Mathematical Biology

Systemic chemotherapy is one of the main anticancer treatments used for most kinds of clinically diagnosed tumors. However, the efficacy of these drugs can be hampered by the physical attributes of the tumor tissue, such as tortuous vasculature, dense and fibrous extracellular matrix, irregular cellular architecture, metabolic gradients, and non-uniform expression of the cell membrane receptors. This can impede the transport of therapeutic agents to tumor cells in quantities sufficient to exert the desired effect. In addition, tumor microenvironments undergo dynamic spatio-temporal changes during treatment, which can also obstruct the observed drug efficacy. To examine ways to improve drug delivery on a cell-to-tissue scale (single-cell pharmacology), we developed the microscale pharmacokinetics/pharmacodynamics modeling framework “microPKPD”. I will present how this framework can be used to design optimal schedules for various treatments and to investigate the development of drug-induced resistance.

The talk describes a substantial extension of the Middle East Respiratory Syndrome (MERS) model constructed, analyzed and simulated in Al-Asuoad et. al. BIOMATH 5 (2016)1, Al-Asuoad, Oakland University Dissertation (2017), and Al-Asuoad and Shillor, BIOMATH 7(1)(2018)2 to the case of the current COVID-19 Respiratory Syndrome pandemic that is sweeping the globe. It is caused by the new SARS-CoV-2 coronavirus that has been identified in December 2019 and since then outbreaks have been reported in all parts of the world. To help predict the dynamics and possible controls of the pandemic we developed a mathematical model for the pandemic. The model has a compartmental structure similar but more complex to the SARS and MERS models. It is a coupled system of nonlinear ordinary differential equations (ODEs) and a differential inclusion for the contact rate parameter. The talk will describe the model in detail, mention some of its analysis, and describe our computer simulations of the pandemic in South Korea. The main modeling novelties are in taking into account the shelter-in-place directives, the rates at which the populations obey them and the observed changes in the infectiveness of ‘contact number’ of the SARS-CoV-2 virus. The model predictions are fitted to some of the data from the outbreak in South Korea. Since the DFE (in South Korea) is found to be asymptotically stable, the pandemic will eventually die out (as long as some control measures remain in place). And, indeed, the model simulations show that the COVID-19 will in the near future be contained. However, the containment time and the severity of the outbreak depend crucially on the contact coefficients and the isolation or shelter-in-place rate constant. The simulations show that when randomness is added to the model coefficients the model captures the pandemic dynamics very well. Finally, the model highlights the importance of isolation of infected individuals and may be used to assess other control measures. It is general and will be used to analyze outbreaks in other parts of the world.

*with Aycil Cesmelioglu and Anna M. Spagnuolo

1 http://dx.doi.org /10.11145/j.biomath.2016.12.141
2 http://dx.doi.org/10.11145/j.biomath.2018.02.277

It is common in SIR models to assume that the infection rate is proportional to the product S*I of susceptible and infected individuals. This form is motivated by the law of mass action from chemistry. While this assumption works at the onset of the outbreak, it needs to be modified at higher rates such as seen currently in much of the world (as of June 2020). We propose a physics-based model which leads to a simple saturation formula based on first principles incorporating the spread radius and population density. We then apply this modified SIR model to coronavirus and show that it fits much better than the ``classical'' law of mass action.

Rapid development of a QSP model to support novel COVID-19 therapies. We intend to publish this model quickly to encourage community feedback. The simulated dynamics of immune response are modeled by describing viral activation of innate and adaptive immune processes involving both pro-inflammatory mediators regulating viral clearance and cell damage (e.g. neutrophils and cytotoxic lymphocytes) as well as counter-regulatory immune suppressive mediators (e.g. Treg cells and IL-10).

The primary distinction between severe and mild COVID-19 infections is the immune response. Disease severity and fatality has been observed to correlate with lymphopenia (low blood lymphocyte count) and increased levels of inflammatory cytokines and IL-6 (cytokine storm), damaging dysregulated macrophage responses, and T cell exhaustion due to limited recruitment. The exact mechanism driving the dynamics that ultimately result in severe COVID-19 manifestation remain unclear. Over the past two months, we have been working on developing tissue- and systemic-level models of the immune response to SARS-CoV-2 infection with the goal of pinpointing what may be causing dysregulated immune dynamics in severe cases. At the tissue level, we been working as part of an international collaboration to build a computational framework to study SARS-CoV-2 in the tissues. This platform is based upon PhysiCell, an open-source computational cell-based software. With this model, we have been investigating how the level of pro-inflammatory cytokines influence immune cell recruitment into the infected tissue and how this correlates with tissue damage. In parallel, we have constructed a systemic, within-host delay-differential equation model that accounts for the interactions between immune cell subsets, cytokines, lung tissue, and virus to help understand differential responses in COVID-19. While this work is still ongoing, this talk will address how a variety of mathematical and computational techniques contribute to the ongoing study of SARS-CoV-2 infections, helping to increase our understanding of COVID-19 severity.

* with Sofia Alfonso (McGill University), Rosemary Aogo (University of Tennessee Health Science Center), Courtney Davis (Pepperdine University), Amber M. Smith (University of Tennessee Health Science Center), Morgan Craig (Université de Montréal, CHU Sainte-Justine Research Centre)