Comment les maths ont pr\u00e9dit le Covid en Afrique \u2014 MathsVivantes<\/title>\n<link href=\"https:\/\/fonts.googleapis.com\/css2?family=Playfair+Display:ital,wght@0,400;0,700;1,400&family=Source+Serif+4:ital,wght@0,300;0,400;0,600;1,300;1,400&display=swap\" rel=\"stylesheet\">\n<style>\n :root {\n --noir: #0f0e0b;\n --creme: #f5f0e8;\n --vert-sombre: #0d3320;\n --vert: #1a6b3c;\n --vert-clair: #27ae60;\n --rouge: #c0392b;\n --orange: #e67e22;\n --gris: #6b6560;\n --gris-clair: #e8e2d8;\n --ocre-clair: #f0c060;\n }\n\n * { margin: 0; padding: 0; box-sizing: border-box; }\n\n body {\n background-color: var(--creme);\n color: var(--noir);\n font-family: 'Source Serif 4', Georgia, serif;\n font-weight: 300;\n line-height: 1.8;\n }\n\n .hero {\n background: #051a0d;\n color: var(--creme);\n padding: 80px 40px 70px;\n position: relative;\n overflow: hidden;\n }\n .hero::before {\n content: \"S I R \u03b2 \u03b3\";\n position: absolute;\n top: 20px; right: 40px;\n font-size: 5rem;\n opacity: 0.05;\n font-family: 'Playfair Display', serif;\n letter-spacing: 0.3em;\n }\n .hero-inner { max-width: 760px; 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}\n .code-keyword { color: #7ec8e3; }\n .code-string { color: #f0c060; }\n .code-number { color: #a8d8a8; }\n .code-function { color: #c8e6c9; }\n\n .exercice {\n background: #051a0d;\n color: var(--creme);\n padding: 36px 40px;\n margin: 3em 0;\n }\n .exercice h3 { color: var(--ocre-clair); font-style: normal; margin-top: 0; margin-bottom: 1em; font-size: 1.1rem; }\n .exercice p { color: rgba(245,240,232,0.82); font-size: 0.97rem; margin-bottom: 0.8em; }\n .exercice .correction { border-top: 1px solid rgba(39,174,96,0.4); margin-top: 1.5em; padding-top: 1.2em; }\n .exercice .correction-titre { font-size: 0.7rem; letter-spacing: 0.18em; text-transform: uppercase; color: var(--ocre-clair); margin-bottom: 0.8em; font-weight: 600; }\n\n blockquote {\n padding: 0 0 0 32px;\n margin: 2em 0;\n position: relative;\n font-style: italic;\n font-size: 1.12rem;\n color: #3a3530;\n }\n blockquote::before {\n content: \"\\201C\";\n position: absolute;\n left: 0; top: -10px;\n font-size: 3.5rem;\n color: var(--vert-clair);\n font-family: 'Playfair Display', serif;\n line-height: 1;\n }\n blockquote cite { display: block; font-size: 0.82rem; font-style: normal; letter-spacing: 0.1em; text-transform: uppercase; color: var(--gris); margin-top: 8px; }\n\n .conclusion { border-top: 2px solid var(--noir); margin-top: 3em; padding-top: 2em; }\n\n @media (max-width: 600px) {\n .hero { padding: 50px 20px 40px; }\n .article-body { padding: 40px 20px 60px; }\n .exercice { padding: 24px 20px; }\n .sir-grid { grid-template-columns: 1fr; }\n .code-block { padding: 16px; font-size: 0.78rem; }\n }\n<\/style>\n<\/head>\n<body>\n\n<div class=\"hero\">\n <div class=\"hero-inner\">\n <div class=\"article-rubrique\">Mod\u00e9lisation \u00b7 Article #7<\/div>\n <h1>Comment les maths<br>ont pr\u00e9dit le Covid<br><em>en Afrique<\/em><\/h1>\n <p class=\"hero-intro\">Mars 2020. Le premier cas de Covid-19 est d\u00e9tect\u00e9 au B\u00e9nin. Quelques jours plus tard, des \u00e9pid\u00e9miologistes africains sortent des chiffres : combien de personnes seront infect\u00e9es, quand le pic arrivera, combien de lits d’h\u00f4pitaux pr\u00e9parer. Ces pr\u00e9dictions venaient des math\u00e9matiques. D’un mod\u00e8le vieux de plus de 100 ans.<\/p>\n <div class=\"meta\">Par Odilon AKOWANOU \u00b7 MathsVivantes \u00b7 Lecture : 12 min<\/div>\n <\/div>\n<\/div>\n\n<article class=\"article-body\">\n\n <p>\n En mars 2020, quand le Covid-19 a commenc\u00e9 \u00e0 se propager en Afrique, les gouvernements avaient un besoin urgent : <strong>pr\u00e9voir<\/strong>. Combien de personnes vont tomber malades ? En combien de temps ? Faut-il confiner ? Fermer les fronti\u00e8res ? Pr\u00e9parer des h\u00f4pitaux de campagne ?\n <\/p>\n\n <p>\n Pour r\u00e9pondre \u00e0 ces questions de vie ou de mort, les \u00e9pid\u00e9miologistes ont utilis\u00e9 un outil math\u00e9matique invent\u00e9 en <strong>1927<\/strong> par deux chercheurs \u00e9cossais \u2014 William Kermack et Anderson McKendrick. Un mod\u00e8le si puissant et si \u00e9l\u00e9gant qu’il est encore aujourd’hui au c\u0153ur de toute mod\u00e9lisation \u00e9pid\u00e9mique : le <strong>mod\u00e8le SIR<\/strong>.\n <\/p>\n\n <h2>L’histoire : une id\u00e9e n\u00e9e d’une \u00e9pid\u00e9mie de peste<\/h2>\n\n <div class=\"encadre-histoire encadre\">\n <span class=\"encadre-titre\">\ud83d\udcdc Naissance du mod\u00e8le SIR \u2014 1927<\/span>\n <p>\n En 1927, l’Inde britannique est ravag\u00e9e par des \u00e9pid\u00e9mies r\u00e9currentes de peste et de chol\u00e9ra. William Kermack, un chimiste, et Anderson McKendrick, un m\u00e9decin militaire, cherchent \u00e0 comprendre <em>pourquoi<\/em> certaines \u00e9pid\u00e9mies s’\u00e9teignent d’elles-m\u00eames avant d’infecter toute la population.\n <\/p>\n <p>\n Leur id\u00e9e r\u00e9volutionnaire : mod\u00e9liser une population comme un syst\u00e8me de <strong>flux entre trois compartiments<\/strong>. Les individus passent d’un groupe \u00e0 l’autre selon des taux math\u00e9matiques pr\u00e9cis. Avec ce mod\u00e8le, ils parviennent \u00e0 reproduire fid\u00e8lement les courbes \u00e9pid\u00e9miques observ\u00e9es \u2014 et \u00e0 pr\u00e9dire leur \u00e9volution.\n <\/p>\n <p>\n Presque un si\u00e8cle plus tard, ce m\u00eame mod\u00e8le \u2014 adapt\u00e9 et enrichi \u2014 a guid\u00e9 les d\u00e9cisions sanitaires de 54 pays africains face au Covid-19. Les math\u00e9matiques de 1927 ont sauv\u00e9 des vies en 2020.\n <\/p>\n <\/div>\n\n <h2>Le mod\u00e8le SIR : trois compartiments, une population<\/h2>\n\n <p>\n L’id\u00e9e centrale est simple : \u00e0 chaque instant, chaque individu d’une population appartient \u00e0 l’un de ces trois groupes :\n <\/p>\n\n <div class=\"sir-grid\">\n <div class=\"sir-bloc sir-S\">\n <span class=\"sir-lettre\">S<\/span>\n <div class=\"sir-nom\">Susceptibles<\/div>\n <div class=\"sir-desc\">Personnes saines pouvant \u00eatre infect\u00e9es<\/div>\n <\/div>\n <div class=\"sir-bloc sir-I\">\n <span class=\"sir-lettre\">I<\/span>\n <div class=\"sir-nom\">Infect\u00e9s<\/div>\n <div class=\"sir-desc\">Personnes malades et contagieuses<\/div>\n <\/div>\n <div class=\"sir-bloc sir-R\">\n <span class=\"sir-lettre\">R<\/span>\n <div class=\"sir-nom\">R\u00e9tablis<\/div>\n <div class=\"sir-desc\">Gu\u00e9ris ou immunis\u00e9s (ou d\u00e9c\u00e9d\u00e9s)<\/div>\n <\/div>\n <\/div>\n\n <p>\n La population totale N reste constante : <strong>S + I + R = N<\/strong>. Les individus ne font que circuler entre ces trois groupes \u2014 jamais dans l’autre sens.