Chaos, Solitons & Fractals

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.

Malaria is still one of the most dangerous causes of morbidity and mortality in tropical and subtropical regions even though it has been actively combated for many years. In 2023, there were approximately 263 million malaria cases and 597,000 deaths from this disease on a global scale, with sub-Saharan Africa being the region most affected by it [24]. Climate factors affect mosquito biology, including their abundance, survival, and biting rates, as well as parasite development, while human awareness plays a crucial role in adopting preventive measures and effective treatments. Despite the progress in both climate- and awareness-based malaria modelings, few studies integrate these factors in one comprehensive model that involves the detailed mechanisms of transmission processes. The current study develops a deterministic climate-driven SEAIR-SEI malaria transmission model that includes the impact of temperature, rainfall, and humidity on mosquito biology and endogenous community awareness. The model was proven to be well-posed by showing the positivity and boundedness of its solution and through the demonstration of the existence and uniqueness of its solution. The malaria-free equilibrium was determined, and the basic reproduction number was calculated using the next-generation matrix method. The model underwent local and global stability analyses to characterise the diseases persistence in the population. Additionally, a normalized forward sensitivity analysis was conducted, revealing the mosquito biting rate as the key force driving malaria transmission. Four time-dependent malaria interventions, namely, long-lasting insecticidal nets, community awareness campaigns, indoor residual spraying, and prompt treatment, were included in the model through optimal control theory and analysed using Pontryagins Maximum Principle. The numerical results for the optimal control problem showed that employing all four interventions leads to the best outcome by decreasing the objective functional value by 88.17%, reducing the total number of infected humans by 92.49%, and minimizing the total number of infectious mosquitoes by 93.87%. Interestingly, combining two interventions, indoor residual spraying, and prompt treatment, also yielded nearly optimal results. Therefore, the designed control strategy can serve as an efficient and affordable framework for malaria control in sub-Saharan Africa.

Many people which are affected by drop foot syndrome, have to face difficulty while walking which leads to pathological gait. This type of syndrome is treated by means of an external artificial stimulation known as functional electric stimulator (FES). In this paper we are designing an online feedback control system which optimize the strength of a FES given to paretic muscle which results in correction of pathological gait of the patient in a tolerable domain. Different phases of gait are identified using inertial measurement unit (IMU) as a feedback sensor mounted on the foot. Data is collected form 8 different healthy subjects and average of collected data is used as a reference template. Different trajectories of drop foot patients are simulated (due to unavailability of patients) and corrected according to the reference template.

Finite element (FE) head models are virtual tools to study brain biomechanics and their predictions must be numerically convergent. Previous convergence studies focused on the influence of mesh size, but the potential effect of element formulation on model convergence was often ignored. To address this, one original model with brain mesh size as 6.4 {+/-} 1.9 mm was modified to generate three derivatives with the same mesh topology but different element sizes, i.e., a coarse model (mesh size: 12.2 {+/-} 3.9 mm), a medium model (mesh size: 3.2 {+/-} 1.0 mm), and a fine model (mesh size: 1.6 {+/-} 0.5 mm). Three commonly used element formulations, i.e., reduced integration, selectively reduced (S/R) integration, and full integration, were implemented to the brain elements. These models were subjected to rotational loadings along the axial, coronal, and sagittal axes, respectively. The maximum relative displacement at representative sites and 95th percentile maximum principal strain at the whole brain level were used to evaluate mesh convergency. The results showed that the S/R integration yielded a 5% difference between the original and medium meshes, while the reduced and full integration revealed a difference over 5% even between the medium and fine meshes. This study verified that the mesh convergence of FE brain models is affected by the choice of element formulation and the S/R integration contributes to the fastest convergence behavior than the reduced and full integrations. It provided practical information on how to develop numerically convergent and computationally efficient FE brain models. HighlightsO_LIThis study verifies that the choice of element formulation affects the mesh convergence behavior of finite element brain models C_LIO_LIThis study finds the selectively reduced integration yields the fastest convergence behavior than the reduced and full integration C_LIO_LIThis study provides practical guidance on the choice of mesh density and element formulation on how to develop numerically convergent and computationally efficient finite element brain models. C_LI

