Mathematical Biosciences

Canine rabies persists in NDjamena (Chad) despite vaccination campaigns exceeding 70% coverage, suggesting a role for dog mobility and spatial heterogeneity. We propose a metapopulation SEIR model incorporating distance-modulated dog movements and an explicit vaccinated class. Analysis of the isolated patch establishes global stability of the disease-free equilibrium via a Lyapunov function. For the metapopulation, a composite Lyapunov function shows that elimination is governed by a reproduction number [R]v. Calibrated with field data (2012-2022), simulations reveal that uniform vaccination of both patches reduces [R]v by 46% (from 2.84 to 1.52) but does not achieve elimination, while targeted strategies are less effective. These results demonstrate that exhaustive vaccination coverage across the entire urban network and increased vaccination intensity are necessary to eliminate canine rabies in NDjamena. Our model provides a quantitative framework for planning effective control strategies.

Ordinary differential equation models of biochemical reactions are often formulated as stoichiometric systems in which the dynamics arise from a collection of interacting processes. A central challenge is that the functional form of each process is rarely known a priori and may be difficult to infer from data. We propose biochemically informed neural ordinary differential equations (BINODEs), a neural-ODE framework that retains the stoichiometric structure of mechanistic models while representing individual processes by neural networks. In BINODEs, the outputs of neural network processes (NNPs) are mapped to state derivatives through a linear layer analogous to a stoichiometric matrix. This architecture allows biological side information, such as process-specific inputs, sign constraints, and monotonicity assumptions, to be built directly into the model. We characterize the approximation properties of NNPs for several standard biochemical rate laws and show that the proposed framework recovers both trajectories and process-level structure in Monod, Lotka-Volterra, pharmacokinetic, and ultradian endocrine models. These results suggest that BINODEs offer a useful compromise between mechanistic interpretability and data-driven flexibility for modeling partially known biochemical or biological dynamical systems.

This work focuses on the global and partial identification problem for fractional differential equations. We provide a general numerical procedure based on global and local optimization algorithms with two refinements for biological systems that ensure solution positivity and homogeneous parameter units. The method is applied to a new fractional model of Dengue outbreak called the Fractional Homogeneous Nishiura (FHN) model, calibrated using data of newly infected people in Cape Verde. We show that our identification method yields a better fit between data and model solutions than previous approaches and that our FHN model captures the dynamics of Dengue more closely than existing systems.

Parameter estimation in nonlinear biological dynamical systems is a difficult inverse problem because the governing equations are often stiff or oscillatory, the data are sparse and noisy, and the objective landscape is non-convex. Physics-informed neural networks (PINNs) offer an alternative to purely simulation-based calibration by representing state trajectories with neural networks while penalizing violations of the governing equations. This paper studies the empirical reliability of PINNs for recovering the parameters of the repressilator, a synthetic genetic oscillator formed by three cyclically repressive genes. We use synthetic time-series generated from the standard ordinary differential equation model and train inverse PINNs to estimate the production parameter {beta} and the Hill coefficient n. The study varies observation noise, partial observation of repressors, sampling density, sensitivity to initial parameter guesses, and the difference between stable and oscillatory regimes. The results show that PINNs can reconstruct trajectories accurately when the model structure is correct and the three repressors are observed, but parameter recovery is more fragile than trajectory fitting. Noise, sparse sampling, unobserved variables, and unfavorable initial guesses increase the risk of biased estimates. The stable regime is easier to reconstruct, whereas the oscillatory regime provides richer information but also exposes optimization sensitivity. These findings support PINNs as a useful reverse-engineering tool for small gene-regulatory ODE models, while highlighting the need for repeated runs, uncertainty reporting, and experimental designs that improve identifiability.

In the dynamic interplay between hosts and pathogens, hosts may produce a defense compound that acts as a toxin to deter pathogen attack. Conversely, pathogens may evolve to produce a counter-defense enzyme, neutralizing the hosts toxin. This evolutionary arms race incurs costs for both parties, prompting adaptations and strategic shifts. We conceptualize this interaction as an asymmetric game, with hosts and pathogens as players, and their potential responses - defense, counter-defense, or inaction - as their strategic options. In this scenario, if the pathogens counter-defense enzyme is entirely effective, then the hosts toxin is rendered obsolete. However, should the host cease toxin production, the pathogens enzyme becomes redundant, ironically reinstating the toxins utility. This interaction leads to potential red-queen cycles in defense and counter-defense strategies under certain conditions, or a balanced, optimal production of toxin and enzymes by hosts and parasites, respectively. To explore this, we introduce a game-theoretical model incorporating replicator dynamics to examine temporal shifts in strategy from active (counter-)defense to non-(counter-)defense and back. In addition, we analyze compromise strategies and interpret them as bet-hedging-like. We provide a deterministic illustration of how partial defense and counter-defense generate a fitness-buffering structure in unpredictable environments and increase the geometric mean fitness of the population. In conclusion, our analysis supports the notion of continuous periodic adjustments in strategies, notably in the levels of defensive and counter-defensive measures.