\n <\/p>\n\n <div class=\"schema-container\">\n <div class=\"schema-titre\">\ud83d\udd04 Flux entre les compartiments du mod\u00e8le SIR<\/div>\n <svg width=\"100%\" viewBox=\"0 0 520 140\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\">\n <rect width=\"520\" height=\"140\" fill=\"white\"\/>\n\n \n <rect x=\"30\" y=\"40\" width=\"110\" height=\"60\" rx=\"4\" fill=\"#2980b9\"\/>\n <text x=\"85\" y=\"68\" font-size=\"22\" fill=\"white\" text-anchor=\"middle\" font-family=\"Playfair Display, serif\" font-weight=\"700\">S<\/text>\n <text x=\"85\" y=\"88\" font-size=\"10\" fill=\"rgba(255,255,255,0.85)\" text-anchor=\"middle\">Susceptibles<\/text>\n\n \n <line x1=\"140\" y1=\"70\" x2=\"195\" y2=\"70\" stroke=\"#c0392b\" stroke-width=\"2.5\" marker-end=\"url(#arrowRed)\"\/>\n <defs>\n <marker id=\"arrowRed\" markerWidth=\"8\" markerHeight=\"8\" refX=\"6\" refY=\"3\" orient=\"auto\">\n <path d=\"M0,0 L0,6 L8,3 z\" fill=\"#c0392b\"\/>\n <\/marker>\n <marker id=\"arrowGreen\" markerWidth=\"8\" markerHeight=\"8\" refX=\"6\" refY=\"3\" orient=\"auto\">\n <path d=\"M0,0 L0,6 L8,3 z\" fill=\"#27ae60\"\/>\n <\/marker>\n <\/defs>\n <text x=\"167\" y=\"60\" font-size=\"11\" fill=\"#c0392b\" text-anchor=\"middle\" font-style=\"italic\">\u03b2\u00b7S\u00b7I\/N<\/text>\n\n \n <rect x=\"195\" y=\"40\" width=\"110\" height=\"60\" rx=\"4\" fill=\"#c0392b\"\/>\n <text x=\"250\" y=\"68\" font-size=\"22\" fill=\"white\" text-anchor=\"middle\" font-family=\"Playfair Display, serif\" font-weight=\"700\">I<\/text>\n <text x=\"250\" y=\"88\" font-size=\"10\" fill=\"rgba(255,255,255,0.85)\" text-anchor=\"middle\">Infect\u00e9s<\/text>\n\n \n <line x1=\"305\" y1=\"70\" x2=\"360\" y2=\"70\" stroke=\"#27ae60\" stroke-width=\"2.5\" marker-end=\"url(#arrowGreen)\"\/>\n <text x=\"332\" y=\"60\" font-size=\"11\" fill=\"#27ae60\" text-anchor=\"middle\" font-style=\"italic\">\u03b3\u00b7I<\/text>\n\n \n <rect x=\"360\" y=\"40\" width=\"110\" height=\"60\" rx=\"4\" fill=\"#27ae60\"\/>\n <text x=\"415\" y=\"68\" font-size=\"22\" fill=\"white\" text-anchor=\"middle\" font-family=\"Playfair Display, serif\" font-weight=\"700\">R<\/text>\n <text x=\"415\" y=\"88\" font-size=\"10\" fill=\"rgba(255,255,255,0.85)\" text-anchor=\"middle\">R\u00e9tablis<\/text>\n\n \n <text x=\"85\" y=\"120\" font-size=\"10\" fill=\"#6b6560\" text-anchor=\"middle\">Population totale N = S + I + R<\/text>\n <\/svg>\n <\/div>\n\n <h2>Les \u00e9quations : le langage math\u00e9matique de l’\u00e9pid\u00e9mie<\/h2>\n\n <p>\n Le mod\u00e8le SIR est d\u00e9crit par un syst\u00e8me de trois <strong>\u00e9quations diff\u00e9rentielles<\/strong> \u2014 des \u00e9quations qui d\u00e9crivent comment chaque compartiment varie au fil du temps :\n <\/p>\n\n <div class=\"encadre\">\n <span class=\"encadre-titre\">\ud83d\udcd0 Les \u00e9quations du mod\u00e8le SIR<\/span>\n <p style=\"font-family:'Playfair Display',serif; font-size:1.1rem; text-align:center; margin: 8px 0;\">dS\/dt = \u2212\u03b2 \u00d7 S \u00d7 I \/ N<\/p>\n <p style=\"font-family:'Playfair Display',serif; font-size:1.1rem; text-align:center; margin: 8px 0;\">dI\/dt = \u03b2 \u00d7 S \u00d7 I \/ N \u2212 \u03b3 \u00d7 I<\/p>\n <p style=\"font-family:'Playfair Display',serif; font-size:1.1rem; text-align:center; margin: 8px 0;\">dR\/dt = \u03b3 \u00d7 I<\/p>\n <p style=\"margin-top:12px\"><strong>\u03b2 (b\u00eata)<\/strong> = taux de transmission (vitesse \u00e0 laquelle S devient I)<\/p>\n <p><strong>\u03b3 (gamma)<\/strong> = taux de gu\u00e9rison (vitesse \u00e0 laquelle I devient R)<\/p>\n <p><strong>dX\/dt<\/strong> = variation de X par unit\u00e9 de temps (ici, par jour)<\/p>\n <\/div>\n\n <p>\n Note que dS\/dt + dI\/dt + dR\/dt = 0 \u2014 ce qui confirme que la population totale est constante. Ce qui sort de S entre dans I, ce qui sort de I entre dans R.\n <\/p>\n\n <h2>Le nombre R\u2080 : le param\u00e8tre qui change tout<\/h2>\n\n <p>\n Du mod\u00e8le SIR \u00e9merge un nombre fondamental qui r\u00e9sume la dangerosit\u00e9 d’une \u00e9pid\u00e9mie : le <strong>nombre de reproduction de base R\u2080<\/strong> (prononc\u00e9 \u00ab\u00a0R z\u00e9ro\u00a0\u00bb).\n <\/p>\n\n <div class=\"formule\">R\u2080 = \u03b2 \/ \u03b3<\/div>\n\n <div class=\"definition\">\n <strong>R\u2080<\/strong> repr\u00e9sente le nombre moyen de personnes qu’un individu infect\u00e9 contaminera dans une population enti\u00e8rement susceptible. C’est <em>le<\/em> param\u00e8tre cl\u00e9 de toute \u00e9pid\u00e9mie.\n <\/div>\n\n <div class=\"encadre\">\n <span class=\"encadre-titre\">\ud83d\udd11 Interpr\u00e9tation de R\u2080<\/span>\n <p><strong>R\u2080 < 1<\/strong> \u2192 Chaque infect\u00e9 contamine moins d’une personne \u2192 l’\u00e9pid\u00e9mie s’\u00e9teint naturellement \u2705<\/p>\n <p><strong>R\u2080 = 1<\/strong> \u2192 L’\u00e9pid\u00e9mie est stable \u2014 ni croissance ni d\u00e9clin \u26a0\ufe0f<\/p>\n <p><strong>R\u2080 > 1<\/strong> \u2192 Chaque infect\u00e9 contamine plus d’une personne \u2192 l’\u00e9pid\u00e9mie se propage \u274c<\/p>\n <p style=\"margin-top:8px\"><strong>Pour le Covid-19 en Afrique :<\/strong> R\u2080 estim\u00e9 entre 2 et 3 selon les pays et les variants.