Biomathematics
Department of Mathematics and Statistics
Texas Tech University
Zebra and quagga mussel are among the world's notorious invasive species because of their large and widespread ecological and economic effects. Although these two species have similar life histories and share many ecological traits, they have some significant ecological differences and impacts. Understanding their long-term population dynamics is critical to determining impacts and effective management. To investigate how the population reproduction rates, intraspecific and interspecific competitions, as well as dispersal abilities affect the population persistence and spatial distributions of the two species in a spatially heterogeneous environment, we developed a dynamic model that describes the competitive interactions between zebra and quagga mussels in multiple patches. The dynamic analysis of the model yields some sufficient conditions that lead to population persistence, extirpation, as well as competitive exclusion and coexistence. By the numerical solutions of a two-patch model, we examine how the interplay between the local population dynamics in each patch and the individual dispersal between patches affects the competition outcomes of the two species in a spatially variable system.
This Biomath seminar may be attended Monday the 14th at 4:00 PM CST (UT-6) via this Zoom link. Meeting ID: 839 9465 7333 Passcode: BfriM6
A complex adaptive system (CAS) is a system that is complex in that it is a dynamic network of interactions, but the behavior of the ensemble may not be predictable according to the behavior of the components. Social insect colonies; the brain; the immune system; and human social group-based endeavors are excellent examples of complex adaptive. Mathematical models are powerful tools that can provide us quantitative approaches to
elucidate complicated ecological and evolutionary processes on the numerous spatial, temporal and hierarchical scales at which CAS such as social insect colonies and/or human groups operate. In this talk, I will review some of our recent collaborative work with biologists and psychologists regarding important and interesting questions of CAS such as how information spreads in the social insect colonies? How may we define and model trust dynamics in human and robotic teaming? I will particularly present our modeling work through ODE, SDE and evolutionary models on addressing ...
1. How do task organization and work performance scale with colony size and metabolism?
2. How does environment impact task allocation?
3. What are co-evolutionary dynamical outcomes of the interaction of social parasite and host?
This Biomath seminar may be attended Monday the 21st at 4:00 PM CST (UT-6) via this Zoom link. Meeting ID: 839 9465 7333 Passcode: BfriM6
The use of mathematical models to make predictions about tumor growth and response to treatment has become increasingly prevalent in the clinical setting. The level of complexity within these models ranges broadly, and the calibration of more complex models requires detailed clinical data. This raises questions about how much data should be collected and when, in order to minimize the total amount of data used and the time until a model can be calibrated accurately. To address these questions, we propose a Bayesian information-theoretic calibration protocol for experimental design, using a gradient-based score function to identify optimal times at which to collect data for informing treatment parameters. We illustrate this framework by calibrating a simple ordinary differential equation model of tumor response to radiotherapy to a set of synthetic data.
This Biomath seminar may be attended Monday the 28th at 4:00 PM CST (UT-6) via this Zoom link. Meeting ID: 839 9465 7333 Passcode: BfriM6
Breast cancer is the second most commonly diagnosed cancer in women worldwide. MCF-7 cell line is an extensively studied human breast cancer cell line. This cell line expresses estrogen receptors, and the growth of MCF-7 cells is hormone dependent. In this talk, I will propose a mathematical model which governs MCF-7 cell growth with interaction among tumor cells, estradiol, natural killer (NK) cells, cytotoxic T lymphocytes (CTLs) or CD8+ T cells, and white blood cells (WBCs). Experimental data are used to determine functional forms and parameter values. Breast tumor growth is then studied using the mathematical model. The results obtained from numerical simulation are compared with those from clinical and experimental studies. The system has three coexisting stable equilibria representing the tumor-free state, a microscopic tumor, and a large tumor. These three equilibrium states are similar to the three phases of immunoediting. Numerical simulation shows that a healthy immune system is able to effectively eliminate a small tumor or produce long-term dormancy. The cytotoxicity of CTLs plays an important role in immune surveillance. The association between the circulating estradiol level and cancer risk is not significant.
