
MBW:Bubonic plague: a metapopulation model of a zoonosisFrom MathBioContentsExecutive SummaryIn the article by Keeling and Gilligan, Bubonic plague: A metapopulation model of a zoonosis, the authors begin by using a coupled system of SIR type models for humans, rats, and fleas to show how epidemics in the human population are preceded by an epidemic within the rat population. Keeling and Gilligan sought to have the most realistic model possible so they incorporated real data, and parameter sensitivity analysis when choosing the values for their parameters. They then performed an exhaustive analysis of this model, and conclude that it was adequate for shortterm dynamics but something more robust would need to be developed for longterm analysis. Keeling and Gilligan then shifted their focus towards a stochastic model that was more biologically realistic. By combining the results of the model in question with modern biological data surrounding prevalence of common rodent species the risk of modern bubonic plague outbreaks were estimated probabilistically. All of this analysis provided them with a way to describe what a modern outbreak of the plague would look like (assuming that no measures were taken to combat the disease). BackgroundThe bubonic plague is one of the most prolific and devastating diseases throughout global history. Spread by the bacterium Yersinia pestis, commonly found in flea populations, this disease has recurred for millenia. The bubonic plague is considered a zoonosis, meaning it is a disease that can be transmitted from animals to humans. The first well documented epidemic was “Great Plague of Justinian”, occurring in the midsixth century this outbreak decimated Europe and Western Asia, spreading from Constantinople to what is now the United Kingdom. Likely the most wellknown instance of the disease is the Black Death. Taking place in Europe during the 14th century this pandemic was responsible for the death of nearly half the population of Europe and lasted for over 100 years. A more contemporary outbreak occurred over large portions of Asia in the late 19th and early 20th centuries, this epidemic persisted in India for about 50 years, claiming over 10 million lives before eventually arriving in San Francisco via stowaways on shipping routes. These devastating outbreaks, and the knowledge that disease persists at a low level today show the need to have accurate and available models to determine not just the spread of the disease but, more importantly, the risks to humans and the ways the disease can outbreak into human populations. With the disease models readily available the persistence of Yersinia pestis can be modeled, but to gain a high degree of accuracy in relation to historically seen attributes, multiple models must be extended and combined. The SIR model originated with the KermackMcKendrick model put forth in the early twentieth century as continuous compartment model with susceptible, infective, and removed classes. This model provides the framework for the expanded system derived by Keeling and Gilligan, where they use the typical classes for both rat and human populations, as well creating two classes for flea populations: average number of fleas living on an individual rat, and number of fleas searching for a host – a major driving force for the disease. As aforementioned, the termination of rats drives the fleas to find new hosts, often ending up on humans or pets, thereby transmitting the disease to a new susceptible population. Biosystem StudiedThe system studied was the spread of the bubonic plague from fleas and rats to humans. This was done through looking at fleas, rats, and humans in terms of their state of infection and their population. Mathematics UsedMathematics used include differential equations that show the number of susceptible, infectious, and recovered number of humans in a population, the number of susceptible, infectious, and recovered number of rats in a population, and the number of fleas on a rat and fleas searching for a new host. Flea population models were added to gain realism to the model. Parameters on these equations are used to show how the overall number of fleas searching for hosts and susceptible rats can cause a human outbreak of the Bubonic plague. Type of ModelThe model used in this study is an SIRlike model. This is a "suseptible, infectious, recovered" model, and measures these values for humans and rats. This is a stochastic model. A deterministic model is also created to link the humans, rats, and fleas. The ModelKeeling and Gilligan developed an SIR like model to describe the disease dynamics within the human population, within the rat population and within the flea population. The results of their development are shown below:
Brief Description of the terms in the model
First equation  Susceptible Humans:
Second Equation  Infective Humans:
Third Equation  Recovered Humans:
First Equation  Susceptible Rats:
Second Equation  Infective Rats:
Third Equation  Recovered Rats:
First Equation  Average Number of Fleas on a Rat:
Second Equation  Infective Fleas Searching for a Host:
Keeling and Gilligan noted that the human population dynamics do not have an affect on the other two populations so they instead focused on the number of fleas who failed to find a host in conjunction with the total rat population. They argued that these free, searching fleas could feed on and subsequently infect a human host, or pass the disease to other animals cohabiting with humans. They then derive the value for the potential force of infection (it is apparent that the true force of infection would be less because not all fleas will actually infect a human). This term is given below: Determining Parameter ValuesAccording to Keeling and Gilligan most of the parameters were taken from their literature, experiments and observations. The parameters that they did not have information on were chosen to be within what they called, “biologically reasonable bounds”. These values are listed in the following table:
Sensitivity of Parameters Keeling and Gilligan then carried out a sensitivity analysis on these parameters. Namely they tracked the change in three model outputs as the parameter values were varied. These three outputs were the equilibrium number of rats, the period of the first epizootic cycle, and the potential force for that first cycle. This measure S was defined as follows:
Deterministic ResultsThe motivation for developing this model was to determine how the rat and flea population could lead to an outbreak of the plague in the human population, and the result is summarized in the following plot:
We can see that the force of infection for a given number of free fleas changes exponentially as we vary the number of rats, and that for a given number of rats the force of infection changes linearly as we change the number of free fleas. We can also observe that the force of infection is greatest when there are a lot of free searching fleas, but only a handful of rat hosts. This result is logical, as a large searching flea population with few rats to find as hosts have a greater potential to find human hosts. The force of infection and total number of infective rats were plotted as a function of time by Keeling and Gilligan. This plot is analogous to findings from the computational analysis of the purported model, where the force of infection was found and plotted as a function of time. See the model below to the left, and the found results below to the right (note :both plots are logarithmic in the yaxis).