<\/p>\n <\/div>\n\n <h2>Application au B\u00e9nin : simulons une \u00e9pid\u00e9mie<\/h2>\n\n <p>\n Appliquons le mod\u00e8le SIR \u00e0 une ville fictive de 100 000 habitants au B\u00e9nin, avec des param\u00e8tres inspir\u00e9s des donn\u00e9es Covid-19 africaines :\n <\/p>\n\n <div class=\"table-container\">\n <table>\n <tr>\n <th>Param\u00e8tre<\/th>\n <th>Valeur<\/th>\n <th>Signification<\/th>\n <\/tr>\n <tr><td>N (population)<\/td><td>100 000<\/td><td>Habitants de la ville<\/td><\/tr>\n <tr><td>I\u2080 (infect\u00e9s initiaux)<\/td><td>10<\/td><td>10 cas au jour 0<\/td><\/tr>\n <tr><td>S\u2080 (susceptibles initiaux)<\/td><td>99 990<\/td><td>Tout le monde sauf les 10<\/td><\/tr>\n <tr><td>R\u2080 (r\u00e9tablis initiaux)<\/td><td>0<\/td><td>Personne n’est encore immunis\u00e9<\/td><\/tr>\n <tr><td>\u03b2 (taux transmission)<\/td><td>0,3<\/td><td>30% de chances de transmettre par contact<\/td><\/tr>\n <tr><td>\u03b3 (taux gu\u00e9rison)<\/td><td>0,1<\/td><td>Gu\u00e9rison en ~10 jours en moyenne<\/td><\/tr>\n <tr><td><strong>R\u2080 = \u03b2\/\u03b3<\/strong><\/td><td><strong>3,0<\/strong><\/td><td>Chaque infect\u00e9 en contamine 3<\/td><\/tr>\n <\/table>\n <\/div>\n\n <h2>Le code Python : simulons l’\u00e9pid\u00e9mie<\/h2>\n\n <div class=\"code-block\">\n <span class=\"code-label\">Python<\/span>\n<span class=\"code-comment\"># Mod\u00e8le SIR – Simulation \u00e9pid\u00e9mie Covid au B\u00e9nin<\/span>\n<span class=\"code-keyword\">import<\/span> numpy <span class=\"code-keyword\">as<\/span> np\n<span class=\"code-keyword\">from<\/span> scipy.integrate <span class=\"code-keyword\">import<\/span> odeint\n<span class=\"code-keyword\">import<\/span> matplotlib.pyplot <span class=\"code-keyword\">as<\/span> plt\n\n<span class=\"code-comment\"># — Param\u00e8tres —<\/span>\nN = <span class=\"code-number\">100000<\/span> <span class=\"code-comment\"># Population totale<\/span>\nI0 = <span class=\"code-number\">10<\/span> <span class=\"code-comment\"># Infect\u00e9s au jour 0<\/span>\nR0_init = <span class=\"code-number\">0<\/span> <span class=\"code-comment\"># R\u00e9tablis au jour 0<\/span>\nS0 = N – I0 – R0_init <span class=\"code-comment\"># Susceptibles au jour 0<\/span>\n\nbeta = <span class=\"code-number\">0.3<\/span> <span class=\"code-comment\"># Taux de transmission<\/span>\ngamma = <span class=\"code-number\">0.1<\/span> <span class=\"code-comment\"># Taux de gu\u00e9rison<\/span>\nR0 = beta \/ gamma\n<span class=\"code-function\">print<\/span>(<span class=\"code-string\">f\u00a0\u00bbR\u2080 = {R0} \u2192 \u00e9pid\u00e9mie {‘croissante \u26a0\ufe0f’ if R0 > 1 else ‘d\u00e9croissante \u2705’}\u00a0\u00bb<\/span>)\n\n<span class=\"code-comment\"># — \u00c9quations diff\u00e9rentielles SIR —<\/span>\n<span class=\"code-keyword\">def<\/span> <span class=\"code-function\">modele_SIR<\/span>(y, t, N, beta, gamma):\n S, I, R = y\n dS = -beta * S * I \/ N\n dI = beta * S * I \/ N – gamma * I\n dR = gamma * I\n <span class=\"code-keyword\">return<\/span> [dS, dI, dR]\n\n<span class=\"code-comment\"># — R\u00e9solution —<\/span>\nt = np.<span class=\"code-function\">linspace<\/span>(<span class=\"code-number\">0<\/span>, <span class=\"code-number\">200<\/span>, <span class=\"code-number\">200<\/span>) <span class=\"code-comment\"># 200 jours<\/span>\ny0 = [S0, I0, R0_init]\nsolution = <span class=\"code-function\">odeint<\/span>(modele_SIR, y0, t, args=(N, beta, gamma))\nS, I, R = solution.T\n\n<span class=\"code-comment\"># — R\u00e9sultats cl\u00e9s —<\/span>\npic_jour = t[np.<span class=\"code-function\">argmax<\/span>(I)]\npic_infectes = <span class=\"code-function\">int<\/span>(<span class=\"code-function\">max<\/span>(I))\n<span class=\"code-function\">print<\/span>(<span class=\"code-string\">f\u00a0\u00bbPic \u00e9pid\u00e9mique : jour {pic_jour:.0f}\u00a0\u00bb<\/span>)\n<span class=\"code-function\">print<\/span>(<span class=\"code-string\">f\u00a0\u00bbMaximum d’infect\u00e9s simultan\u00e9s : {pic_infectes:,}\u00a0\u00bb<\/span>)\n<span class=\"code-function\">print<\/span>(<span class=\"code-string\">f\u00a0\u00bbTotal infect\u00e9s en fin d’\u00e9pid\u00e9mie : {int(R[-1]):,}\u00a0\u00bb<\/span>)\n\n<span class=\"code-comment\"># — Graphique —<\/span>\nplt.<span class=\"code-function\">figure<\/span>(figsize=(<span class=\"code-number\">10<\/span>, <span class=\"code-number\">6<\/span>))\nplt.<span class=\"code-function\">plot<\/span>(t, S, <span class=\"code-string\">‘#2980b9’<\/span>, linewidth=<span class=\"code-number\">2<\/span>, label=<span class=\"code-string\">‘Susceptibles (S)’<\/span>)\nplt.<span class=\"code-function\">plot<\/span>(t, I, <span class=\"code-string\">‘#c0392b’<\/span>, linewidth=<span class=\"code-number\">2.5<\/span>, label=<span class=\"code-string\">‘Infect\u00e9s (I)’<\/span>)\nplt.<span class=\"code-function\">plot<\/span>(t, R, <span class=\"code-string\">‘#27ae60’<\/span>, linewidth=<span class=\"code-number\">2<\/span>, label=<span class=\"code-string\">‘R\u00e9tablis (R)’<\/span>)\nplt.<span class=\"code-function\">axvline<\/span>(x=pic_jour, color=<span class=\"code-string\">‘#e67e22’<\/span>, linestyle=<span class=\"code-string\">‘–‘<\/span>, label=<span class=\"code-string\">f’Pic (jour {pic_jour:.0f})’<\/span>)\nplt.<span class=\"code-function\">xlabel<\/span>(<span class=\"code-string\">‘Jours’<\/span>)\nplt.<span class=\"code-function\">ylabel<\/span>(<span class=\"code-string\">‘Nombre de personnes’<\/span>)\nplt.<span class=\"code-function\">title<\/span>(<span class=\"code-string\">‘Mod\u00e8le SIR – Simulation Covid, ville b\u00e9ninoise (100 000 hab.)’<\/span>)\nplt.<span class=\"code-function\">legend<\/span>()\nplt.<span class=\"code-function\">grid<\/span>(alpha=<span class=\"code-number\">0.3<\/span>)\nplt.<span class=\"code-function\">show<\/span>()\n <\/div>\n\n <div class=\"encadre\">\n <span class=\"encadre-titre\">\ud83d\udcbb R\u00e9sultats attendus<\/span>\n <p><strong>R\u2080 = 3.0 \u2192 \u00e9pid\u00e9mie croissante \u26a0\ufe0f<\/strong><\/p>\n <p><strong>Pic \u00e9pid\u00e9mique : environ jour 70<\/strong><\/p>\n <p><strong>Maximum d’infect\u00e9s simultan\u00e9s : ~23 000 personnes<\/strong><\/p>\n <p><strong>Total infect\u00e9s en fin d’\u00e9pid\u00e9mie : ~94 000 personnes<\/strong><\/p>\n <p style=\"margin-top:8px; color:#c0392b\">Sans intervention, pr\u00e8s de 94% de la population est touch\u00e9e !<\/p>\n <\/div>\n\n <h2>L’impact des mesures sanitaires : les maths le montrent<\/h2>\n\n <p>\n C’est ici que le mod\u00e8le devient un outil de <strong>d\u00e9cision politique<\/strong>. Les mesures sanitaires (masques, distanciation, confinement) agissent directement sur \u03b2 \u2014 elles r\u00e9duisent le taux de transmission.