Bulletin of Mathematical Biology

Sadhu, Gopinath; Jain, Paras; George, Jason Thomas; Jolly, Mohit Kumar

doi: 10.1007/s11538-025-01591-2pmid: 41533050

Tumor growth and angiogenesis drive complex spatiotemporal variation in micro-environmental oxygen levels. Previous experimental studies have observed that cancer cells exposed to chronic hypoxia retained a phenotype characterized by enhanced migration and reduced proliferation, even after being shifted to normoxic conditions, a phenomenon which we refer to as hypoxic memory. However, because dynamic hypoxia and related hypoxic memory effects are challenging to measure experimentally, our understanding of their implications in tumor invasion is quite limited. Here, we propose a novel phenotype-structured partial differential equation modeling framework to elucidate the effects of hypoxic memory on tumor invasion along one spatial dimension in a cyclically varying hypoxic environment. We incorporated hypoxic memory by including time-dependent changes in hypoxic-to-normoxic phenotype transition rate upon continued exposure to hypoxic conditions. Our model simulations demonstrate that hypoxic memory significantly enhances tumor invasion without necessarily reducing tumor volume. This enhanced invasion was sensitive to the induction rate of hypoxic memory, but not the dilution rate. Further, shorter periods of cyclic hypoxia contributed to a more heterogeneous profile of hypoxic memory in the population, with the tumor front dominated by hypoxic cells that exhibited stronger memory. Overall, our model highlighted the complex interplay between hypoxic memory and cyclic hypoxia in shaping heterogeneous tumor invasion patterns.

Fog, Agner

doi: 10.1007/s11538-025-01584-1pmid: 41547707

It is often debated whether group selection can explain altruistic behaviors that lower the fitness of individual organisms for the benefit of their group. Several models of group selection are simulated here with more details and fewer simplifying assumptions than previous quantitative studies. The simulated models include island models with selective extinction, selective dispersal, selective migration, outsider exclusion, conformity, altruistic punishment, haystack model, and a new model with floating group territories. The simulations are repeated with different parameter sets in order to map the parameter areas that lead to either fixation of altruism, fixation of egoism, or stable polymorphism. This can help decide whether a particular behavior can be explained by genetic group selection. The conditions for group selection to override counteracting individual selection are found to be very restrictive. These conditions are met for eusocial insects, some parasites, and a few other species. The necessary conditions are unlikely to have been met in the evolutionary history of humans and most other group-living animals. Altruistic behaviors in humans could not have evolved without involving cultural mechanisms, including norms, rewards and punishments, reputation, and leadership. A comprehensive open-source simulation program is provided to facilitate further research.

Yang, Xiyan; Wang, Zihao; Shi, Changhong; Wu, Yahao; Zhang, Jiajun

doi: 10.1007/s11538-025-01581-4pmid: 41489699

Gene expression is inherently stochastic, and transcription initiation is a key source of variability across cells. While classical promoter models often assume linear state transitions, emerging evidence suggests more flexible promoter architectures. Here we introduce a generalized cyclic promoter model and compare it with the standard linear model using exact analytical solutions for initiation-time and nascent RNA distributions. Our results reveal that linear promoters produce only monotonic initiation-time statistics and a limited set of RNA expression patterns, whereas cyclic promoters generate non-monotonic initiation-time distributions and richer RNA profiles, including multimodal cases not achievable with linear architectures. We further show that cyclic promoters consistently buffer variability in initiation timing and RNA output, providing tighter control over transcriptional noise. Within the cyclic model, the number of exit pathways serves as a tunable parameter that shifts distributions from bimodal to unimodal and reduces noise, offering a potential mechanism for balancing robustness with flexibility in gene regulation. This framework highlights promoter topology as a critical determinant of transcriptional heterogeneity, bridges initiation dynamics with RNA-level variability, and generates testable predictions that can guide single-cell experiments probing promoter structure.

