Nonlinear Dynamics

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

Growing interest to describe the electrical behavior of glial cells, mainly astrocytes, in intact brain tissue poses more and more challenges to commonly accepted belief they only respond in a linear manner in uptake of the excess of extracellular potassium and maintenance of their network equipotentiality. Their highly conductive mutual interconnections via gap junction (GJ) connections introduce yet another class of nonlinear elements. As more studies report nonlinearities in membrane voltage Vm dependence of both, the membrane and junctional conductances, the need to formulate minimal dynamical models of their transient behavior is getting more acute. Since ODE models of coupled cells, even in simplest 1-d arrays, require simplified descriptions and small set of parameters, rare quantitative studies on glia makes the task even more difficult. This study attempts to qualify a self-coupled cell, or a glial cell coupled to fixed voltage as useful system for detecting the nature of instabilities and transitions coming from coupling. In a novel biophysical model of coupled astrocyte, we introduce nonlinear kinetics of deactivation for large junctional voltages for the first time. We found that N-shaped nonlinearities and corresponding fold structure in the vector field of isolated cell serves as a baseline on top of which coupling nonlinearities enrich the bifurcation picture. Numerical simulations of 1-d array of coupled astrocytes show that coupling increases the propensity of astrocytic Vm to bistability and front propagation. We believe that presented illustrations of possible effects of coupling nonlinearities will motivate neurobiologists to further explore their impact in disease. Significance statementTransient changes in membrane voltage of glial cells may produce significant transient voltage difference between directly coupled cells. Nonlinear steady-state conductance of their interconnection elements, the gap junctions, introduce nonlinear current profiles which are very difficult to measure and quantitate using the available methods due to marked permeability of the junctions and leakiness of glial membrane in general. We propose a minimal model of glial membrane extended with a self-coupled feedback loop, which under realistic simplifying assumptions could serve for qualitative analysis of the impact of coupling, on the stability of resting membrane voltage. Neuronal cells of the brain and spinal cord cannot exist and function without supportive and neuromodulatory functions of the diverse population of glial cells. This applies to virtually all physiological processes on cell level - from cell development, metabolic support, membrane signaling, slow molecular signal transduction, ion homeostasis, neurovascular coupling, myelination, to mention only a few, manifest neuro-glial interaction. Even though all glial cell types are interconnected, the most abundant ones, the astrocytes are massively interconnected by gap junctions to form ordered networks. Electrically, astrocytic networks display membrane voltage equipotentiality, which is considered system-wide resting state for given neuro-glial circuit or unit. With molecular and cellular substrates of glial connectivity being slowly elucidated, network science and dynamical modeling are slowly "invading" that area with many important issues left open. In this study using classical dynamical systems approaches we give indications how nonlinear intercellular coupling between astrocytes affects physiological resting state and its instabilities compared to isolated, uncoupled cell. We strongly believe the suggested minimal model could fill the gap in ODE modeling of neuro-glial circuits, within broadest scope of hypothesis-driven research in cell-level neuroscience.

A novel mathematical framework to define the threshold of action potentials in excitable cells is presented. Unlike previously applied methods that rely on approximations or specific fixed-point bifurcations, the approach focuses on the geometry of membrane potential trajectories. Specifically, the focus is on the concavity changes during the upstroke of an electrical pulse. These changes in concavity form a curve of inflection points that defines a region in phase space crossed by all the action potentials in the system, and containing no non-action potential trajectories. Such region is called the excitability region and its size can be measured, thus providing a measure for the excitability of a dynamical system, and a way to compare the excitability between systems representing different biological phenotypes and stimulus conditions. The work transforms the traditionally vague physiological concept of excitability into a rigorous analytical description applicable across continuous, single compartment models of electrical excitability.

We develop a mathematical model of Rift Valley Fever integrating mosquito vectors, ruminants, and humans, based on an SEIR-type structure with vertical transmission in vectors. Local data from the Sudanian and especially the Sahelian zones are used to capture the impact of climatic variations on mosquito population dynamics. The mathematical analysis establishes the models positivity, determines the basic reproduction number R0, and demonstrates the local and global stability of the disease-free equilibrium. Sensitivity analysis (PRCC) highlights the most influential parameters, while the stochastic approach using a continuous-time Markov chain confirms the major role of seasonal rainfall. Numerical simulations reveal a peak in animal and human infections around the 9th month, correlating with periods of heavy rainfall. This model provides a relevant tool for surveillance and prevention within a "One Health" approach in Chad.

