Journal of Theoretical Biology

Understanding and predicting extinction risk is a central challenge in population biology. Mathematical models incorporating Allee thresholds are commonly used to understand population dynamics and to assess extinction risks. Inaccurate predictions can have serious consequences for conservation management. In this simulation study, we develop a likelihood-based inference and prediction workflow to estimate parameters, including the Allee threshold and population diffusivity parameters, using noisy count data generated using a well-defined discrete model. Although parameters are identifiable according to commonly used criteria, the accuracy of resulting predictions depends strongly on the quantity, quality, collection time and spatial resolution of the data. Our workflow demonstrates that seemingly reliable parameter estimates can lead to inaccurate predictions, highlighting the need for careful consideration of data quality and quantity to guide extinction-risk modelling and prediction. Open source software is provided on GitHub to replicate and extend all results considered.

Cancer therapy balances between two competing objectives - treatment efficacy against the tumour and the risk of treatment related severe adverse events, including patient death. Most existing optimal control theory (OCT) formulations rely on optimising heuristic cost functionals that lack direct clinical interpretability. In clinical practice treatment efficacy and patient tolerability are primarily assessed through survival metrics and adverse event rates. Here we introduce the Continuous Lifetime Payoff (CLP), a novel OCT objective functional that directly links treatment decisions to patient survival. It explicitly incorporates tumour dynamics, tumour eradication, and patient mortality from tumour progression, drug-related toxicity and age. We fit age-related mortality from life tables and infer parameters from simulated survival data. The CLP provides a clinically grounded framework for optimising chemotherapy regimens.

Neutrophil swarming is a critical immune response in mammals and fish, in which neutrophils are recruited to inflammatory sites where they coordinate into a swarm that neutralizes pathogens. While excessive swarming can drive prolonged inflammation, a quantitative understanding of swarming dynamics remains limited. We developed a one-dimensional radial reaction-diffusion model of neutrophil swarming with two kinetic parameters, in order to capture the self-limiting swarming dynamics in both murine and human neutrophils in response to different inflammatory stimulus sizes. To ensure that the inverse problem is well-posed, we first performed sensitivity and identifiability analyses. We then developed a physics-informed neural network (PINN) to infer the key parameters governing swarm expansion and self-limitation. To account for uncertainty in noisy experimental measurements, we further extended this framework to a Bayesian PINN (B-PINN), which provides credible intervals for the inferred parameters. Both models were validated against synthetic data generated by numerical simulation and subsequently applied to in vitro experimental data from human and murine neutrophils in response to three bioparticle cluster sizes. The PINN-inferred dynamics show that larger bioparticle clusters are associated with greater cumulative recruitment and larger swarms in both species. The models further reveal species-specific differences in both the amplitude of initial recruitment and the timescale on which it self-limits. Additionally, the B-PINN posterior distributions quantify uncertainty in these species- and cluster size-dependent trends and identify where additional measurements would be most informative. To our knowledge, this is the first application of physics-informed machine learning to model neutrophil swarming dynamics. This framework provides a starting point for systematically comparing recruitment dynamics between human and murine neutrophils and offers guidance for future experimental design.

Mathematical models, e.g. differential equations and stochastic processes, have gained considerable attention for understanding evolution of antibiotic resistance. However, most existing models assume standing genetic variation and do not consider the possibility of random or drug-induced mutation of reference bacterial strains. Therefore, we propose a pharmacokinetics/pharmacodynamics (PK/PD)-based continuous-time Markov chain considering the competition and mutation between sensitive and resistant bacterial within an infected host during treatment. The proposed model is approximated as a generalized birth-death process with immigration, allowing for explicit derivation of the probability resistant population establishes during treatment. Besides capturing the stochasticity of de novo emergence of a resistant bacterial strain, we explore the effects of different antibiotic modes of action, horizontal gene transfer, nutrient availability and drug pharmacokinetics on antibiotic resistance. We find that replication-targeting (biostatic) drugs suppress resistance more than death-targeting (biocidal) drugs. Like prior works, we obtain maximized resistance at intermediate drug concentrations, however the consideration of de novo mutation magnifies the superiority of higher doses in preventing resistance emergence.

