Methods
The Epidemiologic Volatility Index
The epidemiologic volatility index is calculated for a rolling window of time series epidemic data (i.e. the number of new cases per day). At each step, the observations within the window that are analyzed are obtained by shifting the window forward over the time series data one observation at a time. Let \(y_i=\left\{y_1,\ y_2,...,y_n\right\}\) be a time series of length \(N\). The rolling window size - that is the number of consecutive observations per rolling window - is \(m\). With \(0<m\le m_{\max}\) and \(0\le m_{\max}\le N\), there are \(t=N-m+1\) consecutive rolling windows.
At each of the \(t\) steps, \(EVI\) uses the standard deviation \(\left(\sigma_t\right)\) of the newly reported cases \(\left(y_{j_t}=\left\{y_{1_t},\ y_{2_t},...\ ,y_{m_t}\right\}\right)\) within the specified \(m\):\[\sigma_t=\sqrt{\frac{1}{m}\sum_{i_t=1}^m\left(x_{i_t}-\overline{\mu}_t\right)}\]
with \(\overline{\mu_t}\) the mean of the \(t^{th}\) window. Subsequently, \(EVI\) is calculated as the relative change of \(\left(\sigma_t\right)\) between two consecutive rolling windows:
\[EVI_{t-1,t}=\frac{\sigma_t-\sigma_{t-1}}{\sigma_t}\]
We expect an increase in the future number of cases, if \(EVI_{t-1,t}\) exceeds a threshold \(c\) \(\left(c\in\left[0,1\right]\right)\) and the observed cases at time point \(t,\left(y_t\right)\) are higher than the average of the reported cases in the previous week:
\[Ind_{EVI_{t-1,t}} = \begin{cases}
1 \;\;\; if \;\;\; EVI_{t-1,t}>=c \;\; \wedge \;\; y_t>=\overline{\mu}_{t:t-7} \\
0 \;\;\; \text{otherwise}
\end{cases} \]