The Ebola contagion and forecasting virus: Evidence from four African countries

Nadhem, Selmi; Nejib, Hachicha D. Article The Ebola contagion and forecasting virus: Evidence from four African countries Health Economics Review Provided in Cooperation with: Springer Nature Suggested Citation: Nadhem, Selmi; Nejib, Hachicha D. (2015) : The Ebola contagion and forecasting virus: Evidence from four African countries, Health Economics Review, ISSN 2191-1991, Springer, Heidelberg, Vol. 5, Iss. 16, pp. 1-15, https://doi.org/10.1186/s13561-015-0047-5 This Version is available at: https://hdl.handle.net/10419/150496 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. Sofern die Verfasser die Dokumente unter Open-Content-Lizenzen (insbesondere CC-Lizenzen) zur Verfügung gestellt haben sollten, gelten abweichend von diesen Nutzungsbedingungen die in der dort genannten Lizenz gewährten Nutzungsrechte. Terms of use: Documents in EconStor may be saved and copied for your personal and scholarly purposes. You are not to copy documents for public or commercial purposes, to exhibit the documents publicly, to make them publicly available on the internet, or to distribute or otherwise use the documents in public. If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. http://creativecommons.org/licenses/by/4.0/ RESEARCH Open Access The Ebola contagion and forecasting virus: evidence from four African countries Selmi Nadhem * and Hachicha D Nejib Abstract This paper is focused on examining the number of deaths’increases participation in the propagating the Ebola virus during the period ranging from March to October 2014. An application of the MGARCH-DCC model regressions on four countries has led to discover that the finding that human contact play a significant role in transmitting the Ebola virus. Our findings also reveal that Guinea has already suffered from a spread-like virus originating from Sierra Lione and Liberia. Noteworthy also, other countries are now liable to such a risk; for instance, Nigeria is a country vulnerable to the propagation of this virus. Consequently, we undertake to conduct our forecasts for EGARCH model estimates implements; which has estimated a decrease in the Ebola virus incurred number of deadly Ebola virus over the two months following the November and December. Keywords: Ebola 2014; Contagion; Virus; EGARCH; MGARCH-DCC Background The Ebola virus belongs to a family of viruses called filovirus a , commonly carrying the Marburg virus. Discovered in 1976 in the western equatorial province of Sudan and in a nearby region of northern Zaire (now Democratic Republic of Congo (DRC)) [1,2], the Ebola virus had affected several African countries between 1976 and 2008 (Centers for Disease Control and Prevention, 2014; [3,5]; Leroy et al., 2004; [6-12] and [13]. Previous Ebola virus modeling papers have examined its 1995 outbreak in Kikwit, DRC [14-16] and [17,18]), and the in Uganda in 2000–2001 (see [16,19] and [17]). Currently, however, the Ebola has also had certain implications in Guinea, Sierra Leone and Liberia. Indeed, a small number of cases are reported in Nigeria (21 deaths) and a single case in Senegal (travelers arriving from Guinea). This consists, actually, in is the contagious Ebola virus spread among countries. Until September 8, 2014, 4366 cases had been reported, including 2218 deaths. The Ebola virus is discovered to be the most dangerous epidemic, recently resulting in a remark able increasing in the number of patients and deaths it’ssince the outbreak out of this virus. The first Ebola outbreak among human beings occurred in 1976 [20], with a spread contagion due mainly to travelling. An infected person travelled for instance from Guinea to Liberia may be carrying the infection, which would in turn infect some other people there (The institute of tropical medicine, 2013 [21]). Based on the Ministry Health reported information (September 2014), Guinea recorded 1040 confirmed deaths highly exceeding the confirmed cases. Figures relevant to the Ebola cases and deaths have been provided by WHO, with the most prominent number being registered in Liberia, (more than 1760), while, 2450 probable deaths have been reported with respect to Sierra Leone. These data are based on official information reported by the Ministry of Health up the October 5th for both Guinea and Sierra Leone and October 4th for Liberia. Concerning Guinea, one might well note that the number of confirmed cases is important in respect of the probable and suspected cases. Still, the Liberian probable cases remain the most significant in relation to the entirety of: confirmed and suspected cases. As for Sierra Lione confirmed cases, they constitute the highest proportion as compared to the other three remaining countries. Finally, Nigeria exhibits the weakest level of cases regarding the three indicators involving confirmed, probable and suspected cases. Actually, five species of Ebola have been detected to prevail in Bundibugyo, Côte d’Ivoire, Reston b [22], Sudan and Zaire. The outbreaks of Ebola virus associated of * Correspondence: [email protected] Department of Quantitative Methods, Faculty of Economics and Management of Sfax, Sfax, Tunisia © 2015 Nadhem and Nejib. