Bulletin of Mathematical Biology

Francis, Andrew; Steel, Mike

doi: 10.1007/s11538-023-01157-0pmid: 37097343

Phylogenetic networks are mathematical representations of evolutionary history that are able to capture both tree-like evolutionary processes (speciations) and non-tree-like ‘reticulate’ processes such as hybridization or horizontal gene transfer. The additional complexity that comes with this capacity, however, makes networks harder to infer from data, and more complicated to work with as mathematical objects. In this paper, we define a new, large class of phylogenetic networks, that we call labellable, and show that they are in bijection with the set of ‘expanding covers’ of finite sets. This correspondence is a generalisation of the encoding of phylogenetic forests by partitions of finite sets. Labellable networks can be characterised by a simple combinatorial condition, and we describe the relationship between this large class and other commonly studied classes. Furthermore, we show that all phylogenetic networks have a quotient network that is labellable.

Zhang, Yuyuan; Wang, Kaiqun; Du, Yaoyao; Yang, Huiyuan; Jia, Guanjie; Huang, Di; Chen, Weiyi; Shan, Yanhu

doi: 10.1007/s11538-023-01158-zpmid: 37142885

Tumor immunotherapy aims to maintain or enhance the killing capability of CD8+ T cells to clear tumor cells. The tumor–immune interactions affect the function of CD8+ T cells. However, the effect of phenotype heterogeneity of a tumor mass on the collective tumor–immune interactions is insufficiently investigated. We developed the cellular-level computational model based on the principle of cellular Potts model to solve the case mentioned above. We considered how asymmetric division and glucose distribution jointly regulated the transient changes in the proportion of proliferating/quiescent tumor cells in a solid tumor mass. The evolution of a tumor mass in contact with T cells was explored and validated by comparing it with previous studies. Our modeling exhibited that proliferating/quiescent tumor cells, exhibiting distinct anti-apoptotic and suppressive behaviors, redistributed within the domain accompanied by the evolution of a tumor mass. Collectively, a tumor mass prone to a quiescent state weakened the collective suppressive functions of a tumor mass on cytotoxic T cells and triggered a decline of apoptosis of tumor cells. Although quiescent tumor cells did not sufficiently do their inhibitory functions, the possibility of long-term survival was improved due to their interior location within a mass. Overall, the proposed model provides a useful framework to investigate collective-targeted strategies for improving the efficiency of immunotherapy.

Barril, Carles; Bliman, Pierre-Alexandre; Cuadrado, Sílvia

doi: 10.1007/s11538-023-01159-ypmid: 37156965

The final infection size is defined as the total number of individuals that become infected throughout an epidemic. Despite its importance for predicting the fraction of the population that will end infected, it does not capture which part of the infected population will present symptoms. Knowing this information is relevant because it is related to the severity of the epidemics. The objective of this work is to give a formula for the total number of symptomatic cases throughout an epidemic. Specifically, we focus on different types of structured SIR epidemic models (in which infected individuals can possibly become symptomatic before recovering), and we compute the accumulated number of symptomatic cases when time goes to infinity using a probabilistic approach. The methodology behind the strategy we follow is relatively independent of the details of the model.

Rapti, Z.; Cuevas-Maraver, J.; Kontou, E.; Liu, S.; Drossinos, Y.; Kevrekidis, P. G.; Barmann, M.; Chen, Q.-Y.; Kevrekidis, G. A.

doi: 10.1007/s11538-023-01152-5pmid: 37166513

Metapopulation models have been a popular tool for the study of epidemic spread over a network of highly populated nodes (cities, provinces, countries) and have been extensively used in the context of the ongoing COVID-19 pandemic. In the present work, we revisit such a model, bearing a particular case example in mind, namely that of the region of Andalusia in Spain during the period of the summer-fall of 2020 (i.e., between the first and second pandemic waves). Our aim is to consider the possibility of incorporation of mobility across the province nodes focusing on mobile-phone time-dependent data, but also discussing the comparison for our case example with a gravity model, as well as with the dynamics in the absence of mobility. Our main finding is that mobility is key toward a quantitative understanding of the emergence of the second wave of the pandemic and that the most accurate way to capture it involves dynamic (rather than static) inclusion of time-dependent mobility matrices based on cell-phone data. Alternatives bearing no mobility are unable to capture the trends revealed by the data in the context of the metapopulation model considered herein.

