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The Mathematics of Invasions in Ecology and Epidemiology
Fred Brauer (University of British Columbia),
James Watmough (University of New Brunswick),
Overview of the Field
This summer school was a continuation of the MITACS summer school series on disease modelling. Pre-vious schools have been held at BIRS (2004), York University (2006), Xian Jiaotong University (2006), theAtlanta Center for Disease Control (2007), the University of Edmonton (2008) and the University of Ot-tawa (2009). In contrast to these schools, which focused on the mathematics of epidemiology and publichealth, this school focused on the dynamics of invasions and evolution. The school was organized jointlyby two MITACS research groups: a group of researchers working on mathematical models of infectious dis-eases (www.liam.yorku.ca/research/MADI/) and a group of researchers working on mathematical models ofbiological invasions and dispersal (www.unb.ca/bid).
Historically, the fields of mathematical ecology and the dynamics of evolution have developed separately
and it is only recently that work has been done to begin to bridge these two fields. Models for the evolutionof populations assumed slowly changing or constant populations, and models for ecological populations as-sumed evolution took a much longer time scale than population dynamics. Recent theoretical work has begunto bridge these two approaches allowing population traits to change on the same timescale as population size.
This advance is necessary for a theoretical framework for pathogen evolution in many systems. The influenzavirus provides a pressing example. The timescale of viral evolution is similar to the rate of spread of the virusthough the host population. Any control measures, such as vaccines or antiviral medications, must take intoaccount the rapid appearance of drug resistant strains. Other examples presented in lectures include weedyspecies , HIV and vector-borne parasites such as malaria.
The lectures were divided roughly along two lines: the evolution of pathogens; and the spread of an
invading pathogen. The mathematical foundations of pathogen evolution were outlined in the lectures of Dayand Gomulkiewicz, and applied in the lectures of Gilchrist, Reisberg and Reid. The lectures of Allen andBrauer introduced the basic stochastic and deterministic models for pathogen invasion and spread, while thelectures of Arino and Gourley covered spatial approaches to modelling, first in the context of metapopulations,and the latter in the continuous space. Nelson applied these methods to the spread of a forest insect andintroduced additional modelling techniques for structured host populations. The lecture of Zou extended themodels introduced by Gourley to include delays.
Recent Developments and Open Problems
Several key open problems were introduced in the lectures.
1. A single theoretical framework for pathogen evolution and spread has yet to be developed.
2. New modelling paradigms are necessary for pathogen spread and evolution that incorporate both the
scales of within-host and between-host into a single theoretical framework .
3. Key public concerns surrounding treatment and vaccination require models for the evolution of resis-
tance, such as a resistance to treatment or vaccine in a human infectious disease.
4. Analytical tools are needed to address the spatial component of resistance evolution. For example,
Chloroquine resistance in mosquitoes seems to arise in areas of low transmission , which suggeststhat control measures must take spatial dynamics and evolution of the pathogen and vector into account.
5. Much more analytical work is needed to understand the path of spread of a pathogen through a struc-
tured population. Two examples given were the global spread of novel human pathogens  and thespread of the mountain pine beetle . In the first case, the host population is spatially structured, andin the second, the population is structured by host defenses.
6. Most diseases involve a delay between infection and onset. This leads to many open mathematical
As with previous schools, there were many more applications than could be accepted. In attendance were 35students from mathematics and ecology of which 8 were from the US and 4 from outside north america.
The format for the school consisted of a series of short courses, case studies and student group projects.
Short courses were an important component, providing students with the basic theory on a topic. The partic-ipants were able to apply the theory in group projects. Each short course consisted of one 90 minute lectureand one 60 minute tutorial. They covered topics on basic modelling, both deterministic and probabilistic,and specialized modelling topics such as metapopulations, evolution and dispersal. Case studies consisted of90 minute lectures by leading international researchers. This summer school covered relevant mathematicalbackground and recent progress in the fields of biological invasions in ecology and epidemiology
The highlight of the meeting was the presentations by the students. These students varied from upper
level undergraduates to Postdoctoral and young researchers. The projects were assigned on the second day ofthe workshop, giving a mere five days for development and analysis of an appropriate model. As in the past,the projects were well done, with some exceptional presentations.
The student projects identified six open problems:
• the role of disease in the decline of amphibian species, specifically the role of Chytridiomycosis
• the role of a pathogen in the spread of an invasive forest or agricultural insect;
• the evolution of virulence in the spread of syphilis;
• the reasons for spectacular failures of biological control using exotic species;
• the evolution of a species in a changing habitat;
• the spread of a novel strain of influenza.
Scientific Progress Made
The lectures and case studies presented techniques, ideas and problems at the leading edge of mathematicsand its applications to ecology and evolution. Most of the case studies covered applications of mathematicsfrom an ecological perspective, encouraging all participants to think about new mathematical approaches toproblems. Some of the project reports covered new ground, and we encourage the students to continue theircollaboration wih the goal of publishing their results in a peer reviewed journal.
Outcome of the Meeting
In summary, the school brought together students and researchers in evolution, ecology and epidemiologyusing a variety of modelling and analytical techniques, including game theoretic, statistical, pde, ode anddynamical systems. We hope that many students will continue to work in this emerging research area oftheoretical epidemiology at the interface of mathematics, ecology and evolution.
 Andrew F. Read and Silvie Huijben, Evolutionary biology and the avoidance of antimicrobial resistance,
Evolutionary Applications 2
 Troy Day and Stephen R. Proulx, A General Theory for the Evolutionary Dynamics of Virulence, Amer-
ican Naturalist 163
 Daniel Coombs, Michael A. Gilchrist and Colleen L. Balla, Evaluating the importance of within- and
between-host selection pressures on the evolution of chronic pathogens, Theoretical Population Biology
 Khan, Kamran and Arino, Julien and Hu, Wei and Raposo, Paulo and Sears, Jennifer and Calderon,
Felipe and Heidebrecht, Christine and Macdonald, Michael and Liauw, Jessica and Chan, Angie and
Gardam, Michael, Spread of a Novel Influenza A (H1N1) Virus via Global Airline Transportation, N Engl
J Med 361
(July 9, 2009), 212–214
 William A. Nelson and Mark A. Lewis, Connecting host physiology to host resistance in the conifer-bark
beetle system Theor Ecol 177
 Jing Li and Xingfu Zou, Modeling spatial spread of infectious diseases with a fixed latent period in a
spatially continuous domain, Bulletin of Mathematical Biology
, (accepted March 2009).
 Nolan C. Kane and Loren H. Rieseberg, Genetics and evolution of weedy Helianthus annuus
adaptation of an agricultural weed, Molecular Ecology 17
Federal Register / Vol. 76, No. 159 / Wednesday, August 17, 2011 / Notices The OMB-approved ABCs MRSA form take about 5 minutes and the telephone patients to be contacted for a telephone Daniel Holcomb, Reports Clearance Officer, Centers for Disease [FR Doc. 2011–20919 Filed 8–16–11; 8:45 am] SUPPLEMENTARY INFORMATION: In 1984, BILLING CODE 4163–18–P Congress enacte
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