Triangles are abundant in real-world networks but rare in standard null models for sparse graphs. Existing explanations typically rely on explicit triadic closure mechanisms or geometry-based connection rules. We propose an alternative hypothesis: the frequent appearance of triangles may arise naturally from the requirement of dynamic stability that maintains coexistence of species in Lotka-Volterra systems with equal competitive interactions. To evaluate this idea, we prove that, across all possible interaction graphs, coexistence is guaranteed whenever the coupling strength is below the reciprocal of the graph's maximum degree, and guaranteed not to occur when the coupling strength exceeds 1. This leaves a large gap that is unexplained by the graph degrees alone. We notice that the lower and upper bounds are achieved for star and complete graphs respectively and to investigate further what structural properties of the interaction graph control the critical coupling within the gap, we optimise networks algorithmically while keeping the degree sequence fixed. We find that networks supporting stronger interaction strengths consistently exhibit higher clustering coefficients in several network models. Moreover, in real-world grassland plant networks, we observe higher clustering and stronger stability than those expected from a configuration model with the same degree sequence. Our result suggests that triangles, and clustering in general, may emerge as a structural signature of stabilising competition.

Understanding island biodiversity requires studying the processes driving its assembly and examining how these processes vary over time, across lineages, and across different islands. The DAISIE (Dynamic Assembly of Island biota through Speciation, Immigration, and Extinction) framework has been developed for this purpose, and has advanced our understanding of island community assembly by estimating, from phylogenetic data, the contribution of the processes of colonization, speciation and extinction to island community assembly. However, the model assumes uniform colonization and diversification rates across lineages, ignoring potential variation in these rates caused by, for example, lineage-specific traits. This assumption thus restricts the model's capacity to capture complex colonization and diversification dynamics, and may consequently bias inference of colonization and diversification dynamics on islands. An extension of the framework behind these more complex dynamics is therefore desired. However, for this state-dependent colonization and diversification extension, the current computation of the likelihood of the model given phylogenetic data is computationally prohibitive. In this study, we therefore propose an alternative approach to computing the DAISIE likelihood, under the assumption of diversity-independent colonization and diversification. Our novel approach is based on the pruning algorithm that has been used for computing the likelihood of SSE (State-dependent Speciation and Extinction) models, which traces lineage history backward in time from the present day to the root of the tree. We demonstrate that our alternative approach reproduces DAISIE's predictions under diversity-independent colonization and diversification. This provides an additional layer of support to DAISIE in diversity-independent settings, but in doing so also gives more confidence in the results under diversity-dependence. Furthermore, our alternative approach offers greater computational efficiency. Finally, it provides a flexible framework for incorporating state-dependent dynamics in colonization, speciation, and extinction processes.

Medication non-adherence is recognized as a critical determinant of therapeutic efficacy and safety. However, its specific impact on the concentration fluctuations of chiral drugs remains poorly understood. By incorporating dual stochasticity in dosing intervals and dosages, we developed a stochastic pharmacokinetic model for chiral drugs comprising bio-active and bio-inactive enantiomers under multiple intravenous bolus administrations. Utilizing characteristic functions and second-type Volterra integral equations, we derived explicit expressions for the expectation and variance of both the active moiety and total measured concentrations, as well as their discrepancy. Moreover, leveraging Kesten theorem and ergodic theory, the existence and the geometric ergodicity of the stationary probability distribution were mathematically demonstrated. Taking ibuprofen as a case study, we simulated various medication non-adherence scenarios to quantify the enantiomeric randomness. The results demonstrated that stochastic concentrations not only cause significant deviation from the ideal levels under perfect adherence, but also induce non-negligible discrepancy between the effective and total measured concentrations. These findings of the work provide a rigorous theoretical guidance for the characterization of two enantiomers variability of chiral drugs induced by medication non-adherence, offering new insights for enhancing therapeutic efficacy and safety.

