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\title{The~Epidemic Volatility Index: an early warning tool for~ epidemics~}



\author[1]{Polychronis Kostoulas}%
\author[1]{Eletherios Meletis}%
\author[1]{Konstantinos Pateras}%
\author[2]{Paolo Eusebi}%
\author[3]{Theodoros Kostoulas}%
\author[4]{Niko Speybroeck}%
\author[5]{Matthew Denwood}%
\author[6]{Suhail A.R. Doi}%
\author[7]{Christian L. Althaus}%
\author[5]{Carsten Kirkeby}%
\author[8]{Pejman Rohani}%
\author[9]{Navneet K. Dhand}%
\author[10]{José L. Peñalvo}%
\author[11]{Luis Furuya-Kanamori}%
\author[12]{Lehana Thabane}%
\author[13]{Slimane BenMiled}%
\author[14]{Hamid Sharifi}%
\author[12]{Stephen Walter}%
\affil[1]{Faculty of Public Health, University of Thessaly, Greece.}%
\affil[2]{Department of Medicine and Surgery, University of Perugia.}%
\affil[3]{Department of Information and Communication Systems Engineering, University of the Aegean, Greece}%
\affil[4]{Research Institute of Health and Society (IRSS), Université Catholique de Louvain, 1200, Brussels, Belgium.}%
\affil[5]{Department of Veterinary and Animal Sciences, University of Copenhagen, Denmark}%
\affil[6]{Department of Population Medicine, College of Medicine, QU Health, Qatar University, Doha, Qatar.}%
\affil[7]{Institute of Social and Preventive Medicine, University of Bern, Bern, Switzerland.}%
\affil[8]{Center for the Ecology of Infectious Diseases, University of Georgia, Athens, GA 30602, USA.}%
\affil[9]{Sydney School of Veterinary Science, The University of Sydney, Camden, NSW, Australia.}%
\affil[10]{Unit of Noncommunicable Diseases, Department of Public Health, Institute of Tropical Medicine, Antwerp, Belgium.}%
\affil[11]{UQ Centre for Clinical Research, Faculty of Medicine, The University of Queensland, Herston, Australia.}%
\affil[12]{Department of Health Research Methods, Evidence, and Impact, McMaster University, Hamilton ON, Canada.}%
\affil[13]{Pasteur Institute, University of Tunis el Manar, Tunis, Tunisia.}%
\affil[14]{HIV/STI Surveillance Research Center, and WHO Collaborating Center for HIV Surveillance, Institute for Futures Studies in Health, Kerman University of Medical Sciences, Kerman, Iran.}%


\vspace{-1em}



  
  \date{May 01, 2026}


\begingroup
\let\center\flushleft
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\selectlanguage{english}
\begin{abstract}
\textbf{Background}. This paper presents, for the first time, the
Epidemic Volatility Index (EVI), a conceptually simple, early warning
tool for emerging epidemic waves. ~

\textbf{Methods}. EVI is based on the volatility of the newly reported
cases per unit of time, ideally per day, and issues an early warning
when the rate of the volatility change exceeds a threshold.

\textbf{Results}. Results from the COVID-19 epidemic in Italy and New
York are presented here, while daily updated predictions for all world
countries and each of the United States are available online.

\textbf{Interpretation}. EVI's application to data from the current
COVID-19 pandemic revealed a consistent and stable performance in terms
of detecting oncoming waves. The application of EVI to other epidemics
and syndromic surveillance tasks in combination with existing early
warning systems will enhance our ability to act fast and~optimize
containment of outbreaks.%
\end{abstract}%



