Papers

The idea that emotions in biological systems are analogous to intrinsic colored noise is advanced and justified. A model describing the dynamics of operation of biological networks under the influence of colored noise is suggested. The agents of a biological network can be represented either by biological species, such as humans and animals, or by neurons of the brain, or by the nodes of a neural network. Operational actions, or decisions, in a biological noisy network are based not only on the evaluation of utility of alternatives, but also on the agents emotions. The model is probabilistic, with the choice of alternatives characterized by the related probabilities. At the initial step, the agents make decisions individually and then start exchanging information with each other and imitating the actions of other agents, thus forming a biological network of interacting agents. Numerical simulations are accomplished for a heterogeneous society consisting of three groups of agents, one group possessing long-range memory, the other group, short-range memory, and the third group of super-rational agents acting strictly on the basis of utility, being deprived of emotions. Dynamics of opinions in different groups can be smooth, oscillatory, or chaotic. Altogether, eight types of operation are found, depending on the dynamics of group decisions. Under strong imitation effect, there appears chaotic motion in the evolution of decision choice. It is shown how the Ellsberg paradox can be resolved and how the exchange of information influences the dynamics of this paradox.

Cancer progression often requires multiple driver mutations, but the same drivers may be acquired in different orders. How these pathways jointly shape tumor clonal structure remains unclear. We develop a multitype branching-process model in which malignant transformation requires two driver mutations, distinguishing malignant cells by mutation order and the independent transformation event that founded their clone. Under a successive exponential approximation, we establish point-process limits for pathway-specific clone sizes and derive a closed-form expression for the limiting expected Simpson's index of the combined malignant population. When both mutation orders yield malignant cells with the same net growth rate, the index decomposes into effective pathway weights, determined by mutation rates and birth-death dynamics at preceding stages, and within-pathway concentration terms, determined by intermediate-to-malignant growth-rate ratios. A driver's effect on heterogeneity thus depends critically on when it is acquired. A strong driver acquired early expands the intermediate lineage and increases the supply of independent malignant founders, whereas the same driver acquired last strengthens the growth and age advantage of early-founded malignant clones. Under additive fitness effects, these opposing mechanisms can produce a non-monotone relationship between selective advantage and clonal concentration. Threshold-like non-additive fitness effects can generate highly concentrated malignant populations, while order-dependent terminal fitness causes the faster-growing pathway to dominate asymptotically. These results show how mutation order, mutational accessibility, selection, and epistasis jointly determine lineage-level intratumor heterogeneity.

We characterize tree-child phylogenetic networks in terms of their mu-representations. First, we give a structural characterization of tree-child networks by means of ordered tree-path decompositions. We then translate this decomposition into a set of purely vectorial conditions on finite subsets M in N^n. We prove that such a set M is the mu-representation of a tree-child phylogenetic network if and only if it is tree-child mu-compatible. This provides a feasibility criterion for tree-child mu-representations which can be used as a basis for reconstruction and further algorithmic applications. Note that this paper presents results arising from ongoing research on tree-child networks and that the results will be further developed and placed into proper context in subsequent versions.

The transition from clinical latency to active HIV infection is a stochastic process. Traditional deterministic models based on ordinary differential equations (ODEs) fail to capture the extinction boundary and the subsequent rebound of the virus because they neglect demographic noise in the small-population limit. We map the resulting Master Equation onto a coherent-state path integral using the Doi-Peliti second-quantization formalism and verify that the stochastic Hamiltonian reproduces the correct mean-field ODEs on the classical manifold $\bar{\phi}_i = 1$. In the semiclassical (large system-size) limit we derive eight coupled Hamilton-Jacobi equations, identify the virus-free and endemic fixed points, and obtain the basic reproduction number $R_0$ for the four-species network. For the analytically tractable case of single-virion bursting ($n=1$) we show that the stochastic Hamiltonian factorizes into two bilinear terms, exposing a non-trivial zero-energy surface that constitutes the instanton trajectory. A quasi-steady-state reduction then yields a first-order linear ODE whose closed-form solution gives the instanton action $S_{\rm inst}$ in terms of biological parameters. Calibrating to clinical data—infected-cell half-life $\delta_I^{-1}\approx 2\,\rm d$, viral burst size $N\approx 10^3$, clearance rate $\delta_V\approx 23\,\rm d^{-1}$, reactivation rate $\eta\approx 10^{-3}\,\rm d^{-1}$, and latent reservoir size $L_0\approx 10^6$ cells—we obtain a mean first-passage time (MFPT) to viral rebound of $\approx 6$ months for a typical patient, with a range of days to years depending on reservoir size. These predictions are qualitatively consistent with observed post-interruption rebound timescales.

