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          "en" => "<p id="spar0035" class="elsevierStyleSimplePara elsevierViewall">The observed data during the training is depicted in circles while the observed data of the testing is shown as full circles&#46; The solid and broken lines in the middle are the predictions during the training and testing&#44; respectively&#46; The broken lines are the 95&#37; confidence interval of the prediction&#46;</p>"
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    "textoCompleto" => "<span class="elsevierStyleSections"><span id="sec0005" class="elsevierStyleSection elsevierViewall"><span class="elsevierStyleSectionTitle" id="sect0035">Introduction</span><p id="par0005" class="elsevierStylePara elsevierViewall">Covid-19 local epidemic which originated from Wuhan has become a pandemic with severe socio- economic&#44; health and environmental consequences affecting countries all over the world&#46;<a class="elsevierStyleCrossRefs" href="#bib0130"><span class="elsevierStyleSup">1&#8211;3</span></a> In order to better manage this pandemic&#44; many institutions develop different models for predicting&#58; temporal evolution of Covid-19 cases&#44; hospital capacity for treating COVID-19 patients&#44; and case fatality rate of the disease&#46;<a class="elsevierStyleCrossRefs" href="#bib0145"><span class="elsevierStyleSup">4&#8211;7</span></a> Recently&#44; however&#44; some of these predictions are shown to be unreliable and undergone frequent revisions&#46;<a class="elsevierStyleCrossRefs" href="#bib0165"><span class="elsevierStyleSup">8&#44;9</span></a> Inadequacy in both rigorous model testing and forecast verification is considered amongst factors responsible to this failure<a class="elsevierStyleCrossRefs" href="#bib0175"><span class="elsevierStyleSup">10&#44;11</span></a> and the use of a fat-tailed probability distribution is recommended for pandemic forecasting&#46;<a class="elsevierStyleCrossRefs" href="#bib0185"><span class="elsevierStyleSup">12&#44;13</span></a></p><p id="par0010" class="elsevierStylePara elsevierViewall">In this work&#44; we first developed a fat-tailed model that uses the classic Bose&#8211;Einstein energy for predicting up to 14-days in advance Covid-19 confirmed cases&#46; The model prediction is then tested and verified against three data sets&#58; simulated data and data from two Covid-19 epicenters&#58; New York &#40;USA&#41; and DKI Jakarta &#40;Indonesia&#41;&#46; There are two prediction skill metrics for verifying the predictions&#44; i&#46;e&#46; predictability &#40;<span class="elsevierStyleItalic">R</span><span class="elsevierStyleSup">2</span>&#41; and RMSE &#40;root-mean-squared error&#41;&#46; The time variation of these metrics is then used for making inference about whether or not a social restriction order still be implemented for Covid-19 containment at the epicenters&#46;</p></span><span id="sec0010" class="elsevierStyleSection elsevierViewall"><span class="elsevierStyleSectionTitle" id="sect0040">Methods</span><span id="sec0015" class="elsevierStyleSection elsevierViewall"><span class="elsevierStyleSectionTitle" id="sect0045">Study sites</span><p id="par0015" class="elsevierStylePara elsevierViewall">This COVID-19 modelling and prediction work is using three data sets&#58; simulation data&#44; and confirmed COVID-19 cases from New York and DKI Jakarta&#46;</p></span><span id="sec0020" class="elsevierStyleSection elsevierViewall"><span class="elsevierStyleSectionTitle" id="sect0050">Data acquisition</span><p id="par0020" class="elsevierStylePara elsevierViewall">All data sources are publicly available &#40;Supplementary data&#58; COVID-19 data sets and model verification results&#46; The COVID-19 New York and DKI Jakarta data sets&#44; Tables 1 and 2&#44; respectively&#41;&#46; New York had its first day of the Covid-19 epidemic on the 1st March 2020&#46; It started recording the cases on 13th March and the Stay-at-Home order on were set on 22nd March&#44; i&#46;e&#46; the 22nd day of the outbreak&#46; The last recorded data for New York in this analysis was on the 9th June&#46; DKI Jakarta announced its first cases on 2nd March and applied the so-called PSBB &#40;Large Social Social Restriction&#41; order on 10th April&#44; i&#46;e&#46; day 40 of its outbreak&#46; The last recorded data for DKI in this analysis was on June the 13<span class="elsevierStyleSup">th</span>&#46; It would be interesting to show the impact of these two different timing of the social restriction on COVID-19 cases&#46; The resulting forecast verifications for these simulation data&#44; New York and DKI Jakarta cases are also presented for public uses &#40;Supplementary data&#58; Tables 3&#8211;5&#44; respectively&#41;&#46;</p></span><span id="sec0025" class="elsevierStyleSection elsevierViewall"><span class="elsevierStyleSectionTitle" id="sect0055">Data simulation and processing</span><p id="par0025" class="elsevierStylePara elsevierViewall">The Covid-19 model developed here uses the following Bose&#8211;Einstein &#40;BE&#41; energy distribution &#91;<a class="elsevierStyleCrossRef" href="#bib0195">14&#44;equation 4&#46;21</a>&#93; as follows&#58;<elsevierMultimedia ident="eq0005"></elsevierMultimedia><elsevierMultimedia ident="eq0010"></elsevierMultimedia></p><p id="par0030" class="elsevierStylePara elsevierViewall">Here the three parameters are&#58; <span class="elsevierStyleItalic">B</span><span class="elsevierStyleInf">1</span><span class="elsevierStyleHsp" style=""></span><span class="elsevierStyleItalic">&#61;</span><span class="elsevierStyleHsp" style=""></span><span class="elsevierStyleItalic">8 &#960;hc</span>&#44; <span class="elsevierStyleItalic">B</span><span class="elsevierStyleInf">2</span><span class="elsevierStyleHsp" style=""></span>&#61;<span class="elsevierStyleHsp" style=""></span>5&#44; and <span class="elsevierStyleItalic">B</span><span class="elsevierStyleInf">3</span><span class="elsevierStyleHsp" style=""></span>&#61;<span class="elsevierStyleHsp" style=""></span><span class="elsevierStyleItalic">hc</span>&#47;<span class="elsevierStyleItalic">kT</span>&#46; The <span class="elsevierStyleItalic">B</span><span class="elsevierStyleInf">1</span>&#47;<span class="elsevierStyleItalic">B</span><span class="elsevierStyleInf">3</span> ratio is proportional to the energy <span class="elsevierStyleItalic">E</span>&#46;</p><p id="par0035" class="elsevierStylePara elsevierViewall">The BE distribution in <a class="elsevierStyleCrossRef" href="#eq0010">&#40;2&#41;</a> is presented in <a class="elsevierStyleCrossRef" href="#fig0005">Fig&#46; 1</a>&#46; It describes several important properties of the BE distribution to be related to an epidemic curve&#46; First&#44; it has a single peak and decaying period after the peak&#46; Second&#44; the shape of the distribution is determined by the magnitude of the parameters&#46; More specifically&#44; the higher the <span