Theoretical Population Biology

A sudden reduction in population size increases the rate of genetic drift, reducing variability and increasing the mean level of homozygosity. The resulting increased exposure of recessive or partially recessive, strongly deleterious alleles to selection against homozygotes may lead to their being purged from the population, potentially allowing mean fitness to increase after an initial decline, and accelerating the decline in inbreeding depression associated with reduced variability. However, detailed population genetic theory on the effects of population bottlenecks on mean fitness and inbreeding depression remains limited. We develop a theoretical framework for small, randomly mating populations founded from a large population near mutation-selection-drift equilibrium, using both simulations and approximate analytical predictions. These provide quantitative predictions for the dynamics of the populations mean fitness and level of inbreeding depression following a bottleneck. In particular, we derive an approximate expression for the time needed for mean fitness to recover after an initial decline; such a recovery requires selection to be sufficiently strong relative to drift and mutations to be sufficiently recessive. In contrast, weakly deleterious mutations cause reductions in mean fitness and inbreeding depression that are similar in size to those predicted from increases in neutral homozygosity.

Darwinian fitness is equated here with invasion fitness and defined as the quantity determining the fate--certain extinction or possible spread--of a single mutant type. We derive it, together with its phenotypic derivative, for evolution in group-structured populations under limited genetic mixing, where the demography of the focal species and its environment is modeled as a discrete-time stochastic process. Reproduction, physiological development, dispersal, and survival are influenced by interactions within and between groups and by environmental fluctuations within and across generations. Using multitype branching processes in random environments, we show that invasion fitness is predicted by a stochastic growth rate that can be represented biologically in two meaningful genealogical ways. First, as the long-term geometric mean of the expected per-capita number of mutant copies produced per time step by a representative member of the mutant lineage. Second, as the the long-term geometric mean of the expected reproductive-value-weighted per-capita number of mutant copies produced by such an individual. This latter representation is useful for computing the phenotypic directional derivative of invasion fitness. Moreover, this derivative can be written as an actor-centered inclusive-fitness effect derived from properties of the resident population process. This effect depends on class-specific fitness differentials, relatedness, reproductive values, and class frequencies. However, unless generation- and class-specific fitness defines a stochastic matrix, the derivative does not separate stochastic reproductive values from relatedness and class frequencies, and must be evaluated by simulations. In summary, we formalize invasion fitness biologically quite generally and show how Hamiltons marginal rule is deduced from it.

Interest in quantifying linkage disequilibrium (LD, non-random associations of alleles at different loci) has skyrocketed in recent years as researchers have focused on use of LD in genome-wide association studies (GWAS), for studying historical demography, and for estimating effective population size (Ne). The most widely used LD metric is r2 = the squared correlation of alleles at a pair of loci. Despite a half century of efforts, developing an unbiased expectation of r2 as a function of the many factors that can affect it (physical linkage, genetic drift, selection, migration, mutation, mating systems) remains elusive. Furthermore, even when all of these other factors are absent, empirical estimates of r2 are upwardly biased by sampling a finite number (S) of individuals, and that must be accounted for if one wants to focus on the desired signal of LD. Previous approaches to estimate [Formula] have been shown to be biased to greater or lesser degrees. The purpose of this short paper is to demonstrate that a simple and apparently exact expression for [Formula] does exist for the special case where sampling error is the only factor contributing to r2, in which case [Formula] = 1/(S - 1). When other factors contribute heavily to LD, [Formula] shrinks toward 0 as empirical r2 [->] 1. However, for estimating contemporary Ne with unlinked markers, empirical r2 will generally be small and 1/(S - 1) will provide a robust estimate of [Formula].

Minimum viable populations (MVPs) are population levels large enough to surmount risk from demographic, environmental, and genetic stochasticity. MVPs are estimated by biologists to guide conservation practices. However, MVPs are generally estimated for a target population without regard for how they interact with intra- and inter-species population dynamics in the broader ecological community. Thus, how and why population dynamics interact with MVPs imposed by conservation biologists remain unclear. When MVPs are imposed on a continuous population model, traditional analyses fail to capture the range of possible outcomes those MVPs create. Here, we describe viability space decomposition (VSD) as a mathematical tool to systematically analyze the potential crossing of MVPs during population dynamics. We demonstrate that different extinction and survival outcomes can be recovered from a model with imposed MVPs using three VSD concepts in junction with a traditional phase portrait: mortality manifolds which separate conditions that lead to different existential outcomes, ordering manifolds which determine the order of extinction events for multiple populations, and collapse manifolds which determine the survival or extinction of one species given the loss of another. We employ these methods with a standard consumer-resource model, and the methods can be scaled to systems with more species. VSD is a useful tool for conservation biologists and community ecologists concerned with boundary crossing problems in any dynamical system.

