{"id":789,"date":"2021-02-08T21:59:04","date_gmt":"2021-02-08T20:59:04","guid":{"rendered":"https:\/\/www.andremalo.be\/blog\/?p=789"},"modified":"2022-02-11T16:31:04","modified_gmt":"2022-02-11T15:31:04","slug":"une-exponentielle-cest-dangereux","status":"publish","type":"post","link":"https:\/\/www.andremalo.be\/blog\/2021\/02\/08\/une-exponentielle-cest-dangereux\/","title":{"rendered":"Une exponentielle, c&rsquo;est dangereux ? C&rsquo;est explosif ?"},"content":{"rendered":"\n<div class=\"wp-block-media-text alignwide is-stacked-on-mobile is-vertically-aligned-top\" style=\"grid-template-columns:38% auto\"><figure class=\"wp-block-media-text__media\"><img decoding=\"async\" loading=\"lazy\" width=\"1024\" height=\"683\" src=\"https:\/\/www.andremalo.be\/blog\/wp-content\/uploads\/2021\/02\/Nenu-1024x683.jpeg\" alt=\"\" class=\"wp-image-794 size-full\" srcset=\"https:\/\/www.andremalo.be\/blog\/wp-content\/uploads\/2021\/02\/Nenu-1024x683.jpeg 1024w, https:\/\/www.andremalo.be\/blog\/wp-content\/uploads\/2021\/02\/Nenu-300x200.jpeg 300w, https:\/\/www.andremalo.be\/blog\/wp-content\/uploads\/2021\/02\/Nenu-768x512.jpeg 768w, https:\/\/www.andremalo.be\/blog\/wp-content\/uploads\/2021\/02\/Nenu-600x400.jpeg 600w, https:\/\/www.andremalo.be\/blog\/wp-content\/uploads\/2021\/02\/Nenu.jpeg 1200w\" sizes=\"(max-width: 1024px) 100vw, 1024px\" \/><\/figure><div class=\"wp-block-media-text__content\">\n<p class=\"has-text-align-left\"><em><strong>Imaginons un n\u00e9nuphar plant\u00e9 dans un grand lac qui aurait la propri\u00e9t\u00e9 h\u00e9r\u00e9ditaire de produire, chaque jour, un autre n\u00e9nuphar.<\/strong><\/em><br><em><strong><br>Au bout de trente jours, la totalit\u00e9 du lac est couverte et l\u2019esp\u00e8ce meurt \u00e9touff\u00e9e, priv\u00e9e d\u2019espace et de nourriture.<\/strong><\/em><br><em><strong><br>Au bout de combien de jours les n\u00e9nuphars ont-ils couvert la moiti\u00e9 du lac ?<\/strong><\/em> <span class=\"footnote_referrer\"><a role=\"button\" tabindex=\"0\" onclick=\"footnote_moveToReference_789_1('footnote_plugin_reference_789_1_1');\" onkeypress=\"footnote_moveToReference_789_1('footnote_plugin_reference_789_1_1');\" ><sup id=\"footnote_plugin_tooltip_789_1_1\" class=\"footnote_plugin_tooltip_text\">[1]<\/sup><\/a><span id=\"footnote_plugin_tooltip_text_789_1_1\" class=\"footnote_tooltip\">Albert Jacquard, L&rsquo;\u00e9quation du n\u00e9nuphar, les plaisirs de la science, Calmann-L\u00e9vy, 1998<\/span><\/span><script type=\"text\/javascript\"> jQuery('#footnote_plugin_tooltip_789_1_1').tooltip({ tip: '#footnote_plugin_tooltip_text_789_1_1', tipClass: 'footnote_tooltip', effect: 'fade', predelay: 0, fadeInSpeed: 200, delay: 400, fadeOutSpeed: 200, position: 'top center', relative: true, offset: [-7, 0], });<\/script><\/p>\n<\/div><\/div>\n\n\n\n<p><\/p>\n\n\n\n<p>Vous avez compris : le lac \u00e9tait \u00e0 moiti\u00e9 couvert au bout de 29 jours.<br>C&rsquo;est une surprise &#8230; ? R\u00e9fl\u00e9chissons en mettant la marche arri\u00e8re.<br>Si nous sommes le 29\u00e8me jour, le lac est \u00e0 moiti\u00e9 couvert, donc, demain, le 30\u00e8me jour, la surface aura doubl\u00e9 et les n\u00e9nuphars couvriront l&rsquo;enti\u00e8ret\u00e9 du lac.<br>De m\u00eame, si nous sommes le 28\u00e8me jour, un quart du lac est recouvert, et le lendemain, la moiti\u00e9 le sera &#8230; etc.<br>Donc, lorsqu&rsquo;un huiti\u00e8me de la surface est couverte, cela para\u00eet rassurant, mais en fait nous sommes \u00e0 trois jours de la couverture compl\u00e8te &#8230;<br>C\u2019est un exemple d\u2019exponentielle de base 2 : la quantit\u00e9 double chaque jour.