Spreading Patterns of the Influenza A (H1N1) Pandemic Sergio de Picoli Junior1*, Jorge Juarez Vieira Teixeira2, Haroldo Valentin Ribeiro1,3, Luis Carlos Malacarne1, Ricardo Paupitz Barbosa dos Santos1, Renio dos Santos Mendes1,3 1 Departamento de F´ısica, Universidade Estadual de Maringa´, Maringa´, Parana´, Brazil, 2 Departamento de Ana´lises Cl´ınicas e Biomedicina, Universidade Estadual de Maringa´, Maringa´, Parana´, Brazil, 3 National Institute of Science and Technology for Complex Systems, Rio de Janeiro, Rio de Janeiro, Brazil Abstract We investigate the dynamics of the 2009 influenza A (H1N1/S-OIV) pandemic by analyzing data obtained from World Health Organization containing the total number of laboratory-confirmed cases of infections - by country - in a period of 69 days, from 26 April to 3 July, 2009. Specifically, we find evidence of exponential growth in the total number of confirmed cases and linear growth in the number of countries with confirmed cases. We also find that, i) at early stages, the cumulative distribution of cases among countries exhibits linear behavior on log-log scale, being well approximated by a power law decay; ii) for larger times, the cumulative distribution presents a systematic curvature on log-log scale, indicating a gradual change to lognormal behavior. Finally, we compare these empirical findings with the predictions of a simple stochastic model. Our results could help to select more realistic models of the dynamics of influenza-type pandemics. Citation: Picoli Junior Sd, Teixeira JJV, Ribeiro HV, Malacarne LC, Santos RPBd, et al. (2011) Spreading Patterns of the Influenza A (H1N1) Pandemic. PLoS ONE 6(3): e17823. doi:10.1371/journal.pone.0017823 Editor: Paulo Ho, Instituto Butantan, Brazil Received November 8, 2010; Accepted February 14, 2011; Published March 31, 2011 Copyright: 2011 Picoli Junior et al. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited. Funding: This work was supported by CNPq (Conselho Nacional de Pesquisa) and CAPES (Coordenac¸a˜ode Aperfeic¸ oamento de Pessoal de Nı´ vel Superior) - Brazilian agencies. Competing Interests: The authors have declared that no competing interests exist. * E-mail: junior@dfi.uem.br Introduction The spread of infectious diseases is a threat to the world public health and a subject of great scientific interest. Influenza viruses, for example, circulate around the world every year and seasonal influenza is one of the most worrying respiratory infections of humans [1]. From time to time new strains of influenza virus emerge and cause large-scale global pandemics – such as the 2009 influenza A (H1N1/S-OIV). The dynamics of influenza-type pandemics has been the focus of several scientific works which may provide necessary information to deal with future pandemic events [2–9]. However, conventional modeling techniques are usually only able to provide a general idea of how a pandemic might evolve, since crucial information concerning model parameters is generally unavailable. Methods Here, we investigate the dynamics of influenza-type pandemics using techniques of statistical physics typically applied in the study of complex systems. We analyze a database - from World Health Organization - containing laboratory-confirmed cases of influenza A (H1N1/S-OIV) infections around the world [10]. Specifically, we analyze the total number of laboratory-confirmed cases - by country - in a period of 69 days, from 26 April to 3 July, 2009. We search for patterns of spread for the virus influenza A (H1N1/S-OIV). Basically, we investigate the growth in the total number of confirmed cases, the growth in the number of countries with confirmed cases and the cumulative distribution of cases among countries. We also propose a simple stochastic model which reproduces the main empirical findings obtained from the analysis of the database on the 2009 influenza A (H1N1/S-OIV) pandemic. Results First, we consider the time evolution of the global number of confirmed cases, Y(t). At a given time t, the total number of cases Y is given by the sum of y over all countries. In Fig. 1A we show Y(t) in comparison with an exponential curve, given by the equation dY ~lY , with l~0:05 (compare with results reported in ref. dt [11]). This result suggests that the total number of confirmed cases exhibits an exponential growth. However, a systematic deviation of the exponential behavior can be observed for smaller times (t , 20 days). In addition, Fig. 1B shows the time evolution of the total number of countries with confirmed cases, W(t). Observe that W(t) exhibits approximately a linear growth in all the period considered. Next, we investigate the distribution of confirmed cases among countries and its time evolution. In order to reduce statistical fluctuations, it is common to consider the cumulative distribution R(y)~ Ð P(y’)dy’. Here, y is the number of confirmed cases in a given country and P(y) is the probability distribution function (PDF) of y. The empirical cumulative distribution R(y), for initial times, is shown in Fig. 2A. In this range, R(y) presents linear behavior on log-log scale being well described by a power law decay, R(y)*y{a, 1 ð Þ where a is the power law exponent. The time evolution of the power law exponent a is shown in Fig. 2B. Observe that a stays approximately constant, 0.35,a,0.040, for 21 days. For larger times, R(y) gradually deviates from the power law behavior (Supplementary Figure). In this range, R(y)has a modest negative curvature everywhere (see Fig. 3A) and is in good PLoS ONE | www.plosone.org 1 March 2011 | Volume 6 | Issue 3 | e17823