class="elsevierStyleItalic">B</span><span class="elsevierStyleInf">1</span>&#47;<span class="elsevierStyleItalic">B</span><span class="elsevierStyleInf">3</span> ratio the steeper the distribution is and vice versa&#46; This is in accordance to that shown in <a class="elsevierStyleCrossRef" href="#bib0195">&#91;14&#44; Fig&#46; 10&#93;</a>&#46; In epidemics&#44; any intervention such as &#8216;Stay-at-Home&#8217; order or PSBB is meant to flatten the epidemic curve&#44; i&#46;e&#46; changing <a class="elsevierStyleCrossRef" href="#fig0005">Fig&#46; 1</a>a into becoming <a class="elsevierStyleCrossRef" href="#fig0005">Fig&#46; 1</a>b&#46; Third&#44; there exists accelerated and decelerated phases in the distribution&#46; These phases are shown by the different spacing of the circles&#46; Acceleration occurs when crowded circles become into more separated circles&#44; while deceleration happens when sparse circles turn into more dense circles&#46; These phases are more pronounced before the distribution peaks&#46;</p><elsevierMultimedia ident="fig0005"></elsevierMultimedia><p id="par0040" class="elsevierStylePara elsevierViewall">Equation <a class="elsevierStyleCrossRef" href="#eq0010">&#40;2&#41;</a> is used to generate simulated data and together with COVID-19 data from New York and DKI Jakarta&#44; their features are drawn in <a class="elsevierStyleCrossRef" href="#fig0010">Fig&#46; 2</a>&#46; It is important that the COVID-19 data is smoothed using a 14-day moving average using a MATLAB programming<a class="elsevierStyleCrossRef" href="#bib0200"><span class="elsevierStyleSup">15</span></a> before using it for making predictions&#46; The smoothed data is important in reducing the noise for further data processing&#44; i&#46;e&#46; calculating its derivatives&#46; This process of differentiation needed for obtaining the rate of infection add more noise to the data as can be seen in <a class="elsevierStyleCrossRef" href="#fig0010">Fig&#46; 2</a>d and f&#46; <a class="elsevierStyleCrossRef" href="#fig0010">Fig&#46; 2</a> presents important features&#46; First&#44; the presence of both acceleration and deceleration phases are clearer by inspecting the first derivatives in <a class="elsevierStyleCrossRef" href="#fig0010">Fig&#46; 2</a>b&#44; d and f&#46; Second&#44; there is a contrast between New York and DKI Jakarta infection rates&#46; The New York infection rate &#40;<a class="elsevierStyleCrossRef" href="#fig0010">Fig&#46; 2</a>d&#41; perfectly resembles the BE derivative &#40;<a class="elsevierStyleCrossRef" href="#fig0010">Fig&#46; 2</a>b&#41; while that of DKI curve is still crossing the green line in many occasions&#46; The difference has an important implication in the issue of epidemic containment discussed later on&#46;</p><elsevierMultimedia ident="fig0010"></elsevierMultimedia></span><span id="sec0030" class="elsevierStyleSection elsevierViewall"><span class="elsevierStyleSectionTitle" id="sect0060">Model prediction and skill assessment</span><p id="par0045" class="elsevierStylePara elsevierViewall">In this work&#44; we use different number of inputs to predict a fixed number of outputs&#46; Here&#44; we choose to produce up to 14 values in advance out-of-sample prediction&#46; Therefore&#44; for a daily data&#44; the model predicts up to 14 days ahead&#46; The first prediction use the first eight inputs of <span class="elsevierStyleSmallCaps">X</span>&#44; the second prediction uses nine inputs&#44; etc&#46; while maintaining the same forecasting horizon of 14 days in advance for each prediction P&#46; This is called a rolling forecast scheme&#46;<a class="elsevierStyleCrossRef" href="#bib0205"><span class="elsevierStyleSup">16</span></a> To do the prediction&#44; we first generate two different data sets&#44; i&#46;e&#46; training and testing data sets&#46; For the training data set&#44; we then apply a nonlinear fitting with initial value for the parameters <span class="elsevierStyleItalic">B</span><span class="elsevierStyleInf">1</span><span class="elsevierStyleHsp" style=""></span>&#61;<span class="elsevierStyleHsp" style=""></span>0&#46;001&#44; <span class="elsevierStyleItalic">B</span><span class="elsevierStyleInf">2</span><span class="elsevierStyleHsp" style=""></span>&#61;<span class="elsevierStyleHsp" style=""></span>7 and <span class="elsevierStyleItalic">B</span><span class="elsevierStyleInf">3</span><span class="elsevierStyleHsp" style=""></span>&#61;<span class="elsevierStyleHsp" style=""></span>0 to map between input&#47;output pairs for the training data set&#46; The <span class="elsevierStyleSmallCaps">X</span> versus P mapping results in the final model parameters&#46; The nonlinear fitting subroutine used is called NLINFIT from MATLAB with its robust weight function called &#8216;bi-square&#8217;&#46; The final parameters are then fed into the model using the 14 testing inputs data set to give the 14-day predictions&#46; The prediction skill of the model using simulated data&#44; New York cases and DKI Jakarta cases are measured using predictability &#8211; the coefficient of determination <span class="elsevierStyleItalic">R</span><span class="elsevierStyleSup">2</span> and RMSE &#40;root-mean-squared-error&#41; of&#46;<a class="elsevierStyleCrossRef" href="#bib0210"><span class="elsevierStyleSup">17</span></a></p></span></span><span id="sec0035" class="elsevierStyleSection elsevierViewall"><span class="elsevierStyleSectionTitle" id="sect0065">Result and discussion</span><p id="par0050" class="elsevierStylePara elsevierViewall">Two examples of model predictions for each of the three data sets are presented In <a class="elsevierStyleCrossRef" href="#fig0015">Fig&#46; 3</a>&#46; They are&#58; Simulation &#40;<a class="elsevierStyleCrossRef" href="#fig0015">Fig&#46; 3</a>a and b&#41;&#44; New York cases &#40;<a class="elsevierStyleCrossRef" href="#fig0015">Fig&#46; 3</a>c and d&#41; and DKI Jakarta cases &#40;<a class="elsevierStyleCrossRef" href="#fig0015">Fig&#46; 3</a>e and f&#41;&#46; We only want to pay attention to the out-of-sample predictions of the three data sets&#46; Model predictions for the simulation data set before the peak are poor&#46; Significant over predictions occur as can be seen when the full circles are much lower beyond the broken lines&#46; After the peak&#44; predictions improve&#46; The broken lines start to come closer to the full circles&#46; Model prediction for New York is significantly better than that of the simulation data&#46; The observed cases &#40;full circles&#41; observed before and after the peak are well within the broken lines&#46; The model prediction for the DKI Jakarta cases has different features&#46; Before the peak&#44; the 95&#37; confidence interval predictions represented by the broken lines are too wide&#46; After&#44; the peak&#44; predictions become poorer&#46; The majority of observed cases &#40;full circles&#41; are out of bound&#46;</p><elsevierMultimedia ident="fig0015"></elsevierMultimedia><p