<\/p>\n\n\n\n<p>Les math\u00e9maticiens aiment mettre cela en formule : <br><p class=\"ql-center-displayed-equation\" style=\"line-height: 18px;\"><span class=\"ql-right-eqno\"> &nbsp; <\/span><span class=\"ql-left-eqno\"> &nbsp; <\/span><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/www.andremalo.be\/blog\/wp-content\/ql-cache\/quicklatex.com-c83bbd56ed37abe6bc58551108011bfe_l3.png\" height=\"18\" width=\"50\" class=\"ql-img-displayed-equation quicklatex-auto-format\" alt=\"&#92;&#91;&#121;&#61;&#50;&#94;&#120;&#92;&#93;\" title=\"Rendered by QuickLaTeX.com\"\/><\/p><br>o\u00f9 <img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/www.andremalo.be\/blog\/wp-content\/ql-cache\/quicklatex.com-49b85cfbdabd6b01284e0b4f939b89d9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"9\" style=\"vertical-align: -4px;\"\/> est le nombre de plants de n\u00e9nuphars et <img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/www.andremalo.be\/blog\/wp-content\/ql-cache\/quicklatex.com-6ce5261fb6842547b48c39330b7254c0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"10\" style=\"vertical-align: 0px;\"\/> le nombre de jours depuis le premier. Le premier jour est bien le jour <img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/www.andremalo.be\/blog\/wp-content\/ql-cache\/quicklatex.com-7b198fe49320c03cf54c843278d0c4af_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"9\" style=\"vertical-align: 0px;\"\/>, il y a un seul plan de n\u00e9nuphar, donc <img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/www.andremalo.be\/blog\/wp-content\/ql-cache\/quicklatex.com-324511d584747b321c66c141af5ec97b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"43\" style=\"vertical-align: 0px;\"\/> et <img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/www.andremalo.be\/blog\/wp-content\/ql-cache\/quicklatex.com-24041290a5e7367a5fa95ddaf51d7b18_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#94;&#120;&#61;&#50;&#94;&#48;&#61;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"89\" style=\"vertical-align: 0px;\"\/>.<\/p>\n\n\n\n<p><p class=\"ql-center-picture\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/www.andremalo.be\/blog\/wp-content\/ql-cache\/quicklatex.com-01d266303a6796b44945d700c186fe34_l3.png\" height=\"511\" width=\"731\" class=\"ql-img-picture quicklatex-auto-format\" alt=\"Rendered by QuickLaTeX.com\" title=\"Rendered by QuickLaTeX.com\"\/><\/p><\/p>\n\n\n\n<p>C&rsquo;est joli, des n\u00e9nuphars, mais quel rapport avec la Covid 19 ?<br><br>Imaginons une situation : septembre 2020, un h\u00f4pital imaginaire, si standard qu&rsquo;il correspond exactement aux statistiques de Sciensano, un service Covid, 46 places, une partie en soins intensifs. La cheffe du service consulte quotidiennement les statistiques de Sciensano <span class=\"footnote_referrer\"><a role=\"button\" tabindex=\"0\" onclick=\"footnote_moveToReference_789_1('footnote_plugin_reference_789_1_2');\" onkeypress=\"footnote_moveToReference_789_1('footnote_plugin_reference_789_1_2');\" ><sup id=\"footnote_plugin_tooltip_789_1_2\" class=\"footnote_plugin_tooltip_text\">[2]<\/sup><\/a><span id=\"footnote_plugin_tooltip_text_789_1_2\" class=\"footnote_tooltip\">Site des statistiques de Sciensano : <a href=\"https:\/\/epistat.wiv-isp.be\/covid\/covid-19.html\"><span class=\"footnote_url_wrap\">https:\/\/epistat.wiv-isp.be\/covid\/covid-19.html<\/span><\/a><\/span><\/span><script type=\"text\/javascript\"> jQuery('#footnote_plugin_tooltip_789_1_2').tooltip({ tip: '#footnote_plugin_tooltip_text_789_1_2', tipClass: 'footnote_tooltip', effect: 'fade', predelay: 0, fadeInSpeed: 200, delay: 400, fadeOutSpeed: 200, position: 'top center', relative: true, offset: [-7, 0], });<\/script>, attentive au doublement du nombre de personnes hospitalis\u00e9es.