\n <\/p>\n\n <div class=\"table-container\">\n <table>\n <tr>\n <th>Sc\u00e9nario<\/th>\n <th>\u03b2<\/th>\n <th>R\u2080<\/th>\n <th>Pic (jour)<\/th>\n <th>Max infect\u00e9s<\/th>\n <th>Total touch\u00e9s<\/th>\n <\/tr>\n <tr>\n <td>\u274c Sans mesures<\/td>\n <td>0,30<\/td>\n <td>3,0<\/td>\n <td>Jour 70<\/td>\n <td>~23 000<\/td>\n <td>~94 000 (94%)<\/td>\n <\/tr>\n <tr>\n <td>\u26a0\ufe0f Mesures partielles<\/td>\n <td>0,15<\/td>\n <td>1,5<\/td>\n <td>Jour 120<\/td>\n <td>~8 000<\/td>\n <td>~58 000 (58%)<\/td>\n <\/tr>\n <tr>\n <td>\u2705 Mesures strictes<\/td>\n <td>0,08<\/td>\n <td>0,8<\/td>\n <td>\u2014<\/td>\n <td>~200<\/td>\n <td>~500 (0,5%)<\/td>\n <\/tr>\n <\/table>\n <\/div>\n\n <p>\n La diff\u00e9rence est saisissante. Passer de R\u2080 = 3 \u00e0 R\u2080 < 1 transforme une catastrophe sanitaire en \u00e9pid\u00e9mie ma\u00eetris\u00e9e. C’est ce calcul \u2014 math\u00e9matiquement pr\u00e9cis \u2014 qui a justifi\u00e9 les confinements et les mesures barri\u00e8res en Afrique en 2020.\n <\/p>\n\n <blockquote>\n Les \u00e9pid\u00e9mies ne se combattent pas seulement avec des m\u00e9dicaments. Elles se combattent d’abord avec des \u00e9quations.\n <cite>\u2014 Principe de l’\u00e9pid\u00e9miologie math\u00e9matique<\/cite>\n <\/blockquote>\n\n <h2>Les limites du mod\u00e8le SIR<\/h2>\n\n <div class=\"encadre\">\n <span class=\"encadre-titre\">\u26a0\ufe0f Ce que le mod\u00e8le ne capture pas<\/span>\n <p><strong>1. Les variants :<\/strong> Chaque nouveau variant a son propre \u03b2 et son propre \u03b3 \u2014 le mod\u00e8le doit \u00eatre recalibr\u00e9.<\/p>\n <p><strong>2. L’h\u00e9t\u00e9rog\u00e9n\u00e9it\u00e9 :<\/strong> Le mod\u00e8le suppose que tout le monde se m\u00e9lange uniform\u00e9ment \u2014 pas r\u00e9aliste dans les grandes villes africaines avec des quartiers denses et d’autres dispers\u00e9s.<\/p>\n <p><strong>3. La vaccination :<\/strong> Le mod\u00e8le SIR de base ne prend pas en compte les vaccins \u2014 on utilise alors des versions enrichies (SEIR, SEIRV…).<\/p>\n <p><strong>4. Les comportements humains :<\/strong> Les gens changent de comportement face \u00e0 l’\u00e9pid\u00e9mie \u2014 ce que les \u00e9quations simples ne mod\u00e9lisent pas facilement.<\/p>\n <p><strong>5. Les donn\u00e9es africaines :<\/strong> La sous-d\u00e9claration des cas en Afrique rend le calibrage difficile \u2014 les \u03b2 et \u03b3 r\u00e9els sont durs \u00e0 estimer avec pr\u00e9cision.<\/p>\n <\/div>\n\n \n <div class=\"exercice\">\n <h3>\u270f\ufe0f Exercice corrig\u00e9<\/h3>\n <p>\n Dans un quartier de Porto-Novo de <strong>10 000 habitants<\/strong>, une \u00e9pid\u00e9mie de grippe d\u00e9bute avec <strong>5 personnes infect\u00e9es<\/strong>. On estime \u03b2 = 0,25 et \u03b3 = 0,125.\n <\/p>\n <p><strong>a)<\/strong> Calcule R\u2080. L’\u00e9pid\u00e9mie va-t-elle se propager ?<\/p>\n <p><strong>b)<\/strong> Calcule dI\/dt au jour 0 (avec S\u2080 \u2248 N). L’\u00e9pid\u00e9mie cro\u00eet-elle rapidement ?<\/p>\n <p><strong>c)<\/strong> Pour stopper l’\u00e9pid\u00e9mie, \u00e0 quelle valeur maximale faut-il ramener \u03b2 (en gardant \u03b3 = 0,125) ?<\/p>\n <p><strong>d)<\/strong> Quelle mesure concr\u00e8te permet de r\u00e9duire \u03b2 ?<\/p>\n\n <div class=\"correction\">\n <div class=\"correction-titre\">\u25b8 Correction<\/div>\n <p><strong>a)<\/strong> R\u2080 = \u03b2\/\u03b3 = 0,25\/0,125 = <strong>2,0<\/strong><\/p>\n <p>R\u2080 = 2 > 1 \u2192 <strong>Oui, l’\u00e9pid\u00e9mie va se propager<\/strong>. Chaque malade en contamine 2 en moyenne.<\/p>\n <p><strong>b)<\/strong> dI\/dt = \u03b2\u00d7S\u00d7I\/N \u2212 \u03b3\u00d7I = 0,25\u00d710000\u00d75\/10000 \u2212 0,125\u00d75<\/p>\n <p>= 0,25\u00d75 \u2212 0,125\u00d75 = 1,25 \u2212 0,625 = <strong>+0,625 cas\/jour<\/strong><\/p>\n <p>L’\u00e9pid\u00e9mie cro\u00eet lentement au d\u00e9but \u2014 c’est la phase exponentielle initiale.<\/p>\n <p><strong>c)<\/strong> Pour stopper : R\u2080 < 1 \u2192 \u03b2\/\u03b3 < 1 \u2192 \u03b2 < \u03b3 = 0,125<\/p>\n <p>Il faut ramener \u03b2 en dessous de <strong>0,125<\/strong> (r\u00e9duire la transmission de plus de 50%).<\/p>\n <p><strong>d)<\/strong> Mesures concr\u00e8tes : port du masque, lavage des mains, distanciation physique, a\u00e9ration des espaces \u2014 tout ce qui r\u00e9duit la fr\u00e9quence et l’efficacit\u00e9 des contacts entre personnes. \u2705<\/p>\n <\/div>\n <\/div>\n\n <div class=\"conclusion\">\n <h2>Ce qu’on retient<\/h2>\n <p>\n Le mod\u00e8le SIR nous rappelle une v\u00e9rit\u00e9 profonde : les math\u00e9matiques ne sont pas abstraites. Elles sont <strong>vivantes<\/strong>, ancr\u00e9es dans la r\u00e9alit\u00e9, capables de sauver des vies. Des \u00e9quations \u00e9crites en 1927 dans un laboratoire \u00e9cossais ont guid\u00e9 des d\u00e9cisions sanitaires au B\u00e9nin, au S\u00e9n\u00e9gal, en C\u00f4te d’Ivoire et dans toute l’Afrique en 2020.\n <\/p>\n <p>\n Et si tu regardes les courbes de l’\u00e9pid\u00e9mie Covid en Afrique \u2014 ce pic, cette descente, cette asymptote \u2014 tu vois maintenant ce que peu de gens voient : <strong>des \u00e9quations diff\u00e9rentielles en action<\/strong>.\n <\/p>\n <p>\n Dans le prochain article, nous allons explorer un autre pilier des math\u00e9matiques modernes : <strong>les statistiques et la loi normale<\/strong> \u2014 cette courbe en cloche qui d\u00e9crit les notes des \u00e9l\u00e8ves, la taille des adultes b\u00e9ninois, et presque tout ce qui varie dans la nature.\n <\/p>\n <\/div>\n\n<\/article>\n\n<\/body>\n<\/html>\n","protected":false},"excerpt":{"rendered":"<p>Comment les maths ont pr\u00e9dit le Covid en Afrique \u2014 MathsVivantes Mod\u00e9lisation \u00b7 Article #7 Comment les mathsont pr\u00e9dit le […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_uag_custom_page_level_css":"","site-sidebar-layout":"default","site-content-layout":"","ast-site-content-layout":"default","site-content-style":"default","site-sidebar-style":"default","ast-global-header-display":"","ast-banner-title-visibility":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","ast-disable-related-posts":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","astra-migrate-meta-layouts":"default","ast-page-background-enabled":"default","ast-page-background-meta":{"desktop":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"ast-content-background-meta":{"desktop":{"background-color":"var(--ast-global-color-4)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"var(--ast-global-color-4)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"var(--ast-global-color-4)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"footnotes":""},"categories":[8],"tags":[],"class_list":["post-87","post","type-post","status-publish","format-standard","hentry","category-fondements"],"yoast_head":"\n<title>Covid et le mod\u00e8le SIR - AfrikIntelligent<\/title>\n<meta name=\"description\" content=\"Comment les math\u00e9matiques ont pr\u00e9dit le covid en Afrique . 