) and acceleration (median R 2 = 0.92 with smooth factor of 3) (Figure 2) ; (iv) it is very robust to departures from sigmoidal curves; and (v) it does not rely on observations of the whole curve to produce instantaneous growth rate and acceleration estimates, and thus can produce such estimates in near real time. Argument (v) is especially relevant to the analysis of COVID-19 data since the pandemic is ongoing and each country will be at a different stage of the growth curve as time passes. A clear disadvantage of MR is that it may over-fit the growth curve to the data, especially if the selected smooth factor is small (say < 3), in which case accurate prediction of new cases of COVID-19 is limited to very few days in the future. Still, even single-day predictions can be of great use during a pandemic if reasonably accurate. In the ECDC data set, a forward validation showed that single-day predictions were sufficiently accurate (R² ~ 0.95) (Figure 3) .