Dans les articles précèdents nous avons vu qu’un modèle mathématique de Gompertz pouvait simuler la pandémie de Covid19.
La courbe de Gompertz permet d’obtenir le nombre de décès du au coronavirus.
G(t) = N exp(− exp(− (t − T)/ U))
N est le maximum de décès de la courbe
T est je jour du point d’inflexion où la dérivée seconde de G(t) s’annule et la dérivée première est maximun
U donne la courbure de la courbe valeur en jours
Exemple sur le nombre de décès en France. Le point départ dit de la « seconde vague » est pris au 8 Juillet 2020 on a alors 29940 décès. (x=-50, y =29940)
Au 16 Octobre 2020 33303 décès.
Dans 60 jours nous devrions atteindre un plateau, soit à la mi-décembre.
Plateau 35920 décès au 16 Octobre 33303 décès.
Basic properties of the Gompertz functions and its logarithms. The Gompertz function is an exponential of an exponential written as G(t) = Ne−e−(t−T)/U or G(t) = N exp(− exp(− (t − T)/ U)), and defined by three parameters N, T & U, each with clear physical meaning. Parameter N is the asymptotic number, the maximum plateau value that G(t) reaches after a long time, t. Parameter T, is the point of inflection, which is the time in days at which the second-derivative of G(t) is zero and its first derivative is a maximum. It is a natural mid-point of the function where the value of G(T)=N/e=0.37N. The Parameter U, is the most important as it changes the shape of the curve; it is a time-constant measured in days.
Given the double exponential nature of G(t), one might expect to use a double logarithm to simplify it. The function G(t) itself has the expected S-shape of saturating growth function. Taking the logarithm once gives ln(G(t))=ln(N)–exp(−(t−T)/U), where ln is the natural logarithm or loge; it is shown in dashed line increasing very rapidly at first but curving steadily to become horizontal at saturation. Rearranging as ln(N)–in(G(t))=exp(−(t−T)/U)and taking the logarithm a second time gives Y(t) = −ln[ln(N)−in(G(t))]=−ln[ln(N/G(t))] = (t–T)/U. This function is shown in the dotted line to be a simple straight line. This is hugely significant as extrapolation of a straight-line is trivial: just keep going straight. As we show in the text, the function Y(t) is always a straight line for the Gompertz function. More generally, Y(t), tends to a straight line for a very general class of saturating functions
(b) Illustrating how the linearity of the Y(t)=−ln(ln(N/G(t)) depends on the value of N. The linearity shown in (a) has an apparent weakness, namely the line is only straight when the value of N is the correct saturation value and this value will be unknown until the epidemic is over. This weakness is in fact a strength. One can try different values of N and find the one that gives a straight line. In fact, “straighten the line” is much more relevant than the saying “flatten the curve” popularly applied to COVID19.
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