From the FPM model in viral spreading to predictions of CO2 concentration in the atmosphere

Through a sufficiently developed introduction and whose cohesion has been an aim, this article begins by recalling and specifying the FPM (Fractional Power Model), or model in tm, namely A+Btm with positive t and m: thus, the model essence and its specificities both in terms of representativity and predictivity are the subject of a synthetic presentation. Concerning the model representativity, its capacity to take into account an indefinite number of internal dynamics, of different rapidity, such that those of complex phenomena, is proven by the decomposition of tm into its internal dynamics, and this, by using the synthesis transmittance of the integrator of non-integer order, m, that conditions tm. As for the model predictivity, its capacity to take into account all the past, by weighting it as appropriate, is proven by the predictive form with long memory of tm such that any predicted value is a function of all the past values. This predictive form is established, here, by taking into account first an integer power, then by extending the results obtained to a non-integer one: that is to say a new approach which makes it possible to highlight the interest of a non-integer power in the consideration of the past, comparatively to an integer power. Beyond these fundamental aspects and although the FPM model was originally conceived for a viral spreading, the model is used here to represent and predict the evolution of CO2 concentration in the terrestrial atmosphere. Such an application to CO2 concentration is justified by a common approach, of countable nature, that unifies these complex phenomena through cumulative data. In order to lead a comparative study of predictivity in this applicative context, the article strives to compare the results obtained in prediction with three models having the same number of parameters: the FPM model, a polynomial model of degree 2 and an exponential model. Whereas the modeling errors obtained with the three models (for different identification periods) are similar in first analysis, the prediction errors (on a prediction horizon belonging to the past) clearly prove to be in favor of the FPM model which well illustrates that the greater the past taken into account, the better the predictivity. Being the most predictive, only the FPM model is then used as a predictor in the case of a prediction horizon truly relating to the future; namely a horizon on 35 years, from 2022 to 2056, where, at the end, the relative dispersion of the predictions remains consistent with a precision of the order of 1% such as verified with real data for a horizon on 35 years, from 1988 to 2022.