AnnotationMitochondrial heteroplasmy --the coexistence of both wild-type and mutant copies of mitochondrial DNA (mtDNA) within a cell--is a key factor in the pathogenesis of mitochondrial diseases. Classical approaches, which rely solely on the scalar fraction of mutant DNA, fail to fully account for threshold effects, the stochastic nature of heteroplasmy dynamics, and tissue specificity. The aim of the work is to construct a complex stochastic model of heteroplasmy dynamics, which for the first time combines the effects of selection, genetic drift, migration of mitochondrial genomes between tissues and threshold mechanisms of pathology development, for a quantitative assessment of the risk of mitochondrial diseases. In this paper, we propose a complex-phase formalism in which the state of a cells mitochondrial genome is described by a complex number Z = a + ib, where a and b are the absolute numbers of normal and mutant mtDNA copies, respectively. This approach naturally combines information on copy number and heteroplasmy level, and the argument{phi} = arctan (b / a) is interpreted as a phase characterizing the mutant load. Based on this formalism, we developed a stochastic model of tissue dynamics that includes the processes of selection, genetic drift, and intertissue migration of mitochondrial genomes. Using Monte Carlo methods (1000 simulations), we demonstrated that neuronal tissues are characterized by high heteroplasmy variability and a significant probability of reaching a pathological threshold even with a relatively low systemic mutant load. Kaplan-Meier survival analysis demonstrates that the development of pathology is probabilistic and can be described as a time -to-event process . The proposed approach enables quantitative assessment of the individual risk of developing mitochondrial diseases and opens the door to personalized prognosis.

In a population model for an infectious disease, we consider the early stochastic dynamics of an emergent 'mutant' strain, appearing and spreading during an epidemic of another 'wildtype' strain. The mutant may not reach establishment in the host population. The time at which the mutant first appears determines its probability of establishment. We calculate this establishment probability with two methods. The first method assumes a classical branching process, with a constant transmission rate. The second method reflects the changing size of the pool of susceptible hosts, due to the dynamics of the wildtype. We find that susceptible depletion can substantially impact the establishment probability. We explore the consequences of this stochastic establishment on the "escape pressure" acting on a pathogen to produce immune escape variants. We find that the overall escape pressure rate depends strongly on the appearance time of the mutant, especially if the establishment probability is itself shaped by the continued spread of the wildtype. In most scenarios, the escape pressure rate (and thus, the risk of new escape variants) peaks slightly earlier than the prevalence of the wildtype strain. Integrating the escape pressure over time, we obtain the cumulative escape pressure generated by the wildtype epidemic. The relationship between the escape pressure and the vaccination coverage depends on the cross-immunity, due to susceptible depletion. For example, with intermediate cross-immunity, the risk of immune escape may be lowest at intermediate vaccination coverages. Thus, these results raise important considerations for vaccination strategies in response to novel outbreaks.

Despite the impacts of within-host disease dynamics on disease outcomes in individual hosts and disease spread among-hosts, generic models of within-host population dynamics have received far less attention than their among-host counterparts. While a number of models have been proposed to explore theoretical eco-evolutionary dynamics, they have yet to be evaluated for estimability, raising questions about their ability to provide reliable inference when confronted with data. We evaluated the estimability of two generic within-host population dynamics models by assessing: (1) parameter estimation, our ability to recover correct values of model parameters from data, (2) the consequences of mis-assigning the underlying mechanistic model on parameter estimation, and (3) the reproduction of qualitative dynamics, or, our ability to use parameter estimates to reproduce observed dynamical behaviours. In some cases, fitting a mis-matched mechanistic model to time series data produced reasonable parameter estimates that were able to reproduce system dynamics, and that when provided the data-generating model, parameter uncertainty can produce substantial behavioural uncertainty. Our findings highlight the impacts of structural, parametric, and behavioural uncertainty on inference, and demonstrate the value of improving system-specific knowledge to prevent the use of incorrect functional forms and of measuring consequential parameters to improve estimability.

