Titles and abstracts of talks are given below. The detailed schedule is available here.

**Thursday Morning Session** (June 27, 9:00-12:00)

Moderator: Bill Nelson

Mathematical models of childhood diseases often employ homogeneous time-dependent transmission rates. These model can provide good agreement with data in the absence of significant changes in population demography or levels of transmission, such as in the case of pre-vaccine era measles in industrialized countries. However, accurate modeling and forecasting of transient dynamics after the start of mass vaccination has proved more challenging. This is true even in the case of measles which has a well understood natural history and a very effective vaccine. Here, we demonstrate how the dynamics of homogeneous and age-structured models can be similar in the absence of vaccination, but diverge after vaccine roll-out. We also propose methods to fit such models to long term epidemiological data with imperfect covariate information.

Source-sink dynamics is a problem in mathematical ecology that originated with a 1969 paper of Levins. Suppose a species lives in an environment that is heterogeneous; some locations are favourable to persistence (sources), while others are unfavourable and can only be populated if there is an inflow of individuals to these locations (sinks). What is the impact on survival or extinction of the species as a whole? This problem has been studied extensively in the case of so-called Levins-type metapopulation models, which count the number of locations (patches) in various states and couple them implicitly.

The case of metapopulation models with explicit movement, where individuals move between locations, is less known. I will discuss the solution to a simple problem set in this context: is there a critical number of source patches ensuring population persistence in the entire system? Providing a positive answer to this question will illustrate the power of linear algebra in this type of large scale problem. This is joint work with Nicolas Bajeux and Steve Kirkland (University of Manitoba).

This paper concerns a question that frequently occurs in various applications: Is any diffusive coupling of stable linear systems, also stable? Although it has been known – and in fact, frequently been exploited! - for a long time that this is not the case, we shall identify a reasonably diverse class of systems for which it is true.

The aim of physiologically structured population models (PSPM), is to enable the theoretical analysis of the relationship between mechanisms at the i-level (i for individual) and phenomena at the p-level (p for population).
In such models, individuals are characterised by (discrete and) continuous variables, like age and size, collectively called their i-state.
The world in which these individuals live is characterised by another set of variables, collectively called the environmental condition.
The model consists of submodels for (i) the dynamics of the i-state, e.g. growth and maturation, (ii) survival, (iii) reproduction, with the relevant rates described as a function of (i-state, environmental condition), (iv) functions of (i-state, environmental condition), like biomass or feeding rate, that integrated over the i-state distribution together produce the output of the population model.
When the environmental condition is treated as a given function of time (input), the population model becomes linear in the state.
Density dependence and interaction with other populations is captured by feedback via a shared environment, i.e., by letting the environmental condition be influenced by the populations' outputs.
This yields a systematic methodology for formulating community models by coupling nonlinear input-output relations defined by state-linear population models.

The first step to analyse such models mathematically, consists of associating a dynamical system with the model.
In order to do so, we lift the concept of state from the i- to the p-level by describing a population by a measure (often represented by a density) over the i-state space.
The dynamical system updates this measure.

Even though this is the conceptually designated way to define a dynamical system, alternatively we can use the history of both the birth rate and the environmental condition to code, a bit implicitly, the current state.
The model specifies the rules for extending these variables into the future (such rules take the form of renewal equations and delay differential equations).
The dynamical system updates the histories, so is defined by shifting along the extended functions.

To analyse the models, we need

- a qualitative stability and bifurcation theory for the corresponding dynamical systems (for instance, the linearized, around a steady state, version of the delay equations yields a characteristic equation, and we want to draw conclusions about the local behaviour of solutions of the nonlinear equations from information about the position in the complex plane of the roots of the characteristic equation)
- numerical bifurcation tools

After a general introduction, the lecture will focus on the state-of-the-art of these two topics and speculate about imminent progress.

**Thursday Afternoon Session** (June 27, 13:00-16:30)

Moderator: Felicia Magpantay

Temperature is a major determinant of vector-borne disease transmission because many of the key processes involved in parasite and vector development are highly temperature-sensitive. One of these processes is the extrinsic incubation period (or EIP), which measures the time it takes for an exposed vector to become infectious. A growing body of evidence from a variety of vector-borne diseases shows that while the mean EIP is sensitive to temperature, so is the distribution of individual variation around that mean. This distribution is critical for understanding the risk of vector-borne diseases because the odds are often stacked against a "typical" parasite in a "typical" vector: average EIP is often as long or longer than average vector lifespans. It is easy to show analytically, using the ecological principle of Jensen's Inequality, that ignoring such variation can lead to underestimating disease risk. Despite this, theoretical studies to date have investigated temperature-dependent disease risks using only the average vector and parasite parameters. Using an individual-based model of dengue virus transmission, we show that EIP variation elevates the probability of disease emergence in a human population because exceptionally rapid, yet biologically realistic extrinsic incubation —ignored by modelling only the average — increases the chance of disease emergence. We show that EIP variation causes the greatest increase in the risk of disease emergence in regions where dengue transmission is least expected: at cooler temperatures where the mean incubation period is longer, and the associated variation larger. Finally, using dengue incidence data from Brazil, we demonstrate that incorporating variation in EIP improves disease risk predictions.

