Mathematical Biosciences

Cancer-associated fibroblasts (CAFs) are a central component of the tumor microenvironment that facilitate a supportive niche for cancer progression and metastasis. Experimental evidence suggests that CAFs can facilitate estrogen-independent tumor growth, thereby reducing the efficacy of anti-hormonal therapies. Understanding and quantifying the complex interactions between tumor cells, hormonal signalling, and the microenvironment are crucial for designing more effective and individualized treatment strategies. We propose a mathematical framework to explore the influence of CAFs on ER+ breast cancer progression and to evaluate strategies to mitigate their impact. We develop a system of nonlinear ordinary differential equations that substantiates the experimental observations by providing a mechanistic basis for the role of CAFs in regulating estrogen-independent growth dynamics. We then employ optimal control theory to evaluate distinct therapeutic approaches involving monotherapy or combinations of: (i) inhibition of tumor-to-CAF signaling, (ii) inhibition of CAF-to-tumor proliferative signaling, and (iii) endocrine therapy. Taken together, our results demonstrate that CAF-targeted strategies can enhance treatment efficacy across various estrogen dosing regimens. Our study provides new insights into the potential of CAF as a therapeutic target that could help to improve existing approaches for endocrine therapies.

Evolutionary therapies regulate heterogeneous populations by altering selective pressures through treatment sequences in cancer and infections. This letter develops an invariant-set framework for treatment-induced containment based on positive triangular invariant sets. For periodically switched systems, sufficient conditions are derived for the existence of such invariant regions. Robustness with respect to mutation is established by showing that the invariant simplex persists under small perturbations of the subsystem matrices. In the two-phenotype case, the analysis yields an explicit mutation threshold that separates regimes in which therapy cycling maintains containment from regimes in which mutation can enable evolutionary escape. Simulations illustrate the geometry of the invariant sets and the role of mutation and dwell time in containment robustness.

Human immunodeficiency virus (HIV) persistence remains a major barrier to cure due to the existence of long-lived latent reservoirs that evade immune clearance and persist despite combination antiretroviral therapy (ART). Although ART effectively suppresses viral replication, treatment interruption often leads to rapid viral rebound originating from these latent reservoirs. In this study, we develop a deterministic mathematical model describing the in vivo dynamics of HIV infection incorporating uninfected CD4+ T cells, infected cells, latent reservoirs, deep latent reservoirs, and infectious and non-infectious virions, while explicitly accounting for the therapeutic effects of reverse transcriptase inhibitors (RTIs), protease inhibitors (PIs), and Tat transcription inhibitors. Analytical results establish positivity and boundedness of solutions and derive the effective reproduction number Re using the next-generation matrix approach. Stability analysis shows that the virus-free equilibrium is locally asymptotically stable when Re < 1, while viral persistence occurs when Re > 1. Numerical simulations were performed to investigate therapy interactions, viral rebound following treatment interruption, and the impact of drug efficacy on viral set-points and latent reservoir dynamics. To further explore therapy interactions, three-dimensional viral set-point surfaces and heat maps were generated to examine how combinations of infection inhibition, viral production inhibition, and transcriptional inhibition influence viral dynamics. The simulations reveal that Tat inhibition suppresses viral transcription, thereby reducing the transition of infected cells into productive infection and limiting viral propagation when combined with conventional ART mechanisms. The therapy parameter planes further demonstrate that strong transcriptional inhibition promotes the transition of infected cells into deep latency, supporting the emerging block-and-lock strategy for functional HIV cure. In addition, a three-dimensional eradication boundary surface and therapy cube were constructed to identify regions of parameter space where Re < 1, corresponding to successful viral control. These visualizations show that viral eradication is unlikely when therapies act independently but becomes achievable when multiple therapeutic mechanisms act simultaneously. Overall, the results highlight the critical role of transcriptional inhibition through Tat-targeting therapies in complementing existing ART regimens. By simultaneously suppressing viral replication and promoting deep latency, Tat-based combination strategies may significantly reduce viral rebound and contribute to long-term functional control of HIV infection.

