Mathematical Biosciences

Auxins are a family of plant hormones involved in various processes across plant tissues and species. The Nuclear Auxin Pathway (NAP) consists of interacting transcription factors (ARFs) and repressors (Aux/IAAs), which govern an individual cells response to changes in auxin concentration. These components are present in all land plants, and many species possess multiple copies of each signalling component. We present a general framework for ODE-based models of NAP submodules with the flexibility to model the promotion and repression of target genes by any combination of transcriptional regulators. We analyse published data and show that auxin treatment in Arabidopsis thaliana roots triggers a range of characteristically distinct temporal response profiles--for both target genes and the signalling components themselves. Using our modelling framework, we recapitulate aspects of this behaviour by presenting examples of real and theoretical NAP subnetworks, and by analysing the effect that these network dynamics have on auxin-mediated transcriptional responses. This work demonstrates the utility of our modelling framework as a general-purpose tool for understanding the function of certain protein-protein and protein-DNA interactions through their effects on the NAP. This exploration of the rich dynamics of more complex signalling pathways promises to advance our understanding of the NAP.

Persistent viral infections arise from complex interactions between viral replication, host cell responses, and ongoing viral evolution. A general framework linking viral fitness to persistence dynamics is lacking. Here, we develop for the first time a mathematical model of viral persistence that takes into consideration viral fitness variations. Essential to the model is the partition of the classical fitness parameter into three components: replicative, infective, and dispersal fitness. The model was initially inspired by a new experiment on hepatitis C virus (HCV) persistence, established in human hepatoma cells, also reported in this work. This experiment documents two strikingly different viral trajectories depending on the initial replicative fitness of the viral population used to establish persistence. The dynamical model describes the interactions among uninfected cells, infected cells, and infectious virions, and it incorporates, through a continuum, two alternative mechanisms of viral release from cells: budding and lysis. Analysis of the model reveals that viral fitness parameters organise infection outcomes into distinct dynamical regimes. Low replicative and dispersal fitness values lead to viral extinction, whereas high values enable persistence through either stable coexistence or recurrent infection waves. The space of fitness discloses a hierarchy among these parameters, with replicative and dispersal fitness being able to trigger important shifts in the outcome of the infection as opposed to infective fitness. The model identifies trade-offs between replication and dispersal that shape viral production and predicts slow dynamical regimes in which infection may persist despite low detectable viral loads. These dynamical transitions provide candidate mechanisms capable of generating the persistence patterns observed experimentally. Our results establish a computational framework linking multidimensional viral fitness to persistence dynamics and suggest general principles by which evolving RNA viruses transition between extinction (cell curing) and sustained persistence.

Modern medicine implicitly assumes that physiological responses to intervention are predictably determined by administered treatments. However, physiological systems containing intrinsic delays between the detection of a stimulus and the biological response may violate this assumption. We investigate the human glucose-insulin system as described by the Ultradian model and mathematically demonstrate that clinically relevant forcing protocols-such as pulsatile insulin delivery and step-wise glucose infusion, both commonly used in intensive care units (ICUs)-can induce sustained temporal chaos that may hamper accurate prediction of the physiological response. If not accounted for, these chaotic dynamics could create difficulties in achieving optimal dosing and timing when administering glucose and insulin in clinical or home care settings. This phenomenon, termed delay-induced uncertainty (DIU), arises from the interaction between physiological delay, intrinsic shear near a limit cycle, and external forcing. Using the Ultradian glucose-insulin model, we compute top Lyapunov exponents to quantify predictability. Across a range of pulsatile and step-wise forcing regimes, including stochastic amplitudes drawn from Markov processes, we observe positive Lyapunov exponents, indicating sustained chaos. Our results suggest that delayed endocrine regulation may fundamentally limit the predictive value of the models used to develop glycemic management strategies, with implications for clinical protocols in the ICU.

Much of the literature and intuition associated with mathematical epidemiology is driven by deterministic models, which are a reasonable assumption when the population size is large. Stochastic models, especially individual based models, are however considered vital when dealing with small population sizes, especially at times of invasion or extinction. The overwhelming majority of these models (both deterministic and stochastic) assume that the underlying parameters are fixed (or follow a regular seasonal pattern). Here, we consider an analytic framework for dealing with randomly varying parameters through the use of stochastic differential equations - thereby capturing the action of external noisy processes such as weather. In particular, we focus on when the transmission rate, {beta}, varies as the solution to a Cox-Ingersoll-Ross Model, such that {beta} is gamma distributed with autocorrelation. We consider the impact of this parameter variation on a stochastic version of the Susceptible-Infected-Recovered model, and for this 'double-stochastic' model show through simulation and analytical results that exogenous noise increases the impact of stochasticity, potentially leading to more early extinctions, wider variations in the number of cases at equilibrium, but that early growth rate can be faster or slower depending on the precise parameters.