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.

Mark-recapture analyses on the delineation of natural populations between areas often assume random sampling, with a between/within (B/W) area resighting ratio that declines towards zero as the population components of two areas become more-and-more isolated from one another, with fewer-and-fewer individuals mixing between areas. I use an individual based population model split in two areas to simulate this result, analysing also for the potential effects of the space-time fidelity of the mark-recapture sampling in the areas. I find that small B/W resighting ratios--that traditionally is taken as evidence of population isolation--can easily be observed within a completely mixing population if a random sampling scheme is restricted in space and/or time. Random sampling within restricted areas and time windows is not sufficient to estimate mixing rates and population isolation between areas, unless the resighting rates are analysed by a method that accounts both for the space-time fidelity of the scientific sampling scheme and the space-time fidelity of the distributional behaviour of the individuals in the population.

The basic reproduction number R0 is central to malaria epidemiology, yet it is typically treated as a static quantity derived under memoryless assumptions for mosquito demography. In natural systems, however, mosquito populations are shaped by delayed processes such as larval development and density-dependent feedback, introducing biological memory into vector dynamics. We develop a minimal delay-based framework that incorporates this memory into the Ross-Macdonald model by describing adult mosquito abundance with a retarded differential equation. This formulation induces a time-dependent transmission potential R0(t). Using complex analysis and the argument principle, we derive an explicit stability threshold [Formula], which separates stable from oscillatory transmission regimes. Near this threshold, delayed feedback produces slow relaxation times and sustained transient oscillations, implying that transmission potential may vary intrinsically even in the absence of external forcing. To account for ecological variability, we extend this deterministic condition into a probabilistic framework and define the stability probability as [Formula]. Numerical simulations and global sensitivity analysis show that recruitment and developmental delays are the primary drivers of instability, while adult mortality has a weaker stabilizing effect. These results indicate that malaria interventions may influence not only the magnitude of malaria transmission but also its dynamical stability. By linking delay dynamics, transmission theory, and uncertainty quantification, this framework provides a basis for stability-aware modeling and interpretation of malaria transmission under ecological variability. Author summaryMalaria transmission is often summarized by a single number, R0, treated as a fixed indicator of whether transmission will increase or decline. This assumes mosquito populations respond instantly to environmental conditions. In reality, mosquitoes develop through stages where larval conditions, such as crowding, nutrition, or temperature, affect adult populations only after a delay. This creates biological memory: todays mosquitoes reflect past environments. We show that this memory can fundamentally reshape transmission dynamics. When developmental delays are included, transmission potential is no longer constant but can fluctuate over time, even in stable environments. These fluctuations can persist or amplify depending on the balance between mosquito growth, mortality, and delay. As a result, variability in mosquito abundance or malaria transmission may arise from intrinsic dynamics rather than external drivers alone. Under ecological variability, stability becomes probabilistic, allowing estimation of how likely transmission is to remain stable. Interventions that reduce larval productivity or increase adult mortality may therefore both lower transmission and make it more predictable, improving interpretation and control strategies.

Gene drives, genetic constructs that can spread deleterious alleles in wild populations, have the potential to address some of the major pressing challenges of the Anthropocene such as invasive species, spread of disease vectors, and agricultural pests. However, responsible and effective deployment of gene drive requires taking into account the complex nature of real-world population connectivity networks. In particular, it is unclear how the topological position of the deployment site affects the spread process and its final outcome. Here we develop a framework for modeling gene drive spread in population connectivity networks, and study the eco-evolutionary dynamics of gene drive spread under complex population structures. We investigated the relationship between the position of the deployment site in the topology of the network and whether the gene drive is eventually lost, fixed, or maintained at an intermediate frequency. We identified network centrality measures of deployment sites that are highly correlated with the outcome of deployment for different gene drive designs and across diverse network topologies. We also show that there is a trade-off between the time-to-fixation and the final outcome, implying that multiple centrality measures of the deployment site would need to be considered when aiming to achieve rapid and successful population control using gene drives.