A well-known phenomenon in statistical analyses of populations of phylogenetic trees in the Billera-Holmes-Vogtmann space is that the topology of the Fréchet mean tree can contain multifurcations (i.e., internal nodes with more than two children), which raises the practical question of whether this reflects a population-level branching structure (hard polytomy) or merely sampling variability in the data (soft polytomy). This is an instance of the more general phenomenon of "stickiness" in non-Euclidean statistics, whereby the sample Fréchet mean in certain non-positively curved stratified spaces becomes permanently trapped in a lower-dimensional stratum. In this work, we identify a particular multidimensional random walk embedded within the Fréchet mean process, and we show that the time at which stickiness occurs is determined by the largest last-passage time above zero of the coordinates of this random walk. Using this representation, we develop a fully nonparametric procedure for estimating the probability that trifurcations in a sample Fréchet mean tree will bifurcate at some future time if more observations are collected. Lastly, we apply our methodology to a problem in phylogenetics where we consider whether an observed trifurcation in the species tree of primates, glires, and tree shrews is genuinely trifurcated at the population level.

Genomic epidemiology has several methods for identifying which sampled individuals are linked by close proximity in the transmission chain, whether by grouping them into clusters under a genetic distance threshold or by identifying probable direct transmission pairs. Individuals found to have no sampled neighbours—singletons—are usually set aside. We argue that they are informative. Whether any two sampled individuals prove to be linked depends on the size of the sampling frame and the number of independent lineage introductions into it, and thus the balance of linked and unlinked individuals is itself data about these quantities. We formalise this by treating the intersection of a transmission tree with a sampling frame as a labelled rooted forest, from which a fixed number of nodes are sampled uniformly at random. Using the all-minors matrix-tree theorem, we derive a closed-form expression for the number of labelled rooted $k$-forests on $N$ nodes in which a specified set of nodes is independent. From this we obtain, again in closed form, the probability that a sample of a given size contains no linked individuals at all, and the likelihood of an arbitrary observed configuration of clusters, both when the transmission structure within each is known, and when only the cluster sizes are. The approach is closer to a survey, or to mark-recapture, than to conventional phylodynamic model fitting: the information comes from the linkage structure of a single sample alone with no assumptions regarding pathogen dynamics. We outline three applications: power calculations for prospective studies, a test of the uniform sampling assumption that underlies the reading of clusters as transmission hotspots, and demographic inference itself. We finally set out the assumptions that a more flexible implementation would need to relax.

An animal's ability to perceive its surroundings, as well as the ecological interactions it experiences, can be strongly influenced by the sleep patterns of both itself and surrounding species. Animals with limited sensory capabilities may struggle to navigate their habitat at night, while in a predator-prey system, it may be detrimental for prey to be inactive while predators are awake and hunting. Here, we develop a general mathematical framework to study these interactions using partial differential equations, incorporating activity cycles and daylight-dependent perception into existing models of animal movement. We show that both the dominant sensory mode and the length of daylight are key factors determining the strength of aggregation in social species. Extending this framework to a two-species predator-prey system, we use game-theoretic approaches to show that the equilibrium sleep strategy of each species is shaped not only by its own sensory capabilities, but also by those of its ecological opponent. Depending on the sensory capabilities of the two species, equilibrium strategies may consist of a single sleep pattern or a combination of multiple sleep patterns.