\sloppy


\section*{}

{\label{874460}}

\section*{Introduction}

{\label{874460}}

Early warning tools are crucial for the timely application of
intervention strategies and the mitigation of the adverse health, social
and economic effects associated with epidemics. Sentinel networks in
combination with information technology infrastructures in public
health~\hyperref[csl:1]{(Heffernan et al., 2004)} provide data for the detection of spatial and
temporal aberrations~in the expected number of cases for groups of
clinical signs and symptoms\hyperref[csl:2]{(Brett and Rohani, 2020)}. Several modelling
frameworks exist for the analysis of such data. For example, the moving
epidemic method, an approach used to monitor, among others, the start of
the flu epidemic~\hyperref[csl:3]{(Vega et al., 2013)}. Further, methods based on
seasonality patterns, the link between pathogens and meteorological
parameters~\hyperref[csl:4]{(Abeku et al., 2004)} and/or the measurement of vector indices
for vector-borne pathogens~\hyperref[csl:5]{(Chang et al., 2015)} are also available.~

~~~~Once an epidemic erupts,~growth models can be used to predict the
course of the outbreak and quantify its consequences. The advantages and
limitations of these methods have been extensively
discussed~\hyperref[csl:6]{(Chowell et al., 2016)}. Machine learning algorithms have also been
utilized with the most recent application being in the current COVID-19
pandemic~\hyperref[csl:7]{(Wang et al., 2020)}. Correlating the number of COVID-19 cases
with parameters obtained using big data approaches can predict future
rise in the number of cases. For example, monitoring of digital data
streams can provide an early indication of a rise in the COVID-19 cases
and deaths in the next 2 to 3 weeks~\hyperref[csl:8]{(Kogan et al., 2021)}. All models have
limitations arising from the imperfect nature of the data. The need for
open, better, detailed data is imperative for the deployment of models
with improved accuracy, models that will have better predictive ability
and will be more useful for the timely application of appropriate
control measures for the COVID-19 pandemic~\hyperref[csl:9]{(Vespignani et al., 2020)}.

~~~~Our work introduces the Epidemic Volatility Index (EVI), which is
inspired by the use of volatility indices in the stock
market~\hyperref[csl:10]{(Fernandes et al., 2014}; \hyperref[csl:11]{Brenner and Galai, 1989)}.~EVI is based on the moving standard deviation
of the newly reported cases during an epidemic. First we present the
rationale of EVI and then provide an example application
with~COVID-19~data from Italy and New York. Daily updated predictions -
with a 48 hour lag for confirmation purposes - are available online for
all world countries and each of the United States.~Results revealed a
firm and consistent ability of EVI to predict the COVID-19
epidemic~waves,~in all instances.