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited. Nadhem and Nejib Health Economics Review (2015) 5:16 DOI 10.1186/s13561-015-0047-5 hemorrhagic fever has taken place mainly in Africa, with a death rate comprised 25% and 90%. As a matter of fact, the Ebola virus is transmitted to human beings from wild animals, and then spreads from human-to-human. The 2014 Ebola epidemic is the most prominent in history, affecting aerial several countries. Since the beginning of 2014, some cases of Ebola virus infection have been reported in several sub-Saharan African countries. These countries, health authority’s line with the World Health Organization (WHO) and its different partners, have all been mobilized to prevent the virus transmission to other countries. Travelers to or from these countries can find relevant health care related information on the Ministry of Foreign Affairs and International Development website. In effect, Ebola is a communicative disease, as it can be transmitted by either contact with blood, body fluids or tissues of infected subjects, or sick or dead animals. So for airborne transmission has been indicated to occur. The incubation period (the time between the infection and symptoms’onset) ranges from 2 to 21 days. A person who bears no symptoms is not contagious. Most often, the risk degree of the Ebola virus contamination is considered too low during the disease the early stages but contagion of the Ebola virus makes the symptoms worsen or improve. People are infectious as long as their blood and secretions bear traits of the virus. The outbreak of the Ebola virus tends to occur at irregular intervals in a medical care environment. The first trigger was reported in the Forest region of Guinea. The Ebola outbreaks in Liberia, Sierra Leone and Nigeria were contained [23]. Until recently, the Ebola virus has been raging mainly in central and eastern Africa. Contagion occurred between 1976 and 2012, in the Democratic Republic of Congo and Sudan along with the three other countries which had remarkably been affected by serious epidemics’, namely; Gabon, Uganda and Congo. Although Ebola has resulted in a high mortality rate, it’s the real varies between 25% and 90%, depending on the strain. According to the WHO (October 2014), Ebola currently stands within an average of 54% of total deaths [13]. Based on Figure 1, and for an effective combating of the epidemics some conditions and resources need be considered and ensured, namely case management, contact tracing and monitoring, laboratory quality, safe burials along with social mobilization. The community is involved in the outbreak control as early-stage awareness of the Ebola virus infection and the possible protective measures are likely to be successful. In fact, they have greatly helped in reducing contamination among humans. Messages about risk reduction should rather focus on social mobilization and community-based awareness campaigns to help reduce the disease propagation risk [24]. Indeed, the WHO intends to reduce the Ebola virus propagation risk between animals and human beings, about all contact with monkeys. In fact animals should be handled with gloves as well as other appropriate protective clothes. More importantly, animal resource based foods must be thoroughly cooked before eating them. Other various factors may well contribute greatly in increasing or decreasing contamination risk [25]. In this respect, the present study attempts to provide certain suggestions whereby contamination risk to human beings can be reduced, a risk occurring mainly through direct contact with similar subjects, and, predominately with their body fluids. Normally, people need to wear gloves and appropriate personal protective equipment when taking care of an affected patient at home. To prevent such a contagion, people also have to wash out hands after visiting patients in hospitals. The various measures necessary for containing the Ebola outbreak include corpses ‘rapid burial and identifying subjects likely to be in contact with an infected person. Concerning fight against infection in health care facilities, health workers should always apply the standard precautions on taking care of patients, regardless of the presumed diagnosis. These precautions pertain to the basic rules regarding hygiene including, wearing personal protective equipment (to protect oneself against splashing or direct contact with infected materials), safety injections as well as funeral rites. Health workers responsible for taking care of the Ebola virus infection suspected or confirmed cases must avoid any contact with the patient’s blood or body fluids and with such contaminated surfaces as clothing and bed linen materials. On standing too close too sick person (say, i.e. within a distance of less than a meter), one has to should wear face protection (face shield or surgical mask and goggles), a clean sterile gown and gloves (sterile medical procedures). In him laboratory staff might well face the same risk. The Ebola samples taken from human beings or 0 500 1000 1500 2000 2500 3000 confirmed propable suspected Figure 1 Confirmed probable, and suspected of Ebola virus cases. Nadhem and Nejib Health Economics Review (2015) 5:16 Page 