Some organisms exhibit declining mortality and increasing fecundity following sexual maturity, a demographic pattern known as negative senescence. According to life history theory, ageing occurs because resources are preferentially allocated to reproduction over somatic maintenance. Models connecting indeterminate growth to negative senescence exist, but none integrate somatic maintenance as a competing allocation decision alongside growth and reproduction. We formulate a life history model in which an individual allocates energy among reproduction, somatic growth, and somatic maintenance and mortality rate depends on both body size and somatic damage. We show that negative actuarial senescence, whereby mortality declines with age, occurs when the proportional change in reproductive value exceeds the proportional change in fitness returns from current investments into reproduction and soma. We derive the necessary conditions for an uninvadable allocation strategy using invasion analysis and Pontryagin's maximum principle, and examine biologically relevant cases numerically. We show that both negative senescence and indeterminate growth arise together as uninvadable outcomes even when maintenance competes for the same resources as growth and reproduction. We show that both diminishing returns to reproduction and diminishing returns to growth can give rise to negative senescence. These results extend the disposable soma theory to organisms with indeterminate growth, in which mortality decreases with size, and identify key mechanisms for the empirically observed association between indeterminate growth and non-senescent demographic trajectories.

Spatial population structure and seasonality are both central to the spread of many infectious diseases of plants, animals and humans. While seasonal forcing in transmission often plays an important role in epidemiological models of a wide range of infectious disease, and we now have some theoretical understanding of the dynamical impacts of spatial structure, the combined effects of these two ubiquitous processes has not been examined in detail. Here, we develop a novel model to explore the combined influence of spatial structure and temporal variability on disease dynamics. Spatial structure is represented using a lattice-based approach with near-neighbour interactions, while temporal variability is included through regular, seasonal, variation of the transmission rate. We use bifurcation analysis of a pair approximation of the full spatial model to identify the parameter regimes associated with qualitatively distinct dynamical behaviours. The model exhibits a remarkably wide range of complex dynamics, including limit cycles, quasi-periodic cycles, multi-year cycles, chaotic dynamics and bistability between these different states. In particular, complex dynamics occur when reproduction is predominantly local, with the dynamics depending critically on the amplitude of the seasonal transmission rate. We show how high transmission rates, high birth rates and in particular low recovery rates are requirements for complex dynamics. We predict that SI-type disease interactions in plant pathogen systems will show complex dynamics even with relatively global transmission dynamics.

The coexistence of competing pathogen strains is shaped by cross-immunity, the cross-protection that infection with one strain confers against another. Although cross-immunity is often asymmetric between strains, this asymmetry is often neglected in the literature on multi-strain coexistence. The effect on coexistence-exclusion outcomes of waning immunity\textemdash which is particularly relevant for antigenically evolving pathogens\textemdash is also poorly understood. To understand how these factors affect strain coexistence, here we analyze a status-based two-strain SIRS model with asymmetric cross-immunity and strain-specific rates for transmission, recovery, and waning of immunity. We derive explicit invasion thresholds that also determine the feasibility and local stability of a unique coexistence equilibrium. Thus, these thresholds allow us to characterize the region of stable strain coexistence, as a function of the cross-immunities and rates of waning immunity. We also obtain closed-form expressions for the strain prevalences at the coexistence equilibrium, showing that the total prevalence may vary non-monotonically as the basic reproduction number of one strain increases. Finally, we show that a transient reduction in transmission can move a coexisting strain pair across an invasion boundary, driving the weaker strain extinct. Applying this result to influenza B, our analysis offers a parsimonious explanation for the disappearance of the Yamagata lineage during the COVID-19 pandemic.

Cancer progression is driven by the interplay between cancer cell phenotypic variability regulated by EMT/MET, and cell-cell and cell-matrix interactions. Starting with an individual-based model, in which every cell is characterised by its position, velocity and a continuously varying epithelial-mesenchymal phenotype, we derive, via a kinetic description and a mean-field limit, two alternative macroscopic formulations: an Euler-like system that couples mass and momentum to the phenotypic variable, and a single advection-aggregation-diffusion equation (AADE) for the cancer cell density. Both macroscopic models retain the non-local adhesion-repulsion forces, haptotactic response to an evolving ECM, and phenotype-dependent transition dynamics driven by TGF-{beta}. Numerical experiments in two spatial dimensions indicate that the macroscopic equations reproduce key scenarios obtained at the individual scale. In particular, a microscopic-macroscopic comparison shows that the AADE density reproduces acurately both the spatial localisation and the phenotypic decomposition of the individual cell population. We also demonstrate that varying only the steepness of the TGF-{beta} switch function, changes the EMT response from an almost binary epithelial-mesenchymal (EM) separation to a partial EM phenotypes. This study provides a systematic bridge from stochastic, heterogeneous cell dynamics to continuum descriptions, for investigating phenotype driven tumour invasion and supporting the choice of macroscopic models in large-scale simulations and analytical studies.