We study signal prediction and smoothing in hidden Markov models whose latent signal is a multi-type Wright-Fisher diffusion and whose discrete-time observations are categorical. We consider a repeated-sampling regime with finitely many inspection times and large within-time samples. Under parent-independent mutation, the Bayesian predictive and smoothing laws admit exact representations as mixtures of Dirichlet distributions. We show that the one-step predictive and local smoothing laws converge in total variation to the corresponding Wright-Fisher transition and bridge laws determined by the neighboring population frequencies, thereby justifying plug-in approximations based on empirical proportions. For prediction, we derive explicit finite-sample bounds for this approximation and for bounded predictive functionals such as diversity, together with a finite-sample concentration bound for the marginal smoother. The joint posterior at the inspection times concentrates at the target frequencies and satisfies a Bernstein-von Mises limit with distinct interior and boundary regimes, with Gaussian and Gamma limits on different scales. The results allow arbitrary dependence across inspection times. A key technical ingredient is a fixed-interval tail bound for the coalescent-based dual death process.

We present a didactic treatment of compartmental disease spread modeling centered on the SIR framework, aimed at mathematics education. Our focus is on how modeling choices encode transmission mechanisms and data-collection practices. We begin by stating the structural hypotheses of compartmental models, including population partitioning, homogeneous mixing, flow closure, and Markovian transitions. We then derive the basic SIR dynamic equations in both continuous-time (differential equations) and discrete-time (difference equations) settings. A key goal is to clarify the distinction between density-dependent and frequency-dependent transmission. We do this by constructing the force of infection from the contact rate, prevalence, and per-contact transmissibility while tracking units and interpretations. We go beyond the usual mass-balance SIR model by introducing the discrete Kermack-McKendrick perspective through the cumulative force of infection. This naturally leads to discrete SIR updates with an exponential escape probability term. We also derive Bernoulli-based exponential formulations for both density- and frequency-dependent transmission, interpret these probabilistically over observation intervals, and obtain the associated discrete infectivity kernel. This highlights how geometric infectious survival emerges from per-step removal. The paper includes reproducible, introductory-level Python code to implement the basic discrete SIR model, the continuous SIR model, and the exponential discrete variants. We provide instructors with an example of parameter estimation from prevalence data using least squares, along with guidance on interpretation and limitations. This framework is designed to help teachers integrate model assumptions, units, simulation, and inference in a coherent introduction to infectious disease modeling.

Phylogenetic networks provide a flexible framework for representing reticulate evolutionary processes, such as hybridization, introgression, recombination, and horizontal gene transfer. However, their combinatorial complexity makes even basic enumeration problems difficult. Building on our previous work for networks with up to three reticulations, we derive an explicit closed-form formula for the number of unrestricted rooted binary phylogenetic networks with four reticulations on \(n\) labeled taxa. Our approach is based on tree-component graphs. We classify the 79 possible component graphs corresponding to networks with four reticulations into ten groups. We then enumerate the networks associated with each group by combining known counts of one-component networks, forests, and networks with fewer reticulations. Summing these contributions yields the desired formula. This result extends the exact enumeration of unrestricted binary phylogenetic networks to four reticulations and further demonstrates the effectiveness of component graphs for systematically organizing and counting increasingly complex network classes.

Best match graphs (BMGs) were introduced in mathematical phylogenetics to describe the concept of closest relatives for related genes (leaves of rooted tree) in different organisms (defining leaf colors). We generalize this concept here to leaf-colored rooted networks, where least common ancestors are in general neither unique nor comparable. We characterize BMGs of rooted networks as those vertex-colored digraphs that are properly colored and satisfy an easy-to-check condition that we call the sicor-in-hub property. BMGs can be recognized in linear time and an explaining network can be constructed in quadratic time. Analogous results are obtained for reciprocal best match graphs (RBMGs), where an edge $\{x,y\}$ corresponds to pairs of vertices with different color that are mutually closest relatives.