\par\null

\section*{Materials and Methods}\label{materials-and-methods}

\subsection*{The Epidemic Volatility
Index}\label{the-epidemic-volatility-index}

~~~~EVI is based on the calculation of the rolling standard deviation
for a time series of epidemic data (i.e. the number of new cases per
day). The number of consecutive observations used for this calculation,
per rolling window, is the rolling window size -- \(m\). At
each time step, for a rolling window of size~\(m\), the
observations within the window are obtained by shifting the window
forward, over the time series data, one observation at a time (Figure
1).\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/Figure1-1m-1start/Figure-1}
\caption{{A time series~\(\left(y_1,y_2,...,y_T\right)\) of epidemic data with a rolling window
of size -\(m=4\).
{\label{909380}}%
}}
\end{center}
\end{figure}

Then, for each rolling window the standard deviation of the newly
reported cases is calculated and EVI is estimated as the relative change
of the standard deviation between two consecutive rolling windows. A
warning signal is issued if this relative change exceeds a
threshold~\emph{\(c\)\(\left(c\in\left[0,1\right]\right)\)} and the observed
cases at the current time point are higher than the average of the
reported cases in the previous week.

\subsection*{Case definition and desired
accuracy}

{\label{421982}}

The accuracy of EVI is measured by its sensitivity~\(\left(Se\right)\)
(i.e., the probability of correctly issuing an early warning for an
upcoming epidemic wave) and its specificity~\(\left(Sp\right)\) (i.e.,
the probability of not signaling a false alarm in the absence of
upcoming waves) and depends on the case definition of what constitutes a
noteworthy rise in the expected number of cases to be considered as the
indicative of an epidemic wave. For example, a case definition can be,
as in the example application that follows, a rise in the mean number of
cases between two consecutive weeks higher than 20\%.

For a specified case definition, the accuracy of EVI depends on the
window size~\emph{m} and the threshold~\emph{c}, which should be
selected in a way to achieve a desired accuracy target. One option is
the selection of~\emph{m} and~\emph{c} values that lead to the
simultaneous optimization of~\emph{Se} and~\emph{Sp} for EVI and the
maximization of the Youden index~\(\left(J=Se+Sp-1\right)\)\hyperref[csl:12]{(Fluss et al., 2005)}
and, hence, the overall minimization of the false results (i.e. both
false positive and false negative early warnings). Another approach
could be to select~\(m\) and~\(c\) such that
the highest~\emph{\(Se\left(or\ Sp\right)\)} is achieved
with~\emph{\(Sp\left(or\ Se\right)=1\)} or not dropping below a critical value
(e.g. 95\%). Advanced Receiver Operating Characteristic curve analysis
can also be performed~\hyperref[csl:13]{(Zweig and Campbell, 1993)} and selection of critical
values can be based on indices that quantify the relative cost of false
positive (i.e., falsely predicting an upcoming epidemic wave) to false
negative (i.e., failing to predict an upcoming epidemic wave) warnings,
like the misclassification cost term (MCT).

\subsection*{Generation of an early
warning}

{\label{636120}}

For a defined case definition, the optimal \emph{\(m\)} and
\(c\) for a desired accuracy target are selected through an
iterative process, every time a new time point \emph{\(t\)}
is observed. Briefly:

1. Observed cases up to~\(t\) are analyzed for all possible
window sizes~\(\left(m\right)\) and thresholds \(\left(c\right)\).

2. For each of the~\emph{\(m\)} and \(c\)
combinations, the \emph{\(Se_{t_{m,c}}\)} \textsubscript{} and
\(Sp_{t_{m,c}}\) \textsubscript{} are estimated for the defined case
definition.

3. The~~\emph{\(m'\)} and~\(c'\) that give the
best~\emph{\(Se_{t_{m',c'}}\)}\textsubscript{~}and~\(Sp_{t_{m',c'}}\)\textsubscript{~}combination
are selected.

4. Based on ~\emph{\(m'\)} and~\(c'\), EVI is
calculated at the most recent time point~\(t\) and a
decision is made on whether a warning signal is issued or not.

The graphical representation of the EVI calculation is given in Figure
2, while the extensive technical details are described in the Appendix.

\par\null\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/Figure-2-no-legend/Figure2-210718}
\caption{{Graphical representation of the Epidemic Volatility Index (EVI)
model.~\(T\) denotes the end of the time
series,~\(t\) a time point of the
time-series.~\(Se\) and~\(Sp\) stand for the
sensitivity and specificity of the testing procedure calculated at each
step of the algorithm. Solid lines are explanatory; at each time point
dashed lines represent the iterative optimization process while the bold
solid line denotes the end of the algorithm.~
{\label{775050}}%
}}
\end{center}
\end{figure}

\subsection*{Overall Accuracy and Predictive
Values}

{\label{269223}}

~~~ It is possible, at each time point~\emph{t}, to calculate the
positive and negative predictive values, defined as the probability of
observing a rise or drop in the future number of cases, given that an
early warning was issued or not, respectively. Finally, once the entire
time series data has been observed the overall
\emph{Se}\textsubscript{EVI}~ and \emph{Sp}\textsubscript{EVI} can be
estimated.

\subsection*{Sensitivity analysis}

{\label{135315}}

~~~ The accuracy of EVI depends on the specified case definition.
Ideally, in the presence of historical data, various case definitions
should be explored to identify which are suitable for the optimal
monitoring of an epidemic.

\subsection*{Example application}

{\label{997248}}

The current most serious threat to global health and
economy\hyperref[csl:14]{(Fauci et al., 2020)} is the COVID-19 pandemic that begun in China
and was first reported to the WHO China Country Office on December 31,
2019\hyperref[csl:15]{(Organization and others, 2020)}. Data on the confirmed cases of COVID-19 were
retrieved from
the~\href{https://github.com/CSSEGISandData/COVID-19}{COVID-19 Data
Repository}, which is maintained by the Center for Systems Science and
Engineering (CSSE) at Johns Hopkins University\hyperref[csl:16]{(Dong et al., 2020)}. The
number of daily~confirmed new cases of COVID-19, for each country, from
January 22, 2020 until April 13, 2021 were analyzed.~Due to unnatural
variability in the reported cases between working days and weekends, the
7-day moving average rather than the actual observed cases were
analyzed. For the analysis,~\(m_{max}\) was restricted to 30 days
in order to avoid the effect of potentially higher volatility from
previous epidemic waves on the volatility estimates of the most recent
data and the predictive ability of ~EVI for upcoming and perhaps milder
epidemic waves.