2 of 15 animals should be handled by specialized trained staff and processed in suitable equipped laboratories. The WHO aims to prevent Ebola outbreaks by providing disease control and helping countries at-risk to develop special safe guarding measures and preparing plans. In this context, this present paper is dedicated to provide some general guidance regarding the fight against outbreaks caused by these two viruses. Both of the Ebola disease and Marburg virus outbreak and curbing stages are: preparation, warning, control and evaluation. Whenever an outbreak is detected, the WHO is involved in providing assistance through monitoring, supporting patients, mobilizing laboratory services, research contacts, infection control, logistic support training and assistance with safe burial practices. In This regards, the WHO has developed detailed preventive information to curb prevent and fight against with Ebola virus contamination. Since December 2013, an Ebola virus originating hemorrhagic fever epidemic has predominantly prevailed in West Africa (Guinea, Liberia, Sierra Leone, and Nigeria are the major affected countries in addition to a confirmed case in Senegal). For the first time, such in West African countries as Congo, DRC and Uganda have been affected by with the Ebola virus. On the interest, international SOS organization is monitoring the situation closely and updates the site with the latest available information. In this respect, the present work constitutes a case study concerning of the Ebola virus transmission among the most affected countries by the virus (Guinea, Sierra Leone, Liberia and Nigeria). This paper is designed to check whether these countries can be actually constitute a source of recent contagion judging by the number of Ebola virus incurred deaths. More particularly, this work is focused on showing whether the Ebola virus can produce contagious effects within the studied countries’in conditional volatilities over the period ranging between March and October 2014. Previous contagion related studies have somewhat failed to take account for the important distinction between the concepts of interdependence and contagion c . Indeed, the conditional test results’dynamic correlation reveal well that the contagion effects appear to be multidirectional since the Ebola virus emanating return shocks can actually sweeping across the entirely of the markets. Yet, the contagion in-volatility effects are mostly driven by the Ebola virus originating negative return shocks. Such an empirical finding reveals well that the number of death shocks can turn out to be contagious not only at the volatility level, but also at the mean level, indicating that Ebola outbreak can be a major source of contagion over the period March-October 2014. In fact, the objective of the paper lies in highlighting that deaths’number does actually constitute a contagion channel. In a first step, this study is concerned to test the persistence of shocks and the stylized facts of these countries’ considered Deaths returns number through EGARCH process [4]. In effect, the presence of structural changes in the series of conditional variances as detected via the ICSS algorithm heave motivated us to study Conditional Correlation Dynamics [26] in a second stage as a procedure to determine the contagion effect across markets. Finally, this study serves to help predict the number of Deaths’ volatility across the EGARCH process. In this study, the cited process involves four major steps. On a first stage, the Iterated Cumulative Sums of Squares algorithm (ICSS) of [27] is applied to detect the presence of structural breaks in deaths number relevant to the studied countries. On a second stage, and structural breaks and asymmetry to be taken into estimation, the study undertakes to analyze the univariate EGARCH model and bring the structural breaks’dummy variables into variance equation. Compared to the standard specification, the EGARCH framework exhibits several advantages namely: no need is imposed to artificially implement a non-negative constraint on the model parameters, while asymmetries are allowed under the EGARCH formulation. Dubbed dynamic conditional correlations such a methodology has actually been developed by [26]. In reality, the major advantage lying implementing the DCC-MGARCH models consist in detecting of plausible changes noticeable in the relationships governing the variables remains prevalent in the underlying studied data. The remainder of this paper is structured as follow. Section Methods is devoted to highflying the surveyed econometric methodology. The relevant data and major empirical results are discussed in Section Empirical results. As for the ultimate section it comprises provides the pertinent concluding remarks, along with the retained economic and political implications. Methods For the purpose of the determining whether the number of deaths’volatility dynamics does actually differ from that of the countries, we, firstly, undertakes to expose the exact local Whittle and structural break tests, and, then, discuss the GARCH specifications so as to be able to capture the possible conditional dynamic dependencies likely to be noted in the number of death return