Deterministic dynamical models are widely used in infectious disease modelling, but they often become systematically biased when key drivers such as seasonality and random environmental variation are simplified or omitted. This paper proposes a practical way to account for structural bias while preserving the underlying mechanistic model. We model the observed time series as the sum of (i) a deterministic transmission component given by a reduced Ross malaria model and (ii) a latent stochastic seasonal component that captures unresolved seasonal forcing and other unmodelled variability. The seasonal component is defined as a mean-reverting stochastic differential equation with periodic forcing, which can be interpreted as a seasonally forced Ornstein-Uhlenbeck (OU) process and, at the same time, as a dynamically constrained model-discrepancy term. The combined model forms an additive Bayesian state-space system. A key challenge in additive decompositions is identifiability: many combinations of the deterministic and stochastic components can explain the same observations. We therefore use informative priors to stabilise this decomposition, together with a non-centred parameterisation (a reparameterisation that improves MCMC efficiency by sampling standardised noise terms instead of states directly) of the latent stochastic differential equation (SDE) states that enables efficient joint inference with Hamiltonian Monte Carlo (NUTS) in Stan. We validate the approach in three steps: (1) an OU example that contrasts parameter-only inference with latent state-space inference, (2) synthetic experiments that have a good agreement with the observed signal while highlighting the expected negative posterior dependence between components, and (3) an application to five years of monthly malaria case reports from Delta State, Nigeria. In the real-data analysis, the proposed additive ODE-SDE model produces a close fit with coherent uncertainty quantification and a flexible seasonal reconstruction, while keeping the mechanistic transmission model interpretable. Overall, the framework provides a flexible and transferable method for accounting for structural model discrepancy in misspecified dynamical models using a structured stochastic discrepancy.

Vaccine-related communication can be harnessed to curb infection; yet, in practice, it often undermines vaccine uptake and sustains transmission even when effective vaccines are available. Our goal is to understand how vaccine information dynamics may shape infection spread within a coupled information-infection modeling framework. We present a coupled information-infection modeling framework that links a standard SVIRS infection-spread model with an integrate-and-fire-inspired information-dissemination model. Vaccine-positive and vaccine-critical active groups disseminate competing information that shapes non-active/hesitant individuals' vaccine attitudes through direct peer influence, threshold-based acceptance, and persistence of engagement, while infection prevalence can feed back by amplifying caution. The evolving vaccine attitudes modulate vaccination uptake, and the model tracks the joint evolution of information dynamics and epidemic trajectories. Our analysis shows that, under the assumed coupling, information dynamics can shift the system among qualitatively distinct infection outcomes. In particular, reducing resistance to vaccine-positive information and sustaining vaccine-positive engagement can move the system toward lower-endemic regimes more reliably than changes focused only on weakening vaccine-critical engagement, for the parameter ranges considered here. These findings highlight the potential importance of information dynamics in epidemic modeling and suggest that sustained vaccine-positive engagement can be an important qualitative mechanism for reducing long-term infection burden.

Ecological interactions shape the dynamics of natural populations in the wild. Density-dependent processes are widespread and may change the respective effects of populations on one another, for instance by shifting interactions from mutualistic to parasitic relationships. Here, we develop a general deterministic model of two interacting populations, assuming density-dependent costs and benefits for the interacting individuals within and between species. This framework aims at generalizing pre-existing population dynamics models involving competition, predation, mutualism and parasitism, by allowing ecological interactions to transition when the respective densities of interacting species change. Through ordinary differential equations and phase portrait analysis, we derive general principles governing these systems, identifying constraints on the organization of equilibria and sufficient conditions for the emergence of certain dynamic behaviors. In particular, we show that equilibrium indices alternate along isoclines under broad geometric assumptions, and that limit cycles can arise when interactions include mutualistic and parasitic phases, while they cannot be generated locally in strictly mutualistic regions where the relevant interaction signs remain fixed. This framework provides a general approach for characterizing the population dynamics of interacting species and highlights the effect of the density dependent transitions in ecological interactions.

Lonafarnib (LNF) is an investigational drug targeting hepatitis delta virus (HDV) but not hepatitis B virus (HBV), providing a unique opportunity to model HDV kinetics and how changes in HDV affect HBV. We performed a detailed kinetic analysis and developed a mathematical model to explain serum HBV DNA, HDV RNA and hepatitis B surface antigen (HBsAg) kinetics in 15 HBV/HDV coinfected patients receiving LNF-based treatment. After a delay of 0-2 days, patients experienced a rapid 1st-phase HDV-decline followed by either a viral plateau, 2nd slower-decline phase, or viral breakthrough (VB). LNF monotherapy led to a flat-partial-response (often followed by VB), while LNF combination therapy with ritonavir or pegylated interferon-α (PEG-IFN α) was associated with a biphasic HDV decline (without VB). All treatments except LNF + PEG-IFN α had at least one patient experiencing an increase in HBV on-treatment. Our model successfully reproduced the observed HDV and HBV kinetics. We estimated an HDV RNA half-life of 1.26 days [95% confidence interval, CI 1.05-1.47] in serum and treatment efficacy of 94% in inhibiting HDV RNA production across all treatments [95% CI 89-97%], as reflected by the 1st phase HDV decline. The 2nd phase of HDV decline was explained by a time-dependent increase in efficacy, reaching a maximum of 98.9%. The model explained the increase in serum HBV DNA by a median four-fold [interquartile range, IQR: 1-28] increase in HBV DNA production rate when HDV declined below an inhibitory threshold. The stability of serum HBsAg was explained by a constant number of HBsAg-producing cells.

