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Infectious Disease Modelling Group Cameron Davidson-Pilon (Wilfrid Laurier), Preeyantee Ghosh (Hyderabad), Yueh- Ning Lee (National Taiwan); Supervisor: Jianhong Wu (York) After a disease outbreak, we can plot a curve for the cumulative number of infected individuals and fit it to a curve. Often, and with enough frequency to raise curiosity, the outbreak will follow the Richard’s model, which is the solution to the differential equation a ⎛ ⎛ C ′( ( t) ⎞ ⎞ C t) = rC ( t) ⎜1− ⎜ ⎟ ⎟ ⎜ ⎝ K ⎠ ⎟ ⎝ ⎠ where C ( t) is the cumulative number of infectives at time t. The model is very similar to the logistic model but possesses the parameter a, an index of the inhibitory effect of the deviation of growth from the exponential relationship. The problem presented to us was to mathematically derive the Richard’s model from first principles. In particular, we were to give epidemiological reasons for the characteristic growth of the accumulated infected. We started by reading the current literature on disease models. The most popular model is the deterministic, compartmental SIR model or a derivative of it. The most important feature of the standard SIR model is it gives relations between the number of infected and susceptibles and their rate of change in the form of differential equations. We started our task by abandoning this assumption and left the rate as an unknown and then assumed a priori that the cumulative total infected, followed the Richard’s model. Using a very creative idea, Yueh- Ning derived a closed form expression for the incidence rate. An alternate novel approach to the problem, derived by Cameron, was to look at an outbreak as a branching process. Using a dynamic expected number of secondary infections per infection, Cameron could derive the expected number of accumulated total infected in terms of the mean duration of infection and a certain integration of the expected number of infectives up to the time of interest. Assuming the accumulated total follows the Richard’s model, we could numerically examine how the disease spread. Both methods show a defining characteristic of a disease following the Richard’s model. Since the Richard’s model often fits real life outbreaks, we decude with confidence that the outbreak will have a constant number of secondary infections, but then quite rapidly decrease to almost zero. Preeyantee’s work focused on the appearance of multiple strains or species of diseases in a single host, called a superinfection. She noticed that during treatment, it is possible to mistake resistance with super-infection, where the non-vanishing symptoms are caused by the other diseases and not the treated one. This idea led to a system of differential equations examining how resistance may evolve under such circumstances. This idea arose from discussions with several participants of the 2010 Summer Thematic Program on Mathematics for Drug Resistance of Infectious Disease that took place during our summer research program. We are trying to put everything together and submit it for possible publication. —Cameron Davidson-Pilon 6 FIELDSNOTES | FIELDS INSTITUTE Research in Mathematical Sciences Cancer Stem Cell Modelling Group Abhishek Deshpande (IIIT Hyerabad), Joy Jing Liu (Ottawa), Philip Marx (Tulane), Tian An Wong (Vassar); Supervisors: Matthew Scott (Waterloo) Mohammad Kohandel (Waterloo), Sivabal Sivaloganathan (Waterloo) On the first day, we met with our advisers, who introduced us to two potential paths for research— modelling of hydrocephalus and stochastic simulations of brain tumour growth. As both were novel concepts to all of the group members, we spent the first few days reading papers to improve our understanding and decide which topic to pursue. After choosing the brain tumour problem, we read recent biology papers in the field, with the goal of formulating simple yet descriptive mathematical models, then testing them using MATLAB and C++. Testing included setting parameters to fit the given data and, in the case that conclusions in the biology papers were inexplicable through the mathematical models, the addition and testing of additional assumptions about cell hierarchies and dynamics. We ended up focusing on a specific paper, A Hierarchy of Self-Renewing Tumor-Initiating Cell Types in Glioblastoma by Chen et. al, which provided its own hypothesis on the still poorly understood tumour cell hierarchy. We focused on mathematically replicating the paper’s data on neurosphere formation (through stochastic simulations) and CD133 percentages (using the system of deterministic equations). In order to explain the high CD133 percentages reported in the paper, we then inserted an element of dynamic dedifferentiation, in which progenitor cells are able to “de-differentiate” to cells with stem-like properties, for example through the process of epithelialmesenchymal transition (EMT). In a field where the cell hierarchy is still not well understood, this could yield interesting insight on current models and provide ideas for future biological experiments. The dynamic evolution of the population can be described deterministically or stochastically. Stochastic models are appropriate for small cell populations during earlier stages of tumour growth. In later periods, when the cell number increases, stochastic fluctuations are minimized allowing a deterministic description of the dynamics. The great advantage of a deterministic framework is its analytic tractability allowing a larger perspective of the population growth and a facilitated estimate of parameter-sensitivity in the model. —Philip Marx and Tian An Wong