Dengue transmission is shaped by multiple viral serotypes, temporary cross-immunity (TCI), antibody-dependent enhancement (ADE), and repeated exposure in endemic populations. Classical multi-strain models usually assume lifelong protection against reinfection with the same serotype. However, recent evidence suggests that homologous dengue reinfections, although rare, can occur. Their population-level consequences remain poorly understood. We extend a two-infection, two-strain dengue model with TCI and ADE-mediated transmission differences to include homologous reinfections. Homologous reinfection is represented by two exploratory parameters: relative susceptibility to reinfection with the same serotype and relative infectiousness during homologous reinfection. Using equilibrium analysis, bifurcation diagrams, simulations, and phase-space projections, we examine how these parameters affect dengue dynamics and interact with TCI duration and seasonal forcing under intermediate and long TCI durations, with and without seasonality. The extended model shows that qualitative dynamics characteristic of endemic dengue transmission are reproduced mainly when susceptibility to homologous reinfection is low, so that homologous reinfections remain rare but dynamically influential. Longer TCI broadens regions of complex oscillatory dynamics, while seasonality shifts the bifurcation structure and makes torus bifurcations a central route to complex behavior. Although backward bifurcation can occur when homologous susceptibility exceeds the biologically meaningful range, this result should be interpreted as a mathematical mechanism rather than a realistic dengue scenario. These results indicate that rare homologous reinfection pathways can influence long-term dengue dynamics when interacting with immune history, TCI, ADE-mediated transmission differences, and seasonal variation. Incorporating such pathways may improve understanding of recurrent outbreaks and irregular incidence patterns in highly exposed populations.

The global potential of a chemical reaction network has many applications and is closely related to the stochastic detailed balance. However, many fundamental questions concerning stochastic detailed balance remain unresolved, such as whether it depends on the system volume and how to construct new systems that satisfy it. In this paper, we show that stochastic detailed balance may depend on the system volume. We therefore introduce four types of stochastic detailed balance according to their dependence on volume and rate constants, and systematically investigate the relationships among them. Our results distinguish detailed balance arising from particular choices of volume and parameters from that enforced by network structure, and identify conditions under which detailed balance at one volume extends to all volumes. We further obtain a class of networks satisfying stochastic detailed balance for every volume and every positive choice of rate constants, and construct new systems whose global potentials exhibit double-well structures.

Non-exponential residence times give rise to history-dependent or memory effects in epidemic dynamics, yet their mathematical formulations remain fragmented. Here we develop a unified framework for a history-dependent SEIR model with general latent- and infectious-period distributions. We first establish the equivalence between the stage-age and renewal formulations and show that both admit a unified transition-flow representation. We then introduce the concept of a transition-flow multiplier, which provides a direct link between residence-time distributions and transition flows. When the multiplier is a weighted sum of shifted power functions, the model admits an exact representation in terms of Riemann-Liouville or tempered Riemann-Liouville fractional operators, providing a first-principles foundation for fractional epidemic models. Finally, using Euler-Lotka equations and stochastic-order arguments, we characterize how the latent- and infectious-period distributions shape epidemic growth and decay. A longer latent period slows both early growth and late decay, while, at fixed means, greater variability in the latent period accelerates early growth but slows late decay, whereas greater variability in the infectious period slows both.

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.

Host immunity acquired through prior infections is a key factor shaping the spread of emerging variants. However, accounting for complete infection histories in mathematical models is challenging because the number of possible histories grows exponentially with the number of variants. Here, we present a flexible framework for modeling the co-circulation of multiple variants emerging at arbitrary times while incorporating cross-immunity and variant-specific characteristics. We introduce three tractable approximations: truncation, which retains only infection histories up to a given number of strains; one variant at a time, which is exact when variants circulation does not overlap; and two variants at a time, which is exact when at most two strains co-circulate. We systematically evaluate these approximations across a range of cross-immunity levels and inter-variant emergence delays. Truncation approximations perform best under short emergence delays and strong cross-immunity, whereas the other approaches outperform them at longer delays. When applied to SARS-CoV-2 variant dynamics, the second-order truncation and the two variants at a time approximation accurately capture incidence patterns, whereas the other approaches perform well only for specific subsets of variants. These results identify the parameter regimes in which simplifying assumptions remain valid, provide guidance for model selection, and highlight the key role of infection history in shaping epidemic trajectories.

