The parameter region of multistationarity of a reaction network contains all the parameters for which the associated dynamical system exhibits multiple steady states. Describing this region is challenging and remains an active area of research. In this paper, we concentrate on two biologically relevant families of reaction networks that model multisite phosphorylation and dephosphorylation of a substrate at n sites. For small values of n, it had previously been shown that the parameter region of multistationarity is connected. Here, we extend these results and provide a proof that applies to all values of n. Our techniques are based on the study of the critical polynomial associated with these reaction networks together with polyhedral geometric conditions of the signed support of this polynomial.

In this article, we present a computational study of the Conductance-Based Adaptive Exponential (CAdEx) integrate-and-fire neuronal model, focusing on its multiple timescale nature, and on how it shapes its main dynamical regimes. In particular, we show that the spiking and so-called delayed bursting regimes of the model are triggered by discontinuity-induced bifurcations that are directly related to the multiple-timescale aspect of the model, and are mediated by canard solutions. By means of a numerical bifurcation analysis of the model, using the software package COCO, we can precisely describe the mechanisms behind these dynamical scenarios. Spike-increment transitions are revealed. These transitions are accompanied by a fold and a period-doubling bifurcation, and are organised in parameter space along an isola periodic solutions with resets. Finally, we also unveil the presence of a homoclinic bifurcation terminating a canard explosion which, together with the presence of resets, organises the delayed bursting regime of the model.

Drugs exhibiting nonlinear pharmacokinetics hold significant importance in drug research and development. However, evaluating drug exposure accurately is challenging with the current formulae established for linear pharmacokinetics. This article aims to investigate the steady-state drug exposure for a one-compartment pharmacokinetic (PK) model with sigmoidal Hill elimination, focusing on three key topics: the comparison between steady-state drug exposure of repeated intravenous (IV) bolus ( ) and total drug exposure after a single IV bolus ( ); the evolution of steady-state drug concentration with varying dosing frequencies; and the control of drug pharmacokinetics in multiple-dose therapeutic scenarios. For the first topic, we established conditions for the existence of , derived an explicit formula for its calculation, and compared it with . For the second, we identified the trending properties of steady-state average and trough concentrations concerning dosing frequency. For the third, we developed formulae to compute dose and dosing time for both regular and irregular dosing scenarios. As an example, our findings were applied to a real drug model of progesterone used in lactating dairy cows. In conclusion, these results provide a theoretical foundation for designing rational dosage regimens and conducting therapeutic trials.

In vivo observations show that oxygen levels in tumours can fluctuate on fast and slow timescales. As a result, cancer cells can be periodically exposed to pathologically low oxygen levels; a phenomenon known as cyclic hypoxia. Yet, little is known about the response and adaptation of cancer cells to cyclic, rather than, constant hypoxia. Further, existing in vitro models of cyclic hypoxia fail to capture the complex and heterogeneous oxygen dynamics of tumours growing in vivo. Mathematical models can help to overcome current experimental limitations and, in so doing, offer new insights into the biology of tumour cyclic hypoxia by predicting cell responses to a wide range of cyclic dynamics. We develop an individual-based model to investigate how cell cycle progression and cell fate determination of cancer cells are altered following exposure to cyclic hypoxia. Our model can simulate standard in vitro experiments, such as clonogenic assays and cell cycle experiments, allowing for efficient screening of cell responses under a wide range of cyclic hypoxia conditions. Simulation results show that the same cell line can exhibit markedly different responses to cyclic hypoxia depending on the dynamics of the oxygen fluctuations. We also use our model to investigate the impact of changes to cell cycle checkpoint activation and damage repair on cell responses to cyclic hypoxia. Our simulations suggest that cyclic hypoxia can promote heterogeneity in cellular damage repair activity within vascular tumours.

Orchard and tree-child networks share an important property with phylogenetic trees: they can be completely reduced to a single node by iteratively deleting cherries and reticulated cherries. As it is the case with phylogenetic trees, the number of ways in which this can be done gives information about the topology of the network. Here, we show that the problem of computing this number in tree-child networks is akin to that of finding the number of linear extensions of the poset induced by each network, and give an algorithm based on this reduction whose complexity is bounded in terms of the level of the network.

There is growing empirical evidence that animal movement patterns depend on population density. We investigate travelling wave solutions in reaction-diffusion models of animal range expansion in the case that population diffusion is density-dependent. We find that the speed of the selected wave depends critically on the strength of diffusion at low density. For sufficiently large low-density diffusion, the wave propagates at a speed predicted by a simple linear analysis. For small or zero low-density diffusion, the linear analysis is not sufficient, but a variational approach yields exact or approximate expressions for the speed and shape of population fronts.

