Co-Authors: Katie Ericson, Julia Goyco, Renae Rhode, Lauren Smith
Advisor: Emek Kose
Mathematical Modeling - Fall 2019
The Black Death was one of the most devastating diseases to spread through Europe and Asia. In current times, it is treatable; however, in the 1300’s, it was a death sentence. Today, it is known that better hygienic practices could have prevented the plague from being as horrible as it was. In this paper, we examine the effect of hygiene on the plague through agent-based modeling simulations, using historical data to produce as accurate results as possible. We also examine the possibility of the plague being completely eradicated, constantly present, and completely wiping out the population.
The bubonic plague has long since been thought largely eradicated in the 21st century in most developed countries, with on average 1-17 cases appearing in the southwestern United States each year (“Frequently Asked Questions”, 2018). There are very few cases of the plague outside of underdeveloped countries, with the rare instances of appearance in developed countries occurring in those whom have recently returned from travel abroad or in places with high levels of rodents (generally urban or extremely rural environments). This is due in large part to hygienic upkeep, both personally and publicly. In countries where personal hygiene is difficult (due to lack of resources) and public hygiene is not enforced (that is the cleaning of litter and bodily expulsions such as fecal matter), the plague is more likely to be observed in the population. A rise in homelessness in places such as California and the subsequent decrease in the upkeep of public hygiene, has led to an increase in the observance of medieval diseases in cities such as San Francisco (DeVore, 2019). Recently, in 2019 there has been a reemergence of the black plague in China, with 4 cases having been reported (Feng & Cheng, 2019).
During the mid-1300s, the bubonic plague spread throughout Europe, carried by lice and fleas on rats and people. The bubonic plague, also referred to as the Black Death or the Black Plague, is caused by Yersinia pestis bacteria (Ross, D., N.d.). The initial symptoms for this plague are fever, headache, chills, weakness, and growths, called buboes, that can be filled with a puss-like substance (“Frequently Asked Questions”, 2018). Those who contracted the Black Death normally succumbed within 2-6 days. In total, roughly 30-40% of Europe’s population was wiped out within 2 years (Ross, D., N.d.).
In the 1300s, there was no known cure or prevention for the Black Death, which led to the incredibly high mortality rate. Today, we have the medical supplies and knowledge to survive the disease, such as washing our hands or popping the buboes. In medieval times, this knowledge of medicine and hygiene was absent, resulting in the continued spread of the disease. However, as the spread of the disease progressed and more people died, the understanding of hygienic practices developed and Europe was able to survive the Black Death.
The goal of this paper is to observe the Black Death operating under various parameters in order to determine the factors that would most affect the spread of the disease and to develop a model that is historically accurate. Specifically, we are examining the importance of hygiene and its effect on the spread of the disease and to control various factors and perceive their results. We are focusing on the UK (that is, England, Wales, and Scotland ignoring Northern Ireland) in order to have a more isolated population and the factors that we are looking at are: maximum distance covered in one day (individual), the chance of recovery from the plague, and the infectiousness of the plague from one individual to another. A major question we hope to answer is whether or not it was possible for the Black Death to be completely eradicated or if there was a possibility of the plague to be a constant threat, but never strong enough to eliminate a majority of the population.
In our original examination of the Black Death of the mid-1300s, we employed an SIR model. Over the course of developing and testing the original models, we discovered multiple limitations which are expanded upon in the next section. These limitations led to the decision to look into expanding our model through agent-based modeling; which utilizes “autonomous decision-making entities called agents” (Bonabeau, 2002), where these agents act individually but have similar characteristics and behavior. In the model we developed, the agents we examine are humans who have the ability to contract the plague, die from the plague, infect other agents while alive, infect other agents while dead, and recover from the plague.
We had originally planned on incorporating rats as another type of agent in our model, as an alternative carrier to humans. However, further research into the historical data showed that there is doubt in the role of rats – at least in Great Britain – in the Black Plague as a “rodent-related outbreak” (Davis, 1986). The species of rats largely responsible for carrying fleas that spread the disease, rattus rattus, were “rare or absent… where Black Death spread” (Davis, 1986) in the 1300s. This is supported through an extremely limited amount of written evidence of the presence of rats in Europe, in addition to doubt that rats would have spread the disease as slowly as it did, if rats were in fact involved (Davis, 1986). It is more likely that the disease was spread through human-to-human contact (through touching an infected individual or body) and flea-to-human contact (a flea from an infected human or body moves to another host). Therefore, based on this information, we chose to remain focused on humans as our only agent within the model, as both methods of spreading are based on an individual's proximity to another.