We have studied the emergence of slow relaxation oscillations in next generation neural mass models with spike frequency adaptation. Relaxation oscillations connect low firing state (Down state) to high firing state (Up state) via the slow adaptation. In the examined cases, the orbit relaxes towards the Up State via a sequence of collective damped oscillations (peaks of activity), thus revealing population bursting dynamics. The slower is the adaptation time scale the higher is the complexity (number of peaks) displayed by the relaxation oscillations. In particular, a chaos-induced spike-adding mechanism regulates the increase in the number of peaks. In analogy to what found in the Hidmarsh-Rose neuron model, two different types of chaotic behaviors have been identified: Population Spiking and Population Bursting Chaos. The increase of the adaptation strength leads to shorter (longer) Up (Down) state durations somehow mimicking the effect of charbachol in in vitro experiments, where spontaneous slow waves are observed. Indeed, the scenario depicted in [1], where an increase of the concentration of carbachol induces a transition from anesthesia-like to sleep-like dynamics is consistent with our results based on the variation of the adaptation strength. HighlightsO_LISpike Frequency Adaptation (SFA) promotes the emergence of Slow Relaxation Oscillations C_LIO_LISpike-adding mechanisms, controlled by SFA, lead to Relaxation Oscillations of increasing complexity C_LIO_LITwo types of chaotic behaviours: Population Spiking and Population Bursting Chaos C_LIO_LISFA regulates Up and Down States durations and their correlation C_LI

Violent conflicts increasingly involve multiple armed actors competing for influence over shared civilian populations, creating complex dynamics that challenge conventional security analysis and policy design. We present a framework that adapts epidemiological methods informed by the conflict landscape in Nigeria to model multi-actor violent conflict as an epidemic process. We derive a basic insecurity reproduction number ($R_0$), identify violence-free and persistent-violence equilibria, and introduce a novel Civilian Harm Index (CHI) to quantify humanitarian impact. Sensitivity analyses identify recruitment, ideological support from civilian populations, and abduction as the key drivers of conflict persistence and civilian harm. The framework reveals several counterintuitive findings. Interventions that most effectively suppress violence transmission are not necessarily those that minimise civilian harm, demonstrating that epidemic control and humanitarian protection may require distinct optimisation criteria. Likewise, interventions effective against one armed actor may be ineffective, or even counterproductive, when applied uniformly across groups. In addition, prisoner exchange and ransom payments increase violence persistence and civilian harm. Although developed as an illustrative rather than predictive framework, our results show that epidemiological methods provide quantitative metrics for evaluating intervention priorities and trade-offs in complex multi-actor conflicts.

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 bifurcations, the approach focuses on the geometry of membrane potential trajectories. The changes in concavity during the upstroke of an action potential can be directly obtained from a time series of voltages. The concavity criterion is then extended to models based on autonomous dynamical systems where the changes in concavity can be obtained analytically from a curve of inflection points in phase space. The inflection point manifold defines a region required for excitability: all the orbits that cross it contain action potentials, and all the trajectories that contain action potentials are in it. This analytical principle can then be used to define excitability in a dynamical system, and also a measure of excitability that enables quantification and comparisons of excitability across dynamical system. The measure provides a way to compare the excitabilities of systems that model neurons with different electrophysiological phenotypes and consider different stimulus conditions. The traditionally vague physiological concept of electrical excitability is transformed into a rigorous analytical description by considering the time-dependent curvature of the membrane potential. The criterion is robust across smooth, single compartment models of electrical excitability and can be can be extended to single compartment models in higher dimensions, and multicompartment models as well.

Classification of motor imagery (MI) tasks through EEG is valuable in brain-computer interfacing and rehabilitation engineering. EEG channels selection for MI task classification is well discussed problem and is challenging due to its combinatorial nature. Most of the existing methods are subject and task-dependent. This paper introduces a subject-independent EEG channel selection. The proposed approach consists of two stages. First, we rank channels based on their divergence from a reference channel Cz. We hypothesize that channels less divergent from Cz are more relevant for MI task classification. In the second stage, we employ a three-stage feature selection and classification model to evaluate the selected channels. It consists of a bandpass filter, followed by common spatial pattern (CSP) filter and three classifiers viz. SVM, 1-NN and 5-NN. Two publicly available datasets viz. PhysioNet and BCI Competition III IVa datasets have been used to assess the method. It performs 15.21\% more than 3Cs and just 2.91\% less than all-channels accuracy with as few as 20/118 channels on BCI Competition data and 19.64\% more than 3Cs on the PhysioNet dataset with 16/64 channels. Empirical comparison implies that the method performs better than classical models such as CSP Rank, fishers rank, and normalized mutual information, significantly. Results support that our hypothesis that divergence between channels and a reference channel Cz can be used as a ranking measure for channel selection.

