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Abstract: In the early stages of an outbreak, changes in the exponential growth rate of cases serve as crucial indicators for evaluating the effectiveness of control measures. Establishing a suitable likelihood function is essential for accurate growth rate estimation. To address this, we derive the probability generating function for new cases using a linear stochastic SEIR model and obtain formulas for its mean and variance. We approximate its distribution as a negative binomial and, by comparing this approximation with the probability distribution of simulated data, find that the negative binomial distribution fits the data well. Although the negative binomial distribution provides a good approximation for the distribution of new cases, selecting the most appropriate model for estimating the growth rate remains a challenge. We evaluate the performance of the negative binomial regression model and the hidden Markov model (HMM) in estimating the initial epidemic growth rate. Using simulated daily new cases with known parameters, we fit these models via the Markov Chain Monte Carlo (MCMC) method. Our results show that the confidence intervals generated by the HMM exhibit better coverage probabilities and narrower widths.

ZOOM: https://uvic.zoom.us/j/5826187847?pwd=RlVrb0RoU0xDWTlLUDVkZW54ZThyQT09

Abstract: The solution to the heat equation with non-negative, non-zero initial data is strictly positive. This property generalizes to most parabolic PDEs, but not necessarily to stochastic PDEs. The solution to a heat equation with multiplicative noise may be a compactly supported function, depending on the regularity of the noise coefficient. I will first discuss some classical theorems of this type when the equation has white Gaussian noise, and then discuss a recent result which proves the compact support property for solutions to a class of stochastic heat equations with white stable noise. Along the way we will develop some heuristics for why this property holds, sketch some proof techniques, and discuss connections with superprocesses.

Abstract: Short-term wind forecasting is essential for effectively utilizing wind energy. This talk presents an overview of research projects aimed at improving short-term wind speed forecasting. Initially, the benefits of incorporating atmospheric information using traditional time series models are discussed. Building on this foundation, a new modelling framework is introduced, along with several novel correlation functions designed to account for space- and time-asymmetry in empirical cross-correlations. Several case studies are presented to demonstrate the advantages of the proposed models.

Abstract: Non-local interaction plays a crucial role in various biological systems, ranging from the behaviour of individual cells to the movements of entire organisms. In recent years, there has been a growing focus on mathematically modelling these non-local interactions, often employing advection-diffusion equations with integral terms capturing non-local effects. In this presentation, we propose a natural extension of existing models by introducing a reaction term, specifically addressing birth and death processes in biological systems. We first present some results regarding the well-posedness of this non-local reaction-advection-diffusion model. Then, we will shift our focus to travelling wave solutions, which exemplify biological invasions. We use a combination of perturbation methods, exponential dichotomy, Fredholm operator theory and fixed point argument to prove the existence of travelling wave solutions. Finally, we will end our presentation with some numerical investigations.

Joint work with Dr. Slim Ibrahim and Dr. Mark Lewis.

See abstract (PDF).