Biomathematics
Department of Mathematics and Statistics
Texas Tech University
A discrete-time model of tumor-immune system interactions based on a published model is derived. The tumor is assumed to grow exponentially while the immune system can either slow down or shrink tumor growth. However, cancerous tumor cells have many mechanisms to impair the immune system. The mathematical model therefore incorporates immunotherapy of immune checkpoint inhibitors to study tumor dynamics. It is proven that the tumor can grow to unboundedly large if its intrinsic growth rate exceeds a critical value or if its i size is beyond a threshold when first detected. In addition, a region of initial conditions for which the tumor will eventually reach an unbounded size is derived when the tumor growth rate is smaller than the critical value. As a result, the immunotherapy fails to control the aggressive tumor. There are two positive equilibrium points when the tumor growth rate is smaller than the critical value. We verify that a saddle-node bifurcation occurs at the critical tumor growth rate whereas the equilibrium with the larger tumor size is always unstable. The equilibrium with the smaller tumor size may undergo a subcritical Neimark-Sacker bifurcation in certain parameter regimes. Parameter values derived from various clinical studies are simulated to illustrate our analytical findings.
Bacterial swimming mediated by flagellar rotation is one of the most ubiquitous forms of cellular locomotion that plays a major role in many biological processes. Here, a typical swimming path of flagellated bacteria looks like a random walk with no purpose, but the random movement becomes modified as environmental conditions change. Modified random movement is particularly characterized by their motility pattern or a combination of their swimming modes. We present several distinct motility patterns exhibited by bacterial species and discuss a recently reported swimming mode – some bacterial species can swim backward by wrapping their flagella around the cell body. By introducing a mathematical model of a polarly flagellated bacterium through the fluid, we investigate what mechanism differentiates swimming modes and how a combination of the swimming modes affects their swimming pattern. Furthermore, we suggest benefits of their swimming pattern in complex native habitats. This is a joint work with S. Lim (Univ. of Cincinnati), Y. Kim (Chung-Ang Univ.), and W. Lee (National Institute for Mathematical Sciences, South Korea).
Please virtually attend this week's Biomath seminar via this Zoom link. Passcode: BfriM6
A mathematical model of tumor-immune system interactions with an oncolytic virus therapy for which the immune system plays a twofold role against cancer cells is derived. The immune cells can kill cancer cells but can also eliminate viruses from the therapy. In addition, immune cells can either be stimulated to proliferate or be impaired to reduce their growth by tumor cells. It is shown that if the tumor killing rate by immune cells is above a critical value, the tumor can be eradicated for all sizes, where the critical killing rate depends on whether the immune system is immunosuppressive or proliferative. For a reduced tumor killing rate with an immunosuppressive immune system, bistability exists in a large parameter space. The tumor can either be eradicated or grow near to its carrying capacity depending on the tumor size. However, reducing the viral killing rate by immune cells always increases the effectiveness of the viral therapy. This reduction may be achieved by manipulating certain genes of viruses via genetic engineering.
The limited supply of the COVID-19 vaccine and its inequitable distribution pose a public health concern contributing to worsening health disparity. This study explores the public health — and economic impact of four possible vaccination strategies — fixed-dose interval (S1), prioritization of the first dose (S2), and screen-and-vaccinate those with the COVID-19 infection history with fixed-dose interval (S3) or first-dose prioritization (S4). Using mathematical modelling, we quantified the number of quarantine and hospitalization days and deaths averted from each strategy, as well as the associated cost. The model parameters and initial conditions are based on Canada, and the simulation ran over 365 days starting from June 1st, 2021. Net monetary benefit (NMB) and incremental cost-effectiveness ratio were calculated from a societal perspective. In addition, sensitivity analysis explored how each strategy reacts to different conditions of daily vaccine supply, the initial proportion of the recovered, and the initial coverage of the first dose. The findings suggest the potential benefits of alternative vaccination strategies that can save lives and costs. Our study's framework and findings can inform policymakers to explore the optimal COVID-19 vaccination strategy under their unique settings.
Please virtually attend this week's Biomath seminar at 3:30 PM (UT-5) on Tuesday the 12th via this Zoom link. Meeting ID: 839 9465 7333 Passcode: BfriM6
We present some optimal control work on mosquito-borne diseases: Malaria and West-Nile Virus. First, a malaria transmission model with SEIR (susceptible-exposed-infected-recovered) classes for the human population, SEI (susceptible-exposed-infected) classes for the wild mosquitoes and an additional class for sterile mosquitoes is formulated. We derive the basic reproduction number of the infection. We formulate an optimal control problem in which the goal is to minimize both the infected human populations and the cost to implement two control strategies: the release of sterile mosquitoes and the usage of insecticide-treated nets to reduce the malaria transmission. Adjoint equations are derived, and the characterization of the optimal controls are established. We quantify the effectiveness of the two interventions aimed at limiting the spread of Malaria. A combination of both strategies leads to a more rapid elimination of the wild mosquito population that can suppress Malaria transmission.
Secondly, we consider a West-Nile Virus transmission model that describes the interaction between bird and mosquito populations (eggs, larvae, adults) and the dynamics for larvicide and adulticide, with impulse controls. We derive the basic reproduction number of the infection. We reformulate the impulse control problems as nonlinear optimization problems to derive adjoint equations and establish optimality conditions. We formulate three optimal control problems which seek to balance the cost of insecticide applications (both the timing and application level) with (1) the benefit of reducing the number of mosquitoes, (2) the benefit of reducing the disease burden, or (3) the benefit of preserving the healthy bird population. Numerical simulations are provided to illustrate the results of both models.
Please virtually attend this week's Biomath seminar at 3:30 PM (UT-5) on Tuesday the 26th via this Zoom link. Meeting ID: 839 9465 7333 Passcode: BfriM6
Abstract pdf
This Biomath seminar may be attended Tuesday the 9th at 3:30 PM CST (UT-6) via this Zoom link. Meeting ID: 839 9465 7333 Passcode: BfriM6
In this talk, I will present two models of tumor volume dynamics describing individual patient response to radiotherapy and how we use these models to explore the potential for treatment personalization.
I will introduce the clinical problem we are addressing, the two models, a framework for forecasting individual patient responses, and two mathematical explorations of treatment schedule and dose personalization.
Please virtually attend this week's Biomath seminar at 3:30 PM (UT-5) on Tuesday the 30th via this Zoom link.
Meeting ID: 839 9465 7333 Passcode: BfriM6