Title: Parameter identifiability for PDE models of fluorescence microscopy experiments
Abstract: The dynamics of intracellular proteins is key to many cellular functions, and is often investigated using microscopy experiments such as FRAP (fluorescence recovery after photobleaching). These experiments generate time-series data that average out spatial information about protein localization. Partial differential equations (PDE) models are often used to model the fluorescence dynamics and to infer parameters such as diffusion coefficients or reaction rates. However, it is not known whether these parameters can be identified based on such averaged time-series data. Here, we limitations of known methods in assessing parameter identifiability for PDE models, and propose custom methods for analyzing the dynamics of RNA binding proteins in the development of frog eggs.

Title: Mathematical Model of Immune-inflammatory Response in COVID-19 Patients
Abstract: Investigating the immune-inflammatory response characteristics in COVID-19 patients can predict disease severity and progression. There is evidence that severity of SARS-CoV-2 infection is linked to the dysregulation of the inflammatory immune response. Previous studies have shown the neutrophil-to-CD$8^+$T cell ratio (N8R) can be used in predicting disease severity in COVID-19 patients. Specifically, N8R increases as COVID-19 disease severity worsens. We developed a mathematical model (system of ordinary differential equations) to describe the inflammatory response to SARS-CoV-2 infection. Specifically, the model incorporates cellular and cytokine populations of neutrophils, macrophages, pro-inflammatory mediators, anti-inflammatory mediators, and CD$8^+$T cells. The model is fit to longitudinal data from COVID-19 patients confirmed to have either mild or severe disease. We identify and classify steady states, and conduct bifurcation analysis to determine parameters required for (1) the resolution of inflammation, (2) the persistence of inflammation (chronic inflammation), and (3) alterations in the neutrophil-to-CD$8^+$T cell ratio (used as a predictor of disease severity). This mathematical model can be used to identify markers for severe COVID-19 and give insight into effective treatment protocols and interventions directed at reducing inflammation and the neutrophil-to-CD$8^+$T cell ratio.

Title: Cholera Transmission Dynamic Model with Environmental Impacts of Plankton Reservoirs
Abstract: Cholera is an acute disease that is a global threat to the world and can kill people within a few hours if left untreated. In the last 200 years, seven pandemics occurred, and, in some countries, it remains endemic. The World Health Organization (WHO) declared a global initiative to prevent cholera by 2030. Cholera dynamics are contributed by several environmental factors such as salinity level of water, water temperature, presence of plankton especially zooplankton such as cladocerans, rotifers, copepods, etc. Vibrio cholerae (V. cholerae) bacterium is the main reason behind the cholera disease and the growth of V. cholerae depends on its host in the water reservoir which is the zooplankton because they share a symbiotic relationship. Investigating plankton bloom could be one of the key indicators for predicting cholera outbreaks. Though there are lots of models for cholera transmission dynamics, there are few existing models focused on the environmental impacts of plankton reservoirs. In this work, we have formulated a model of cholera transmission dynamics with the environmental impacts of plankton reservoirs. We have derived the basic reproduction number and discussed various alternative threshold parameters using the next generation matrix approach. Next, we have considered the existence and stability of the disease-free and positive equilibria. Our model analysis could be helpful for scientists to better understand the impact of environmental factors on cholera outbreaks and eventually for a possible prediction of the timing and location of the next cholera outbreak.