Aryee et al. BMC Public Health         (2018) 18:1292 
https://doi.org/10.1186/s12889-018-6221-z
RESEARCH ARTICLE Open Access
Estimating the incidence of tuberculosis
cases reported at a tertiary hospital in
Ghana: a time series model approach
George Aryee1* , Ernest Kwarteng2, Raymond Essuman1, Adwoa Nkansa Agyei2, Samuel Kudzawu3,
Robert Djagbletey1, Ebenezer Owusu Darkwa1 and Audrey Forson2
Abstract
Background: The incidence of Tuberculosis (TB) differs among countries and contributes to morbidity and mortality
especially in the developing countries. Trends and seasonal changes in the number of patients presenting with TB
have been studied worldwide including sub-Saharan Africa. However, these changes are unknown at the Korle-Bu
Teaching Hospital (KBTH). The aim of this study was to obtain a time series model to estimate the incidence of TB
cases at the chest clinic of the Korle-Bu Teaching hospital.
Methods: A time series analysis using a Box-Jenkins approach propounded as an autoregressive moving average
(ARIMA) was conducted on the monthly TB cases reported at the KBTH from 2008 to 2017. Various models
were stated and compared and the best was found to be based on the Akaike Information Criterion and
Bayesian Information Criterion.
Results: There was no evidence of obvious increasing or decreasing trend in the TB data. The log-transformed of
the data achieved stationarity with fairly stable variations around the mean of the series. ARIMA (1, 0, 1) or ARMA (1,1)
was obtained as the best model. The monthly forecasted values of the best model ranged from 53 to 55 for the year
2018; however, the best model does not always produce the best results with respect to the mean absolute and mean
square errors.
Conclusions: Irregular fluctuations were observed in the 10 -year data studied. The model equation to estimate the
expected monthly TB cases at KBTH produced an AR coefficient of 0.971 plus an MA coefficient of − 0.826 with a
constant value of 4.127. The result is important for developing a hypothesis to explain the dynamics of TB
occurrence so as to outline prevention programmes, optimal use of resources and effective service delivery.
Keywords: Tuberculosis, Time series, Incidence, Estimate, Forecast
Background tobacco use have much higher risk of falling ill” [2]. The
Tuberculosis (TB) is one of the infectious diseases dis- incidence of Tuberculosis varies among different coun-
tressing many countries widely and transmitted by the tries worldwide. It is estimated that one-third(1/3) of the
bacterium known as Mycobacterium Tuberculosis [1]. world’s population has been plague-ridden with the M.
According to the World Health Organisation (WHO), tuberculosis, particularly in the developing countries, as
“persons with TB bacteria have a 5-15% lifetime risk of a major cause of morbidity and mortality worldwide
falling ill with TB [2]; however, persons with compro- [3, 4]. An estimated 9 million new cases of tubercu-
mised immune systems such as people living with losis arise annually with an estimated 1.7million deaths
HIV(PLWH), malnutrition or diabetes, and those with globally [5]. In 2015, the highest number of new TB cases
occurred in Asia (61%) whilst 26% in Africa and usually
infect adults in their fecund years [2]. In Ghana, it is
* Correspondence: garyee43@gmail.com
1Department of Anaesthesia, School of Medicine and Dentistry, University of estimated that each year over 46,000 new cases of TB
Ghana, Legon, Ghana occur [6].
Full list of author information is available at the end of the article
© The Author(s). 2018 Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0
International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and
reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to
the Creative Commons license, and indicate if changes were made. The Creative Commons Public Domain Dedication waiver
(http://creativecommons.org/publicdomain/zero/1.0/) applies to the data made available in this article, unless otherwise stated.