We study an in-silico substrate in which every pixel of a two-channel cellular-automata grid carries a tiny neural network (an agent) that senses its Moore neighborhood. A cell persists only by self-replication: a living neighbor is cloned and its weights are mutated by a uniform perturbation, so that phenotype (cell state) is driven entirely by genotype (network weights). From a handful of seeded founders the system grows into a spatially organized ecosystem of coexisting, competing and dominating species. Our main contribution is a battery of coarse-grained diversity metrics that make such growth measurable at two scales: four phenotypic tools based on cellular-type frequency, entropy and cell variance, and two genotypic tools that colour each agent by a hash of its full weight vector versus a sparse random-weight probe. Across a five-fold sweep of 1680 small runs and 24 long (1000-generation, 200 x 200) runs, the substrate is persistent and self-maintaining in 20 of the 24 long configurations and exposes a clear phenotype-genotype diversity trade-off: raising phenotypic diversity collapses genotypic diversity and vice versa. Full-genome hash colouring further reveals lineage structure that a random-weight probe systematically misses. Code, data and animations are released as supplementary material.

Crises and population responses can form coupled dynamical systems, with crisis conditions shaping protective behaviours and collective responses altering the crisis trajectory. Existing models rarely capture both evolving crisis conditions and the reinforcement-dependent spread of competing behaviours. We propose a complex contagion with feedback (CCF) model that couples competing complex contagion with crisis dynamics through bidirectional feedback. Agents stochastically switch between competing states according to social reinforcement and adoption complexity, which is influenced by crisis conditions and information campaigns, while population-level behavioural change feeds back into the crisis. We formulate and analyse the model for a well-mixed population, characterising its equilibria and associated stability conditions. As a case study for model validation, we integrate CCF model with a census-calibrated agent-based model of COVID-19 transmission in Australia to represent social distancing adoption and discontinuation. We compare the CCF model with an existing opinion dynamics model and find that, despite using fewer parameters, it achieves comparable performance in reproducing recurrent infection waves. The resulting framework is parsimonious, analytically tractable, and can be adapted to different types of crises.

Organisms' foraging strategies are shaped by a trade-off between search area and local capture efficiency. This trade-off leads individuals to adapt their foraging area to an optimal value, impacting population dynamics. Here we study the effect of multiple foraging areas in a predator-prey model composed of three trophic levels. The interactions between predators and prey occur only within a limited neighborhood of the predators, where adaptation can occur over generations. We assume a trade-off where local predation efficiency is inversely proportional to the foraging area. These dynamics were implemented computationally via cellular automata and analytically via Master Equations with mean-field and pair approximations. Unlike the mean-field approximation, the pair approximation reproduced the dependence of population density on foraging area observed in the simulations. However, the simulations showed that the optimal foraging area does not maximize population density. Moreover, we found that a polymorphic population emerged, where not a single optimal strategy but a range of optimal strategies can coexist. Using the framework of Adaptive Dynamics, we confirm that the range of optimal areas is not the one that maximizes population size, but the Evolutionary Stable Strategy that can invade a population and not be invaded.

Some aspects of AI development resemble a population process in which models are specialised, retrained on the output of peers, or combined by averaging weights. These practices lead to generations of models, in the biological sense studied by population genetics. Here, I develop this parallelism and interpret multigenerational model populations in terms of sexual and asexual reproduction, formally recombining the two fields. I test these analogies in an exact inheritance model, in trained networks (recurrent, feedforward and variational autoencoder generators) and in large language models, and show that they hold generally, with some measurable architecture-specific biases. Training recursively on model output is known to lead to model collapse, a process previously described as akin to genetic drift; I develop all that follows. A minimal model of a learner retrained on its parent's output reproduces the Wright-Fisher process exactly; verified real data added to each generation play the role of immigration, with the surprising finding that the absolute number of real data samples matters, not their share, exactly as in population genetics. Training a child on the average of its parents' outputs cancels the benefit of having several parents, matching blending inheritance (and reviving Jenkin's objection to Darwin), whereas combining parents so that each keeps its strongest contribution preserves it; merged language-model specialists exceeded every parent across seeds (the Fisher-Muller effect); and lineages become reproductively isolated, losing the ability to merge at all, when they have learned conflicting conventions and not when they have merely drifted apart. As AI societies become societies in time as well as in space, a mathematical framework for their inheritance acquires predictive power. Remarkably, that framework can be adapted almost wholesale from biology.

