Mathematical Biology

Fix a positive integer $n$ and a graph $F$. A graph $G$ with $n$ vertices is called $F$-saturated if $G$ contains no subgraph isomorphic to $F$ but each graph obtained from $G$ by joining a pair of nonadjacent vertices contains at least one copy of $F$ as a subgraph. The saturation function of $F$, denoted $\mathrm{sat}(n, F)$, is the minimum number of edges in an $F$-saturated graph on $n$ vertices. This parameter along with its counterpart, i.e. Turan number, have been investigated for quite a long time.

We review known results on $\mathrm{sat}(n, F)$ for various graphs $F$. We also present new results when $F$ is a complete multipartite graph or a cycle graph. The problem of saturation in the Erdos-Renyi random graph $G(n, p)$ was introduced by Korandi and Sudakov in 2017. We survey the results for random case and present our latest results on saturation numbers of bipartite graphs in random graphs.

The many ways to model an infectious disease go from simple predator-prey Lotka-Volterra compartmentalised models to highly dimensional models. These models are also commonly expressed as the solution to a system of deterministic differential equations. One issue with models that are highly parametrised, which makes them unsuitable for the early stages of an outbreak, is that estimation with a few data points may be impractical. In terms of sampling, small populations are peculiar, e.g., one may find very effective contact tracing along quite noisy data collection and management due to the lack of resources, and a scarcity of methodological developments crafted for those populations. In this presentation, I will argue that in small jurisdictions, stochastic branching and self-exciting processes or variations of basic compartmentalised models are more relevant because of the volatile nature of the disease dynamics, particularly at early stages of an outbreak. Then, we will focus on continuous-time Markov chain compartmentalised models and their parameter estimation through the likelihood. Finally, we comment on the connection of SIR-like models with Hawkes processes. For those unable to attend in person, you can join via Zoom using the link below.

The rainbow and the brain have in common that frequencies are produced. In both cases there is a function of frequency, f, called the power spectral density (PSD). In both cases invasive investigation spoils the investigated object. This talk will describe using noninvasive electroencephalography (EEG) to evaluate the PSD of the brain, via stochastic modelling of associated brain structure. We explore the popular question: does the human brain manifest the mysterious property called "1/f"? Is the PSD of the brain proportional to the function "f to the power -a", for some a > 0, and hence scale-free? What would that mean about the brain? Independent of these fascinating questions, the exponent, a, has many successful applications as a diagnostic of brain disorders and treatments.

Following a vaccine inoculation or disease exposure an immune response develops in time, where the description of its time evolution poses an interesting problem in dynamical systems. The principal goal of theoretical immunology is to construct models capable of describing long term immunological trends from the properties and interactions of its elementary components. In this talk I will give a brief description of the human immune system and introduce a simplified version of its elementary components. I will then discuss our contributions to the field achieved through my postdoctoral work with Dr. Jane Heffernan at York University. I will focus on our mechanistic modelling work describing vaccine-generated SARS-CoV-2 immunity and applications of our work towards understanding vaccination responses in people living with HIV. Finally, I will discuss our on-going work towards developing a machine learning public health platform capable of predicting immune response outcomes from repeated-dose immunological data.

Viral infections are social processes. Viral replication requires shared gene products that can be used by multiple viral genomes within the same cell, and hence act as public goods. This gives rise to viral cheats, a type of molecular parasite formed by large deletions, that spread by exploiting public goods encoded by full-length viruses. Cheats exist across the viral universe, arise frequently in laboratory infections, and reflect the emergence of evolutionary conflict at the molecular level. In this talk, I will explore two evolutionary consequences of viral cheating that play out at different timescales. Firstly, we will consider the evolution of multipartite viruses, in which the genome is fragmented, and each fragment must separately infect a host. This genome structure comes with clear costs, but has nevertheless evolved multiple times, and today accounts for nearly 40% of known plant viral species. Previous explanations for the evolution of multipartitism have focused on group benefits, but typically require unrealistic rates of coinfection, especially for multipartite viruses with more than two segments. We will argue that cheating provides a contrasting explanation. By combining evolutionary game theory models with agent-based simulations, we will show that the invasion of mutually complementing viral cheats can drive the evolution of multipartitism under far more permissive conditions, including transitions to highly multipartite viruses. This framework shows that multipartitism need not be a group-level adaptation, but can instead emerge as the evolutionary endpoint of the tragedy of the commons. Secondly, we will consider the evolution of cheat-driven extinction in viruses. Cheats emerge spontaneously in laboratory infections of almost all known viruses, driving drastic reductions in viral population sizes. As a result, virologists have long argued that viral infections may be ‘self-limiting’, a claim supported by recent discoveries of cheats in natural viral infections. However, it is unclear whether viral infections provide enough time for viral cheats to emerge, spread, and drive cooperator extinction. Here, we present a birth-death model that incorporates mutation, demographic noise, and a frequency-dependent selective advantage to cheating. We identify qualitatively different dynamical regimes and the timescales under which they lead to viral extinction. We further show that our model can produce characteristic signatures of selection, opening the door to evolutionary biomarkers for predicting the outcome of viral infections from sequencing data. This approach argues that cheating may not only be relevant over long evolutionary timescales, but may also shape viral dynamics in clinically relevant ways, analogous to the emergence of cancer in multicellular organisms.