volatilities. The Exact Local Whittle method The classes of semi-parametric frequency domain estimators follow the local Whittle approach as suggested by [28] and analyzed by [29] (dubbing it Gaussian semiparametric estimator). The analysis applied process is the following: yt¼μþX t−1 j¼0 φj;dxt−j;t¼1…Tð2:1Þ Nadhem and Nejib Health Economics Review (2015) 5:16 Page 3 of 15 As for the Local Whittle estimator, it defined as the maximization of the local Whittle likelihood purpose, such as: Qg;dðÞ¼log 1 mX m j¼1 λ2d jIyλj  () −2d mX m j¼1 log λj ð2:2Þ where: m=m(T) denotes a bandwidth number tending to infinity T→∞except at a slower speed than T;IλðÞ¼1 2πTX T t¼1 eitλ  2 , represents the periodogram of X t ,andg x (λ)thespectraldensityofX t ,λj¼2πj n,and j=1,…,n. A notable disadvantage as compared to log-periodogram estimation is that a statistical optimization is highly. Still, this estimator underlying assumptions are weaker than those pertaining to the log-periodogram regression (LPR) estimator. In this regard, [29] have show that while d∈− 1 2;1 2  ; ffiffiffiffi m p^ dLW −d  → dN0;1=4ðÞ ð2:3Þ Hence, the asymptotic distribution turns about to be extremely simple, which facilitates easy asymptotic inference. More particularly, this estimator is discovered to be more efficient than the LPR one. The reliability and asymptotic normality ranges concerning the Local Whittle estimator have explicitly been demonstrated by [30] and [31] to equate those associated the LPR estimator. This exact LW procedure as frequency labelled, implies replacing λ2d jIλj in (2.1) by IΔdyλj  , and is only valid if μ= 0 in (2.2). Since the relevant means are different from zero, [32] suggests demeaning {y t } with an appropriate estimator ^ μ, and computing the exact LW estimator starting from the demeaned data. So the objective function to be minimized turns out to be: REm;dðÞ¼log 1 mX m j¼1 IΔdy−^ μðÞ λj  () −2d mX m j¼1 log λj  ð2:4Þ Where: IΔdy−^ μðÞ λj is the periodogram of Δdy−^ μðÞ.For fractional differences, to be determined, it is assumed that {y t } is given by a process similar to equation (2.1). It turns out that the first sample observation y 1 is a reliable mean estimator in the case of large values of d, while the usual arithmetic mean ӯhelps ensure a significant task for small coefficient values of d.Inthisway,[32]suggestsputting forward the subsequent weighted estimator, such as: For the purpose of attaining, a feasible procedure, he considers two necessary steps, the first of which serves to determine an estimator of ^ dindependent from μin order to get an estimator of the constant: ^ μ¼^ μ ^ d  . As for the second step, the slope and Hessian of R E (m,d) are used to compute the feasible estimator as follows: ^ d2ELW ¼ ^ d− R′ Em; ^ d  R″ Em; ^ d  ð2:7Þ Besides, [32] demonstrates shows that the two-step ELW estimator (2ELW) proves to be consistent registering the same limiting distribution as the LW and ELW estimators under −0.5 < d< 2. Similarly, and as indicated as shown by [33], if an unknown mean (initial value) appears to undergo certain change by its sample average, simulations suggest that the ELW estimator is inconsistent for d> 1. It is actually for this reason that we undertake to apply the 2ELW. In addition, [33] resort to modify the ELW objective function in a bid to estimate the mean by means of combining two estimators: the sample average and the first observation. He indicates the resulting estimator as being a two Stage Exact Local Whittle (2ELW). Applying the tapered estimator of [30] in its first stage, the 2ELW estimator, bears the same N0;1 4  limit distribution for N−1 2;2  and is consistent when d>1 2. Furthermore, the 2ELW estimator finite sample performance appears to inherit the 2ELW estimator, desirable properties. Moreover, it can also be computed with prior data de-trending (2ELWd) as in [33]. Structural breakpoints detection As developed by [27] the ICSS algorithm has been applied to detect the structural breakpoints on 4 series over the study period. As a starting point, the stock return for market ion day tcan be written as: ri;t¼logPit −logPit−1 ðÞ100 ð2:8Þ where: (P i,t ) is the closing number of deaths: Next, we proceed by defining ai;t¼ri;t−μið2:9Þ where {a i,t } is with zero mean and unconditional variance σ2 t,μ i denotes country i average return. Let Ck¼X k t¼1 a2 t;k¼1;…;Tbe the cumulative sums of ^ μdðÞ¼vdðÞ  yþ1−vdðÞðÞy1ð2:5Þ vdðÞ¼ 1;d≤0:5 1þcos 4πdðÞ 2;0:5<d<0:75 0d≥0:75 8 > > < > > : ð2:6Þ Nadhem and Nejib Health Economics Review (2015) 5:16 Page 4 of 15 squares of {a t } series, then D k statistic can be calculated as follows: Dk¼Ck CT  −k T;k¼1; ::::; T and D0¼DT¼0ð2:10Þ The ICSS algorithm is adopted to detect the multiple breaks in the unconditional variance of {a i,t } series. Thus, the D k statistics based ICSS algorithm would initiate with testing the structural breaks over the whole sample. In doing so, the ICSS helps depict any significant break, by applying the new statistic to examine the break for each of the two sub-samples (defined by the break). The algorithm proceeds in such a way as the statistics become insignificant for the entirety of sub-samples defined by any significant break. On the last stage, a dummy