Population-level heterogeneities, combined with temporal fluctuations in sexual partnerships, shape the structure of sexual contact networks and can substantially influence the spread of sexually transmitted infections (STIs). Traditional static network models, which assume fixed attributes of partnerships, such as count and duration, may not adequately capture the effects of partnerships on STI transmission. In contrast, agent-based dynamic network models offer a flexible framework for incorporating individual and population-level heterogeneities. We developed an agent-based dynamic network model in which partnership formation and dissolution probabilities, stratified by age, sex, and sexual orientation (including bisexual individuals), govern the formation of monogamous and concurrent partnerships and their dissolution via a duration-dependent hazard. Partnership statistics from the National Survey of Sexual Attitudes and Lifestyles (NATSAL-3) were used as model calibration targets, and Latin Hypercube Sampling (LHS) was used to generate candidate parameter combinations. Parameter estimation was performed by selecting the combination that produced the lowest Mean Squared Error (MSE) between the model outputs and the calibration targets. Our study addresses three questions: (1) how well can the observed characteristics of sexual partnerships in NATSAL-3 be reproduced using an agent-based model; (2) how does concurrency shape the structure of dynamic sexual contact networks; and (3) how do concurrent partnerships affect the dynamics of STI transmission. In this study, we find that interactions between individual characteristics such as age, sex, and sexual orientation, and partnership attributes such as count, duration, and concurrency play a critical role in shaping the population-level sexual contact network and, in turn, the dynamics of STI transmission.

Molecular sequence data are used to reconstruct evolutionary relationships among taxa, but reconstruction accuracy depends not only on the tree-building method but also on how pairwise sequence relationships are represented. We evaluated sequence-to-affinity representations in a recursive normalized-cut (Ncut) framework whose graph-partitioning subproblems were formulated as quadratic unconstrained binary optimization (QUBO) models and solved using Simulated Bifurcation. Using simulated amino-acid and nucleotide datasets spanning multiple tree-generation settings and evolutionary divergence, we compared normalized bit-score affinities with representations derived from transformed sequence similarities and evolutionary distances, examined post-swap refinement, and used neighbor joining (NJ) as a distance-based comparator. Affinity representation substantially affected internal split recovery, particularly for nucleotide data. JC69-based local affinities maintained comparatively high accuracy as divergence increased, whereas normalized bit-score and BLAST-derived kernel representations declined more markedly. Post-swap refinement generally improved recovery, but not consistently across individual reconstructions. NJ achieved higher mean split recovery than corresponding recursive Ncut reconstructions for WAG and JC69 distances across all evaluated conditions, whereas recursive Ncut outperformed NJ for BLAST-derived logarithmic distances under some conditions. These results show that graph construction is an important determinant of recursive Ncut-based phylogenetic reconstruction. A representation that performs well within Ncut does not necessarily provide the most accurate use of the underlying pairwise distances. Pairwise representation, affinity transformation, optimization, and recursive tree construction should therefore be evaluated jointly.

The pace of evolution depends on how rapidly new phenotypes arise. We show that neutral Wright-Fisher evolution exhibits anomalous first passage despite diffusive mutations. The mean time for the first individual to reach a prescribed phenotypic distance scales approximately as $(\sigma^2)^{-3/2}$ with mutation variance $\sigma^2$. Two crossovers bound this regime, with inverse-variance scaling on either side. Combining coalescent theory with Kardar-Parisi-Zhang (KPZ) fluctuations at the dilute population edge, we develop an edge-KPZ theory of all three regimes. The anomaly persists under weak selection.