Tumor progression is often modeled as a passive response to external therapy or immune pressure, but tumor populations may also exhibit population-level regulation of proliferation and apoptosis. We develop a continuous-time Markov decision framework in which a controlled birth--death process represents a tumor population modulating the balance between proliferation and susceptibility to apoptosis in the presence of extrinsic death pressure. We examine threshold and quadratic costs, an unbounded linear reward, and constrained linear and quadratic formulations to determine how objective structure shapes optimal policies and induced population drift. Threshold and quadratic penalties generate restoring dynamics, with transitions from growth to suppression and regions of near-neutral drift associated with regulated or near-dormant behavior. An unbounded linear reward instead produces sustained or near-neutral growth without a restoring regime. Under constraints, a linear reward expands the region of positive drift as capacity increases, whereas a quadratic reward generates restoring, logistic-like drift around an interior population scale. These results show that regulated tumor dynamics depend on how growth incentives, extrinsic death pressure, penalties, and constraints scale with population size.

Biological processes, from molecules diffusing to their regulatory destinations to whales pursuing food or mates, are widely exploratory. Such behaviors are effective when some verifiable functional state or outcome can be reached regardless of initial conditions. In these cases, systems repeatedly undergo distinct and abortive trajectories; the process only ends when the system finds 'right' outcomes. This dominance of the terminal condition (and indifference to the initial state) provokes a very different perspective than the conventional 'dynamical systems' framework that has been a centerpiece of the quantitative sciences for centuries. We hypothesize that many of these problems defy the initial-condition driven or gradients on landscapes so useful in problems ranging from mechanics to electrodynamics to chemical kinetics to mass and heat transport. We examine several mathematical frameworks that capture and unify key aspects of exploratory dynamics. One powerful way of thinking of such processes is the geometric distribution, where repeated failures are punctuated by a successful trajectory. We develop intuitions for how random walks with resets accelerate search processes. We highlight fresh and surprising behaviors of search under drift, cues, and checkpoints. Last, appreciating the probability of trajectories conditioned on satisfying macroscopic final outcomes reveals a language for the potency of variation sculpted by selection in their broadest forms. This can give the fictitious appearance of the future making itself known in the present, but we view this as conceptually similar to the way 'fictitious forces' arise in non-inertial reference frames. These approaches stress unity, open questions, and applications across a range of biological phenomena.

Immune checkpoint blockade (ICB) therapy has revolutionized cancer treatment, though it is still effective in only about 20-40% of patients. To increase the efficacy of ICB therapy, combinations of ICB antibodies such as anti-PD1 and anticoagulants (e.g., thrombin inhibitors) have been studied in preclinical mouse models and clinical trials. We developed a Bliss-type analysis to quantify the synergy between anti-PD1 and dabigatran etexilate using published tumor growth data from a mouse model. We then developed a minimal mechanistic computational model to quantitatively study the synergy between anti-PD1 and the thrombin inhibitor dabigatran etexilate in a published study (Metelli et al.) of a preclinical mouse model of colon cancer. Our model included tumor cells, CD8+ T cells, and the pleiotropic cytokine TGF{beta}, whose production in platelets is influenced by thrombin, and described a potential mechanism of interplay among these components in the tumor microenvironment. We performed nonlinear mixed-effects modeling to capture mouse-to-mouse variation in tumor growth under different treatment conditions. The estimated parameter values pointed to several underlying mechanisms of synergy, including increased expansion of CD8+ T cells in the tumor microenvironment in the presence of anti-PD1 and dabigatran etexilate. These predictions can be further validated in future experiments. Thus, the combination of longitudinal tumor growth data, mechanistic population dynamics modeling, and nonlinear mixed-effects modeling can be used to study synergy between ICB and other drugs in controlling tumor growth.

