Title: Identifiability and Parameter Estimation of Within-Host Model of HIV with Immune Response.

Abstract: This study examines a within-host HIV model that describes the interactions between healthy target cells, infected cells, viral particles, and immune cells. The model has two equilibrium points: an infection-free equilibrium and an infection equilibrium. Stability analysis shows that the infection-free equilibrium is locally asymptotically stable when R0 < 1 and unstable when R0 > 1, while the infection equilibrium is locally asymptotically stable when R0 > 1. The structural and practical identifiability of the model were investigated using differential algebra techniques and Monte Carlo simulations. The practical identifiability of the model was assessed by observing the dynamics of uninfected and infected target cells, immune cells, and viral load. The results emphasize that increasing the frequency of data collection can significantly improve the identifiability of all parameters. 

Title: Computing stable and unstable manifolds using parametrization method.  

Abstract: Our objective is to be able to compute high-order polynomial approximations of stable and unstable manifolds attached to long periodic orbits for maps. Finding the long periodic orbits numerically, by considering the composition map, leads us to significant challenges. Traditional numerical methods are highly sensitive to initial conditions and trajectories accumulate errors over time. Traditional numerical methods are effective for short periodic orbits. In other words, it’s helpful for short term predictions. However, numerical methods can struggle to provide validations for long periodic orbits. Therefore, to ensure the existence and properties of computed orbits, we use Computer Assisted Proof (CAP) by combining numerical approximations with rigorous mathematical verification. In our work we develop standard multiple shooting scheme for periodic orbits without considering compositions of the map. 

Title: Topological representations of distributive lattices

Abstract: The interplay between topology and order is a central theme in the study of distributive lattices. Priestley duality sets up a functorial bridge from algebra to topology, revealing how lattice-theoretic properties manifest in the behavior of ordered topological spaces. We examine how conditions like compactness, disconnectedness, and separation reflect across this duality, and explore examples drawn from particular classes of lattices that illustrate the power of the approach.