~~~~The case definition was an increase in the mean of expected cases,
between two consecutive weeks, equal or higher than twenty
percent~\(\left(r\le\frac{1}{1.2}\right)\). For sensitivity analysis, the detection of
an increase in the mean of expected cases equal or higher than 50
percent\(\left(r\le\frac{1}{1.5}\right)\)was considered. Data were analyzed separately
for each country and for each of the states of the United States of
America that had experienced a total number of cases higher than 20,000,
until April 13, 2021.

\par\null

\subsection*{Statistical software}\label{statistical-software}

~~~~All models were run in R\hyperref[csl:17]{({R Core Team}, 2020)}. The packages
readxl\hyperref[csl:18]{(Wickham et al., 2019)}, ggplot2\hyperref[csl:19]{(Wickham, 2011)},
cowplot\hyperref[csl:20]{(Wilke et al., 2019)} and readr\hyperref[csl:21]{(Wickham et al., 2015)} were used.

\section*{Results}\label{results}

~~~~Results for Italy, one of the most severely affected EU
countries\hyperref[csl:22]{(Livingston and Bucher, 2020)}, and New York, which was in the epicenter of
the pandemic in the U.S.\hyperref[csl:23]{(Thompson et al., 2020)}, are presented in the main
manuscript. Daily updated results for all world coutnries and each of
the United States are available online at
\url{http://83.212.174.99:3838}.

~~~~Confirmed COVID-19 cases for Italy and New York State, from January
22, 2020, until April 13, 2021, are in Figures 1 and 2, respectively.
Red dots correspond to time points~that an early warning was issued
according to~\(Ind_{EVI_{t,t-1}}\), while grey dots to time points without
an early warning indication. Further, the positive and negative
predictive values at each time point are in Figures 3 and 4,
respectively.~

\par\null\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/Italy1/Italy1}
\caption{{Daily confirmed cases of COVID-19 in Italy, from January 22, 2020 until
April 13, 2021. Red dots correspond to dates that, according to the
Epidemic Volatility Index (EVI), an early warning was issued indicating
that a rise in the COVID-19 cases is expected. Data are presented on the
original scale (1a) and the logarithmic scale (1b), which facilitates
the comparison of the steepness of the epidemic curve between the
different waves.%
}}
\end{center}
\end{figure}\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/NewYork11/NewYork11}
\caption{{Daily confirmed cases of COVID-19 in New York, from January 22, 2020
until April 13, 2021. Red dots correspond to dates that, according to
EVI, an early warning was issued indicating that a rise in the COVID-19
cases is expected. Data are presented on the original scale (1a) and the
logarithmic scale (1b), which facilitates the comparison of the
steepness of the epidemic curve between the different waves.
{\label{354486}}%
}}
\end{center}
\end{figure}\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/Italy21/Italy21}
\caption{{Positive and negative predictive values (PPV in 3a and NPV in 3b), for
Italy, depending on whether or not an early warning was issued. Higher
color intensity corresponds to predictive values closer to the value of
1.~%
}}
\end{center}
\end{figure}\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/NewYork21/NewYork21}
\caption{{Positive and negative predictive values (PPV in 4a and NPV in 4b), for
New York, depending on whether or not an early warning was issued.
Higher color intensity corresponds to predictive values closer to the
value of 1.
{\label{499052}}%
}}
\end{center}
\end{figure}

~~~~For Italy, the overall sensitivity for EVI was 0.82 (0.75; 0.89) and
the specificity was 0.91 (0.88; 0.94). For New York, the corresponding
values were 0.55 (0.47; 0.64) and 0.88 (0.84; 0.91).~

~~~ Sensitivity analysis results for Italy, under
alternative~\(r\) specification (i.e., \(r=\frac{1}{1.5}\)),
are in Figure 5. The overall sensitivity and specificity,
for~\(r=\frac{1}{1.5}\), were 0.75 (0.66; 0.85) and 0.93 (0.91; 0.96),
respectively.

\par\null\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/Italy-sa1/Italy-sa1}
\caption{{Daily confirmed cases of COVID-19 in Italy, from January 22, 2020 until
April 13, 2021. Analysis with~\(r=\frac{1}{1.5}\). Red dots correspond to
dates that, based on EVI, an early warning was issued indicating that a
rise in the COVID-19 cases is expected. Data are presented on the
original scale (1a) and the logarithmic scale (1b) which facilitates the
comparison of the steepness of the epidemic curve between the different
waves.
{\label{360889}}%
}}
\end{center}
\end{figure}