Keeling and Gilligan noticed that the force of infection lagged behind the number of rat cases by four weeks, and this agreed with the literature that they had been referencing. They also noticed that the human epidemics were damped much faster than the rodent epidemics, and this means that for there to be human cases there must have been a large epidemic in the rodent population. The deterministic model can accurately describe the short term behavior of the plague, but as can be seen in the plot we only observe fixed point behavior. This is an obvious flaw because throughout history there have been seemingly spontaneous outbreaks of the plague, and this model would never predict such outbreaks. This motivated Keeling and Gilligan to seek another description that would allow eruptions of the disease after the disease becomes endemic in the rat population.
Student Evaluation of the model To evaluate this model a set of initial conditions needed to be determined. In order to gain relevance and realism a scaled version of populations from Boulder County, Colorado were used. Taking population data from the 2014 U.S. Census Bureau's population estimates and using diseasecarrying rodent densities of North America from a study by Gary Witmer and Gilbert Proulx approximations for local human and rodent populations were formulated and scaled. Given a rodent density of 300 per hectare and the population estimates for Boulder the initial population sizes become 313,000 humans, and 1,880,000 rodents. Based on flea to rat ratios taken from Keeling and Gilligan a reasonable estimate of 5,000,000 free fleas was used. These numbers were then reduced by a factor of 1000, giving 313 humans, 1880 rats, and 5000 fleas. A scenario was chosen in which only one of the scaled "humans" would carry the disease, and 20% of the rat population was infective initially. These initial conditions can be seen summarized in the following table:
Greater Biological RealismStandard assumptions have historically been that plague outbreaks were triggered by a specific event, be that infected rats transferred via ships, or an infected human entering a local population, which could accurately be shown by the deterministic model described above. The following model puts forth an alternative scenario, one which has been seen in past outbreaks, where the bubonic plague can erupt randomly. Keeling and Gilligan hypothesize that outbreaks such as these arise from movements in the rat population, where the disease is endemic. To gain increased realism the authors introduce three forms of heterogeneity: temporal forcing, individuality, and spatially distinct subpopulations. The latter of these allows the total population to be discretized and coupled via random movements of rats and fleas. The stochasticity of the model now allows for random fluctuations in the system further explaining much of the historically seen behaviors of the bubonic plague. One important result from this extension is that outbreaks of the plague do not necessarily need to arise from a specific triggering event as previously assumed. In larger populations the disease can become endemic in the rat population, posing little to no threat to humans, but following the right series of random events, the force of infection could rise and another wave of human cases could erupt. Namely, following a surge in the susceptible rat population the disease has the capability of jumping between species and posing serious health risks to the human population, which has been seen historically, but is not effectively shown by the deterministic model. Analysis/InterpretationThe overall purpose of Keeling and Gilligan’s work was to provide an accurate model in order to not only qualify previous episodes of bubonic plague outbreak but to evaluate risk for potential outbreaks in current human populations. Through sensitivity analysis and the introduction of stochasticity into the model one parameter combination was determined to be of supreme importance when assessing the risk to human health from bubonic plague, this is the term aKr where a is the searching efficiency of fleas, and Kr is the rat (or more generally: rodent) carrying capacity. Since the searching efficiency and the carrying capacity should move inversely to one another as the size of a subpopulation changes their product, aKr will remain (close to) constant. What this leads to, interpretively, is a scenario where a large population of susceptible rats in a subpopulation can become infective rapidly, where the mass expiration of rats in the population leaves infective fleas to search for new hosts thereby transmitting the disease to new populations. Once spread to human populations the disease moves through the population via the differential equations shown above with a force determined by the total number of rats in the system, and the number of free infected fleas. As shown in the stochastic model this can arise from random fluctuations in the endemic state of bubonic plague among the rat population, and does not need to be driven by an outside force. This is a significant improvement to the deterministic model, as the introduction of stochasticity shows behaviors seen historically but not shown by the original model, namely the spontaneous eruption of disease in isolated populations of susceptible humans. Subsequent ResearchA 2017 study used a modified version of this model to simulate the 1801 bubonic plague outbreak in Cairo, Egypt. This study (X. Didelot, L. K. Whittles, I. Hall) modified the model to achieve three goals: to be able to analyze a shorter time scale, to avoid parameters that with high uncertainty, and to balance the zoonotic vs interhuman infections. This model looks at the plague outbreak in human and in rats, eliminating fleas from the equations. The rate of humans infecting humans was added to the model. With a short time scale, the normal birth and death rates of humans were considered equal. There are 6 ordinary differential equations that define their model. When running the stochastic simulation, rat populations were not historically accurate. To avoid needing exact rat populations, the study used proportions of the rat class (susceptible, infectious, dead and infectious) in the simulation. A large rat population caused the model to act deterministically. References[1] Keeling, M. J., and C. A. Gilligan. "Bubonic Plague: A Metapopulation Model of a Zoonosis." Proceedings of the Royal Society B: Biological Sciences 267.1458 (2000): 2219230. Web. [2] Witmer, Gary and Gilbert Proulx. Rodent outbreaks in North America. (2010. Pages 253267 in Grant R. Singleton, Steve R. Belmain, Peter R. Brown, and Bill Hardy, editors. Rodent outbreaks: ecology and impacts. Los Banos (Philippines): International Ride Research Institute. [3] Annual Estimates of the Resident Population: April 1, 2010 to July 1, 2014 Source: U.S. Census Bureau, Population Division Release Dates: For the United States, regions, divisions, states, and Puerto Rico Commonwealth, December 2014. For counties, municipios, metropolitan statistical areas, micropolitan statistical areas, metropolitan divisions, and combined statistical areas, March 2015. For Cities and Towns (Incorporated Places and Minor Civil Divisions), May 2015. [4]Ligon, B. Lee. 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