The epidemics of donations: logistic growth and power-laws

Abstract

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.

Figure 1. Daily number (blue) and amount (red) of donations shown as a fraction of the total number/amount over a period of one year (mid of 2004 until mid of 2005, time series DH, see Table 1).

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

Figure 2. Probability distribution, P(x), estimated from the relative frequencies to find donations of an amount of x or larger, for both time intervals before (blue) and after (red) the tsunami (time series DH, see Table 1).

Figure 3. Fraction of the total number of donations (inset: relative growth of amount of donations) over time, after the disaster (time series DH).

The blue curves results from fits of Eqs. 3, 4 with μ = 8.05±0.07, τ = 1.98±0.06 (inset: μ = 7.38±0.14, τ = 1.68±0.07 )

Figure 4. Decay of the parameter 1/τ obtained from empirical data of time series DH (red), together with the fit (blue) for 1/τ (Eq. 5) resulting in a = 0.08±0.01, b = 2.52±0.33, c = −1.27±0.38.