Mathematical models

Guest blog: Multi-disciplinary interactions under the Mexican sun – reflections from the ISVEE conference

Guest blog by Tiziana Lembo and Liliana Salvador with contributions from Rowland Kao and Louise Matthews.

What is the role of scientific conference? Is it to present our research and expound upon our scientific philosophies? Is it to hear people talking about the interesting research that they are developing? Or is it to meet old friends and make new ones while also traveling to interesting places? All of these aims and more were fulfilled when a group of us left umbrellas and raincoats behind to travel to sunny and warm Mérida, Mexico, for two stimulating weeks of ISVEE 14 (International Society for Veterinary Epidemiology and Economics) conference and workshops. The sun, heat, margaritas, “Jarana” tunes and dances, and the colourful decorations of the “Día de Muertos” created an ideal atmosphere for productive and enlightening scientific interactions.

Dia de muertos

The beginning of ISVEE 14 coincided with the “Dia de Muertos” (Day of the Dead), an ancient Mexican celebration to remember ancestors, family members and friends who have died. Traditionally, altars (“ofrendas”) are built that are laden with decorations, and favourite foods and beverages of teh departed. Above an altar dedicated to the famous Mexican painter Frida Kahlo de Rivera at La Casa Azul in Mexico City, where Frida Kahlo lived and worked most of her life (photo: Tiziana Lembo)

Every three years, ISVEE provides opportunities for academics from a range of different disciplines, policy-makers and stakeholders from the private sector to come together to share their expertise in innovative research, technological developments, and policy health agendas. By blending a wide range of disciplines to address health issues of global importance, the Boyd Orr Centre has a major role to play in all these discussions. What were our contributions to the ISVEE agenda this year?


Let’s start with our research on the very topical bovine Tuberculosis (bTB), caused by Mycobacterium bovis. Rowland Kao discussed a subject that is very close to his heart – the transformative role of Whole-Genome-Sequencing (WGS) in elucidating complex transmission dynamics and disease maintenance patterns in multi-host systems. He provided examples of how the approach has been used by the Glasgow team and collaborators to expose the role of wildlife in the maintenance and transmission of bTB to cattle in different parts of the world, including Great Britain, the United States, and New Zealand. He contrasted currently available data with optimal data and listed some of the key features of an ideal dataset for WGS approaches, most importantly dense, representative sampling across all important hosts; representative samples across populations, but also the way that evolutionary analyses and model-based epidemiological approaches complement each other in interpreting these data.


Joseph Crisp and Liliana Salvador provided examples of the use of WGS to tackle bTB in New Zealand and US wildlife and cattle populations. Joe showed that the evolutionary substitution rates of M. bovis in his study populations, including cattle, possums and other wildlife are higher than previously thought and that non-cattle reservoirs were heavily involved in the maintenance of M. bovis in the sampled population. Liliana focused on bTB transmission amongst elk, deer and cattle in Michigan, US, and demonstrated that elk is the only one of these species with spatial and temporal clustering of M. bovis. In addition, for the available data, she showed that there is no evidence of transmission between elk and cattle and that cross-species transmission in Michigan is likely due to deer.


Liliana also presented her work on surveillance of bTB in Low Risk Areas (LRAs) in England. She showed that larger herds and herds that receive a high number of animals from high-risk areas are most exposed to infection. She also demonstrated that in LRAs there is no clear advantage of testing herds for bTB more frequently, since it would give no increase in the number of detected breakdowns, but the number of false positive would rise considerably. However, adopting risk-based surveillance, where herds that are at higher risk of infection are targeted, can improve the efficiency of the testing regime by increasing the number of identified cases and reducing the number of herds tested.


An entire session of the conference was dedicated to foot-and-mouth disease (FMD) with a focus on the latest research efforts in Africa, Asia and Latin America. The work of the Boyd Orr Centre on endemic FMD in Tanzania featured prominently. Findings from micro-econometric studies investigating impacts of FMD outbreaks on individual livestock-owning communities were presented by Tom Marsh, a collaborator from Washington State University. These analyses have revealed that FMD outbreaks in cattle cause reductions in milk production, traction capacity and income from livestock sales, and that households would spend more on child education if they were not affected by milk losses due to FMD. Tiziana Lembo’s talk focused on epidemiological studies investigating temporal and spatial FMD virus dynamics in East Africa to devise appropriate control strategies. She showed that four different serotypes (A, O, SAT 1 and SAT 2) are responsible for FMD outbreaks in cattle in northern Tanzania, and that there is a pattern of serotypic dominance over time across Tanzania and Kenya, which allows us to predict the timing of epidemics of specific serotypes. The implications are that livestock vaccination could target given serotypes ahead of expected outbreaks, using monovalent vaccines, which are much more readily available than polyvalent vaccines needed to cover all of the wide range of serotypes circulating in these areas.


