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Covid19 Pas d’exponentielle pas de vagues

Jordi -

On entend souvent les termes d’exponentielle et de vagues pour la pandémie Covid19. Ces termes sont mal appropriés, mais pourquoi?

Benjamin Gompertz (5 mars 1779 – 14 juillet 1865, Londres, Angleterre) était un mathématicien britannique de formation autodidacte. On lui avait refusé l’entrée à l’université à cause de ses origines juives.

On lui doit le modèle de Gompertz (1825).

Le modèle de propagation d’épidémie (dit modèle SIR), (Kermack et McKendrick 1927) S = Sains, I = Infectés, R = Recovered (immunisés). Il peut s’agir d’un population sujette à une infection : S˙ = −βIS + γR (1) ˙I = βIS − νI (2) R˙ = νI − γR (3)

reprend ses travaux et sa fameuse courbe en forme de «S» ou sigmoïde.

La fonction de Gompertz décrit bien certains types de croissance biologique, y compris les épidémies. L’équation elle-même est la solution à une équation différentielle spécifique qui a une certaine plausibilité en ce qui concerne la croissance des cas dans une épidémie. La deuxième justification plus empirique est que les auteurs chinois ont constaté que l’équation de Gompertz semblait bien décrire la croissance des cas de COVID-19 dans certaines parties de la Chine (Jia et al. ArXiv: 2003.05447v2 [q-bio.PE]).

Une épidémie de type Sars Cov2 suit donc une courbe de type Gompertz, ce n’est pas une exponentielle. Au début, on constate un développement rapide d’allure exponentielle, qui s’atténue, passe par un point d’inflexion, puis s’atténue encore jusqu’à la disparition du phénomène, phase ultime qui peut durer assez longtemps. On a une courbe dite « en cloche » si l’on trace le nombre d’infectés ou de morts par jour. On dit que c’est une épidémie de type « saisonnier ». Ce terme « saisonnier » porte à confusion, ce n’est pas seulement l’été et la chaleur qui font disparaitre l’épidémie. Il faut comprendre le sens de « saison » comme une période épidémique.

La modélisation des maladies transmissibles est un outil qui permet :
 d’expliquer la dynamique d’une épidémie,
 d’estimer son ampleur,
 d’anticiper l’efficacité des interventions sanitaires.

Il existe deux grands types de modèles mathématiques pour l’étude de la dynamique épidémique d’une maladie t r a n s m i s s i b l e: les modèles déterministes et les modèles stochastiques. Les modèles déterministes, que l’on vient de décrire a propos du modèle SEIR, reposent sur des systèmes d’équations différentielles non linéaires dont la programmation et la résolution, devenues aisées grâce a l’informatique, permettent de disposer très rapidement de résultats. Ils présentent cependant le grand inconvénient de ne pas prendre en compte des événements aleatoires inéluctables lors des démarrages épidémiques.
Nous ne détaillerons pas ici les nombreuses techniques de modélisation stochastique, mais elles reposent toutes sur des simulations effectuées sur de puissants calculateurs où l’on peut programmer les contacts entre chaque individu d’une population avec un maximum de précision sur les probabilités de contacts inter-individuels. Ces probabilités ainsi que d’autres paramètres du modèle (durée d’incubation, durée des symptômes, délais de prise en charge par le système de soin) sont souvent variables et les programmes permettent pour chaque simulation de tenir compte de cette variabilité en la modélisant.

Dans l’article précédent de ce Blog, j’avais montré qu’un simple programme permettait de retrouver l’équation de Gompertz, en modélisant le phénomène de contagion. Mais cette modélisation et sa projection se heurte aux actions prises, confinement, masques, lavage des mains…

Contrôle d’une épidémie
 Réduction de la proportion de susceptibles : S(t)
 Immunité collective (« herd immunity ») ou immunité grégaire.
 Si une vaccination existe :
 proportion d’individus à vacciner pour contrôler l’épidémie :
 P > (1 – 1 / R0)
• pour la rougeole (R0 = 15 – 20) : p = 93-95 %
• pour la grippe (R0 = 1,5 – 2,5) : p = 33-60 %

Tant que le pourcentage d’infectés n’est pas atteint le virus va circuler. Quel est le pourcentage à atteindre? On dit souvent 60% car c’est le taux de vaccination pour la grippe, mais il semble que pour la Covid19 il faudrait atteindre de 20 à 30% de la population. Un exemple intéressant à suivre est celui de la Suède qui n’a pas fait de confinement et n’a pas préconisé le port du masque; en Suède le gouvernement a déclaré que l’épidémie était finie.

Nombre de décès COVID19 en Suède

On retrouve la fameuse équation de Gompertz!

En France on a préféré le confinement pour éviter la saturation des hôpitaux et « aplanir la courbe », mais on ne peut pas tout avoir et donc la propagation du virus est repartie.

Pourquoi parle t-on de « deuxième vague »?

Pour la grippe espagnole, une première vague eu lieu en 1918.
La première vague de la pandémie qui s’achève en juillet 1918 touche inégalement différents pays.

La deuxième vague commence elle aussi aux Etats-Unis, dans un campement militaire en banlieue de Boston, en septembre 1918.

C’est aussi la vague la plus létale de la pandémie : elle « apparaît en même temps en Europe, aux Amériques et en Asie, et constitue la réelle pandémie de grippe de 1918″
Il est probable que l’une des sources du virus responsable de la grippe espagnole ait muté et ait adopté une forme plus virulente, plus sélective. Ce qui a provoqué l’hécatombe à travers tout l’hémisphère nord entre septembre et novembre 1918 .

Mais le virus de la Covid19 n’a rien à voir avec celui de la grippe. Si il ne mute pas il n’y aura pas de deuxième vague. Pourquoi un rebond?

On devrait dire plutôt réplique comme dans les séismes. Le virus circulant, on repart sur une courbe de type Gompertz. Sera t’il aussi fort qu’en Mars, non car c’est la même vague, le point maximum sera moindre qu’en Mars.

Pour faire une projection, on peut reprendre une courbe de Gompertz et regarder le nombre de personnes hospitalisées et de décès. Le nombre d’infectés est difficile à interpréter car le nombre de tests PCR n’a plus rien à voir avec le mois de mars.

la bonne nouvelle pour la France c’est que le point d’inflexion vient d’être passé, nous aurons donc pendant quelques jours une augmentation puis une descente lente et ceci quelque soit les mesures prises.

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