variables set is set up for the normalized return volatility to be captured. This section is devoted to provide a through description of the wavelet transform as applied to the number of the death data decompositions, together with the multivariate GARCH model as used in our proper analysis. As countries co-movement is but an outcome of transmissions emitted from each country, the global country transmission is most often represented by the number of deaths occurring in transferred to other countries. For the sake of accounting for such interdependencies, we consider appealing to an MGARCH model including the EGARCH structural changes associated with the variances introduced by [4]. In what follows is a presentation of the dynamic model EGARCH (1, 1). The EGARCH model The above GARCH specification helps restrict any shock effect on the conditional variance to be symmetric and within the same size, be it positive or negative. Noteworthy however, such a shock effect should disappear geometrically over time. Still it well-become a known fact notably with the help of death data, the different country reactions in accordance with to the shock size and sign. To overcome such limitations, several asymmetrical GARCH models, such as the Exponential GARCH model [34], have been introduced in this respect. Formally, an EGARCH (1, 1) model corresponds to the following: ln ht ðÞ¼δ0þδ1ln ht−1 ðÞþγ0 εt−1 ht−1þγ1 εt−1 jj ht−1ð2:11Þ With an EGARCH specification, positivity constraints on parameters are no longer required. Furthermore, it is henceforth possible to apprehend asymmetry in volatility reaction toward external shocks. Indeed, if γ 0 > 0 (respectively γ 0 < 0), a positive shock on the lagged conditional variance implies an increase (respectively a decrease) in current volatility. Parameter γ 1 helps capture the asymmetry effect associated with the shock size εt−1 jj ht−1.Ifγ 1 = 0, a positive innovation would them have the same effect on conditional variance as would a negative innovation, while for γ 1 > 0, a high size shock should have more effect on conditional variance than should a low size shock. So for the Ebola virus contagion to be determined, the MGARCH-DCC model will be applied so that the virus propagation effect can be determined concerning four African countries (Guinea, Sierra Leone, Liberia and Nigeria). The multivariate GARCH -DCC model In this section, the dynamic conditional correlations’twostage model will be treated as proposed by [26]. Let us for instance, consider a vector consisting of any two variables Y t =[y 1t y 2t ]’. Each variable constitutes a constant function along with its own past values. Thus, the autoregressive process reduced form is written as: ALðÞYt¼cþεtavec εt→N0;Ht ðÞ;∀t ¼1;2;…;Tð2:12Þ where: A(L) is the polynomial delay, and ε t =[ε 1t ε 2t ]’is a vector of residuals training from the estimation auto regression process for each variable whose variance-covariance matrix is described by H t ={h i } t with s. Actually, the DCC-MGARCH model can be easily apprehended by rewriting the matrix of variance-covariance H t such as: Ht¼DtRtDt where: Dt¼diag ffiffiffiffiffi hit p  is the standard deviations diagonal matrix, a variable temporally different from the two previous equations’estimation with a univariate GARCH process; R t ={ρ ij,t } representing the conditional correlation coefficients matrix. The D t contained elements are generated into a GARCH (P,Q) process, which can be formulated as: hit ¼wiþX P p¼1 αipε2 it−pþX Q q¼1 βiqhit−qð2:13Þ In addition, [26] considers adopting a GARCH-type structure while modelling the correlations’dynamics. Thus, an (M, N) order DCC process can be described by: Rt¼Qt ðÞ −1QtQt ðÞ −1 Qt¼1−X M m¼1 am−X N n¼1 bn !  Q þX M m¼1 amξt−mξ0 t−m  þX N n¼1 bnQt−nð2:14Þ Nadhem and Nejib Health Economics Review (2015) 5:16 Page 5 of 15 where: ξt¼εit =ffiffiffiffiffi hit p  is the vector englobing the standardized residuals derived from the univariate GARCH model estimation, as a matrix of these standardized residuals’conditional variance-covariance, whereas Q t ={q ij,t } represents the unconditional variance-covariance matrix, which are temporally invariant. The parameters (a m ;b n ) are supposed to respectively intercept the shock effects and delay the dynamic correlations at the level of recent contemporary. As for Q * t , it stands for a diagonal matrix containing the square root of the main diagonal elements of Q t . Regarding to our example, this matrix is written as: Qt¼ffiffiffiffiffiffiffi q11 p 0ffiffiffiffiffiffiffi q22 p  ð2:15Þ with: ρ12;t¼q12;t ffiffiffiffiffiffiffiffiffiffiffiffiffi q11;tq22;t pdenoting the dynamic conditional correlations i.e. the matrix elements R t whose main diagonal consists of 1. The model parameters are estimated via the maximum likelihood DCC method. In this regard, [35] have show that the log-likelihood function can be expressed as: L¼−1 2X T t¼1 2 log 2πðÞþ2 log Dt jj þlog Rt jj þξtR−1 tξt  ð2:16Þ The estimation process involves two steps, the first of which consists in substituting an identity matrix by a matrix R t within the log-likelihood the function. The advantage of such a method is that it allows for getting the likelihood function sum of the GARCH univariate