Life and evolution require precise yet imperfect transmission of genetic information across generations. Accurate transmission maintains the genetic blueprint for phenotypes that succeed in current conditions, while some transmission error is necessary to generate heritable diversity that can further increase fitness and hedge against future environmental change. This background noise, however, risks excessive mutation accumulation that degrades fitness ("Muller's ratchet") or even destroys genetic information beyond recovery ("error catastrophe"). Across the history of life these competing pressures resolve into two regimes: during environmental fluctuation, a higher mutation rate aids survival; during prolonged stability, populations converge toward an evolutionarily stable state (ESS) in which mutations can only reduce fitness. We model this tension with a Fisher Information framework in which genetic inheritance is a noisy two-state channel. The nontrivial eigenvalue of the channel's symmetric circulant transition operator governs the decay of allelic contrast across generations; selection modifies this eigenvalue by suppressing the channel's switching (mutation) probability, acting as an anti-depolarizing force. We show this compensation is bounded: an instability boundary at p ( 1 - p ) = 1 / 9 marks the point beyond which selection can no longer maintain informational stability - a boundary that coincides with the loss of quantum coherence in the formally equivalent depolarizing channel and with thermodynamic fine-graining failure, suggesting that genetic stability, regulatory fidelity, and structural order share a common information-geometric limit. We apply this framework to cancer, framing carcinogenesis as an informational transition: a shift from host-regulated, high-fidelity transmission that maintains tissue homeostasis to a regime in which cells retain only the genetic and epigenetic changes that maximize their own proliferation - including loss of functions that serve the host and gain of functions that improve competition for space and nutrients and evasion of the predator-like immune response. Because cancer cells operate farther from this information-geometric limit than the normal cells around them, we propose that they may pursue an informational "niche construction" strategy: producing a mutagenic, acidic, hypoxic microenvironment that pushes neighboring host cells past the same collapse boundary, driving loss of function (e.g., "T cell exhaustion") in infiltrating immune cells. We present empirical support for this framework together with specific, experimentally testable predictions.

Polyploidy occurs in plants and animals, and is an important force in speciation and genome evolution. The main focus of this paper is the following fundamental question that was recently posed by Huber and Maher: Given the ploidy numbers of a collection of extant species, or their ploidy profile, what is the smallest number of hybridizations needed in any evolutionary history for these species to completely represent these numbers? In this paper, we shall show that this question can be rephrased in terms of addition chains and the closely related addition sequences, which have been studied for over a century in mathematics and computer science. These are sequences of natural numbers that start with 1, so that each number in the sequence larger than 1 is the sum of two other numbers arising earlier in the sequence. In our first main result, we show that finding the smallest number of hybridization events to explain a ploidy profile, or the hybrid number, is equivalent to solving the so-called addition sequence problem. This immediately implies that computing the hybridization number is computationally intractable. Even so, it also leads to new connections to representing polyploid evolution using networks. More specifically, in our second main result we show that ploidy profiles representable by tree-child networks are exactly the addition chains, implying a polynomial-time algorithm for identifying these profiles. We then consider beaded tree-child networks, which permit the representation of autopolyploidy events, and in our third main result we provide a greedy polynomial-time algorithm to decide whether a given profile can be realized by such a network. We expect that our results can be leveraged in future work through, for example, making use of known algorithms for computing short addition sequences to give bounds for the hybrid number, and in guiding network reconstruction for polyploid species.

Cancer mortality remains high in part because tumor-immune interactions can be unpredictable and may exhibit multistability, making malignant progression difficult to anticipate. We develop and analyze a tumor-immune model incorporating chemotherapy-induced toxicity and a Norton-Simon type tumor response, and show via deterministic analysis and continuation that increasing toxicity can generate hysteresis and bistability separating tumor dormancy from an uncontrolled full-growth state, so that small perturbations may precipitate abrupt progression. To capture uncertainty, we study environmental and demographic stochasticity and observe noise-induced switching in both cases; however, demographic noise sustains higher resilience of the tumor-dominant state under bistability, whereas stronger environmental noise tends to suppress uncontrolled tumor growth. We further assess early-warning signals using single and composite rolling-window indicators and find that they can provide advance warning of transitions from dormancy to full growth, with composite measures offering greater robustness across noise levels. Finally, we formulate an optimal control problem for combination immunotherapy and radiotherapy and demonstrate that appropriately timed treatment can substantially reduce tumor burden when initiated from either dormancy or full-growth conditions, highlighting how stochasticity-aware monitoring and optimized interventions may help prevent catastrophic tumor progression.