The diversity of brain size and organization across mammals reflects how evolutionary and ecological pressures have shaped neural architecture through time. Yet, how these selective pressures translated into measurable variation in the proportions and overall shape of the brain across species remains poorly understood. In this study, we used high-resolution virtual endocasts from 41 musteloid species (e.g. otters, skunks, raccoons) to quantify variation in endocranial composition and shape. We analyzed both the complete endocast and its principal subdivisions (olfactory bulbs, cerebrum, and hindbrain) using a combination of compositional data analysis and 3D geometric morphometrics. Phylogenetically informed regressions (PGLS) and model comparisons were used to assess the effects of endocranial size, locomotion, diet, and taxonomy on both the regional proportions and the overall shape of the endocranial cast. Musteloid brain organization is mostly driven by size-dependent scaling: larger brains have enlarged cerebra, reduced olfactory and hindbrain regions, and rounder endocasts. However, some lineages markedly deviate from this largely conserved allometry: otters (Lutrinae) exhibit proportionally enlarged cerebra and strongly reduced olfactory bulbs, whereas skunks (Mephitidae) show the reverse pattern, emphasizing enhanced olfactory investment. Feeding ecology also contributes to variation in brain organization, as piscivores show disproportionately smaller olfactory bulbs compared with other feeding guilds. In contrast, locomotor mode has little explanatory power for either endocranial composition or shape. Brain evolution in musteloids is therefore not uniform: a common scaling pattern dominates, but lineage-and diet-specific adjustments modify how brain regions are proportioned.

The parasitic mite Varroa destructor poses a severe existential threat to global honey bee Apis mellifera populations. In this paper, we present a dynamical systems model of hive-mite interactions incorporating a eusocial Allee effect to evaluate the efficacy of chemical interventions. We partition treatments into “soft” miticides (targeting phoretic mites) and “harsh” miticides (penetrating the capped brood to target reproductive mites). Our bifurcation analysis reveals a fundamental trade-off: while harsh treatments effectively eradicate the protected mite reservoir by shifting the transcritical bifurcation boundary, they impose sublethal toxicity on the bees. We analytically show that exceeding a critical dosage threshold culminates in a catastrophic saddle-node bifurcation that guarantees colony collapse. To navigate this toxicity limit, we formulate an optimal control problem using Pontryagin's Maximum Principle. Numerical solutions reveal that a dynamic “shock and maintain” cocktail strategy optimally balances reservoir clearance with hive viability. Expanding the model to include treatment-resistant strains demonstrates that single-chemical reliance forces the hive onto a highly toxic, nearly unsustainable chemical treadmill. Finally, our mathematical framework yields a striking ecological prediction: because the eradication boundary scales intimately with the colony's carrying capacity, unchecked mite pressure will likely exert evolutionary forces that select for smaller, naturally resistant hives over the massive colonies favored by commercial agriculture.

Large brains are metabolically costly, and associations with changing environments do not imply they evolved there, as the Cognitive Buffer Hypothesis (CBH) would suggest. They may instead evolve in stable conditions and later facilitate colonization of changing environments. Using neuro-evolution in an artificial seasonal foraging task, we compared agents evolving exclusively in changing environments to agents first evolved in static environments before transitioning. Results show that larger neural networks in dynamic environments arise mainly from prior static evolution, achieving superior performance under unpredictable changes. Our results challenge strict CBH predictions, provide agent-based (computational) support for a colonization-based account and highlight the role of evolutionary history in brain size evolution.

We study the evolution of the speed of life history in populations competing for a single growth-limiting resource subject to demographic stochasticity and mass mortality events. We focus on a quasi-neutral regime in which competing types have equal resource-use efficiency but differ in life-history speed. In the large-carrying-capacity scaling limit, we reduce the model to a one-dimensional jump-diffusion supported on the manifold of ecological equilibria. The drift and diffusion components capture the joint effects of demographic stochasticity and density regulation, while the jump component is driven by the catastrophic mortality events. Using this limiting process, we derive a first-order approximation for the fixation probability of an invading type that differs slightly in life history speed from the resident population. Calculations show that small demographic events tend to favor slower life histories, whereas catastrophic mortality events create transient periods of resource abundance that benefit faster types. The interaction of these two evolutionary forces allows for the existence of a nontrivial evolutionarily attractive life-history speed, which is an increasing function of the frequency and intensity of the catastrophic mortality events. These results provide a rigorous mathematical framework for some classical r/K-selection arguments, and furthermore demonstrate how mass mortality events can maintain selection for faster life histories even in resource-constrained populations.