Variability in gene expression among single cells and growing cell populations can arise from the stochastic nature of protein synthesis, which often occurs in random bursts. This study investigates the variability in the expression of a growth-sustaining protein, whose concentration is regulated by a negative feedback loop due to cell growth-induced dilution. We model the distribution of protein concentration using a Chapman-Kolmogorov equation for single cells and a population balance equation for growing cell populations. For single cells, we derive an explicit solution for the protein concentration distribution in state space and represent it as a Bessel function in Laplace space. For growing populations, we find that the distribution satisfies a Heun differential equation with singular boundary conditions. By addressing the central connection problem for the Heun equation, we quantify the population-level protein distribution and determine the Mathusian parameter, which characterizes population growth. This work provides a comprehensive analytical framework for understanding how stochastic protein synthesis impacts gene expression variability and population dynamics.

Mass-conserved reaction-diffusion systems are used as mathematical models for various phenomena such as cell polarity. Numerical simulations of this system present transient dynamics in which multiple stripe patterns converge to spatially monotonic patterns. Previous studies indicated that the transient dynamics are driven by a mass conservation law and by variations in the amount of substance contained in each pattern, which we refer to as "pattern flux". However, it is challenging to mathematically investigate these pattern dynamics. In this study, we introduce a reaction-diffusion compartment model to investigate the pattern dynamics in view of the conservation law and the pattern flux. This model is defined on multiple intervals (compartments), and diffusive couplings are imposed on each boundary of the compartments. Corresponding to the transient dynamics in the original system, we consider the dynamics around stripe patterns in the compartment model. We derive ordinary differential equations describing the pattern dynamics of the compartment model and analyze the existence and stability of equilibria for the reduced ODE with respect to the boundary parameters. For a specific parameter setting, we obtained results consistent with previous studies. Moreover, we present that the stripe patterns in the compartment model are potentially stabilized by changing the parameter, which is not observed in the original system. We expect that the methodology developed in this paper is extendable to various directions, such as membrane-induced pattern control.

Cerebral blood flow (CBF) control is essential for normal brain function and is disrupted in pathological conditions. Arterial diameters are tightly regulated to provide on demand increases in blood flow in regions of neuronal activity. Pericytes (PCs) exhibit robust myogenic tone and may also respond to neuronal activity to fine-tune local resistance and blood flow. Thus, mural control of microcirculatory resistance may extend beyond arteries and arterioles. Yet, PCs electrophysiology and contractility have not been thoroughly characterized, and this prohibits an integrated view of brain blood flow control. In this study, we develop a detailed mathematical model of mural cell electrophysiology, Ca2+ dynamics and biomechanics. The model is informed by electrophysiological data in smooth muscle cells (SMCs) or PCs and predictions are compared against pressure-induced responses in isolated arterioles and capillaries, respectively. Simulations recapitulate myogenic constrictions and examine differences in contractile dynamics as we move from arterioles to proximal and distal capillaries. In arteriole-to-capillary transitional (ACT) zone PCs, increased mechanosensitivity, more Ca2+ influx through non-selective cation (NSC) channels and/or a higher sensitivity of the contractile apparatus to Ca2+ can compensate for reduced L-type voltage-operated (VOCC) Ca2+ influx and allow for robust constrictions at the lower operating pressures of capillaries relative to the arterioles. A significant Ca2+ influx through NSC relative to VOCC, however, can decouple the PCs contractile apparatus from electrical signaling. Vasoactivity to chemomechanical stimuli along the arteriole to capillary axis is progressively driven by VOCC-independent Ca2+ influx and Ca2+ sensitization with slow kinetics. The proposed cell model can form the basis for detailed multiscale and multicellular models that will examine physiological function at a single vessel or vascular network levels and investigate CBF control in health and in disease. Key pointsO_LIA mural cell model of electrophysiology, calcium (Ca2+) dynamics and biomechanics is informed by data and adapted for modeling cerebral arteriole smooth muscle cells and capillary pericytes. C_LIO_LIIon channel activities are characterized by patch-clamp electrophysiology in isolated cerebral smooth muscle cell and pericytes, and capillary and arteriole electromechanical responses to transmural pressure changes are assessed using novel ex vivo preparations. C_LIO_LIMyogenic constrictions in arterioles can be reproduced by pressure-induced non-selective cation channel (NSC) activation that depolarizes the cell, opens L-type Ca2+ channels (VOCCs) and increases Ca2+ influx. C_LIO_LIRobust myogenic constrictions in arteriole-to-capillary transition (ACT) zone pericytes may reflect significant Ca2+ influx through NSC, increased mechanosensitivity, or higher sensitivity of the contractile apparatus to Ca2+, potentially compensating for reduced VOCC density relative to arteriolar smooth muscle. C_LIO_LIA significant contribution of NSC relative to VOCC in Ca2+ influx, can decouple the contractile apparatus from electrical signaling. C_LIO_LIThe model shows how gradients in ionic activities, mechanosensitivity and/or Ca2+ sensitivity can alter contractile phenotype and electromechanical coupling along the arteriole to capillary continuum. C_LIO_LIThe proposed model can form the basis for detailed multiscale and multicellular models that will investigate cerebral blood flow control in health and in disease. C_LI