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}\n .code-keyword { color: #7ec8e3; }\n .code-string { color: #f0c060; }\n .code-number { color: #a8d8a8; }\n .code-function { color: #c8e6c9; }\n\n .exercice {\n background: #051a0d;\n color: var(--creme);\n padding: 36px 40px;\n margin: 3em 0;\n }\n .exercice h3 { color: var(--ocre-clair); font-style: normal; margin-top: 0; margin-bottom: 1em; font-size: 1.1rem; }\n .exercice p { color: rgba(245,240,232,0.82); font-size: 0.97rem; margin-bottom: 0.8em; }\n .exercice .correction { border-top: 1px solid rgba(39,174,96,0.4); margin-top: 1.5em; padding-top: 1.2em; }\n .exercice .correction-titre { font-size: 0.7rem; letter-spacing: 0.18em; text-transform: uppercase; color: var(--ocre-clair); margin-bottom: 0.8em; font-weight: 600; }\n\n blockquote {\n padding: 0 0 0 32px;\n margin: 2em 0;\n position: relative;\n font-style: italic;\n font-size: 1.12rem;\n color: #3a3530;\n }\n blockquote::before {\n content: \"\\201C\";\n position: absolute;\n left: 0; top: -10px;\n font-size: 3.5rem;\n color: var(--vert-clair);\n font-family: 'Playfair Display', serif;\n line-height: 1;\n }\n blockquote cite { display: block; font-size: 0.82rem; font-style: normal; letter-spacing: 0.1em; text-transform: uppercase; color: var(--gris); margin-top: 8px; }\n\n .conclusion { border-top: 2px solid var(--noir); margin-top: 3em; padding-top: 2em; }\n\n @media (max-width: 600px) {\n .hero { padding: 50px 20px 40px; }\n .article-body { padding: 40px 20px 60px; }\n .exercice { padding: 24px 20px; }\n .sir-grid { grid-template-columns: 1fr; }\n .code-block { padding: 16px; font-size: 0.78rem; }\n }\n<\/style>\n<\/head>\n<body>\n\n<div class=\"hero\">\n <div class=\"hero-inner\">\n <div class=\"article-rubrique\">Mod\u00e9lisation \u00b7 Article #7<\/div>\n <h1>Comment les maths<br>ont pr\u00e9dit le Covid<br><em>en Afrique<\/em><\/h1>\n <p class=\"hero-intro\">Mars 2020. Le premier cas de Covid-19 est d\u00e9tect\u00e9 au B\u00e9nin. Quelques jours plus tard, des \u00e9pid\u00e9miologistes africains sortent des chiffres : combien de personnes seront infect\u00e9es, quand le pic arrivera, combien de lits d’h\u00f4pitaux pr\u00e9parer. Ces pr\u00e9dictions venaient des math\u00e9matiques. D’un mod\u00e8le vieux de plus de 100 ans.<\/p>\n <div class=\"meta\">Par Odilon AKOWANOU \u00b7 MathsVivantes \u00b7 Lecture : 12 min<\/div>\n <\/div>\n<\/div>\n\n<article class=\"article-body\">\n\n <p>\n En mars 2020, quand le Covid-19 a commenc\u00e9 \u00e0 se propager en Afrique, les gouvernements avaient un besoin urgent : <strong>pr\u00e9voir<\/strong>. Combien de personnes vont tomber malades ? En combien de temps ? Faut-il confiner ? Fermer les fronti\u00e8res ? Pr\u00e9parer des h\u00f4pitaux de campagne ?\n <\/p>\n\n <p>\n Pour r\u00e9pondre \u00e0 ces questions de vie ou de mort, les \u00e9pid\u00e9miologistes ont utilis\u00e9 un outil math\u00e9matique invent\u00e9 en <strong>1927<\/strong> par deux chercheurs \u00e9cossais \u2014 William Kermack et Anderson McKendrick. Un mod\u00e8le si puissant et si \u00e9l\u00e9gant qu’il est encore aujourd’hui au c\u0153ur de toute mod\u00e9lisation \u00e9pid\u00e9mique : le <strong>mod\u00e8le SIR<\/strong>.\n <\/p>\n\n <h2>L’histoire : une id\u00e9e n\u00e9e d’une \u00e9pid\u00e9mie de peste<\/h2>\n\n <div class=\"encadre-histoire encadre\">\n <span class=\"encadre-titre\">\ud83d\udcdc Naissance du mod\u00e8le SIR \u2014 1927<\/span>\n <p>\n En 1927, l’Inde britannique est ravag\u00e9e par des \u00e9pid\u00e9mies r\u00e9currentes de peste et de chol\u00e9ra. William Kermack, un chimiste, et Anderson McKendrick, un m\u00e9decin militaire, cherchent \u00e0 comprendre <em>pourquoi<\/em> certaines \u00e9pid\u00e9mies s’\u00e9teignent d’elles-m\u00eames avant d’infecter toute la population.\n <\/p>\n <p>\n Leur id\u00e9e r\u00e9volutionnaire : mod\u00e9liser une population comme un syst\u00e8me de <strong>flux entre trois compartiments<\/strong>. Les individus passent d’un groupe \u00e0 l’autre selon des taux math\u00e9matiques pr\u00e9cis. Avec ce mod\u00e8le, ils parviennent \u00e0 reproduire fid\u00e8lement les courbes \u00e9pid\u00e9miques observ\u00e9es \u2014 et \u00e0 pr\u00e9dire leur \u00e9volution.\n <\/p>\n <p>\n Presque un si\u00e8cle plus tard, ce m\u00eame mod\u00e8le \u2014 adapt\u00e9 et enrichi \u2014 a guid\u00e9 les d\u00e9cisions sanitaires de 54 pays africains face au Covid-19. Les math\u00e9matiques de 1927 ont sauv\u00e9 des vies en 2020.\n <\/p>\n <\/div>\n\n <h2>Le mod\u00e8le SIR : trois compartiments, une population<\/h2>\n\n <p>\n L’id\u00e9e centrale est simple : \u00e0 chaque instant, chaque individu d’une population appartient \u00e0 l’un de ces trois groupes :\n <\/p>\n\n <div class=\"sir-grid\">\n <div class=\"sir-bloc sir-S\">\n <span class=\"sir-lettre\">S<\/span>\n <div class=\"sir-nom\">Susceptibles<\/div>\n <div class=\"sir-desc\">Personnes saines pouvant \u00eatre infect\u00e9es<\/div>\n <\/div>\n <div class=\"sir-bloc sir-I\">\n <span class=\"sir-lettre\">I<\/span>\n <div class=\"sir-nom\">Infect\u00e9s<\/div>\n <div class=\"sir-desc\">Personnes malades et contagieuses<\/div>\n <\/div>\n <div class=\"sir-bloc sir-R\">\n <span class=\"sir-lettre\">R<\/span>\n <div class=\"sir-nom\">R\u00e9tablis<\/div>\n <div class=\"sir-desc\">Gu\u00e9ris ou immunis\u00e9s (ou d\u00e9c\u00e9d\u00e9s)<\/div>\n <\/div>\n <\/div>\n\n <p>\n La population totale N reste constante : <strong>S + I + R = N<\/strong>. Les individus ne font que circuler entre ces trois groupes \u2014 jamais dans l’autre sens.