Cassidy, Tyler; Johnston, Stuart T.; Plank, Michael; Botha, Imke; Flegg, Jennifer A.; Murphy, Ryan J.; Hamis, Sara

doi: 10.1007/s11538-025-01583-2pmid: 41530617

Mathematical modelling is a widely used approach to understand and interpret clinical trial data. This modelling typically involves fitting mechanistic mathematical models to data from individual trial participants. Despite the widespread adoption of this individual-based fitting, it is becoming increasingly common to take a hierarchical approach to parameter estimation, where modellers characterize the population parameter distributions, rather than considering each individual independently. This hierarchical parameter estimation is standard in pharmacometric modelling. However, many of the existing techniques for parameter identifiability do not immediately translate from the individual-based fitting to the hierarchical setting. In this work, we propose a nonparametric approach to study practical identifiability within a hierarchical parameter estimation framework. We focus on the commonly used nonlinear mixed effects framework and investigate two well-studied examples from the pharmacometrics and viral dynamics literature to illustrate the potential utility of our approach.

Kosmacher, Gabriel K.; Max, Dillon; Rapti, Zoi; Cáceres, Carla E.; Stewart Merrill, Tara E.

doi: 10.1007/s11538-025-01582-3pmid: 41530616

In both human and wildlife disease systems, temporal shifts in host immunity may shape the timing and severity of epidemics. Yet, immune responses, as well as seasonal patterns in their expression, are difficult to measure. Rather, field studies collect phenomenological data on infection outcomes. Pairing epidemic data of multiple outbreaks with models that directly parameterize immune metrics can be a powerful approach for exploring the role of time-varying immunity on disease. Field data can be used to determine how well a parameterized model can reproduce trends and differences observed among outbreaks.Previous work in the Daphnia dentifera-Metschnikowia bicuspidata focal host-fungal pathogen disease system has not taken full advantage of coupling patterns in nature with mechanisms predicted by theory. Here, we study a mathematical model accounting for host immunity in the form of resistance to and recovery from M. bicuspidata infections and temporal variation in key aspects of the system’s epidemiology and ecology. Specifically, host population birth, predation and transmission rates, the fraction of recovering hosts, as well as the fungal spore yield were allowed to vary within the epidemic season. Modifying the system’s carrying capacity produces good correspondence between observed and model-estimated densities. Adjusting the transmission rate, spore yield, and the fraction of recovering hosts, captures the timing of disease outbreaks, as well as other qualitative features of outbreaks, such as the disparity between the prevalence of early- and late-stage infections. Our findings suggest that host immunological parameters are an important within-host constraint on disease dynamics.

Iwasa, Yoh; Hayashi, Rena; Hara, Akane; Matsuo, Kosei

doi: 10.1007/s11538-025-01569-0pmid: 41547663

In mammals, the immune system recognizes and combats pathogens while retaining a memory of prior encounters. In the thymus, naïve T cells capable of recognizing specific antigens are generated through random gene rearrangement, ensuring a diverse immune repertoire. However, the production rate of naïve T cells declines with age, typically following an exponential or power-law function—a phenomenon known as thymic involution, which is often regarded as a deterioration of biological function (immunosenescence). In this paper, we propose a novel theory suggesting that thymic involution may represent an adaptive strategy. As individuals age, repeated exposure to diverse pathogens leads to the accumulation of memory T cells, thereby reducing the need for newly generated naïve T cells to combat infections. Moreover, naïve T cells can persist in the periphery and retain the capacity to initiate immune responses against novel antigens. Using Pontryagin’s Maximum Principle, we calculate the optimal schedule of naïve T cell production. The results show that the production rate peaks during a brief period shortly after birth, followed by an exponential decline throughout life, eventually reaching a phase in which naïve T cell production ceases. If peripheral naïve T cells decay very slowly, the optimal strategy may consist of producing all cohorts at birth, with no subsequent production.

Jiménez-Sánchez, Juan; Ortega-Sabater, Carmen; Maini, Philip K.; Pérez-García, Víctor M.; Lorenzi, Tommaso

doi: 10.1007/s11538-025-01561-8pmid: 41533262

Evolvability is defined as the ability of a population to generate heritable variation to facilitate its adaptation to new environments or selection pressures. In this article, we consider evolvability as a phenotypic trait subject to evolution and discuss its implications in the adaptation of populations of asexual individuals. We explore the evolutionary dynamics of an actively proliferating population of individuals, subject to changes in their proliferative potential and their evolvability, through mathematical simulations of a stochastic individual-based model and its deterministic continuum counterpart. We find robust adaptive trajectories that rely on individuals with high evolvability rapidly exploring the phenotypic landscape and reaching the proliferative potential with the highest fitness. The strength of selection on the proliferative potential, and the cost associated with evolvability, can alter these trajectories such that, if both are sufficiently constraining, highly evolvable populations can become extinct in our individual-based model simulations. We explore the impact of this interaction at various scales, discussing its effects in undisturbed environments and also in disrupted contexts, such as cancer.