We determine statistics of allele-frequency trajectories across replicate populations. Defining $R = 2N_e|s|$ where $N_e$ is the effective population size and $s$ is the additive selection coefficient of a focal allele, we focus on strong selection which corresponds to $R \gg 1$. Strong selection does not require large selection coefficients: in populations of realistic size, it can readily arise from small selection coefficients. Our analysis concerns a biallelic locus over timescales on which mutation can be neglected. Every allele-frequency trajectory belongs to one of two absorbing classes: fixation or loss. When $R$ is large, individual allele-frequency trajectories can appear nearly deterministic. This does not imply that an approximation of allele-frequency statistics can be based on a single deterministic trajectory. For a beneficial allele, the deterministic trajectory approaches fixation, yet a non-negligible fraction of stochastic trajectories may be lost. Ignoring trajectories that achieve loss can produce severely erroneous statistics. We address this by conditioning separately on the eventual loss and eventual fixation of the focal allele. This yields an outcome-conditioned representation of a statistic. In the strong-selection regime, we apply a small-noise expansion that reasonably captures transient behaviours of statistics while recovering their exact long-time limits. We illustrate results with numerical examples for the mean allele frequency and its variance. The biological interpretation of this work is that, even under strong selection, both trajectory types are of importance. We provide a simple framework for calculating time-dependent allele-frequency statistics relevant to experimental evolution, longitudinal population-genomic data, and the dynamics of alleles driving selective sweeps.

The basic reproduction number $R_0$ is the central dimensionless quantity in mathematical epidemiology, characterizing the threshold for disease outbreak and the early growth rate of an epidemic. The algebraic form of $R_0$ varies widely across models of distinct transmission mechanisms, and its interpretation can yield further biological insight into transmission dynamics and disease intervention. We present a structural taxonomy of $R_0$ for compartmental ordinary differential equation models spanning a range of disease transmission features, including staged progression, differential infectivity, competing strains, vector-borne transmission, sexually transmitted infection, recurrent infection, maternal transmission, and gene drive inheritance. For each structural class, we provide a worked example, deriving $R_0$ via the next-generation matrix approach and interpreting the resulting algebraic expression in terms of the underlying transmission pathway. By unifying disparate derivations under a single structural framework, we clarify when and why particular algebraic forms of $R_0$ arise. This review serves both as a research synthesis and as a self-contained pedagogical reference for students and researchers entering the field.

In this paper, we study a stochastic Susceptible-Infected-Removed (SIR) model where the infection and the recovery rates depend on individual covariates for susceptibility and infectiousness of the infector and the infectee. Such models allow explicit nonlinearity in the incidence term. They are also important from a practical perspective, as they allow for the incorporation of individual heterogeneity into the epidemic process. Statistical estimates for crucial epidemiological parameters, such as the basic reproduction number, herd immunity threshold, could be vastly different, and even biased, when the population heterogeneity is ignored in the mathematical model. We describe our epidemic model as an Interacting Particle System (IPS) of Stochastic Differential Equations (SDEs) driven by Poisson Random Measures. Our main mathematical contributions are a Functional Law of Large Numbers (FLLN), which approximates the empirical random measure of the IPS by means of a deterministic measure-valued function, and the propagation of chaos phenomenon, which establishes asymptotic independence of the particles as the population size goes to infinity with an explicit construction of McKean–Vlasov type Kac's “nonlinear process”. We also briefly mention how the propagation of chaos phenomenon leads to a product-form likelihood function, which forms the basis of the so-called Dynamic Survival Analysis (DSA) method for parameter inference based on sparse data.

We study the evolution of sexually reproducing populations subjected to sudden environmental changes. Using an individual based model we consider scenarios where abrupt changes can affect the carrying capacity, the mating range or the strength of assortativity, that controls the minimum similarity between individuals that allows mating to occur. We show that a boost in carrying capacity can either lead to new speciation events, thereby increasing biodiversity, or to widespread extinctions, causing the system to become dominated by a single species. A similar effect is observed with respect to the mating range. We show that the key factor determining the outcome of the process is genome length, i.e., the number of genes controlling reproduction and assortativity. We also show that when mating becomes more restrictive, increasing the degree of assortativity, a burst of speciation occurs. The explosion lasts only for a few generations, converging to values larger than before the change, but not as high as at the peak.