This Biomath seminar may be attended Monday the 21st at 4:00 PM CST (UT-6) via this Zoom link. Meeting ID: 839 9465 7333 Passcode: BfriM6
Obesity has become a global epidemic due to an increase in unhealthy eating habits and sedentary lifestyles. Since excess weight gain can be considered a disease transmitted through social influence, understanding its interpersonal dynamics is crucial for effective intervention and prevention programs. This project proposes a compartment model by considering the social effects on weight gain. We study the local stability of disease-free equilibrium and derive a closed-form formula for the occurrence of backward bifurcation. Moreover, applying fluctuation lemma to prove the global stability of disease-free equilibrium provides a parameter condition to complete elimination of the overweight and obesity epidemic. Due to the fact that the overweight and obesity prevalence in the United States appears to be leveled off, we analyze the global stability of the non-trivial equilibrium by a geometric approach to establish a condition for the plateau of the overweight and obesity epidemic. Numerical simulations support the analytical results and show that the pro-posed model adequately describes complex epidemic patterns of overweight and obesity epidemic.
Please virtually attend this week's Biomath seminar at 4:00 PM Monday the 28th via this zoom link
Meeting ID: 839 9465 7333
Passcode: BfriM6
We formulate and analyze a differential equation model for the population dynamics of feral cats. The model includes three categories: kittens, adult females, and adult males. Feral cats are subject to various animal control measures including impounding, adoption, and euthanasia. The feral cat population also interacts with a fixed population of domestic house cats, some of which experience abandonment. We attempt to classify all equilibrium points and describe their stability.
The Biomath seminar may be attended Monday the 4th at 4:00 PM CDT (UT-5) via this Zoom link. Meeting ID: 839 9465 7333 Passcode: BfriM6
Solid tumours develop much like a fortress, acquiring characteristics that protect them against invasion. A common trait observed in solid tumours is the synthesis of excess collagen which traps therapeutic agents, resulting in a lack of dispersion of treatment within the tumour mass. In most tumours, this results in only a localised treatment. Often the tumour quickly recovers and continues to invade surrounding regions. Anti-tumour viral therapy is no exception to this rule. Experimental results show collagen density affects virus diffusion and inhibits cell infection; therefore, accurately modelling virus dispersion is an important aspect of modelling virotherapy. Mathematical models generally focus on the interaction between cancer cells and collagen. In this project, we aim to accurately capture virotherapy outcome in relation to collagen density in the tumour environment. Beginning with a random walk, we derive a novel non-Fickian diffusion term for virus dispersion and show that this diffusion term captures virus dispersion in dense collagen. Applying this diffusion term to a system of reaction-diffusion equations, we validate our model against experimental results and show that our model can predict treatment outcome in different collagen structures. The results demonstrate that collagen density is an important predictor of tumour response to therapy, and that standard Fickian diffusion cannot sufficiently capture virus spread in collagen-dense tumours.
The Biomath seminar may be attended Monday the 25th at 4:00 PM CDT (UT-5) via this Zoom link. Meeting ID: 839 9465 7333 Passcode: BfriM6
Cancer immunotherapy is intended to reactivate the body's own immune system to fight cancers. However, it is difficult to predict tumor responses, long term benefits, and remission via experiments or clinical trials, since it is such a new treatment. Thus, in this work, we combine mathematical modeling and biological experiments to help overcome these problems. In this talk, I will focus on two types of treatments: immune-checkpoint inhibitor and cytokine. For these treatments, we constructed PDE models to help clarify some controversial issue, design treatment protocols to reduce side-effect, and predict the long term benefits.
This Biomath seminar may be attended Monday the 2nd at 4:00 PM CDT (UT-5) via this Zoom link. Meeting ID: 839 9465 7333 Passcode: BfriM6