Host-parasite relationships are defined by transmission. Transmission takes diverse forms, with horizontal transmission modes generally characterized as density- or frequency-dependent. However, many observed parasites display mixed-mode transmission using both density- and frequency-dependent routes. To investigate how mixed-mode transmission impacts epidemics, we assessed infection prevalence under single- and mixed-mode transmission when there was a linear trade-off between the probability of infection by the two modes. Our results show that the prevalence of a parasite with mixed-mode transmission can be greater than one with a single-mode only when density- and frequency-dependent routes are associated with different effects of the parasite on host fitness. This work shows that mixed transmission modes per se may not necessarily increase disease prevalence, and that substituting two transmission modes for a single one may result in higher prevalence under limited conditions.

Background: Tuberculosis remains a persistent infectious disease whose control is complicated by latent infection, delayed treatment, incomplete recovery, reinfection, and continuing transmission from infectious individuals. Although treatment is central to tuberculosis management, it is frequently represented only as a transition parameter in mathematical models rather than as a separate epidemiological state. In this study, treatment was therefore incorporated explicitly as an independent compartment so that its influence on transmission, recovery, disease-induced mortality, and long-term disease persistence could be evaluated. Methods: A deterministic nonlinear compartmental model was formulated by dividing the total population into susceptible, exposed, actively infected, treated, and recovered classes. Reinfection of recovered individuals, progression from latent infection to active disease, movement of infectious individuals into treatment, treatment-associated recovery, natural mortality, and disease-induced mortality were included. Positivity and boundedness of the solutions were examined to establish biological validity. The basic reproduction number, R0, was derived through the next-generation matrix approach. Disease-free and endemic equilibria were determined, and their local and conditional global stability properties were investigated using Jacobian analysis, the Routh-Hurwitz criterion, center manifold theory, Lyapunov functions, and LaSalles invariance principle. Normalized sensitivity indices, Latin hypercube sampling, partial rank correlation coefficients, and numerical simulations were also applied. Results: The disease-free equilibrium was shown to be locally asymptotically stable when ,R0<1 whereas sustained transmission and a unique endemic equilibrium were associated with R0>1. Under the stated reduced-model assumptions, stability of the endemic equilibrium was established. Transmission-related parameters were identified as the strongest positive contributors to disease persistence. In contrast, treatment and recovery parameters were found to reduce the reproduction number and infectious burden. Numerical simulations indicated that stronger treatment implementation and reduced transmission opportunities produced substantial reductions in active tuberculosis cases. Conclusion: Treatment was shown to function as both a clinical pathway and an epidemiological control mechanism. The proposed framework may support the design of treatment-centered strategies for reducing tuberculosis prevalence and preventing long-term endemic persistence.

For decades, population geneticists have relied upon a formula derived by Malecot and by Kimura to estimate the fixation probability of mutant alleles ({phi}). Among other things, this formula leads to the conclusion that in sufficiently large populations{phi} asymptotically approaches 2s(Ne/N ) (for small s), where s is the relative selective advantage of the mutant allele, and Ne and N respectively denote the effective and absolute (haploid) population sizes. In contrast, in sufficiently small populations,{phi} = 1/N, with the two domains of behavior being separated at the point where s [bsime] 1/(2Ne). These results hold when s, N, and Ne remain constant during the fixation process, but require modification when populations are changing in size. Here, we show that if there are ecological effects associated with competing alleles, such that the genetic composition of the population influences the total population size, the fixation probability of a beneficial allele can substantially deviate from the levels suggested above, even in the domain of effective neutrality (i.e., | s | [lsim]1/(2Ne)). We obtain analytical results for a variety of frequency-dependent functions for the overall population size, and show that the overall effect is largely a function of the response at low mutant-allele frequency. We also derive expressions for times to fixation and for fixation probabilities of deleterious alleles, and show how our results relate to prior work for the situation in which population sizes change in frequency-independent manners.