First Montreal Spring School in Graph Theory MUCH OF MATHEMATICS IS DRIVEN BY conjectures, and this is particularly true of graph theory. Two of the great conjectures, which drove, stretched, and teased a generation of graph theorists, were Wagner’s Conjecture and The Strong Perfect Graph Conjecture (due to Berge). In the past decade both have been proved, the former by Robertson and Seymour and the latter by Chudnovsky, Robertson, Seymour and Thomas. Both proofs (and this is especially true of the former) represent the culmination of a grand project of research that has built up a whole structural theory surrounding the required result. For a new generation of graph theorists, these results may be taken as given. However, that does not mean their proofs should be ignored. The wealth of knowledge and techniques built up in proving these results is a bounty that the new generation is lucky to inherit. The First Montreal Spring School in Graph Theory was an opportunity for young researchers from Canada and around the world to learn of this bounty from three top academics in the field. In all we had over 50 participants at the school, coming from 15 countries. In 1937, Wagner proved that a graph G is planar if and only if it contains neither K5 nor K as a minor. He then 3, 3 considered the more abstract problem: does there exist, for each surface Σ, a similar characterization (a finite list of excluded minors) for graphs embeddable in Σ? Wagner noticed that to ensure the existence of such characterizations, it suffices to prove that in every infinite sequence of graphs there is one that is a minor of another. It is this latter statement that became known as Wagner’s Conjecture. Wagner’s Conjecture was proved by Robertson and Seymour as part of their grand project on graph minors. Results from this entire project were covered in the lecture course Structural results obtained from excluding graph minors given by Bruce Reed. Bruce is currently in the final stages of completing a book on the graph minors project. The key structural theorem of the course states (approximately) that all graphs without a fixed graph H as a minor can be obtained by gluing together (in an appropriate way) certain ‘topologically simple’ graphs. At the end of the course we turned to Wagner’s Conjecture. It is by no means trivial to deduce Wagner’s Conjecture from the structure theorem; however, the structure theorem does allow us to get a grip on the problem, which is essential to its resolution. One colours a graph by assigning a colour to each vertex in such a way that no two adjacent vertices receive the same colour. If there is a set of ω vertices which are all mutually adjacent (a clique), then it is clear that at least ω colours will be needed to colour the graph. It is a somewhat interesting property of a graph if this number of colours suffices (i.e. if the number of colours needed is equal to the size of the largest clique). It is much more interesting still if this property holds not only for G, but also for all induced subgraphs (graphs that can be obtained from G by deleting vertices). If this is the case then we say G is perfect. It is easy to find graphs that are not perfect. For example, odd cycles of length at least five are imperfect, and so are their complementary graphs (i.e. those obtained by switching edges and non-edges). Thus, for a graph to be perfect it is certainly necessary that it contains neither an odd cycle of length at least five, Clockwise from top left: Maria Chudnovsky; Paul Seymour; workshop participants. (Photos by Juraj Stacho) nor the complement of such a cycle, as an induced subgraph. The Strong Perfect Graph Conjecture, due to Berge (1960), states that this trivial necessary condition is also sufficient. The other course in the school, Structural results obtained by excluding induced subgraphs, included a proof of the Strong Perfect Graph Conjecture. The course was taught jointly by Maria Chudnovsky (Columbia) and Paul Seymour (Princeton). In Paul’s lectures we were introduced to a structure theorem for graphs (now known as Berge graphs) that have neither odd cycles of length at least five nor their complements as induced subgraphs. The structure theorem shows that such graphs either belong to one of a few families of basic graphs or admit one of a few types of decomposition. Since the families of basic graphs are known to be perfect, and graphs admitting such decompositions cannot be minimal imperfect graphs, the proof of the Strong Perfect Graph Conjecture follows. Maria’s lectures focused on newer results and open conjectures, including recent advances related to the Erdős- Hajnal Conjecture (a conjecture that is still open). Simon Griffiths (McGill) FIELDS INSTITUTE Research in Mathematical Sciences | FIELDSNOTES 7
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Page 3 and 4: This summer, 20 undergraduate stude
Page 5: Combinatorial Games Group #2 Matthe
Page 9 and 10: Extended Workshop on Groups and Gro
Page 11 and 12: 010 ections in Geometry and Physics
Page 13 and 14: Directed Polymers and Random Growth
Page 15 and 16: Optimization and Data Analysis in B
Page 17 and 18: ics of the continuum mechanics fram
Page 19 and 20: Richard Cerezo: From my understandi
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Page 23 and 24: FIELDS ACTIVITIES Bridging Research

SCIENTIFIC ACTIVITIES - Fields Institute - University of Toronto

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