We introduce GrassTop, a genome representation that integrates Grassmann manifolds and algebraic topology for viral classification and phylogenetic analysis. The framework begins by constructing multiscale topological and spectral descriptors of (k)-mer positional patterns. It then extracts a low-rank subspace that summarizes variation across the filtration and compares genomes using a Grassmannian distance. Although the reported implementation uses the chordal distance, the framework is not restricted to this particular choice. We evaluate GrassTop on four families of viral classification datasets, four phylogenetic clustering datasets, and a sequence perturbation experiment. Under the reported 5-nearest-neighbor protocol, GrassTop achieves higher scores than five published alignment-free reference methods across all reported classification metrics and datasets. Its UPGMA (unweighted pair-group method using arithmetic averages) trees achieve an average label purity of 1.0 on every phylogenetic dataset. The perturbation experiment provides a more nuanced result: the subspace representation differs most clearly from direct comparison of the unprojected feature matrices for SARS-CoV-2, whereas the differences are smaller or non-monotonic for the other datasets. Overall, these results support GrassTop as an effective topological-geometric representation for viral classification and phylogenetic analysis.

Clustering systems provide a natural way to encode structural information contained in phylogenetic networks. In this note, we study the clustering systems of normal and tree-child networks through an overlap-based property of set systems, called not-overlap-covered (NOC). We show that the NOC property is equivalent to inclusion-visibility, a memberwise formulation of the strict-compatibility condition previously used for tree-child clustering systems. We characterize normal networks as precisely the semi-regular networks whose clustering systems satisfy NOC. Consequently, a clustering system is realized by a normal network if and only if it satisfies NOC, or equivalently, if every one of its clusters is inclusion-visible. In this case, the Hasse diagram provides a canonical normal realization. These are exactly the clustering systems realized by tree-child networks. The NOC formulation yields a sharp quadratic upper bound on the number of distinct clusters of tree-child and normal networks and a direct polynomial-time recognition algorithm. Finally, we explore several consequences of the NOC perspective beyond the phylogenetic setting. These include connections to the enumeration of normal networks, an order-theoretic interpretation of inclusion-visibility, structural properties of NOC set systems, and a tractable special case of Minimum Set Cover, which is NP-hard in general.

Statistical models of ecological and evolutionary dynamics often include latent variables that are either continuous (e.g., average body size) or discrete (e.g., numerical abundance). Mixed-type hierarchical models containing both are typically fitted using Markov chain Monte Carlo (MCMC), which can be prohibitively slow for large models. Here, we introduce an alternative in the R package RTMB that automates the sequential reduction of small groups of related discrete variables, allowing them to be efficiently marginalized. Sequential reduction is combined with automatic differentiation and the Laplace approximation to estimate parameters and predict both continuous and discrete variables. We demonstrate speed and flexibility using demographic examples (occupancy, dynamic occupancy, N-mixture, and open dynamic N-mixture models), benchmarking RTMB against JAGS and unmarked. We then develop two novel applications. The first is a multi-site open N-mixture model with a spatial latent variable governing site-specific initial abundance and recruitment, which shows that continuous Gaussian Markov random fields can be estimated jointly with discrete abundance dynamics in under a minute. The second is phylogenetic trait imputation for a published data set of female Liolaemus lizards, where we jointly impute a binary trait (viviparity), estimate its state-switching rates, and estimate its effect on a continuous trait (body size) during ancestral state reconstruction. This indicates that phylogenetic comparative methods can estimate linkages among discrete and continuous traits. We envision that intuitive and efficient specification of mixed-type models will allow more expressive representation of ecological and evolutionary dynamics.

In evolutionary biology, factors like lateral gene transfer complicate the tree of life, making inference of a species tree more difficult. The phenomenon of lateral gene transfer allows genetic material to be passed between organisms as opposed to inheritance, causing the tree for a particular gene to differ from the species tree. In this work, we define a model of lateral gene transfer on site patterns. In the case where lateral gene transfer is restricted to occur only between closely related species, we show that the unrooted topology of the species tree is identifiable from SNP data on the taxa. Our proof involves showing a connection between our lateral gene transfer model and graphical models on a related tree, and uses ranks of flattenings to identify splits in the tree. We also report on results of using the singular value decomposition on flattening matrices to identify the unrooted topology in simulated data.