id="par0055" class="elsevierStylePara elsevierViewall">Over predictions and excessively wide confidence interval of a prediction obtained in this study have also been reported earlier&#46;<a class="elsevierStyleCrossRefs" href="#bib0165"><span class="elsevierStyleSup">8&#8211;11&#44;18</span></a> Over predictions occurring in the real data set could be explained by the presence of intervention in reducing social contact amongst infected and healthy people&#46;<a class="elsevierStyleCrossRefs" href="#bib0220"><span class="elsevierStyleSup">19&#8211;22</span></a> However&#44; over prediction and excessively wide confidence interval in the simulation data set needs to be addressed further by using other robust fitting algorithms and exploring different fat-tail distributions related to a pandemic&#46;<a class="elsevierStyleCrossRefs" href="#bib0185"><span class="elsevierStyleSup">12&#44;13</span></a></p><p id="par0060" class="elsevierStylePara elsevierViewall">Calculated parameters and prediction skill of the three data sets are presented in <a class="elsevierStyleCrossRef" href="#fig0020">Fig&#46; 4</a> &#40;simulation data&#41;&#44; <a class="elsevierStyleCrossRef" href="#fig0025">Fig&#46; 5</a> &#40;New York cases&#41; and <a class="elsevierStyleCrossRef" href="#fig0030">Fig&#46; 6</a> &#40;DKI Jakarta cases&#41;&#46; <a class="elsevierStyleCrossRef" href="#fig0020">Fig&#46; 4</a> shows that before the peak of BE distribution&#44; all calculated parameters wildly fluctuate&#46; This leads to low prediction skill&#46; But as the <span class="elsevierStyleSmallCaps">X</span> inputs pass <span class="elsevierStyleSmallCaps">X</span><span class="elsevierStyleHsp" style=""></span>&#61;<span class="elsevierStyleHsp" style=""></span>0&#46;02 onward&#44; the calculated parameters stabilize to the values given for simulating the data set&#46; As a consequence&#44; the model prediction skill metrics for the simulation data are close to 100&#37; predictability &#40;<span class="elsevierStyleItalic">R</span><span class="elsevierStyleSup">2</span>&#41; and RMSE equals to 0&#46; Similar finding is obtained for the New York cases described in <a class="elsevierStyleCrossRef" href="#fig0025">Fig&#46; 5</a>&#46; At prediction has low skill before the peak due to the spurious calculated parameters <span class="elsevierStyleItalic">B</span>&#39;s&#46; After day 60&#44; however&#44; the skill improves with predictability is near 100&#37; and RMSE is very small&#46; The DKI Jakarta cases presented in <a class="elsevierStyleCrossRef" href="#fig0030">Fig&#46; 6</a> are quite different&#46; The prediction skill metric wildly oscillate up to the end of the series&#46; Both metrics do not behave as expected&#44; i&#46;e&#46; converge into high predictability and low RMSE&#46; If we recall back&#44; this different behaviour is also appeared in the rate of infection &#40;<a class="elsevierStyleCrossRef" href="#fig0010">Fig&#46; 2</a>f&#41;&#46;</p><elsevierMultimedia ident="fig0020"></elsevierMultimedia><elsevierMultimedia ident="fig0025"></elsevierMultimedia><elsevierMultimedia ident="fig0030"></elsevierMultimedia><p id="par0065" class="elsevierStylePara elsevierViewall">DKI Jakarta is still unable to contain the virus spread amongst its population&#46; If we compare <a class="elsevierStyleCrossRefs" href="#fig0025">Figs&#46; 5 and 6</a> and focus the timing of applying the social restriction order&#44; New York started the order much earlier than that of DKI Jakarta&#46; The importance of applying earlier social restriction in reducing virus spread has been demonstrated&#46;<a class="elsevierStyleCrossRefs" href="#bib0240"><span class="elsevierStyleSup">23&#8211;25</span></a></p></span><span id="sec0040" class="elsevierStyleSection elsevierViewall"><span class="elsevierStyleSectionTitle" id="sect0070">Conclusions</span><p id="par0070" class="elsevierStylePara elsevierViewall">A simple model is developed to predict up to 14 days in advance Covid-19 cases&#46; The model is obtained through a nonlinear curve fitting of the BE distribution&#46; The out-of-sample rolling prediction has been validated extensively against three data sets&#46; The skill of the model is poor when predicting the early progress of the epidemic but the skill improves significantly toward the end of the epidemic&#46; The model is capable of providing an early warning in deciding whether or not to continue the social restriction order for containing an epidemic&#46;</p></span><span id="sec0045" class="elsevierStyleSection elsevierViewall"><span class="elsevierStyleSectionTitle" id="sect0075">Supplementary data</span><p id="par0075" class="elsevierStylePara elsevierViewall">The supplementary data &#40;Covid-19 data sets and model verification results&#41; can be found on this GitHub repository&#58; <a href="https://github.com/Andika9807/Data_ModelCovidHalmar">https&#58;&#47;&#47;github&#46;com&#47;Andika9807&#47;Data&#95;ModelCovidHalmar</a></p></span><span id="sec0050" class="elsevierStyleSection elsevierViewall"><span class="elsevierStyleSectionTitle" id="sect0080">Conflicts of interest</span><p id="par0080" class="elsevierStylePara elsevierViewall">The authors declare no conflict of interest&#46;</p></span></span>"
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          "titulo" => "Abstract"
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              "titulo" => "Conclusion"
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          "titulo" => "Introduction"
        ]
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          "titulo" => "Methods"
          "secciones" => array:4 [
            0 => array:2 [
              "identificador" => "sec0015"
              "titulo" => "Study sites"
            ]
            1 => array:2 [
              "identificador" => "sec0020"
              "titulo" => "Data acquisition"
            ]
            2 => array:2 [
              "identificador" => "sec0025"
              "titulo" => "Data simulation and processing"
            ]
            3 => array:2 [
              "identificador" => "sec0030"
              "titulo" => "Model prediction and skill assessment"
            ]
          ]
        ]
        4 => array:2 [
          "identificador" => "sec0035"
          "titulo" => "Result and discussion"
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        5 => array:2 [
          "identificador" => "sec0040"
          "titulo" => "Conclusions"
        ]
        6 => array:2 [
          "identificador" => "sec0045"
          "titulo" => "Supplementary data"
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        7 => array:2 [
          "identificador" => "sec0050"
          "titulo" => "Conflicts of interest"
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        8 => array:2 [
          "identificador" => "xack576343"
          "titulo" => "Acknowledgments"
        ]
        9 => array:1 [
          "titulo" => "References"