Helen J. Wearing^{1}, Amalie McKee^{2} and Rebecca C. Christofferson^{3}

^{1}Department of Mathematics and Statistics; Department of Biology, The University of New Mexico,
Albuquerque, NM, USA

^{2}Department of Biology, The University of New Mexico, Albuquerque, NM, USA

^{3}Department of Pathobiological Sciences, School of Veterinary Medicine, Louisiana State University,
Baton Rouge, LA, USA

Models are increasingly being used to assess the potential for mosquito-borne pathogens to emerge in novel geographic regions.
In particular, characterizing how temperature affects estimates of the basic reproductive number, *R _{0}*, is an important component of understanding the risk of transmission in a warming world.
As we incorporate parameter dependence on temperature and other environmental variables into our predictions of

*R*, we should also (re)consider the structural assumptions of the underlying model framework. Most models of mosquito-borne disease are derivatives of the Ross-Macdonald framework, which assumes that the age at which a mosquito becomes infected does not impact its ability to transmit the pathogen. In addition, a mosquito in such a model could, in theory, live infinitely long and bite infinitely much. In this talk, we discuss a framework that relaxes these assumptions by structuring the female adult mosquito population by the number of bites taken and incorporating a gamma-distributed extrinsic incubation period. We compare output from this model to those from models without biting structure and show that predictions of

_{0}*R*are reduced. Consequently, fitting a model to match observed values of

_{0}*R*without considering model structure may result in incorrect estimates of parameters. Using data on chikungunya virus, we also demonstrate that the predicted optimal temperature for viral transmission depends on model structure.

_{0}Tick-borne disease transmission dynamics is governed by the contacts between various competent hosts and stage-structured ticks through a range of tick-ecological activities (questing, attaching, engorging, and reproducing) and two distinct epidemiological processes, systematic and co-feeding transmissions. In comparison with intensive modeling research on the systematic transmission dynamics, modeling frameworks have been developed only recently to describe the impact of co-feeding transmission on the persistence/extinction patterns of tick-born pathogen spread. As co-feeding transmission requires the co-occurrence of some susceptible ticks at one physiological stage in the proximity of infected ticks at other stages, the development delay and the potential diapause, both depending on the weather conditions, can play important roles in the effectiveness of co-feeding transmission to ensure pathogen persistence in the natural tick-host cycle. The talk summarizes some of the recent development from the LIAM research group and its collaborators on the tick-borne disease spread patterns by incorporating co-feeding and diapause in a structured epidemiological model.

Distributed Compute Labs (DCL) is a Canadian educational nonprofit organization responsible for developing and deploying the Distributed Compute Protocol (DCP), a lightweight, easy-to-use, idiomatic, and powerful computing framework built on modern web technology, that allows any device — from smartphones to enterprise web servers — to contribute otherwise idle CPU and GPU capacity to secure and configurable general-purpose computing networks. By leveraging existing devices and infrastructure — a university’s desktop fleet, for example — a large supply of latent computational resources becomes available at no additional capital cost. DCP makes it possible for everyone — from a student on the Gaspé Coast to a large enterprise in Vancouver — to have access to large quantities of cost-effective computing resources. In summary, the Distributed Compute Protocol democratizes access to digital infrastructure, reduces barriers, and unleashes innovation.

*Dan Desjardins is the Founder and CEO of Distributed Compute Labs based in Kingston ON, Canada.
He is also an Assistant Professor of Physics and Space Science at the Royal Military College of Canada.*

**Friday Morning Session** (June 28, 9:00-12:00)

Moderator: Troy Day

Understanding the biological mechanisms underlying episodic outbreaks of infectious diseases is one of mathematical epidemiology's major goals. Historic records are an invaluable source of information in this enterprise. Pertussis (whooping cough) is a re-emerging infection whose intermittent bouts of large multiannual epidemics interspersed between periods of smaller-amplitude cycles remain an enigma. It has been suggested that recent increases in pertussis incidence and shifts in the age-distribution of cases may be due to diminished natural immune boosting. I will discuss a model that incorporates this mechanism and how it may account for a unique set of pre-vaccine-era data from Copenhagen. Under this model, immune boosting induces transient bursts of age-stratified large amplitude outbreaks. In the face of mass vaccination, the boosting model predicts larger and more frequent outbreaks than do models with permanent or passively-waning immunity. The renewal-equations emphasize the importance of understanding the mechanisms responsible for maintaining immune memory for pertussis epidemiology using an age-structural approach.

