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The Mathematical Modelling of Bubonic Plague Robert Williams Collingwood College

Department of Mathematics University of Durham

Submitted for the Degree of Master of Mathematics 路 April 2009 路

Acknowledgements First and foremost, my sincere thanks go to my supervisors Dr James Blowey and Professor Brian Straughan, for their assistance with all matters mathematical and computational. The project has required a large amount of historical research, and I am immensely grateful to Dr Ben Dodds of the University of Durham and Thomas Hughes, both of whom were invaluable in directing me through the sea of material available and in convincing me that the subject matter was worthwhile. Emma Blamey, of the Tropical Disease Centre at the University of Liverpool, kindly spent a considerable period of time correcting my GCSE knowledge of pandemics, and pointing me in the direction of several useful references. My parents, Brian and Jacqueline Williams, spent a huge amount of time going through the final draft, and I thank them for their guidance and comments, even if I feel that our views on the usage of the semicolon are now irrevocably different. Michelle Melvin and my housemates Adam, Kieran and Marcus have dealt with my year-long obsession with all things rat with sympathy and understanding, and I think we can now put the issue of supersonic rats to bed for the last time. Finally, and despite the best efforts of those mentioned above, any remaining mistakes within the text, be they grammatical, mathematical, historical or biological, are mine and mine alone.


Robert Williams

Between October 1347 and January 1351, the Black Death spread over 4000km from Sicily to Norway and killed upwards of 70 million individuals, affecting all strata of the population from peasants to royalty and thwarting any attempts to halt its progress. No conclusive proof has been found as to the nature of the disease, and the standard interpretation of a pandemic of bubonic plague (Yersinia pestis) has come under attack primarily due to the rapid spread and high mortality not being thought feasible, given the primary vector of the infection is the slow moving black rat. Using an original differential equation model similar to Fisher’s Equation, the spread of a colony of healthy rats via travelling waves was studied analytically, before adapting the equation to allow for infection using a standard incidence form of transferral and an SI framework. Numerical simulations carried out on the diseased model suggested that bubonic plague would be unable to spread significantly faster than one kilometre a year within a population of rats. This closely matches data gathered during the Third Pandemic of bubonic plague between 1900 and 1950, but is markedly different to the several hundred miles annually recorded during the Black Death, adding further evidence to the argument against bubonic plague as the probable culprit. However, the author believes that there is a strong possibility that human parasites may be capable of sustaining an epidemic without the presence of black rats, and this theory is discussed in detail in the final chapter.

Contents 1 A Question to be Answered?


2 The Beginnings of a Second Pandemic 2.1


The Black Death in Europe . . . . . . . . . . . . . . . . . . . . .



Arrival and Spread in Europe . . . . . . . . . . . . . . . .



The Nature of the Black Death Infection . . . . . . . . . .



The Black Death as a Pandemic of Bubonic Plague . . . . . . . .



The Intentions of this Project . . . . . . . . . . . . . . . . . . . .


3 The Construction of a Differential Equation Model 3.1


Biology and Epidemiology . . . . . . . . . . . . . . . . . . . . . .



The Black Rat; Rattus rattus . . . . . . . . . . . . . . . .



The Oriental Rat Flea; Xenopsylla cheopis

. . . . . . . .



Bubonic Plague; Yersina pestis . . . . . . . . . . . . . . .



Other Species . . . . . . . . . . . . . . . . . . . . . . . . .


The Defining Characteristics and Assumptions Leading to a Model 26

4 The Analysis of a Model for a Healthy Rat Population 4.1




Fisher’s Model and a Transformation of Variables . . . . . . . . .



Introduction to Fisher’s Equation . . . . . . . . . . . . . .



Transformation of Variables of (3.1) . . . . . . . . . . . .


The Analytical Solution of Fisher’s Equation . . . . . . . . . . .




Finding a Travelling Wave Solution . . . . . . . . . . . . .


4.2.2 4.3


The Occurrence of a Travelling Wave . . . . . . . . . . . .


Finding Variables for the Healthy Model . . . . . . . . . . . . . .



Biologically Derived Parameters . . . . . . . . . . . . . .



Numerical Simulations of the Healthy Model . . . . . . .


The Diffusion Paradox . . . . . . . . . . . . . . . . . . . . . . . .




The Hyperbolic Fisher’s Equation . . . . . . . . . . . . .

5 Modelling Diseased Rats



The Diseased Model . . . . . . . . . . . . . . . . . . . . . . . . .



Numerical Simulations of the Diseased Model . . . . . . . . . . .



Summary of Findings from the Diseased Model . . . . . . . . . .


6 Conclusions and Further Work



Initial Conclusions and Limitations of Mathematical Models . . .



The Role of Human Parasites . . . . . . . . . . . . . . . . . . . .



Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . .


Appendix: A Introduction to the Mathematical Modelling of Epidemics


A.1 An Introduction to Mathematical modelling of Disease Transmission . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .


A.1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . .


A.1.2 Mathematical Epidemiology: An Overview . . . . . . . .


A.1.3 Terminology and Notation . . . . . . . . . . . . . . . . . .


A.2 Six Standard Epidemic Models . . . . . . . . . . . . . . . . . . .


A.2.1 The Simple Epidemic (SI) Model . . . . . . . . . . . . . .


A.2.2 The SIR Model . . . . . . . . . . . . . . . . . . . . . . . .


A.2.3 The SIRS Model . . . . . . . . . . . . . . . . . . . . . . .


A.2.4 The SIVS Model . . . . . . . . . . . . . . . . . . . . . . .


A.2.5 The SEIR Model . . . . . . . . . . . . . . . . . . . . . . .


A.2.6 The MSEIR Model . . . . . . . . . . . . . . . . . . . . . .



A.3 Additional Modelling Factors . . . . . . . . . . . . . . . . . . . .


B The Keeling and Gilligan Model for Bubonic Plague








List of Figures 2.1

The Spread of Black Death throughout Europe between 1347 and 1351 [1] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .


Triumph of Death by Pieter Bruegel the Elder. The skeletal figure of Death gallops through the dying population . . . . . . . . . .




SIDR Model. This could be used to model the temporarily infectious nature of a rat carcass, as its infectious fleas leave its body in search of a new host. . . . . . . . . . . . . . . . . . . . . . . .


Phase portraits for the system (4.14), drawn using John Polking’s ‘PPLANE 2005.10’ [2].


. . . . . . . . . . . . . . . . . . . . . . .


Sketch of travelling waves generated by Fisher’s Equation, redrawn from solution by Towers [3] . . . . . . . . . . . . . . . . .




Numerical simulation of a healthy rat population diffusing through space. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .



Snapshot of healthy rat diffusion, taken at t = 20.



Initial and boundary conditions for simulations of the diseased model.


. . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .


Numerical simulations for the diseased model. The first shows the profile of the wavefront at t = 52, while the second shows the spread of the wave for t between 0 and 208. Note the vertical axes, which have been resized.


. . . . . . . . . . . . . . . . . . .


A.1 Graphical representation of classes and transferral in basic MSEIR model [4].

. . . . . . . . . . . . . . . . . . . . . . . . . .


A.2 Susceptible and infectious populations varying with time, redrawn from Jan Medlock’s Mathematical Modelling of Epidemics [5]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .


A.3 Phase portrait for SIR model without vital statistics [4]. . . . . .


A.4 Bifurcation diagram for equilibrium points in SIR model with vital dynamics [4] . . . . . . . . . . . . . . . . . . . . . . . . . . .



Chapter 1

A Question to be Answered? The Black Death swept across the globe between 1300 and 1700, killing up to a third of the world’s population and permanently changing the structure of society. Despite the tradional assumption that a pandemic of the bacterial infection bubonic plague was to blame, growing evidence suggests that this could not be possible. This raises the question; if bubonic plague was not to blame, what was? Various other theories have been put forwards, including typhus, anthrax, or some airborne form of viral haemorrhagic fever (VHF). The major concern is that, although bubonic plague can be treated with prompt and aggressive use of antibiotics and fluids, techniques for tackling viral infections are still very much in their infancy. Treatment for the common cold has not progressed in hundreds of years, influenza still requires annual immunisation of vulnerable individuals to combat the ever-changing strains of the virus, and Ebola, arguably the most studied form of VHF, maintains a 95% death rate in those it infects. More worryingly, no convincing hypothesis has been put forward for the gradual disappearance of The Black Death, raising the possibility that it may return and thus increasing the importance of the cross-disciplinary


search for evidence on the clinical nature of the infection. This project seeks to briefly summarise current theories from biological, historical and mathematical perspectives, before moving on to investigate the role of differential equations in modelling the spatial spread of the infection. It is divided into the following sections: • Chapter 1 provides an introduction to the Black Death, including a basic history of the infection, some comments regarding the clinical nature of bubonic plague and a summary of the main arguments for and against bubonic plague as the cause of the Black Death. This section also summarises the previous mathematical work in this area, before moving on to discuss the information required to build a new mathematical model. • Chapter 2 contains a more in-depth analysis into the various factors within a bubonic plague epidemic, including the bacterium itself, humans, rodents and fleas. A justification is given for the sole consideration of rat infection dynamics as a measure of the spatial spread of bubonic plague, and an original model is created for a healthy population of rats. • Chapter 3 analyses analytically the healthy model from the previous chapter, with an emphasis on finding a travelling wave solution for the spread of a population of rats from a certain point. Various parameters are determined for the healthy model, and numerical simulations are carried out to check biological consistency. Finally, an outline of a hyperbolic healthy model and its travelling wave solution is given, but not taken further within this project due to the increased complexity. • Chapter 4 begins by adjusting the healthy to incorporate the spread of infection within the rat population, and than various numerical simulations are carried out to test the effect of various parameter adjustments. The results are summarised, and conclusions drawn, including suggestions for the direction of possible future work.


• Chapter 5 introduces the modifications to the healthy model to allow for the simulation of bubonic plague infection. Several numerical simulations are discussed, and a brief discussion of the robustness of the model to parameter variation is included. • Chapter 6 draws conclusions from all parts of the project, and details possible avenues for further work. • Appendix A consists of a brief summary of various methods of modelling infection using differential equations, including an in-depth look at the Susceptible-Infectious-Removed (SIR) model. • Appendix B discusses the Keeling and Gilligan model which is introduced in Chapter 2.



Chapter 2

The Beginnings of a Second Pandemic Alas! our ships enter the port, but of a thousand sailors hardly ten are spared. We reach our homes; our kindred . . . come from all parts to visit us. Woe to us for we cast at them the darts of death! Going back to their homes, they in turn soon infected their whole families, who in three days succumbed, and were buried in one common grave. Priests and doctors visiting . . . from their duties ill, and soon were . . . dead. O death! cruel, bitter, impious death! . . . Lamenting our misery, we feared to fly, yet we dared not remain . . . Gabriele de’Mussi, 1348

2.1 2.1.1

The Black Death in Europe Arrival and Spread in Europe

The Great Mortality that later became known as the Black Death is thought to have been endemic within Central Asia for a number of years, before spreading rapidly across the majority of Asian civilisations to start the pandemic. AbÂŻ u 11

al-Ward¯i, writing in 1348, describes the plague as being current in the ‘land of darkness’ for fifteen years, before sweeping across China, India, Uzbekistan and Transoxiana1 , then reaching the Persian Empire and spreading to the Crimea, Cyprus and ‘the islands’ [6]. In all places, the mortality was devastating, described as a storm; a scourge; sitting ‘like a king on a throne...killing daily one thousand or more and decimating the population’ [6] and leaving ‘of seventy men only seven’. It is usually assumed that the pandemic first reached Europe at Caffa on the Black Sea, carried by the invading Mongol army of Jani Beg2 . The Italian city-states, in particular Genoa and Venice, had strong trading connections with many settlements,


Caffa, along the Crimean coast; bringing back grain, slaves and fur for trade.