        ]
      ]
    ]
    "pdfFichero" => "main.pdf"
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    "fechaRecibido" => "2021-06-28"
    "fechaAceptado" => "2021-07-30"
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          "clase" => "keyword"
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          "palabras" => array:3 [
            0 => "Bose&#8211;Einstein energy"
            1 => "COVID-19"
            2 => "Social restrictions"
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      ]
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    "resumen" => array:1 [
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        "titulo" => "Abstract"
        "resumen" => "<span id="abst0005" class="elsevierStyleSection elsevierViewall"><span class="elsevierStyleSectionTitle" id="sect0010">Objective</span><p id="spar0005" class="elsevierStyleSimplePara elsevierViewall">Global society pays huge economic toll and live loss due to COVID-19 &#40;Coronavirus Disease 2019&#41; pandemic&#46; In order to have a better management of this pandemic&#44; many institutions develop their own models to predict number of COVID-19 cases&#44; hospitalizations and mortalities&#46; These models&#44; however&#44; are shown to be unreliable and need to be revised on a daily basis&#46;</p></span> <span id="abst0010" class="elsevierStyleSection elsevierViewall"><span class="elsevierStyleSectionTitle" id="sect0015">Methods</span><p id="spar0010" class="elsevierStyleSimplePara elsevierViewall">Here&#44; we develop a Bose&#8211;Einstein &#40;BE&#41;-based statistical model to predict daily COVID-19 cases up to 14 days in advance&#46; This fat-tailed model is chosen based on three reasons&#46; First&#44; it contains a peak and decaying phase&#46; Second&#44; it also has both accelerated and decelerated phases which are similarly observed in an epidemic curve&#46; Third&#44; the shape of both the BE energy distribution and the epidemic curve is controlled by a set of parameters&#46; The BE model daily predictions are then verified against simulated data and confirmed COVID-19 daily cases from two epidemic centres&#44; i&#46;e&#46; New York and DKI Jakarta&#46;</p></span> <span id="abst0015" class="elsevierStyleSection elsevierViewall"><span class="elsevierStyleSectionTitle" id="sect0020">Result</span><p id="spar0015" class="elsevierStyleSimplePara elsevierViewall">Over- predictions occur at the earlier stage of the epidemic for all data sets&#46; Models parameters for both simulated and New York data converge to a certain value only at the latest stage of the epidemic progress&#46; At this stage&#44; model&#39;s skill is high for both simulated and New York data&#44; i&#46;e&#46; the predictability is greater than 80&#37; with decreasing RMSE&#46; On the other hand&#44; at that stage&#44; the DKI&#39;s model&#39;s predictability is still fluctuating with increasing RMSE&#46;</p></span> <span id="abst0020" class="elsevierStyleSection elsevierViewall"><span class="elsevierStyleSectionTitle" id="sect0025">Conclusion</span><p id="spar0020" class="elsevierStyleSimplePara elsevierViewall">This implies that New York could leave the stay-at-home order&#44; but DKI Jakarta should continue its large-scale social restriction order&#46; There remains a great challenge in predicting the full course of an epidemic using small data collected during the earlier phase of the epidemic&#46;</p></span>"
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        "texto" => "<p id="par0085" class="elsevierStylePara elsevierViewall"><span class="elsevierStyleSmallCaps">I</span> express my gratitude to WORLDOMETER for providing the daily New York Covid-19 cases and the DKI Jakarta province for sharing the public of its daily Covid-19 cases through their websites&#46; <span class="elsevierStyleSmallCaps">I</span> also thank Mr Andika for type-setting the equations and archiving the Supplementary data&#46;</p>"
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Predicting COVID-19 confirmed cases in New York and DKI Jakarta by nonlinear fitting of a Bose–Einstein energy distribution and its implications on social restrictions
Halmar Halide
Geophysics Department, FMIPA, Universitas Hasanuddin, Makassar, Indonesia
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    "titulo" => "Predicting COVID-19 confirmed cases in New York and DKI Jakarta by nonlinear fitting of a Bose&#8211;Einstein energy distribution and its implications on social restrictions"
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          "en" => "<p id="spar0035" class="elsevierStyleSimplePara elsevierViewall">The observed data during the training is depicted in circles while the observed data of the testing is shown as full circles&#46; The solid and broken lines in the middle are the predictions during the training and testing&#44; respectively&#46; The broken lines are the 95&#37; confidence interval of the prediction&#46;</p>"
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    "textoCompleto" => "<span class="elsevierStyleSections"><span id="sec0005" class="elsevierStyleSection elsevierViewall"><span class="elsevierStyleSectionTitle" id="sect0035">Introduction</span><p id="par0005" class="elsevierStylePara elsevierViewall">Covid-19 local epidemic which originated from Wuhan has become a pandemic with severe socio- economic&#44; health and environmental consequences affecting countries all over the world&#46;<a class="elsevierStyleCrossRefs" href="#bib0130"><span class="elsevierStyleSup">1&#8211;3</span></a> In order to better manage this pandemic&#44; many institutions develop different models for predicting&#58; temporal evolution of Covid-19 cases&#44; hospital capacity for treating COVID-19 patients&#44; and case fatality rate of the disease&#46;<a class="elsevierStyleCrossRefs" href="#bib0145"><span class="elsevierStyleSup">4&#8211;7</span></a> Recently&#44; however&#44; some of these predictions are shown to be unreliable and undergone frequent revisions&#46;<a class="elsevierStyleCrossRefs" href="#bib0165"><span class="elsevierStyleSup">8&#44;9</span></a> Inadequacy in both rigorous model testing and forecast verification is considered amongst factors responsible to this failure<a class="elsevierStyleCrossRefs" href="#bib0175"><span class="elsevierStyleSup">10&#44;11</span></a> and the use of a fat-tailed probability distribution is recommended for pandemic forecasting&#46;<a class="elsevierStyleCrossRefs" href="#bib0185"><span class="elsevierStyleSup">12&#44;13</span></a></p><p id="par0010" class="elsevierStylePara elsevierViewall">In this work&#44; we first developed a fat-tailed model that uses the classic Bose&#8211;Einstein energy for predicting up to 14-days in advance Covid-19 confirmed cases&#46; The model prediction is then tested and verified against three data sets&#58; simulated data and data from two Covid-19 epicenters&#58; New York &#40;USA&#41; and DKI Jakarta &#40;Indonesia&#41;&#46; There are two prediction skill metrics for verifying the predictions&#44; i&#46;e&#46; predictability &#40;<span class="elsevierStyleItalic">R</span><span class="elsevierStyleSup">2</span>&#41; and RMSE &#40;root-mean-squared error&#41;&#46; The time variation of these metrics is then used for making inference about whether or not a social restriction order still be implemented for Covid-19 containment at the epicenters&#46;</p></span><span id="sec0010" class="elsevierStyleSection elsevierViewall"><span class="elsevierStyleSectionTitle" id="sect0040">Methods</span><span id="sec0015" class="elsevierStyleSection elsevierViewall"><span class="elsevierStyleSectionTitle" id="sect0045">Study sites</span><p id="par0015" class="elsevierStylePara elsevierViewall">This COVID-19 modelling and prediction work is using three data sets&#58; simulation data&#44; and confirmed COVID-19 cases from New York and DKI Jakarta&#46;</p></span><span id="sec0020" class="elsevierStyleSection elsevierViewall"><span class="elsevierStyleSectionTitle" id="sect0050">Data acquisition</span><p id="par0020" class="elsevierStylePara elsevierViewall">All data sources are publicly available &#40;Supplementary data&#58; COVID-19 data sets and model verification results&#46; The COVID-19 New York and DKI Jakarta data sets&#44; Tables 1 and 2&#44; respectively&#41;&#46; New York had its first day of the Covid-19 epidemic on the 1st March 2020&#46; It started recording the cases on 13th March and the Stay-at-Home order on were set on 22nd March&#44; i&#46;e&#46; the 22nd day of the outbreak&#46; The last recorded data for New York in this analysis was on the 9th June&#46; DKI Jakarta announced its first cases on 2nd March and applied the so-called PSBB &#40;Large Social Social Restriction&#41; order on 10th April&#44; i&#46;e&#46; day 40 of its outbreak&#46; The last recorded data for DKI in this analysis was on June the 13<span class="elsevierStyleSup">th</span>&#46; It would be interesting to show the impact of these two different timing of the social restriction on COVID-19 cases&#46; The resulting forecast verifications for these simulation data&#44; New York and DKI Jakarta cases are also presented for public uses &#40;Supplementary data&#58; Tables 3&#8211;5&#44; respectively&#41;&#46;</p></span><span id="sec0025" class="elsevierStyleSection elsevierViewall"><span class="elsevierStyleSectionTitle" id="sect0055">Data simulation and processing</span><p id="par0025" class="elsevierStylePara elsevierViewall">The Covid-19 model developed here uses the following Bose&#8211;Einstein &#40;BE&#41; energy distribution &#91;<a class="elsevierStyleCrossRef" href="#bib0195">14&#44;equation 4&#46;21</a>&#93; as follows&#58;<elsevierMultimedia ident="eq0005"></elsevierMultimedia><elsevierMultimedia ident="eq0010"></elsevierMultimedia></p><p id="par0030" class="elsevierStylePara elsevierViewall">Here the three parameters are&#58; <span class="elsevierStyleItalic">B</span><span class="elsevierStyleInf">1</span><span class="elsevierStyleHsp" style=""></span><span class="elsevierStyleItalic">&#61;</span><span class="elsevierStyleHsp" style=""></span><span class="elsevierStyleItalic">8 &#960;hc</span>&#44; <span class="elsevierStyleItalic">B</span><span class="elsevierStyleInf">2</span><span class="elsevierStyleHsp" style=""></span>&#61;<span class="elsevierStyleHsp" style=""></span>5&#44; and <span class="elsevierStyleItalic">B</span><span class="elsevierStyleInf">3</span><span class="elsevierStyleHsp" style=""></span>&#61;<span class="elsevierStyleHsp" style=""></span><span class="elsevierStyleItalic">hc</span>&#47;<span class="elsevierStyleItalic">kT</span>&#46; The <span class="elsevierStyleItalic">B</span><span class="elsevierStyleInf">1</span>&#47;<span class="elsevierStyleItalic">B</span><span class="elsevierStyleInf">3</span> ratio is proportional to the energy <span class="elsevierStyleItalic">E</span>&#46;</p><p id="par0035" class="elsevierStylePara elsevierViewall">The BE distribution in <a class="elsevierStyleCrossRef" href="#eq0010">&#40;2&#41;</a> is presented in <a class="elsevierStyleCrossRef" href="#fig0005">Fig&#46; 1</a>&#46; It describes several important properties of the BE distribution to be related to an epidemic curve&#46; First&#44; it has a single peak and decaying period after the peak&#46; Second&#44; the shape of the distribution is determined by the magnitude of the parameters&#46; More specifically&#44; the higher the <span class="elsevierStyleItalic">B</span><span class="elsevierStyleInf">1</span>&#47;<span class="elsevierStyleItalic">B</span><span class="elsevierStyleInf">3</span> ratio the steeper the distribution is and vice versa&#46; This is in accordance to that shown in <a class="elsevierStyleCrossRef" href="#bib0195">&#91;14&#44; Fig&#46; 10&#93;</a>&#46; In epidemics&#44; any intervention such as &#8216;Stay-at-Home&#8217; order or PSBB is meant to flatten the epidemic curve&#44; i&#46;e&#46; changing <a class="elsevierStyleCrossRef" href="#fig0005">Fig&#46; 1</a>a into becoming <a class="elsevierStyleCrossRef" href="#fig0005">Fig&#46; 1</a>b&#46; Third&#44; there exists accelerated and decelerated phases in the distribution&#46; These phases are shown by the different spacing of the circles&#46; Acceleration occurs when crowded circles become into more separated circles&#44; while deceleration happens when sparse circles turn into more dense circles&#46; These phases are more pronounced before the distribution peaks&#46;</p><elsevierMultimedia ident="fig0005"></elsevierMultimedia><p id="par0040" class="elsevierStylePara elsevierViewall">Equation <a class="elsevierStyleCrossRef" href="#eq0010">&#40;2&#41;</a> is used to generate simulated data and together with COVID-19 data from New York and DKI Jakarta&#44; their features are drawn in <a class="elsevierStyleCrossRef" href="#fig0010">Fig&#46; 2</a>&#46; It is important that the COVID-19 data is smoothed using a 14-day moving average using a MATLAB programming<a class="elsevierStyleCrossRef" href="#bib0200"><span class="elsevierStyleSup">15</span></a> before using it for making predictions&#46; The smoothed data is important in reducing the noise for further data processing&#44; i&#46;e&#46; calculating its derivatives&#46; This process of differentiation needed for obtaining the rate of infection add more noise to the data as can be seen in <a class="elsevierStyleCrossRef" href="#fig0010">Fig&#46; 2</a>d and f&#46; <a class="elsevierStyleCrossRef" href="#fig0010">Fig&#46; 2</a> presents important features&#46; First&#44; the presence of both acceleration and deceleration phases are clearer by inspecting the first derivatives in <a class="elsevierStyleCrossRef" href="#fig0010">Fig&#46; 2</a>b&#44; d and f&#46; Second&#44; there is a contrast between New York and DKI Jakarta infection rates&#46; The New York infection rate &#40;<a class="elsevierStyleCrossRef" href="#fig0010">Fig&#46; 2</a>d&#41; perfectly