We introduce a spectral existence criterion for the evolution of cooperation in the form of the inequality{lambda} maxb > c, where{lambda} max is the leading eigenvalue of an interaction operator encoding population structure, and b and c represent benefit and cost tradeoffs, respectively. Nowaks five rules for the evolution of cooperation correspond to cases in which the cooperation condition reduces to a scalar assortment coefficient. These results follow from the Price equation, which sheds light on a long-standing debate on the role of inclusive fitness and evolutionary dynamics in explaining the evolution of cooperation.

Infectious disease dynamics are strongly shaped by human mobility, social structure, and heterogeneous contact patterns, yet many epidemic models do not jointly capture these features. This study develops a spatial metapopulation epidemic model incorporating recurrent group-switch interactions to represent real-world transmission processes. Building on the Movement-Interaction-Return framework, the model integrates household structure, age-stratified contacts, and mobility between locations within a single SEIR framework. Using UK demographic, mobility, and social contact data, the model quantifies how within- and between-group interactions, mobility rates, and location connectivity influence epidemic spread. Both deterministic and stochastic simulations are implemented to analyse outbreak dynamics, variability, and fade-out probabilities for COVID-19-like and Ebola-like infections. Results shows that highly connected locations drive faster transmission, earlier epidemic peaks, and greater difficulty in containment, whereas larger but less connected locations tend to produce slower, more localised outbreaks despite their population size. Comparative analysis reveals that COVID-19-like infections spread rapidly and remain difficult to control even under interventions, while Ebola-like infections exhibit slower dynamics and are more effectively contained, particularly under targeted measures. Non-pharmaceutical interventions, particularly widespread closures, substantially reduce infections, hospitalisations, and deaths, although effectiveness depends on timing and pathogen characteristics. These findings highlight the importance of integrating mobility, clustering, and demographic heterogeneity to inform targeted and effective epidemic control strategies.

The number of offspring produced per reproductive cycle varies widely across animals and is constrained by the number of ovarian follicles that proceed to ovulation. In vertebrates, this phenomenon has been explained by a luteinizing hormone receptor (LHR)-threshold model, in which only follicles expressing sufficient levels of LHR respond to the LH surge and proceed to ovulation. Here we propose a novel mechanism that explains the difference between ovulatory (F1) and non-ovulatory (F2) follicles using the cloudy catshark as a model. The cloudy catshark possesses a hierarchical ovary and produces only two eggs per reproductive cycle. Both F1 and F2 follicles are capable of receiving and responding to LH, as evidenced by their comparable expression of lhr and the downregulation of lhr following LH surge. Nevertheless, LH stimulation selectively activates transcriptional programs associated with the ovulatory process exclusively in F1 follicles. These include progesterone production via star2 upregulation, as well as cancer-associated transcriptional pathways, including transcription factors runxs, prostaglandin-related genes (ptgs2 and ptger1), and matrix metalloproteinases. These results indicate that ovulatory and non-ovulatory follicles may exhibit qualitatively distinct transcriptional responses to the LH surge, potentially challenging the prevailing LHR-threshold model in vertebrates, in which LHR expression is considered a key determinant of ovulatory competence.