~~~~A common and consistent finding in the results from all countries
was that repetitive early warnings are linked to the start of a new
epidemic wave, while the absence of warnings indicates a stable course
or a future drop in the number of new COVID-19 cases (Fig. 1, 2
and~\url{http://83.212.174.99:3838/}).~

\section*{Discussion}\label{discussion}

~~~~EVI is an efficient and easy to implement early-warning tool for an
upcoming rise in the number of new cases. The performance of~EVI, as
expressed by its overall~\(Se\) and~\(Sp\), was,
in all instances, high. A more important aspect lies in the fact that
repetitive issuance of early warnings indicates the beginning of an
epidemic wave. This is a consistent and remarkably stable finding across
all countries and each of the United States (Fig. 1, 2
and~\url{http://83.212.174.99:3838/}). In a similar manner, the absence
of a series of early warnings implies that the number of new cases will
remain stable or drop. The latter was also a consistent finding.
Additionally, false early warnings (i.e. false positives) were isolated
instances and did not occur in a consecutive series. There were few
occasions with a consecutive absence of early warnings despite a
continuing rise in the number of cases (i.e. false negatives).
Nevertheless, such series of false negatives were always close to the
peak of a wave. This finding is reasonable and could be interpreted as
an early sign of reaching the peak because EVI depends on volatility and
the increase in the number of new~cases decelerates when approaching the
peak of an epidemic wave. Positive and negative predictive values that
are calculated at each time point can also be used to assess the
probability that an early warning, or its absence, is true. In all
instances, predictive values were high with the exception of few
instances at the beginning of the time series due to the absence of
enough data.~

~~~~Work on the SIR and SIS models has revealed that moving-window
estimates of the variance increase while approaching the emergence of a
pathogen as well as during the elimination phase and that it can be used
as an early warning tool~\hyperref[csl:24]{(O'Regan and Drake, 2013)}. EVI is based on the
relative rather than the absolute change of the standard deviation
because the latter depends on the underlying prevalence at each time
point of the epidemic. Hence, a low threshold would be efficient in
detecting a surge in the new cases at the beginning of an epidemic, when
the baseline prevalence is low, but would have failed to do so for
subsequent epidemic waves that commence from a higher baseline
prevalence.~ On the other hand, a high absolute threshold would have
failed to capture waves at the beginning of the epidemic. EVI is based
on the relative increase in volatility, which implicitly adjusts for the
baseline prevalence at each point of the time series.

~~~~In general, the ability of EVI to provide valid predictions does not
seem to be affected by the fact that sampling and testing schemes for
COVID-19 are mainly based on passive surveillance systems.~EVI performed
equally well among different countries with different control
strategies, testing intensity and reporting accuracy and despite the
fact that even within countries sampling and testing has changed over
time and/or differs between regions~\hyperref[csl:25]{(Brynildsrud, 2020}; \hyperref[csl:26]{Middelburg and Rosendaal, 2020)}. Restriction of
the maximum window size~\(\left(m_{\max}\right)\) to one month plays a key
role, because reporting bias is expected to remain similar over short
time periods. This form of non-differential misclassification leads to
reporting rates that, though biased, do not have a significant impact on
volatility, EVI and its predictive ability. Crucially, it is important
that the data do not exhibit strong artifacts of recording bias, as
there is no way for the method to distinguish between a trend due to
underlying epidemic patterns and an observed trend due to changes in
reporting practices or increased testing capacity or
effort\hyperref[csl:27]{(Halasa et al., 2020)}. This could for instance happen when a country
changes its general testing regime, experiences local outbreaks and
focus testing in a specific area, or targets other subgroups of the
population than previously. Thus, EVI should preferably be evaluated for
use in smaller geographical regions, such as counties or municipalities,
if sufficient, high quality data are available. Undoubtedly,~ all models
are prone to limitations due to imperfect data~\hyperref[csl:9]{(Vespignani et al., 2020)} but~
the continuing enhancement of active and passive surveillance systems -
as the testing regimes and methods also improve - will lead to improved
data quality.~