Chronic spontaneous urticaria (CSU) is a typical example of an intractable skin disease with no clear cause and significantly affects daily life of patients. Because CSU is a human-specific disease and lacks proper animal model, there are many questions regarding its pathophysiological dynamics. On the other hand, most clinical symptoms of urticaria are notable as dynamic appearance of skin eruptions called wheals. In this study, we explored dynamics of wheal by dividing it into three phases using a mathematical model: onset, development, and disappearance. Our results suggest that CSU onset is critically associated with endovascular dynamics triggered by basophils positive feedback. In contrast, the development phase is regulated by mast cell dynamics via vascular gap formation. We also suggest a disappearance mechanism of skin eruptions in CSU through an extension of the mathematical model using qualitative and quantitative comparisons of wheal expansion data of real patients with urticaria. Our results suggest that the wheal dynamics of the three phases and CSU development are hierarchically related to endovascular and extravascular pathophysiological networks.

The observed time evolution of a population is well approximated by a logistic growth function in many research fields, including oncology, ecology, chemistry, demography, economy, linguistics, and artificial neural networks. Initial growth is exponential, then decelerates as the population approaches its limit size, i.e., the carrying capacity. In mathematical oncology, the tumor carrying capacity has been postulated to be dynamically evolving as the tumor overcomes several evolutionary bottlenecks and, thus, to be patient specific. As the relative tumor-over-carrying capacity ratio may be predictive and prognostic for tumor growth and treatment response dynamics, it is paramount to estimate it from limited clinical data. We show that exploiting the logistic function's rotation symmetry can help estimate the population's growth rate and carry capacity from fewer data points than conventional regression approaches. We test this novel approach against published pan-cancer animal and human breast cancer data, achieving a 30% to 40% reduction in the time at which subsequent data collection is necessary to estimate the logistic growth rate and carrying capacity correctly. These results could improve tumor dynamics forecasting and augment the clinical decision-making process.

The two main components of the planktonic ecosystem are phytoplankton and zooplankton. Fungal parasites can infect zooplankton and spread between them. In this paper, we construct a dynamic model to describe the spread of fungal parasites among zooplankton. Basic reproduction number for fungal parasite transmission among zooplankton are rigorously derived. The dynamics of this system are analyzed including dissipativity and equilibria. We further explore the effects of ecological factors on population dynamics and the relationship between fungal parasite transmission and phytoplankton blooms. Interestingly, our theoretical and numerical results indicate that a low-light or oligotrophic aquatic environment is helpful in mitigating the transmission of fungal parasites. We also show that fungal parasites on zooplankton can increase phytoplankton biomass and induce blooms.

The mechanism of cytoplasmic incompatibility (CI) is important in the study of Wolbachia invasion in wild mosquitoes. Su et al. (Bull Math Biol 84(9):95, 2022) proposed a delay differential equation model by relating the CI effect to maturation delay. In this paper, we investigate the dynamics of this model by allowing the same density-dependent death rate and distinct density-independent death rates. Through analyzing the existence and stability of equilibria, we obtain the parameter conditions for Wolbachia successful invasion if the maternal transmission is perfect. While if the maternal transmission is imperfect, we give the ranges of parameters to ensure failure invasion, successful invasion and partially suppressing, respectively. Meanwhile, numerical simulations indicate that the system may exhibit monostable and bistable dynamics when parameters vary. Particularly, in the bistable situation an unstable separatrix, like a line, exists when choosing constant functions as initial values; and the maturation delay affects this separatrix in an interesting way.

Continuous-time predator-prey models admit limit cycle solutions that are vulnerable to the phenomenon of phase-sensitive tipping (P-tipping): The predator-prey system can tip to extinction following a rapid change in a key model parameter, even if the limit cycle remains a stable attractor. In this paper, we investigate the existence of P-tipping in an analogous discrete-time system: a host-parasitoid system, using the economically damaging forest tent caterpillar as our motivating example. We take the intrinsic growth rate of the consumer as our key parameter, allowing it to vary with environmental conditions in ways consistent with the predictions of global warming. We find that the discrete-time system does admit P-tipping, and that the discrete-time P-tipping phenomenon shares characteristics with the continuous-time one: Both require an Allee effect on the resource population, occur in small subsets of the phase plane, and exhibit stochastic resonance as a function of the autocorrelation in the environmental variability. In contrast, the discrete-time P-tipping phenomenon occurs when the environmental conditions switch from low to high productivity, can occur even if the magnitude of the switch is relatively small, and can occur from multiple disjoint regions in the phase plane.