Alzheimers disease (AD) is characterized by the accumulation of amyloid-{beta} (A{beta}), yet the specific link between plaque burden and cognitive decline remains a subject of intense investigation. This paper presents a mathematical model that simulates the coupled dynamics of A{beta} monomers, soluble oligomers, and fibrillar species in the brain tissue. By modifying existing moment equations to include a dedicated conservation equation for A{beta} monomers, the model explores how various microscopic processes, such as primary nucleation, surface-catalyzed secondary nucleation, fibril elongation, and fragmentation, contribute to macroscopic disease progression. Central to this study is the concept of "accumulated neurotoxicity" as a surrogate marker of biological age, defined as the time-integrated concentration of soluble A{beta} oligomers. Unlike plaque burden, accumulated neurotoxicity cannot be reversed, and the harm it causes depends critically on the sequence of events that produced it. Numerical results demonstrate that while plaque burden and neurotoxicity both increase over time, their relationship is non-linear and highly sensitive to the efficiency of protein degradation machinery. Specifically, impaired degradation leads to a rapid advancement of biological age relative to calendar age. The model further identifies oligomer dissociation and fibril fragmentation as potential protective mechanisms that can counterintuitively reduce neurotoxic burden by diverting monomers away from the soluble oligomer pool. These findings provide a quantitative framework for understanding why individuals with similar plaque burdens may experience vastly different cognitive outcomes, underscoring the importance of targeting soluble oligomers early in therapeutic interventions.

The basic reproduction number of an infectious disease is known to depend on the structure of contacts between individuals in a population. This relationship has been explored mathematically through two well-known models: one which depends on a matrix of contact rates between different demographic groups, and another which depends on the variability of contact rates over the population. Here we introduce a model that combines and generalises these two approaches. We derive a formula for the basic reproduction number and validate it through comparisons to simulated outbreaks. Applying this method to contact survey data collected in Belgium between 2020 and 2022, we find that our model produces higher estimates of the basic reproduction number and larger relative changes over periods when social contact behaviour was changing during the COVID-19 pandemic. Our analysis suggests some practical considerations when using contact data in models of infectious disease transmission.

Clavicle fractures often exhibit markedly different clinical outcomes: some patients recover acceptable function despite shortening or displacement, whereas others with apparently similar deformity develop persistent pain, functional loss, or poor healing. To explain this distinction, we propose a minimal nonlinear mechanical model for prognostic analysis of clavicle fractures. The model describes the interaction between fracture-related shortening and compensatory shoulder-girdle posture through a reduced equilibrium equation incorporating stiffness, geometric nonlinearity, and shortening-posture coupling. Within this framework, we analyze equilibrium branches, local stability, and the emergence of critical thresholds. We show that post-fracture destabilization can be interpreted as a fold bifurcation, while more complex parameter dependence gives rise to cusp-type structures and multistability. These bifurcation mechanisms provide a mathematical explanation for sudden deterioration after injury or treatment, as well as for strong inter-individual variability. We further introduce an optimization principle based on a utility functional to guide treatment planning. The analysis predicts that the optimal safe correction should lie strictly below the bifurcation threshold, thereby generating a natural safety margin. Although the model is simplified and has not yet been calibrated against patient data, it nevertheless provides a theoretical framework for understanding why fracture prognosis may deteriorate abruptly near critical mechanical conditions and offers a dynamical-systems interpretation of empirical treatment thresholds used in clinical practice.

Cells must maintain stable protein levels despite the inherently stochastic nature of gene expression, as excessive fluctuations can disrupt cellular function and impair the reliability of decision-making. Regulatory mechanisms, such as negative feedback, buffer protein fluctuations. Yet, it remains unclear how fluctuations are affected by delays that depend on a molecules specific state. Here, we develop a stochastic model in which proteins are produced in bursts as inactive molecules and pass through a series of intermediate steps before becoming active. The duration of such activation delays depends on the current level of active protein, creating a state-dependent feedback loop. Our model provides explicit analytical expressions relating the delay structure and feedback strength to the variability of active protein levels, quantified using the Fano factor, and shows that state-dependent delays can reduce fluctuations below the baseline expected from simple bursty production. Stochastic simulations confirm these predictions, and incorporating negative feedback in burst production further decreases variability while keeping system behavior predictable. These results reveal how temporal and state-dependent regulation stabilizes protein expression, offering guidance for understanding natural cellular control and designing robust synthetic gene circuits.

