A key challenge is the limited availability of reliable data for both sexes, particularly regarding prevalence. While female-specific data is abundant, male data is often overlooked in existing models, despite men playing a crucial role in HPV transmission. Addressing this gap could enhance the accuracy of models and lead to more effective public health interventions. Expanding data collection efforts to better represent males is essential for robust modelling.

The study will develop and calibrate an age- and sex-stratified mathematical model using Bayesian inference methods and MATLAB. This model will incorporate complex contact patterns and disease dynamics to simulate various vaccination scenarios and assess their long-term impacts on HPV transmission and health outcomes. The research will also estimate the potential reduction in cervical cancer cases resulting from the optimal vaccination strategy, providing quantitative evidence of its effectiveness.

Ultimately, this study aims to inform public health policy by identifying the most effective vaccination strategies for controlling HPV and preventing cervical cancer, while advocating for more comprehensive data collection to improve future modelling efforts.

Joint work with Tanjima Akhter and Michael Y Li from University of Alberta.

Our goal was to see how far we could push the modular method to avoid using linear forms of logarithms. By using only the modular method in combination with an expanded set of coupled generalized Fermat equations which are derived using descent, we explain how to asymptotically solve Sch\"affer's conjecture for select $k$, additionally using the multi-Frey technique and theorem of Darmon-Merel.

In 2003, Hassett constructed a weighted variant of $\overline{M_{0,n}}(\mathbb{R})$: For each of the $n$ labels, we assign a weight between 0 and 1; points can coincide if the sum of their weights does not exceed one. We seek combinatorial presentations for the fundamental groups of Hassett spaces with certain restrictions on the weights. In particular, we express the Hassett space as a blow-down of $\overline{M_{0,n}}$ and modify the cactus group to produce an analogous short exact sequence. The relations of this modified cactus group involves extensions to the braid relations in $S_n$. To establish the sufficiency of such relations, we consider a certain cell decomposition of these Hassett spaces, which are indexed by ordered planar trees.

Our research was focused on determining explicit formulas and polynomial time algorithms for the broadcast independence number of various types of graphs. This parameter is difficult to compute for graphs in general, so we restrict the problem to specific classes of graphs to make use of their special structural properties. One type of graph that we examined in our research was split graphs. Split graphs are defined to have a partition of its vertices into a clique and an independent set, a property which is specific to this class of graphs. Additionally, all split graphs have diameter two or three. Using these special properties, we determined explicit formulas for the broadcast independence number of special types of split graphs. We also showed that the broadcast independence number is polynomial time solvable for all split graphs.

In this project, we devise a procedure to obtain a $(2+c)$-regular graph of minimum order from a given cycle graph $C_n$, where $c,n\in\mathbb{Z}^+$ and $n\ge3$. We employ the use of cases to determine the minimum number of vertices that must be added to $C_n$ such that the resulting graph $R$ is $(2+c)$-regular. The results of this project demonstrate that if $c\le n-3$, then our desired graph $R$ can be obtained by adding at most 1 vertex. Additionally, if $c>n-3$, our findings indicate that $R$ can be obtained by adding $3+c-n$ vertices. Some additional results regarding the size and Hamiltonian property of $R$ are also presented at the end of this project.

We extend the definition of $2$-CFF to include a graph($G$), called $\overline{G}$-CFF, where the edges of $G$ specify the pair of subsets whose union must not cover any other subset. We denote by $t(G)$ as the minimum $t$ for which there exists a $\overline{G}$-CFF. Thus, $t(K_n) = t(2, n)$. We will discuss some classical results on CFFs, along with constructions of $\overline{G}$-CFFs. We prove that for a graph $G$ with $n$ vertices, $t(1, n) \leq t(G) \leq t(2, n)$ and for an infinite family of star graphs with $n$ vertices, $t(S_n) = t(1, n)$. We also provide constructions for $\overline{P_n}$-CFF and $\overline{C_n}$-CFF using a mixed-radix Gray code. This yields an upper bound for $t(P_n)$ and $t(C_n)$ that is smaller than the lower bound of $t(2, n)$ mentioned above.