resembles the BE derivative &#40;<a class="elsevierStyleCrossRef" href="#fig0010">Fig&#46; 2</a>b&#41; while that of DKI curve is still crossing the green line in many occasions&#46; The difference has an important implication in the issue of epidemic containment discussed later on&#46;</p><elsevierMultimedia ident="fig0010"></elsevierMultimedia></span><span id="sec0030" class="elsevierStyleSection elsevierViewall"><span class="elsevierStyleSectionTitle" id="sect0060">Model prediction and skill assessment</span><p id="par0045" class="elsevierStylePara elsevierViewall">In this work&#44; we use different number of inputs to predict a fixed number of outputs&#46; Here&#44; we choose to produce up to 14 values in advance out-of-sample prediction&#46; Therefore&#44; for a daily data&#44; the model predicts up to 14 days ahead&#46; The first prediction use the first eight inputs of <span class="elsevierStyleSmallCaps">X</span>&#44; the second prediction uses nine inputs&#44; etc&#46; while maintaining the same forecasting horizon of 14 days in advance for each prediction P&#46; This is called a rolling forecast scheme&#46;<a class="elsevierStyleCrossRef" href="#bib0205"><span class="elsevierStyleSup">16</span></a> To do the prediction&#44; we first generate two different data sets&#44; i&#46;e&#46; training and testing data sets&#46; For the training data set&#44; we then apply a nonlinear fitting with initial value for the parameters <span class="elsevierStyleItalic">B</span><span class="elsevierStyleInf">1</span><span class="elsevierStyleHsp" style=""></span>&#61;<span class="elsevierStyleHsp" style=""></span>0&#46;001&#44; <span class="elsevierStyleItalic">B</span><span class="elsevierStyleInf">2</span><span class="elsevierStyleHsp" style=""></span>&#61;<span class="elsevierStyleHsp" style=""></span>7 and <span class="elsevierStyleItalic">B</span><span class="elsevierStyleInf">3</span><span class="elsevierStyleHsp" style=""></span>&#61;<span class="elsevierStyleHsp" style=""></span>0 to map between input&#47;output pairs for the training data set&#46; The <span class="elsevierStyleSmallCaps">X</span> versus P mapping results in the final model parameters&#46; The nonlinear fitting subroutine used is called NLINFIT from MATLAB with its robust weight function called &#8216;bi-square&#8217;&#46; The final parameters are then fed into the model using the 14 testing inputs data set to give the 14-day predictions&#46; The prediction skill of the model using simulated data&#44; New York cases and DKI Jakarta cases are measured using predictability &#8211; the coefficient of determination <span class="elsevierStyleItalic">R</span><span class="elsevierStyleSup">2</span> and RMSE &#40;root-mean-squared-error&#41; of&#46;<a class="elsevierStyleCrossRef" href="#bib0210"><span class="elsevierStyleSup">17</span></a></p></span></span><span id="sec0035" class="elsevierStyleSection elsevierViewall"><span class="elsevierStyleSectionTitle" id="sect0065">Result and discussion</span><p id="par0050" class="elsevierStylePara elsevierViewall">Two examples of model predictions for each of the three data sets are presented In <a class="elsevierStyleCrossRef" href="#fig0015">Fig&#46; 3</a>&#46; They are&#58; Simulation &#40;<a class="elsevierStyleCrossRef" href="#fig0015">Fig&#46; 3</a>a and b&#41;&#44; New York cases &#40;<a class="elsevierStyleCrossRef" href="#fig0015">Fig&#46; 3</a>c and d&#41; and DKI Jakarta cases &#40;<a class="elsevierStyleCrossRef" href="#fig0015">Fig&#46; 3</a>e and f&#41;&#46; We only want to pay attention to the out-of-sample predictions of the three data sets&#46; Model predictions for the simulation data set before the peak are poor&#46; Significant over predictions occur as can be seen when the full circles are much lower beyond the broken lines&#46; After the peak&#44; predictions improve&#46; The broken lines start to come closer to the full circles&#46; Model prediction for New York is significantly better than that of the simulation data&#46; The observed cases &#40;full circles&#41; observed before and after the peak are well within the broken lines&#46; The model prediction for the DKI Jakarta cases has different features&#46; Before the peak&#44; the 95&#37; confidence interval predictions represented by the broken lines are too wide&#46; After&#44; the peak&#44; predictions become poorer&#46; The majority of observed cases &#40;full circles&#41; are out of bound&#46;</p><elsevierMultimedia ident="fig0015"></elsevierMultimedia><p id="par0055" class="elsevierStylePara elsevierViewall">Over predictions and excessively wide confidence interval of a prediction obtained in this study have also been reported earlier&#46;<a class="elsevierStyleCrossRefs" href="#bib0165"><span class="elsevierStyleSup">8&#8211;11&#44;18</span></a> Over predictions occurring in the real data set could be explained by the presence of intervention in reducing social contact amongst infected and healthy people&#46;<a class="elsevierStyleCrossRefs" href="#bib0220"><span class="elsevierStyleSup">19&#8211;22</span></a> However&#44; over prediction and excessively wide confidence interval in the simulation data set needs to be addressed further by using other robust fitting algorithms and exploring different fat-tail distributions related to a pandemic&#46;<a class="elsevierStyleCrossRefs" href="#bib0185"><span class="elsevierStyleSup">12&#44;13</span></a></p><p id="par0060" class="elsevierStylePara elsevierViewall">Calculated parameters and prediction skill of the three data sets are presented in <a class="elsevierStyleCrossRef" href="#fig0020">Fig&#46; 4</a> &#40;simulation data&#41;&#44; <a class="elsevierStyleCrossRef" href="#fig0025">Fig&#46; 5</a> &#40;New York cases&#41; and <a class="elsevierStyleCrossRef" href="#fig0030">Fig&#46; 6</a> &#40;DKI Jakarta cases&#41;&#46; <a class="elsevierStyleCrossRef" href="#fig0020">Fig&#46; 4</a> shows that before the peak of BE distribution&#44; all calculated parameters wildly fluctuate&#46; This leads to low prediction skill&#46; But as the <span class="elsevierStyleSmallCaps">X</span> inputs pass <span class="elsevierStyleSmallCaps">X</span><span class="elsevierStyleHsp" style=""></span>&#61;<span class="elsevierStyleHsp" style=""></span>0&#46;02 onward&#44; the calculated parameters stabilize to the values given for simulating the data set&#46; As a consequence&#44; the model prediction skill metrics for the simulation data are close to 100&#37; predictability &#40;<span class="elsevierStyleItalic">R</span><span class="elsevierStyleSup">2</span>&#41; and RMSE equals to 0&#46; Similar finding is obtained for the New York cases described in <a class="elsevierStyleCrossRef" href="#fig0025">Fig&#46; 5</a>&#46; At prediction has low skill before the peak due to the spurious calculated parameters <span class="elsevierStyleItalic">B</span>&#39;s&#46; After day 60&#44; however&#44; the skill improves with predictability is near 100&#37; and