The average life expectancy in Great Britain the mid-1300s was around 28 years, averaged from men and women (Mellinger, 2006). In order to limit the movement and scope of individual agents in our model and to manage for population size, we chose only to examine the plague as it was in Great Britain (Cesaretti, et.al, 2016). Shifting our focus from all of Europe, as we examined in our previous SIR model, to Great Britain allowed for us to more accurately represent the spread of the disease within the population. This accuracy is derived from the small amount of travel done by individuals in Great Britain, as opposed to mainland Europe where individuals were more likely to travel. We also assumed that there was little entry from outside the population reproducing, as immigration to Great Britain did not rise until a decade or two after the Black Death had ended in 1351.
Our initial model did not produce a realistic result, as the entirety of the population of Europe died within a few hours of the start of the plague. Our second model produced results more in line with our expectations of the doomsday approach, with the population of Europe dying in a few years (94 weeks). The significant change between the two models was the introduction of a capping variable to reduce the amount of interaction between individuals. This change was due to the realization that our initial model suggested that one person was able to infect (or interact) with every individual within the population.
The sensitivity analysis of our model yielded that the two most sensitive parameters were body-to-human contact and the capping variable, which limited the chance of interaction (one person could not interact and infect all of Europe). An increase in body-to-human contact results in a decrease in the amount of time it takes for the plague to spread. This means that the more infected bodies there are, the more interactions between susceptibles and the infected (both living and dead) will occur and therefore increase the overall rate of infection. When the capping variable decreases, the spread of the disease increases; this is due to the fact that the capping variable reduces the amount of interaction between individuals. If the capping variable decreases, then more people interact and increase the likelihood of a susceptible individual coming into contact with an infected individual.
For the SIR model, we wanted to emphasize simplicity and determine how long, with as few mitigating factors as possible, it would take for Europe to reach the doomsday result with everyone in the population dying. Due to the desire for simplicity, we limited the amount of variables and parameters considered, and only used differential equations to model the scenario.
Some of the variables and parameters we did not include in our SIR model: entry/exit into the population, the transfer of the disease via rats/lice, travel and semi-isolated populations, hygiene, and recovery of infected individuals. A few of the parameters mentioned would be incredibly complex if considered in a differential equations model, which is why we decided to utilize agent-based modeling using NetLogo to create a more accurate and realistic model to demonstrate the spread of the plague in Europe.
The purpose of the model is to create both a visual and a data-driven representation of the Black Death on the isle of Great Britain during the years 1347 and 1351, with the major years of the plague between 1348 and 1350. Our model will be based off of what we completed for our SIR model, but will become more realistic using the agent-based modeling approach. We believe this model is important for a couple of reasons. The first, being that there is a lot of pre-existing data, so it is a good project for novice modelers, and the second is that there has been a recent resurgence of illnesses from the middle ages, including the bubonic plague. It is important to understand how these illnesses spread, and what could hinder an epidemic event.
This brings us to the specific questions we will be attempting to answer: 1) Does our model accurately match the actual spread of the black plague during the middle ages using our base parameters; 2) How do hygiene levels affect the spread of the plague; and 3) Is there a specific collection of parameters that will not lead to an epidemic, but will keep the plague from being eradicated entirely? The ideal model should follow the same patterns as the Black Death did in the mid-1300s. We plan to adjust the sliders in our model to explore different possibilities, but, as listed in subsection 4.5., there will be an ideal parameter values for the most realistic scenario. Due to the amount of data regarding the plague that is available, there should be minimal issue in confirming that our results are fairly accurate.
For our agent-based model, have various entities that will be represented in the system. The entities include the agents in the model and the patches they travel on. The agents in the models are humans which have four possible phases of life that we represent by different colors. Healthy, non-infected humans will be green in color, infected humans will be red, dead humans will be yellow and immune humans will be grey. Dead humans still have the ability to infect humans, but they have no movement (unlike the other human agents). Both healthy, non-infected humans and infected humans will move randomly on the map.
