Modeling the Spread of the Black Plague
In the 14th century, the Black Plague epidemic swept from across Central Asia into Europe, wiping out upwards of 70 million people. The disease was carried by infected fleas on rats, and facilitated by exchanges on trading routes like the Silk Road, which allowed for greater contact between neighboring and distant regions. However, as the article discusses, recent research suggests that the reason the Black Plague was able to spread so quickly and kill so many people within a few years was that the disease was also airborne. A lethal pneumatic strain of the disease, transmittable person-to-person via coughs, sneezes, and even breathing, made the plague dangerously infectious. In addition to symptoms of fever, trembling, and black-blue blood “swellings,” the airborne version of the Black Plague also induced coughing and parched throat. To understand how this disease spread with such deadly reach and impact, we can examine its propagation through both a simple branching process model and a more realistic SIR model.
From the branching process model, if we consider each node (person) to have had k neighbors in the next wave of people she comes in contact with and a probability p of passing the contagion to each neighbor, we can develop a basic reproductive number R0= pk, for the expected number of people infected, given you have the Black Plague. In this model, one explanation for the spread of the plague was the inability to curtail it early in its onset, leading to a snowball effect of a highly contagious disease spreading to many people (often in crowded cities like Florence or London); as more and more people carry the disease, there is greater likelihood of the disease spreading to succeeding waves of people. While quarantine measures were attempted to reduce k, the number of contacts by a person, it was difficult with the Black Plague, where an infected person could carry the disease for many days before symptoms even showed. However, given that the disease died out after a few years, we can hypothesize R0= some value < 1, such that with probability 1, the Black Plague dies out in a finite number of steps. With modern advancements in antibiotics and hygiene, it is even likelier that Black Plague will eventually completely die out.
The SIR model can also be employed to more realistically model the spread of the Black Plague, with nodes being in susceptible, infectious, or removed states. While infected (I), nodes have a probability p of infecting each susceptible (S) neighbor, and those infected nodes are eventually removed (R), either because they recovered or because they died. In the case of 14th-century Black Plague, before the advent of antibiotics and a “cure,” it is likely that most nodes (people) switched from the infectious state to the removed state via death, which was good because they could no longer infect others, but bad because, well, they died. With the SIR model, it is harder to accurately model the spread of a disease like Black Plague, especially given considerations like changing contact networks between people over time, but it provides a network-level framework to think about how such a disease could have spread so quickly across so many people.
Sources:
http://www.history.com/news/medieval-black-death-was-airborne-scientists-say
https://web.cn.edu/kwheeler/black_plague.html