It is assumed

that merchants, possibly fleeing the Mongol advance, carried the infection into Western Europe by ‘galleys horribly infected and heavily laden with

Figure 2.1: The Spread of Black Death throughout Europe between 1347 and 1351 [1]

spices’ [7], although the number and destination of the galleys and the health of their crews is uncertain. Louis Sanctus, writing from the Papal court at Avignon, mentions three ships, with crews which survived long enough to berth at Genoa having been ‘forcibly expelled from eastern parts’, before being driven from port to port, spreading infection as they went and finally docking at Marseille. The Franciscan friar Michael da Piazza (writing a 1 Ancient name for an area of central Asia covered by modern Uzbekistan, Tajikistan and southwest Kazakhstan, and generally taken to be the area between the Amu Darya and Syr Darya rivers. 2 Jani Beg is alleged to have ordered that his plague-sticken army catapult the corpses of infected soldiers over the walls of Caffa in an attempt to infect the inhabitants.


decade after the event) claims ‘twelve Genoese galleys...entered the harbour of Messina’, whereas Giovanni Villani, writing in 1348, suggests that the myriad of vessels travelling across the Mediterranean were equally to blame for the spread, citing only as an example ‘of eight Genoese galleys that were stationed in the Black Sea...only four of them returned, full of the sick and the dying’. Regardless of the culprit, plague appeared in the spring of 1347 in Constantinople [8], on the banks of the Bosphorus, through which which all Italian shipping from the Black Sea had to pass. By late summer, plague had reached large parts of Turkey and southern and central Greece, accelerated by both land and sea trade and Greek nationals fleeing the pandemic [9]. Deaths were recorded in Messina, Sicily in the beginning of October 1347, although it is likely that deaths had been occurring unrecorded since the second week of August, as this better fits in with subsequent spread. As people fled Sicily, the infection reached Sardinia, Corsica and the island of Elba. By the end of 1347, infection had crossed the narrow Strait of Messina and was present in the south of mainland Italy, and deaths were being reported in Genoa. By mid-January 1348, Black Death had crossed into Pisa on the Tuscan coast, which served as the main port for Florence. Presence in Venice was noted on 25 January 1348. Somewhat surprisingly, Marseille recorded plague-related deaths on 1 November 1347, seemingly out of sequence with the current spread. However, this correlates with the comment from Sanctus mentioned above, and also with the fact that plague is recorded in Aix-en-Provence, 25km to the north, around a month later. From Marseille, plague-bearing ships landed in Mallorca and Roussillon, subsequently followed by a spree of outbreaks on the Iberian Peninsula, with the first recorded death on Spanish soil occuring at the end of March [9]. Throughout May, deaths were occurring across most of the eastern Spanish mainland, and the Bishop of Cadiz succumbed to plague on the 1 June. By this point most of northern and central Italy were infected, along with southern France.



The Nature of the Black Death Infection

The previous section clearly shows the speed with which the Black Death could move and the devastation it left in its wake. After arriving in Messina, the pandemic took a mere three years to spread as far as Scandinavia, even more remarkable given that human transport at the time was limited to the speed of a horse or sailing vessel. In terms of effect on society, within the fourteenth century the Black Death is estimated to have killed between 75200 million individuals, or 20-45% of the global population at that time. The dominance of the Roman Catholic Church was massively reduced, and a new culture of ‘living for the moment’ began to take hold. In the absence of any obvious reason for this apparent divine judgement,

the general populace

turned to astrological forces, Godly displeasure and the poiFigure 2.2: Triumph of Death by Pieter Bruegel the Elder. The skeletal figure of Death gallops through the dying population

soning of wells by Jews as possible causes. Strict religious orders blossomed in the gulf left by the Catholic Church, with

fanatics preaching salvation through flogging and mutilation. Minority groups such as lepers, beggars, pilgrims and those with skin diseases were vigorously persecuted and the extermination of the Jewish communities of Mainz and Cologne in August 1349 demonstrated the full brutality of which the terrified populace was capable. Also remarkable about the Black Death is the persistence of the infection. Ignoring the standard scattering of cases per year, between roughly 1300 and 1700 at least one part of the world was suffering on a full epidemic scale, with 14

most areas retaining the Black Death as endemic throughout this period. However, widespread infections generally failed to strike a community in consecutive years, with most outbreaks occuring at five to fifteen year intervals. This trend was successfully reproduced in Keeling and Gilligan’s comprehensive model [10], which suggested that it was possible for bubonic plague to remain endemic in dispersed colonies of rats without affecting the human population, until the percentage of susceptible rats grew to such a level that the infection broke out amongst rats and subsequently grew into a localised human epidemic. As was mentioned in the introduction, no convincing explanation has been given for the gradual disappearance of the Black Death. Although many infections become gradually less deadly to their hosts over time (as a dead host is of no use to an infectious parasite), the fact that bubonic plague still exists in a form which appears similar to that mentioned in contemporary accounts suggests either this decrease in potency was not the case, or that Black Death was not bubonic plague. This question is discussed in greater detail in the following section.


The Black Death as a Pandemic of Bubonic Plague

Despite the vast effect on medieval life, the epidemiological nature of the Black Death is uncertain, to the point where even the disease itself is in question. While the standard view is that the pandemic was one of bubonic plague, or Yersina pestis, relatively recent works have suggested that this may have been a somewhat unjustified judgement made off the back of the successful identification of Y pestis as the primary cause of the Third Pandemic which killed over 12 million people in India and China alone between 1855 and 1959 [11]. There are significant differences between both current medical knowledge on Y pestis and observations made about the Third Pandemic, compared to those recorded during the Black Death. The main issues are summarised below. 15

Lack of unique clinical symptoms for Y pestis The most commonly cited effect of a Black Death infection is swollen lymph nodes, resulting in characteristic ‘buboes’ in the groin, armpit or neck. Termed lymphadenopathy, this is by no means unique to Y pestis infections, and can be caused by diseases ranging from throat infections and severe measles to toxoplasmosis, gonorrhea and tuberculosis [12]. Many contemporary sources also made reference to petechiae; small haemorrhagic spots on the skin referred to as ‘God’s Tokens’ and generally occurring shortly before death. These occur significantly in viral haemorrhagic fevers (VHF) such as Ebola and Marburg Fever, and have led to suggestions, primarily by Scott and Duncan [13], that the Black Death was an outbreak of an unknown VHF capable of droplet transmission3 . Such fevers also tend to cause a breakdown of the capillary blood vessels in the lungs and throat, which could also explain the ‘spitting of blood’ referred to by several sources, formally known as haemoptysis. Other symptoms such as fever, headaches, vomiting, delirium and general malaise can be ascribed to most serious infections, including epidemic typhus. However, the fact that other diseases also fit the recognised symptoms is not a reason to discount Y pestis. Very few known diseases have unique symptoms, and no infection fits the clinical facts quite so well as plague. Bubonic plague does cause particularly marked buboes in almost all cases, which would explain the prevalence in historical sources, and also causes all the other above symptoms, including haemorrhagic ‘tokens’ near the end of infection. The ‘spitting of blood’ mentioned by, amongst others, Micheal da Piazza and Louis Sanctus is always described as a somewhat rarer event, but one that causes the individual to ‘die so suddenly’. Compare this with pneumonic plague, which arises in approximately five per cent of plague infections4 , often causes haemoptysis 3 Droplet transmission refers to the transmission of infection from one individual to another via contaminated mucous, primarily through coughs and sneezes. Most of the most contagious infections known can be transmitted in this manner, such as the common cold, influenza and measles. No known viral haemorrhagic fevers are able to transmit in this manner; the main reason that VHF epidemics have not had a much more devastating effect in Africa and South America. 4 Pneumonic plague is contracted from inhalation of aerosolised infective droplets, gener-


and has a mortality rate approaching 100%. Septicaemic plague, also arising naturally during epidemics of bubonic plague, would help to explain the small percentage of cases with neither buboes nor haemorrhagic fever.

Y Pestis is not sufficiently lethal to have caused the Black Death It has been strongly argued that bubonic plague is not sufficiently lethal to have caused such widespread devastation in the medieval world. This conclusion is primarily based on evidence gathered from the Third Pandemic and in modern laboratories and then compared with contemporary estimates of Black Death mortality rates. For example, the Communicable Disease Centre of the USA states a mortality rate of 50-60% in modern cases of untreated bubonic plague [14], whereas Francesco Petrarch wrote in 1349 of ‘empty houses, derelict cities, ruined estates, fields strewn with cadavers, a horrible and vast solitude encompassing the whole world’ [15]. This is by no means a unique view, with many figures of deaths given for an area which approach (or in some cases, exceed) the estimates of total population of the time. However, it has been well documented that medieval authors frequently used figures of biblical proportions to convey a sense of drama. Add to this the somewhat shaky numeracy of the majority of the populace, it can be understood that a writer may refer to empty streets and piles of corpses, instead of quoting a figure of 200 deaths a day. Benedictow estimates that 60% of the European populace may have died5 , but mentions that ‘a significant...amount was due to secondary catastrophe events’, such as famine, reduced trade, and exposure [9]. Finally, as mentioned above, it should be noted that the mortality of an infection often reduces over long periods of time, simply due to evolution favouring a strain of the disease which allows the host to live slightly longer and thus infect further individuals. ated after bubonic or septicaemic plague spreads into an individual’s lungs. While highly contagious, pneumonic plague rapidly renders an infectious individual bed-bound, and thus tends not to result in a significant percentage of plague deaths in an epidemic. 5 Note that this is a percentage of the total population, rather than a proportion of those infected and, due to the fact that not every single individual in Europe will have been infected, still points to a significantly higher percentage than the CDC’s estimate.


The issue of a vector As will be discussed in the following chapter, Y pestis is primarily transmitted to humans via infected flea bites, and the fleas themselves require rats to host and infect them. Many historians and archeologists have argued that there is insufficient evidence of a sufficiently dense rat population across Europe to support an epidemic, firstly citing a lack of rodent remains in archaeological digs and secondly an inhospitable climate to both the black rat and the oriental rat flea. The climate issue is easily dealt with; the black rat, while preferring warmer temperatures, is quite capable of thriving in much colder conditions. Successful colonies are present on Macquarie Island, halfway between Australia and the Antarctic, where air temperature averages around three degrees Celsius and regularly drops below freezing [16]. The lack of archeological remains is a possible problem, especially in areas such as Iceland which suffered two major epidemics without any evidence of a rat population. However, it should be mentioned that the techniques needed to find rat bones require a very delicate sieving of debris, and as such, evidence could easily be missed. Medieval and Roman writers frequently made reference to the presence of rodents


and thus

this project will proceed by assuming that black rats were capable of surviving in Europe at the time. Finally, it has frequently been proposed that in order to sustain an epidemic of Black Death, there is a requirement for a disease-resistant sector of the rodent population to act as a reservoir for the infection [13]. While this is not the case with the vast majority of infections, it has been suggested via mathematical modelling by Christakos et al. [17] and Keeling and Gilligan [10] that this is also not the case with Y pestis. Due to the large population size of rats and fleas, it is quite possible for the disease to remain endemic within small populations of rodents, with epidemic waves occuring as the population builds to a critical point. 6 The distinction between mice and rats was often overlooked, with many authors simply using the terms small mouse and large mouse


The contagious nature of the Black Death The final issue to be discussed here is one drawn extensively upon by Scott and Duncan in their proposition for Black Death as a haemorrhagic plague [13]. Several historical sources describe the Black Death as being highly contagious, for example da Piazza writes of the infection in Messina ‘not only did everyone die who spoke with the victims, but also anyone who bought from them, touched them, or had any kind of intercourse with them...’ [18]. Scott and Duncan claim that such a statement could not be made of bubonic plague, as there does not exist a method for transmission between humans. In a brief defense, firstly such accounts cannot be credited as being completely accurate; if a disease was as universally lethal as this comment suggests then it would leave no survivors at all. Secondly, descriptions of this severity of infectiousness were often directed only at casualties suffering from haemoptysis, and as such could be attributed to the few cases of pneumonic plague expected in any outbreak of Y pestis. Finally, there also exists evidence which strongly supports the flea-borne transmission theory, and which could not be explained by a disease which is only transmitted in aerosol form. Boccaccio, writing between 1349 and 1351 in Florence, tells of a pair of pigs which came upon the rags of a pauper who had died from the disease. ‘The pigs first of all gave the rags a thorough mauling with their snouts after which they took them between their teeth and shook them against their cheeks. And within a short time they began to writhe as though they had been poisoned, then they both dropped dead to the ground’. It seems likely that the pigs had been infected by fleas residing between the rags.