RMSE is very small&#46; The DKI Jakarta cases presented in <a class="elsevierStyleCrossRef" href="#fig0030">Fig&#46; 6</a> are quite different&#46; The prediction skill metric wildly oscillate up to the end of the series&#46; Both metrics do not behave as expected&#44; i&#46;e&#46; converge into high predictability and low RMSE&#46; If we recall back&#44; this different behaviour is also appeared in the rate of infection &#40;<a class="elsevierStyleCrossRef" href="#fig0010">Fig&#46; 2</a>f&#41;&#46;</p><elsevierMultimedia ident="fig0020"></elsevierMultimedia><elsevierMultimedia ident="fig0025"></elsevierMultimedia><elsevierMultimedia ident="fig0030"></elsevierMultimedia><p id="par0065" class="elsevierStylePara elsevierViewall">DKI Jakarta is still unable to contain the virus spread amongst its population&#46; If we compare <a class="elsevierStyleCrossRefs" href="#fig0025">Figs&#46; 5 and 6</a> and focus the timing of applying the social restriction order&#44; New York started the order much earlier than that of DKI Jakarta&#46; The importance of applying earlier social restriction in reducing virus spread has been demonstrated&#46;<a class="elsevierStyleCrossRefs" href="#bib0240"><span class="elsevierStyleSup">23&#8211;25</span></a></p></span><span id="sec0040" class="elsevierStyleSection elsevierViewall"><span class="elsevierStyleSectionTitle" id="sect0070">Conclusions</span><p id="par0070" class="elsevierStylePara elsevierViewall">A simple model is developed to predict up to 14 days in advance Covid-19 cases&#46; The model is obtained through a nonlinear curve fitting of the BE distribution&#46; The out-of-sample rolling prediction has been validated extensively against three data sets&#46; The skill of the model is poor when predicting the early progress of the epidemic but the skill improves significantly toward the end of the epidemic&#46; The model is capable of providing an early warning in deciding whether or not to continue the social restriction order for containing an epidemic&#46;</p></span><span id="sec0045" class="elsevierStyleSection elsevierViewall"><span class="elsevierStyleSectionTitle" id="sect0075">Supplementary data</span><p id="par0075" class="elsevierStylePara elsevierViewall">The supplementary data &#40;Covid-19 data sets and model verification results&#41; can be found on this GitHub repository&#58; <a href="https://github.com/Andika9807/Data_ModelCovidHalmar">https&#58;&#47;&#47;github&#46;com&#47;Andika9807&#47;Data&#95;ModelCovidHalmar</a></p></span><span id="sec0050" class="elsevierStyleSection elsevierViewall"><span class="elsevierStyleSectionTitle" id="sect0080">Conflicts of interest</span><p id="par0080" class="elsevierStylePara elsevierViewall">The authors declare no conflict of interest&#46;</p></span></span>"
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        "resumen" => "<span id="abst0005" class="elsevierStyleSection elsevierViewall"><span class="elsevierStyleSectionTitle" id="sect0010">Objective</span><p id="spar0005" class="elsevierStyleSimplePara elsevierViewall">Global society pays huge economic toll and live loss due to COVID-19 &#40;Coronavirus Disease 2019&#41; pandemic&#46; In order to have a better management of this pandemic&#44; many institutions develop their own models to predict number of COVID-19 cases&#44; hospitalizations and mortalities&#46; These models&#44; however&#44; are shown to be unreliable and need to be revised on a daily basis&#46;</p></span> <span id="abst0010" class="elsevierStyleSection elsevierViewall"><span class="elsevierStyleSectionTitle" id="sect0015">Methods</span><p id="spar0010" class="elsevierStyleSimplePara elsevierViewall">Here&#44; we develop a Bose&#8211;Einstein &#40;BE&#41;-based statistical model to predict daily COVID-19 cases up to 14 days in advance&#46; This fat-tailed model is chosen based on three reasons&#46; First&#44; it contains a peak and decaying phase&#46; Second&#44; it also has both accelerated and decelerated phases which are similarly observed in an epidemic curve&#46; Third&#44; the shape of both the BE energy distribution and the epidemic curve is controlled by a set of parameters&#46; The BE model daily predictions are then verified against simulated data and confirmed COVID-19 daily cases from two epidemic centres&#44; i&#46;e&#46; New York and DKI Jakarta&#46;</p></span> <span id="abst0015" class="elsevierStyleSection elsevierViewall"><span class="elsevierStyleSectionTitle" id="sect0020">Result</span><p id="spar0015" class="elsevierStyleSimplePara elsevierViewall">Over- predictions occur at the earlier stage of the epidemic for all data sets&#46; Models parameters for both simulated and New York data converge to a certain value only at the latest stage of the epidemic progress&#46; At this stage&#44; model&#39;s skill is high for both simulated and New York data&#44; i&#46;e&#46; the predictability is greater than 80&#37; with decreasing RMSE&#46; On the other hand&#44; at that stage&#44; the DKI&#39;s model&#39;s predictability is still fluctuating with increasing RMSE&#46;</p></span> <span id="abst0020" class="elsevierStyleSection elsevierViewall"><span class="elsevierStyleSectionTitle" id="sect0025">Conclusion</span><p id="spar0020" class="elsevierStyleSimplePara elsevierViewall">This implies that New York could leave the stay-at-home order&#44; but DKI Jakarta should continue its large-scale social restriction order&#46; There remains a great challenge in predicting the full course of an epidemic using small data collected during the earlier phase of the epidemic&#46;</p></span>"
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        "etiqueta" => "&#9734;"
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          "en" => "<p id="spar0025" class="elsevierStyleSimplePara elsevierViewall">The BE energy distribution plotted with the black circles &#8216;o&#8217; with different parameters &#40;a&#41; <span class="elsevierStyleItalic">B</span><span class="elsevierStyleInf">1</span><span class="elsevierStyleHsp" style=""></span>&#62;<span class="elsevierStyleHsp" style=""></span><span class="elsevierStyleItalic">B</span><span class="elsevierStyleInf">3</span>&#44; &#40;b&#41; <span class="elsevierStyleItalic">B</span><span class="elsevierStyleInf">1</span><span class="elsevierStyleHsp" style=""></span>&#60;<span class="elsevierStyleHsp" style=""></span><span class="elsevierStyleItalic">B</span><span class="elsevierStyleInf">3</span>&#46; The blue horizontal line is the level to the 10&#37; of peak value and the intersections between these lines and the distributions are plotted with the red crosses &#8216;&#215;&#8217;&#46;</p>"