The ecological and evolutionary dynamics of populations, including viral populations, are known to be jointly shaped by deterministic and stochastic processes. While the impact of stochastic processes has been rigorously explored for viral dynamics at the level of the host population, most dynamic models for acutely-infecting respiratory viral pathogens at the within-host scale remain deterministic in their formulation. While this may be reasonable for identifying key processes shaping their within-host viral population dynamics, recent studies indicate that stochastic processes need to be invoked for understanding patterns of within-host viral evolution. Specifically, several studies have shown that viral allele frequencies can change dramatically over the time course of days in acute infections. Here, we use stochastic dynamic models to explore the role of environmental noise in shaping observed patterns of virus evolution in acute respiratory virus infections. We summarize ways in which environmental stochasticity can be biologically realized in these acute viral infections and describe within-host models that can be implemented to jointly yield viral population dynamics and evolutionary dynamics. We further develop a statistical approach to estimate the extent of environmental noise from observed within-host allele frequency changes. We test this approach on simulated data and apply it to existing influenza A virus and SARS-CoV-2 within-host data. With these applications, we show that environmental stochasticity can parsimoniously reproduce key features of empirically observed allele frequency changes without needing to invoke demographic stochasticity or to adopt Wright-Fisher model formulations with a constant effective population size. Finally, we show that purifying selection and positive selection can both still contribute to within-host viral evolution in the context of a noisy environment, providing theoretical support for studies that have found purifying and positive selection in acutely-infecting respiratory virus populations.

Whole genome duplication is a common mutation in eukaryotes with far-reaching phenotypic effects. The resulting morphological, physiological, and fitness consequences and how they affect the survival probability of newly polyploid lineages are intensively studied, but very little is known about the effect of genome doubling on the short-term evolvability of populations. Understanding the effect of polyploidization on the adaptive potential of populations is of crucial importance to predict the future of polyploid populations. In this paper, I investigate the immediate consequences of genome doubling on the genetic variance of populations. To do so, I performed numerical iterations and simulations of how the genetic variance of a quantitative trait changes after polyploidization, under different genetic architectures (additivity, dominance, and epistasis). I found that genetic variance generally decreases after genome doubling. Non-additive gene actions can make autotetraploid populations genetically more diverse than their diploid progenitors in rare cases, notably with overdominance and directional epistasis. By collecting estimates from the agronomic literature, I found that both dominance and epistatic variance contribute to the genetic variance of polyploid populations. These results bring new insights into the adaptive potential of newly formed tetraploid populations, and call for further experimental investigations of how polyploidization is associated with a short-term decrease in evolvability.

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.

The fungal crop pathogen Blumeria hordei, causal agent of powdery mildew on barley, presents life-history and epidemiological characteristics, as well as and selective pressures due to modern agriculture leading to expected sweepstakes reproduction, that is highly skewed offspring distributions. Using genome-wide polymorphism data and population genomics inferences, we aim to 1) infer the past demographic history and the strength of sweepstakes reproduction in B. hordei, and 2) quantify the contributions of these selective and neutral processes in the genome. An new inference method based on Neural Posterior Estimation and diversity and linkage disequilibrium statistics was developed and tested on simulated and B. hordei genomic data. We confirm that B. hordei exhibits a moderate sweepstakes reproduction (-parameter of 1.6). We highlight that the Site Frequency Spectrum (SFS) appears sensitive to the joint occurrence of sweepstakes and recent demographic changes, which may caution on the reliability of the SFS to infer sweepstakes reproduction. We then scan the genome for selective sweeps, adjusting the significance thresholds of the methods for demographic history and sweepstakes reproduction, thereby yielding a counterintuitive result. When conditioning the significance threshold for sweep detection on simulations under sweepstakes and demography, a very large number of putatively selected regions is found (11.6% of the genome). We suggest that sweepstakes reproduction in B. hordei is due to 1) neutrality (clonal/sexual phases and Boom-and-Bust cycles) generating a genome-wide level of background noise in the coalescent genealogies, and 2) selective sweepstakes due to pervasive positive selection. Our findings have important implications for both population genomic methodology and our understanding of pathogen evolution.