Human Papillomavirus (HPV) infection among men who have sex with men (MSM) has become a significant public health concern, particularly in countries where male vaccination is unavailable. Given the high susceptibility of MSM to HPV and anal cancer, and the unavailability of HPV vaccination for males in low- and middle-income countries (LMICs), there is a need to identify alternative interventions for reducing disease transmission and burden in this population. The novel mathematical model presented in this article couples smoking behavior dynamics with HPV transmission and anal cancer progression among MSM. Smoking reduction is introduced as an intervention to assess its effects on disease transmission and burden. The basic reproduction number (R0) is derived using the next-generation matrix method, and a global sensitivity analysis is performed using partial rank correlation coefficients (PRCC) to identify the influence of model parameters on RR0. Further, the theoretical analysis of the model reveals a backward bifurcation, implying that RR0 < 1 is necessary but not sufficient to eradicate the disease. The study finds that smoking reduction among MSM reduces HPV infection and anal cancer burden relative to baseline projections without intervention. The joint effect of smoking reduction and vaccination shows that the critical vaccination coverage needed to achieve RR0 <1 decreases as the level of smoking reduction increases. A similar outcome is observed for contact reduction. These findings highlight the importance of concurrent interventions, which can significantly curtail the spread of HPV and reduce disease burden in both the high-risk group and the general population.

Background: Globally and in Bangladesh, high-risk fertility behaviour (HRFB) continues to be a significant public health issue, contributing to negative health outcomes for both mothers and children. So, this study aimed to evaluate the trends, prevalence, determinants, and socio-economic inequalities associated with HRFB among presently married women of reproductive age by utilising data from the Bangladesh Demographic and Health Survey (BDHS). Methods: We analysed data from 19,060 currently married women aged 15-49 years. HRFB was defined as the presence of any of the following: maternal age (<18 or >34 years), short birth intervals (<24 months), or high birth order ([&ge;]4). Two outcome variables were constructed: a binary indicator for any HRFB (yes/no) and a three-category variable indicating no, single, or multiple HRFBs. Bivariate analysis was conducted to determine the prevalence of HRFB, and multilevel mixed-effect logistic and multinomial regression models were applied to identify determinants, accounting for the complex survey design. Socioeconomic inequalities were examined using concentration indices and concentration curves. Results: Overall, 58% of women experienced at least one HRFB, with 33% exhibiting a single HRFB and 25% multiple HRFBs. Women who married after age 18 years, had higher education, were exposed to media, or had educated husbands were significantly less likely to experience HRFB. Higher odds of HRFB were associated with rural residence, lower household wealth, and regions such as Mymensingh, Barishal, and Chattogram. Significant inequalities were observed, with HRFB disproportionately concentrated among women with lower wealth (CIX = -0.427, p<0.001) and no education (CIX = -0.268, p<0.001). Conclusion: In Bangladesh, HRFB remains prevalent and is unevenly distributed across socio-economic and geographic groups. Targeted interventions aimed at delaying early marriage, improving educational attainment for women and their partners, expanding mass media outreach, and increasing access to reproductive healthcare- especially among disadvantaged and rural populations- are essential for reducing HRFB and improving maternal health outcomes.

This paper introduces a sensorimotor learning framework for a corticocerebellar network, grounded in the perspective of population dynamics. Using an optimal control theory approach, the cerebellum model enhances preparatory activity through premotor input, allowing the motor cortex to reach the desired initial conditions for movement more efficiently. Unlike traditional motor learning approaches that focus on acquiring new skills, this paradigm emphasizes automatization of already executable behaviors through repetition driven by intrinsic motivation. The proposed model is evaluated using a center-out reaching task, demonstrating that the role of the cerebellum is to shorten the preparatory period required for the successful execution of the movement. These findings suggest that corticocerebellar interactions play a crucial role in optimizing motor preparation, offering insight into the neural mechanisms underlying movement efficiency.