In the face of an epidemic where a population behaviour both influences disease transmission and reacts to it, social processes can generate norms to support socially beneficial behaviour. Most mathematical models of coupled behaviour disease dynamics treat norms as pre-existing rather than explaining how they are maintained. Here, we investigate whether altruistic punishment can sustain a social distancing norm when individuals may defect, cooperate without punishing, or cooperate while paying a cost to punish defectors. We couple a transmission model to an imitation model for these three strategies. Disease prevalence affects behavioural payoffs, while the behavioural composition modifies transmission. We also compare this coupled system with a control where behavioural decisions respond to a fixed prevalence. We find a wide parameter regime corresponding to the establishment of an injunctive social norm in support of social distancing, where the punisher strategy is widespread. Persistence may occur through stable states where punishers or dominant. Disease behaviour feedback can also create oscillations (where the three strategies succeed one another in response to epidemic waves) or tipping points (sharp transitions between all-defector and cooperative states). These effects do not occur in the uncoupled model, although there are still broad parameter regimes where a social norm persists. Our findings show that costly peer punishment can support persistence of social norms that mitigate disease transmission. More broadly, endogenous epidemic feedback can qualitatively change the conditions under which cooperation and punishment are sustained, producing tipping points and long-term behavioural epidemiological cycles that fixed-payoff models cannot capture.

Avian Influenza viruses (AIVs) infect a broad host range despite having a natural reservoir in wild aquatic birds. Whilst most strains stay within their host species, some break the species barrier through genetic adaptations. We are most concerned about zoonotic cases, where a human becomes infected. Despite these events being rare, they are associated with high mortality and introduce the risk of onward human-to-human transmission of AIV. As a novel pathogen within the human population, this could have pandemic potential. Using genetic composition features for 8 AIV proteins drawn from viral sequence data, we employ machine-learning algorithms to classify AIV cases as zoonotic or not. These genetic features encode host 'signatures' which can indicate zoonosis and include frequency measures such as dipeptide composition and amino acid physiochemical properties. We consistently find XGBoost to outperform all other algorithms. We optimise parameters for ten classification models: one for each of the 8 proteins and two combined models. Following this, we show that a multi-model approach gives the best performing prediction for AIV zoonosis. We have identified all 8 proteins as having a role in predicting zoonotic transmission. Of particular importance is the PB2 and HA proteins, with specific amino acid physiochemical properties such as charge, secondary structure and hydrophobicity amongst the most indicative features in our combined models. Our alignment-free computational study can identify AIV cases still within avian hosts which are genetically closest to zoonotic AIV cases, thereby identifying the cases most likely to cross the species barrier. In a resource limited environment, our model could be used to quickly identify high priority cases for further investigation.

In this study, we develop a mathematical model of the hypothalamic pituitary adrenal (HPA) axis and systematically investigate the model complexity required to reproduce observed hormone dynamics. The model is fitted to ACTH and cortisol measurements from five healthy volunteers reported in the literature. We show that simultaneous reproduction of both hormone profiles requires a delay differential equation model with at least five state variables among the model architectures considered. Following fitting the volunteer data, we investigate the response of the model to sustained stress using bifurcation analysis. The coupled system exhibits bifurcations over a range of stress intensities. This analysis reveals bifurcations whose critical stress thresholds depend strongly on the modulatory state of the system. These results provide a mathematical framework for investigating transitions between physiological and dysregulated HPA axis dynamics, which may be related to states of health and disease.

Blooms of the toxic dinoflagellate Ostreopsis cf. ovata can cause respiratory irritation in beach users, yet their environmental drivers and temporal dynamics remain poorly understood. Here, we use a dynamical-systems approach to investigate how environmental conditions shape bloom dynamics from long-term time series of algal abundance and abiotic variables. Using the global modelling framework GPoM, we reconstruct nonlinear ordinary differential equation systems and identify candidate models capable of reproducing key features of the observed dynamics, including chaotic behaviour. Topological and statistical analyses reduce the candidate models to two systems that best capture the observed distributions and dynamical patterns. These models reveal distinct combinations of environmental conditions associated with bloom occurrence and amplitude, providing a framework for interpreting bloom trajectories and a potential basis for short-term prediction.

We propose a minimal mathematical framework to describe the go-or-grow dynamics of tumor cells comprising two phenotypically distinct populations. One population is migratory and undergoes constant diffusion, while the other proliferates in an oxygen-dependent manner. The local oxygen concentration governs transitions between these phenotypes. We then ask whether the time scale of cyclic hypoxia, where oxygen is spatially homogeneous but fluctuates between normoxic and hypoxic levels over time, alters the dynamics of the reduced generalist model in the fast phenotype-switching regime compared with the full go-or-grow model. We observe that the reduced generalist phenotype model fails to capture the go-or-grow dynamics under rapid oxygen fluctuations. Our results predict that the tumor growth rate increases in the generalist-phenotype model under rapid oxygen fluctuations compared with tumors of the proposed go-or-grow model. We then explore whether oxygen consumption, which leads to spatial heterogeneity in oxygen concentration, plays a role in distinguishing the dynamics of the reduced model from the proposed go-or-grow model. We demonstrate that the tumor in the proposed go-or-grow model exhibits a higher growth rate than the tumor in the reduced generalist-phenotype model under a slow oxygen consumption rate.