Based on the chance of infection in the model, when healthy agents encounter either infected humans or dead humans they will become infected with the bubonic plague. At each tick the agents will move randomly among the patches. The patches in the model have no characteristics as they are all identical.
The state variables in this model are the infection status (healthy, infected, dead or immune), how much time of immunity they have remaining, how much time dead (on the map) they have remaining, infection time (how long an agent has been infected) and age (agents have a life expectancy of 28 years).
At each step, the agents move randomly and the status of each human’s infection can updated. If a healthy agent (green) becomes infected at a step by encountering an infected agent (red) or a dead agent (yellow), their color will be updated to red (i.e. the agent will become an infected agent). If an infected agent reaches the duration of the disease (without recovering) their color will be updated to yellow (representing a dead agent). The dead agent will no longer move as the steps progress. An infected agent has a chance of recovery so at a given step they may recover and become an immune agent (grey). A dead agent will remain on the same patch at each step, and will disappear after 10 weeks or remain on the map permanently if dead-remain-infectious is switched on.
The basic principles behind the model are those of the spread of the disease: that there are those who are susceptible, infected, deceased, and recovered (temporary before returning to susceptible population). When susceptible individuals interact with infected agents and deceased agents, the agent becomes infected. Recovered agents are immune for a period of time before they return to the susceptible population. These behaviors are incorporated at the submodel level. The model initializes at a scaled level, using real-world parameters and utilizes previously established dynamics. As the model does not include any adaptive or learning behaviors for the agents within the model, there are no emergent results to understand and interpret and agents do not develop any predictions.
Agents interact with one another at random, but are limited in the size and scope of their movement (they cannot move over the entire map) and therefore do not interact with all other agents in the model. The agents do form groups and act in a collective manner, depending on the parameters at certain levels of “chance-recover.”
The data observed from the model is that of the distribution amongst the population of those whom are susceptible, infected, deceased, or recovered in order to determine the various scenarios resulting from changing parameter values. This is to obtain an idea of what is realistic (that is, closely resembles the actual Black Death in the 1300s) and what might occur were any of the parameter values, such as infectiousness, to change.
Many of the initial parameters were taken from our original SIR model; however, we refined our project to be centered around Great Britain (England, Scotland, and Wales) instead of all of Europe. This changed the target population from 85 million to around 6 million, with a projected 2 million casualties (Ross, n.d.) in the two years of the plague. For our model, we scaled this down to 600 individuals, as 6 million would be too many to fit on the map.
The duration of the disease (from initial infection of an individual to either their death or recovery) was at most 10 days, as already determined from our previous paper. If an individual has recovered from the plague, they were immune for a period of 14 weeks (Pfeifer & et.al, 2008), before they returned to the susceptible population. This is also built-in to the model. The way our model is constructed, the initial number of sick individuals is determined at random out of 50 of the initial 600 agents in the population. This randomization and uncertainty is one of the reasons it is important to run the model multiple times, in order to ensure that it is accurate.
The other initial values are meant to be changed by the observer while experimenting with the model. This includes the infectiousness, chance-recover, and dead-remain-infectious variables. The infectiousness parameter is initially set at 99%, however it was interesting to alter to see what the effects would be if lowered to certain values. While researching, we did not find a specific parameter value to initialize the model, as multiple sources included anything from 60-100% or only mentioned that the plague was “highly infectious.” Therefore we determined that the initial value of infectiousness should be set at 99%. The chance-recover parameter actually includes several factors. Hygiene, the disease’s natural recovery chance, and immune systems at the time were all included (Keeling & Gilligan 2000). The most realistic number we found for an individual’s chance at recovery was 10%. We believed the dead-remain-infectious parameter should have a switch to turn it on or off in order to test multiple scenarios. We found that if this was variable turned on, and the dead were able to infect the living indefinitely, this would possibly skew our results (which is expanded upon in the results section). When turned off, we had the dead remain infectious for a period of 10 weeks (Raloff, 2015).
There is no input data for this model.