While this is by no means an extensive summary of the points put for and against the Black Death being an epidemic of bubonic plague, it shows that with the current evidence available it appears unlikely that a conclusive answer can be reached. This opens the door for a variety of mathematical techniques, some of which are suggested in the following section. 19


The Intentions of this Project

There are many aspects in the study of the Black Death for which mathematical modelling can play an important part. Work has been done by Christakos et al in creating a general (stochastic) model for the Black Death and comparing findings with historical evidence. They concluded that ‘the findings of advanced stochastic modelling and spatiotemporal mapping support the view that Black Death was a different kind of epidemic than bubonic plague’, although they were unable to find a conclusive proof of this. Keeling and Gilligan used a model, covered in Appendix B, to investigate the risks the bubonic plague posed to the modern world, and in doing so produced a wave-like epidemic nature which matched fairly accurately the outbreaks in Europe within the 1600’s. While both Christakos et al and Keeling and Gilligan sought to encapsulate as much of the behaviour of bubonic plague as possible, this leads to very complex models, involving a great many variables and many linked equations. In contrast, this project will address just a single aspect of the epidemiological nature of the Black Death; supposing the pandemic was caused by bubonic plague, can the rapid spatial spread of the epidemic be adequately explained mathematically? If not, to what extent do the parameters require adjustment before the recorded historical behaviour is adequately represented? In order to build a model which could answer such a question, a more detailed look at the factors involved in a bubonic plague epidemic is necessary. The following chapter covers the relevant biology and epidemiology, and the construction of a suitable model .


Chapter 3

The Construction of a Differential Equation Model Mathematics is biology’s next microscope, only better. Conversely, mathematics will benefit increasingly from its involvement with biology. Joel E Cohen, 2005

The aim of this chapter is to construct a model which could be used to model the movement of an infection of bubonic plague in Italy at the time of the first outbreak of the Black Death. The following section looks at the nature of the biological components; the black rat, the rat flea and bubonic plague itself. Section 3.2 examines the assumptions required to create a model, and isolates the defining characteristics of the components, before assembling these into a model.



Biology and Epidemiology

In order to construct any model, it is necessary to study the relevant behaviour of species involved in transmitting the disease. In the case of bubonic plague, this consists of the bacterium itself, a rodent (in this case the black rat) and a rodent flea, commonly the oriental rat flea.


The Black Rat; Rattus rattus

The black rat, also known as the ship or roof rat, is found on all continents and is most common in coastal areas. Although it thrives in tropical areas, it is capable of surviving in much colder environments, and would likely have flourished in medieval Italy. R rattus is a strong climber and capable of swimming, therefore was likely present in large numbers on most medieval ships, exploiting the supplies of food, relative shelter and lack of predators [19]. Black rats tend to live in polygynous groups of up to several hundred individuals, which defend an area usually less than 100 square metres. Studies on rat behaviour have shown that most groups stay within a 20m radius territory, although as the rat population approaches their biotope’s carrying capacity, rats will leave and search out new habitats [20]. The movement of adult rats appears to be an automatic mechanism to balance body fat, and thus rats will frequently explore their own territories, resulting in interactions (and subsequent cross-infection) with rats in surrounding colonies. In female rats, this movement peaks every four days in line with the estrous cycle. In a healthy colony, females produce litters mainly in the summer, although if the habitat can support substantially more rats, does can produce up to five litters annually, with an average of eight rats in each. The gestation period is 21-29 days, and newborns of both sexes are independent of their mother after four weeks and become sexually active within 3-5 months [19]. The fertility of females has been observed to decrease in crowded conditions. Like most rodents, black rats are omnivores, but prefer hard grains such


as wheat and barley. However, they are happy eating raw meat, eggs, waste human food, and most foodstuffs that could be expected to be around human settlements. The average individual will live up to a year in the wild, with an annual mortality of around 95%.


The Oriental Rat Flea; Xenopsylla cheopis

X cheopis is found worldwide in association with its primary hosts, Rattus, although it prefers warmer climates. They have a four stage life cycle (holometabulous), from egg to larva to pupa to adult. Adult females can lay up to 50 eggs a day, although this tends to only occur in temperatures of 65 to 80◦ C and around 70% humidity, and higher or lower temperatures can inhibit egg laying. These eggs usually incubate for two to twelve days, before hatching into larvae which feed mainly on dead skin cells. The larval stage usually lasts between nine and fifteen days, but can last up 200 days in unfavourable conditions [21], before the larva spins a silk cocoon. The pupation stage can be as short as a single week, or up to a year if inhibited by temperature or humidity. Upon emerging from the cocoon, an adult X cheopis can expect to live for around 100 days in temperatures of 7 to 10◦ C, but can survive up to 376 days in an ideal environment. Disease can be passed vertically between adult females and offspring, thus an infected female can lay a large number of infected eggs which can delay developing into adults for almost two years. Adult fleas are nidiculous parasites, living in the host’s nest unless attached to their host while feeding. Rat fleas will take a rat host whenever the option exists, thus bites to other species tend only to occur when a large number of rats die [22].


Bubonic Plague; Yersina pestis

Although much is known about Y pestis, not a great deal of it is relevant from a modelling perspective; it is sufficient to understand the reactions of the bacterium with the host’s body in the various species it infects. 23

Flea Vector Transmission via fleas of Y pestis is well documented. Initial acquisition occurs as the flea feeds on an infected host, introducing the bacterium into the flea’s digestive tract. The bacteria begin to secrete several proteins, including Yersinia murine toxin (Ymt) and hemin storage (Hms), with the latter aggregating within the oesophagus and piercing mouthparts of the flea. Eventually this forms a plug preventing the flea from ingesting blood. As the flea starves, it bites more frequently, often using several hosts, and regurgitates a mass of bacteria into each bite. This blockage is almost universally fatal; however, if the ambient temperature exceeds 28◌ C, infected fleas can often clear these gut blockages, and the disease fails to reach epidemic proportions [23]. Transmission from flea to another vector can also occur through ingestion of flea faecal matter or dead remnants of an infected flea.

Rodent Vector Although many different rodent species can support Y pestis, this project will focus on the black rat, which would have been the primary host in medieval Italy. Rats contract Y pestis primarily through infected flea bites, and are poisoned rapidly by Ymt secreted by the bacteria. It has been shown that some individuals in a rodent population are likely to have increased resistance to Ymt, and thus may act as carriers for some time, even if they eventually succumb to infection [24].

Plague in Humans Transmission from rats to humans will peak when large numbers of rats die, leaving an excess of fleas without hosts. Modern untreated bubonic plague is known to have a mortality rate of around 60%, but it is possible that around 1350 this value may have been higher, due to poorer individual hygiene and health. Even with modern-day medicine the rates for pneumonic and septicaemic plague verge on 95%. Cross infection between humans of bubonic and 24

septicaemic plague is highly unlikely, but the pneumonic strain is highly contagious, approaching measles in terms of virulence. Epidemics of airborne plague are prevented by the exceptionally rapid onset of illness in victims, which removes them from situations where they would be likely to pass on the infection. Bubonic plague has a somewhat longer incubation period of 2-6 days while the bacteria reproduces within the lymph nodes, before spreading throughout the rest of the body.


Other Species

Although 3.1.1 and 3.1.3 cover the prime species involved in an outbreak of bubonic plague, some other species have been observed to have an effect in several outbreaks, as detailed below.

The Human Flea; Pulex irritans Sharing much the same lifecycle as X cheopis, the human flea also infests cats, dogs and many other domestic animals, particularly the pig [25]. Although not common in modern domestic dwellings, fleas and lice were widespread in all classes of people in the medieval world, and Father Antero Mario wrote in 1657 ‘I have to change my clothes frequently if I do not want to be devoured by fleas, armies of which nest in my gown...the lice would feast on my flesh; they vie with the fleas’. P irritans could catch plague off infected animals exactly as X cheopis, and thus would stand a good chance of infecting any individual it fed off subsequently. Although the effects of human fleas on plague transmission tend to be dismissed out of hand, there is no real reason why they could not have played a major part. They would certainly have contributed to the chances of infection of any human living in the same house as an infected family member and when the host died, their fleas would move off in search of new hosts in exactly the same way as X cheopis leaving a dead rat. If human fleas are considered a candidate, then thought must also be given to the wide variety of different insect parasites which lived off livestock, but which were also quite 25

capable of taking humans as hosts.

Other Animal Vectors Although minimal research has been carried out, no species which played a part in medieval life has demonstrated that it would be unsuitable as a vector in much the same way as rats. Pigs are certainly susceptible to plague infection, as are dogs and cats.


The Defining Characteristics and Assumptions Leading to a Model

Attempting to combine all the above information into a model would result in an incredibly unwieldy system and would give almost no potential for analytical work. Although Keeling and Gilligan [10] succeed in performing analysis on a system of eight coupled equations briefly described in Appendix B, they acknowledge that the human equations do not affect the dynamics of disease spread, due to the lack of human-to-human transmission. In addition, their paper was aimed at creating a model to encapsulate as much of the behaviour of bubonic plague as possible, whereas this project is only investigating a single aspect; the speed of spatial dispersal. Thus several simplifications with regard to disease behaviour can be introduced. 1. The model will solely concern the spread of infection in rats. Human to human transmission is only possible in roughly five per cent of infections when an infection reaches the individual’s lungs, by which point they have often already been removed from the susceptible population due to illness. In an environment where rats and humans coexist, infected rats will pass the infection on to humans fairly rapidly as fleas flee their dead hosts. Thus the spatial spread of bubonic plague within a large human population can be well estimated by the spread of the disease within rats. The behaviour


of fleas has also been ignored as the independent movement of a flea is minimal compared to when it is attached to its host. 2. As fleas will not be modelled, a different mathematical representation for transmission is required. Biologically, this tends to occur when fleas from a recently deceased rat leave its corpse and bite nearby susceptible rats. As mentioned above, fleas are not able to move long distances without assistance, so this model can be simplified to infection being passed directly from a recently diseased rat to a susceptible rat by a simple mass action principle. Dead rats do not continue to infect the population; after a short period all their fleas will have left the body and the rat ceases to play a part in the system. This would suggest a model of the form shown in Fig 3.1, with dead rats in class D progressing to a removed class R after a certain period of time, and the force of infection being represented by a term asd, where a is a positive constant and s and d are the fractions of the rat population in the susceptible and dead classes respectively. However, such a model is again becoming too complicated for the purpose of calculating spatial diffusion, and reducing the force of infection to a more typical asi eases analysis considerably, with i referring to the still-living infectious rats. This can be justified as an infectious rat survives only for a few days. The period of time a rat continues to host fleas after its death is also of a similar order, and almost all infectious rats die from the disease. Thus the terms asi and asd are likely to be fairly similar.

Figure 3.1: SIDR Model. This could be used to model the temporarily infectious nature of a rat carcass, as its infectious fleas leave its body in search of a new host.

3. A fully detailed rat model would include the four months the average


newborn rat spends in the nest, possibly by incorporating a class of young rats, say Y , which has no diffusion term and does not contribute to the rate of reproduction of the population, and a transfer term µy into the susceptible class. However, as young rats can also be infected, this would require a similar class Y 0 which consisted of infectious rats which do not reproduce and with no diffusion term, with impact on infection for all susceptible classes. For simplicity, this stage of the lifecycle is omitted. This assumption is a serious one, as the period until sexual maturity can be up to a third of a standard rat’s life, but it eases the computation considerably. 4. It is assumed that all infected rats die from their infection. Before beginning to model the effects of disease, a model for a rat population free from bubonic plague must be constructed. The behaviour of healthy rat colonies is well understood, with data available for the rate of growth and spread of a group with ample resources. If a model can be constructed which well approximates this known behaviour, conclusions drawn after introducing the effects of disease should be more reliable. The key assumptions are: 1. The above omission of the time period before rat sexual maturity. 2. It is assumed that rats are born at a rate proportional to the total population of rats, say au and the death rate du is similarly proportional (although not necessarily equal), with u being the fraction of reproducing rats with respect to the entire population. However, a given biotope is not capable of supporting an infinite number of rats, so a logistical term should be incorporated, allowing the population to reach a limiting value, say k. 3. As a rat colony begins to reach its biotope’s carrying capacity, rats begin to diffuse at a much greater rate. As mentioned in Barnett’s The Rat:A study in behaviour [20], this diffusion tends to peak slightly before the


population limit is reached, ideally modelled by a term such as 

u+ω k


∂2u ∂x2

where ω, f , k are positive constants. Again, the u included in the diffusion coefficient takes the model out of reach of analytical methods; thus it is assumed that the coefficient for diffusion is a constant, D. Combining all these assumptions leads to the following model for a healthy population  ∂u u ∂2u − du + D 2 = au 1 − ∂t k ∂x


Before this model is elaborated to include the effects of disease, it would be useful to check that (3.1) exhibits behaviour which well approximates the known expansion of rat colonies. The following chapter covers an analytical solution to this model.