\n <\/p>\n\n <div class=\"schema-container\">\n <div class=\"schema-titre\">\ud83d\udd04 Flux entre les compartiments du mod\u00e8le SIR<\/div>\n <svg width=\"100%\" viewBox=\"0 0 520 140\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\">\n <rect width=\"520\" height=\"140\" fill=\"white\"\/>\n\n \n <rect x=\"30\" y=\"40\" width=\"110\" height=\"60\" rx=\"4\" fill=\"#2980b9\"\/>\n <text x=\"85\" y=\"68\" font-size=\"22\" fill=\"white\" text-anchor=\"middle\" font-family=\"Playfair Display, serif\" font-weight=\"700\">S<\/text>\n <text x=\"85\" y=\"88\" font-size=\"10\" fill=\"rgba(255,255,255,0.85)\" text-anchor=\"middle\">Susceptibles<\/text>\n\n \n <line x1=\"140\" y1=\"70\" x2=\"195\" y2=\"70\" stroke=\"#c0392b\" stroke-width=\"2.5\" marker-end=\"url(#arrowRed)\"\/>\n <defs>\n <marker id=\"arrowRed\" markerWidth=\"8\" markerHeight=\"8\" refX=\"6\" refY=\"3\" orient=\"auto\">\n <path d=\"M0,0 L0,6 L8,3 z\" fill=\"#c0392b\"\/>\n <\/marker>\n <marker id=\"arrowGreen\" markerWidth=\"8\" markerHeight=\"8\" refX=\"6\" refY=\"3\" orient=\"auto\">\n <path d=\"M0,0 L0,6 L8,3 z\" fill=\"#27ae60\"\/>\n <\/marker>\n <\/defs>\n <text x=\"167\" y=\"60\" font-size=\"11\" fill=\"#c0392b\" text-anchor=\"middle\" font-style=\"italic\">\u03b2\u00b7S\u00b7I\/N<\/text>\n\n \n <rect x=\"195\" y=\"40\" width=\"110\" height=\"60\" rx=\"4\" fill=\"#c0392b\"\/>\n <text x=\"250\" y=\"68\" font-size=\"22\" fill=\"white\" text-anchor=\"middle\" font-family=\"Playfair Display, serif\" font-weight=\"700\">I<\/text>\n <text x=\"250\" y=\"88\" font-size=\"10\" fill=\"rgba(255,255,255,0.85)\" text-anchor=\"middle\">Infect\u00e9s<\/text>\n\n \n <line x1=\"305\" y1=\"70\" x2=\"360\" y2=\"70\" stroke=\"#27ae60\" stroke-width=\"2.5\" marker-end=\"url(#arrowGreen)\"\/>\n <text x=\"332\" y=\"60\" font-size=\"11\" fill=\"#27ae60\" text-anchor=\"middle\" font-style=\"italic\">\u03b3\u00b7I<\/text>\n\n \n <rect x=\"360\" y=\"40\" width=\"110\" height=\"60\" rx=\"4\" fill=\"#27ae60\"\/>\n <text x=\"415\" y=\"68\" font-size=\"22\" fill=\"white\" text-anchor=\"middle\" font-family=\"Playfair Display, serif\" font-weight=\"700\">R<\/text>\n <text x=\"415\" y=\"88\" font-size=\"10\" fill=\"rgba(255,255,255,0.85)\" text-anchor=\"middle\">R\u00e9tablis<\/text>\n\n \n <text x=\"85\" y=\"120\" font-size=\"10\" fill=\"#6b6560\" text-anchor=\"middle\">Population totale N = S + I + R<\/text>\n <\/svg>\n <\/div>\n\n <h2>Les \u00e9quations : le langage math\u00e9matique de l’\u00e9pid\u00e9mie<\/h2>\n\n <p>\n Le mod\u00e8le SIR est d\u00e9crit par un syst\u00e8me de trois <strong>\u00e9quations diff\u00e9rentielles<\/strong> \u2014 des \u00e9quations qui d\u00e9crivent comment chaque compartiment varie au fil du temps :\n <\/p>\n\n <div class=\"encadre\">\n <span class=\"encadre-titre\">\ud83d\udcd0 Les \u00e9quations du mod\u00e8le SIR<\/span>\n <p style=\"font-family:'Playfair Display',serif; font-size:1.1rem; text-align:center; margin: 8px 0;\">dS\/dt = \u2212\u03b2 \u00d7 S \u00d7 I \/ N<\/p>\n <p style=\"font-family:'Playfair Display',serif; font-size:1.1rem; text-align:center; margin: 8px 0;\">dI\/dt = \u03b2 \u00d7 S \u00d7 I \/ N \u2212 \u03b3 \u00d7 I<\/p>\n <p style=\"font-family:'Playfair Display',serif; font-size:1.1rem; text-align:center; margin: 8px 0;\">dR\/dt = \u03b3 \u00d7 I<\/p>\n <p style=\"margin-top:12px\"><strong>\u03b2 (b\u00eata)<\/strong> = taux de transmission (vitesse \u00e0 laquelle S devient I)<\/p>\n <p><strong>\u03b3 (gamma)<\/strong> = taux de gu\u00e9rison (vitesse \u00e0 laquelle I devient R)<\/p>\n <p><strong>dX\/dt<\/strong> = variation de X par unit\u00e9 de temps (ici, par jour)<\/p>\n <\/div>\n\n <p>\n Note que dS\/dt + dI\/dt + dR\/dt = 0 \u2014 ce qui confirme que la population totale est constante. Ce qui sort de S entre dans I, ce qui sort de I entre dans R.\n <\/p>\n\n <h2>Le nombre R\u2080 : le param\u00e8tre qui change tout<\/h2>\n\n <p>\n Du mod\u00e8le SIR \u00e9merge un nombre fondamental qui r\u00e9sume la dangerosit\u00e9 d’une \u00e9pid\u00e9mie : le <strong>nombre de reproduction de base R\u2080<\/strong> (prononc\u00e9 \u00ab\u00a0R z\u00e9ro\u00a0\u00bb).\n <\/p>\n\n <div class=\"formule\">R\u2080 = \u03b2 \/ \u03b3<\/div>\n\n <div class=\"definition\">\n <strong>R\u2080<\/strong> repr\u00e9sente le nombre moyen de personnes qu’un individu infect\u00e9 contaminera dans une population enti\u00e8rement susceptible. C’est <em>le<\/em> param\u00e8tre cl\u00e9 de toute \u00e9pid\u00e9mie.\n <\/div>\n\n <div class=\"encadre\">\n <span class=\"encadre-titre\">\ud83d\udd11 Interpr\u00e9tation de R\u2080<\/span>\n <p><strong>R\u2080 < 1<\/strong> \u2192 Chaque infect\u00e9 contamine moins d’une personne \u2192 l’\u00e9pid\u00e9mie s’\u00e9teint naturellement \u2705<\/p>\n <p><strong>R\u2080 = 1<\/strong> \u2192 L’\u00e9pid\u00e9mie est stable \u2014 ni croissance ni d\u00e9clin \u26a0\ufe0f<\/p>\n <p><strong>R\u2080 > 1<\/strong> \u2192 Chaque infect\u00e9 contamine plus d’une personne \u2192 l’\u00e9pid\u00e9mie se propage \u274c<\/p>\n <p style=\"margin-top:8px\"><strong>Pour le Covid-19 en Afrique :<\/strong> R\u2080 estim\u00e9 entre 2 et 3 selon les pays et les variants.<\/p>\n <\/div>\n\n <h2>Application au B\u00e9nin : simulons une \u00e9pid\u00e9mie<\/h2>\n\n <p>\n Appliquons le mod\u00e8le SIR \u00e0 une ville fictive de 100 000 habitants au B\u00e9nin, avec des param\u00e8tres inspir\u00e9s des donn\u00e9es Covid-19 africaines :\n <\/p>\n\n <div class=\"table-container\">\n <table>\n <tr>\n <th>Param\u00e8tre<\/th>\n <th>Valeur<\/th>\n <th>Signification<\/th>\n <\/tr>\n <tr><td>N (population)<\/td><td>100 000<\/td><td>Habitants de la ville<\/td><\/tr>\n <tr><td>I\u2080 (infect\u00e9s initiaux)<\/td><td>10<\/td><td>10 cas au jour 0<\/td><\/tr>\n <tr><td>S\u2080 (susceptibles initiaux)<\/td><td>99 990<\/td><td>Tout le monde sauf les 10<\/td><\/tr>\n <tr><td>R\u2080 (r\u00e9tablis initiaux)<\/td><td>0<\/td><td>Personne n’est encore immunis\u00e9<\/td><\/tr>\n <tr><td>\u03b2 (taux transmission)<\/td><td>0,3<\/td><td>30% de chances de transmettre par contact<\/td><\/tr>\n <tr><td>\u03b3 (taux gu\u00e9rison)<\/td><td>0,1<\/td><td>Gu\u00e9rison en ~10 jours en moyenne<\/td><\/tr>\n <tr><td><strong>R\u2080 = \u03b2\/\u03b3<\/strong><\/td><td><strong>3,0<\/strong><\/td><td>Chaque infect\u00e9 en contamine 3<\/td><\/tr>\n <\/table>\n <\/div>\n\n <h2>Le code Python : simulons l’\u00e9pid\u00e9mie<\/h2>\n\n <div class=\"code-block\">\n <span class=\"code-label\">Python<\/span>\n<span class=\"code-comment\"># Mod\u00e8le SIR – Simulation \u00e9pid\u00e9mie Covid au B\u00e9nin<\/span>\n<span class=\"code-keyword\">import<\/span> numpy <span class=\"code-keyword\">as<\/span> np\n<span class=\"code-keyword\">from<\/span> scipy.integrate <span class=\"code-keyword\">import<\/span> odeint\n<span class=\"code-keyword\">import<\/span> matplotlib.pyplot <span class=\"code-keyword\">as<\/span> plt\n\n<span class=\"code-comment\"># — Param\u00e8tres —<\/span>\nN = <span class=\"code-number\">100000<\/span> <span class=\"code-comment\"># Population totale<\/span>\nI0 = <span class=\"code-number\">10<\/span> <span class=\"code-comment\"># Infect\u00e9s au jour 0<\/span>\nR0_init = <span class=\"code-number\">0<\/span> <span class=\"code-comment\"># R\u00e9tablis au jour 0<\/span>\nS0 = N – I0 – R0_init <span class=\"code-comment\"># Susceptibles au jour 0<\/span>\n\nbeta = <span class=\"code-number\">0.3<\/span> <span class=\"code-comment\"># Taux de transmission<\/span>\ngamma = <span class=\"code-number\">0.1<\/span> <span class=\"code-comment\"># Taux de gu\u00e9rison<\/span>\nR0 = beta \/ gamma\n<span class=\"code-function\">print<\/span>(<span class=\"code-string\">f\u00a0\u00bbR\u2080 = {R0} \u2192 \u00e9pid\u00e9mie {‘croissante \u26a0\ufe0f’ if R0 > 1 else ‘d\u00e9croissante \u2705’}\u00a0\u00bb<\/span>)\n\n<span class=\"code-comment\"># — \u00c9quations diff\u00e9rentielles SIR —<\/span>\n<span class=\"code-keyword\">def<\/span> <span class=\"code-function\">modele_SIR<\/span>(y, t, N, beta, gamma):\n S, I, R = y\n dS = -beta * S * I \/ N\n dI = beta * S * I \/ N – gamma * I\n dR = gamma * I\n <span class=\"code-keyword\">return<\/span> [dS, dI, dR]\n\n<span class=\"code-comment\"># — R\u00e9solution —<\/span>\nt = np.<span class=\"code-function\">linspace<\/span>(<span class=\"code-number\">0<\/span>, <span class=\"code-number\">200<\/span>, <span class=\"code-number\">200<\/span>) <span class=\"code-comment\"># 200 jours<\/span>\ny0 = [S0, I0, R0_init]\nsolution = <span class=\"code-function\">odeint<\/span>(modele_SIR, y0, t, args=(N, beta, gamma))\nS, I, R = solution.T\n\n<span class=\"code-comment\"># — R\u00e9sultats cl\u00e9s —<\/span>\npic_jour = t[np.<span class=\"code-function\">argmax<\/span>(I)]\npic_infectes = <span class=\"code-function\">int<\/span>(<span class=\"code-function\">max<\/span>(I))\n<span class=\"code-function\">print<\/span>(<span class=\"code-string\">f\u00a0\u00bbPic \u00e9pid\u00e9mique : jour {pic_jour:.0f}\u00a0\u00bb<\/span>)\n<span class=\"code-function\">print<\/span>(<span class=\"code-string\">f\u00a0\u00bbMaximum d’infect\u00e9s simultan\u00e9s : {pic_infectes:,}\u00a0\u00bb<\/span>)\n<span class=\"code-function\">print<\/span>(<span class=\"code-string\">f\u00a0\u00bbTotal infect\u00e9s en fin d’\u00e9pid\u00e9mie : {int(R[-1]):,}\u00a0\u00bb<\/span>)\n\n<span class=\"code-comment\"># — Graphique —<\/span>\nplt.<span class=\"code-function\">figure<\/span>(figsize=(<span class=\"code-number\">10<\/span>, <span class=\"code-number\">6<\/span>))\nplt.<span class=\"code-function\">plot<\/span>(t, S, <span class=\"code-string\">‘#2980b9’<\/span>, linewidth=<span class=\"code-number\">2<\/span>, label=<span class=\"code-string\">‘Susceptibles (S)’<\/span>)\nplt.<span class=\"code-function\">plot<\/span>(t, I, <span class=\"code-string\">‘#c0392b’<\/span>, linewidth=<span class=\"code-number\">2.5<\/span>, label=<span class=\"code-string\">‘Infect\u00e9s (I)’<\/span>)\nplt.<span class=\"code-function\">plot<\/span>(t, R, <span class=\"code-string\">‘#27ae60’<\/span>, linewidth=<span class=\"code-number\">2<\/span>, label=<span class=\"code-string\">‘R\u00e9tablis (R)’<\/span>)\nplt.<span class=\"code-function\">axvline<\/span>(x=pic_jour, color=<span class=\"code-string\">‘#e67e22’<\/span>, linestyle=<span class=\"code-string\">‘–‘<\/span>, label=<span class=\"code-string\">f’Pic (jour {pic_jour:.0f})’<\/span>)\nplt.<span class=\"code-function\">xlabel<\/span>(<span class=\"code-string\">‘Jours’<\/span>)\nplt.<span class=\"code-function\">ylabel<\/span>(<span class=\"code-string\">‘Nombre de personnes’<\/span>)\nplt.<span class=\"code-function\">title<\/span>(<span class=\"code-string\">‘Mod\u00e8le SIR – Simulation Covid, ville b\u00e9ninoise (100 000 hab.)’<\/span>)\nplt.<span class=\"code-function\">legend<\/span>()\nplt.<span class=\"code-function\">grid<\/span>(alpha=<span class=\"code-number\">0.3<\/span>)\nplt.<span class=\"code-function\">show<\/span>()\n <\/div>\n\n <div class=\"encadre\">\n <span class=\"encadre-titre\">\ud83d\udcbb R\u00e9sultats attendus<\/span>\n <p><strong>R\u2080 = 3.0 \u2192 \u00e9pid\u00e9mie croissante \u26a0\ufe0f<\/strong><\/p>\n <p><strong>Pic \u00e9pid\u00e9mique : environ jour 70<\/strong><\/p>\n <p><strong>Maximum d’infect\u00e9s simultan\u00e9s : ~23 000 personnes<\/strong><\/p>\n <p><strong>Total infect\u00e9s en fin d’\u00e9pid\u00e9mie : ~94 000 personnes<\/strong><\/p>\n <p style=\"margin-top:8px; color:#c0392b\">Sans intervention, pr\u00e8s de 94% de la population est touch\u00e9e !<\/p>\n <\/div>\n\n <h2>L’impact des mesures sanitaires : les maths le montrent<\/h2>\n\n <p>\n C’est ici que le mod\u00e8le devient un outil de <strong>d\u00e9cision politique<\/strong>. Les mesures sanitaires (masques, distanciation, confinement) agissent directement sur \u03b2 \u2014 elles r\u00e9duisent le taux de transmission.\n <\/p>\n\n <div class=\"table-container\">\n <table>\n <tr>\n <th>Sc\u00e9nario<\/th>\n <th>\u03b2<\/th>\n <th>R\u2080<\/th>\n <th>Pic (jour)<\/th>\n <th>Max infect\u00e9s<\/th>\n <th>Total touch\u00e9s<\/th>\n <\/tr>\n <tr>\n <td>\u274c Sans mesures<\/td>\n <td>0,30<\/td>\n <td>3,0<\/td>\n <td>Jour 70<\/td>\n <td>~23 000<\/td>\n <td>~94 000 (94%)<\/td>\n <\/tr>\n <tr>\n <td>\u26a0\ufe0f Mesures partielles<\/td>\n <td>0,15<\/td>\n <td>1,5<\/td>\n <td>Jour 120<\/td>\n <td>~8 000<\/td>\n <td>~58 000 (58%)<\/td>\n <\/tr>\n <tr>\n <td>\u2705 Mesures strictes<\/td>\n <td>0,08<\/td>\n <td>0,8<\/td>\n <td>\u2014<\/td>\n <td>~200<\/td>\n <td>~500 (0,5%)<\/td>\n <\/tr>\n <\/table>\n <\/div>\n\n <p>\n La diff\u00e9rence est saisissante. Passer de R\u2080 = 3 \u00e0 R\u2080 < 1 transforme une catastrophe sanitaire en \u00e9pid\u00e9mie ma\u00eetris\u00e9e. C’est ce calcul \u2014 math\u00e9matiquement pr\u00e9cis \u2014 qui a justifi\u00e9 les confinements et les mesures barri\u00e8res en Afrique en 2020.