Genome-scale metabolic network reconstructions contain extremely detailed and valuable information regarding cellular metabolism. For many applications such as finding genetic engineering targets and reduced kinetic model construction, metabolic network analysis techniques exist. Yield spaces based on the extreme rays of solution cones related to the metabolic network are frequently constructed for these types of analyses. However, for genome-scale networks, full enumeration of these extreme rays is not computationally feasible. In this work, a novel direct generation method for yield spaces is presented. This allows the application of many metabolic network analysis techniques to even the most recent genome-scale metabolic networks. Inspired by principles from multi-objective optimization algorithms, the proposed method performs highly efficient recursive exploration but specifically adapted to the mathematical properties of yield spaces. Two case studies showcase both the efficiency of the method and its applicability for analysis of genome-scale metabolic networks.

Interdependent multicellular circuits must maintain stable coexistence despite competition for shared environmental resources. Fibroblast-macrophage circuits represent a conserved signaling architecture in which fibroblasts produce colony-stimulating factor 1 (CSF) to support macrophages, whereas macrophages produce platelet-derived growth factor (PDGF) to support fibroblasts. Previous analytical models proposed receptor-mediated endocytosis as a stabilizing negative-feedback mechanism, but these formulations assumed spatial homogeneity and independently assigned carrying capacities. Here, we constructed a spatial agent-based fibroblast-macrophage circuit model using PhysiCell to investigate how PDGF and CSF endocytosis regulate circuit stability under explicit competition for shared oxygen and space. Fibroblasts and macrophages competed for common environmental resources supplied by spatially distributed capillary sources, allowing carrying capacity to emerge dynamically from local resource competition. Across nine enhancer conditions spanning fourfold variation in PDGF and CSF signaling strength, heterotypic coexistence remained broadly achievable regardless of endocytic activity. In contrast, endocytosis strongly suppressed stochastic circuit failure. This stabilization depended critically on macrophage CSF uptake, whereas broad ranges of fibroblast PDGF uptake produced comparable outcomes, generating a sloppy stabilization landscape along the PDGF uptake axis. Mechanistically, excessive CSF signaling drove macrophage overexpansion, depletion of shared resources, and eventual fibroblast extinction. Importantly, despite fundamentally different carrying-capacity assumptions from previous analytical models, both frameworks converged on the same systems-level conclusion: stabilization of the macrophage-supporting CSF axis is substantially more critical than stabilization of the PDGF axis. These results identify endocytosis as a robustness mechanism that suppresses catastrophic failure in interdependent multicellular circuits under shared-resource competition without requiring precise parameter tuning.

Cooperative and competitive interactions among individuals harvesting resources can shape environmental states, such as prey abundance. In turn, environmental conditions feed back to influence strategic interactions. Eco-evolutionary game theory studies how these feedbacks shape the co-evolution of behavior and environment. Existing models typically assume deterministic, noise-free environmental dynamics. However, real environments are inherently stochastic, for example due to finite resources, and noise can qualitatively alter social outcomes. Here, we incorporate stochastic environmental dynamics into eco-evolutionary game theory. When environmental change is slow relative to strategy updates, we show that behavior reflects a mixture of the games associated with low and high environmental states, often yielding outcomes qualitatively distinct from deterministic predictions. In particular, environmental stochasticity can eliminate bistability and enforce dominance of a single behavior. When environmental dynamics are faster, populations have less opportunity to track fluctuations, and behavior converges toward strategies that are optimal on average. Stochasticity can even causes persistent oscillations in the tragedy of commons, in regimes where classical models predict stability. Our framework provides a tractable approach for analyzing social behavior linked to environmental dynamics how noise shapes long-term eco-evolutionary outcomes.

