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

In order to automate the process of growth stage classification, we built a HMM that uses acceleration data obtained from MR as input. Considering a as the n-dimensional vector of estimated growth accelerations across n recorded days, we first compute z = sign(a), where sign(.) is a modified sign function which retrieves -1 for a < -c, +1 for a > c and 0 otherwise. Scalar c is defined as an acceleration cutoff, which is treated here as a free parameter. The objective of the HMM was to generate a sequence of states K = (k 1 , k 2 , …, k n ) where each element k i takes one of the following values: "lagging", "exponential", "deceleration" or "stationary". The initial probabilities for these hidden states were set to 1, 0, 0 and 0, respectively, assuming that all growth curves start from a lagging stage. Now let T be a 4 x 4 matrix of transition probabilities between hidden states and E be a 4 x 3 matrix of emission probabilities that models the probability of each hidden state producing a z value of -1, 0 or +1. We adopted: The selected values in T only permitted transitions lagging → exponential, exponential→ deceleration or deceleration → stationary. Values in E made z = 0 more likely to be produced by either the lagging or stationary stages, z = +1 more likely to be produced by the acceleration stage and z = -1 more likely to be produced by the deceleration stage. For the atypical transition deceleration → exponential, the described model would generate a short and intermediate stationary step between these two stages. In these cases, the spurious stationary step was replaced by an exponential classification after the HMM has been fitted to the data. The Viterbi algorithm implemented in the HMM v1.0 package (15) in R (9) was used to estimate the sequence K. After prediction of growth stages, stationary classifications were confronted against growth rates. If a given stationary stage presented a median growth rate greater than the maximum growth rate of the lagging phase, it was re-classified as a "linear" stage.