May 31, 2020 Figure Open Access

Pagano Angelo; Pagano Emanuele V.

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        "affiliation": "INFN", 
        "name": "Pagano Angelo", 
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        "affiliation": "INFN", 
        "name": "Pagano Emanuele V.", 
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    "description": "<p>1. Introduction&nbsp;</p>\n\n<p>Measurements of the number of infected population in Italy occurring in the period February -May&nbsp;2020&nbsp;have shown an increasing behavior with the time similar to the one expected in Infectious Diseases of Humans [1].&nbsp; However, the model used in the analysis predicts a saturation level of the infected around 220.000 units.&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].</p>\n\n<p>Statistical analysis Covid-19 data: from 20/02/2020 to 23/05/2020 by&nbsp; A.Pagano (infn ct) ; Emanuele V. Pagano (LNS-CT) &ndash; Entropic&nbsp; model adapted from economic process [5] &ndash; The Number (y) of infected humans is solution of the differential equation: <em>y&rsquo;= by-cy^2.</em></p>\n\n<p>The parameters: K,&nbsp;<em>b </em>and <em>c</em> are determined by fitting the curve of infected humans (Fig. 2- rate of infection&nbsp;) from 20-02-2020 to 19-03-2020.&nbsp; The parameter<em> b</em> measures the rate of the infection process for unitary time. The parameter <em>c</em> takes into account for an empirical entropy-balance.&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) &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&nbsp; the available data from 20-02-2020 to 19-03-2020.&nbsp;Fig. 1 and Fig.3 are , respectively,&nbsp;the time integral&nbsp; of curve in Fig.2&nbsp;and the ratio between the rate (derivative) and the cumulative number (Fig. 1), with no further adjustments.&nbsp;&nbsp;The agreement of Fig.1, Fig. 2 and Fig. 3 with experimental data is very good.&nbsp;</p>\n\n<p>In the simplest&nbsp; model [2],&nbsp; &nbsp;it is highly desirable &nbsp;to obtain quantitative indications about two main parameters: &nbsp;<em>beta</em> and <em>gamma</em>&nbsp;, roughly indicating the power of the infectious process.&nbsp; The parameter <em>beta</em>&nbsp;is the transmission coefficient from &ldquo;non-infected&rdquo; individuals (S) to infected individuals (I), following the symbolic transition :&nbsp; S &gt; I; it depends on the social structure and the intrinsic property of the virus. The inverse of the second parameter: &nbsp;1/ gamma&nbsp; gives the average time &nbsp;of permanence of an individual in the &ldquo;infected state&rdquo; I, following the symbolic transition: I &gt; S.&nbsp; They enter in the time-depending differential equation, describing the time-evolution of the a-priori probability <em>p</em> (i.e., given by the ratio between infected individuals and the total population) associated to the infectious process [2]:</p>\n\n<p><em>&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; p&#39; = (beta -gamma)p -beta p^2 .</em>&nbsp;&nbsp;</p>\n\n<p>In order to obtain realistic evaluation of the two parameters (or others, such the number of infected including &ldquo;asymptomatic&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. &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% .&nbsp;</p>\n\n<p>References:</p>\n\n<ol>\n\t<li>Morens D.M., Folker G.K., Fauci A.S., &ldquo; The challenge of emerging and re-emerging infectious diseases&rdquo;, Nature, 439, (2004), pp. 242-249.</li>\n\t<li>Carlo Piccardi, &ldquo;Reti Sociali e Diffusione di Epidemie&rdquo;, Lettera Matematica, n.86 Pristeam Univ. Bocconi, Springer (2013), pp. 30-37</li>\n\t<li>Anderson R., May R., &ldquo;Infectious Diseases of Humans: Dynamics and Control&rdquo;, Oxford University Press, (1991).</li>\n\t<li>Anderson R., &ldquo;The Application of Mathematical Models in Infectious Disease Research&rdquo;, <a href=\"http://www.ph.ucla.edu/epi/faculty/olsen/200B2010/ANDERSON.FITL.2001.pdf\">http://www.ph.ucla.edu/epi/faculty/olsen/200B2010/ANDERSON.FITL.2001.pdf</a></li>\n\t<li>G. Amata &ndash; S.Notarrigo, &ldquo;Energia e Ambiente: una ridefinizione della teoria economica&rdquo;, C.U.E.C.M. Catania (1987)</li>\n</ol>\n\n<p>&nbsp;</p>", 
    "doi": "10.15161/oar.it/23701", 
    "keywords": [
      "Covid -19", 
      "Statistical Analysis", 
      "Entropic model", 
      "Total Infected"
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    "notes": "working in progress", 
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