<br><br>Elle \u00e9crit sur le tableau du service : <br>\u00abNombre de personnes hospitalis\u00e9es :\u00bb<br>\u00abmardi 22 septembre 2020 &#8211; Sciensano : 505 &#8211; nous : 3 <img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/www.andremalo.be\/blog\/wp-content\/ql-cache\/quicklatex.com-ca7b58cd01f89769b0a62f6ed5733f72_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#110;&#101;&#97;&#114;&#114;&#111;&#119;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"18\" style=\"vertical-align: -4px;\"\/> (en hausse).\u00bb<br><br>15 jours apr\u00e8s, elle \u00e9crit : <br>\u00abmercredi 7 octobre 2020 &#8211; Sciensano : 1050 &#8211; nous : 6 <img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/www.andremalo.be\/blog\/wp-content\/ql-cache\/quicklatex.com-ca7b58cd01f89769b0a62f6ed5733f72_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#110;&#101;&#97;&#114;&#114;&#111;&#119;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"18\" style=\"vertical-align: -4px;\"\/>.\u00bb<br>La veille, le communiqu\u00e9 officiel du 6 octobre titrait : \u00abLe Comit\u00e9 de concertation d\u00e9cide de mesures plus strictes et d\u00e9signe un commissaire COVID-19\u00bb.<br><br>9 jours plus tard : <br>\u00abvendredi 16 octobre 2020 &#8211; Sciensano : 2099 &#8211; nous : 12 <img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/www.andremalo.be\/blog\/wp-content\/ql-cache\/quicklatex.com-ca7b58cd01f89769b0a62f6ed5733f72_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#110;&#101;&#97;&#114;&#114;&#111;&#119;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"18\" style=\"vertical-align: -4px;\"\/>.\u00bb<br><br>7 jours apr\u00e8s : <br>\u00abvendredi 23 octobre 2020 &#8211; Sciensano : 4056 &#8211; nous : 24 <img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/www.andremalo.be\/blog\/wp-content\/ql-cache\/quicklatex.com-ca7b58cd01f89769b0a62f6ed5733f72_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#110;&#101;&#97;&#114;&#114;&#111;&#119;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"18\" style=\"vertical-align: -4px;\"\/><br>On suit le mouvement, mais  \u00e7a double de plus en plus vite.\u00bb<br>Se souvenant de ses cours de math\u00e9matiques, elle pense : \u00abc&rsquo;est pire qu&rsquo;une exponentielle de base 2, o\u00f9 les intervalles de temps seraient \u00e9gaux. Notre professeur nous avait parl\u00e9 de n\u00e9nuphars &#8230;\u00bb<br>Arrive le directeur financier de l\u2019h\u00f4pital : \u00abTout va bien, le service est occup\u00e9 \u00e0 \u2026 (il calcule vite sur son smartphone) 24 : 46 x 100 =\u2026 52,2 %, il reste presque la moiti\u00e9 de lits disponibles !\u00bb<br>Mais la r\u00e9partie est imm\u00e9diate : \u00abPas du tout, vous n&rsquo;y \u00eates pas! Si \u00e7a continue ainsi, vendredi prochain, nous serons complets et nous aurons d\u00fb trouver un autre h\u00f4pital pour deux malades !