Understanding how complex systems self-organize, exhibit emergent properties beyond their constituent elements remains a challenge across physics, biology, and cognitive science. In resource-constrained neuronal systems, existing theoretical approaches, including gauge theoretic formulations, statistical physics-inspired methods, dynamical population models, and variational principles such as the Free Energy Principle, address important aspects of this problem but do not fully specify the physical conditions and thermodynamic costs under which self-organizing behavior occurs. Here, we introduce Dynamic Resource Theory (DRT) as a general physical framework for describing self-organization under constrained resource availability. DRT formalizes complexity as a physical property of self-organizing systems arising from coupled mechanisms of resource allocation and dynamic reallocation of internal resources. This framework provides a thermodynamic and variational account of how stability is preserved while adaptive reconfiguration remains possible, consistent with stationary action and thermodynamic constraints. DRT is formulated within a gauge theoretic setting and directly incorporates the energetic costs associated with maintaining structure and enabling system-level reconfiguration. Within DRT, baseline resource allocation preserves system stability, while internal and external demands perturb the system, driving self-organization through dynamic resource reallocation across a coupled free energy landscape without assuming subsystem separability. We then develop Neural Resource Theory (NRT) and Cognitive Resource Theory (CRT) as principled specializations of DRT, illustrating how this structure is instantiated in resource constrained neuronal and cognitive systems. We conclude by discussing the broader implications of DRT for understanding how complexity, emergence, and adaptive capacity arise over time through thermodynamically permissible reallocation processes across scales.

Multicellular organisms regularly encounter microbes, which are, however, only rarely pathogenic. Our understanding of this phenomenon is currently restricted due to lacking theory on evolutionary transitions between non-pathogenic and pathogenic microbial lifestyles. Here we addressed this gap by investigating a mathematical model of host-microbe interactions that is based on the danger theory of immunology, which states that danger signals related to host tissue damage play a key role in activating immune responses. We formally implemented this idea by assuming that immune activation increases with costs that microbes cause to their host, and we compared this to scenarios in which immune activation depends only on the presence or load of infecting microbes. Our model analysis revealed that cost-based - but not presence or load-based - immune activation favours the evolution of avirulence and associated non-pathogenic microbial lifestyles. Based on our results, we propose the danger hypothesis of virulence evolution which states that evolution towards avirulence and intermediate virulence are both possible - depending on whether hosts can accurately assess costs generated by microbes. The idea that basic host immune responses can select for avirulence offers a new explanation for why most microbes are not pathogenic to a given host.

The Hajj is an annual pilgrimage made by millions of Muslims to Mecca in the Kingdom of Saudi Arabia (KSA). The large number of international attendees at the Hajj increases the risk of global infectious disease spread. However, we know very little about the benefits, costs, and cost-effectiveness of testing and quarantining strategies to contain epidemic spread during mass gathering events. In this work we developed a stochastic discrete-time compartmental metapopulation model to simulate international epidemics of infectious pathogens and their potential importation into KSA during the Hajj. We used the model and an epidemic simulation study to evaluate the impact and cost-effectiveness of three testing and quarantining strategies for arriving pilgrims: randomly testing 99% of pilgrims, 80% of pilgrims, or using a symptom-based screening strategy. The simulations lasted 100 days, covering the 30 days before the Hajj and 65 days after the Hajj. Under the conditions assumed in our simulation study, there was strong evidence that testing and quarantining strategies are cost-effective measures for controlling epidemic threats at the Hajj. The median net monetary benefits of intervention strategies ranged from Intl$-41.89M [95% quantile range Intl$-42.37M to Intl$3.18B] to Intl$12.68B [Intl$-8.70B to Intl$13.82B] across scenarios with different pathogen characteristics (based on the natural histories of SARS-CoV-2 and H1N1 Influenza) and epidemic seed locations. Our results were sensitive to the data sources that were used to estimate the number of pilgrims travelling to KSA by origin country, with flight passenger statistics providing biased estimates of pilgrim numbers. Our work provides an adaptable tool to inform infectious disease risk assessments and evaluate the cost-effectiveness of possible disease control measures for the Hajj, and could be extended to other mass gathering events.