A hosts diet can alter the course of parasite infection. This is especially true of trophic parasites, which a host acquires through feeding. While a large body of work attests to the role of diet in the spread of disease within-hosts, diet can also impact host density and encounter rate with parasites, both of which are expected to modify disease dynamics. When parasites are acquired through feeding, epidemics may be larger and more severe on high-quality diets if these diets support a higher density of hosts that feed more and thus ingest more parasites. Alternately, epidemics may be more severe on low-quality diets if malnourishment decreases hosts ability to resist disease. To differentiate these hypothesized effects of diet on disease, we quantified individual infections and epidemic dynamics for the natural microsporidian parasite Nematocida ironsii infecting its nematode host Caenorhabditis elegans. We measured feeding rate, parasite transmission, and host fitness across three bacterial diets that vary in quality and elicit distinct feeding behaviors in C. elegans. We found that low-quality diets reduced feeding rate, which corresponded to reduced acquisition of parasite spores. However, these diet-mediated differences in parasite acquisition did not directly map onto fitness consequences: hosts eating the poor-quality diet had similar reductions in fitness to those on higher quality diets. During epidemics, a combination of increased parasite acquisition and higher population growth rates resulted in higher parasite abundance for hosts on high-quality diets. Our work underscores the importance of considering both individual- and population-level impacts acting in concert to determine how diet affects the spread of infectious disease.

Elucidating how habitat degradation facilitates extinction is critical for effective conservation efforts. Here, we propose integrating physiologically-structured population models into stochastic population viability analyses to assess how differing consequences of habitat degradation interact to drive extinction dynamics in a focal population. Using the isolated spectacled caiman Caiman crocodilus population/ecomorph from the Apaporis River as a case study, we find that threatening the resource base, which individuals increasingly rely upon, to outgrow vulnerable size ranges and mature accelerates extinction. We also found that when habitat degradation impacts both the primary adult and juvenile resource bases, this can have marked synergistic effects on threatening population viability. By contrast, destroying nesting sites has only a small effect on accelerating the impact of deteriorating prey availability. Through integrating community-level feedback between habitat degradation/change and population dynamics/structure, our approach provides a comparative framework for assessing the relative importance of distinct mechanisms through which habitat degradation ultimately drives extinction risk.

Epistasis makes the fitness effect of a mutation depend on the genetic background in which it occurs, thereby shaping the accessibility and reversibility of evolutionary trajectories. Along adaptive walks, this path-dependent epistasis can take distinct forms: contingency, when a mutation requires prior substitutions to be beneficial; entrenchment, when later substitutions make its reversion increasingly deleterious; and diminishing returns, when successive beneficial mutations reduce one anothers effects. Although these regimes have been documented experimentally, the conditions under which each predominates remain poorly understood. Here we use Fishers geometric model to derive a general framework for path-dependent epistasis under stabilizing selection on a multidimensional phenotype. We show that epistasis between substitutions has a simple geometric interpretation: contingency and entrenchment arise when the collateral effects of mutations, orthogonal to the direction of the optimum, are compensated by the preceding or subsequent adaptive path, whereas diminishing returns arise when successive substitutions remain strongly aligned with the same direction of selection. Analytical results and simulations reveal a transition controlled by a single composite parameter combining phenotypic complexity, mutation size, and distance to the optimum. Far from the optimum, adaptive walks are dominated by diminishing returns epistasis. As populations approach the optimum, or as phenotypic complexity increases, antagonistic pleiotropy generates systematic contingency and entrenchment. At mutation-selection-drift equilibrium, these effects become strong, rapidly established, and increase with phenotypic complexity. These results show that contingency and entrenchment do not require specific molecular interactions between residues: they emerge generically from nonspecific epistasis produced by stabilizing selection on pleiotropic traits. Fishers geometric model thus unifies diminishing returns, contingency, and entrenchment as distinct regimes of the same underlying geometry of adaptation.