models. In other words, this initial step, series to achieve we obtain the parameters of equation (2.13). As for the second step, it consists in estimating the equation (2.14) parameters via adopting the original likelihood function as described by equation (2.16). This procedure ensures maintaining the dynamic correlations among the studied variables. Results Data and descriptive statistics The empirical data comprise total number of daily d deaths caused by the Ebola virus as calculated by “OMS”of four African countries. Actually, the number of deaths recorded exclusively in Guinea, Sierra Leone, Liberia and Nigeria has been selected as subject of study as these four countries are the most affected with the Ebola virus. The sample ranges over the period comprised between March 2014 and the end of October 2014 e , yielding 206 observations for each series. Table 1 presents a wide range of descriptive statistics concerning the four series under investigation during the period (March-October 2014). The null hypothesis of no ARCH effects is rejected at a significance level of 1%, suggesting that the GARCH parameterization might well fit for the conditional variance processes. In Table 1, the sample size and the unconditional mean indicate that the number of average daily deaths is fixed at 3.9 for Guinea, 5.7 for Sierra Leone, 11.9 for Liberia and 0.03 for Nigeria. With respect to Liberia and Sierra Leone, the sample provides a higher average return with a higher unconditional variance, highlighting the prevalence of a strong dispersion around the Ebola virus incurred deaths in these two countries. It is actually, this high dispersion feature which provides us with significant breaks through the conditional variance. Furthermore, distribution of the number of the virus related deaths seems to be symmetrical and leptokurtic, implying rejection of the normality null hypothesis. The mean and variance will, conditionally, be modelled. Therefore, the most interesting statistics, such as the skewness and the excess kurtosis, can be used for the purpose of testing whether the empirical distribution does have kurtosis and skewness just like a normal distribution. This undertaking has already been applied done with the Jarque-Bera test which rejected the null hypotheses, indicating that the sample distribution stems actually from a normal distribution set to a 1% significance level. To note, LB (24) statistics are employed for the take of testing the persistence of a high autocorrelation in the results’first and second moments. The LB (24) statistics figured on Table 1 display just the existence of a highorder autocorrelation for Guinea, Sierra Leone, Liberia and Nigeria. These LB (24) statistics are supposed to be significant, reflecting the existence of a noticeable interdependence among second moments of returns. This also indicates that the returns’heteroscedasticity should change with time. Such as result highlights well the use of estimation and variance of the autoregressive conditional heteroscedasticity (ARCH) model of [35]. As for the ARCH test, the null hypothesis of homoscedasticity is not accepted suggesting a further conditional timevarying in the number of death return dynamics. Table 1 Summary of descriptive statistics Guinea Sierra Leone Liberia Nigeria T 206 206 206 206 Moy 3.951 5.742 11.932 0.038 Var 40.085 309.694 565.89 0.0037 T-stat 8.957* 4.683* 7.199* 2.877* Skew 4.164* 7.951* 3.283* 4.809* Kurt 22.687* 73.768* 16.933* 21.333* J-B 5013.42* 48879.33* 2831.32* 4700.51* ARCH 21.54* 9.35* 8.76* 14.39* LB(24) 43.25* 38.807* 45.48** 37.33* Notes: *indicate the significance level at 5%. **indicate the significance level at 1%. Nadhem and Nejib Health Economics Review (2015) 5:16 Page 6 of 15 Long memory dependency and the Ebola virus In a first place, the Robinson estimators of long memory parameters relevant to Table 2 reported periods, prove to be lower than 0.5 for the entirely series. Such a result highlights well that long memory dependency within the period turns out to be critically important. This might well be due to the recurrent shocks and breaks recently occurring in the African countries. A possible explanation for this lies in the fact that the invasive occurrences in the number of Ebola virus related deaths respective to Guinea, Sierra Leone, Liberia and Nigeria, during the studied period corresponding have lasted for extended periods and increased the long memory property as a mean process which responded asymmetrically and gradually to such shocks and breaks as already pointed out by [36]. The plot of Figure 2 depicts the Sample Partial Autocorrelation Function clearly highlighting the prevalence of a significant autocorrelation. The long memory process captures a very low frequency cycle in the number of Ebola virus incurred deaths, by permitting a slowly declining autocorrelation of a hyperbolic shape in the long horizon (see Figures 1 and 3 relevant to Guinea and Sierra Leone). Inversely, however, the short memory process is characterized by a fast exponential decline in autocorrelations as depicted by Figures 2 and 4 pertaining to Liberia and Nigeria. For the purpose of verifying the persistence of the Ebola virus, pertinent real shocks, we