We present a quantum-enhanced version of the classic k-Nearest Neighbors (kNN) classification algorithm, applied to the prediction of arterial hypertension. The traditional Euclidean distance metric of the kNN algorithm is replaced with a Fidelity-derived quantum dissimilarity measure to evaluate the similarity between data samples. We map classical real-world clinical and ECG-derived data features into quantum states via the Dense-Angle Encoding, which efficiently utilizes parameterized rotation gates to pack multiple features into minimal qubits while maintaining pure states. We evaluate the performance of the dissimilarity measure using both the noiseless state vector Simulator and the IBM Qiskit Estimator primitives. The quantum circuit demonstrates robust predictive capabilities comparable to the classical model. While it does not claim computational supremacy over the classical baseline, the framework proves that fidelity-based similarity is a physically meaningful and efficient approach for hybrid quantum classical classification.

Classical neuronal cable theory relies on quasi-static electric field approximations and neglects magnetic induction, Lorentz force coupling, and transient electromagnetic currents, limiting its ability to fully characterize action potential propagation within geometrically branched axons and dendrites. This work develops a coupled Maxwell-electromagnetic cable framework by integrating finite-difference time-domain (FDTD) solutions of Maxwells equations with extended Hodgkin-Huxley and Fitzhugh-Nagumo membrane dynamics, incorporating magnetic gating perturbations, electromagnetic trans-membrane currents IEM, and nanoscale quantum corrections for thin neural segments. Controlled propagation experiments are designed to quantify deviations from standard cable predictions across asymmetric and symmetric axonal bifurcation geometries. Numerical results demonstrate that inductive magnetic effects lower the critical branch radius for junction conduction failure and break symmetric action potential invasion in geometrically identical child branches under external transverse magnetic fields. An electromagnetic corrected geometric ratio GREM is proposed to revise impedance-matching conditions at branch points, accounting for size-dependent axial current imbalance induced by magnetic and displacement currents. Parent axon conduction velocity deviates substantially from the canonical [Formula] scaling law when electromagnetic feedback and quantum charge distributions are included, triggering early signal blockage at large cable diameters. Collectively, this study establishes that quasi-static cable models underestimate electromagnetic corrections to propagation speed, waveform shape, and bifurcation transmission fidelity; the coupled Maxwell-cable framework provides a comprehensive multi-physics tool for modeling electrodynamic signal behavior in complex neuronal architectures.

Repetitive transcranial magnetic stimulation (rTMS) is an established treatment for major depressive disorder (MDD), yet variability in treatment response remains a significant challenge. Network control theory provides a framework to quantify how brain networks facilitate state transitions, but prior work has focused primarily on node level metrics. Here, I investigate whether edge based controllability of the structural connectome is associated with rTMS outcomes. Twenty five patients with treatment-resistant depression underwent diffusion MRI prior to a five week course of high frequency rTMS targeting the dorsolateral prefrontal cortex. Structural connectomes were constructed using MRtrix3 and the Destrieux atlas, and edge based controllability metrics were computed at baseline. Controllability of specific middle frontal gyrus centered edges showed significant associations with changes in HAMD-24 scores, including connections to the superior frontal gyrus, hippocampus, angular gyrus, and orbital gyrus (r = 0.470-0.597, p < 0.05). These findings suggest that edge based controllability captures circuit level properties relevant to treatment response and may inform personalized neuromodulation strategies.

Approximately 50% of the population in India is estimated to experience sleep-related disorders. Sleep deprivation is a prevalent condition that adversely impacts cognitive performance, neural functioning, and overall health. Electroencephalography (EEG) offers an objective means of capturing neural alterations associated with sleep loss, making it well-suited for automated detection frameworks. In this study, we explore the application of a Quantum Support Vector Machine and Hybrid Quantum Neural Networks to classify sleep-deprived and well-rested states using resting-state EEG signals. A comprehensive feature extraction pipeline is employed, incorporating spectral band power, band ratios, Hjorth parameters, and functional connectivity measures. These features are subsequently encoded into quantum states to construct a quantum kernel, which is then utilized for classification. Model performance is evaluated under both epoch-level and subject-level data partitioning schemes. The Hybrid Quantum Neural Network (HQNN) achieves the highest performance across both evaluation settings, attaining an accuracy of 96.88% at the epoch level and 81.25% at the subject level. The QSVM model achieves accuracies of 93.75% and 75.00% for epoch-level and subject-level evaluations, respectively. At subject-level and epoch -level evaluation, HQNN outperforms previously reported results (68.23% and 95.72%). Overall, these findings highlight the potential of quantum machine learning as a competitive approach for EEG-based sleep deprivation detection, with promising implications for real-world biomedical applications.