The set up section of the code creates the environment. The agents are created, the initial conditions are set up, the agents are set in the environment, and the ticks reset to zero. The set up turtles section assigns the agents to a random location in the environment and gives them conditions, like random lifespan, immunity, and healthiness. It randomly assigns the illness to certain agents.
The get-sick section assigns conditions for when agents get sick, like immunity or death. The get-healthy section assigns health to the agent, saying they haven’t died so they keep moving. The become-immune section gives temporary immunity to those agents that haven’t died of the plague; so they continue to move in the environment without being susceptible for a certain amount of time.
To become-dead means the agent died of the plague. This means the agent stops moving. They could have died from old age or of the plague. The ones that died of the plague remain infectious, but all agents that died disappear after 4 weeks.
The setup-constraints gives the exact variables. It names the lifespan, names the maximum amount of people in the environment, length of immunity, reproduction possibility, and the disappearance of dead bodies. The go section tells the agents start interacting and moving around their environment. If they die they can still infect other agents. Each movement is determined by the ticks.
The update-global variables counts the percentages of agents that are sick, immune, and dead. The update-display creates a chart that maps the numbers of infected, healthy but susceptible, dead, or immune. The get-older section sets the conditions for aging for the agents. When it hits the maximum age, the agents die but aren’t infectious. Dead agents don’t age, but disappear. The move sections tells the agents to move randomly. The to infect section tells the agents that are susceptible that if they come into contact with an infected (sick) agent, they also become sick.
The recover-or-die section gives the random chance of a sick agent recovering and become healthy and susceptible. If the agents do not recover, they die from the plague. The reproduce section tells the agents to reproduce if the number of agents hit below the carrying capacity. The to-report immune sets up the immunity and the startup section sets up the constraints.
Realistically, the plague lasted for roughly 4 years in Great Britain, with the majority of the deaths and infection occurring in a period of one to two years (DeWitte & Kowaleski, 2017). During the time of the plague, it is believed that about 2 million of England’s 6 million population died (Ross, n.d.). Since our model was scaled down to 600 agents, if we have around 200 deaths in that two year period that would match historical records. For our initial testing, we used the parameters that are listed in section 4.5 (ODD-Initialization). For each one of the questions we sought to answer, we tested the model at least three times to determine the patterns of the model and to ensure that there were no outliers. The results of our most realistic model produced the following graph:
Figure 1: Historical Model
Multiple tests of the model produced similar results to the one above. The plague death count can be confusing, because the bodies remain infectious for a period of time, and are thus recounted, so we will be referring to the max plague deaths. In this model, the plague was completely eradicated in a little over a year, and deaths due to the plague were slightly less than 300. In comparison to the data that we found while researching, this is accurate. Notice also that the plague deaths have a positively skewed distribution, whereas the number of sick have a small spike, then slowly decrease. This is not something we expected to observe in the model and was very interesting to see.
Next, we wanted to test if increasing hygiene would lead to the plague being eradicated more quickly. To do this, we simply increased the chance-recover parameter to 50% from its initial 10%. We kept the other parameters at the same values as their initializations as a control. The results are displayed below:
Figure 2: Increased chance-recover Model
The results were contrary to our original expectations that the plague would be eradicated in a shorter amount of time. Instead, though the number of plague deaths did not peak as high as it had previously, it remained in the population for over 50 years. This particular image shows roughly 60 years, though we ran other simulations for longer periods of time (some over 100 years) and the plague never disappeared from the population. This may be due to the fact that with more of the population dead, this meant that people were less likely to become infected and could in part avoid the dead or sick. However, when there is a larger population, individuals are more likely to have a chance at encountering the sick or dead. Watching the simulation, groups of sick and infected emerged in the population. Interestingly, those groups would travel to different areas of the population and not stay in the same space, however it would always affect them in definitive groupings. Lowering the chance of recovery answered our third question: the plague was not eradicated and remained a presence within the population without ever reaching an epidemic state (when a disease spreads rapidly through a population and many people are infected at the same time).