Despite sharing the same genes and the same environment, individuals often develop substantial phenotypic differences. While this pattern has been documented across diverse species and traits, the processes giving rise to this "stochastic" or non-shared environmental variation remain unclear. Recent mathematical models of development in which phenotypes are gradually constructed may offer some clues. These models show that imperfect environmental cues can generate striking variation in developmental trajectories and adult phenotypes. At the population level, such imperfect cues produce increasing stability of individual differences across ontogeny (e.g. animal personality) and patterned distributions of mature phenotypes (e.g. normal or skewed) that resemble those observed in real organisms. Our paper synthesizes existing models in which stochastic phenotypic variation arises solely as a by-product of mechanisms missing their phenotypic targets because of imperfect cues. We then link these models to related, but independent, mathematical theory exploring the environmental conditions under which stochastic phenotypic variation is favoured by natural selection. Our integration shows that stochastic sampling is often favoured over classic bet-hedging strategies involving non-plastic generalist or specialist strategies. Our findings provide new directions of research on stochastic sampling as a mechanism for adaptive stochastic variation within and across generations.

Stressful environments can pose a threat to microbial populations, but resistant individuals can emerge and avoid extinction. Adaptation to stress is classically studied in isolated microbial species, ignoring ecological interactions, a key component of natural ecosystems. A growing body of experimental work has shown that community context can affect resistance evolution due to a large variety of mechanisms. Here we set out to identify the minimal components needed to predict the likelihood of acquiring resistance in a focal species embedded within a simple community. To achieve this, we developed a mathematical model based on evolutionary rescue theory and validated it with two experimental systems: Escherichia coli evolving on exposure to the antibiotic nitrofurantoin alone or with one of 14 bacterial isolates from urinary tract infections, and Microbacterium liquefaciens evolving in ampicillin alone or with ampicillin-degrading Comamonas testosteroni. One key factor that emerged from our analyses - the relative strength of competition versus protection - could explain whether a focal species is more or less likely to evolve resistance in the presence of a partner species. While competition always hinders the emergence of resistance, protection can rescue the focal species in two ways: (i) ecological rescue, when the partner species completely removes the antibiotic and favors the survival of the susceptible population, or (ii) evolutionary rescue, when the partner only lowers antibiotic concentrations and favors the emergence of resistant variants, a previously overlooked evolutionary consequence of detoxification. Overall, by integrating theory and experiments, we propose a framework that clarifies how ecological interactions favor or hinder the evolution of resistance to antibiotics or potentially other stressors. SignificanceBacteria can rapidly adapt to resist stressors, such as antibiotics. While resistance evolution in single populations or species is well understood, it remains unclear how ecological interactions with other species influence this process. We develop a mathematical framework to predict what interactions should favor resistance evolution and validate it with two sets of experiments where bacteria adapt to antibiotics in small communities. Our work demonstrates that interactions with other species shape the probability of evolving resistance in a predictable way, determined by the balance between competition and protection against the stressor. By identifying the key factors that drive these dynamics, our work helps explain how bacteria adapt to environmental challenges within species-rich ecosystems.

Most species are geographically structured, leaving characteristic signatures in neutral regions of the genome. These signatures can be distorted when neutral regions are linked to deleterious mutations. In such regions, purifying selection can reduce genetic diversity through Background Selection (BGS) or, for recessive mutations, increase diversity through Associative Overdominance (AOD). While the effect of BGS and AOD are well characterized in panmictic populations, their effects remain largely unexplored in structured populations. Here, we investigated an Isolation with Migration model using forward simulations across a range of migration, selection, dominance, and recombination parameters. We first used a genotype-based approach to quantify the effects of deleterious mutations on standard summary statistics ({pi}, dxy, FST, DAFi). We then showed that an Ancestral Recombination Graph-based (ARG) approach, tracking tree sequences from a sample of one diploid per deme, recovers the same patterns while directly relating genetic variation to the underlying coalescent processes. When recombination is sufficiently low, we found a BGS-driven regime for weakly codominant mutations, characterized by lower diversity and increased genetic differentiation (FST). For recessive mutations, we first identified an AOD-driven regime, characterized by increased diversity and lower FST values followed by a transition to a subsequent BGS-driven regime. Genealogies were similarly impacted by deleterious mutations: BGS shrunk coalescent times and produced a shift towards lineage sorting topologies, while AOD stretched coalescent times and produced a shift toward incomplete lineage-sorting topologies. These patterns were weakened by gene flow, with FST and topologies remaining close to expected under neutrality, while diversity and coalescence times remained robust to demography. Our results provide clear evidence of BGS, AOD, and of their transition in a structured model with gene flow. Importantly, these processes leave distinct and interpretable signatures on gene trees, highlighting the potential of ARG-based approaches for inferring linked selection and dominance in structured populations. Author summaryCharacterizing how demography and selection jointly shape genomic variation is a central question in population genetics. As deleterious mutations reduce fitness, they are continuously removed from populations by purifying selection. Through linkage, this affects nearby regions of the genome, leaving signatures of selection on linked neutral genetic diversity. While these effects are well understood in random mating populations, much less is known in structured populations. Specifically, the occurrence of Background Selection (BGS), which reduces diversity, and Associative Overdominance (AOD), which increases diversity, remains underexplored. Here, we used simulations to investigate how deleterious mutations shape genomic variation in a structured two-population isolation with migration model. By combining standard population genetic analyses with a genealogical approach based on Ancestral Recombination Graphs (ARGs), we showed that BGS and AOD leave distinct and interpretable signatures on common summary statistics and the underlying genealogies. We identified clear signatures of BGS and AOD when recombination was low and revealed a transition from AOD to BGS for recessive mutations, as the strength of selection increased. Our results highlight the importance of jointly considering demography and linked selection when interpreting genomic data and demonstrate the potential of ARGs to jointly infer demography, selection, and dominance from genomic data.