Humans are just another species on Earth, but modern telecoupled societies and their socioeconomies impose immense consumption demands on the biosphere, detaching from common ecological rules. Starting from a simple ecological consumer-resource model, with humans as the consumers and terrestrial organic carbon (i.e., the biosphere) as the resource, we assume that technology modulates both human carrying capacity,{nu} 0, and the rate of biosphere consumption, 0. Three different functional-relation scenarios were tested, modulated by parameter a. In all three scenarios, equilibria and stability directly depended on the relative role that technology played in the model parameters, or the compound technological impact ({epsilon} {equiv} 0{nu}0). Moreover, two of the three scenarios showed Hopf bifurcations and regions with no equilibrium. The models were parameterized and fitted to actual data using a trajectory of more than 150 years. These analyses suggest that we are currently in a stable oscillatory spiral with no immediate Hopf bifurcation threat, but within a trajectory that continuously depletes the biosphere and approaches a collapse in human population size if no changes are made in the relationship that technology has with growth (i.e.,{nu} 0) versus consumption (i.e., 0) dynamics. Because our predatory dynamics also appear to have shifted from regular predator- prey dynamics toward a supply-demand scenario, with persistently increasing values, the threat of a Hopf bifurcation is now present in our trajectory: changes in the stability of the coexistence equilibrium may arise. This simple model warns that we must pay closer attention to the predatory relations that our technologies are creating with bio-sphere dynamics, in a way that goes beyond population numbers and technological development alone.

Network-based epidemic models have been instrumental in understanding how contact structure shapes infectious disease dynamics, yet widely used frameworks such as Erd[o]s-Renyi, configuration-model, and stochastic block networks do not explicitly capture the combination of fully accessible (saturated) within-group interactions and constrained between-group connectivity characteristic of many real-world settings. Here, we introduce the Multi-Clique (MC) network model, a generative framework in which individuals are organised into fully connected cliques representing stable contact groups (e.g., households, classrooms, or workplaces), with a limited number of external connections governing inter-group transmission. Using stochastic susceptible-infectious-recovered (SIR) simulations on degree-matched networks, we compare epidemic dynamics on MC networks with those on classical random graph models. Despite having an identical mean degree, MC networks exhibit systematically distinct behaviour, including slower epidemic growth, reduced peak prevalence, increased fade-out probability, and delayed time to peak. These effects arise from rapid within but constrained between clique transmission, creating structural bottlenecks that standard models do not capture. The MC framework provides an interpretable, data-driven representation of recurrent contact structure, with parameters that map directly to observable quantities such as household and classroom sizes. By isolating the role of intergroup connectivity, the model offers a basis for evaluating targeted intervention strategies that reduce between-group mixing while preserving within-group interactions. Our results highlight the importance of explicitly representing the real-life clique-based network structure in epidemic models and suggest that classical degree-matched networks may systematically overestimate epidemic speed and intensity in structured populations.

We describe an approach for analyzing biological networks using rows of the Krylov subspace of the adjacency matrix. Specifically, we explore the scenario where the Krylov subspace matrix is computed via power iteration using a non-random and potentially non-uniform initial vector that captures a specific biological state or perturbation. In this case, the rows the Krylov subspace matrix (i.e., Krylov trajectories) carry important functional information about the network nodes in the biological context represented by the initial vector. We demonstrate the utility of this approach for community detection and perturbation analysis using the C. Elegans neural network.

