Biomathematics
Department of Mathematics and Statistics
Texas Tech University
The progression of HIV infection to AIDS is unclear and under examined. Many mechanisms have been proposed, including a decline in immune response, increase in replication rate, involution of the thymus, syncytium inducing capacity, activation of the latently infected cell pool, chronic activation of the immune system, and the ability of the virus to infect other immune system cells. The significance of each mechanism in combination has not been studied. We develop a simple HIV viral dynamics model incorporating proposed mechanisms as parameters that are allowed to vary. In the entire parameter space, we derive two formulae for the basic reproduction number (R0) by considering the infection starting with a single infected CD4 T cell and a single virion, respectively. We show that both formulae are equivalent. We derive analytical conditions for the occurrence of backward and forward bifurcations. To investigate the influence of the proposed mechanisms to the HIV progression, we perform uncertainty and sensitivity analysis for all parameters and conduct a bifurcation analysis on all parameters that are shown to be significant, in combination, to explore various HIV/AIDS progression dynamics. PDF available
The electrocardiogram (ECG) has long been used to monitor cardiac health. With each feature of the ECG corresponding to a part of the cardiac cycle, the automated identification of such features can play a role in developing decision-support tools that provide further insight regarding a patient’s clinical status. Although algorithms for carrying out such feature identification are well-studied in adult populations, suitable algorithms are lacking for the pediatric congenital heart disease population. This work presents a framework for identifying representative subsets of ECG beats from large data sets and subsequent steps for automated feature identification. ECG subset selection is performed using methods related to the discrete empirical interpolation method (DEIM), and the approach to feature identification relies on techniques more commonly seen in the informatics and data science communities. With methods that generalize beyond the pediatric congenital heart disease population, both subset selection and feature identification aspects of this work can be used toward real-time analyses for the clinical setting. Watch online on Tuesday the 8th at 3:30 PM via this Zoom link.
Locusts gather in large numbers to feed on crops, destroying agricultural fields. Wingless juveniles marching together through a field demonstrate collective behavior that forms a coherent front of advancing insects. We examine this front through two models: an agent-based model and a set of partial differential equations. We construct the agent-based model using observations of individual behavior from the biological literature. The PDE model yields insight into collective behavior of the front. We demonstrate that resource-dependent behavior can explain the density distribution observed in locust hopper bands.
Prostate cancer is a common cancer among males in the United States and is frequently treated by intermittent androgen deprivation therapy. This therapy requires a patient to alternate between periods of androgen suppression treatment and no treatment. Prostate-specific antigen levels are used to track relative changes in tumor volume of prostate cancer patients undergoing intermittent androgen deprivation therapy. During this therapy, there is a pause between treatment cycles. We use dynamic equations to estimate prostate-specific antigen levels and construct a novel time scale model to account for both continuous and discrete time simultaneously. This allows us to account for breaks between treatment cycles. Using empirical data sets of prostate-specific antigen levels, a known biomarker of prostate cancer, across multiple patients, we fit our model and use least squares to estimate two parameter values. We compare our model to the data and find a resemblance on treatment intervals similar to our time scale. We then use several different time scales to construct variations of our novel model.
This seminar is held in conjuction with the Applied Math seminar group.
Nearly 13.7 million (18.5%), or 1 in 5, children and adolescents were considered obese in the United States according to the Centers for Disease Control and Prevention (CDC). The progression of overweight and obesity among children and adolescents is complex, dynamic, and involves nonlinear interactions among many factors at the biological (genetics and physiology), behavioral, social, and environmental levels. In school environments, nutrition education and peer programs have key roles in shaping the health behaviors and health outcomes of young individuals. Many school-based policies have been implemented to improve diet and physical activity behaviors to address childhood obesity. However, variability in school policies and measurement errors in health research make program evaluation challenging. This session will give a high-level overview of applications of statistical and mathematical modeling methods to obesity and nutrition research in the context of health policy and interventions.
With travel becoming more frequent across the world, it is important to understand how spatial dynamics impact the spread of diseases. Human movement plays a key part on how a disease can be distributed as it enables a pathogen to invade a new environment and helps the persistence of a disease in locations that would otherwise be isolated. In this project, we explore how spatial heterogeneity combines with mobility network structure to influence vector-borne disease dynamics by using cellphone data from Namibia. In addition, we derived an approximation for the domain reproduction number for a n-patch SIR Ross-MacDonald model using a Laurent series expansion. Lastly, we will analyze the sensitivity equations with respect to the domain reproduction number to determine which parameters should be targeted for intervention strategies. PDF available
Immunity following natural infection or immunization may wane, increasing susceptibility to infection with time since infection or vaccination. Symptoms, and concomitantly infectiousness, depend on residual immunity. We quantify these phenomena in a model population composed of individuals whose susceptibility, infectiousness, and symptoms all vary with immune status. We also model age, which affects contact, vaccination and possibly waning rates. The resurgences of pertussis that have been observed wherever effective vaccination programs have reduced typical disease among young children follow from these processes. As one example, we compare simulations with the experience of Sweden following resumption of pertussis vaccination after the hiatus from 1979 to 1996, reproducing the observations leading health authorities to introduce booster doses among school-aged children and adolescents in 2007 and 2014, respectively. Because pertussis comprises a spectrum of symptoms, only the most severe of which are medically attended, accurate models are needed to design optimal vaccination programs where surveillance is less effective.
Watch online Tuesday the 17th at 3:30 PM via this zoom link
See this PDF