Spreading Patterns of the Influenza A Figure 1. Temporal evolution of cases and countries with confirmed cases. (A) The world total number of influenza A (H1N1) laboratory-confirmed cases, Y(t), on mono-log scale, for the period of 69 days, from 26 April to 3 July, 2009. The solid line is given by y(t)~2825exp(lt), with l~0:05. (B) The total number of countries with confirmed cases, W(t), for the same period. A linear least-square fit to the data (solid line) suggests W (t)~1:02z1:69t. doi:10.1371/journal.pone.0017823.g001 Figure 2. Cumulative distribution of cases among countries: power law behavior. (A) Empirical cumulative distribution, R(y), of laboratory-confirmed cases y for distinct times (few specific days are displayed for better visualization). The solid lines are power laws, given by Eq. (1), with exponents following Fig. 2B. (B) Temporal evolution of the power law exponent a, obtained by least-square fits to the data. Observe that 0.35,a,0.40 for a period of about 20 days. doi:10.1371/journal.pone.0017823.g002 Figure 3. Cumulative distribution of cases among countries: log-normal behavior. (A) Empirical cumulative distribution, R(y), laboratory-confirmed cases y, for larger times (for better visualization, we display few specific days). The systematic curvature on log-log scale indicates a gradual deviation from power law behavior. The solid lines are given by Eq. (2), with parameters shown in Fig. 3B. (B) Temporal evolution of parameters a (circles) and b (squares) obtained by least- square fits of Eq. (2) to the data. doi:10.1371/journal.pone.0017823.g003 agreement with log-normal curves given by R(y)~c exp h {a ln y{b ðln yÞ2i, 2 ð Þ Where c is a normalization constant and a and b are the parameters. Notice that R(y) behaves as a power law, R(y)*y{a, when b?0. The time evolution of the parameters a and b of the log-normal curves are shown in Fig. 3B. Discussion The exponential growth shown in Fig. 1A is consistent with typical results in epidemiology. Well known models for the spread of epidemics, such as SEIR model, predicts exponential growth in the number of infections at early stages of the epidemic. A possible explanation of the deviation of exponential behavior for smaller times, also shown in Fig. 1A, may be found in Fig. 1B. The growth of individual countries is added to the growth of the number of countries with new cases. For initial times, when the total number of cases is small, this combination may generate a deviation of the exponential behavior. For larger times, when the number of cases is large, the effect of the entry of new countries with few infections may be much less intense. According to results shown in Fig. 2, the distribution of cases among countries exhibits power law behavior - indicating scale invariance. The presence of power law behavior suggests that the system self-organizes into a scale-free state. This phenomenon is a remarkable characteristic of a wide range of complex systems - from physics to biology and medicine [12]. Scale invariance in the global spread of influenza pandemic for initial stages is a PLoS ONE | www.plosone.org 2 March 2011 | Volume 6 | Issue 3 | e17823