2.2.2. Power-Laws and Scale-Free Networks

Power-laws are mathematical functions such as *xq*, where *x* is the independent variable and *q* is a negative or positive constant. They are common in physical laws; for example, the gravitational force between two bodies decreases with distance *d* according to a power law proportional to d−2. Similarly, the electric force between two charged bodies falls with d<sup>−</sup>2.

In microbiology, we also find power laws. For example, the mutation rate per replication per nucleotide of DNA-based microorganisms decreases with the genome size according to *<sup>μ</sup>* <sup>≈</sup> *<sup>c</sup>*·*G*−*β*. Because *<sup>β</sup>* ≈ −1, this equation means that the mutation rate per replication per nucleotide is inversely proportional to the genome size (*μ*·*G* ≈ *c*) so that the mutation rate per replication per genome is a constant (*c* ≈ 0.003): Drake's rule [35–37]. Another example concerns the death rate of persister bacterial populations in the presence of a bactericidal antibiotic. Some authors have argued that the death rate of persister populations of some strains follows a power law with an exponent close to −2 [38,39]. A power-law decay is slower than an exponential decay, which means that bacterial cells under antibiotic exposure decay according to a power law can persist alive for longer, sometimes causing health problems and persistent food contamination (Figure 2B).

Concerning the spread of microorganisms through human-contact networks (i.e., with people as nodes), it is relevant to know if the networks are scale-free, that is, if the proportion of people with a *k* connection follows a power-law distribution. That would mean that a non-negligible proportion of people have many connections, as we have seen with the web of sexual contacts [31]. However, there are more examples of scale-free networks relevant to epidemiology.

In 2006, Brockmann et al. found something striking concerning the dispersal of banknotes in the USA. They studied banknote dispersion as a proxy of people traveling. As intuitively expected, most banknotes travel less than 10 km in four days; also, according to our intuition, the number of banknotes detected further away decreases when the distance increases. Gaussian or Exponential distributions would mostly predict that none or very few banknotes travel more than a few hundred kilometers. However, contrary to common intuition, many banknotes travel thousands of kilometers in those four days. Banknote traveling follows a decreasing power law [40]. The transactions of these banknotes may be twofold in their relevance to epidemiological studies: (i) banknotes move between physically close people, enabling cross-contagion with microorganisms; (ii) banknotes may carry microorganisms, so a person may contaminate another one without being physically close.

Tracking the position of 100,000 mobile phones for six months provides a similar distribution [41]. Most mobile phones only travel a few kilometers, and the proportion of phones traveling decays when the distance increases. A non-negligible number of cell phones traveled hundreds of kilometers. As we have seen for banknotes, the overall traveling of mobile phones follows a decreasing power law. These long-distance traveling people (measured through their banknotes and mobile phones) may constitute relevant microorganism spreaders.

As we have seen, networks where the proportion of people with *k* connections decreases according to *<sup>p</sup>*(*k*) <sup>≈</sup> *<sup>k</sup>*−*α*, where *<sup>α</sup>* is a positive fixed number are epidemiologically relevant. However, concerning the diseases' spread, some are even more relevant such as those power laws where the parameter *α* is between two and three. We have seen above that a suitable parameter commonly discussed in epidemiological studies is that of *R0*, which informs us how many people a single infected person will transmit the infection to on average in a fully susceptible population. Strikingly, there is no such threshold in networks whose connections between people follow a decreasing power-law distribution with 2 < *α* < 3. Therefore, an epidemic spread may occur even with low rates of disease transmission between the hosts [42,43]. Networks with *α* < 3 have very high standard deviations in terms of the number of connections to each node. Therefore, in the context of the equation *R*<sup>0</sup> = *β*·*c*·*d*, the number of contacts between infected and susceptible individuals per time unit, *c*, may also be extremely high.

If the human population network structure (small word, sometimes following a powerlaw distribution) somewhat facilitates microorganism spread, why do novel pathogens not almost instantly infect humans worldwide? In the case of scale-free networks, the *α* parameter mentioned above is sometimes lower than two or above three. For example, we have seen above that, in the case of sexual partners in the previous year, that *α* is slightly higher than three [31,32]. With the *α* parameter outside that interval, the network is still a small world, but there is an epidemic transition value [34]. Moreover, real-life scale-free networks are finite (i.e., have a limited number of people), which implies that, even if the *α* parameter falls between two and three, there is a non-null epidemic threshold.

Moreover, humans do not become infectious immediately after contagion, which may take a few days, depending on the disease. Furthermore, people are not permanently in contact with each other, particularly if they feel ill. Additionally, people, medical doctors, and the government commonly implement measures to halt disease spread. Even so, we have seen that with, for example, the COVID-19 pandemic, and despite arduous efforts employed by the governments of several countries, two and a half months (between December 2019 and the first days of March 2020) were sufficient to spread the SARS-CoV-2 virus to most countries worldwide. Governments employed compulsory confinements and other demanding measures because COVID-19 would kill many people and cause morbidity to many others [44].