Evolution of dispersal is a fascinating topic at the intersection of ecology and evolutionary dynamics that has generated many challenging problems in the analysis of reaction-diffusion equations. Early results indicated that lower random diffusion rates are generally beneficial. However, in riverine environments with downstream drift, high diffusion may be optimal, depending on downstream boundary conditions. Most of these results were obtained from modeling a single river reach, yet many rivers form intricate tree-shaped networks. We study the evolution of dispersal on a metric graph representing the simplest such possible network: two upstream segments joining to form one downstream segment. We first show that the shape of the positive steady state of a single population depends crucially on the geometry of the network, here considered as the relative length of the three segments. We then study the evolution of dispersal by considering the possibility of "invasion" of a second type (invader) at the steady state of the first type (resident). We show that the geometry of the network determines whether higher or intermediate dispersal is favored.

There is extensive evidence that network structure (e.g., air transport, rivers, or roads) may significantly enhance the spread of epidemics into the surrounding geographical area. A new compartmental modeling framework is proposed which couples well-mixed (ODE in time) population centers at the vertices, 1D travel routes on the graph's edges, and a 2D continuum containing the rest of the population to simulate how an infection spreads through a population. The edge equations are coupled to the vertex ODEs through junction conditions, while the domain equations are coupled to the edges through boundary conditions. A numerical method based on spatial finite differences for the edges and finite elements in the 2D domain is described to approximate the model, and numerical verification of the method is provided. The model is illustrated on two simple and one complex example geometries, and a parameter study example is performed. The observed solutions exhibit exponential decay after a certain time has passed, and the cumulative infected population over the vertices, edges, and domain tends to a constant in time but varying in space, i.e., a steady state solution.

Cancers are typically fueled by sequential accumulation of driver mutations in a previously healthy cell. Some of these mutations, such as inactivation of the first copy of a tumor suppressor gene, can be neutral, and some, like those resulting in activation of oncogenes, may provide cells with a selective growth advantage. We study a multi-type branching process that starts with healthy tissue in homeostasis and models accumulation of neutral and advantageous mutations on the way to cancer. We provide results regarding the sizes of premalignant populations and the waiting times to the first cell with a particular combination of mutations, including the waiting time to malignancy. Finally, we apply our results to two specific biological settings: initiation of colorectal cancer and age incidence of chronic myeloid leukemia. Our model allows for any order of neutral and advantageous mutations and can be applied to other evolutionary settings.

The paranasal sinuses are a group of hollow spaces within the human skull, surrounding the nose. They are lined with an epithelium that contains mucus-producing cells and tiny hairlike active appendages called cilia. The cilia beat constantly to sweep mucus out of the sinus into the nasal cavity, thus maintaining a clean mucus layer within the sinuses. This process, called mucociliary clearance, is essential for a healthy nasal environment and disruption in mucus clearance leads to diseases such as chronic rhinosinusitis, specifically in the maxillary sinuses, which are the largest of the paranasal sinuses. We present here a continuum mathematical model of mucociliary clearance inside the human maxillary sinus. Using a combination of analysis and computations, we study the flow of a thin fluid film inside a fluid-producing cavity lined with an active surface: fluid is continuously produced by a wall-normal flux in the cavity and then is swept out, against gravity, due to an effective tangential flow induced by the cilia. We show that a steady layer of mucus develops over the cavity surface only when the rate of ciliary clearance exceeds a threshold, which itself depends on the rate of mucus production. We then use a scaling analysis, which highlights the competition between gravitational retention and cilia-driven drainage of mucus, to rationalise our computational results. We discuss the biological relevance of our findings, noting that measurements of mucus production and clearance rates in healthy sinuses fall within our predicted regime of steady-state mucus layer development.

The kinetics of intravenous (IV) fluid therapy and how it affects the movement of fluids within humans and animals is an ongoing research topic. Clinical researchers have in the past used a mathematical model adopted from pharmacokinetics that attempts to mimic these kinetics. This linear model is based on the ideas that the body tries to maintain fluid levels in various compartments at some baseline targets and that fluid movement between compartments is driven by differences between the actual volumes and the targets. Here a nonlinear pressure-based model is introduced, where the driving force of fluid movement out of the blood stream is the pressure differences, both hydrostatic and oncotic, between the capillaries and the interstitial space. This model is, like the linear model, a coarse representation of fluid movement on the whole body scale, but, unlike the linear model, it is based on some of the body's biophysical processes. The abilities of both models to fit data from experiments on both awake and anesthetized cats was analyzed. The pressure-based model fit the data better than the linear model in all but one case, and was deemed statistically significantly better in a third of the cases.

Dispersal of propagules (seeds, spores) from a geographically isolated habitat into an uninhabitable matrix can play a decisive role in driving population dynamics. ODE and integrodifference models of these dynamics commonly feature a "dispersal success" parameter representing the average proportion of dispersing propagules that remain in viable habitat. While dispersal success can be estimated by empirical measurements or by integration of dispersal kernels, one may lack resources for fieldwork or details on dispersal kernels for numerical computation. Here we derive simple upper bounds on the proportion of propagule loss-the complement of dispersal success-that require only habitat area, habitat perimeter, and the mean dispersal distance of a propagule. Using vector calculus in a probabilistic framework, we rigorously prove bounds for the cases of both symmetric and asymmetric dispersal. We compare the bounds to simulations of integral models for the population of Asclepias syriaca (common milkweed) at McKnight Prairie-a 14 hectare reserve surrounded by agricultural fields in Goodhue County, Minnesota-and identify conditions under which the bounds closely estimate propagule loss.

