2018 Spring Math Colloquia
February 21
Michael Otunuga (Marshall University)
Global stability for a (2n+1)-dimensional HIV/AIDS epidemic model with treatments
Smith Hall 518, 4:00pm
Abstract: In this work, we derive and analyze a (2n+1)-dimensional deterministic differential equation modeling the transmission and treatment of HIV (Human Immunodeficiency Virus) disease. The model is extended to a stochastic differential equation by introducing noise in the transmission rate of the disease. A theoretical treatment strategy of regular HIV testing and immediate treatment with Antiretroviral Therapy (ART) is investigated in the presence and absence of noise. By defining R(0,n), R(t,n) and R(t,n) as the deterministic basic reproduction number in the absence of ART treatments, deterministic basic reproduction number in the presence of ART treatments and stochastic reproduction number in the presence of ART treatment, respectively, we discuss the stability of the infection-free and endemic equilibrium in the presence and absence of treatments by first deriving the closed form expression for R(0,n), R(t,n) and R(t,n). We show that there is enough treatment to avoid persistence of infection in the endemic equilibrium state if R(t,n)=1. We further show by studying the effect of noise in the transmission rate of the disease that transient epidemic invasion can still occur even if R(t,n)<1. This happens due to the presence of noise (with high intensity) in the transmission rate, causing R(t,n)>1. A threshold criterion for epidemic invasion in the presence and absence of noise is derived. Numerical simulation is presented for validation.
April 19
Carl Mummert (Marshall University)
The number TREE(3), and counting down in base infinity
Smith Hall 518, 4:00pm
Abstract: The motivation of this talk is a peculiar situation from computer science. In some cases, we know that a program will eventually stop, but we have no way to concretely describe or even bound the number of steps the program will take. For one such program, the number of steps is a number TREE(3) so large that there is no concrete way to describe it or bound it from above.
This talk will introduce TREE(3) and the related result known as Kruskal’s theorem. We will look at some simpler versions of the theorem, leading us to a “base infinity” number system. This system is like base ten, but each digit can be arbitrarily large. We will see that counting down to 1 from a base infinity number is not as easy as it sounds.
The work on base infinity numbers is joint research with mathematics major Samantha Colbert.