Chapter 4

The Analysis of a Model for a Healthy Rat Population Fate is not an eagle, it creeps like a rat . . . Elizabeth Bowen, 1935

The model (3.1) was proposed in the previous section as a model for a healthy rat population, including potentially unequal birth and death rates, incorporating a logistical behaviour as the population approached some limiting value, and allowing for spatial diffusion.  ∂u u ∂2u = au 1 − − du + D 2 ∂t k ∂x This chapter aims to find an analytical solution to this model, and thus determine values for the parameters such that the solution closely resembles known rat colony behaviour. Once a good model for a healthy population has been achieved, the conclusions drawn from a diseased model based upon the modifications to this initial model, for which no biological data exists, should be more 31



Fisher’s Model and a Transformation of Variables

(3.1) appears sufficiently simple for an analytical solution to be possible, yet it contains enough complexity for this solution not to be readily apparent. There does, however, exist a detailed solution for Fisher’s Equation, worked through in detail by Murray [26]. It is possible, by transformation of variables, to switch (3.1) into Fisher’s Equation, and thus find a solution.


Introduction to Fisher’s Equation ∂u ∂2u = ku(1 − u) + D 2 ∂t ∂x


Fisher’s or the Fisher-Kolmogorov Equation was proposed by Ronald Fisher in 1937 as a deterministic version of a stochastic model for the spatial spread of an advantageous gene within a population. It is perhaps the simplest version of the family of reaction-diffusion equations ∂u = ∆u + f (u) ∂t Equation (4.1) contains both the logistical and diffusion terms within (3.1), but does not have the separate birth and death terms.


Transformation of Variables of (3.1)

By transforming u and t it is possible to shift (3.1) into (4.1). First, as it is assumed that the birth and death rates of rats does not vary spatially or


temporally, a and d are constants and can be combined into one value, say α α := a − d ∂u au2 ∂2u = αu − +D 2 ∂t k ∂x


By a similar redefining of k, (3.1) begins to resemble the Fisher Equation

κ := k/a  ∂u ∂2u αu  +D 2 = αu 1 − ∂t κ ∂x


To absorb the coefficient α/κ, define u ˜ as

u ˜ :=

αu κ


so that (3.1) now reads κ ∂u ˜ ακ κD ∂ 2 u = u ˜(1 − u ˜) + α ∂t α α ∂x2


Similarly, defining t˜ and adjusting the constant D to absorb extra coefficients gives a final equation of 2 ˜ ∂u ˜ ˜∂ u = κ˜ u(1 − u ˜) + D ∂x2 ∂ t˜


where α := (a − d) κ := k/a αu κ αt t˜ := κ κD ˜ := D α u ˜ :=




The Analytical Solution of Fisher’s Equation

The solution to Fisher’s Equation was first presented by Kolmogoroff, Petrovsky and Piscounoff in 1937, and was subsequently clearly laid out in section 11 of Murray’s Mathematical Biology [26]. The following section includes a reworked copy of Murray’s solution, paying particular attention to the speed of the travelling wave.


Finding a Travelling Wave Solution

Take the equation 2 ∂u ˜ ˜ ˜∂ u = κ˜ u(1 − u ˜) + D ∂x2 ∂ t˜


and redefine x and t˜ as the following

t = κt˜,

x =x

κ ˜ D

 12 (4.9)

Omitting the asterisks and tildes for notational simplicity, (4.8) becomes ∂u ∂2u = u(1 − u) + ∂t ∂x2


In a situation where rats are evenly distributed in space (spatially homogeneous), it can clearly be seen that equilibrium solutions occur for u = 0 and u = 1. The former gives an unstable steady state, while the latter is stable, suggesting that travelling wavefront solutions may exist for 0 ≥ u ≥ 1. Negative u values can be discounted, as they would make no biological sense. To find a travelling wave solution, set

u(x, t) = U (z),

z = x − ct


where c is the wavespeed and is assumed to be a positive constant. Substituting


into (4.10) and rearranging gives

U 00 + cU 0 + U (1 − U ) = 0


where the prime notation represents differentiation with respect to z. A wavefront typically links two steady states of U , which are achieved separately as z tends to positive or negative infinity. From above, (4.12) has two steady states, giving an eigenvalue problem to determine values of c such that there exists a non-negative solution for U satisfying

lim U (z) = 0,

lim U (z) = 1




Note that there is not yet a reason for the system in (4.10) with initial conditions u(x, o) = u0 (x) to evolve into a travelling wave solution. For now, however, consider the phase plane (U, V ) where V := U 0 . Substitution into (4.12) yields the system U0 = V (4.14) V 0 = −cV − U (1 − U ) By inspection, this system has the critical points (U, V ) = (0, 0) and (1, 0), corresponding to the equilibrium states of (4.10). To examine the stability of the states, linearise the system to U0 = V (4.15) V 0 = −cV − U Analysis is carried out on the matrix A, defined by   

dU dt dV dt

  U   = A  V



The critical point (U, V ) = (0, 0) gives   0 A= −1

 1   −c


The eigenvalues of this matrix, λ1,2 are given by c λ± = − ± 2

c2 − 4 2


As c ≥ 0 by definition, this gives two separate cases, c < 2 and c ≥ 2. The former results in a pair of complex eigenvalues, which due to the negative, nonzero real part indicates a stable focus. The latter results either in two identical eigenvalues, or two negative, distinct real values. Both indicate a stable node. A similar method is used for (U, V ) = (1, 0), resulting in the real, opposite sign eigenvalues c λ± = − ± 2

c2 + 4 2


and indicating a saddle point. The manifolds for this saddle are given by Es = V = λ− U − 1 (4.20) Eu = V = λ+ U − 1 where Es is the stable mainfold, Eu is unstable and the −1 accounts for the translation of the critical point from the origin to (1, 0). Thus sketches of the phase plane (U, V = U 0 ) can be made for varying values of c, as shown in Fig.4.1. From these phase portraits and using continuity arguments, it can be seen that for any value of c ≥ cmin = 2 there exists a trajectory from the critical point at (1, 0) to the origin, such that the trajectory lies entirely within the area bounded by U 0 ≤ 0 and 0 ≤ U ≤ 1. This represents a travelling wave solution between U (z1 ) = 1 and U (z2 ) = 0, moving at speed c. Note that, although solutions exist for values of c < 2, the focal nature of the critical point at the origin causes the trajectory to oscillate between positive and negative values


Figure 4.1: Phase portraits for the system (4.14), drawn using John Polking’s ‘PPLANE 2005.10’ [2].

for U , which again makes no biological sense. Obtaining a value for c in the context of (3.1) requires reversing the substitutions made in this section and section (4.2.1)

z = x∗ − ct∗   12 κ =x − cκt˜ ˜ D ˜ 12 t˜ = x − c(κD)

Now, substitute in the coefficients from (4.7)  z =x−c

κ2 D α


αt κ


= x − c(αD) 2

Finally, allowing for u ˜ and using the value for cmin found in (4.18) gives the 37

result that travelling wave solutions for (3.1) exist for all wavespeeds c such that  c ≥ cmin = 2κ


D α

 12 t


The Occurrence of a Travelling Wave

Although this shows that travelling wave solutions exist for Fisher’s Equation, it does not explain which initial conditions might evolve into such a wave, and what speed above the minimum cmin they travel at. Murray cites several references, but the main result this section will draw on is that proved by Kolmogoroff et al in 1937. If u(x, 0) has compact support

u(x, 0) =

   u0 (x) ≥ 0  



x ∈ (x1 , x2 )


x∈ / (x1 , x2 )


and u0 (x) is continuous in x1 < x < x2 , then the solution u(x, t) of Fisher’s Equation evolves into a travelling wavefront solution U (z) with z = x − cmin t. Due to the invariance of Fisher’s Equation to a change of sign of x, a similar travelling wave moving in the opposite direction evolves if the initial conditions are U (−∞) = 0, U (∞) = 1. Thus if at t = 0, U (z) = 0 along the real line except for a finite domain about the origin (where 0 < U (z) ≤ 1), a wavefront solution develops moving out from the origin at a speed of cmin , as shown in Fig 4.2.


Finding Variables for the Healthy Model

Using the speed of the travelling wave found in the previous section, it is possible to begin to fit parameters to the model. Some parameters can be drawn from biological characteristics set out in Chapter Two, while others can be deduced by attempting to match the travelling wave to the known expansion rate of a rat colony.


Figure 4.2: Sketch of travelling waves generated by Fisher’s Equation, redrawn from solution by Towers [3]



Biologically Derived Parameters

The original form of (3.1) is  ∂u u ∂2u = au 1 − − du + D 2 ∂t k ∂x with a, d and k all directly referring to biological characteristics of the black rat, namely the birth rate, the death rate and the maximum population density a given biotope is capable of supporting. It is known that the average female black rat will give birth to 30 young per year, thus assuming a population split equally between male and female rats, the average increase in population is 15 rats per individual per year. The lifespan of a rat is assumed to be a single year. Rats are capable of surviving with population densities as high as three rats per square metre, although a slightly lower value would represent an average environmental saturation point. If the model takes a scale of x = 100m and t = one week, this gives the following parameter values: • a = 1.3: Birth rate of fifteen rats per individual per year equates to approximately 0.3 rats per individual per week. Thus with every timestep, 39

u increases by a factor of 1.3. • d = 0.02: If one rat dies after 52 weeks, then approximately 0.02 rats die per week. • k = 200: Allowing a limit of two rats per square metre and allowing for the step size of x = 100m. In terms of the transformed coefficients κ and α, this gives • α = 1.28 • κ = 154 Finally, the model requires a value for the rate of diffusion parameter D. Looking ahead to the diseased model and considering the aim of the project, this should not be thought of as the speed with which a rat colony physically expands, but the speed with which an infection moves through varous colonies. This is by far the most difficult parameter to predict using the biological facts laid out in the second chapter; however, an informed estimate can still be made. It is known that most rats will visit most points on the edge of their colony’s territory at least once a day, and that rat colonies tend to sit adjacent to one another. An encounter with another rat from a neighbouring colony tends to result in a fairly physical ‘wrestle’, during which it seems likely that fleas could easily be dislodged and switch hosts, spreading infection. Given the number of individuals within each colony, and allowing for the incubation period of bubonic plague within each rat, the author finds it hard to believe that an infection could cross more than three ten-metre-wide colonies a day. Thirty metres a day implies 210 metres a week, or 2.1x, and substituting this into the equation for the speed of a travelling wave found in (4.21), gives a value for D of  D=

210 2.154

2 ∗ 1.28

= 5.95 ∗ 10−5


For simplicity, it is useful to count u in units of 200 rats, allowing k = 1 and the final model ∂u ∂2u = 1.3u (1 − u) − 0.02u + 5.95 ∗ 10−5 2 ∂t ∂x


Numerical Simulations of the Healthy Model

Figure 4.3: Numerical simulation of a healthy rat population diffusing through space.

Using Maple, it is possible to produce numerical simulations for the Healthy Model. Fig 4.3 shows the diffusion of a population of rats, with initial conditions given by the Heaviside function such that u(x, 0) = Heaviside(1 − x). From this simulation, several points can be made: • The population of rats spreads into the area with no rats at a constant speed, and grows until it reaches the limiting density (although possibly after a brief peak above this sustainable limit). • The irregular peaks on the surface of the advancing wavefront, and the initial large peak, are assumed to be noise generated by the numerical


method used to solve the model. â&#x20AC;˘ An unexpected result is that the rat population sits slightly below the limiting density, clearly shown in Fig 4.4. This is likely to be a result of the inexact nature of the numerical simulation, but is not unwelcome as it matches biological evidence that a spreading population of rats tends to fail to reach its limiting density until it has finished spatially diffusing.

Figure 4.4: Snapshot of healthy rat diffusion, taken at t = 20.

In summary, whilst much more could be made of this model, it is sufficient to say that it closely resembles the selected characteristics of a biological system of rats upon which it was based, and is thus suitable for development into the diseased model, which is covered in the following chapter.