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          "en" => "<p id="spar0030" class="elsevierStyleSimplePara elsevierViewall">The BE energy distribution &#40;a&#41;&#44; the New York &#40;c&#41; and DKI Jakarta epidemic curves &#40;e&#41;&#46; The cases are depicted as blue circles&#46; The first derivatives w&#46;r&#46;t &#40;with respect to X&#41; of the BE distribution and the first derives w&#46;r&#46;t time &#40;the rate of infection IR&#41; of the smoothed epidemic curves &#40;solid blue lines&#41; for New York and DKI Jakarta are plotted in &#40;b&#41;&#44; &#40;d&#41; and &#40;f&#41;&#46; The dotted red line is the location of the peak of the BE distribution&#44; while the dashed red lines are the time when the social restriction began&#46; The green horizontal line is the line associated with d<span class="elsevierStyleItalic">E</span>&#47;d<span class="elsevierStyleItalic">X</span><span class="elsevierStyleHsp" style=""></span>&#61;<span class="elsevierStyleHsp" style=""></span>0 or IR<span class="elsevierStyleHsp" style=""></span>&#61;<span class="elsevierStyleHsp" style=""></span>0&#46;</p>"
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          "en" => "<p id="spar0035" class="elsevierStyleSimplePara elsevierViewall">The observed data during the training is depicted in circles while the observed data of the testing is shown as full circles&#46; The solid and broken lines in the middle are the predictions during the training and testing&#44; respectively&#46; The broken lines are the 95&#37; confidence interval of the prediction&#46;</p>"
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          "en" => "<p id="spar0040" class="elsevierStyleSimplePara elsevierViewall">Calculated parameters <span class="elsevierStyleItalic">B</span><span class="elsevierStyleInf">1</span>&#44; <span class="elsevierStyleItalic">B</span><span class="elsevierStyleInf">2</span>&#44; <span class="elsevierStyleItalic">B</span><span class="elsevierStyleInf">3</span> the <span class="elsevierStyleItalic">B</span><span class="elsevierStyleInf">1</span> to <span class="elsevierStyleItalic">B</span><span class="elsevierStyleInf">3</span> ratio and prediction skill metrics <span class="elsevierStyleItalic">R</span><span class="elsevierStyleSup">2</span> and RMSE obtained from the simulation data&#46; The vertical red dots lines represent the peaks&#46;</p>"
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          "en" => "<p id="spar0045" class="elsevierStyleSimplePara elsevierViewall">Calculated parameters <span class="elsevierStyleItalic">B</span><span class="elsevierStyleInf">1</span>&#44; <span class="elsevierStyleItalic">B</span><span class="elsevierStyleInf">2</span>&#44; <span class="elsevierStyleItalic">B</span><span class="elsevierStyleInf">3</span> the <span class="elsevierStyleItalic">B</span><span class="elsevierStyleInf">1</span> to <span class="elsevierStyleItalic">B</span><span class="elsevierStyleInf">3</span> ratio and prediction skill metrics <span class="elsevierStyleItalic">R</span><span class="elsevierStyleSup">2</span> and RMSE obtained from the New York cases&#46; The vertical red dashed lines represent the beginning of the Stay-at-Home order&#46;</p>"
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      "titulo" => "References"
      "seccion" => array:1 [
        0 => array:2 [
          "identificador" => "bibs0015"
          "bibliografiaReferencia" => array:25 [
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            ]
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                          "etal" => true
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                            0 => "S&#46; Cheval"
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                        0 => array:2 [
                          "etal" => false
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                      "doi" => "10.1371/journal.pone.0231236"
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                            "url" => "https://www.ncbi.nlm.nih.gov/pubmed/32231392"
                            "web" => "Medline"
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                      "titulo" => "Estimated demand for US hospital inpatient and intensive care unit beds for patients with COVID-19 based on comparisons with Wuhan and Guangzhou&#44; China"
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                        0 => array:2 [
                          "etal" => true
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                            0 => "R&#46; Li"
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                  ]
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                      "doi" => "10.1001/jamanetworkopen.2020.14661"
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                            "url" => "https://www.ncbi.nlm.nih.gov/pubmed/33030549"
                            "web" => "Medline"
                          ]
                        ]
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                        0 => array:2 [
                          "etal" => true
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                            0 => "H&#46; Salje"
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                          ]
                        ]
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                          "etal" => false
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                      "autores" => array:1 [
                        0 => array:2 [
                          "etal" => false
                          "autores" => array:2 [
                            0 => "Y&#46; Ben-Haim"
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                          ]
                        ]
                      ]
                    ]
                  ]
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                        0 => array:2 [
                          "etal" => true
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                            0 => "S&#46; Sahin"
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                        ]
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                    0 => array:1 [
                      "Libro" => array:3 [
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                          "etal" => false
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                        ]
                      ]
                    ]
                  ]
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                    0 => array:1 [
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        "texto" => "<p id="par0085" class="elsevierStylePara elsevierViewall"><span class="elsevierStyleSmallCaps">I</span> express my gratitude to WORLDOMETER for providing the daily New York Covid-19 cases and the DKI Jakarta province for sharing the public of its daily Covid-19 cases through their websites&#46; <span class="elsevierStyleSmallCaps">I</span> also thank Mr Andika for type-setting the equations and archiving the Supplementary data&#46;</p>"
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