Type 2 diabetes is characterized by progressive aggregation of islet amyloid polypeptide (IAPP) within the islets of Langerhans, a process strongly implicated in beta-cell dysfunction and loss. Although oligomeric IAPP intermediates are widely considered the principal cytotoxic species, the relative contributions of the many biological and kinetic processes governing their formation, clearance, and conversion into fibrils remain poorly quantified. Here, a mathematical model of IAPP aggregation is developed that incorporates the physiology of beta-cell secretion and the microanatomy of the islet, including capillary-mediated clearance, enzymatic degradation, and the kinetics of oligomer and fibril formation within a well-mixed control volume. Building on the hypothesis that oligomers are the major cytotoxic species, the concept of accumulated cytotoxicity is introduced, defined as the time integral of the oligomer concentration, and a systematic sensitivity analysis of this quantity with respect to all model parameters is performed. The results reveal a striking hierarchy: only two parameters, the basal rate of IAPP monomer secretion and the rate constant for spontaneous oligomer dissociation, exert a first-order influence on long-term accumulated cytotoxicity, with dimensionless sensitivities approaching +1 and -1, respectively, while the effect of all other parameters remains subordinate and decays at long times. The model further shows that capillary clearance, owing to the physical exclusion of oligomers from fenestrated capillaries, selectively reduces fibril accumulation and amyloid deposition without affecting oligomer-mediated cytotoxicity, indicating that amyloid area fraction, the standard histological metric of disease severity, may not be a reliable surrogate for cytotoxic burden. The model predicts that approximately 48% of the islet area is replaced by amyloid after 30 years, broadly consistent with histological observations of advanced disease. These findings identify monomer secretion and oligomer dissociation as the most promising therapeutic targets to limit cytotoxic damage in type 2 diabetes and provide a quantitative framework for evaluating candidate intervention strategies.

Many proposed mechanisms for the evolution of cooperation among unrelated individuals rely on relatively demanding cognitive abilities that are not widespread across taxa. In contrast, individual heterogeneity is a pervasive feature of animal groups, encompassing differences in personality as well as physical and cognitive traits. Such heterogeneity can promote the evolution of cooperation, yet its role has received comparatively little attention, particularly as a source of variation giving rise to social organization such as leadership. A specific form of leadership can emerge under unstable environmental conditions, when some individuals become better suited than others to initiate action and influence the behavior of their peers. Unlike fixed dominance hierarchies, emergent leadership can rapidly adjust to changing environmental conditions, thereby reshaping group organization. Because it does not require the maintenance of stable hierarchies, this form of leadership can arise even in species that do not have the cognitive capabilities to sustain complex social structures. In this work, we investigate the combined effects of individual heterogeneity and emergent leadership on the evolution of cooperation using an evolutionary game-theoretic model in which individuals may assume the roles of leaders or followers according to their strength, representing individual differences in suitability to prevailing environmental conditions. We examine different levels of population heterogeneity together with increasingly complex strategy sets requiring progressively greater informational requirements, allowing individuals to condition cooperation on their own strength, leadership role, or both. Our results show that the interplay between leadership and heterogeneity promotes the evolution of cooperation, particularly when only a small fraction of individuals act as leaders. Under these circumstances, cooperation evolves even when individuals employ the simplest possible strategies. Under harsher ecological conditions, cooperation can be sustained by more sophisticated strategies, specifically by conditional strategies that prescribe cooperation when individuals are strong or leading and defect when acting independently. Author summaryIn this study, we propose that emergent leadership mediated by individual diversity can boost the evolution of cooperation in animal groups. Building on growing evidence on the heterogeneity of animal capabilities and personalities, we focus on the fleeting leadership that emerges in animal groups when facing rapidly changing environmental conditions. We suggest that this type of leadership that emerges from individual differences in strength--a generic quality encompassing those characteristics that make an individual more fit to lead in a given situation--does not require complex cognitive capabilities from the animals and represents a valid alternative to more demanding strategies proposed in the past to explain the evolution of cooperation. Using an evolutionary game theory model, we show that if a population includes a few strong players, these can become influential leaders and guide the actions of their peers to achieve cooperation. Although the naive strategy of always cooperating is sufficient for cooperation to evolve, the introduction of more complex strategies leads players to cooperate only when they are more likely to be recognized as influential leaders. These strategies are more effective in promoting cooperation under unfavorable ecological conditions and are also more robust against exploitation by defectors.