The Diffusion Paradox

Before moving on, a comment can be made on one of the undesired characteristics of a standard diffusion model, such as Fisherâ&#x20AC;&#x2122;s Equation, namely that the travelling waves generated by the standard Fisher Equation exhibit the so-


called ‘paradox of diffusion’. In this context, this is equivalent to the fact that immediately after the dispersion of a population begins, u is greater than zero at all points along the real line, irrespective of distance from the origin. In many models, such a small value for u can be regarded as negligible and the issue ignored. However, in this case, the highly virulent nature of bubonic plague may present a problem. When the model is adapted and the travelling wave shifts to model diseased rats, the highly virulent nature of bubonic plague could result in outbreaks of infection stemming from these ‘errors’ in rat numbers, before the actual disease front has arrived. To avoid this, it would be desirable to adjust (3.1) to create a precise wave with no seepage forward of the front. One method of doing this is to use the hyperbolic Fisher’s Equation.


The Hyperbolic Fisher’s Equation

It is possible to write Fisher’s Equation as the system ∂u ∂v + = ku(1 − u/us ) ∂t ∂x ∂u v = −D ∂x

(4.23) (4.24)

where the second equation denotes the flux, in this case Fick’s diffusion, also known as Fourier’s law. By changing the flux term, similar systems can be generated with different diffusion characteristics. In particular, using ∂v ∂u = −c2∞ ∂t ∂x


   u u ∂  ∂     +A   = aub ∂t ∂x v v


results in the system 



  0 A= c2∞

 1  T  , b = (1 − u/us , 0) 0


Jordan conducts an in-depth analytical study of this system in Growth, decay and bifurcation of shock amplitudes under the type-II flux law [27], beginning by observing that the system is strictly hyperbolic. However, although he reaches an analytical solution for a travelling wave, it is complicated and beyond the scope of this project, especially as none of the feared ‘spontaneous outbreaks’ occurred during the simulations of the following chapter. Thus this model is not developed further, although it remains a possibility if a similar model is used for an even more virulent disease, such as measles.


Chapter 5

Modelling Diseased Rats They died by the hundreds, both day and night, and all were thrown in ... ditches and covered with earth. And as soon as those ditches were filled, more were dug. And I, Agnolo di Tura ... buried my five children with my own hands ... And so many died that all believed it was the end of the world. . . . Agnolo di Tura, 1348

In this section, the healthy model is adapted to include the spread of disease through the population, and several numerical simulations are carried out to test various values of parameters.


The Diseased Model  ∂u u ∂2u = au 1 − − du − γui + D 2 ∂t k ∂x ∂i ∂2i = γui − δi + D 2 ∂t ∂x


(5.1) (5.2)

This shows the adjustments made to the healthy model to allow the simulation of infection. The new model takes the form of a basic SI model, with susceptible and infectious classes, which is discussed in greater detail in section A.2.1 of Appendix A. However, the new terms are discussed here. The transfer term γui is a simple mass action term, where the rate at which susceptible rats are infected is represented by the rate at which susceptible rats encounter infectious rats. Although this simplifies the flea infection dynamics considerably, it was previously justified in section 3.2. This term is tricky to estimate, as there is no biological research on which to draw. An initial value of γ = 0.8 was chosen, which is quite high for a model of the sort. However, as is shown in the following section, the simulations appear fairly robust to this parameter. The term δi is the rate at which rats infected with bubonic plague die. Similar to humans, most individuals succumb to their infection within a week, suggesting δ = 1, given the timestep of one week used. However, as the assumption that all infected rats will die from their infection is perhaps a little strong, a value of δ = 0.8 may be more appropriate. Again, running multiple simulations suggests that small variation of this parameter does not affect the outcome in any significant way. It has been assumed that infected rats continue to move around at a rate equal to healthy rats, that is to say; the value of the diffusion coefficient for each model, D, is the same. This has been justified as the incubation period of Y. pestis combined with the period that rats are infected, but well enough to move around, is markedly larger than the period at the end of infection when an infected rat is too sick to move.


Numerical Simulations of the Diseased Model

Simulations for the diseased model (5.1) were carried out in Maple, using much the same methods as the simulations at the end of the preceding chapter for


Figure 5.1: Initial and boundary conditions for simulations of the diseased model.

the healthy model. The selection of biologically reasonable initial and boundary conditions proved challenging and eventually a variation on the Heaviside function setup previously used was most effective. The full conditions used on the range x ∈ (0, 20) were

u(0, t) = 0.5


u(20, t) = 0.5


u(x, 0) = 0.5


i(0, t) = 0.5


i(20, t) = 0


i(x, 0) = 0.5 ∗ Heaviside(1 − x)


This gives the biological equivalent of a (one dimensional) environment with a saturated population of healthy rats, apart from near the origin where 10% of the population is infected. This was designed to model the initial appearance of plague in a virgin population of rats, and could perhaps be a simplified model for the passage of plague down a busy trade route. These initial conditions are shown graphically in Fig 5.1.



Summary of Findings from the Diseased Model

The model gave somewhat unexpected results. Fig 5.2 shows the population of infected rats both at a snapshot in time of t = 52, and as a graphic of the spread up to t = 208. Possibly the most surprising result was the exceptionally high values of t required to see any sort of spread of infection. The snapshot for t = 52 gives a dispersal rate of approximately one kilometre a year, which is nowhere near the hundreds of miles covered by the Black Death in a single year. It is, however, remarkably similar to the recorded statistics for the Third Pandemic of Black Death, which was estimated to travel between one and three miles a year. The second point is that at no point is the survival of a large number of healthy rats in doubt. As before, the total population of rats settles slightly below 1, but the fraction of infected rats (after the initial wavefront of infection has passed) stabilises at approximately 10% of the total population.

Figure 5.2: Numerical first shows the second 0 and 208. resized.

simulations for the diseased model. The the profile of the wavefront at t = 52, while shows the spread of the wave for t between Note the vertical axes, which have been


Chapter 6

Conclusions and Further Work A wise man that had it for a by-word, when he saw men hasten to a conclusion; ‘Stay a little, that we may make an end the sooner’ . . . Gabriele de’Mussi, 1348


Initial Conclusions and Limitations of Mathematical Models

Before going into depth about the interpretation of the results, it is worth bearing in mind the limitations of any mathematical model. No model can hope to encompass all the factors which might affect the spread of a pandemic in the real world, but by trying to encapsulate the key factors and making reasonable assumptions on other aspects, useful results can be obtained. During the period of the Black Death a huge amount of ‘abnormal’ human factors, such as famine, flight from main population centres, wars and innumberable other aspects, threaten the accuracy of a model. This project has attempted to avoid the majority by focus on a single aspect of the infection (spatial spread) and 49

then only within rats, which should hopefully remain relatively unaffected by a large proportion of the human factors. Nonetheless, the assumptions made in section 3.2 should not be ignored. Subject to that mentioned above, the results from Chapter 5 would appear to add further evidence to that outlined in section 2.2 that the Black Death was unlikely to have been a pandemic solely of bubonic plague, primarily due to the massive difference between the actual spread of the Black Death from Corsica to Norway in three years and the simulated spread in the model of a single kilometre per year. However, the similarity with the more reliable recorded data for the Third Pandemic between 1900 and 1950 is especially useful, as it gives some support for the reliability of the model in accurately capturing the behaviour of a bubonic plague infection. The maintenance of a healthy population of rats as infection spreads was unexpected, especially given that the values selected for the parameters were based upon a â&#x20AC;&#x2DC;worst caseâ&#x20AC;&#x2122; scenario for lethality. It is possible that this is due to a quirk of the boundary conditions, although several options were tried, including Neumann conditions, without any significant effects. Nevertheless, any further work would certainly begin by extensively testing these possibilities, to ensure that the results are due to the model, rather than the restrictions imposed during numerical simulation.


The Role of Human Parasites

However the current evidence against bubonic plague as the cause for the Black Death should be strongly tempered by the high possibility, in the authorâ&#x20AC;&#x2122;s opinion, that humans may have been capable of transmitting plague via their own parasites. If this premise is valid, a large number of issues associated with the rat-based theory become irrelevant. In particular, the lack of rats in Iceland is no longer an issue, as humans themselves are capable of habouring the infected fleas. When merchants are capable of carrying the infection between human settlements in the lice infesting their clothing, the slow speed of diffusion of


infection within rats becomes less important, and there is subsequently no requirement for infected rats to hitch rides on trade carts. Effectively, this would be equivalent to bubonic plague operating as a contagious disease between humans. Testing this hypothesis may be challenging, although there are some possibilities: • To the best of the author’s knowledge and research, there is no biological reason why the human flea Pulex irritans would be unable to carry Y pestis in exactly the same way as the rat flea (see sections 3.1.2 and 3.1.4). However, a study of the reaction to Y pestis of the various lice and parasites which lived on and in close proximity to the typical Medieval human, would indicate the extent to which human parasites would be capable of maintaining a sustained pandemic. Certainly it is known that there would be no shortage of human parasites throughout the period in which the Black Death affected Europe. Section 3.1.4 included Father Mario’s referral to the ‘fleas, armies of which nest in my gown’; throughout the Middle Ages it was common practise to suspend expensive clothes in garderobes, over toilets, to fumigate out lice; and research by the Universit´e de la M´editerran´ee suggests that a large proportion of the casualties in Napoleon’s invasion of Russia in 1812 were due to lice-borne diseases [28]. • It is possible that areas with a lower population of rats, such as Iceland, would suffer more violent outbreaks of plague in humans. Without the underlying rodent population to maintain the infection, the disease would either sweep rapidly through the population, or die out. Similarly, such areas would be expected to have fewer recurrences of infection, as the infection reservoir action of rats suggested by Keeling and Gilligan (see Appendix B) is not possible, and a fresh introduction of infection to the human population is required to start each new epidemic. A study of the records of infection in Iceland would demonstrate if this was the case. • From a mathematical point of view, the model within this project could


be reworked to try and simulate this new method of infection between humans, to test if the speed of disease spread more closely matches the recorded data. Again, to the best of the author’s knowledge, this is not an angle which has been previously considered, and it certainly should not be dismissed out of hand.


Future Work

To return to the rat model, it would be useful to simulate the effects of the introduction of infected individuals at points distant from the origin as t increased. To extend the analogy suggested in section 5.2 of a single trade route and plague spreading through the rat populations along the route, this modification would hopefully allow for the ‘leapfrogging’ of the standard diffusion mechanism by groups of infected rats carried by carts. The movement of rats by human transport has often been cited by those defending the bubonic plague theories as a possible way that the speed of spread of the disease could be increased, and it would be enlightening to see if the model supported this. Finally, it should be noted that the model is by no means restricted to modelling bubonic plague. Due to the simplicity of the mass action transferral term, the model would be biologically reasonable (after adjustment of the parameters) for representing various other human illness, including Lassa Fever, Weil’s Disease and hantavirus; infections where rodents are the principle transmitting agents.


Appendix A

Introduction to the Mathematical Modelling of Epidemics I simply wish that, in a matter which so closely concerns the wellbeing of the human race, no decision shall be made without all the knowledge which a little analysis and calculation can provide . . . Daniel Bernoulli, 1760


An Introduction to Mathematical modelling of Disease Transmission



In 1967 the US Surgeon General, William Stewart, famously claimed that it was â&#x20AC;&#x2DC;time to close the books on infectious diseasesâ&#x20AC;&#x2122;. Developments in antibiotics, vaccines and basic medical care had improved the treatment of disease


symptoms, pesticides such as DDT


were being used to control populations

of many traditional disease vectors, water chlorination and improved sewerage reduced the risk of many waterborne infections such as typhoid and dysentery, and a general improvement in the standard of living in the western world helped build the image that disease was set to become a problem of the past. Medical attention should now be turned towards chronic diseases such as cancer and heart disease. Nine years previously in 1958, David Carr had already become the first confirmed victim of Gay-Related Immunodeficiency Disease (GRID), later termed Acquired Immune Deficiency Syndrome (AIDS), an infectious disease set to sweep across the world in a manner not seen since the Black Death, resulting in an estimated 2.4 million deaths worldwide in 2007 and a current infected population of over 35 million [29]. In addition, throughout the latter half of the twentieth century, outbreaks of viral haemorrhagic fevers with no obvious treatment and with mortality rates approaching 95% have occurred in Bolivia, Democratic Republic of the Congo, Angola and other countries. A near-pandemic of Severe Acute Respiratory Syndrome, or SARS, between November 2002 and July 2003 resulted in 8096 known infections and 774 deaths, a large proportion of which occurred in societies with a relatively modern medical infrastructure. Helped by the unrestricted use of broad-spectrum antibiotics after the Second World War, several previously manageable infections returned in new strains capable of resisting treatment, notably Staphylococcus aureus in the guise of MRSA2 which accounted for 37% of all blood poisoning deaths in the UK in 1999, and resulted in a wave of media headlines such as ‘Superbug Deaths Soaring’ and ‘Exposed: Hospital Rampage of Killer Bug’ in the UK press alone. The H5N1 strain of avian influenza, more deadly and capable of infecting more species than any previously known influenza virus strain, caused a leading expert to claim ‘The world is teetering on the edge of a pandemic that could kill a large fraction of the human population’ [30] and has resulted in billions of dollars being spent 1 Dichloro-Diphenyl-Trichloroethane 2 Methicillin-resistant

Staphylococcus aureus


worldwide in research, with no significant results to date. Mathematics has innumerable roles in this developing environment, such as the statistical analysis of new drugs in trials, modelling the spread of a disease as it moves between organs in a single individual, and attempting to predict the most effective way to implement vaccines.