In her talk, Louise Matthews tackled the question of whether farmers would adopt a new diagnostic test for early detection of sheep scab at the subclinical stage. The advantages of using the test are that it would allow farmers to detect and treat the disease before clinical signs, reducing production losses, and also reducing transmission to other sheep and flocks. However, the farmers would need to pay for the test and may be reluctant to do so if they believe their flock to be at low risk of infection or if their neighbour is using the test, therefore not posing a transmission risk. These advantages and disadvantages can be assessed using a game theory framework that predicts whether farmers will adopt the test and how that uptake depends on test cost. The outcome was uptake of the test when farmers are at high risk (i.e. when they had experienced clinical sheep scab in the previous year), leading in the long term to a reduction in the proportion of infected farms by around 50%.


Harriet Auty, a collaborator from Scotland’s Rural College, presented research on human African trypanosomiasis caused by Trypanosoma brucei rhodesiense in Tanzania. She talked about the relative importance of different wildlife species in the reservoir community for human trypanosomiasis in multi-host populations of the Serengeti National Park. She showed that species such as bushbuck, reedbuck and impala, which are frequently infected with T. brucei, might play an important role in the reservoir community, even though they are not regular food sources for the tsetse vector. Conversely, elephant or giraffe are frequently fed on but rarely infected, indicating they may play a role in dampening transmission, and suggesting how changes in wild species composition could impact on human disease risk.


As always, ISVEE also provided a forum for conference delegates to update and strengthen their skills in a number of methods and topics through workshops run by academic colleagues from around the world. For instance, Tiziana Lembo benefited from training and discussions in data management and analyses in R organised by the Swedish National Veterinary Institute, as well as in the use of economics for animal health decision-making coordinated by the Royal Veterinary College and collaborators.

Back to the rain and grey skies now, we have many memories and knowledge to treasure from the land of revolution, music and art.


Street mural depicting Emiliano Zapata in Tepoztlan, State of Morelos, Mexico during the Mexican Revolution. Zapata remains an iconic figure in Mexico to this day (Photo: Tiziana Lembo).










ode to life.png

An ode to life (“Viva la Vida”) by the Mexican painter Frida Kahlo de Rivera, La Casa Azul, Mexico City (Photo: Tiziana Lembo).

The research presented and our attendance were funded by: BBSRC / DFID / Scottish Government (Combating Infectious Diseases of Livestock for International Development initiative), MSD Animal Health, and Scottish Universities Life Sciences Alliance (Tiziana Lembo); BBSRC/DFID (Zoonoses in Emerging Livestock Systems) and EPIC (The Scottish Centre of Expertise in Animal Disease Outbreaks); Defra, NSF/BBSRC

Sherlock Holmes and the deductive paradigm of forensic epidemiology.

No blogs for ages, and then two in one week …

Deductive logic at its finest

Earlier this year, Dom Mellor was giving a talk to the epidemiology group at Glasgow, where he started by saying that, in his view, Sherlock Holmes represented the perfect example of forensic epidemiology. In a sense he was right, and at least some of you will know that it is commonly believed that the Holmesian forensic technique was based on Conan Doyle’s experiences as an Edinburgh medical student, where the medical doctor and University Professor Joseph Bell impressed the young student in his lectures. It was said that “all Edinburgh medical students remember Joseph Bell – Joe Bell – as they called him. Always alert, always up and doing, nothing ever escaped that keen eye of his. He read both patients and students like so many open books. His diagnosis was almost never at fault.” Sherlock Holmes most famous quote, taken from the Sign of the Four: “When you have eliminated the impossible, whatever remains, however improbable, must be the truth” is the iconic expression of deductive logic, and it could be said to be the ultimate goal of forensic epidemiology. I recall a colleague saying to me “Identify and eliminate the source of Infection, and you eliminate the epidemic”, apparently quoting from the highly respected veterinary epidemiologist, Prof. Mike Thrusfield at the Dick Vet School in Edinburgh, though I cannot comment on the accuracy of the quote. Of course, it is also well recognised that it would usually be impossible to be so sure as Sherlock Holmes in real life, but this nevertheless represents a sort of platonic ideal of forensics.