The aim of this paper is to model homeless accommodation settings to investigate how vaccination mitigates the outbreaks, highlighting the importance of vaccination in vulnerable settings. We estimate the daily per capita contact rate with wider community, the internal transmission rate, and the achieved vaccine coverage. We present stochastic simulation of the final size of disease outbreaks given choices of internal and external transmission. We conclude that vaccine that has effect in reducing transmission will mitigate the outbreak in homeless hostels but it will have better results when the household population has large vaccination coverage, which may lead to more cost from the health economic perspective.

This brief work discusses potential superspreading events that may occur during the World Cup in Mexico. The study is particularly focused on the city of Guadalajara due to a large recent outbreak in January and February and insufficient vaccine coverage prior to 2026. Keywords: Superspreading; measles outbreak; branching process; individual reproduction number; World Cup

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.

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.

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.

Coinfection of COVID-19 and malaria in endemic regions may generate complex epidemiological interactions that influence susceptibility patterns, disease burden, and outbreak risk. Although malaria-acquired immunity has been hypothesized to modulate host responses to other infections, its population-level implications for COVID-19 transmission under uncertainty remain insufficiently understood. In this study, we develop a deterministic-stochastic compartmental model for the coupled dynamics of COVID-19, malaria, and their co-infection. Malaria-acquired partial immunity is incorporated through a relative susceptibility parameter that reduces the risk of COVID-19 infection among malaria-recovered individuals. For the deterministic system, we establish positivity, boundedness, an invariant feasible region, and basic reproduction numbers for the COVID-19-only and malaria-only subsystems. We then use numerical simulations to examine how immunity-mediated reductions in susceptibility may influence COVID-19 incidence, peak burden, hospitalization, and cumulative mortality. To account for environmental and transmission variability, we extend the deterministic model to an Ito stochastic differential equation framework and use repeated realizations to characterize uncertainty in epidemic trajectories, peak distributions, and outbreak risk. In addition, global sensitivity analysis based on partial rank correlation coefficients (PRCCs) is performed to identify the parameters with the greatest influence on COVID-19 outcomes. Our results suggest that, under the assumed modeling framework, malaria-acquired partial immunity may reduce the peak infectious burden and cumulative mortality associated with COVID-19. The stochastic simulations further show substantial variability around deterministic trajectories and indicate a non-negligible probability of large outbreak events that are not fully captured by mean-field predictions alone. Overall, the proposed framework provides an uncertainty-aware, mechanistic basis for studying COVID-19-malaria co-dynamics and for assessing how interacting disease processes may shape epidemic outcomes in endemic settings.

In this paper, we propose a new model to estimate the impact of an intervention on human hosts of a vector-borne infection, such as dengue, which occurs in yearly outbreaks of different magnitudes. The model applies to these outbreaks and, in fact, is independent of their intensity, that is, it does not require the steady-state assumption. The model takes as input the officially reported age-dependent number of cases of a vector-borne infection. It is deterministic and does not account for stochasticity. Our objective is to estimate the impact of the intervention (the efficacy), and we rely on the observed fact that the age distribution of the proportion of cases of the infections transmitted by the same vector is independent of both the intensity of transmission and the geographic area studied, at least for Brazilian regions. This finding is highlighted in the main text and forms the basis of our calculations. A hypothetical intervention is simulated using a dengue vaccine, which allows the determination of the optimal strategy for a vaccination campaign.

The basic reproduction number R0 confounds pathogen biology with adaptive human contact behavior. Earlier epidemiological--economic theory predicted a forward-looking behavioral contact response but could not test it in the absence of appropriate behavioral data. Using directly measured mobility as an observable proxy for contact, we (i) estimate the behavioral response function directly from data; (ii) show that the biology/behavior decomposition and hence the behavioral correction to R0 is not identified from an epidemic trajectory, the apparent constant-contact R0 being one endpoint of an observational-equivalence class that fits the factual curve identically yet diverges under counterfactual; and (iii) characterize that divergence ("what R0 deletes") as state-dependent, unimodal in counterfactual severity and vanishing when behavior saturates. We then show that, across US jurisdictions, the correction is empirically bounded because risk-responsiveness and behavioral non-saturation are confounded (r=-0.57, n=51): where behavior could compensate, it was already maximal, and where it was not maximal it did not respond. What R0 deletes is thus real and structurally characterizable yet empirically modest here, for reasons the framework itself supplies.