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.

The classical concept of Critical Community Size (CCS) as formulated by Bartlett defines the minimum host population required for a pathogen to persist endemically without stochastic extinction. While this framework successfully described directly transmitted childhood infections in relatively isolated populations, it is increasingly inadequate for modern urban systems characterized by strong connectivity between cities. Pathogens circulating in highly connected urban networks can repeatedly re-emerge through spatial reintroduction even when local transmission temporarily fades out. In such systems, persistence is inherently probabilistic and influenced simultaneously by population size, environmental suitability, and network connectivity. In this study, we develop a generalization of the CCS concept, the Empirical Persistence Threshold (EPT), and apply it to three of the main arboviruses circulating in Brazil--dengue, chikungunya, and Zika--over the period 2017-2024. The Empirical Persistence Threshold generalizes the classical notion of critical community size by replacing a single deterministic threshold with a probabilistic, datadriven measure. Instead of asking for the minimum population at which persistence is guaranteed, EPT characterizes the lower tail of the population distribution among municipalities that empirically sustain transmission. Using weekly incidence data from thousands of municipalities, we transform temporal incidence series into binary sequences describing the presence or absence of reported transmission. For each municipality, we characterize persistence through the empirical distribution of run lengths of consecutive weeks with reported cases. Distances between run-length distributions are computed using the Wasserstein-1 metric, allowing a geometrically meaningful comparison between persistence profiles, and municipalities are grouped into epidemiological regimes using hierarchical clustering methods. Across all three arboviruses, we identify two robust regimes: one exhibiting sporadic and recurrent epidemic transmission, and the other exhibiting sustained persistent transmission. We then estimate the population scales associated with each persistence regime. The analysis is further extended to evaluate how persistence thresholds vary across climate regimes (Koppen classification) and urban hierarchy levels (REGIC). This framework allows the estimation of probabilistic persistence thresholds analogous to CCS, but adapted to connected urban systems. We define the Empirical Persistence Threshold as lower quantiles of the population distribution among municipalities in the persistent regime, and additionally estimate persistence thresholds based on regime membership probabilities. Results reveal strong interactions between population size, climate, and urban connectivity. Dengue exhibits the lowest persistence thresholds, Zika intermediate thresholds, and chikungunya the highest thresholds. These findings demonstrate that pathogen persistence in modern urban systems cannot be described by a single deterministic population threshold. Instead, persistence emerges from the joint effects of demographic scale, environmental suitability, and network position within metapopulation systems. Author SummaryInfectious diseases often require a minimum population size to persist locally, a concept known as the critical community size (CCS). This idea was developed for relatively isolated populations, but modern cities form highly connected networks where diseases can repeatedly reappear even after local transmission disappears. In this study, we introduce the Empirical Persistence Threshold (EPT), a data-driven approach that replaces the idea of a single fixed threshold with a probabilistic description of persistence. Instead of focusing on case counts, we analyze how long transmission persists over time in each municipality. Using weekly data for dengue, chikungunya, and Zika across Brazil from 2017 to 2024, we identify distinct patterns of transmission persistence and estimate the population levels associated with sustained transmission. We also examine how these thresholds vary with climate and urban structure. Our results show that persistence depends not only on population size, but also on environmental conditions and the position of cities within the urban network.