“Balance of probabilities, little brother” Mycroft Holmes, Hearse, Sign and Vow (from

Move forward a century and more, and the hugely popular TV series ‘Sherlock’ presents a modern updating of the old stories, an updating which, to my great surprise, I have thoroughly enjoyed. In the third series, in the episode ‘Hearse, Sign and Vow’, Sherlock and his older, more intelligent brother Mycroft are engaged in a contest to characterise a man from only his woolly hat. In this contest Sherlock queries one of Mycroft’s “deductions”, when Mycroft replies “Balance of probabilities, little brother.” Now this statement is decidedly un-Holmesian – in the world of Arthur Conan Doyle’s Sherlock Holmes, probabilities have nothing to do with it. This statement is in fact, one of inductive logic. And it could be argued that the mathematical and statistical modelling of infectious diseases lies very much more in this inductive tradition. Not so much concerned with identifying the single chain of transmission, modelling traditionally concentrates on the identification of general, population level principles of transmission, and an overall ‘balance of probabilities’ of getting the right pattern.

These two traditions – that of the forensic epidemiologist and the mathematical/statistical epidemiologist do not sit easily together, and indeed it could be argued that much of the controversy over the 2001 Foot-and-mouth disease (FMD) epidemic in Great Britain can be attributed to precisely that clash of cultures.

Phylodynamic reconstruction of a foot-and-mouth disease (FMD) epidemic. (A) Identified likelihood that a particular infected premises was the source of another infected premises based on a space–time–genetic model. Circle size is proportional to the relative likelihood of that event. (B) Spatial relationships among premises in the dataset. Reproduced from Morelli et al. PLoS Pathogens 2012.

Phylodynamic reconstruction of a cluster of cases from the 2001 FMD epidemic in Great Britain. (A) Identified likelihood that a particular infected premises was the source of another infected premises based on a space–time–genetic model. Circle size is proportional to the relative likelihood of that event. (B) Spatial relationships among premises in the dataset. Adapted from Morelli et al. PLoS Pathogens 2012.

Now however, the integration of rapid high throughout sequencing of pathogens allows us to trace to a very fine scale the movement of pathogens from place-to-place, and even from individual-to-individual. Combined with mathematical models, this can often lead to very precise identification of likely sources of infection. The figure here is taken from a paper by Marco Morelli while he was working with Dan Haydon at Glasgow, illustrating precisely that kind of analysis using data from the 2001 FMD epidemic. Of course the most likely source under one model of transmission is not necessarily proof that the relationship is the true one (e.g. what if another model gives an equally strong but different prediction?) and there are many challenges still to be addressed. Despite these issues, the future is bright and it is just possible that, through these new technologies and approaches, we can at last approach that Holmesian ideal.

The Goldsboro Incident really happened! Nonlinearity, mathematical and statistical models


Opening up the paper today, I was pleased to see this story on the front page of the Guardian, about the Goldsboro incident in November 1961. Why pleased? Well for years the Goldsboro incident has been my analogy of choice for explaining the difference between linearity and nonlinearity, based on an interpretation of nonlinearity inspired by George Sugihara on physical vs. biological noise. I’ve always prefaced this analogy by saying that it was unconfirmed but useful – and now it appears to be true! So what happened in Goldsboro? From the companion piece in the Guardian:

The document, obtained by the investigative journalist Eric Schlosser under the Freedom of Information Act, gives the first conclusive evidence that the US was narrowly spared a disaster of monumental proportions when two Mark 39 hydrogen bombs were accidentally dropped over Goldsboro, North Carolina on 23 January 1961. The bombs fell to earth after a B-52 bomber broke up in mid-air, and one of the devices behaved precisely as a nuclear weapon was designed to behave in warfare: its parachute opened, its trigger mechanisms engaged, and only one low-voltage switch prevented untold carnage.


