Title: Stability conditions for linear neutral delay differential equations with distributed delays
Abstract: We discuss stability conditions, of both delay-independent and delay-dependent type, for systems of linear neutral delay equations with distributed delays. Some examples and numerical results will be given.

Title: Modeling Immunosuppression in Renal Transplant Recipients
Abstract: In 2021, a new record was set in the United States for the number of kidney transplant operations performed in a year with over 24,000 procedures taking place, more than half of all solid organ transplants that year. As of 2018, more than 200,000 Americans are renal allograft hosts and this number grows while continuing to set records annually. The recipients of donor kidneys are met with two difficult biological challenges: 1) in almost all cases, allograft tissues are targeted and attacked by the host’s immune system thereby requiring suppression of the host’s immune system and 2) immunosuppression treatments weaken the host’s defense allowing latent pathogens to proliferate, such as the BK virus which infects cells within the nephrons of the kidney causing permanent damage and irreversible loss of kidney function. These patients require a balanced immunosuppression regimen which still allows the immune system to protect the body while preventing acute rejection of the allograft. The treatment protocols for kidney recipients vary among the over 250 renal transplant programs while an optimal treatment regimen has yet to be determined. In a collaboration between the Center for Research in Scientific Computation (NCSU) and the Duke Transplant Center, we are developing a mathematical model to assist in determining personalized optimal immunosuppression treatments for individual renal transplant recipients. A system of ordinary differential equations describe interactions between the allograft, BK virus and the immune system through six state variables including creatinine, a biomarker for kidney health and functioning. Determining optimal drug efficacy levels is performed by a receding horizon controller, and to personalize the treatment, Kalman filtering is utilized as a feedback mechanism to introduce current individual patient data into the model. Our initial efforts demonstrate the utility of taking this modeling approach.

Title: Population model for the decline of the invasive pest Homalodisca vitripennis while in the presence of the parasitoid Cosmocomoidea ashmeadi
Abstract: The glassy-winged sharpshooter, Homalodisca vitripennis, is an invasive pest which presents a major economic threat to the grape industries in California by spreading a disease-causing bacteria, Xylella fastidiosa. Recently a common enemy of H. vitripennis, certain mymarid parasitoid species including Cosmocomoidea ashmeadi and Cosmocomoidea morrilli, have been studied to use in place of insecticides as a control method. We create a time and temperature dependent mathematical model to analyze data and answer the question: Does the implementation of C. ashmeadi as a biological control method cause a significant decrease in the population of H. vitripennis?