A fundamental biological process of transcription occurs in the cell nucleus, which is a complex medium that also contains multiple heterogeneous structures known as biomolecular condensates. Interestingly, some of these condensates contain mRNA molecules in addition to proteins, suggesting an important cellular role in transcription that is not yet well understood. In this work, we develop a minimal theoretical framework for quantitative investigation of the role of reversible mRNA condensation in transcription. Our discrete-state stochastic approach accounts for the most relevant processes, allowing us to explicitly evaluate the properties of the system and clarify the effects of condensation. Analytical calculations supported by computer simulations suggest that reversible mRNA condensation influences the transcription processes by maintaining a constant level of free mRNA in the nucleoplasm while lowering the degree of stochastic noise and increasing the robustness against external perturbations. Physicochemical arguments are presented to explain these observations. The proposed theoretical framework elucidates important microscopic aspects of transcription, providing a convenient quantitative tool for investigating complex biological phenomena.

Diffuse gliomas are deadly because the individual tumor cells invade - they travel far from the imageable mass, so it is impossible to remove the tumor completely. On the cellular level, glioma cells seem to be in either a "go" state (in which they do not divide) or a "grow" state (in which they do not migrate). We investigate what this tiny choice has to say about the large-scale speed of the invasion front and whether the implication is sufficiently strong to rule out the classical description of the Fisher-Kolmogorov-Petrovsky-Piskunov (Fisher-KPP) type, in which a single phenotype migrates and proliferates. We derive a two-phenotype reaction-diffusion model with density-dependent switching, and we prove the cooperative (quasi-monotone) structure and the associated comparison principle and study travelling-wave solutions of the model. A leading-edge linearization gives minimal front speed as minimizer of an explicit dispersion relation, and direct simulation verifies the predicted speed. In the experimentally relevant fast switching limit, we find a closed-form expression for the speed, that is, we obtain an effective Fisher-KPP equation with rescaled diffusivity and growth rate, with the fractions of the phenotypes. The "go-or-grow" (GoG) front can move at a maximum speed of half the Fisher speed for the same single-cell motility $D$ and proliferation rate $r$, which occurs only when the cells divide their time equally between the two phenotypes. This bound is directly testable: measurement of the front speed, plus independent determination of $D$ and $r$, discriminates the two hypotheses, and in the GoG case, yields recovery of the phenotype balance. We then extend the result to anisotropic (DTI-informed) invasion along white-matter tracts and discuss implications for understanding clinical measurements of growth rate.

Beta-band (13-30 Hz) oscillations are increasingly understood to occur as transient "bursts" rather than sustained rhythms, with altered burst dynamics, specifically increased duration and power alongside reduced burst rates, in patients with Parkinsons disease (PD). In this study, we utilise resting state magnetoencephalography (MEG) data from healthy adults to quantify the temporal fluctuations in the beta-band, and examine the distributions of burst statistics. We then fit a stochastic next-generation neural mass model to these empirical statistics using a Genetic Algorithm. Systematic parameter sweeps reveal that reducing background drive to excitatory and inhibitory neuronal populations reproduces the altered burst statistics observed in PD. Crucially, we show that strengthening synaptic coupling can counteract these deficits and restore healthy bursting dynamics. Together, this work establishes a computational framework linking cellular-level mechanisms to macroscale burst statistics, and highlights potential targets for therapeutic neuromodulation in movement disorders. Author summaryBrain activity is comprised of rhythmic electrical patterns called "brain waves." Traditionally, these waves were viewed as smooth and continuous, but recent evidence reveals that they actually occur in brief, intense bursts. In conditions such as Parkinsons disease, these bursts become altered--lasting longer, growing stronger, and occurring less frequently. In this study, we developed a mathematical model of brain tissue to understand what drives these burst patterns. Using real brain scans from healthy human volunteers, we tuned our model with an optimisation algorithm until its simulated bursts closely matched real human brain activity. We then systematically varied the models settings to investigate how abnormal bursting arises in disease. We discovered that reducing the background signals to the brain cells reproduces the burst alterations seen in Parkinsons disease. Importantly, our simulations showed that strengthening the connections between brain cells can counteract this deficit, restoring healthy burst patterns. By connecting microscopic cell properties to whole-brain rhythms, our work offers new insights into how movement disorders disrupt brain networks and highlights potential cellular targets to guide future brain stimulation therapies or medications.

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 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.