Mrad, Lidia; Lega, Joceline

doi: 10.1007/s11538-025-01577-0pmid: 41533203

This article introduces a computational method, called Recapture of Diffusive Agents & Particle Swarm Optimization (RDA-PSO), designed to estimate the dispersal parameter of diffusive insects in mark-release-recapture (MRR) field experiments. In addition to describing the method, its properties are discussed, with particular focus on robustness in estimating the observed diffusion coefficient in the presence of uncertainty. It is shown that RDA-PSO provides a simple and reliable approach to quantify insect dispersal that can handle low recapture rates and uneven capture site distributions without the need for area corrections. Tests on synthetic data, for which the actual diffusion coefficient is known, show the method outperforms three techniques based on the solution of the diffusion equation, which are also introduced in this work. Examples of application to real field data for the yellow fever mosquito are provided.

Foster, Richard R.; Delaney, Connor; Krusienski, Dean J.; Ly, Cheng

doi: 10.1007/s11538-025-01566-3pmid: 41533177

Visual stimuli with constant temporal frequency input are known to induce peaks in the driving frequency of the power spectrum of the electroencephalogram (EEG) over the visual cortex. While EEG responses with random temporal frequencies (m-sequences) of scenes alternating between two images have been studied, the underlying mechanisms that shape these responses are not fully understood. We analyze our new EEG data from a controlled experiment with m-sequence inputs and model the EEG using statistical time series models: an autoregressive (AR) model, adding exogenous input to AR (ARX), adding moving average terms (ARMAX), and finally adding a seasonality term (SARMAX). We implement computational methods to robustly handle model instabilities induced by this data, fitting these models with the Box-Jenkins methodology and assessing prediction accuracy of some statistical aspects of the EEG for long periods of several seconds out-of-sample. We find in-sample fits are good in all models despite the complexities of the visual pathway, and that all models can predict aspects of EEG: including the distribution of point-wise values in time, the point-wise Pearson’s correlation of EEG and model, and the frequency content. Surprisingly, we find little variation in the performance among these models, with the most sophisticated model (SARMAX) performing comparatively poorly in some instances. Our results suggest the simplest AR model is viable and can outperform more complicated models. Since these models are relatively simple and more transparent than contemporary models with numerous parameters, our study could inform future mechanistic studies of the temporal dynamics of human EEG responses to visual stimuli.

Tan, Yiping; Liu, Suli; Cai, Yongli; Sun, Xiaodan; Yao, Ruoxia; He, Daihai; Peng, Zhihang; Wang, Weiming

doi: 10.1007/s11538-025-01586-zpmid: 41530444

HIV-1 remains a formidable global health challenge, as complete viral eradication is still unattainable despite considerable advances in combination antiretroviral therapy (cART). To address this, we develop a stochastic differential equation (SDE) model that incorporates environmental noise into a classical HIV-1 infection dynamics framework, establishing two key advances. Mathematically, we derive the stochastic basic reproduction number Rs\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\mathcal {R}^s$$\end{document} and establish the corresponding threshold dynamics: when Rs<1\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\mathcal {R}^s<1$$\end{document} (under mild conditions), the infection is almost surely cleared, whereas for Rs>1\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\mathcal {R}^s>1$$\end{document}, the virus persists stochastically, following an ergodic stationary distribution. Epidemiologically, we demonstrate that environmental noise profoundly influences HIV-1 dynamics and reaffirm the central role of cART. Using optimal control theory, we evaluate three intervention strategies: Strategy 1 (cART enhancement), Strategy 2 (immune modulation), and Strategy 3 (a combined cART-immune approach). Both statistical indicators and dynamical outcomes confirm that Strategy 3 provides a clear advantage in promoting rapid viral suppression by integrating cART enhancement with immune modulation. Moreover, we observe that this combined intervention remains highly effective even under stringent cost constraints, and further reductions in intervention cost could improve its cost-efficiency. These results provide not only a novel theoretical framework for understanding HIV-1 infection dynamics, but also actionable clinical insights for optimizing treatment protocols, underscoring the critical importance of cost considerations in HIV-1 management.