Mathematical Epidemiology: An Overview

The first published attempt to use mathematics to model a disease was by Daniel Bernoulli in 1766, with ‘An attempt at a new analysis of the mortality caused by smallpox and of the advantages of inoculation to prevent it’ [31]. At the time, smallpox killed an estimate 400,000 Europeans each year [32], and Bernoulli calculated that approximately three quarters of all living people had been infected at some point in their lives. In 1721 a method of immunisation was introduced to England, termed variolation, conferring good protection, but in a small number of cases resulted in a full blown smallpox infection and the possible death of the patient. In an argument for universal variolation Bernoulli calculated the expected number of individuals saved each year in the event of smallpox eradication, concluding that eradication would ‘increase expectation of life at birth from 26 years 7 months to 29 years 9 months’ [33]. Echoes of his methodology are still being used today to argue for the widespread availability of retrovirals for HIV-infected individuals. Further landmarks occurred in 1911 when Ronald Ross applied differential equations to model the role of mosquitoes in the spread of malaria in his book ‘The Prevention of Malaria’, and in 1927 when Kermack and McKendrick published the General Epidemic Model. This last model is discussed later in subsection A.2.2, and was the first model to allow for infectiousness varying with time in an effort to model the rapid rise and fall of cases in epidemics of the plague and cholera. Most contemporary models are constructed in a similar manner to the General Epidemic Model, drawing from a standard set of classes, with the model


consisting of a subset of these classes and a method of transferral between them. Consider Fig A.1:

Figure A.1: Graphical representation of classes and transferral in basic MSEIR model [4].

Such a model could describe a disease such as measles before the widespread introduction of a vaccine. Consider a situation where measles is endemic within the population and the majority of individuals are exposed to the disease, subsequently developing resistance to it, before the age of 21. As a result, most mothers pass on a temporary passive immunity to their offspring, which lasts for an average of six months [34], resulting in births within the M , or Maternally Acquired Immunity class. As the immunity fades, these individuals transfer to the S, or susceptible class. This class also contains those newborns who, for whichever reason, were not born with passive immunity, be it due to premature birth, an infection of the mother such as AIDS, or simply because the mother had not been exposed to the disease. When a susceptible has an ‘adequate’ contact with an infective (I), they contract the disease. Measles has a latent period of 7-14 days where the individual has contracted the disease, but is not infectious [35], and thus is placed in the exposed class E. As the infection gains hold, the exposed individual becomes infectious, and capable of transmitting measles to susceptibles. At the end of this period, the individual is placed in the removed or recovered class R. As a single infection of measles is usually sufficient to infer lifetime immunity, there is no transferral from the class R back to susceptible. It is common practice, particularly in systems for short epidemics where birth rate is negligible, to treat the R class as ‘removed’ thus also encompassing those who die in any other class. In this way the total


population of the system is kept constant, which simplifies some calculations. To continue with measles, a contemporary model of the western world would be unlikely to be accurate if it could not allow for immunity gained by vaccination, say by including an extra class, V . It is generally agreed that the standard measles vaccine confers around 95% immunity, which could be modeled by a small transfer from V to S. It may also be decided, for example when modelling a short term measles epidemic in a university campus, that the M class is irrelevant, and thus a VSEIR model could be used, where the acronym describes the basic flow between classes. Thus, for many diseases, the basic structure of a model can be simply described using its acronym. Several examples are discussed in section A.2.


Terminology and Notation

When analysing models for epidemics, it is helpful to calculate several threshold quantities. In many instances, the main threshold is the basic reproduction number R0 , defined as the average number of secondary infections produced when one infectious individual is introduced into a virgin population. Thus in most models an epidemic is only possible if R0 > 1. Very similar to the basic reproduction number is the contact number σ, defined as the average number of adequate contacts an infective individual makes with susceptibles during the infectious period. Finally, the replacement number R is the average number of secondary infections produced by a single infective during the entire period of infectiousness. Note that at the start of any model, R0 = σ = R and frequently R0 ≡ σ throughout a model’s timespan. In any model, the total number of individuals within the entire system is termed N , with population at time t being N (t). Similarly, individuals within a class at time t are M (t), S(t), etc. For ease of calculation, assume the population remains constant. Now the susceptible fraction of the population is s(t) = S(t)/N , and likewise for other classes. If β is the average number of adequate contacts of an infective per unit time, then βi is the average number of contacts


with infectives per unit time of one susceptible, and thus βiS = βN is is the number of new cases of infection per unit time within all susceptibles. This is termed the standard incidence form of transferral. Finally, transferral out of classes such as M , E and I and into the next class are often governed by terms such as δM , ηE, and γI in the ordinary differential equations model. It can be shown that these terms correspond to exponentially distributed waiting times in each compartment [36]. For example, the transfer rate δM corresponds to P (t) = e−δt as the fraction that is still within the Passively Immune class t units after entering the class and, from properties of the exponential distribution, this gives an average ‘waiting time’ of 1/γ. Other transferral rates can be used, for example stepwise functions, piecewise continuous functions, or other statistical distributions. The following summary table is redrawn from H W Hethcote’s The Mathematics of Infectious Diseases [4]. M S E I R N m,s,e,i,r β R0 σ R

Passively immune infants Susceptibles Exposed, but not infectious Infected and infectious Removed individuals Total population Fractions of total population in classes Contact Rate Basic Reproduction Number Contact Number Replacement Number

Table A.1: Summary of notation.

The next section puts much of the above into practice by analysing six of the most common models.


A.2 A.2.1

Six Standard Epidemic Models The Simple Epidemic (SI) Model

Consider the simplest possible epidemic model, consisting of the susceptible and infective classes and a transfer rate between them. Note the assumptions required for this model; the population size N is constant and assumed large and there are no vital dynamics (births and deaths), immigration or emigration. From an epidemiological point of view, there is no chance of recovery from infection, and a susceptible becomes infectious as soon as they contract the disease. Mixing within the population is deemed homogeneous (ie no strata have been added to allow for age, social class, or other possible layers in society), and the infection rate will be assumed proportional to the number of infectives using the standard incidental form of transferral; λ = rβI. Although simple, SI models still have applications. Roxana L´opez-Cruz uses SI models with age layering in an attempt to model HIV spread within Peru and the USA, drawing upon the simplicity of the model to allow a more detailed analysis to be carried out [37]. In several computer simulations, it is possible to model forest fire spread by populating a large lattice with ‘susceptible’ trees, before running an SI epidemic through the model, with the infection being the fire [38]. However, returning to the simple model above, it is possible to represent the diagram as a pair of ordinary differential equations dS = −rβI(t)S(t) dt dI = rβI(t)S(t) dt


Using N = S(t) + I(t), this is equivalent to S(t) = N − I(t) dI = rβI(t)(N − I(t)) dt



Taking the second of these, separate and integrate Z t 0

 Z t 1 dI dt = rβdt I(t)(N − I(t)) dt 0  Z I(t)  Z t 1 rβdt du = u(N − u) I(0) 0 I(t)

[ln(u) − ln(N − u)]I(0) = rβN t I(t) =

I(0)N I(0) + (N − I(0))erβN t


with increasing time, as (N − I(0))erβN t → 0, the right hand side tends to I(0)N /I(0). Thus as time tends to infinity, all susceptibles become infected, as shown in Fig A.2.

Figure A.2: Susceptible and infectious populations varying with time, redrawn from Jan Medlock’s Mathematical Modelling of Epidemics [5].

Further analysis for this is limited and, from a biological point of view, few infections can accurately be modeled in such a way. However, the addition of a single extra class to the model expands the possible applications markedly. Consider chickenpox, a frequently endemic infection of the varicella-zoster virus


that mainly affects young children. Even without the use of a vaccine in the UK, the NHS estimates that around 89% of adults have developed immunity to the virus through childhood infection, and such immunity is effectively permanent [39] (although a possible genetic predisposition to reinfection is noteworthy). Such a flow of susceptible竊段nfectious竊段mmune could be represented by an SIR model.


The SIR Model

In this initial analysis of the SIR model, take N to be constant and large. Now consider the assumptions made, in particular regarding a possible model for chickenpox resistance throughout a country. No vital dyamics have been used, meaning there is no flow of newborns into the system, nor deaths from any classes. Constant population could be argued to be reasonable, especially in western cultures where population growth has slowed significantly, but the lack of births is a problem and restricts the model to somewhat artificial environments (although possibly of use for the introduction of chickenpox to a virgin, isolated environment; such as an island). In addition, a chickenpox mass infection would not be expected to have any measurable effect on population, with death resulting from severe complication occurring in only 0.0025% of cases in 2008 according to the US Communicable Disease Centre (CDC). Again, no strata have been allowed for, which is a definite failing in the model as the mixing of adults and school children is not homogeneous. Biologically, chickenpox has a latent period of ten to twenty days, which could be significant. However, assuming that the model is intended to represent behaviour of an infection over several years, the SIR model seems reasonable. A constant recovery rate has also been assumed; again reasonable, as the development of new treatments for such a typically mild childhood infection is unlikely to be viable. In its simplest form such a model could be represented by the following three


ODE’s dS = −βI(t)S(t) dt dI = βI(t)S(t) − αI(t) dt dR = αI(t) dt


Note this is equivalent to dS = −βI(t)S(t) dt dI = βI(t)S(t) − αI(t) dt


R(t) = N − S(t) − I(t) By dividing (A.5) through by N , the scale can be removed from the model and all terms become proportions of the total population3 ds = −βi(t)s(t) dt di = βi(t)s(t) − αi(t) dt


r(t) = 1 − s(t) − i(t) Combining, separating and integrating between t = 0 and t of the latter two equations above leads to ds β = − s(t) dr α


and further integration gives β

s(t) = s(0)e− α (r(t)−r(0)) β

≥ s(0)e− α >0

(A.8) (A.9) (A.10)

3 While not relevant here, if the population size is not constant this avoids the possibility of, for example, I going to infinity as i tends to zero if the population N grows faster than I, or I tending to zero even if i is bounded away from zero if the N goes to zero. Thus using the fractions s, i, r etc provides a more reliable guide to disease behaviour which is independent of the population size.


This is important; for all values of t, the number of susceptibles left in the population is greater than zero, thus not everyone gets infected. Returning to (A.6), combine the first two equations to acquire the following variables separable equation di α − βs = ds βs α −1 = βs


Again, integration between t = 0 and t gives the following conserved quantity for all t s(t) + i(t) −

α α ln s(t) = s(0) + i(0) − ln s(0) β β


This can be rearranged using the fact that i(0) + s(0) = 1 to give an equation for the value of i(t), given s(t)

i(t) = 1 − s(t) +

α s(t) ln( ) β s(o)


Recall from A.1.3 that the contact number was defined as the average number of adequate contacts made by an infectious individual with susceptibles during the infectious period, trivially equivalent to the mean number of adequate contacts per day per infected individual, multiplied by the mean number of days the infection remained. It has already been shown that the average waiting time in the infectious class was 1/α. By definition, β is the average number of adequate contacts per day per infected, hence σ = β/α and (A.13) can be written

i(t) = 1 − s(t) +

1 s(t) ln( ) σ s(o)


Returning to the last line of (A.6), r(t) = 1 − s(t) − i(t), the triangle T in the si phase plane such that

T = {(s, i)|s ≥ 0, i ≥ 0, s + i ≤ 1}



is positively invariant, and unique solutions exist in T for all positive time. Thus the model is both mathematically and biologically well posed. Combining (A.14) and (A.15) gives the following phase portrait, taken from Hethcote’s Mathematics of Infectious Diseases [4], using an arbitrary value of σ = 3

Figure A.3: Phase portrait for SIR model without vital statistics [4].