We extend previous models of cumulative cultural evolution by incorporating structured populations and social networks. We examine how connectivity and network topology shape the accumulation of cultural complexity under unbiased (copy randomly), indirectly biased (copy successful individuals), and directly biased (copy successful traits) transmission. We consider random, scalefree, and small-world networks, as well as the communication structures introduced by Mason and Watts, and derive analytical approximations for the homogeneous case. We find that the effects of social structure depend strongly on the form of transmission bias. Under unbiased transmission, network effects are weak except at very low connectivity. Under indirect bias, cultural complexity increases with connectivity, whereas direct bias shows optimal performance at intermediate connectivity, reflecting a trade-off between diffusion and diversity. Differences across topologies are generally modest once the average degree is fixed. Overall, our results show that no single social structure universally promotes cumulative cultural evolution; instead, its effects depend primarily on the dynamics of learning and innovation.

Infectious disease surveillance systems in tropical countries show that respiratory disease incidence generally manifests as year-round activity with weak fluctuations and irregular seasonality. Previously, using a ten-year time series of influenza-like illness (ILI) collected from outpatient clinics in Ho Chi Minh City (HCMC), Vietnam, we found a combination of nonannual and annual signals driving these dynamics, but with unknown mechanisms. In this study, we use seven stochastic dynamical models incorporating humidity, temperature, and school term to investigate plausible mechanisms behind these annual and nonannual incidence trends. We use iterated filtering to fit the models and evaluate the models by comparing how well they replicate the combination of annual and nonannual signals. We find that a model including specific humidity, temperature, and school term best fits our observed data from HCMC and partially reproduces the irregular seasonality. The estimated effects from specific humidity and temperature on transmission are nonlinearly negative but weak. School dismissal is associated with decreased transmission, but also with low magnitude. Under these weak external drivers, we hypothesize that stochasticity makes a strong sub-annual cycle more likely to be observed in ILI disease dynamics. Our study shows a possible mechanism for respiratory disease dynamics in the tropics. When the external drivers are weak, the seasonality of respiratory disease dynamics is prone to the influence of stochasticity.

Spatio-temporal interactions shape microbial community dynamics. Metabolism, through competition and cross-feeding, is a foundational mechanism of these interactions. Flux balance analysis enables efficient simulation of steady-state metabolism. Integrating these simulations through time, using dynamic flux balance analysis, provides temporal predictions of growth and metabolism. Incorporating spatial context, through partial differential equations, enables spatio-temporal simulation of microbial communities. In this chapter, we step through this sequential process, moving from steady-state, to temporal, to spatio-temporal simulation of microbial community metabolism. We provide an illustrative example using the modeling software COMETS (Computation of Microbial Ecosystems in Time and Space) to simulate interacting bacterial colonies of Bifidobacterium longum subsp. infantis and Anaerobutyricum hallii (previously Eubacterium hallii). Within this simulation, both competition and cross-feeding influenced the production of butyrate leading to an intermediate optimal interaction distance for metabolite production. We outline each step and provide open-source code such that this simulation can serve as a template for future spatio-temporal simulations of microbial community metabolism.

Cancer cell dormancy is characterized by late relapse and therapy resistance, yet the mechanisms that awaken dormant cells remain poorly understood. The tumor microenvironment has emerged as a key driver of these state transitions. Here we present a theoretical framework based on evolutionary game theory in which interactions between cancer and host cells are coupled explicitly to a changing tumor microenvironment. Cancer cells produce a conditioning factor that is cleared by the microenvironment and tolerated only up to a threshold. Through this conditioning factor, the microenvironment feeds back on cancer-host interactions and reshapes their competitive balance. Unlike a model with fixed interactions, this feedback allows dormancy and awakening to emerge as dynamic outcomes of microenvironmental change. We show that this minimal coupling is sufficient to generate distinct long-term regimes. Across these regimes, feedback generates thresholds and history dependence, so that the same cancer population can follow different fates depending on whether the microenvironment is already primed. Our framework reduces these dynamics to experimentally and biologically interpretable parameters linked to conditioning factor production, clearance, tolerance, and microenvironment-dependent changes in cancer-host competition. More broadly, it provides a quantitative basis for testing how collective microenvironmental feedback shapes cancer dormancy and awakening, and for designing experiments to uncover the mechanisms that awaken dormant cancer cells.