An ongoing outbreak of Bundibugyo virus disease (BVD) in the Democratic Republic of the Congo was deemed a public health emergency of international concern in May 2026. To prevent cross-border importation, many countries, including the United States, Canada, India, Thailand, and Kenya have already proposed containment strategies, and others are likely to follow suit. How well (or poorly) are screening and quarantine containment measures are likely to work? We leverage established epidemiological theory and develop a mathematical model of traveler screening and post-arrival quarantine for BVD to answer this question. We find that traveler screening via symptom screening or molecular testing will miss the majority of infected travelers, and should be complemented by post-arrival quarantine and monitoring of sufficient duration to detect those with long incubation periods. Our findings underscore the limitations of border screening and the importance of complementary measures like post-arrival quarantine to prevent local importation of BVD.

The emergence of a hantavirus variant aboard a commercial cruise ship presents a significant public health concern. This study develops a discrete-time stochastic Susceptible-Exposed-Infectious-Recovered-Dead model to estimate transmission dynamics, hidden exposed infections, and outbreak risk among passengers and crew. Epidemiological parameters and latent disease states were inferred using an Ensemble Adjustment Kalman Filter calibrated to reported case data from WHO and ECDC situation reports. The estimated basic reproduction number was 2.76, with a 95% confidence interval of 2.52-2.99, indicating substantial potential for sustained onboard transmission before strict quarantine measures. Simulations further suggest that several exposed individuals may remain unidentified during the early outbreak phase, creating a hidden reservoir that symptom-based surveillance alone may fail to detect. These findings highlight the importance of rapid surveillance, widespread testing, targeted quarantine, and active monitoring of exposed individuals in confined travel settings. The proposed modeling framework can support timely outbreak assessment and intervention planning for infectious-disease events in similarly dense and spatially constrained populations.

1Although resources are typically distributed continuously in space, species distributions often organize into discrete clusters. In his seminal paper [36], Turing demonstrated that such clusters can spontaneously arise in population densities, even when populations evolve in environments with continuously varying conditions. This phenomenon is known as Turing instability. In this work, we focus on two models grounded in population dynamics: a one-dimensional model based on the nonlocal Fisher-KPP equation, and a two-dimensional model involving an environmental gradient. We show that phenotypic clusters (sometimes referred to as "species") emerge in these models. We prove that they do not emerge because of Turing instability, but because of stochasticity, and that they disappear when stochasticity is reduced. First, for both models, we start our simulations with initial populations uniformly distributed in the state space. We show that phenotypic clusters quickly emerge and that the distances between them depend on the population size, that is, on the degree of stochasticity. Next, we start from already clearly defined phenotypic clusters. We identify three regimes in the connection between population size, the initial distances between clusters, and the distances between clusters at equilibrium. Last, on the two-dimensional model, we relax the hypothesis of complete clonality by varying the effective recombination rate, explore its effect on phenotypic clustering, and show that phenotypic clustering decays drastically with slight recombination.

This paper presents a unified six-state Continuous-Time Markov Chain (CTMC) framework for Chronic Kidney Disease (CKD) progression, with CKD stages 1-5 modeled as transient states and death as an absorbing state. Under a non-homogeneous CTMC formulation, we derive integral representations for transition probabilities, state distributions, sojourn times, and survival-related quantities. We then study the homogeneous case as a tractable baseline and provide explicit formulas for key quantities. Although the methodology is rooted in standard multi-state theory, these expressions are often left implicit in applied analyses; here they are written out explicitly within a unified CKD framework. We construct covariate-dependent transition rates through a proportional hazards structure, using the standard identification of cause-specific hazards with CTMC transition rates. We fit the time-homogeneous baseline model to 335,283 longitudinal observations from 21,100 synthetic electronic health record patients by maximum likelihood. In this synthetic cohort, the covariate model improves held-out log-likelihood per transition over the null model, with stable performance across 10-times-repeated 5-fold cross-validation, and reproduces the main population-level prevalence patterns. The transition-specific estimates can also be translated into sojourn-time and survival summaries. The model suggests that male sex is associated with faster progression across nearly all CKD transitions, and that hypertension shows a stage-dependent association, with lower estimated transition rates in early stages but a substantial acceleration of the Stage 4 to Stage 5 transition. Overall, the proposed framework provides a mathematically explicit approach for studying CKD trajectories from longitudinal health records.