Aryee et al. BMC Public Health         (2018) 18:1292 Page 2 of 8
Globally, the annual TB incidence has decreased by an plotted for the period 2008–2017 to identify the various
average of 1.5% since 2000 which needs to increase to a time series components in the data. The data were
4–5% yearly drop to attain the 2020 milestones of the log-transformed and re-plotted. Stationarity was assessed
End TB strategy. Between 2000 and 2015, an estimated and confirmed using the Augmented Dickey-Fuller
49 million lives were rescued as a result of TB diagnosis (ADF) test on the transformed data. The series was
and treatment [2]. While efforts being made in dealing judged stationary with the p-value of the ADF test ≤5%
with the condition leading to a decline through various level of significance. An Autocorrelation Function (ACF)
TB programmes and interventions, trends and seasonal and Partial Autocorrelation Function (PACF) were plot-
models associated with the occurrence of TB have also ted to obtain the orders p and q of AR and MA respect-
been studied extensively [7–12]. In the Ashanti Region ively. Upon determining the order of AR and MA terms,
of Ghana, Gyasi-Agyei and colleagues [12] found that tu- the model was obtained. The autoregressive model equa-
berculosis incidence studied can best be modelled with tion of order (p) is expressed as: Yt =Φ1Yt − 1+2ΦYt − 2 + .
an autoregressive moving average [ARMA (1, 0) or AR . ……… +ΦpYt − p +wt, where Yt represents the current
(1)], and was predicted to be steady between April 2013 value of the series, Yt-1,………Yt-p denotes the prior
and April 2015. However, at the Korle-Bu Teaching Hos- values of the same series whilst wt is the white noise
pital, in the Greater Accra Region, there is a paucity of andΦ1,……., Φpare the regression coefficients of the
information regarding the trends and peaks period of re- model.
ported TB cases referred to the facility. Trends in the in- The moving average model equation of order (q) is
cidence of TB has the propensity to impact significantly also written as:
on planning and more efficient use of the facility’s re- Yt =wt + ϕ1wt − 1 + ϕ2wt − 2 +……… + ϕqwt − q, where Yt
sources as well as public health intervention pro- denotes the current value of the series, wt... ... ... wt-q are
grammes. Therefore, the aim of this study was to obtain the white noise and ϕ1… ϕqare the regression coeffi-
a time series model to estimate the incidence of TB cients of the model.
cases at the chest clinic of the Korle-Bu Teaching Thus, the ARIMA model is given as ϕ (B) (1 ‐ B)dYt
Hospital. = θ (B) ωt where ϕ (B) is the operator for the AR term
given as ϕ (B) = 1 ‐ ϕ1B − ϕ2B
2 −… − ϕ PPB and θ (B) is the
Methods operator for the MA term and is given as θ (B) = 1 + θ1B
Study design and site + θ B22 +… + θ
q
qB . Where p and q represent the respect-
A time series analysis of time-dependent data compris- ive number of lags for the AR and MA terms and d rep-
ing of 120 reported monthly TB cases from 2008 to resents the order of the integration term.
2017 at the chest unit of the Department of Medicine Whilst the ARMA model is a blend of both the AR
and Therapeutics, Korle-Bu Teaching Hospital (KBTH) with order p and MA with order q expressed in the
was conducted. Korle-Bu Teaching Hospital is the lar- equation: Yt =Φ1Yt − 1 +Φ2Yt − 2 + . . ……… +ΦpYt − p +
gest and the premier teaching hospital in Ghana with a wt + ϕ1wt − 1 + ϕ2wt − 2 +……… + ϕqwt − q.