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

An epidemic's expected course is determined by the magnitude and timing of a typical person's infectiousness --- captured, in turn, by the basic reproduction number and the generation-time distribution. These fundamental, population-average quantities can mask individual-level variation that shapes how an epidemic actually unfolds: for example, individual variation in the magnitude of infectiousness (overdispersion) creates superspreading, a key feature of the SARS-CoV-1 and SARS-CoV-2 epidemics. However, the impact of individual variation in infectiousness timing is less well understood. Here, we demonstrate that individual infectiousness timing varies substantially and to different degrees across pathogens. For some common pathogens, including influenza, measles, and SARS-CoV-2, infectiousness is "bursty", or highly concentrated and variably-timed across individuals: for example, the window of appreciable infectiousness for SARS-CoV-2 may last for roughly a day, vs. the 9--12 days usually quoted. We show that bursty infectiousness creates superspreading without inherent superspreaders, makes epidemic timing more variable, amplifies the time-sensitivity of common interventions, and complicates inference of key epidemiological parameters. Together with the reproduction number, the generation-time distribution, and overdispersion, burstiness completes a family of basic parameters that govern how epidemics unfold.

Viral infection can coactivate interferon (IFN)--JAK/STAT1 signalling and the p53--Mdm2 stress-response pathway, two modules that jointly shape antiviral defence and cell-fate decisions. Here, we focus on viral infection contexts capable of inducing genotoxic stress associated with DNA double-strand breaks, thereby triggering oscillatory or sustained p53--Mdm2 dynamics. Whether p53 acts merely as a parallel stress pathway, or actively reshapes how an activated JAK/STAT1 response is temporally decoded and functionally routed, remains unclear. We develop a coupled ordinary [ndash]differential-equation model linking an IFN{gamma}centred JAK/STAT1 core, a p53--Mdm2 module, downstream antiviral and apoptotic effectors, and a coarse-grained viral-burden layer, with p53 regulation placed downstream of STAT1 activation. We find that p53 does not simply increase nuclear STAT1 availability; it redistributes the response towards DNA-bound STAT1 persistence, transcriptional memory and STAT1-driven feedback, producing a persistence--recovery trade-off in which prior p53 stress prolongs the transcriptionally active STAT1 state but delays re-inducibility after repeated IFN stimulation. When IFN and p53-associated stress are both driven by viral burden, p53 is not a uniform amplifier of host defence: p53 preactivation strengthens the upstream memory layer, but downstream effectors buffer rather than mirror this priming. The model further separates antiviral-state engagement from realised viral control: strong effector activation does not guarantee suppression of poorly sensitive viral classes, whereas sensitive viral classes can be cleared before apoptosis. The origin of the stimulus also matters: exogenous IFN or p53 stimulation allows us to assess the host's intrinsic response capacity, whereas virus-induced IFN and p53 stress remain coupled to viral persistence. Persistent viral burden thus emerges as the dynamical link between IFN induction, p53 stress-memory, antiviral maintenance, viral control and the choice between JAK/STAT--IRF1-associated, p53-autonomous or dual apoptotic routing.

The severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) has continuously evolved since its emergence in the human population in 2019. As of 1st August 2025, more than 1,700 Omicron subvariants have been designated by the Pango nomenclature system. The Pango nomenclature system designates a new lineage based on genetic and epidemiological information of SARS-CoV-2 strains. However, there is a possibility that strains that have similar genetic backgrounds and the same phenotype are given different Pango lineage names. In this paper, we propose a new algorithm, called FindPart-w, which can identify groups of viral lineages that share the same relative effective reproduction numbers. We introduced a new lineage replacement model, called the constrained RelRe model, which constrains groups of lineages to have the same relative effective reproduction numbers. The FindPart-w algorithm searches the equality constraints that minimise the Akaike Information Criterion of constrained RelRe models. Using hypothetical observation count data created by simulation, we found that the FindPart-w algorithm can identify groups of lineages having the same relative effective reproduction number in a practical computational time. Applying FindPart-w to actual real-world data of time-stamped lineage counts from the United States, we found that the Pango lineage nomenclature system may have given different lineage names to SARS-CoV-2 strains even if they have the same relative effective reproduction number and similar genetic backgrounds. In conclusion, this study showed that viruses that had the same relative effective reproduction number were identifiable from temporal count data of viral sequences. These findings will contribute to the future development of lineage designation systems that consider both genetic backgrounds and transmissibilities of lineages.