Wolbachia-based mosquito population replacement is a promising biocontrol strategy to reduce arboviral disease transmission by introducing Wolbachia-infected mosquitoes into wild populations. The success of this approach depends on the strain-specific characteristics of the Wolbachia symbiont and the mosquito host. This study presents a mathematical modelling framework to evaluate how Wolbachia strains and mosquito hosts influence the success of population replacement. The model incorporates key biological factors, including incomplete cytoplasmic incompatibility (CI), imperfect maternal transmission (MT), fitness costs, immigration, and potential loss of Wolbachia from mosquitoes. Analytical and numerical simulations demonstrate that strain-specific differences significantly impact the establishment, spread, and long-term persistence of Wolbachia in mosquito populations. Results indicate that high CI and MT levels, combined with low fitness costs and minimal immigration, facilitate successful Wolbachia establishment. To account for environmental stress, the model is extended to a stochastic framework incorporating environmental noise. Analysis reveals that stochasticity can shift invasion thresholds, destabilize equilibria, and induce noise-driven transitions between coexistence states. Numerical simulations show that low noise levels allow deterministic thresholds to remain predictive, whereas stronger fluctuations can undermine replacement success despite favourable deterministic conditions. A stochastic optimal control formulation further indicates that higher and more prolonged releases are necessary in noisy environments to ensure replacement. Collectively, these findings highlight how both strain traits and environmental stress shape the deterministic and stochastic thresholds governing the success of Wolbachia-based mosquito population replacement.

Virtual clinical trials (VCTs) hold significant promise for improving the drug development process, yet their predictive reliability depends critically on design decisions that remain poorly understood. This study examines how model complexity influences VCT outcomes, as well as how the choice of prior parameter distributions and virtual patient inclusion criteria affects those outcomes. Using oncolytic virotherapy treatment of murine tumors as a case study, we compared a relative hierarchy of three mathematical models of increasing complexity under different parameter priors (uniform and normal distributions) and two inclusion methods (accept-or-reject and accept-or-perturb). Our results demonstrate that the simplest model produces a plausible population that inadequately spans the feasible trajectory space, potentially missing critical inter-patient heterogeneity. However, we found diminishing returns beyond intermediate model complexity, as both the intermediate and complex models captured similar ranges of patient responses across dosing protocols. Notably, the accept-or-reject method generated posterior parameter distributions that are more likely to resemble the chosen prior, possibly overly reducing inter-patient variability in treatment responses, particularly at high doses. In contrast, the accept-or-perturb inclusion criteria produced more robust results that were less sensitive to prior assumptions. These findings suggest that VCT design should prioritize models with sufficient biological detail to capture key mechanisms without unnecessary complexity, paired with inclusion criteria that avoid over-constraining plausible populations to match potentially unrealistic prior assumptions.

Systemic sclerosis (SSc) is an autoimmune disease marked by excessive extracellular matrix (ECM) deposited by myofibroblasts. The disease carries the risk of pathologically progressing to internal organs, particularly the lung. Since abnormally large densities of myofibroblasts are associated with SSc, clinical studies consider trials that reduce the density of myofibroblasts: imatinib treatment (N), which promotes apoptosis in myofibroblasts, and SSc by fresolimumab (S), which inhibits TGF- β , a key growth factor of myofibroblasts. In this paper, we develop a mathematical model of SSc by a system of partial differential equations, and use it to explore a range of treatment strategies with N and S . For example, we considered administering S in fractions three weeks apart. In this case, we determined the smallest amount of S such that ρ (t) , the density of the ECM at time t , will continuously and oscillatingly decrease from a disease level 2 ρ 0 , where ρ 0 is the density ρ in health. Since SSc has no cure, ρ (t) cannot decrease below ρ 0 . We found that with the smallest amount of S , ρ (t) decreases over a few months to 1.19 ρ 0 and remains nearly stable thereafter. The results of the paper could be useful in the design of future clinical trials aimed to decrease the excessive extracellular matrix in SSc patients.

We present a new kinetic equation for cell migration driven by mechanical interactions with the substrate, an effect not previously captured in kinetic models, and essential for explaining observed collective behaviors such as those in bacterial colonies. The model introduces an acceleration term that accounts for the dynamics of motile cells undergoing mechanotaxis, where extracellular signals modulate the forces arising from cell-substrate interactions. From this formulation, we derive a family of macroscopic limit equations and analyze their principal properties. In particular, we examine linear stability and pattern formation ability through theoretical analysis, supported by numerical simulations.