bed capacity of 2000 as at 2013. It is a major referral The model obtained was compared to other ARIMA
centre for the whole of Ghana and the West African models. The model with the least Akaike Information
Sub-region. The chest unit caters for patients with chest Criterion (AIC) and Bayesian Information Criterion
diseases such as Tuberculosis. The average number of (BIC) was selected as the best model. Diagnostic tests
patients with TB seen per month is seventy (70). Prior were done on the best-chosen model by performing a re-
permission to use the data was obtained from the Chest sidual analysis to determine the adequacy of the model;
clinic of the Korle-Bu Teaching Hospital. The study did this was done by assessing the normality and independ-
not require ethical review because the used data never ence of the residuals. The normality of the residuals was
had identifiers nor anonymous human biological mate- determined using the Quantile–Quantile (Q-Q) plot and
rials associated with them (The letter explaining this has confirmed by the Shapiro-Wilk’s test. Residual points
been attached to the study). found within the significant bounds of the ACF of the
residual plot determined the independence of the resid-
Data analysis uals and confirmed by the Ljung-Box test. The best
Data was inputted into Microsoft Excel 2013 and ana- model was used to forecast the estimated number of
lysed in R statistical software version 3.3.2. Box-Jenkins monthly TB cases. A p-value ≥5% level of significance of
time series approach put forward as Autoregressive Inte- the Shapiro-Wilk’s and Ljung-Box tests was considered
grated Moving Average (ARIMA) model was employed statistically significant. Forecasting errors such as the
for modelling. The Box-Jenkins methodology comprised Mean Square Error (MSE) and Mean Absolute Error
model Identification, Parameter Estimation, Model Diag- (MAE) of the specified models were determined to as-
nostics and Forecasting [13]. Time series of the data was certain the accuracy of the model for prediction a year
Aryee et al. BMC Public Health         (2018) 18:1292 Page 3 of 8
Fig. 1 Time plot of the series actual data (left Graph) and log-transformed of the actual data (right Graph) over the period 2008–2016
ahead. The model with the minimum errors was consid- number of 7676 cases. The time plot of the data showed
ered accurate for prediction but the best model does fairly the same level from 2008 to 2012 but began to de-
may not necessarily give the best forecasting errors. crease slowly with several irregular fluctuations within
the series (left plot in Fig. 1) with a peak in October and
Results trough in March. However, the log-transformed of the
The monthly TB time-dependent data consisted of 120 time series data achieved quite a number of stable fluc-
data points for the period (2008–2017) with a total tuations as shown in Fig. 1(right plot).
Fig. 2 Correlogram plot of the ACF (left Graph) and PACF (right Graph) for the log-transformed of the actual data at various lags. The horizontal
dash lines in the ACF and PACF are the significant bounds
Aryee et al. BMC Public Health         (2018) 18:1292 Page 4 of 8
Table 1 Comparison between formulated models and Ideal Model Other models formulated were compared to the em-
No. Models AIC BIC pirical model ARIMA (1, 0, 1) as shown in Table 1 using
1. ARIMA(1,0,1) − 32.76 −26.61 their AICs and BICs.
2. ARIMA(1,0,2) − 30.81 − 16.87 From Table 1, the best model was selected based on
the minimum AIC and BIC values. It was found that
3. ARIMA(2,0,1) −30.81 −16.88
ARIMA (1, 0, 1) had the minimum AIC and BIC. Hence
4. ARIMA(3,0,1) − 28.90 −12.17 ARIMA (1, 0, 1), the empirical model was selected as
5. ARIMA(3,0,2) −27.68 −8.17 the best model among the other models formulated.
6. ARIMA(0,0,1) −13.02 −4.66
7. ARIMA(1,0,0) −17.97 −9.6
8. ARIMA(0,0,2) −16.71 −5.56 ARIMA (1, 0, 1) with zero mean model diagnostics
Figure 3 showed that a plot of the model residuals was
9. SARIMA (1,0,1)*(1,0,1)12 −30.55 −13.83 fairly constant.
10. SARIMA (1,0,2)*(0,0,1)12 − 29.19 −12.46 The Q-Q plot in Fig. 4 also showed the model resid-
11. SARIMA (2,0,1)*(1,0,1)12 −28.67 −9.16 uals were normally distributed as most of the residual
12. SARIMA (2,0,1)*(1,0,0)12 −29.07 −12.35 points were closed to the normal line. The Shapiro-
Wilk’s normality test confirmed normality of the resid-
From Fig. 1(right plot), it was found that the uals (W = 0.986, p-value = 0.270).
log-transformed of the series achieved stationarity. The The standardised residuals plot in Fig. 5 (at the top)
Augmented Dickey Fuller test statistically confirmed sta- were random and the lags of the autocorrelation of re-
tionarity of the series (ADF = − 3.84, p-value = 0.020). siduals (middle plot) were all within the significant
The ACF indicated spikes at different lags (i.e 0,1, 2, 3, bounds. The ACF of residuals ranged between − 0.2 and
4 etc.) above the significant bounds and the PACF also 1.0. All the p-values of the Ljung-Box test (Fig. 5 bottom
indicated spikes at lags 1 and 2 above the significant plot) which ranged between 0 and 1 were all above the
bounds (Fig. 2). significant line indicating the residuals were independent
From the plots of the ACF and PACF (Fig. 2), the model (χ2 = 8.951; p-value = 0.984).
