May 31, 2020 Figure Open Access

Pagano Angelo; Pagano Emanuele V.

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    <subfield code="a">&lt;p&gt;1. Introduction&amp;nbsp;&lt;/p&gt;

&lt;p&gt;Measurements of the number of infected population in Italy occurring in the period February -May&amp;nbsp;2020&amp;nbsp;have shown an increasing behavior with the time similar to the one expected in Infectious Diseases of Humans [1].&amp;nbsp; However, the model used in the analysis predicts a saturation level of the infected around 220.000 units.&amp;nbsp; 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].&lt;/p&gt;

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

&lt;p&gt;The parameters: K,&amp;nbsp;&lt;em&gt;b &lt;/em&gt;and &lt;em&gt;c&lt;/em&gt; are determined by fitting the curve of infected humans (Fig. 2- rate of infection&amp;nbsp;) from 20-02-2020 to 19-03-2020.&amp;nbsp; The parameter&lt;em&gt; b&lt;/em&gt; measures the rate of the infection process for unitary time. The parameter &lt;em&gt;c&lt;/em&gt; takes into account for an empirical entropy-balance.&amp;nbsp; 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) &amp;ndash;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&amp;nbsp; the available data from 20-02-2020 to 19-03-2020.&amp;nbsp;Fig. 1 and Fig.3 are , respectively,&amp;nbsp;the time integral&amp;nbsp; of curve in Fig.2&amp;nbsp;and the ratio between the rate (derivative) and the cumulative number (Fig. 1), with no further adjustments.&amp;nbsp;&amp;nbsp;The agreement of Fig.1, Fig. 2 and Fig. 3 with experimental data is very good.&amp;nbsp;&lt;/p&gt;

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

&lt;p&gt;&lt;em&gt;&amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; p&amp;#39; = (beta -gamma)p -beta p^2 .&lt;/em&gt;&amp;nbsp;&amp;nbsp;&lt;/p&gt;

&lt;p&gt;In order to obtain realistic evaluation of the two parameters (or others, such the number of infected including &amp;ldquo;asymptomatic&amp;rdquo; ) 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. &amp;nbsp;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% .&amp;nbsp;&lt;/p&gt;

&lt;p&gt;References:&lt;/p&gt;

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