Computational models of tumor growth are valuable for simulating the dynamics of cancer progression and treatment responses. In particular, agent-based models (ABMs) tracking individual agents and their interactions are useful for their flexibility and ability to model complex behaviors. However, ABMs have often been confined to small domains or, when scaled up, have neglected crucial aspects like vasculature. Additionally, the integration into tumor ABMs of precise radiation dose calculations using gold-standard Monte Carlo (MC) methods, crucial in contemporary radiotherapy, has been lacking. Here, we introduce AMBER, an Agent-based fraMework for radioBiological Effects in Radiotherapy that computationally models tumor growth and radiation responses. AMBER is based on a voxelized geometry, enabling realistic simulations at relevant pre-clinical scales by tracking temporally discrete states stepwise. Its hybrid approach, combining traditional ABM techniques with continuous spatiotemporal fields of key microenvironmental factors such as oxygen and vascular endothelial growth factor, facilitates the generation of realistic tortuous vascular trees. Moreover, AMBER is integrated with TOPAS, an MC-based particle transport algorithm that simulates heterogeneous radiation doses. The impact of radiation on tumor dynamics considers the microenvironmental factors that alter radiosensitivity, such as oxygen availability, providing a full coupling between the biological and physical aspects. Our results show that simulations with AMBER yield accurate tumor evolution and radiation treatment outcomes, consistent with established volumetric growth laws and radiobiological understanding. Thus, AMBER emerges as a promising tool for replicating essential features of tumor growth and radiation response, offering a modular design for future expansions to incorporate specific biological traits.

Collective migration is an important component of many biological processes, including wound healing, tumorigenesis, and embryo development. Spatial agent-based models (ABMs) are often used to model collective migration, but it is challenging to thoroughly predict these models' behavior throughout parameter space due to their random and computationally intensive nature. Modelers often coarse-grain ABM rules into mean-field differential equation (DE) models. While these DE models are fast to simulate, they suffer from poor (or even ill-posed) ABM predictions in some regions of parameter space. In this work, we describe how biologically-informed neural networks (BINNs) can be trained to learn interpretable BINN-guided DE models capable of accurately predicting ABM behavior. In particular, we show that BINN-guided partial DE (PDE) simulations can (1) forecast future spatial ABM data not seen during model training, and (2) predict ABM data at previously-unexplored parameter values. This latter task is achieved by combining BINN-guided PDE simulations with multivariate interpolation. We demonstrate our approach using three case study ABMs of collective migration that imitate cell biology experiments and find that BINN-guided PDEs accurately forecast and predict ABM data with a one-compartment PDE when the mean-field PDE is ill-posed or requires two compartments. This work suggests that BINN-guided PDEs allow modelers to efficiently explore parameter space, which may enable data-driven tasks for ABMs, such as estimating parameters from experimental data. All code and data from our study is available at https://github.com/johnnardini/Forecasting_predicting_ABMs .

Formation of organs and specialized tissues in embryonic development requires migration of cells to specific targets. In some instances, such cells migrate as a robust cluster. We here explore a recent local approximation of non-local continuum models by Falcó et al. (SIAM J Appl Math 84:17-42, 2023). We apply their theoretical results by specifying biologically-based cell-cell interactions, showing how such cell communication results in an effective attraction-repulsion Morse potential. We then explore the clustering instability, the existence and size of the cluster, and its stability. For attractant-repellent chemotaxis, we derive an explicit condition on cell and chemical properties that guarantee the existence of robust clusters. We also extend their work by investigating the accuracy of the local approximation relative to the full non-local model.

In this work, we obtained a general formulation for the mating probability and fertile egg production in helminth parasites, focusing on the reproductive behavior of polygamous parasites and its implications for transmission dynamics. By exploring various reproductive variables in parasites with density-dependent fecundity, such as helminth parasites, we departed from the traditional assumptions of Poisson and negative binomial distributions to adopt an arbitrary distribution model. Our analysis considered critical factors such as mating probability, fertile egg production, and the distribution of female and male parasites among hosts, whether they are distributed together or separately. We show that the distribution of parasites within hosts significantly influences transmission dynamics, with implications for parasite persistence and, therefore, with implications in parasite control. Using statistical models and empirical data from Monte Carlo simulations, we provide insights into the complex interplay of reproductive variables in helminth parasites, enhancing our understanding of parasite dynamics and the transmission of parasitic diseases.