ARIMA (1, 0, 1) was selected with 0.953 and − 0.784 as
the regression coefficients of AR (1) and MA (1) respect-
ively. The estimated intercept of the model was 4.137.
Fig. 4 Quantile-Quantile plot of the model residuals. The data points
Fig. 3 Plot of the standard residuals of the actual data period (2008– around the diagonal line (line of symmetry) in the plot represent the
2017) and the forecasted figures for 2018 of the obtained model model residuals to assess if the model residuals are from a
[ARIMA (1,0,1)] around a horizontal constant line normal distribution
Aryee et al. BMC Public Health         (2018) 18:1292 Page 5 of 8
Fig. 5 Plots of the standardised residuals (at the top), ACF of residuals (at the middle) and Ljung-Box statistic (at the bottom). The data points in
the standardised residuals plot determine the randomness of the residuals for the actual data period (2008–2017) and the forecasted year (2018).
The data points in the ACF of the residuals which ranged from -0.2 to 1.0 at various lags assessed the independence of the autocorrelation
function (ACF). The data points of Ljung-Box statistic which ranged from 0.0 to 1.0 at various lags represent the p-values of the residuals.
The horizontal dash lines in the ACF of the residuals and Ljung-Box statistic are the significant bounds
Forecasting the other models [i.e ARIMA (1,0,1) and SARIMA
Table 2 showed the monthly (January–December) fore- (1,0,1)*(1,0,1)12].
cast of tuberculosis cases for the year 2018 which ranged
from 53 to 55 with their respective 95% confidence Discussion
interval. The monthly forecasted TB cases depicted a The time series plot of the data showed a general lev-
slow steady rise in the incidence of TB cases for the year elled trend although there was a slow downward trend
2018 as shown in the line of the shaded region in Fig. 6. with irregular variations. The series was fairly of the
same level between 2008 and 2012 but began to decrease
Forecasting accuracy slowly thereafter. Thus, there was no clear evidence of a
Table 3 depicts the mean absolute error (MAE) and trend in the series and the mean of the log-transformed
Mean squared error (MSE) which determined the fore- of the series was constant and the variance was fairly
casting accuracy among the competing models. ARIMA stable over time. Both the ACF and PACF tailed off to
(2,0,1) yielded MAE and MSE of 15.08 and 297.25 respect- zero indicating stationarity of the series. A test of sta-
ively which produced the minimum errors compared to tionarity using the log-transformed data showed the
Aryee et al. BMC Public Health         (2018) 18:1292 Page 6 of 8
Table 2 Forecasted values for the year 2018 Table 3 Forecasting error
Month Point forecast 95% Confidence interval Model Mean Absolute Mean squared
January 53 35–79 Error(MAE) Error (MSE)
ARIMA (1,0,1) 15.75 307.92
February 53 36–80
ARIMA(2,0,1) 15.08 297.25
March 53 36–81
SARIMA (1,0,1)*(1,0,1) 15.75 300.25
April 54 36–81 12
May 54 36–81
June 54 36–82 ARMA (1,1) model produced the best model for the
July 54 36–83 10-year TB data.
August 54 36–83 This model was chosen from among the other models
because it has a minimum AIC and BIC compared to
September 55 38–89
the other competing models. The authors found that a
October 55 36–84 plot of the standardized residuals (Fig. 3) was constant
November 55 36–84 over time and the normality test of the residuals(Fig. 4),
December 55 36–85 as well as Ljung-Box test (Fig. 5) depicting the appropri-
ateness of the best-obtained model for forecasting. How-
series was stationary (ADF = − 3.71, p-value = 0.026), im- ever, the best model selected does not necessarily give the
plying that the mean of the TB data is independent of best results as far as the mean absolute and the mean
time. This is an evidence of the lack of apparent trend in square errors are concerned (Table 3). The forecasted
the series of the log-transformed data. values in this study exhibit a monthly marginal steady in-
We found that the ACF showed significant lags at lag crease in TB incidence for the year 2018.
0, lag 1, lag 2, lag 3 etc. and PACF showed significant Various time series models or methods have been
lags at lag 1 and lag 2. However, lag 1 was selected for used in predicting monthly tuberculosis incidence.
both ACF and PACF since they yielded a better estimate ARIMA model has been shown as the best suitable
of the model than other lags. Hence, ARIMA (1, 0, 1) or model for predicting TB cases among other forecast-
ing methods such as Moving Average, Artificial Neural
Network, Decomposition, linear regression and Holt-win-
ters models [14].