~~~~The performance of EVI depends on the specified case definition
and~\(r\), parameters which are epidemic-specific and
country-specific. Modifications to allow for different case definition
and~\(r\), for the different periods of an epidemic, are
rather straightforward to implement. Parameters~\(c\)
and~\(m\) are allowed to vary and take values that would
satisfy the conditions set by the defined case and the desired accuracy.
A point of concern is the selection of~\(m_{\max}\). For an
ongoing epidemic with multiple waves, as is the case with
COVID-19,~\(m_{\max}\) should be limited to a period shorter than
the entire observation period. This prevents excess volatility of past
epidemic waves from affecting the most recent volatility estimates and
the ability of EVI to warn~for upcoming waves that may be smaller and of
lower volatility than previous ones. In our example, we
limited~\(m_{\max}\) to one month. EVI also depends on data
intensity. Detailed data at the lowest time unit (i.e., days rather than
weeks) is preferable in order to detect changes rapidly. In the COVID-19
example the 7-day moving average was analyzed instead of the daily
reported cases because daily data had unnatural variability due to
reporting variations between working days and weekends. Nevertheless,
analysis based on the daily reported cases provided similar results
(data not shown here).~

~~~~Beyond the case of epidemics or rare events, like the COVID-19
pandemic, an important application of EVI can be in the context of
syndromic surveillance~\hyperref[csl:28]{(Henning, 2004)}, not limited to outbreaks from
terrorist attacks, but in its broader sense: the detection of temporal
and spatial aberrations in the expected number of cases for signs and
symptoms. Such systems already exist and utilize state-of-the-art
information technologies within the context of public
health~\hyperref[csl:1]{(Heffernan et al., 2004)} as well as one health~\selectlanguage{ngerman}\hyperref[csl:29]{(Beltrán-Alcrudo et al., 2009}; \hyperref[csl:30]{Gilbert et al., 2014)}. EVI
can provide an additional early warning tool for these systems.

\par\null\par\null

\section*{Appendix}\label{appendix}

\subsection*{The Epidemic Volatility
Index}\label{the-epidemic-volatility-index}

~~~~EVI 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 are obtained by shifting the window forward over the
time series data one observation at a time.~

Let~\(x_i=\left\{x_1,\ x_2,...,x_n\right\}\) 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~\(0<m_{\max}\le N\), there are~~\(t=N-m+1\) consecutive
rolling windows.~~

~~~~At each of the~\(t\) steps, \(EVI\) uses the
standard deviation~\(\left(s_t\right)\) of the newly reported
cases~\(\left(y_{j_t}=\left\{y_{1_t},\ y_{2_t},...\ ,y_{m_t}\right\}\right)\) within the specified~\(m\)\[s_t=\sqrt{\frac{1}{m}\sum_{i_t=1}^m\left(x_{i_t}-\overline{x}_t\right)^2}\]

with~\(\overline{x_t}\) the mean of the~\(t^{th}\) window.
Subsequently, EVI is calculated as the~relative change
of~\(\left(s_t\right)\) between two consecutive rolling windows:

\[EVI_{t-1,t}=\frac{s_t-s_{t-1}}{s_t}\]

~~~~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:

\[Ind_{EVI_{t-1,t}} = \begin{cases}   1  \;\;\; if \;\;\; EVI_{t-1,t} \geq c \;\; \wedge \;\; y_t \geq \overline{\mu}_{t:t-7} \\  0  \;\;\; \text{otherwise} \end{cases}       \]

\par\null

\subsection*{Case definition and desired
accuracy}\label{case-definition-and-desired-accuracy}

~~~ The user should provide the minimum rise in cases that, if present,
should be detected. A case definition can be the rise in mean the number
of cases between two consecutive weeks~that exceeds a
threshold~\(r\):

\[\frac{\overline{\mu}_{t:t-7}}{\overline{\mu}_{t:t+7}}\le r\]

with~\(0\le r\le1\).

