Research

Applied Mathematics
(Leader: )

April 19, 2023

Title: Pertinence of a change in the meningococcal C vaccine schedule in the Community of Valencia. Agent-based modeling
Speaker: Rafael J. Villanueva, Department of Applied Mathematics at the Universitat Politècnica de Valéncia and University Institute of Multidisciplinary Mathematics (UPV), Valéncia, Spain
Time: 3:00pm–4:00pm
Place: Zoom

Abstract

Meningococcal C (MenC) conjugate vaccines have controlled invasive diseases associated with this serogroup in countries where they are included in National Immunization Programs and also in extensive catch-up programs involving subjects up to 20 years of age. Catch-up was important because it prevented disease in adolescents and young adults at risk and decreased transmission of the bacteria since it was in this age group where the organism was circulating. Our objective is to build an agent-based mathematical model to simulate a new vaccination schedule to achieve maximum seroprotection in these groups.

April 5, 2023

Title: Classification of Lattices of Balanced Colorings of Networks
Speaker: Hiroko Kamei, Division of Mathematics, University of Dundee, Scotland UK
Time: 3:00pm–4:00pm
Place: Zoom

Abstract

We consider robust patterns of synchrony (clusters) of a network, which are purely determined by the network structure. Such a cluster is determined by finding a balanced coloring of the nodes of the network, in which each cluster of nodes is labelled with a different color and receives the same numbers of inputs from each color. Balanced colorings also correspond to subspaces that are flow-invariant for all admissible ODEs. We represent all possible patterns of synchrony as a complete lattice by using the hierarchy structure of node partitions. We show that assigning a well-defined non-negative integer index to a lattice leads to synchrony-breaking bifurcation analysis of networks. After classifying lattice structures using the Jordan normal forms of the adjacency matrices of each network, we also show how some lattice structures can be reduced by identifying an equivalence relation which leads to multiple bifurcating branches from a single bifurcation point along these equivalent synchrony subspaces.

Wednesday, March 22, 2023

Title: Projection Methods for Monotone Mappings in Nonsmooth Optimization
Speaker: Oday Hazaimah, Visiting Assistant Professor, Saint Louis University
Time: 3:00pm–4:00pm
Place: Zoom

Abstract

Optimization algorithms are at the core of machine learning models in which gradients and subgradients play a crucial role in nonsmooth optimization and variational inequalities. In this talk, preliminaries and relations between optimization and inclusions are introduced, important and existing methods are presented. The proposed analytical iteration is a natural modification of the classical extragradient algorithm in which it finds the solution of the sum of two monotone operators by evaluating the smooth operator twice per iteration. The convergence and complexity rates are established. To perform the projection process for monotone operators, a suitable separating hyperplane must be found in the spirit of the cutting-plane idea.

Wednesday, March 1, 2023

Title: Creation & annihilation of sinks in gradient dynamics related to statistical learning
Speaker: Mike Field, Research Associate, UC Santa Barbara
Time: 3:00pm–4:00pm
Place: Zoom

Abstract

The talk is, for the most part, introductory with minimal prerequisites. We begin with a review of some ideas from bifurcation theory — illustrated by basic examples. Motivated by a problem in statistical learning, we then look carefully at a specific bifurcation problem involving the symmetric group. The approach here is new and some results are surprising. Discussion of the motivating problem — which originates from neural networks and machine learning — is left to the end of the talk.

The talk is based on joint work with Yossi Arjevani (School of Engineering and Computer Science, The Hebrew University, Israel).

Wednesday, February 15, 2023

Title: Direct and Indirect Methods for Optimal Control of Virus Propagation in Plants
Speaker: Benito Chen-Charpentier, University of Texas at Arlington
Time: 3:00pm–4:00pm
Place: Zoom

Abstract

In many applications of mathematical modeling to biology, economics, social sciences and engineering, the objective is to find optimal solutions. Usually, we want to minimize an objective function depending on a number of functions subject to constraints given, for example, by systems of differential equations. Two main numerical approaches are used to solve these optimal control problems, depending on whether the problem is optimized first and then discretized, or vice versa. Each of these two approaches has its advantages and disadvantages. In this paper we describe both methods an apply them to a plant virus propagation model, where the virus is propagated through a vector that bites the infected plants. The model includes delays due to the time the virus takes to infect the plant and the vector, and seasonality due to the dependence of the behavior on the seasons.

The objective function is the total cost to a farmer of having infected plants and includes the actual cost of a plant plus the cost of the controls which are insecticides and a predator species that preys on the insects. Numerical simulations are done using both methods and comparisons are made.

Wednesday, February 1, 2023

Title: From Calculus to Optimal Control: Optimization for Sustainable Fishery Harvest
Speaker: Suzanne Lenhart, University of Tennessee
Time: 3:00pm–4:00pm
Place: Zoom

Abstract

Marine fisheries are a significant source of protein for many human populations, and models can suggest management policies for natural renewable food resources. We will start with the concept of maximum sustainable yield modeled with one ordinary differential equation including constant proportional harvesting using calculus. Optimal control techniques can be used to design time varying harvest rates in systems of ordinary differential equations. We will illustrate these techniques with an example of a food chain model on the Turkish coast of the Black Sea. Incorporating data from the anchovy landings in Turkey, optimal control of the harvesting rate of the anchovy population in a system of three ordinary differential equations (anchovy, jellyfish, and zooplankton) gives management strategies. Finally, the idea of marine reserves in simple spatial models will be introduced.