Generally, TB is not known to exhibit seasonality just
like malaria, diphtheria, chickenpox, rotavirus, cholera
among others [15], yet several studies have investigated
the seasonal effect of TB using seasonal ARIMA model
showing variations with peaks in the summer, autumn,
winter and spring [16–18]. Longitudinal studies by
Moosazadeh et al. [1, 19, 20] in Iran on diagnosed tuber-
culosis cases using Box-Jenkins time series approach
yielded seasonal ARIMA (0, 1, 1) (0,1,1) 12 with peaks in
spring and summer. These findings were comparable to
a recent national study in China by Wang et al. [21] in-
volving 13 years of monthly TB data which produced the
same seasonal model with TB peaks in spring. Another
study by Willis et al. in the USA using the Decompos-
ition Time Series method indicated seasonality in TB
data with a peak in spring and trough in late fall [22].
Bras et al. [23] used seasonal trend LOESS (STL) to
model trend and seasonality of pulmonary tuberculosis
(PT) in Portugal. Their findings indicated that SARIMA
(2, 1, 0) (0,1,1)12 was the best fit for the data and PT in-
cidence peaked in the early spring and trough in winter.
A study from South Africa also produced similar tuber-
Fig. 6 Plot of the actual data and the forecasted values from ARIMA culosis seasonality [24].
(1,0,1). The data points represent the plot of the data from 2008 to However, in this study, Box-Jenkins time series ap-
2017 and the shaded region shows the forecasted figures for 2018
proach on monthly TB cases produced non-seasonal
Aryee et al. BMC Public Health         (2018) 18:1292 Page 7 of 8
ARIMA (1, 0, 1) model even though a peak was ob- Conclusions
served in the actual data in October and trough in There was no trend nor seasonal changes in the Univariate
March. These findings support a previous study done in time series data of TB cases at the Korle-Bu Teaching
the Ashanti Region of Ghana by Gyasi-Agyei and col- Hospital. Irregular or random fluctuations were observed in
leagues [12]. The authors used aggregated TB cases in the 10-year-data studied. The TB data was best modelled
the region, yet could not determine any seasonal pattern. with ARIMA (1, 0, 1) or ARMA (1, 1). The model equation
Therefore, the data in the region was best modelled with to estimate the expected monthly TB cases at KBTH pro-
ARMA (1, 0) or AR (1). duced an AR coefficient of 0.971 plus an MA coefficient of
Most of the studies predicting seasonal variations − 0.826 with a constant value of 4.127. There was a slow
pertaining to the incidence of TB were done in the de- steady increase in the monthly forecasted values for the
veloped countries. Explanations regarding seasonal varia- year 2018. This is essential for developing a hypothesis to
tions are not well recognised but it has been assumed to explain the dynamics of TB occurrence so as to plan pre-
be attributed to cold weather and living in restricted vention programmes, optimal use of resources and effective
spaces, which could contribute to the differences be- service delivery.
tween previous studies done and the current study. Most
developed countries endure relatively severe cold wea- Additional file
ther during the year compared to developing countries
like Ghana. During such seasons, it has been noted that Additional file 1: Reported monthly TB cases data set for a ten-year
the incidence of TB is high due to the delicateness of period at the chest clinic-Korle-bu teaching hospital. (XLSX 12 kb)
the immune system as a result of low level of Vitamin D
production in winter [25]. A decline in sunlight which Abbreviations
ACF: Autocorrelation Function; ARIMA: Autoregressive Integrated Moving
leads to a drop in Vitamin D may markedly intensify the Average; ARMA: Autoregressive moving average; PACF: Partial Autocorrelation
chances of getting tuberculosis [26]. Also, the chances of Function; PLWH: People Living with HIV; PT: Pulmonary Tuberculosis;
TB transmission upsurge in the winter when there is TB: Tuberculosis
overcrowding, reduced airflow and increased humidity Acknowledgements
from indoor activities [27]. The authors wish to show their appreciation to the records unit of Chest and
Another factor that may have accounted for non-sea- Infectious Diseases, Korle-Bu Teaching Hospital, Accra, Ghana for accepting
to use the data.