consider applying the LW, 2ELW and 2 ELW with trending. Based on Table 2, the Robinson estimators (GS) do actually prove that 0≤ ^ d≤0:5. Indeed, this consists in a long-memory process case through still stationary, with a slow or smooth decay in the catching-up process. Concerning the ARFIMA(1,d,1) process, it has been demonstrated that 0≤ ^ d≤0:5 with respect to Guinea and Sierra Leone, underling a long memory and stationary process. In regard of both Liberian and Nigerian, cases, it has been prove that ^ d≥1. In effect, this corresponds to an explosive to process case and a situation marked by where there is a strong magnification effect, with any initial difference being unexpected to be potentially reversed in the future. In reality, this coincides with the “stochastic divergence” case liable to comparison with the initial deterministic divergence case. Regarding the 2ELW estimator, it has been demonstrated that 0:5≤ ^ d≤1, corresponding to a long memory process case, which is non-stationary through still reverting. In such a case, the process is featured with high persistence, whereby any distant past output difference would still have a long-lasting present inference. With respect to the 2ELWd estimator, it has also been demonstrated that 0:5≤ ^ d≤1 regarding the entirety of studied cases, except for Sierra Leone. What noting also, the number of deaths highest values (0.272 and 0.237, respectively for Guinea and Liberia), highlight well the persistent of shocks in the number of Ebola virus increased deaths and its contagion. In fact, the Ebola virus can persist in the patient’s body for about 2 months and, the sick and dead people are contagious. Following the incubation period ranging between 2 and 21 days (a period with no symptoms and no contagious), the first symptoms f begin to appear, and the affected person becomes contaminated on with the onset of the symptoms. The Ebola virus raised death occurs relatively quickly due to bleeding and unfortunately, there is no treatment or vaccine against the disease is yet available. This might well sound being plausible when the virus affects at fast countries within a small restricted geographic area, but spreads out to cover a wider geographical area and persists underway for several months (between 3 and 4 months). In addition, if the Ebola dynamics were to be modeled in real time, it would be critical to consider possible delays while report some cases and outcomes. The Ebola virus transmission is marked by two major phases. The first started in July 2014, when Guinea, Liberia and Nigeria become affected with the shocks. As for, the second phase, it started in August 2014, when all the four countries become affected by at least a single shock during that month. Concerning Guinea and Nigeria, there were structural changes for two months, September (05/09, 24/09 and 26/09) and August (09/08, 16/08 and 27/08). This fact indicates an approximation of the Ebola virus quiet periods boundaries and times. This denotes that the number of fatal Ebola viruses appears to be is more dispersed during these two months owing to the increasing number of Ebola virus incurred deaths. As a matter of fact, 494 deaths were recorded in August reaching 739 deaths in late September in Guinea. In Liberia, one might well note that the four points marking structural changes through the ICSS algorithm are sited during August and early September. During the end of the period, the number of deaths increased remarkably from 260 to 953. In regard of Nigeria, only two structural change points have been highlighted occurring between late August and early September. This might well has its explanation in the increase noticed in the number of deaths growing up from 1 to eight due to the Ebola virus. Unless it is imperative, the studied Table 2 Estimation of the long memory parameters number of deaths Guinea Sierra Leone Liberia Nigeria GS 0.272 0.180 0.237 0.184 ARFIMA(1,d,1) 0.365(0.00) 0.604(0.00) 1.063(0.00) 0.122(0.17) LW 0.751 0.725 1.065 0.651 2ELW 0.658 0.592 0.912 0.618 2ELWd 0.560 0.322 0.873 0.569 Notes: GS, LW, 2ELW and 2ELWd indicates respectively the Robinson, Local Whittle, 2 Stage Exact Local Whittle and Exact Local Whittle with detrending estimators. The value between (.) indicates the P-value. Nadhem and Nejib Health Economics Review (2015) 5:16 Page 7 of 15 countries are recommended to suspend all their flights towards the countries where the Ebola haemorrhagic fever cases have proven to be remarkably serious, such as Guinea, Sierra Leone, Liberia and Nigeria. Indeed, for five months, the Ebola outbreak was the worst since the emergence of the initially high contagious hemorrhagic fever in 1976. Actually, it was at the origin of 1,145 reported deaths according to the World Health Organization (WHO), reporting 413 cases in Liberia, 380 in Guinea, Sierra Leone and 348 in Nigeria. Such interesting findings appear to be unable to capture the asymmetry of volatility reaction to external shocks. For this reason, an EGARCH (1,1) model is estimated whose empirical results are reported in Table 3. Accordingly, the parameter γ 0 is shown to be negative and significant except for Liberia and Nigeria. This may well further evidence in