Most theoretical models for the evolution of virulence [1,4] assume that infection from one pathogen strain prevent further infections from any other strain. On the other hand, empirical evidence and immunological models [5,6] show that immune memory cannot block forever homologous or heterologous reinfections. Here I show how pathogen evolution can be studied in a model that is stylized after some aspects of [6], where the susceptible population is stratified through the number of times it has been infected. Strain coexistence is then common, and potential evolutionary consequences are explored. The model examined does not allow for immune waning as in [2,3], as this approach does not easily extend to such complex models.

**References**

[1] André, J.-B., & Gandon, S. (2006). Vaccination, within-host dynamics, and virulence evolution. Evolution, 60, 13–23.

[2] Barbarossa, M. V., & Röst, G. (2015). Immuno-epidemiology of a population structured by immune status: a mathematical study of waning immunity and immune system boosting. J. Math. Biol., 71(6–7), 1737–1770.

[3] Diekmann, O., de Graaf, W. F., Kretzschmar, M. E. E., & Teunis, P. F. M. (2018). Waning and boosting : on the dynamics of immune status. J. Math. Biol. 77, 2023-2048

[4] Gilchrist, M. A., & Sasaki, A. (2002). Modeling host-parasite coevolution. J. Theor. Biol., 218, 289–308.

[5] McCaw, J. M., La Gruta, N. L., Laurie, K. L., Cao, P., Quinn, K. M., McVernon, J., … McCaw, J. M. (2016). Modelling cross-reactivity and memory in the cellular adaptive immune response to influenza infection in the host. Journal of Theoretical Biology, 413, 34–49.

[6] Zarnitsyna, V. I., Handel, A., McMaster, S. R., Hayward, S. L., Kohlmeier, J. E., & Antia, R. (2016). Mathematical model reveals the role of memory CD8 T cell populations in recall responses to influenza. Frontiers in Immunology, 7, 1–9.

Over 1.5 billion people are infected with parasitic helminths, and morbidity due to worm infection claims ~5 million "disability-adjusted life years" each year. For some hosts, the association with helminths is brief, as worms are quickly expelled; for others, the association is chronic, lasting months to years. Infection duration has been shown to vary with the genetics of both host and parasite, and also with the environment (e.g., diet), but even when these are tightly controlled, variation in infection duration is the norm. Current mathematical approaches to studying the immune-parasite interaction are incapable of reproducing these results. Inspired by predator-prey theory, infection duration is an assumption, rather than dynamical outcome, of these models. Here I will show how considering the population of immune cells as a structured population, with individual cells characterized by their expression of so-called "master regulator" transcription factors, can provide insight into the causes of variation in infection duration. Specifically, incorporating biologically realistic immune- and parasite-driven feedback processes can lead to either appropriate immune polarization and rapid clearance, or inappropriate polarization and chronic infection, depending on factors like dose, immune priming, and environment. I suggest how these results can help explain the contingent effects of dose observed in empirical systems, and point out the challenges that this additional scale of structure poses for understanding the interactions between within-host and between-host processes in disease systems.

Insects often undergo regular outbreaks in population density that can have devastating impacts on agricultural crops.
For some pests, however, identifying the causal mechanism for outbreaks has proven difficult.
Here we show that outbreak cycles in the tea tortrix *Adoxophyes honmai* can be explained by temperature driven changes in system stability.
Spectral analysis of over 200 outbreaks reveals a distinct threshold in amplitude as a function of temperature.
To study temperature-dependence in more detail, we developed a mathematical model of insect population dynamics that explicitly incorporates the temperature-dependent response of *A. honmai* life history as observed in the laboratory.
The independently parameterized model predicts that insect populations will cross a Hopf bifurcation from stability to sustained cycles as temperature increases.
To test these predictions, we conducted a series of population experiments under a range of constant temperatures and found that temperature can shift the system from a regime where the population is heading to extinction, to a regime with large amplitude cycles.
Our results suggest an alternative explanation for generation cycles in these multivoltine insects.

Coral reef systems can undergo rapid transitions from coral-dominated to macroalgae-dominated states following disturbances, and models indicate that these may sometimes represent shifts between alternative stable states. While several mechanisms may lead to alternate stable states on coral reefs, only a few have been investigated theoretically. I will present model results that illustrates that reduced vulnerability of macroalgae to herbivory as macroalgae grow and mature could be an important mechanism: when macroalgae are palatable to herbivores as juveniles, but resistant as adults, coral-dominated and algae-dominated states are bistable across a wide range of parameter space. These results suggest that managing reefs to reduce chronic stressors that cause coral mortality and/or enhance the growth rates of algae can help prevent reefs from becoming locked in a macroalgae-dominated state.