\u00bb<br><br>Mais, heureusement, elle peut \u00e9crire le vendredi suivant : <br>\u00abvendredi 30 octobre 2020 &#8211; Sciensano : 6439 &#8211; nous : 38 <img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/www.andremalo.be\/blog\/wp-content\/ql-cache\/quicklatex.com-ca7b58cd01f89769b0a62f6ed5733f72_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#110;&#101;&#97;&#114;&#114;&#111;&#119;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"18\" style=\"vertical-align: -4px;\"\/><br><strong>LE POINT D&rsquo;INFLEXION EST D\u00c9PASS<strong>\u00c9<\/strong>, <strong><strong>\u00c7<\/strong>A<\/strong> MONTE TOUJOURS<\/strong><br><strong>MAIS <strong>\u00c7<\/strong>A RALENTI, L&rsquo;EFFET DES MESURES SE FAIT SENTIR,<\/strong><br><strong>ON VA Y ARRIVER !<\/strong>\u00bb <br>Et elle le r\u00e9p\u00e8te \u00e0 tout le monde !!!<br><br>Et le vendredi suivant, on peut lire au tableau : <br>\u00ablundi 2 novembre &#8211; Sciensano : 7233 &#8211; nous : 43 <img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/www.andremalo.be\/blog\/wp-content\/ql-cache\/quicklatex.com-ca7b58cd01f89769b0a62f6ed5733f72_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#110;&#101;&#97;&#114;&#114;&#111;&#119;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"18\" style=\"vertical-align: -4px;\"\/>\u00bb<br>\u00abmardi 3 novembre &#8211; Sciensano : 7461 &#8211; nous : 44 <img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/www.andremalo.be\/blog\/wp-content\/ql-cache\/quicklatex.com-ca7b58cd01f89769b0a62f6ed5733f72_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#110;&#101;&#97;&#114;&#114;&#111;&#119;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"18\" style=\"vertical-align: -4px;\"\/>\u00bb<br>\u00abmercredi 4 novembre &#8211; Sciensano : 7411 &#8211; nous : 44 <img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/www.andremalo.be\/blog\/wp-content\/ql-cache\/quicklatex.com-424987d1ec48e1f03bc8e023c590e490_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#101;&#97;&#114;&#114;&#111;&#119;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"18\" style=\"vertical-align: -4px;\"\/>\u00bb<br>\u00abjeudi 5 novembre &#8211; Sciensano : 7290 &#8211; nous : 43 <img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/www.andremalo.be\/blog\/wp-content\/ql-cache\/quicklatex.com-424987d1ec48e1f03bc8e023c590e490_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#101;&#97;&#114;&#114;&#111;&#119;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"18\" style=\"vertical-align: -4px;\"\/> \u00bb<br>\u00abvendredi 6 novembre &#8211; Sciensano : 7224 &#8211; nous : 43 <img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/www.andremalo.be\/blog\/wp-content\/ql-cache\/quicklatex.com-424987d1ec48e1f03bc8e023c590e490_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#101;&#97;&#114;&#114;&#111;&#119;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"18\" style=\"vertical-align: -4px;\"\/> \u00bb<br>\u00abOn a \u00e9t\u00e9 presque complet, mais maintenant, apr\u00e8s un mois d&rsquo;efforts,<br><strong>ON A GAGN<strong>\u00c9<\/strong>, <strong>\u00c7<\/strong>A DESCEND !<\/strong>\u00bb<\/p>\n\n\n\n<p><p class=\"ql-center-picture\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/www.andremalo.be\/blog\/wp-content\/ql-cache\/quicklatex.com-c1b49bbdadf27852460704bd7d71186a_l3.png\" height=\"863\" width=\"755\" class=\"ql-img-picture quicklatex-auto-format\" alt=\"Rendered by QuickLaTeX.com\" title=\"Rendered by QuickLaTeX.com\"\/><\/p><\/p>\n<div class=\"speaker-mute footnotes_reference_container\"> <div class=\"footnote_container_prepare\"><p><span role=\"button\" tabindex=\"0\" class=\"footnote_reference_container_label pointer\" onclick=\"footnote_expand_collapse_reference_container_789_1();\">Notes<\/span><span role=\"button\" tabindex=\"0\" class=\"footnote_reference_container_collapse_button\" style=\"display: none;\" onclick=\"footnote_expand_collapse_reference_container_789_1();\">[<a id=\"footnote_reference_container_collapse_button_789_1\">+<\/a>]<\/span><\/p><\/div> <div id=\"footnote_references_container_789_1\" style=\"\"><table class=\"footnotes_table footnote-reference-container\"><caption class=\"accessibility\">Notes<\/caption> <tbody> \r\n\r\n<tr class=\"footnotes_plugin_reference_row\"> <th scope=\"row\" class=\"footnote_plugin_index_combi pointer\"  onclick=\"footnote_moveToAnchor_789_1('footnote_plugin_tooltip_789_1_1');\"><a id=\"footnote_plugin_reference_789_1_1\" class=\"footnote_backlink\"><span class=\"footnote_index_arrow\">&#8593;<\/span>1<\/a><\/th> <td class=\"footnote_plugin_text\">Albert Jacquard, L&rsquo;\u00e9quation du n\u00e9nuphar, les plaisirs de la science, Calmann-L\u00e9vy, 1998<\/td><\/tr>\r\n\r\n<tr class=\"footnotes_plugin_reference_row\"> <th scope=\"row\" class=\"footnote_plugin_index_combi pointer\"  onclick=\"footnote_moveToAnchor_789_1('footnote_plugin_tooltip_789_1_2');\"><a id=\"footnote_plugin_reference_789_1_2\" class=\"footnote_backlink\"><span class=\"footnote_index_arrow\">&#8593;<\/span>2<\/a><\/th> <td class=\"footnote_plugin_text\">Site des statistiques de Sciensano : <a href=\"https:\/\/epistat.wiv-isp.be\/covid\/covid-19.html\"><span class=\"footnote_url_wrap\">https:\/\/epistat.wiv-isp.be\/covid\/covid-19.html<\/span><\/a><\/td><\/tr>\r\n\r\n <\/tbody> <\/table> <\/div><\/div><script type=\"text\/javascript\"> function footnote_expand_reference_container_789_1() { jQuery('#footnote_references_container_789_1').show(); 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jQuery('html, body').animate({ scrollTop: l_obj_Target.offset().top - window.innerHeight * 0.2 }, 380); } }<\/script>","protected":false},"excerpt":{"rendered":"<p>Imaginons un n\u00e9nuphar plant\u00e9 dans un grand lac qui aurait la propri\u00e9t\u00e9 h\u00e9r\u00e9ditaire de produire, chaque jour, un autre n\u00e9nuphar.<\/p>\n<p>Au bout de trente jours, la totalit\u00e9 du lac est couverte et l\u2019esp\u00e8ce meurt \u00e9touff\u00e9e, priv\u00e9e d\u2019espace et de nourriture.<\/p>\n<p>Au bout de combien de jours les n\u00e9nuphars ont-ils couvert la moiti\u00e9 du lac ?<\/p>\n","protected":false},"author":3,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_coblocks_attr":"","_coblocks_dimensions":"","_coblocks_responsive_height":"","_coblocks_accordion_ie_support":""},"categories":[36,43],"tags":[41,40,42,39],"_links":{"self":[{"href":"https:\/\/www.andremalo.be\/blog\/wp-json\/wp\/v2\/posts\/789"}],"collection":[{"href":"https:\/\/www.andremalo.be\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.andremalo.be\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.andremalo.be\/blog\/wp-json\/wp\/v2\/users\/3"}],"replies":[{"embeddable":true,"href":"https:\/\/www.andremalo.be\/blog\/wp-json\/wp\/v2\/comments?post=789"}],"version-history":[{"count":577,"href":"https:\/\/www.andremalo.be\/blog\/wp-json\/wp\/v2\/posts\/789\/revisions"}],"predecessor-version":[{"id":2777,"href":"https:\/\/www.andremalo.be\/blog\/wp-json\/wp\/v2\/posts\/789\/revisions\/2777"}],"wp:attachment":[{"href":"https:\/\/www.andremalo.be\/blog\/wp-json\/wp\/v2\/media?parent=789"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.andremalo.be\/blog\/wp-json\/wp\/v2\/categories?post=789"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.andremalo.be\/blog\/wp-json\/wp\/v2\/tags?post=789"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}