sonality in this study could be a delay in diagnosis or
delay in the presentation of the disease. Therefore, the Funding
data used in this study may encompass the incorrect None
time of diagnosis or onset of TB. Most of the previous Availability of data and materials
studies done used aggregated national data, allowing for The data used and/or analysed during the current study have been
higher TB cases which may have revealed the seasonal attached as Additional file 1.
effect in their studies. However, this study was limited to Authors’ contributions
one tertiary referral hospital in Ghana receiving compli- GA and EK developed the concept and writing of the manuscript. GA and
cated TB cases from primary and secondary healthcare RA analysed the data. AF, SK, RD, EOD and ANA contributed to writing and
facilities throughout the country. This may, therefore, review of different sections of the manuscript. Prior to submission, all the
authors read and approved the manuscript.
have had an influence on the non-seasonal behaviour of
the data. Authors’ information
Demographic and co-morbid variables such as age, GA and RE are Senior Research Assistants of the Department of Anaesthesia,
University of Ghana School of Medicine and Dentistry, College of Health,
gender, socioeconomic status, HIV/AIDS and diabetes as Sciences, University of Ghana, Accra.
well as climatic data such as temperature, rainfall and EK is a Research Assistant of the Chest clinic of the Department of Medicine,
humidity associated with TB transmission were not University of Ghana School of Medicine and Dentistry, College of Health,
Sciences, University of Ghana, Accra.
accounted for in this retrospective study. Hence the SK is a Laboratory Technologist at the chest clinic Korle-Bu Teaching Hospital,
forecasted results must be explained with caution but Accra, Ghana.
other variables must be included to allow more robust EOD and RD are consultant Anaesthetists of the Department of Anaesthesia,
University of Ghana School of Medicine and Dentistry, College of Health,
time series models or methods in future studies. The Sciences, University of Ghana, Accra.
study was conducted in Korle-Bu Teaching Hospital, AF and AAN are Consultants in Internal Medicine, University of Ghana School
thus, the results may not be applicable to other settings of Medicine and Dentistry, College of Health, Sciences, University of Ghana, Accra.
in Ghana. However, the results of this study may be Ethics approval and consent to participate
helpful in putting up a proposition to interpret the Ethics approval was not obtained but the data in this manuscript was used
changes of the event noticed in order to establish epi- with prior permission from the Chest Clinic of the Korle-Bu Teaching Hospital.
demiological surveillance, proper allocation and use of Consent for publication
health resources in Ghana. Not Applicable
Aryee et al. BMC Public Health         (2018) 18:1292 Page 8 of 8
Competing interests 21. Wang H, Tian CW, Wang WM, Luo XM. Time-series analysis of tuberculosis
The authors declare that they have no competing interest. from 2005 to 2017 in China. Epidemiol Infect. 2018;146:935–9.
22. Willis MD, Winston CA, Heilig CM, Cain KP, Walter ND, Mac Kenzie WR.
Seasonality of tuberculosis in the United States , 1993 – 2008. Clin Infect
Publisher’s Note Dis. 2012;54(11):1553–60.
Springer Nature remains neutral with regard to jurisdictional claims in 23. Bras AL, Gomes D, Filipe PA, De SB, Nunes C. Trends , seasonality and
published maps and institutional affiliations. forecasts of pulmonary tuberculosis in Portugal. Int J Tuberc Lung Dis. 2014;
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Author details
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Ghana, Legon, Ghana. 2Department of Medicine and Therapeutics, School of Africa , Using a hybrid model. Int J Environ Res Public Heal. 2016;138:757.
Medicine and Dentistry, University of Ghana, Legon, Ghana. 3Department of 25. Wingfield T, Schumacher SG, Sandhu G, Tovar MA, Zevallos K, Baldwin MR,
Chest and Infectious Diseases, Korle-Bu Teaching Hospital, Accra, Ghana. et al. The seasonality of tuberculosis , sunlight , vitamin D , and household
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Received: 9 March 2018 Accepted: 14 November 2018 26. Venturini E, Facchini L, Martinez-alier N, Novelli V, Galli L, De Martino M, et
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