Note the value of smax labeled on the horizontal axis. Recall the definition of the Basic Reproduction Number, R0 , as the average number of secondary infectious resulting from the introduction of a single infectious individual into a virgin population, equivalent to σs(0). Informally, from the above figure, it appears that when R0 ≤ 1 the infectious fraction tends to zero monotonically as time increases, but when R0 > 1, the infectious fraction rises to a peak, before diminishing. This is formalised by Hethcote in Qualitative Analyses of Communicable Disease Models [40] in the following theorem


Theorem A.2.1. Let (S(t), I(t)) be the solutions of (A.4). If R0 ≤ 1 the I(t) decreases to zero as t → ∞; if R0 > 1, then I(t) first increases up to a maximum value equal to 1−σ −1 −[ln σS(0)] σ −1 and then decreases to zero as t → ∞. The susceptible fraction s(t) is a decreasing function, and the limiting value s(∞) is the unique root in (0, σ1 ) of the equation ln 1 − s(∞) +


s(∞) s(0)


i =0

In biological terms, if the Basic Reproduction Number is greater than one, then the infectious fraction increases up to a peak and then decreases to zero; otherwise the infectious fraction decreases monotonically to zero. The infection spread stops because the Basic Reproduction Number becomes less than one when the infectious fraction becomes small. However, note the final susceptible population is not zero. This type of model effectively represents outbreaks of infection into a virgin population, where the epidemic fades in a short space of time. However, when the timespan is longer, a model which cannot model births and deaths begins to develop inaccuracies. The SIR model can be adapted to include vital dynamics as so dS βIS = µN − µS − dt N dI βIS = − αI − µI dt N dR = αI − µR dt


Note how the influx of births into the susceptible class µN is balanced by the deaths from all classes, thus enabling the population to be kept constant.

1 µ

is taken as the mean lifetime of an individual, around 75 years in the western


world. As above, these equations can be divided by N to give; ds = βis + µ − µs dt di = βis − (α + µ)i dt


r(t) = 1 − s(t) − i(t) Again, there exists a triangle T = {(i, s)|s ≥ 0, i ≥ 0, (i + s) < 1} within which solutions to (A.17) are positive invariant, thus the entire system is well-posed, as r(t) is bounded by the last line of the above system. However, the system’s behaviour as time evolves is more complex. Consider first finding the critical points

0 = µ(1 − s) − βis 0 = βis − (α + µ)i

Rearranging the first equation gives an identity for i


µ(1 − s) βs

Now, equating βis between the two equations gives

µ(1 − s) = (α + µ)i


Substition for i gives µ(1 − s) =

(α + µ)µ(1 − s) βs

and further rearrangement yields   µ(α + µ) 0 = (1 − s) µ − βs   α+µ 0 = µ(1 − s) 1 − βs



Thus critical s values are




α+µ β

Note that in the second equation, α + µ/β is equivalent to σ −1 , with the contact number now modified to take account of the deaths from within the infected class. By substitution, critical points are

(s, i, r) = (1, 0, 0)


1 µ(σ − 1) (1 − σ)(µσ − β) (s, i, r) = ( , , ) σ β σβ


Effectively, two equilibrium solutions are possible; a population consisting solely of susceptibles, or a node solution where periodic peaks of infection diminish until an endemic equilibrium is obtained where the disease remains in the population. Stability for these points can be investigated either numerically [41] or via Lyapunov functions [42], with the results shown graphically in Fig A.4. In summary, if σ < 1, the disease free equilibrium at (s, i) = (1, 0) is locally asymptotically stable, and a disease cannot gain a lasting foothold within a susceptible population. If, however, σ > 1, this equilibrium solution is unstable and any path starting with a positive number of infectives gradually oscillates towards the stable endemic equilibrium. Note that any situation where the entire population is susceptible implies that σ ≡ R0 , and thus again the outcome of a particular model setup can be completely specified with prior knowledge of the basic reproduction number R0 .


Figure A.4: Bifurcation diagram for equilibrium points in SIR model with vital dynamics [4]



The SIRS Model

The SIR model can successfully be used to model diseases where infection confers lifetime immunity in a very high percentage of victims, for example; chickenpox, measles and rubella. However, several infections only confer a short term immunity, such as influenza


or otherwise permanent immunity can be

reduced in a population with immunodeficiency (for example, measles amongst HIV sufferers). In such a case, the model needs to be revised to allow for reinfection of individuals within the recovered class. One way this can be achieved is by adding a term to the SIR model to cause a time-delayed transfer from the 4 In

the case of influenza, an otherwise healthy individual is normally at least partially susceptible to re-infection within a few years of recovering, mainly due to the rapid mutation of the virus. Effectively, the immune system memory for antibodies to a strain â&#x20AC;&#x2DC;Aâ&#x20AC;&#x2122; of influenza becomes less effective as A shifts genetically, until eventually lymphocytes produced during the immune response are unable to successfully attack the virus. See [43] for SIRS modelling of influenza.


‘Removed’ class back to ‘Susceptible’ dS(t) = −βS(t)I(t) + αI(t − τ ) dt dI(t) = βS(t)I(t) − αI(t) dt dR(t) = αI − αI(t − τ ) dt


This is the SIR model without vital dynamics (A.6) and with an additional time-delayed transfer term αI(t − τ ) between the R and S classes. This term could be thought of as being the same as the transfer between I and R, but delayed by the temporary immune period of length t = τ . As usual with models of constant population, divide through by N to remove scale, and using N = S(t) + I(t) + R(t) reduce to a system of two equations ds = −βsi + αi(t − ω) dt di = βai − αi(t) dt


In a similar calculation to the SIR model, critical points are obtained when i = 0 or βs = α. Stability is investigated in [44], but in summary the infection-free equilibrium is stable for R0 < 1 and the endemic equilibrium stable for R0 > 1. Note how this is identical to the stability requirements of the SIR model; the effect of recovered individuals becoming reinfected in a situation with no vital dynamics (SIRS) is equivalent to a situation where recovery confers permanent immunity, but births and deaths serve to replenish the susceptible class.


The SIVS Model

A similar situation to the SIRS model can occur when modelling a disease that confers no immunity after recovery (covered by the SIS model), but can be temporarily vaccinated against. This could be used to model influenza and the vaccination of vulnerable individuals using a seasonal ‘flu shot’, which generally offers protection only against the most common strains of influenza prevalent at


the time [45], or one of the many diseases where an attenuated vaccine gives relatively short-lived protection. It could also feasibly be used to model treatment by use of antiserum 5 . Consider the following model (which can be compared with model (A.17), the SIR model with vital dynamics and constant population) ds = µ(1 − s − q) − βsi − ps + γi + [µq + ps(t − τ )]e−µt dt di = βsi − (µ + γ)i dt dv = (µq + ps) − [µq + ps(t − τ )]e−µt − µv dt


s+i+v =1 Most of these terms have been discussed before, but several points are noteworthy. 1. Vital dynamics have been included, as in general the effect of vaccination schemes is seen over a longer time period than a few weeks. However, a constant population has been assumed, thus µ = µ(s + i + r). 2. The fraction of newborns vaccinated per unit time is q(0 ≤ q ≤ 0). 3. The rate of vaccination per susceptible individual is the constant p(p ≥ 0). 4. Any individual without vaccination is susceptible, while vaccination protects entirely against infection until it wears off6 . 5. The vaccine confers an immune period of length τ , before the individual immediately becomes entirely susceptible. 5 Antiserums are used when a vaccine or conventional treatment may not exist, typically in cases of an outbreak of a new disease. It involves extracting the polyclonal antibodies from a ’lucky survivor’ of the infection, and using these to boost the immune response of a susceptible. It confers only short term protection to the susceptible as their immune system is incapable of creating the antibodies itself. It is currently the only known treatment for Ebola, and is discussed in detail in [46]. 6 The second part of this assumption is generally accurate for most vaccines, the majority of which have an efficacy of around 95%.


Note also how, as v does not occur in either of the first two equations of (A.24), the system can be thought of as the first two, plus the line

s+i+v =1

in much the same way as r has been eliminated previously. A full analysis of this model is carried out in [47], but the key points are below. Taking initial conditions of

s(t) = s˜(t) ≥ 0, i(t) = ˜i(t) ≥ 0 for t ∈ [−τ, 0]; s˜(0) > 0, ˜i(0) ≥ 0


it can be shown that 1. If i(0) = 0, then i(t) ≡ 0∀t ≥ 0, and limt→∞ s(t) =

µ[1−q(1−e−µτ )] µ+p(1−e−µτ )

=: s0

2. If i(0) > 0, then i(t) > 0∀t > 0, and D = (s, i)|s > 0, i > 0, s + i < 1 is a positively invariant set for the system 3. If βs0 > γ and 0 < s˜(t) ≤ s0 , t ∈ [−τ, 0], then the solution s = s(t) of the system with initial conditions (A.25) satisfies 0 < s(t), s0 ∀t > 0. Furthermore, let R0 =

βs0 µ+γ


βµ[1−q(1−e−µτ )] (µ+γ)[µ+p(1−e−µτ )] .

Then if R0 ≤ 1, the model −µτ

)] has only the disease free equilibrium P0 = (s0 , i0 ) = ( µ[1−q(1−e µ+p(1−e−µτ ) , 0), which is

globally asymptotically stable for R0 ≤ 1 and unstable if R0 > 1. In addition, if R0 > 1, there also exists the globally asymptotically stable endemic equilibrium (µ+γ)[µ+p(1−e Pe = (se , ie ) = ( µ+γ β , (R0 − 1) βµ




Again, the success of a disease invading the population is determined by whether R0 is greater or less than 1.


The SEIR Model

A class which has not been used by any of the above model is the ‘exposed’ class, used to include the period between an individual contracting the disease and becoming infectious and termed the incubation period in most biological


literature. In almost all cases, it takes a number of days for an invading microbe to reproduce sufficiently within the host to infect another susceptible, particularly if the hostâ&#x20AC;&#x2122;s immune system is healthy. This can vary from several days for cholera to around twelve years for Kuru7 , with the average being around one to two weeks. A commentary on historical methods of modelling incubation is available in [49], but it is now more common to use the exposed class â&#x20AC;&#x2DC;Eâ&#x20AC;&#x2122;, in which an individual who is no longer classed as susceptible inevitably becomes infected, but is not at that time capable of passing on the infection. For example ds dt de dt di dt dr dt

= Âľ(1 â&#x2C6;&#x2019; s) â&#x2C6;&#x2019; βis = βis â&#x2C6;&#x2019; (Âľ + )e = Ď&#x192;e â&#x2C6;&#x2019; (Âľ + Îł)i


= Îłi â&#x2C6;&#x2019; Âľr

s+e+i+r =1 Again, constant population is assumed, and recovery from infection confers permanent immunity. The only new term is e, defined as the rate at which exposed individuals become infectious. This could also be represented as a time delay term, possibly Âľs(t â&#x2C6;&#x2019; Ď&#x201E; ), but requires the assumption that the period of exposure is the same for all individuals. (A.26) can be reduced using the constant population assumption to the first three equations, giving the positively invariant region

T = {(s, e, i)|0 â&#x2030;¤ s, e, i â&#x2030;¤ 1, s + e + i â&#x2030;¤ 1}


This gives two possible equilibrium solutions; the trivial disease-free solution 7 Kuru is a universally fatal disease, similar to Creutzfeldt-Jakob disease (CJD), which persisted endemically amongst several tribes in Papa New Guinea, and which may have been spread by cannibalism (particularly of infected brain tissue) as part of funeral rituals. As with other transmissible spongiform encephalopathies, the incubation period is exceptionally long, with many years elapsing before symptoms including trembling, slurred speech and mental disturbances began to manifest [48]. After intervention by both the Australian Government and various Christian groups, cannibalism ceased to be practised, and the disease has largely been eliminated.


