SIAM News Blog

New Model of Rat-Flea Interactions Seeks to Clarify Bubonic Plague Transmission

By Lina Sorg

Plagues have affected humans for thousands of years and are responsible for killing millions of people in Europe during the Middle Ages. Infections are caused by a bacterium called Yersinia pestis and typically take one of three clinical forms—pneumonic, bubonic, or septicemic—based on the infection’s location in the body. Pneumonic plague occurs in the lungs, bubonic plague is associated with swollen or inflamed lymph nodes (called buboes) in the armpit or groin, and septicemic plague is an infection of the blood. Developments in modern medicine mean that outbreaks are rare in present-day society.

A plague’s type dictates its manner of spread. Pneumonic plague is spread by aspiration of infectious droplets and can thus be demonstrated with a simple susceptible-infected-recovered (SIR) model. Experts generally agree that bubonic plague is transmitted by infected fleas. These fleas, which reside on rats, eventually leave their hosts and transfer the disease to humans in a process called rat-flea transmission. One of the most dominant models for this type of transmission stems from M.J. Keeling and C.A. Gilligan’s 2000 Nature paper. However, another more recent postulated model for bubonic plague names human-ectoparasite interactions—in the form of lice or fleas that affect humans directly without rodent hosts—as the primary source of transmission. Katharine Dean et al. introduced this model, which presents a very different hypothesis for the spread of plague than the traditional narrative, in a 2018 PNAS paper

During a contributed presentation at the 2021 SIAM Annual Meeting, which is taking place virtually this week, Andrew Oster of Eastern Washington University presented a new model for rat-flea-driven transmission of bubonic plague. He utilized historic data from previous plague epidemics to compare the different models and determine which ones best fit the data. “The question is, are plagues driven via the human-ectoparasite interaction or the classical rat-flea vector story?” he asked. “Dean’s paper seems to suggest that the human-ectoparasite model was a far better fit, whereas here we’re going to be contrasting that with our new model.”

Oster began with a quick survey of existing plague models. The classic SIR model effectively captures the effects of pneumonic plague, though Oster uses “deceased” (SID) rather than “recovered” because pneumonic plague has a high mortality rate. The Keeling-Gilligan rat-flea model tracks the number of fleas per rat and the number of infected fleas that are not on rats (these fleas are infecting humans or other rats). The resulting system of differential equations is then coupled with a susceptible-infected-recovered-deceased (SIRD) model for subpopulations of both rats and humans. In contrast, the initial equations in Dean et al.’s human-ectoparasite model are coupled with an SIIRD model based on the amount of bacteria and severity of the ensuing infection. Depending on the infection level (low or high), individuals either recover or are deceased.

Figure 1. Comparison of the standard susceptible-infected-decseased (SID) pneumonic model, the Keeling-Gilligan rat-flea model (rat-flea model 1), Dean et al.’s human-ectoparasite model, and the Lynch-Oster rat-flea model (rat-flea model 2) for the 1490 plague in Barcelona, Spain. The new Lynch-Oster model and the human-ectoparasite model yielded comparable results
Oster then introduced a new logistic-like model of plague transmission that accounts for rat and flea dynamics, like the Keeling-Gillian model. He created this novel model with then-student Ian Lynch. The model tracks infected carriers as well as the total number of fleas, and the death rate accounts for the deaths of rats that had the plague. Variables include the intrinsic birth rate, intrinsic death rate, rat recovery rate, plague death rate, plague infectivity, and carrying capacity. Oster and Lynch then coupled their model of rat-flea interactions with a susceptible-exposed-infected-recovered/deceased (SEIR/D) model. 

Next, Oster considered several different datasets with existing classical data from past plague events. He could not just run the data directly because the parameters are unknown. “There’s no way to measure [them] because it was hundreds of years ago,” Oster said. “Also, those parameters changed in the different situations depending on the culture of the cities where the infections occurred and what sort of protection protocols they had. It changes case by case.” To compensate, he used a Markov chain Monte Carlo (MCMC) simulation.

Oster then compared the standard SID pneumonic model, the Keeling-Gilligan rat-flea model, Dean et al.’s human-ectoparasite model, and his own rat-flea model in the context of various plague scenarios. He started with the 1490 plague in Barcelona, Spain, for which his new model and the human-ectoparasite model yielded comparable results (see Figure 1). “This seems to suggest that the familiar story of a rat-flea transmission is still valid,” Oster said.

Though the human-ectoparasite model fit the 1813 plague in Malta far better than the SID pneumonic model or the traditional rat-flea model, Oster’s model once again offered similar results. The same was true of the 1400 plague in Florence, Italy. Oster’s final comparison was for the 1713 plague in Prague, for which he only considered his rat-flea model and the human-ectoparasite model. Though he has not verified these findings, his new model appeared to fit the data far better (see Figure 2).

Figure 2. Upon initial comparison, the new Lynch-Oster rat-flea transmission model seems to fit data from the 1713 plague in Prague better than the human-ectoparasite model.

Oster validated his outcomes with a Bayesian information criterion (BIC). According to the criterion, the human-ectoparasite model performs better for Barcelona and Malta, while the Lynch-Oster model is superior for Florence. “But I would argue that with all of these, the results are still comparable,” Oster said. “One of them does not vastly outdo the other.” Analysis of the root mean square error agreed with the BIC.

Oster intends to conduct further testing, utilize additional data and expand to more historical datasets, and employ better algorithms for the MCMC simulations to continue this work. “The story of the rat-flea transmission still seems like a valid explanation,” he said. “Neither the new model that we have for the rat-flea transmission nor the human-parasite model of Dean et al. significantly outperforms the other. Both give suitable results.”

Lina Sorg is the managing editor of SIAM News.
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