The Bubonic Plague: A Discussion of Transitivity and Centrality
This past year, the world has seen grave damage at the hands of the COVID-19 pandemic. Countries, small and large, actualized the insidious nature of the virus as it was able to transcend geographical boundaries with insurmountable magnitude and force. Despite its highly publicized nature and seemingly unparalleled scale, COVID-19 is not the first pandemic to have acted in such a brutal and unforgiving manner. Many pandemics have come before COVID-19, however a majority occurred prior to the age of personal technology — thus the lack of attention towards them at this current time. Throughout the progression of research as a whole, society has seen the importance of looking back towards previous occurrences in order to gain insight into the happenings of now. The infamous Black Death Pandemic is an exemplary instance in which we can gain such insight.
The article Network theory may explain the vulnerability of medieval human settlements to the Black Death pandemic, published by Nature magazine, contains a study conducted by researchers who wished to inquire about what factors allowed for the Black Death to be as detrimental as it was. Researchers José M. Gómez & Miguel Verdú propose that the gravely dangerous nature of this pandemic stemmed from two things: transitivity and centrality. These, they claim, were functions of the social and transport networks that existed at that time. During the medieval period, there were vast networks that connected the continent of Europe across its oceanic boundaries. Pilgrimage and trade networks comprised these. In the study conducted by Gómez and Verdú, networks in question “included information on 2084 trade and pilgrimage connections between 1311 Old World cities (including North Africa, Europe and Asia), with 1013 cities being connected by trading routes and 403 cities connected by pilgrimage routes, with some cities appearing in both trade and pilgrimage routes”. Quite evidently, the hyper nature of the pilgrimage and trading activity facilitated an extremely connected global network. Through their studies, they found that the transitivity — “the propensity of nodes of clustering together” — of the overall network that accounted for both pilgrimage and trading networks was significantly larger than that of a randomly selected network. They also found that the centrality — “the number and intensity of connections with other nodes”— demonstrated the extreme connectedness that the cities of this network experienced between them and cities of other networks. Thus, they concluded the idea that a city that experienced high transitivity and centrality rates, experienced more detrimental and grave pandemic results. Through their findings, Gómez and Verdú suggest the theory that the specific location of the city, in terms of distance to other cities in the network, and the amount of connections that particular city has to others deeply implicates that city in terms of vulnerability to reinfection and high spreading rates.
Beyond the discussion of the centrality and transitivity of the networks affected by the medieval Black Death Pandemic, I would like to include a discussion regarding this pandemic in terms of a simple branching network. Generally speaking, contagion occurs with a singular person who we call “patient zero”. Patient zero meets k people, who then meet another set of k people. This network can be infinitely expanded. Alongside these k values, remains an equally important p value, also known as the probability of contagion with each interaction between people. With the context of the Black Death in mind, we have a large branching network that begins with a city that first becomes infected with the disease from the carrier called the Oriental rat flea. This particular city is involved in a large trading, and or pilgrimage, network that connects it to ten other cities. These ten cities represent the k “people” in the generalized branch. The probability here represents the chance that each of the ten cities will be infected by the first city, patient zero. This infinite procedure continues on to each of the cities that these ten cities will individually come in contact with within their very own networks.
These two terms, k and p, comprise what is known as the basic reproductive number of the contagion, Ro, when multiplied together. In essence, the value of Ro determines the fate of the outbreak. If that value remains under 1, it is likely that the outbreak will be contained, not replenishing itself. On the other hand, if this value exceeds 1, it is likely that the outbreak will be more than replenishing. This leads to the interesting question of what was this value during the time of the Black Death Pandemic? Extensive studies from a number of sources have attempted to calculate this value, some placing it somewhere between 2.8 – 3.5 and some placing it even higher; either case, an extremely dangerous zone. Despite these guesses and the information today regarding the scope and scale of the Black Death, there is still not one widely accepted calculation for the probability of transmission from one host to another.
It would be quite intriguing to further explore how exactly network characteristics derived from the physical features of these particular cities interacted with this Ro value. A study that simulated a parallel research question that allowed for the control of certain variables and an experimentation upon both the p-value and the k-value could lead to some more interesting results. Would a lower p-value and higher k-value paired with constant centrality and transitivity yield different rates than the opposing scenario? While we do not yet know the answer to these inquiries, we can hypothesize that the probability of transmission, multiplied with the k-value of the medieval network, had to have worked in conjunction with the unique levels of transitivity and centrality in this network to have ravaged the medieval world at the extent to which it did.
Source: https://www.nature.com/articles/srep43467