Explanation of Final Projects & Illustration of Bayesian estimation

1 fevereiro 2022, 10:30 Erida Gjini

In this class 2 final projects were explained and the students allocated by free choice to one of them. Project 1 involves study of epidemiological model with diffusion in space, project 2 involves infection modelling with antigenic diversity and evolution. 

The second part of the class was dedicated to illustrating an example of dynamic model fitting to data using Bayesian estimation. The example was chosen from the paper: Gjini, E. "Geographic variation in pneumococcal vaccine efficacy estimated from dynamic modeling of epidemiological data post-PCV7."  Scientific reports 7.1 (2017): 1-16. A Matlab code involving use of the mcmcstat package by Haario et al. Haario, Heikki, et al. "DRAM: efficient adaptive MCMC."  Statistics and computing 16.4 (2006): 339-354.was made available.

Invited Lecture: Introduction to the mathematical modeling of blood flow

27 janeiro 2022, 15:30 Erida Gjini

This class included an invited lecture by Jorge Tiago, (Instiuto Superior Tecnico) to provide the students a general introduction to mathematical modeling of blood flow. 

The mathematical investigation of the cardiovascular system has seen vast and clear progress in the last three decades. Such progress
allowed, for the first time, to obtain computational solutions of mathematical models thought to be sufficiently general to be considered realistic. However, a big gap between the state of the art of cardiovascular modeling and its use by the medical community remains open. This is due to the fact that tools based on patient-specific numerical simulations should be sufficiently accurate and reliable to be used by clinicians in medical practice. In this seminar, we will address some techniques that may be used to improve the reliability of computational models of the cardiovascular system.

Discussion of Assignment 3 Solutions

25 janeiro 2022, 10:30 Erida Gjini

This class was dedicated to a discussion of solutions for Assignment 3 (using PDEs in biomedicine). The focus was on traveling waves for reaction-diffusion equations, methods for computation of the wave-speed, applications in wound-healing, cancer, and population dynamics.

Stochastic processes in continuous time: analysis and simulation

20 janeiro 2022, 15:30 Erida Gjini

This class was dedicated to stochastic processes in continuous time, namely the Poisson process, simple birth, simple death and simple birth-death processes as well as general birth-death processes. Several features and properties of these processes were analyzed, focusing on the inter-event times exponential distribution, stationary state distribution, and examples were simulated on the computer. Matlab codes for simulation of stochastic sample paths and numerical evaluation of the stationary probability distributions were elaborated. The Gillespie Algorithm for simulation of a general stochastic process in continuous time was introduced and illustrated on a stochastic SIS epidemiological model. 

Introduction to stochastic processes in biology and biomedicine

18 janeiro 2022, 10:30 Erida Gjini

This class was dedicated to an introduction of stochastic processes with applications in biology, largely based on Chapters 2-3 of "An Introduction to Stochastic Processes with Applications n Biology" by Linda J.S. Allen. Fundamental definitions and properties of Markov chains were covered and several examples presented, focusing on discrete-time Markov chains (DTMC). This included illustration of the stationary distribution of a finite state-space Markov chain, mean recurrence times of each state, probabilities of absorption at two boundaries in the Gambler's ruin problem, time to absorption, and quasi-stationary distributions in the classic birth-death process. Deterministic exponential growth/decay, logistic growth, and epidemiological SIS dynamics can be formulated as special cases of the birth-death stochastic process. It was emphasised that infinitely-many stochastic formulations may exist for the same deterministic model, and they are not all equivalent. Even though they may share the same mean properties, they can have very different variance and stationary distributions. Numerical explorations and simulations of stochastic processes will be performed in the next class.