Microbial growth has often been analyzed under the assumption that microorganisms have evolved to optimize phenotypic characteristics of interest, such as growth rate and growth yield. This assumption has been useful, for example, for the genome-scale modeling of metabolism and the study of the allocation of cellular resources to physiological processes. In many experimental situations of interest, however, microorganisms are found to be suboptimal with respect to phenotypic characteristics that are thought to be favorable in that situation. Whereas a strong theoretical framework exists to mathematically relate cellular resource allocation strategies to Pareto optimality of microbial growth and other biological processes, much less is known about the consequences of Pareto suboptimality. We extend the framework to the latter case and show that a given Pareto suboptimal phenotype can be explained by a range of underlying resource allocation strategies, each corresponding to a different growth physiology and biomass composition. We test the predictions with the help of a coarse-grained model of microbial growth and published experimental data, which relate Pareto suboptimal rate-yield phenotypes of Escherichia coli and the microalga Tisochrysis lutea to the macromolecular composition of the cells (storage metabolite and total protein contents). Changing the focus from Pareto optimality to suboptimality provides interesting leads to exploring the diversity of growth strategies that can support a given phenotype. This change of perspective is of practical interest, because some of the growth physiologies within this range may be important for biotechnological applications

TROP2-targeted antibody-drug conjugates (ADCs) have demonstrated promising clinical activity in triple-negative breast cancer (TNBC) as monotherapies; however, therapeutic benefit varies among patients. Combination strategies pairing TROP2-targeted ADCs with immune checkpoint inhibitors are also being investigated. Elucidating the mechanistic drivers of ADC monotherapy variability and enabling the rational development of combination regimens require computational frameworks that integrate ADC pharmacology with tumor-immune interactions. A quantitative systems pharmacology (QSP) model is presented that incorporates an ADC module into our established immuno-oncology model for TNBC. The module captures ADC and payload pharmacokinetics and pharmacodynamics. TNBC heterogeneity is represented by two tumor cell clones with high and low TROP2 expression, informed by prior characterizations, and differential sensitivity to the ADC payload is incorporated as an intrinsic property of each clone. Although generalizable, the model was applied to the TROP2-targeted ADC sacituzumab govitecan (SG, TRODELVY). A virtual patient cohort was generated using Latin hypercube sampling and calibrated against objective response rate (ORR) data from SG Phase I/II TNBC basket trial. The model predicted an ORR of 33.2% consistent with ASCENT study (NCT02574455). Simulations suggest TROP2-mediated delivery contributes modestly to SG efficacy with tumor exposure driven largely by systemically released SN-38 payload being sufficient to induce cytotoxicity. Tumor heterogeneity emerged as a key determinant of response with ORR increasing as the fraction of payload-sensitive clones increased. Overall, this QSP framework for TROP2-targeted ADCs accounts for TNBC heterogeneity and is extendable to other ADCs and targets enabling interrogation of ADC mechanisms of action in conjunction with tumor-immune interactions.

Non-genetic adaptation enables cancer cells to alter their phenotype under stress without requiring new mutations. However, the mechanisms by which electrical, mechanical, and hypoxic cues combine to shape this process in 3D tissues remain poorly understood. This work presents an agent-based tumor model that integrates vascular oxygen supply, a globally imposed electric field, mechanically mediated crowding and compression cues, phenotype transitions, cell growth, mitosis, death, and inheritance of adaptive memory across division. The simulated tumors exhibit a three-stage trajectory consisting of necrosis onset, transient collapse of live mass, and partial regrowth accompanied by progressive accumulation of adapted cells. Continuous electrical stimulation produces a dose-dependent reduction in live mass while markedly increasing the adapted fraction, with comparatively limited changes in final necrotic burden. This response is strongly conditioned by mechanics and reshapes (and is reshaped by) adaptive capacity. Pulsed stimulation further shows that, in the model, electric field amplitude and temporal schedule jointly determine memory phenomena, phenotypic diversification, and growth recovery. These results show that coupling local oxygen availability, mechanical constraints, electrical forcing, and history-dependent phenotype transitions can generate distinct tissue-level patterns of phenotypic heterogeneity. Both stimulus magnitude and temporal protocol influenced the resulting population structure, suggesting that the history of physical stress may be an important determinant of adaptive dynamics in spatially organized tumor models.