~~~~The accuracy of~EVI, given the specified case definition, depends
on~\(m\)~and~\(c\), which should be selected
in a way to achieve a desired accuracy target. Several strategies are
available. One option is the selection
of~\(m\)~and~\(c\) values that lead to
the~simultaneous optimization of the sensitivity~\(\left(Se\right)\) and
the specificity~\(\left(Sp\right)\) for~EVI, the maximization of the
Youden index \(\left(J=Se+Sp-1\right)\)\hyperref[csl:12]{(Fluss et al., 2005)} and, hence, the
overall minimization of the false results (i.e. both false positive and
false negative early warnings). Another approach could be to
select~\(m\) and~\(c\) such that~the
highest~\(Se\left(or\ Sp\right)\) is achieved with~\(Sp\left(or\ Se\right)=1\) or not
dropping below a critical value (e.g. 95\%). Advanced Receiver Operating
Characteristic curve analysis can also be performed~\hyperref[csl:13]{(Zweig and Campbell, 1993)}
and selection of critical values can be based on indices that quantify
the relative cost of false positive (i.e., falsely predicting an
upcoming epidemic wave) to false negative (i.e., failing to predict an
upcoming epidemic wave) warnings, like the misclassification cost
term~\(\left(MCT\right)\).~

\subsection*{Generation of an early
warning}\label{generation-of-an-early-warning}

~~~~Every time a new time point~\(t\) is observed, the
model uses all of the observed cases up to~\(t\) to decide
whether it should issue an early warning, at time
point~\(t\). The steps are:~

\par\null

\begin{enumerate}
\tightlist
\item
  Observed cases up to~\(t\) are analyzed for all possible
  values of the window size~\(\left(m\in\left[1,m_{\max}\right]\right)\) and
  threshold~\(\left(c\in\left[0,1\right]\right)\).~
\item
  For each of the~\(m\ and\ c\) combinations,
  the~\(Se_{t_{m,c}}\)and~\(Sp_{t_{m,c}}\)are estimated for the
  predefined case definition (Eq. 4).~
\item
  The ~\(m'\) and~\(c'\) that give the best
  \(Se_{t_{m',c'}}\) and~\(Sp_{t_{m',c'}}\) combination are selected.
\item
  For~\(m'\) and~\(c'\), the value
  of~\(Ind_{EVI_{t,t-1}}\) is determined at the most recent time
  point~\(t\) and a decision is made on whether or not a
  warning signal is issued.
\end{enumerate}

\subsection*{Accuracy and Predictive
Values}\label{accuracy-and-predictive-values}

~~~~Further, at each time point~\(t\), the probability of
observing a rise or drop in the future cases, given that an early
warning was issued or not, can be calculated as the
positive~\(\left(PV_t+\right)\) and negative \(\left(PV_t-\right)\) predictive
value, respectively:

\[PV_t+=P(D+\mid T+)=\frac{p_{1:t}Se_{t_{m',c'}}}{p_{1:t}Se_{t_{m',c'}}+\left(1-p_{1:t}\right)\left(1-Sp_{t_{m',c'}}\right)}\]

\[PV_t-=P(D-\mid T-)=\frac{\left(1-p_{1:t}\right)Sp_{t_{m',c'}}}{\left(1-p_{1:t}\right)Sp_{t_{m',c'}}+p_{1:t}\left(1-Se_{t_{m',c'}}\right)}\] where~\(p_{1:t}\) is the proportion of events
satisfying the condition of Eq. 4 up to time point~\(t\).

Once the entire time series data have been observed, the
overall~\(Se_{EVI}\) can be estimated as the fraction of the total
number of occurrences for which an early warning has been issued, given
that the case definition (Eq. 4 ) holds~\((P (T+ \mid D+))\), divided by
the total number of occurrences that the case definition
holds~\((P(D+))\). Similarly, the overall~\(Sp_{EVI}\) is
calculated as the fraction of the total number of occurrences for which
an early warning was not issued given that the expected rise of cases
was not observed, that is, the case definition is not
true,~\((P(T- \mid D- ))\) divided by the total number of occurrences that
the case definition is not true~\(\left(P\left(D-\right)\right)\):

\[Se_{EVI}=\frac{P\left(T+\mid D+\right)}{P\left(D+\right)},\ Sp_{EVI}=\frac{P\left(T-\mid D-\right)}{P\left(D-\right)}\]

\par\null

\subsection*{Sensitivity analysis}\label{sensitivity-analysis}

~~~~The performance of EVI depends on the specified case definition
(i.e.,~\(r\)) and the desired accuracy. Ideally, in the
presence of historical data, various case definitions
and~\(r\) values should be explored to~identify
combinations that provide the optimal monitoring of an epidemic.~

\selectlanguage{english}
\FloatBarrier
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\end{CJK}\end{document}

