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The outbreak of SARS-CoV-2 is thought to have originated in Wuhan, China in late 2019 and has since spread quickly around the world. To date, the virus has infected tens of millions of people worldwide, compelling governments to implement strict policies to counteract community spread. Federal, provincial, and municipal governments have employed various public health policies, including social distancing, mandatory mask wearing, and the closure of schools and businesses. However, the implementation of these policies can be difficult and costly, making it imperative that both policy makers and the citizenry understand their potential benefits and the risks of non-compliance. In this work, a mathematical model is developed to study the impact of social behaviour on the course of the pandemic in the province of Ontario. The approach is based upon a standard SEIRD model with a variable transmission rate that depends on the behaviour of the population. The model parameters, which characterize the disease dynamics, are estimated from Ontario COVID-19 epidemiological data using machine learning techniques. A key result of the model, following from the variable transmission rate, is the prediction of the occurrence of a second wave using the most current infection data and disease-specific traits. The qualitative behaviour of different future transmission-reduction strategies is examined, and the time-varying reproduction number is analyzed using existing epidemiological data and future projections. Importantly, the effective reproduction number, and thus the course of the pandemic, is found to be sensitive to the adherence to public health policies, illustrating the need for vigilance as the economy continues to reopen., Competing Interests: The authors have declared that no competing interests exist.

Glioblastomas are the most common primary brain tumours. They are known for their highly aggressive growth and invasion, leading to short survival times. Treatments for glioblastomas commonly involve a combination of surgical intervention, chemotherapy, and external beam radiation therapy (XRT). Previous works have not only successfully modelled the natural growth of glioblastomas in vivo, but also show potential for the prediction of response to radiation prior to treatment. This suggests that the efficacy of XRT can be optimized before treatment in order to yield longer survival times. However, while current efforts focus on optimal scheduling of radiotherapy treatment, they do not include a similarly sophisticated spatial optimization. In an effort to improve XRT, we present a method for the spatial optimization of radiation profiles. We expand upon previous results in the general problem and examine the more physically reasonable cases of 1-step and 2-step radiation profiles during the first and second XRT fractions. The results show that by including spatial optimization in XRT, while retaining a constant prescribed total dose amount, we are able to increase the total cell kill from the clinically-applied uniform case., Competing Interests: The authors have declared that no competing interests exist.

The unwelcome evolution of malignancy during cancer progression emerges through a selection process in a complex heterogeneous population structure. In the present work, we investigate evolutionary dynamics in a phenotypically heterogeneous population of stem cells (SCs) and their associated progenitors. The fate of a malignant mutation is determined not only by overall stem cell and non-stem cell growth rates but also differentiation and dedifferentiation rates. We investigate the effect of such a complex population structure on the evolution of malignant mutations. We derive exactly calculated results for the fixation probability of a mutant arising in each of the subpopulations. The exactly calculated results are in almost perfect agreement with the numerical simulations. Moreover, a condition for evolutionary advantage of a mutant cell versus the wild type population is given in the present study. We also show that microenvironment-induced plasticity in invading mutants leads to more aggressive mutants with higher fixation probability. Our model predicts that decreasing polarity between stem and non-stem cells' turnover would raise the survivability of non-plastic mutants; while it would suppress the development of malignancy for plastic mutants. The derived results are novel and general with potential applications in nature; we discuss our model in the context of colorectal/intestinal cancer (at the epithelium). However, the model clearly needs to be validated through appropriate experimental data. This novel mathematical framework can be applied more generally to a variety of problems concerning selection in heterogeneous populations, in other contexts such as population genetics, and ecology.

The tumour control probability (TCP) is a formalism derived to compare various treatment regimens of radiation therapy, defined as the probability that given a prescribed dose of radiation, a tumour has been eradicated or controlled. In the traditional view of cancer, all cells share the ability to divide without limit and thus have the potential to generate a malignant tumour. However, an emerging notion is that only a sub-population of cells, the so-called cancer stem cells (CSCs), are responsible for the initiation and maintenance of the tumour. A key implication of the CSC hypothesis is that these cells must be eradicated to achieve cures, thus we define TCPS as the probability of eradicating CSCs for a given dose of radiation. A cell surface protein expression profile, such as CD44high/CD24low for breast cancer or CD133 for glioma, is often used as a biomarker to monitor CSCs enrichment. However, it is increasingly recognized that not all cells bearing this expression profile are necessarily CSCs, and in particular early generations of progenitor cells may share the same phenotype. Thus, due to the lack of a perfect biomarker for CSCs, we also define a novel measurable TCPCD+, that is the probability of eliminating or controlling biomarker positive cells. Based on these definitions, we use stochastic methods and numerical simulations parameterized for the case of gliomas, to compare the theoretical TCPS and the measurable TCPCD+. We also use the measurable TCP to compare the effect of various radiation protocols.

In this manuscript, we use genetic data to provide a three-faceted analysis on the links between molecular subclasses of glioblastoma, epithelial-to-mesenchymal transition (EMT) and CD133 cell surface protein. The contribution of this paper is three-fold: First, we use a newly identified signature for epithelial-to-mesenchymal transition in human mammary epithelial cells, and demonstrate that genes in this signature have significant overlap with genes differentially expressed in all known GBM subtypes. However, the overlap between genes up regulated in the mesenchymal subtype of GBM and in the EMT signature was more significant than other GBM subtypes. Second, we provide evidence that there is a negative correlation between the genetic signature of EMT and that of CD133 cell surface protein, a putative marker for neural stem cells. Third, we study the correlation between GBM molecular subtypes and the genetic signature of CD133 cell surface protein. We demonstrate that the mesenchymal and neural subtypes of GBM have the strongest correlations with the CD133 genetic signature. While the mesenchymal subtype of GBM displays similarity with the signatures of both EMT and CD133, it also exhibits some differences with each of these signatures that are partly due to the fact that the signatures of EMT and CD133 are inversely related to each other. Taken together these data shed light on the role of the mesenchymal transition and neural stem cells, and their mutual interaction, in molecular subtypes of glioblastoma multiforme.

Hypoxia and acidity are two main microenvironmental factors intimately associated with solid tumors and play critical roles in tumor growth and metastasis. The experimental results of Helmlinger and colleagues (Nature Medicine 3, 177, 1997) provide evidence of a lack of correlation between these factors on the micrometer scale in vivo and further show that the distribution of pH and pO(2) are heterogeneous. Here, using computational simulations, grounded in these experimental results, we show that the lack of correlation between pH and pO(2) and the heterogeneity in their shapes are related to the heterogeneous concentration of buffers and oxygen in the blood vessels, further amplified by the network of blood vessels and the cell metabolism. We also demonstrate that, although the judicious administration of anti-angiogenesis agents (normalization process) in tumors may lead to recovery of the correlation between hypoxia and acidity, it may not normalize the pH throughout the whole tumor. However, an increase in the buffering capacity inside the blood vessels does appear to increase the extracellular pH throughout the whole tumor. Based on these results, we propose that the application of anti-angiogenic agents and at the same time increasing the buffering capacity of the tumor extracellular environment may be the most efficient way of normalizing the tumor microenvironment. As a by-product of our simulation we show that the recently observed lack of correlation between glucose consumption and hypoxia in cells which rely on respiration is related to the inhomogeneous consumption of glucose to oxygen concentration. We also demonstrate that this lack of correlation in cells which rely on glycolysis could be related to the heterogeneous concentration of oxygen inside the blood vessels.