Venegas Garcia, Pablo; Wang, Hao

doi: 10.1007/s11538-023-01160-5pmid: 37162598

Due to global warming, interest in sequestering carbon by appropriately managing soils has contributed to studying the dynamic exchange of carbon and nitrogen in soils and atmospheric CO2\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$_2$$\end{document}. The priming effect, or the intensified CO2\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$_2$$\end{document} emissions from soil organic matter (SOM) decomposition in short periods by using labile substrates, has been a topic of interest over the last decades. A combination of two experimentally supported mechanisms explains the priming effect phenomenon, and for the first time, we combine them in a novel stoichiometric model. The model considers the effects of labile substrate utilization in soils during the SOM decomposition and how CO2\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$_2$$\end{document} emissions rates are affected. Laboratory data and a local sensitivity analysis validate the accuracy and robustness of the model. We find an optimized ratio of labile carbon and nitrogen that intensifies SOM decomposition for different soil features. The priming effect is weakened as C/N in SOM increases for nutrient-poor soils and is independent of C/N in SOM for nutrient-rich soils. The time required for microorganisms to decompose SOM at its maximum rate is delayed only for labile carbon treatments and poor-nutrient soils but remains constant otherwise. Finally, the SOM degradation efficiency determines the priming effect’s acceleration or reduction under different soil treatments.

Amiri, Farhad A.; Zhang, Junfeng

doi: 10.1007/s11538-023-01155-2pmid: 37129671

Oxygen transfer in the microvasculature is a complex phenomenon that involves multiple physical and chemical processes and multiple media. Hematocrit, the volume fraction of red blood cells (RBCs) in blood, has direct influences on the blood flow as well as the oxygen supply in the microcirculation. On the one hand, a higher hematocrit means that more RBCs present in capillaries, and thus, more oxygen is available at the source end. On the other hand, the flow resistance increases with hematocrit, and therefore, the RBC motion becomes slower, which in turn reduces the influx of oxygen-rich RBCs entering capillaries. Such double roles of hematocrit have not been investigated adequately. Moreover, the oxygen–hemoglobin dissociation rate depends on the oxygen tension and hemoglobin saturation of the cytoplasm inside RBCs, and the dissociation kinetics exhibits a nonlinear fashion at different oxygen tensions. To understand how these factors and mechanisms interplay in the oxygen transport process, computational modeling and simulations are favorite since we have a good control of the system parameters and also we can access to the detailed information during the transport process. In this study, we conduct numerical simulations for the blood flow and RBC deformation along a capillary and the oxygen transfer from RBCs to the surrounding tissue. Different values for the hematocrit, arteriole oxygen tension, tissue metabolism rate and hemoglobin concentration and affinity are considered, and the simulated spatial and temporal variations of oxygen concentration are analyzed in conjunction with the nonlinear oxygen–hemoglobin reaction kinetics. Our results show that there are two competing mechanisms for the tissue oxygenation response to a hematocrit increases: the favorite effect of the higher RBC density and the negative effect of the slower RBC motion. Moreover, in the low oxygen situations with RBC oxygen tension less than 50 mmHg at capillary inlet, the reduced RBC velocity effect dominates, resulting in a decrease in tissue oxygenation at higher hematocrit. On the opposite, for RBC oxygen tension higher than 50 mmHg when entering the capillary, a higher hematocrit is beneficial to the tissue oxygenation. More interestingly, the pivoting arteriole oxygen tension at which the two competing mechanisms switch dominance on tissue oxygenation becomes lower for higher oxygen–hemoglobin affinity and lower hemoglobin concentration. This observation has also been analyzed based on the oxygen supply from RBCs and the oxygen–hemoglobin reaction kinetics. The results and discussions presented in this article could be helpful for a better understanding of oxygen transport in microcirculation.

