May 31, 2020 Figure Open Access

Monitoring Covid-19 emergence data

Pagano Angelo; Pagano Emanuele V.

1. Introduction 

Measurements of the number of infected population in Italy occurring in the period February -May 2020 have shown an increasing behavior with the time similar to the one expected in Infectious Diseases of Humans [1].  However, the model used in the analysis predicts a saturation level of the infected around 220.000 units.  Monitoring this number by a suitable statistical analysis is an essential step in order to understand the phenomena with the purpose to disentangle among possible existing models [2] for an efficient description of the dynamical Process [3,4].

Statistical analysis Covid-19 data: from 20/02/2020 to 23/05/2020 by  A.Pagano (infn ct) ; Emanuele V. Pagano (LNS-CT) – Entropic  model adapted from economic process [5] – The Number (y) of infected humans is solution of the differential equation: y’= by-cy^2.

The parameters: K, b and c are determined by fitting the curve of infected humans (Fig. 2- rate of infection ) from 20-02-2020 to 19-03-2020.  The parameter b measures the rate of the infection process for unitary time. The parameter c takes into account for an empirical entropy-balance.  From parameter k the total number of infected (population) could be determined. Solution of the equation is given in the inserts of the figures Fig.1 (cumulative infected)-Fig.2 (rate by day of infected) –Fig.3 (relative rate : rate by day of infected over the number of cumulative infected ). Evidently, only the derivative (Fig.2) of the solution in Fig.1 has been fitted with  the available data from 20-02-2020 to 19-03-2020. Fig. 1 and Fig.3 are , respectively, the time integral  of curve in Fig.2 and the ratio between the rate (derivative) and the cumulative number (Fig. 1), with no further adjustments.  The agreement of Fig.1, Fig. 2 and Fig. 3 with experimental data is very good. 

In the simplest  model [2],   it is highly desirable  to obtain quantitative indications about two main parameters:  beta and gamma , roughly indicating the power of the infectious process.  The parameter beta is the transmission coefficient from “non-infected” individuals (S) to infected individuals (I), following the symbolic transition :  S > I; it depends on the social structure and the intrinsic property of the virus. The inverse of the second parameter:  1/ gamma  gives the average time  of permanence of an individual in the “infected state” I, following the symbolic transition: I > S.  They enter in the time-depending differential equation, describing the time-evolution of the a-priori probability p (i.e., given by the ratio between infected individuals and the total population) associated to the infectious process [2]:

                                                              p' = (beta -gamma)p -beta p^2 .  

In order to obtain realistic evaluation of the two parameters (or others, such the number of infected including “asymptomatic” ) it is highly recommended to observe the time evolution on a time scale of the order of two or three months in order to achieve a reasonable statistical accuracy.  As an example, from our fitting procedure of Fig. 1-Fig.3, it seems that the infectious process has involved an average number of infected humans (in Italy) ranging between 1.5 and 10 millions, with a statistical average of about 5.7 millions of infected; the estimated accuracy is about 20% . 

References:

1. Morens D.M., Folker G.K., Fauci A.S., “ The challenge of emerging and re-emerging infectious diseases”, Nature, 439, (2004), pp. 242-249.
2. Carlo Piccardi, “Reti Sociali e Diffusione di Epidemie”, Lettera Matematica, n.86 Pristeam Univ. Bocconi, Springer (2013), pp. 30-37
3. Anderson R., May R., “Infectious Diseases of Humans: Dynamics and Control”, Oxford University Press, (1991).
4. Anderson R., “The Application of Mathematical Models in Infectious Disease Research”, http://www.ph.ucla.edu/epi/faculty/olsen/200B2010/ANDERSON.FITL.2001.pdf
5. G. Amata – S.Notarrigo, “Energia e Ambiente: una ridefinizione della teoria economica”, C.U.E.C.M. Catania (1987)