P0 = (s0 , e0 , i0 ) = (1, 0, 0) and the endemic equilibrium Pe . Their stability is determined by the contact number, Ď&#x192; = (β)â&#x2C6;&#x2019;1 ((Âľ + )(Îł + Âľ)); if Ď&#x192; â&#x2030;¤ 1, P0 is the only equilibrium in TÂŻ and is globally asymptotically stable. If Ď&#x192; > 1, P0 is an unstable saddle, and Pe exists as a globally asymptotically stable solution in the interior of T [47]. The SEIR model is frequently utilised with the nonlinear transfer term βsp iq between the S and E classes, rather than the standard βsi derived from the principle of mass action, in order to more accurately represent the infectious behaviour of a disease. Analysis is carried out in [50, 51], with the conclusion that stability of the subsequent equilibrium solutions depends solely on the value of p, and that a value of p > 1 results in unpredictable behaviour.


The MSEIR Model

This final model is more complicated than any previously discussed, and is more realistic in describing a disease such as mumps or rubella which combines a period of passive immunity for most newborns, a non-negligible incubation period and a lifetime immunity gained after recovery. The requirement for constant population is also dropped, to be replaced by a constant birthrate of b contributed to by the entire population N , and a death rate of d which is equal across all classes. Limitations of this include the assumption that the diseaseâ&#x20AC;&#x2122;s mortality rate is negligible


, and the global birth rate being N b. It could be

strongly argued that b(N â&#x2C6;&#x2019; (M + I)) would be more accurate, as the passively immune are almost entirely infants and individuals are unlikely to engage in sexual activity for the duration of the infectious period. Further epidemiological assumptions include the standard mass action version of transmission, passive immunity being lost at a rate v, the mean incubation period being 1/ and the 8 While deaths directly attributable to a disease may be very low among healthy individuals, consideration must be paid to any other effects the disease might have. For example, rubella infections to pregnant women are particularly dangerous to the developing foetus, causing serious adverse effects in up to 51% of unborn babies and noticeable effects to birthrates during epidemics [52].


mean infectious period being 1/Îł, resulting in the following model dM dt dS dt dE dt dI dt dR dt

= b(N â&#x2C6;&#x2019; S) â&#x2C6;&#x2019; (d + v)M = (b â&#x2C6;&#x2019; d)S + vM â&#x2C6;&#x2019; β =β


SI â&#x2C6;&#x2019; (d + )E N


= E â&#x2C6;&#x2019; (d + Îł)I = ÎłI â&#x2C6;&#x2019; dR

Note how newborns are split across the M and S class, depending on whether the mother was susceptible or not. It is also possible to check the population assumptions by summing all equations, resulting in the exponential result dN = (b â&#x2C6;&#x2019; d)N dt In line with the analysis in [4], one can eliminate the equation for S by taking S = 1 â&#x2C6;&#x2019; M â&#x2C6;&#x2019; E â&#x2C6;&#x2019; I â&#x2C6;&#x2019; R. Further simplification can be achieved by taking b = d + q, substituting the â&#x20AC;&#x2DC;force of infectionâ&#x20AC;&#x2122; Îť for βi, and dividing through by N to give class size as a fraction of the total population dm dt de dt di dt dr dt

= (d + q)(e + i + r) â&#x2C6;&#x2019; vM = Îť(1 â&#x2C6;&#x2019; m â&#x2C6;&#x2019; e â&#x2C6;&#x2019; i â&#x2C6;&#x2019; r) â&#x2C6;&#x2019; ( + d + q)e (A.29) = e â&#x2C6;&#x2019; (Îł + d + q)i = Îłi â&#x2C6;&#x2019; (d + q)r

within the positively invariant domain

D = {(m, e, i, r)|m, e, i, r â&#x2030;Ľ 0, m + e + i + r â&#x2030;¤ 1}

The threshold quantity is again the basic reproduction number R0 = (β)â&#x2C6;&#x2019;1 ((Îł+ d + q)( + d + q)), and for R0 â&#x2030;¤ 1 there exists a globally stable disease-free


equilibrium P0 at s0 = 1. If R0 > 1, P0 is unstable and an endemic equilibrium Pe exists at se = 1/R0 which is locally asymptotically stable [4].


Additional Modelling Factors

While the above models can be successfully used to model infection, assumptions made may sometimes result in unacceptable deviation from the actual nature of the disease. A factor briefly mentioned in A.2.5 is that the principle of mass action used to derive the transferal rate βis may be unsuitable. For example, if the population is ‘saturated’ with infectives and the density of susceptibles becomes less important in the spread of infection, it may be more appropriate to use βsp iq , with a value of p much less than one. It is also likely that the mixing of susceptibles and infectives will vary markedly in different strata, depending on whether the society is ordered by age, social class, prosperity, or innumerable other factors. This would result in a different rate of spread of disease among each layer of society, and could be modeled by entirely separate systems of equations in extreme cases, or by introducing a form of layering into the original model such as that used in [53] to model epidemic cholera. Another area where basic changes can be made is in the structure of waiting times in classes. In all the above models, a linear term, for example γI has been used to model the movement from one class to another, corresponding to exponentially distributed waiting times within the first class such that the fraction still within the first class after t time units is e−γt . It may be more accurate to model waiting times using a series of gamma distributed sequential stages, described in detail for application to the SEIR model in [54]. An ordinary differential equation model can be used to draw conclusions about the spread of disease in time, but often the spatial diffusion of infection is important. The standard example is the SI model for rabies in foxes [26], which can be used to infer details on the overall spread of rabies as foxes are the main carriers. This basic model can be set up by treating the size of each


class as functions of space and time; S(x, t), I(x, t), and incorporating diffusion terms where necessary to create coupled partial differential equations. Due to the highly territorial nature of healthy foxes, the diffusion term is only included for the infected individuals, resulting in the following ∂S = −βSI ∂t ∂I ∂2I = β − dI + D 2 ∂t ∂x This approach has also been used to model hantavirus spread in mice [55]. In this case a further adjustment was required to allow for the temporal nature of disease spread, where the disease can completely disappear in times of adverse environmental conditions, only to reappear (often after a slight lag) when the mouse population recovers. It may also be necessary to include more direct dependence on time when modelling pulse vaccination, such as the annual polio vaccination of all children under five by the Indian authorities. Finally, a model can often be slightly altered to include a particular characteristic of a disease. For example, a mother infected with AIDS may pass the disease on to her offspring. This is very similar to the passing of passive immunity to rubella discussed earlier, except instead of entering the passively immune M class, the newborn enters the infective I class.


Appendix B

The Keeling and Gilligan Model for Bubonic Plague Keeling and Gilligan’s model for bubonic plague was published in volume 267 of the Royal Society’s Procceedings: Biological Sciences and is included here to give an idea of the amount of complexity that can be built into a model for bubonic plague. It builds upon a previous SIR model by the same authors and consists of a system of eight linked equations; three covering rat dynamics, two flea and a further three for the human population. Parameters linked to the rat, human or flea subsystem are indicated by the appropriate subscripts. The rat equations are    dSR TR SR  = rR SR 1 − + rR RR (1 − p) − dR SR − βR F 1 − e(−aTR ) dt KR TR  dIR SR  = βR F 1 − e(−aTR ) − (dR + mR )IR dt TR   TR dRR = rR RR p − + mR gR IR − dR RR dt KR Where • TR = SR + IR + RR , that is; the total number of rats is the sum of all susceptible, infected and removed (either dead or immune) rats. 77

• rR is the reproduction rate of rats. • KR is the flea carrying capacity of an adult rat. • p is the proportion of newborn rats which inherit a resistance to the disease from their parents. Note that this is a factor which was discounted in this project, as the existence of resistance within rat populations is debatable. • βR is the transmission rate from infected fleas to susceptible rats. • mR is the rate infected rats leave the susceptible class (through death or recovery) and gR is the fraction of these which survive to become resistant. The flea subsystem is in terms of N , the number of fleas living on a rat and F , the number of ‘free’ infected fleas searching for a host.    dN N dF  = rF N 1 − + F 1 − e(−aTR ) dt KF TR dF = (dR + mR (1 − gR ))IR N − dF F dt Where • a provides a measure of the searching efficiency of free fleas. • dF is the rate free fleas die from starvation. Finally, the human equations are dSH = rH (SH + RH ) − dH SH − βH SH F e(−aTR ) dt dIH = βH SH F e(−aTR ) − (dH + mH )IH dt dRH = mH gH IH − dH RH dt Note many of these terms are equivalent to terms in the rat subsystem, but with different values. Keeling and Gilligan point out that the human dynamics play no part in the spread of infection, but were included to allow comparison of death rates with historical data. For a less detailed view, the potential maximum 78

force of infection λH = F e(−aTR ) can be used, which measures the number of free infected fleas searching for hosts. The actual force of infection is lower, as not all fleas will find human hosts. A full analysis of numerical simulations for the model is carried out within the paper Bubonic Plague:A Metapopulation Model of a Zoonosis, but the key points are mentioned here. The model showed that it was possible for bubonic plague to remain at endemic levels within populations of resistant rats, until the susceptible population built to sufficent levels to cause a full epidemic to occur. It required violent epizootics within rats to result in human cases, but the periodicity of these outbreaks closely matched medieval data. The outbreak of human infection lagged behind the main volume of rat deaths by approximately four weeks, which matched more recent data from the Third Pandemic. A key point made within the paper regarding modern treatment of plague outbreaks is that when human cases are observed, the subsequent culling of rats dramatically increases the rate of human infection due to the greater number of free fleas. To extend this idea, it is known that in some medieval towns the killing of dogs in an attempt to reduce the effects of plague actually had some limited beneficial effect. It may be that by reducing the number of predators, the rat population was able to increase, and less fleas remained to take humans as hosts.



Glossary This project has used the mathematical modelling interpretation of several words. In cases where this differs significantly from the medical meaning, both have been given below. Non-standard plural forms have been indicated in italics. Biotope: an area of uniform environmental conditions home to a particular community of species, for example a hedgerow within the habitat of a field. Bubo, buboes: swelling of the lymph nodes, typically appearing as egg-sized lumps in the groin, neck or armpit areas. Endemic: an infection is said to be endemic within a population when the infection is retained within that population without the need for external inputs. Epidemic: occurs when the number of new cases of a certain disease occurring in a given human population, during a given period, substantially exceed the expected number, based on recent experience. An epizootic is an analogous circumstance within a given non-human species. Epizootic: see epidemic. Exposed: see incubation period. Habitat: an ecological or environmental area inhabited by a particular species. Haemoptysis: Coughing up of blood, or blood stained sputum, from the lungs. Haemorrhage: bleeding.


Holometabulous: a term applied to insect groups to describe a life cycle of four distinct stages: an embryo, a larva, a pupa (transitional form between pupa and imago) and imago (a final form, in the case of this project; the flea). Incubation period: in the context of mathematical modelling, the period of time between an individual contracting an infection and the same individual becoming capable of transmitting the infection to other susceptibles. Classed as an exposed individual. Note that this differs from â&#x20AC;&#x2DC;clinical latencyâ&#x20AC;&#x2122;, which is the time elapsed between an individual being exposed to a pathogen and the individual developing medical signs from the infection. Infectious: an individual is capable of passing on a given infection to susceptibles when certain conditions are met. Analogous to contagious. M - passively immune infants: the modelling class of newborns with some form off immunity to a given infection, usually obtained from a disease-resistant mother. Nidiculous: term applied to animals which remain in their nest or birthplace for period of time after birth, during which they are dependent on their parent(s) for nutrition. Pandemic: as epidemic, but across a much larger region, typically a continent or worldwide. Petechia, petachiae: a small red or purple spot on the skin, generally between 1-2mm in diameter, caused by minor haemorrhaging or capillary blood vessels beneath the skin surface. Polygynous: a situation where a male individual has more than one female sexual partner. Pulex irritans: The human flea (see section 3.1.4). Rattus rattus: the black rat, and part of the genus Rattus (see section 3.1.1)


Removed: in a mathematical context, an individual who no longer plays a part in the transmission of disease, either through death, immunity, physical barriers from infectives or some other method. This condition may be temporary. Susceptible: an individual who does not have a given infection, but may contract it when certain conditions are met, such as contact with an infectious individual. Vector: in epidemiology, a species which does not directly cause disease, but transmits the infection by conveying pathogens from one host to another. Note that with respect to humans, the black rat is effectively a vector once-removed, as it conveys fleas which themselves carry the infection. Xenopsylla cheopis: The oriental (or common) rat flea (see section 3.1.2). Yersinia pestis: Bacterium of the family Enterobacteriaceae, which causes bubonic, pneumonic and septicaemic plague when it infects humans.



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