Effective pandemic response requires accurate modeling of population compliance with non-pharmaceutical interventions (NPIs), yet most epidemic models treat behavioral change as fixed scenarios rather than an emergent process. Here, we test whether large language model (LLM)-based agents can generate individualized behavioral responses to time-varying NPIs and disease risk. We instantiate demographically representative agents in three U.S. cities (Boston, Denver, San Antonio) and condition them on evolving outbreak conditions and policies during the early COVID-19 pandemic, without fitting to observed mobility data. Across three frontier LLMs and their ensemble, agents generate zero-shot mobility changes across restaurants, retail, and entertainment venues, benchmarked against cellphone-derived foot-traffic records. The simulations recover average mobility trends across cities and venue types but exhibit overly narrow within-city variation. The three LLMs display distinct biases, while an ensemble approach improves robustness and overall performance. These findings establish LLM agents as a promising framework for modeling adherence to NPIs and highlight the need for further fine-tuning and empirical validation before they can support policy analysis.

Interactions between distinct populations of excitatory (E) and inhibitory (I) neurons can produce complex dynamical landscapes, featuring multistability, oscillations, and paradoxical perturbation responses. By employing an elementary model, the threshold-linear network (TLN), we indicate mathematical conditions for each dynamical regime across fundamental microcircuit architectures, thereby mapping previously unrelated systems neuroscience hypotheses to a common reference space and obtaining novel insights on inputs and connectivity. Namely, we compare balancing strategies in inhibition-stabilized E-I networks, we interpret experiments on gamma oscillations in a canonical neocortical E-I-I circuit, and we discuss bistability in hippocampal E-I-I networks. Then, we show that connectivity determines three fundamentally different kinds of interactions between assemblies in E-E-I circuits. Moreover in, E-E-I-I circuits we find that balanced clustering hinders lateral inhibition, while opponent clustering can produce different bistable configurations, even between completely unstructured assemblies. We conclude that TLNs allow to grasp deep and universal aspects of microcircuit dynamics.

Advances in single-cell RNA sequencing (scRNA-seq) and high-resolution imaging techniques, such as single-molecule tracking (SMT) of RNA and transcription factors, allow researchers to quantitatively explore dynamics and variation but have never been integrated into a single coherent model. In this study, we propose a kinetic model that intakes multiple data types, including steady-state and time-resolved datasets, to simulate and fit stochastic models of gene transcription to experimental data. We find that 3-state models provide an essential improvement over the widely used 2-state model for most genes and have the property of kinetic proofreading, which we argue is advantageous in the cellular context. We further identify two dimensionless quantities derived from the rate equations which are broadly conserved across genes. Finally, we extend this model to scRNA-seq datasets to infer kinetic rates under defined perturbations and reveal biochemical insight into the mechanism of action of transcription factors.

Dopaminergic signalling is central to value learning and decision making. It has been observed that multiple pathways with different patterns of connectivity project to midbrain dopaminergic neurons, some involving direct excitatory projections while others involve disinhibition. However, the respective contributions of these patterns to dopamine control, and their computational and functional advantages remain unclear. In the current work we simulate and evaluate two fully spiking neural models of dopaminergic control, based either solely on disinhibition, or solely on direct inhibitory and excitatory projections. We compare these models in terms of their engineering properties, their resulting spiking profiles, and their ability to successfully acquire representations of expected value in a 3-armed bandit task. We find that both models are able to operate at an asynchronous-irregular firing regime, but that the firing profile of the direct integration model is less resilient to disruption and more sensitive to incoming signals. In addition, the disinhibition model performs better in the learning task. We conclude that while the direct model is more parsimonious, disinhibition-based control remains advantageous in the operational context. Our results have implications for the study of decision-making brain circuits as well as for the design of brain-inspired systems.

Reducing pesticide risks while maintaining food production remains a central challenge for sustainable agriculture. Although pesticide reduction is pursued through centralized regulation and farm-level Integrated Pest Management, how these governance pathways translate into pollinator recovery in agroecological systems remains poorly understood. Existing ecological network models often treat pesticide pressure asm external forcing and management actions as fixed parameters, limiting their ability to capture feedbacks among governance decisions, network structure, and population dynamics. Here, we develop a dynamical framework that embeds pesticide management within tripartite pollinator-plant-pest networks using a policy variable and a farm-level adoption variable. Across empirical and synthetic networks, we show that recovery is not determined by pesticide reduction alone, but by how management acts through ecological interaction structure. More modular networks require stronger intervention, and pollinators with similar degrees show different recovery outcomes, indicating that degree alone does not determine recovery potential. Further, increasing policy strength generally expands the persistence domain more than increasing farmer adoption alone. These results show that pesticide reduction does not automatically yield ecological recovery, and effective strategies must match governance scale to ecological condition and network structure.