The conventional interpretation of nonlinearity. Doubling the input either more than doubles (e.g. oversteer in a car) or less than doubles (e.g. understeer) the response.

Our formal understanding of nonlinearity is based on the idea that, if we consider a response to an input, doubling the input will result, if there is a linear response, in a doubling of the response. Thus if I press the accelerator on my car twice as hard, I might expect to travel (approximately) twice as fast. In a nonlinear response, the return is either more than or less than twice.   However, an alternative understanding of nonlinearity is illustrated by the Goldsboro Incident, where the difference between 5 of 6 safeties failing, and 6 of 6, is the difference between an incident quietly swept under the rug for 50 years, and a monumental disaster.

The difference between 5 switches being triggered and 6 is the difference between a hole in the ground and a nuclear explosion.

The difference between 5 switches being triggered and 6 is the difference between a hole in the ground and a nuclear explosion.

This interpretation of nonlinearity can be viewed in terms of the difference between multiplication and addition. We are quite good at predicting additive phenomena; the problem is, we are are less proficient when it comes to multiplication. The recent story of the death of four year old Daniel Pelka (and this is a type of story repeated with tragic Sisyphean regularity) is a case in point. How could this happen? How could so many safety checks fail? How could so many people miss the warning signs? The truth of the matter is likely to be that there are many, many more cases where “the system” almost fails, but with no observable consequence. Overburdened, pressurised staff, sometimes under motivated or under pressure not to raise alarms unnecessarily, may cut corners or make mistakes far more often than we are aware. It is also likely true that because there is no immediate consequence to these actions (the effect of nonlinearity) the potential for disaster is missed. The question may in fact not be, why does this happen, but why does it not happen more often?

And this leads us to the concept of extrapolation and mathematical and statistical models. Statistical models are fantastically valuable tools for rigorously describing relationships in data. However they are fundamentally ontological in nature; that is, built to classify rather than to explain mechanisms, and thus the ultimate arbiter of the quality of a statistical model is the fit to the data. Of course, in designing the statistical model and in interpreting it, a good scientist will be aware of the existence of these underlying mechanisms. This awareness will drive both experimental design and observation, and the interpretation of the statistics. However, these considerations lie outside the statistical model itself. In contrast, mathematical models should be phenomenological, i.e. built to directly describe the often nonlinear relationships between variables, and therefore they are better suited to extrapolate or predict away from the data, rather than interpolate. What is often not understood, is that even very good mathematical models may give an inferior fit to the statistical within close bounds of the data – the aim is not to develop the best fit to the data, but to better be able to predict what may occur, when moving farther away from known data.

Mathematical models can often provide a poor fit the data, but, if formulated to appropriately describe a fundamental aspect of the data, can provide insight into possible trends as we move away from the known data.

Mathematical models can often provide a poor fit the data, but, if formulated to appropriately describe a fundamental aspect of the data, can provide insight into possible trends as we move away from the known data.

Of course, this is at best a caricature of both mathematical and statistical models, with modern quantitative sciences using in various ways combinations of both of them. Nevertheless there is a fundamental difference in models that aim to describe, and models that aim to explain, a difference that must be considered when evaluating the interpretation of any model.

Nonlinearity is a critical concept in ecology, evolution and epidemiology. The emergence of new pathogens is one example of this. For example, in a paper a few years ago, Nim Pathy and Angela McLean used a theoretical model, to ask whether or not a pathogen (in this case, avian influenza) that has caused hundreds of cases but with little transmission indicates that the species barrier cannot be crossed. Another way of looking at this question is to ask which is worse, 4 introductions of avian flu into humans from birds, or a single introduction, where a chain of 4 infections in humans occurs but the disease then fails? Extrapolation from currently observed data requires an insight into the underlying mechanisms that drive the phenomenon to be understood (in this case, the emergence of a new human pathogen). What Pathy and McLean showed using nonlinear mathematical models, was that a lack of demonstrated transmission cannot rule the possibility of adaptability, regardless of how many zoonoses have occurred – thus even when we think we are safe, we are not necessarily so.

Of course, while I am (unsurprisingly) a keen proponent of the use of mathematical models, it must always be kept in mind that prophecy is difficult, and the biblical admonition against following false prophets reflects the popularity of trying to predict the future, the frequency of our failures, and the ease with which we can be led into following those predictions, especially when espoused by recognised experts.