Abstract Purpose This study introduces a novel approach to address unmeasured confounding in terminal event studies using the prior event rate ratio (PERR) method. The proposed approach PERR_{proxy} used a proxy event to replace the original terminal event in the pre-exposure period, enabling the application of PERR in terminal event settings. Additionally, we also applied difference in difference (DID) regression, which is conceptually analogous to PERR to estimate the standard errors and confidence intervals of PERR_{proxy}. Methods We conducted numeric simulations to evaluate the validity of PERR_{proxy} approach and assessed its performance under varying levels of unmeasured confounding effects, baseline hazard ratios, and the correlation between the proxy and terminal events. To demonstrate its practical applicability, we also performed an empirical analysis to investigate the impact of severe hospitalized COVID-19 on circulatory system disease mortality using the PERR_{proxy}. Results In simulation studies, PERR_{proxy} effectively reduced the unmeasured confounding effects compared to the conventional methods. The performance of PERR_{proxy} was influenced by the strength of unmeasured confounding, baseline hazard ratios, and the correlation between the proxy and terminal outcomes. In addition, difference in difference (DID) regression had much faster computational speed for estimating standard errors and confidence intervals compared to bootstrap. In the empirical analysis, PERR_{proxy} identified that severe hospitalized COVID-19 as a significant risk factor for the circulatory system disease mortality and reduced the unmeasured confounding effects. Conclusions The PERR_{proxy} approach extends the applicability of the original PERR method to terminal event studies, offering a promising solution for addressing unmeasured confounding. Additionally, the DID regression framework provides a computationally efficient alternative for parameter estimation in PERR-based studies. However, careful consideration is still required in PERR_{proxy} for proxy events selection and other underlying assumptions of the PERR method to ensure valid results. Keywords: prior event rate ratio, unmeasured confounding, proxy event, terminal event study, observational study, electronic health records

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

Compartmental models of infectious disease transmission make assumptions about human behaviors. Specifically, they parameterize interactions across population groups, assumed to have distinct epidemiologically-relevant behavioral patterns, primarily through contact matrices stratified by demographic variables such as age, gender, or socioeconomic status. Although such demographic characteristics are readily measurable, they may inadequately capture the social and psychological forces that govern protective behaviors. Drawing on 20 waves of a national survey conducted throughout the COVID-19 pandemic in the United States, we show that institutional trust - particularly trust in public health agencies, physicians, and hospitals - is a dominant predictor of protective behavior adoption. For mask wearing during periods of strongest pandemic activity, for example, institutional trust explains more behavioral variance across population groups than age, income, education, and partisan affiliation combined. In unadjusted analyses, the difference in protective behavior adoption between individuals with the highest and lowest trust in the CDC was four- to six-fold larger than the corresponding differences by age, income, or educational attainment, and exceeded the difference between Democratic and Republican respondents. This association was institutionally specific (e.g., the relationship attenuates for trust in banks), and behaviorally specific (e.g., trust in the CDC is associated with protective behaviors but not visiting a doctor). The latter suggests that trust modifies voluntary compliance with public health recommendations rather than access to or use of healthcare. We conclude that compartmental models of disease transmission would be substantially improved by incorporating institutional trust as a stratifying variable. We additionally offer a trust-integrated mathematical modeling framework and recommendations for the data infrastructure needed for its implementation.

Infectious disease time series often show signs of epidemic transitions, such as the peaks and troughs of the time series. In these time series, key system parameters can lead to catastrophic changes in the dynamical system behaviour (often called critical transitions). Modellers have increasingly shown that early warning signals can anticipate these transitions, both critical and non-critical, in infectious disease time series. Existing methods, however, generally focus on univariate time series data, or ignore spatiotemporal patterns that may be present as a disease spreads through a population. Recent ecological literature developments expand existing temporal and spatial methods to consider the covariance matrix of multiple, related time series. However, many of these proposed signals still make an assumption of stationary time series/system equilibrium. Whilst often true in ecological modelling, disease systems are seldom at equilibrium. In this paper, we propose the usage of the eigendecomposition of the non-stationary covariance matrix as a more suitable early warning signal for epidemiological data. We first analyse the expected trends in the eigenvalues and eigenbasis of the covariance matrix on approach to a transition. Next we apply these methods to a spatially-structured susceptible-infectious-recovered model to explore how the eigenbasis may provide extra information to modellers. Finally, we test these methods on SARS-CoV-2 case data during the 2020-2021 pandemic period in England.