\n <\/p>\n\n <blockquote>\n Les \u00e9pid\u00e9mies ne se combattent pas seulement avec des m\u00e9dicaments. Elles se combattent d’abord avec des \u00e9quations.\n <cite>\u2014 Principe de l’\u00e9pid\u00e9miologie math\u00e9matique<\/cite>\n <\/blockquote>\n\n <h2>Les limites du mod\u00e8le SIR<\/h2>\n\n <div class=\"encadre\">\n <span class=\"encadre-titre\">\u26a0\ufe0f Ce que le mod\u00e8le ne capture pas<\/span>\n <p><strong>1. Les variants :<\/strong> Chaque nouveau variant a son propre \u03b2 et son propre \u03b3 \u2014 le mod\u00e8le doit \u00eatre recalibr\u00e9.<\/p>\n <p><strong>2. L’h\u00e9t\u00e9rog\u00e9n\u00e9it\u00e9 :<\/strong> Le mod\u00e8le suppose que tout le monde se m\u00e9lange uniform\u00e9ment \u2014 pas r\u00e9aliste dans les grandes villes africaines avec des quartiers denses et d’autres dispers\u00e9s.<\/p>\n <p><strong>3. La vaccination :<\/strong> Le mod\u00e8le SIR de base ne prend pas en compte les vaccins \u2014 on utilise alors des versions enrichies (SEIR, SEIRV…).<\/p>\n <p><strong>4. Les comportements humains :<\/strong> Les gens changent de comportement face \u00e0 l’\u00e9pid\u00e9mie \u2014 ce que les \u00e9quations simples ne mod\u00e9lisent pas facilement.<\/p>\n <p><strong>5. Les donn\u00e9es africaines :<\/strong> La sous-d\u00e9claration des cas en Afrique rend le calibrage difficile \u2014 les \u03b2 et \u03b3 r\u00e9els sont durs \u00e0 estimer avec pr\u00e9cision.<\/p>\n <\/div>\n\n \n <div class=\"exercice\">\n <h3>\u270f\ufe0f Exercice corrig\u00e9<\/h3>\n <p>\n Dans un quartier de Porto-Novo de <strong>10 000 habitants<\/strong>, une \u00e9pid\u00e9mie de grippe d\u00e9bute avec <strong>5 personnes infect\u00e9es<\/strong>. On estime \u03b2 = 0,25 et \u03b3 = 0,125.\n <\/p>\n <p><strong>a)<\/strong> Calcule R\u2080. L’\u00e9pid\u00e9mie va-t-elle se propager ?<\/p>\n <p><strong>b)<\/strong> Calcule dI\/dt au jour 0 (avec S\u2080 \u2248 N). L’\u00e9pid\u00e9mie cro\u00eet-elle rapidement ?<\/p>\n <p><strong>c)<\/strong> Pour stopper l’\u00e9pid\u00e9mie, \u00e0 quelle valeur maximale faut-il ramener \u03b2 (en gardant \u03b3 = 0,125) ?<\/p>\n <p><strong>d)<\/strong> Quelle mesure concr\u00e8te permet de r\u00e9duire \u03b2 ?<\/p>\n\n <div class=\"correction\">\n <div class=\"correction-titre\">\u25b8 Correction<\/div>\n <p><strong>a)<\/strong> R\u2080 = \u03b2\/\u03b3 = 0,25\/0,125 = <strong>2,0<\/strong><\/p>\n <p>R\u2080 = 2 > 1 \u2192 <strong>Oui, l’\u00e9pid\u00e9mie va se propager<\/strong>. Chaque malade en contamine 2 en moyenne.<\/p>\n <p><strong>b)<\/strong> dI\/dt = \u03b2\u00d7S\u00d7I\/N \u2212 \u03b3\u00d7I = 0,25\u00d710000\u00d75\/10000 \u2212 0,125\u00d75<\/p>\n <p>= 0,25\u00d75 \u2212 0,125\u00d75 = 1,25 \u2212 0,625 = <strong>+0,625 cas\/jour<\/strong><\/p>\n <p>L’\u00e9pid\u00e9mie cro\u00eet lentement au d\u00e9but \u2014 c’est la phase exponentielle initiale.<\/p>\n <p><strong>c)<\/strong> Pour stopper : R\u2080 < 1 \u2192 \u03b2\/\u03b3 < 1 \u2192 \u03b2 < \u03b3 = 0,125<\/p>\n <p>Il faut ramener \u03b2 en dessous de <strong>0,125<\/strong> (r\u00e9duire la transmission de plus de 50%).<\/p>\n <p><strong>d)<\/strong> Mesures concr\u00e8tes : port du masque, lavage des mains, distanciation physique, a\u00e9ration des espaces \u2014 tout ce qui r\u00e9duit la fr\u00e9quence et l’efficacit\u00e9 des contacts entre personnes. \u2705<\/p>\n <\/div>\n <\/div>\n\n <div class=\"conclusion\">\n <h2>Ce qu’on retient<\/h2>\n <p>\n Le mod\u00e8le SIR nous rappelle une v\u00e9rit\u00e9 profonde : les math\u00e9matiques ne sont pas abstraites. Elles sont <strong>vivantes<\/strong>, ancr\u00e9es dans la r\u00e9alit\u00e9, capables de sauver des vies. Des \u00e9quations \u00e9crites en 1927 dans un laboratoire \u00e9cossais ont guid\u00e9 des d\u00e9cisions sanitaires au B\u00e9nin, au S\u00e9n\u00e9gal, en C\u00f4te d’Ivoire et dans toute l’Afrique en 2020.\n <\/p>\n <p>\n Et si tu regardes les courbes de l’\u00e9pid\u00e9mie Covid en Afrique \u2014 ce pic, cette descente, cette asymptote \u2014 tu vois maintenant ce que peu de gens voient : <strong>des \u00e9quations diff\u00e9rentielles en action<\/strong>.\n <\/p>\n <p>\n Dans le prochain article, nous allons explorer un autre pilier des math\u00e9matiques modernes : <strong>les statistiques et la loi normale<\/strong> \u2014 cette courbe en cloche qui d\u00e9crit les notes des \u00e9l\u00e8ves, la taille des adultes b\u00e9ninois, et presque tout ce qui varie dans la nature.\n <\/p>\n <\/div>\n\n<\/article>\n\n<\/body>\n<\/html>\n","protected":false},"excerpt":{"rendered":"<p>Comment les maths ont pr\u00e9dit le Covid en Afrique \u2014 MathsVivantes Mod\u00e9lisation \u00b7 Article #7 Comment les mathsont pr\u00e9dit le […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_uag_custom_page_level_css":"","site-sidebar-layout":"default","site-content-layout":"","ast-site-content-layout":"default","site-content-style":"default","site-sidebar-style":"default","ast-global-header-display":"","ast-banner-title-visibility":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","ast-disable-related-posts":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","astra-migrate-meta-layouts":"default","ast-page-background-enabled":"default","ast-page-background-meta":{"desktop":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"ast-content-background-meta":{"desktop":{"background-color":"var(--ast-global-color-4)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"var(--ast-global-color-4)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"var(--ast-global-color-4)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"footnotes":""},"categories":[8],"tags":[],"class_list":["post-87","post","type-post","status-publish","format-standard","hentry","category-fondements"],"yoast_head":"\n<title>Covid et le mod\u00e8le SIR - AfrikIntelligent<\/title>\n<meta name=\"description\" content=\"Comment les math\u00e9matiques ont pr\u00e9dit le covid en Afrique . 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