Eilertsen, Justin; Schnell, Santiago; Walcher, Sebastian

doi: 10.1007/s11538-023-01150-7pmid: 37101015

We consider reaction networks that admit a singular perturbation reduction in a certain parameter range. The focus of this paper is on deriving “small parameters” (briefly for small perturbation parameters), to gauge the accuracy of the reduction, in a manner that is consistent, amenable to computation and permits an interpretation in chemical or biochemical terms. Our work is based on local timescale estimates via ratios of the real parts of eigenvalues of the Jacobian near critical manifolds. This approach modifies the one introduced by Segel and Slemrod and is familiar from computational singular perturbation theory. While parameters derived by this method cannot provide universal quantitative estimates for the accuracy of a reduction, they represent a critical first step toward this end. Working directly with eigenvalues is generally unfeasible, and at best cumbersome. Therefore we focus on the coefficients of the characteristic polynomial to derive parameters, and relate them to timescales. Thus, we obtain distinguished parameters for systems of arbitrary dimension, with particular emphasis on reduction to dimension one. As a first application, we discuss the Michaelis–Menten reaction mechanism system in various settings, with new and perhaps surprising results. We proceed to investigate more complex enzyme catalyzed reaction mechanisms (uncompetitive, competitive inhibition and cooperativity) of dimension three, with reductions to dimension one and two. The distinguished parameters we derive for these three-dimensional systems are new. In fact, no rigorous derivation of small parameters seems to exist in the literature so far. Numerical simulations are included to illustrate the efficacy of the parameters obtained, but also to show that certain limitations must be observed.

Fritz, Marvin

doi: 10.1007/s11538-023-01151-6pmid: 37081144

In this survey article, a variety of systems modeling tumor growth are discussed. In accordance with the hallmarks of cancer, the described models incorporate the primary characteristics of cancer evolution. Specifically, we focus on diffusive interface models and follow the phase-field approach that describes the tumor as a collection of cells. Such systems are based on a multiphase approach that employs constitutive laws and balance laws for individual constituents. In mathematical oncology, numerous biological phenomena are involved, including temporal and spatial nonlocal effects, complex nonlinearities, stochasticity, and mixed-dimensional couplings. Using the models, for instance, we can express angiogenesis and cell-to-matrix adhesion effects. Finally, we offer some methods for numerically approximating the models and show simulations of the tumor’s evolution in response to various biological effects.

Wilson, Nicole; Drapaca, Corina S.; Enderling, Heiko; Caudell, Jimmy J.; Wilkie, Kathleen P.

doi: 10.1007/s11538-023-01139-2pmid: 37186175

Fractional calculus has recently been applied to the mathematical modelling of tumour growth, but its use introduces complexities that may not be warranted. Mathematical modelling with differential equations is a standard approach to study and predict treatment outcomes for population-level and patient-specific responses. Here, we use patient data of radiation-treated tumours to discuss the benefits and limitations of introducing fractional derivatives into three standard models of tumour growth. The fractional derivative introduces a history-dependence into the growth function, which requires a continuous death-rate term for radiation treatment. This newly proposed radiation-induced death-rate term improves computational efficiency in both ordinary and fractional derivative models. This computational speed-up will benefit common simulation tasks such as model parameterization and the construction and running of virtual clinical trials.

Anwar, Md Nurul; Hickson, Roslyn I.; Mehra, Somya; Price, David J.; McCaw, James M.; Flegg, Mark B.; Flegg, Jennifer A.

doi: 10.1007/s11538-023-01153-4pmid: 37076740

Plasmodium vivax is the most geographically widespread malaria-causing parasite resulting in significant associated global morbidity and mortality. One of the factors driving this widespread phenomenon is the ability of the parasites to remain dormant in the liver. Known as ‘hypnozoites’, they reside in the liver following an initial exposure, before activating later to cause further infections, referred to as ‘relapses’. As around 79–96% of infections are attributed to relapses from activating hypnozoites, we expect it will be highly impactful to apply treatment to target the hypnozoite reservoir (i.e. the collection of dormant parasites) to eliminate P. vivax. Treatment with radical cure, for example tafenoquine or primaquine, to target the hypnozoite reservoir is a potential tool to control and/or eliminate P. vivax. We have developed a deterministic multiscale mathematical model as a system of integro-differential equations that captures the complex dynamics of P. vivax hypnozoites and the effect of hypnozoite relapse on disease transmission. Here, we use our multiscale model to study the anticipated effect of radical cure treatment administered via a mass drug administration (MDA) program. We implement multiple rounds of MDA with a fixed interval between rounds, starting from different steady-state disease prevalences. We then construct an optimisation model with three different objective functions motivated on a public health basis to obtain the optimal MDA interval. We also incorporate mosquito seasonality in our model to study its effect on the optimal treatment regime. We find that the effect of MDA interventions is temporary and depends on the pre-intervention disease prevalence (and choice of model parameters) as well as the number of MDA rounds under consideration. The optimal interval between MDA rounds also depends on the objective (combinations of expected intervention outcomes). We find radical cure alone may not be enough to lead to P. vivax elimination under our mathematical model (and choice of model parameters) since the prevalence of infection eventually returns to pre-MDA levels.