favor of variance reduction after a positive shock and vice versa. Noteworthy, however, the estimated value of γ 0 appears to differ greatly between (Guinea and Sierra Leone) on the one hand, and (Liberia and Nigeria) on the other hand. This suggests that the volatile countries’exposure and liability to risks varies not only among (Guinea and Sierra Leone) and (Liberia and Nigeria), but also among the countries under review. Obviously, this might have an association with the number of deaths which seemed to be lower with respect to (Liberia and Nigeria) than to (Guinea and Sierra Leone), mainly on the onset of the Ebola virus disease. As for parameter δ 1 which serves to measure the conditional volatility persistence, its estimators prove to be significant. The highest value related to this parameter has been recorded for Guinea and Liberia exceeding 0.9, in respect of its value registered in Sierra Leone and Nigeria. It also reveals volatility dynamics which appear to be more persistent with regard to Guinea. Besides, our results appear also to confirm the Figure 2 Contagion of the Ebola virus between wildlife and human beings. Nadhem and Nejib Health Economics Review (2015) 5:16 Page 8 of 15 15. Velasco C. Gaussian semiparametric estimation of non-stationary time series. J Time Ser Anal. 1999;20:87–127. 16. Lee YN, Gao Y, Wang HY. Differential mediation of the Wnt canonical pathway by mammalian Dishevelleds-1,-2,and-3. CellSignalb20, 443452. 2008 CrossRefWeb of Science. 17. Chowell G, Hengartner NW, Castillo-Chavez C, Fenimore PW, Hyman JM. The basic reproductive number of Ebola and the effects of public healthmeasures: the cases of Congo and Uganda. J Theor Biol. 2004;229:119–26. 18. Miranda ME, Ksiazek TG, Retuya TJ. Epidemiology of Ebola (subtype Reston) virus in the Philippines, 1996. J Infect Dis. 1999;179 suppl 1:S115–9. 19. Ferrari MJ, Bjørnstad ON, Dobson AP. Estimation and inference of R0of aninfectious pathogen by a removal method. Math Biosci. 2005;198:14–26. 20. White LF, Pagano M. A likelihood-based method for real-time estimationof the serial interval and reproductive number of an epidemic. Stat Med. 2008;27:2999–3016. 21. Institute of topical medicine in 2013. Remco van de Pasa and Sara van Belle, Ebola, the epidemic that should never have happened, 2015, 95-100. 22. McKinley T, Cook AR, Deardon R. Inference in epidemic models withoutlikelihoods. Int J Biostat. 2009;5:24. 23. World Health Organisation. Disease Outbreak News. Geneva: WHO; 2014. 24. Carol R. Ebola awareness and prevention activities under way for refugees in Liberia, new stories in bahn Refugee, camp Liberia the UN Refugee Agency. Centers for Disease Control and Prevention, 2014. Outbreaks Chronol-ogy: Ebola Hemorrhagic Fever. 25. Bausch DG, Towner JS, Dowell SF. Assessment of the risk of Ebola virus transmission from bodily fluids and fomites. J Infect Dis. 2007;196(2):S142–7. 26. Engle R. Dynamic conditional correlation: A simple class of multivariate generalized autoregressive conditional heteroskedasticity models. J Bus Econ Stat. 2002;20(3):339–50. 27. International Convention. Burke et al. Bulletin of the World Health Organization. 1978;56(2):271–293. 28. Khan AS, Tshioko FK, Heymann DL, Le Guenno B, Nabeth P, Kerstiëns B, et al. The reemergence of Ebola hemorrhagic fever, Democratic Republic of the Congo 1995. J Infect Dis. 1999;179:S76–86. 29. Report of an International Commission. Ebola haemorrhagic fever in Zaire,1976. Bull World Health Organ. 1978;56:271–93. 30. Shimotsu K, Phillips PCB. Exact local Whittle estimation of fractional integration. Annals Statistics. 2006;33:1890–933. 31. Shimotsu K. Exact local Whittle estimation of fractional integration with unknown mean and trend. Econometric Theory. 2010;26:501–40. 32. Shimotsu K. Exact Local Whittle Estimation of Fractional Integration with Unknown Mean and Time Trend. In: Econometric Theory, 2008, forthcoming. 2008. 33. Robinson PM. Log-periodogram regression of time series with long range dependence. Ann Stat. 1995;23:1048–72. 34. National Center for Emerging and Zoonotic Infectious Diseases. Oyok T et al. 2001. 35. Engle RF. Autoregressive conditional heteroskedasticity with estimates of the variance of United Kingdom inflation”. J Monet Econ. 1982;30:987–1007. 36. Andersen TG, Bollerslev T, Diebold FX. Parametric and nonparametric volatility Measurement. In: Hansen ad LP, Ait-Sahalia Y, editors. Handbook of Financial Econometrics. Amsterdam: North-Holland, forthcoming; 2002. 37. Engle RF, Granger CW. Co-integration and error correction: representation, estimation and testing. Econometrica. 1987;55:251–76. 38. Diebold F. Elements of Forecasting. 2nd ed. South Western: Cincinnati; 2001. p. 254. ISBN 0-324-02393-6. 39. Forbes K, Rigobon R. No contagion, only interdependence: measuring stock market co-movements”. J Financ. 2002;5:2223–61. Submit your manuscript to a journal and benefi t from: 7 Convenient online submission 7 Rigorous peer review 7 Immediate publication on acceptance 7 Open access: articles freely available online 7 High visibility within the fi eld 7 Retaining the copyright to your article Submit your next manuscript at 7 springeropen.com Nadhem and Nejib Health Economics Review (2015) 5:16 Page 15 of 15