^{*}

Analyzed the data: RM. Wrote the paper: FS.

The authors have declared that no competing interests exist.

This paper demonstrates that collective social dynamics resulting from individual donations can be well described by an epidemic model. It captures the herding behavior in donations as a non-local interaction between individual via a time-dependent mean field representing the mass media. Our study is based on the statistical analysis of a unique dataset obtained before and after the tsunami disaster of 2004. We find a power-law behavior for the distributions of donations with similar exponents for different countries. Even more remarkably, we show that these exponents are the same before and after the tsunami, which accounts for some kind of universal behavior in donations independent of the actual event. We further show that the time-dependent change of both the number and the total amount of donations after the tsunami follows a logistic growth equation. As a new element, a time-dependent scaling factor appears in this equation which accounts for the growing lack of public interest after the disaster. The results of the model are underpinned by the data analysis and thus also allow for a quantification of the media influence.

The tsunami that infested South-Eastern Asia on 26 December 2004 has not just caused a tremendous death toll and destruction, but also a huge outpouring of donations worldwide to support relief for the affected areas. The fact that both the number and total amount of donations summed up in an unprecedented way was of course induced by the dimension of the disaster. It also benefited from social feedback processes, caused by the massive involvement of the mass media which eventually led to social herding in donating money.

Herding behavior plays an important role in biological but also in social systems. It is governing biological swarming

In this paper, we apply the concept of contageous behavior to the collective dynamics of donations after the tsunami catastrophe. Thanks to the availability of a unique database described below, we are able to quantify these dynamics. The statistical analysis reveals a power-law behavior for the distributions of donations with similar exponents for different countries and, even more remarkably, both before and after the disaster. We further show that the dynamics of donations follow a logistic growth already known from models of epidemic spreading. As a new element of this dynamics, a time-dependent contagion rate appears which describes the mean-field interaction provided by the mass media. The considerable decrease of this influence in time accounts for the growing lack of public interest after the disaster. By deducing it from the data available, we are able to quantify the influence of the media reporting about the tsunami.

Individual donations, as a voluntary act, may depend on individual, cultural, organisational and economic conditions and thus may differ between countries. In order to find out statistical similarities in the distribution of donations, we investigated three different time series from donor organizations in Germany (DH, AH) and Switzerland (GK) summarized in

Donor org | DH | AH | GK |

Time int | 04/07/26–04/12/23 | 03/12/29–04/06/30 | N/A |

_{tot} | 209,928 | 1,587,442 | N/A |

_{tot} | 3,160 | 19,222 | N/A |

α | 1.501 | 1.171 | N/A |

Time int | 04/12/27 –05/06/24 | 04/12/27–05/06/30 | 04/12/27–05/06/17 |

_{tot} | 126,879,803 | 2,649,097 | 225,022,112 |

_{tot} | 1,556,626 | 28,965 | 768,882 |

α | 1.515 | 1.278 | 1.205 |

μ° | 8.055 | 36.367 | 9.972 |

τ° | 1.985 | 27.770 | 3.271 |

μ* | 7.389 | 10.250 | 9.533 |

τ* | 1.687 | 9.480 | 2.822 |

Each data set contains the amount of each individual donation together with the date of donation. gives the total amount of donations (local currency) in the given time interval (top: before, bottom: after the tsunami), the total number of donations, respectively.

For the largest of these time series (DH), we also compared the number and amount of donations for an interval of six months

The inset magnifies the relative growth of number and amount of donations for the half-year period preceeding the earthquake.

From the inset of

Interestingly, we find that the exponents

In order to sketch the time dependent evolution of donations after the disaster shown in

The act of donation is described as a transition of a potential into an actual donator, . This transition may occur at a gross rate that depends on a constant

The blue curves results from fits of Eqs. 3, 4 with

The data shown in

The simple model of donator dynamics assumes that eventually all possible donators have donated once on an individual basis, i.e.

The remarkable findings of our investigations are (i) the statistical similarities in individual donations before and after the tsunami disaster, as well as between two different countries, and (ii) that the collective dynamics of millions of individual donations can be very well described by a simple epidemic model, which has similarities also to the adoption of innovations

The applicability of the SIR dynamics shows that the interaction of the individuals can indeed be modelled by a mean-field interaction which accounts for the dissemination of information by the mass media: the disaster event, broadcasted in the mass media, triggered the first donations, which were then amplified by the mass media again, broadcasting new information both about the disaster and donations received. This resulted in some global feedback dynamics which eventually slowed down both because of a decreasing public interest and a exhausted resource (potential donators). While the latter one sufficiently describes the saturation effect, it was indeed the decreasing public interest and the related influence of the mass media, covered in the model by the time dependent parameter 1/

The authors wish to thank the three donor organizations (DH)–“Deutschland hilft” (Germany), (AH)–“Andheri-Hilfe” (Germany), (GK)–“Glückskette” (Switzerland) for their cooperation in providing the data.