1.Partial Rank Correlation Coefficient (PRCC), usually performed following Latin Hyper-cube Sampling (LHS), is a global sensitivity analysis that quantifies the monotonic relationship between model parameters and the desired output. To carry out this analysis, a range of acceptable parameter values must be known or estimated. However, within a biological context, approximating these values may be difficult. Parameter values and ranges can be taken from different organisms or systems or be estimated to produce qualitative phenomena in the model. Using a mathematical model of the epithelial mesenchymal transition (EMT) as a test case, this work examines how the parameter ranges chosen prior to analysis can influence LHS-PRCC results and shape subsequent analysis interpretations. Previous LHS-PRCC analysis of this model restricted parameters to {+/-}10% of their original value, which limits the scope and interpretability of parameter influence. Such a small range assumes, in the biological sense, that parameters are well-measured with little variability. Here, this work extends the previous analysis and explores several parameter ranges ({+/-}25%, {+/-}50% of the original value). This work also tests whether, within the {+/-}10%, {+/-}25% and {+/-}50% parameter ranges, the bistable switch present in the original model are maintained. Ultimately, this work showcases how a choice made prior to analysis, such as the accepted parameter ranges for biological rates and values in complex dynamical systems can influence sensitivity analysis results and interpretability. Additionally, these choices can have hidden consequences, such as the loss of phenomenological behavior. Thus, explicit prior knowledge about the appropriate parameter values is needed before using analysis to guide future experiments and model development.

Malaria and diabetes represent two globally significant metabolic disorders whose co-occurrence leads to complex, poorly understood pathophysiological interactions. Plasmodium infection disrupts glucose homeostasis through parasite-driven glucose consumption, inflammatory cytokine production, and pancreatic /{beta}-cell dysfunction, while diabetes impairs host immunity and increases malaria susceptibility. To date, no mathematical framework has captured the bidirectional coupling between these systems. Here we extend the insulin-glucose-glucagon (IGG) model of Dalton et al.\ (2026) by introducing a fourth state variable representing parasite load, incorporating malaria-induced insulin suppression, parasite-driven glucose consumption, inflammatory gluconeogenesis, bidirectional glucagon dysregulation, and insulin-dependent immune enhancement of parasite clearance. We establish positivity, boundedness, existence and uniqueness of steady states, local stability via Routh-Hurwitz criteria, global stability via Lyapunov functions, and sensitivity analysis of parameters driving hypoglycemia risk. Numerical simulations characterise the model across healthy, diabetic, and co-infected states. They show that parasite-driven glucose consumption and inflammatory gluconeogenesis act antagonistically on circulating glucose, that insulin-enhanced immunity lowers peak parasitemia through a saturating clearance term, and that increasing the half-life of exogenous insulin raises hypoglycemia risk in all host states. These mechanisms provide testable hypotheses for the clinical management of malaria-diabetes patients and identify potential therapeutic targets (TNF- blockade, glucagon analogues) for mitigating co-infection morbidity.

Animals use long-range signals, such as hormones and neural signals, to coordinate the actions of distant organs. There is no precise, quantitative framework that explains the problems these control systems must solve and thus predicts their behavior under varied conditions. We consider this problem in the context of blood glucose regulation by the hormone insulin, the failure of which produces diabetes. We show that existing mathematical models of glucose regulation admit equivalent control strategies with no hormones at all, and thus cannot explain the need for hormonal regulation. We therefore introduce a minimal model of inter-organ variations in local glucose, and show that control strategies based on local glucose measurements face severe trade-offs between different control objectives. In contrast, we show that hormonal control signals from the pancreas can overcome these limitations. By exposing the benefits of hormonal control, our work paves the way to a detailed understanding of physiological design principles, with possible implications for the engineering of an artificial pancreas.

The general random walk model (GRW) of Hunt (2006) is used to infer directional evolution in mean trait values from sparse fossil data by modeling phenotypic change as the accumulated result of small steps with mean step sizes and step variances. Using simulations and real data cases, Ergon (2026) showed that the step variances can be estimated reasonably well only when the mean trait values have small measurement errors, while for fossil data with realistic measurement errors they appear to be extremely difficult to find, and they are often found to be negative. In the simulations Ergon (2026) assumed that the true phenotypic mean values were known. Here, I essentially repeat these simulations under the assumption that only mean trait values with large measurement errors are known, and based on weighted mean squared error (WMSE) comparisons the conclusion is that weighted least squares (WLS) is a better method than GRW. A second conclusion is that WLS is a better method also in the possibly rare cases with large measurement errors where the GRW parameters are estimated well. The GRW method is simply not flexible enough to handle such cases. A third conclusion is that Akaike Information Criterion (AIC) results for GRW models with large measurement errors relative to the step variance may be overly optimistic.