After answering our initial questions, we wanted to observe how decreasing infectiousness would affect the model. With all other parameters returned to initialization, the following results were the result of infectiousness decreased to 50% from 99%:
Figure 3: Decreasing infectiousness Model
This parameter change provided very similar results to our initial model, however, there were some key differences. Every agent state in this model was, in general, more gradual than in the initial model with a noticeable difference being that the number of plague deaths was not as steep of a bell curve as the first model, and it never crossed over to be higher than the healthy curve. The plague death curve also ended up being more right-skewed than before and took longer to disappear entirely.
With our initial questions answered, another questions arose as to what would happen if the dead constantly remained infectious and were never removed from the population given that all other parameters were returned to initialization. The results are below:
Figure 4: Dead remain infectious Model
Every time the simulation was run, it produced the same results: the doomsday equilibrium (everyone in the population dies) no matter how many times the simulation was run. It also always took about 1.8 or roughly 2 years. Though this is not a realistic scenario, it does demonstrate the importance of avoiding contact with those infected with the plague, but also those who have died of the plague. It is interesting to note that, given these parameters and the results generated, this model most closely resembles the original SIR mode.
The infectiousness and chance-recover parameters were the most sensitive of the three adjustable parameters. As the infectiousness parameter adjusts the chance of infection, by one agent to another, a lower value extends the length of the plague existing in the population. This is due to there being more healthy agents for a longer period of time, therefore, the disease lasts longer. If the infectiousness parameter is increased, then the chance of infection rises and more agents will become infected. This increases the number of infected and decreases the length of the plague in the population. Changes in the chance-recover parameter extend the presence of the plague in the population for even longer than the infectiousness parameter, with the plague lasting for half a century. The duration parameter is relatively sensitive, as the longer an agent is ill, the more opportunities the agent has to pass the illness to another agent. The less time an agent is ill, the less opportunity there is for the disease to spread.
Overall, we observed that changes in certain parameters have a noticeable effect on the outcome of the model. The first question we asked was whether or not it was possible to match our model to historical data. After running the model, the results were extremely close to the data that we found, though the total death counts were different due in part to scaling. The second question asked whether hygiene influence plague severity, whether it impacted the population and duration of the disease. The answer to this question was determined by adjusting the chance of recovery to a higher value, in order to illustrate the difference between knowledge of hygiene and the initial lack of knowledge of hygienic practices in medieval Great Britain. The results were different to our original expectations: instead of the plague disappearing completely in a shorter amount of time, there were a fewer amount of victims, but the plague was still present in the population for over 50 years. In some instances, the plague lasted for over a century in a small portion of the population. Our third and final question was about the complete eradication of the plague and if the plague would remain a constant, but negligible, force within the population. As seen in the results, the plague eventually ends up completely eradicated. The exception to this was when we increased the chance-recover parameter, with the other parameters remaining at the initialization values (mirroring the historical values).
In addition to our original questions, we observed what would happen if dead bodies were to remain infectious (and in the population) forever. This created a doomsday equilibrium in which everyone died, mimicking the linear results of our first SIR model. In addition to the doomsday scenario, the results highlighted the sensitivity of the infectiousness and other parameters.
All models that were generated emphasized the importance of learning – and adapting – during times of an epidemic. The population gaining knowledge of hygiene and disposing of infected bodies, meant that the disease decreased and eventually died out. This reflects what actually happened during the Black Death, as people learned how to handle those who were sick and those who had died. Without these adaptations, a worse outcome than was actually observed in history would have been possible: including the complete disappearance of Great Britain’s population.
There are a multitude of ways that this model could be expanded. Agents in the model are limited in their movement, however during the time period, there were a few individuals who traveled great distances. This could be incorporated by including a variable which randomly generates agents in the population which travel outside the normal limitations of the agents.
Another means of making the model more realistic would be incorporating a separate hygienic variable, not included in the chance-recovery parameter. This variable could either be a randomly generated constant, that assigns each agent has a random level of hygiene, or a constant assigned to all agents that is controlled by the observer through a slider. As the disease is spread through human-to-human contact (direct contact or fleas), another possible method would be for hygiene to be an adaptive trait: agents learn over time to increase their levels of hygiene.
Given additional time, we would include adaptive behaviors, susceptible agents could learn to sense infected and deceased individuals and avoid them in order to increase their likelihood of not becoming infected themselves.
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