Understanding the dynamics underlying fluid transport in tumour tissues is of fundamental importance to assess processes of drug delivery. Here, we analyse the impact of the tumour microscopic properties on the macroscopic dynamics of vascular and interstitial fluid flow. More precisely, we investigate the impact of the capillary wall permeability and the hydraulic conductivity of the interstitium on the macroscopic model arising from formal asymptotic 2-scale techniques. The homogenization technique allows us to derive two macroscale tissue models of fluid flow that take into account the microscopic structure of the vessels and the interstitial tissue. Different regimes were derived according to the magnitude of the vessel wall permeability and the interstitial hydraulic conductivity. Importantly, we provide an analysis of the properties of the models and show the link between them. Numerical simulations were eventually performed to test the models and to investigate the impact of the microstructure on the fluid transport. Future applications of our models include their calibration with real imaging data to investigate the impact of the tumour microenvironment on drug delivery.

An input-output network has an input node [Formula: see text], an output node o, and regulatory nodes [Formula: see text]. Such a network is a core network if each [Formula: see text] is downstream from [Formula: see text] and upstream from o. Wang et al. (J Math Biol 82:62, 2021. https://doi.org/10.1007/s00285-021-01614-1 ) show that infinitesimal homeostasis can be classified in biochemical networks through infinitesimal homeostasis in core subnetworks. Golubitsky and Wang (J Math Biol 10:1-23, 2020) show that there are three types of 3-node core networks and three types of infinitesimal homeostasis in 3-node core networks. This paper uses the theory developed in Wang et al. (2021) to show that there are twenty types of 4-node core networks (Theorem 1.3) and seventeen types of infinitesimal homeostasis in 4-node core networks (Theorem 1.7). Biological contexts illustrate the classification theorems and show that the theory can be an aid when calculating homeostasis in specific biochemical networks.

Respiratory syncytial virus (RSV) is a leading cause of acute lower respiratory tract infection worldwide, resulting in approximately sixty thousand annual hospitalizations of< 5-year-olds in the United States alone and three million annual hospitalizations globally. The development of over 40 vaccines and immunoprophylactic interventions targeting RSV has the potential to significantly reduce the disease burden from RSV infection in the near future. In the context of RSV, a highly contagious pathogen, dynamic transmission models (DTMs) are valuable tools in the evaluation and comparison of the effectiveness of different interventions. This review, the first of its kind for RSV DTMs, provides a valuable foundation for future modelling efforts and highlights important gaps in our understanding of RSV epidemics. Specifically, we have searched the literature using Web of Science, Scopus, Embase, and PubMed to identify all published manuscripts reporting the development of DTMs focused on the population transmission of RSV. We reviewed the resulting studies and summarized the structure, parameterization, and results of the models developed therein. We anticipate that future RSV DTMs, combined with cost-effectiveness evaluations, will play a significant role in shaping decision making in the development and implementation of intervention programs.

We investigate the oscillatory dynamics and bifurcation structure of a reaction-diffusion system with bistable nonlinearity and mass conservation, which was proposed by (Otsuji et al., PLoS Comp Biol 3:e108, 2007). The system is a useful model for understanding cell polarity formation. We show that this model exhibits four different spatiotemporal patterns including two types of oscillatory patterns, which can be regarded as cell polarity oscillations with the reversal and non-reversal of polarity, respectively. The trigger causing these patterns is a diffusion-driven (Turing-like) instability. Moreover, we investigate the effects of extracellular signals on the cell polarity oscillations.

Homeostasis occurs in a control system when a quantity remains approximately constant as a parameter, representing an external perturbation, varies over some range. Golubitsky and Stewart (J Math Biol 74(1-2):387-407, 2017) developed a notion of infinitesimal homeostasis for equilibrium systems using singularity theory. Rhythmic physiological systems (breathing, locomotion, feeding) maintain homeostasis through control of large-amplitude limit cycles rather than equilibrium points. Here we take an initial step to study (infinitesimal) homeostasis for limit-cycle systems in terms of the average of a quantity taken around the limit cycle. We apply the "infinitesimal shape response curve" (iSRC) introduced by Wang et al. (SIAM J Appl Dyn Syst 82(7):1-43, 2021) to study infinitesimal homeostasis for limit-cycle systems in terms of the mean value of a quantity of interest, averaged around the limit cycle. Using the iSRC, which captures the linearized shape displacement of an oscillator upon a static perturbation, we provide a formula for the derivative of the averaged quantity with respect to the control parameter. Our expression allows one to identify homeostasis points for limit cycle systems in the averaging sense. We demonstrate in the Hodgkin-Huxley model and in a metabolic regulatory network model that the iSRC-based method provides an accurate representation of the sensitivity of averaged quantities.

In this paper, a new framework for obtaining personalized optimal treatment strategies in colon cancer-induced angiogenesis is presented. The dynamics of colon cancer is given by a Itó stochastic process, which helps in modeling the randomness present in the system. The stochastic dynamics is then represented by the Fokker-Planck (FP) partial differential equation that governs the evolution of the associated probability density function. The optimal therapies are obtained using a three step procedure. First, a finite dimensional FP-constrained optimization problem is formulated that takes input individual noisy patient data, and is solved to obtain the unknown parameters corresponding to the individual tumor characteristics. Next, a sensitivity analysis of the optimal parameter set is used to determine the parameters to be controlled, thus, helping in assessing the types of treatment therapies. Finally, a feedback FP control problem is solved to determine the optimal combination therapies. Numerical results with the combination drug, comprising of Bevacizumab and Capecitabine, demonstrate the efficiency of the proposed framework.

Likelihood-based methods are widely considered the best approaches for reconstructing ancestral states. Although much effort has been made to study properties of these methods, previous works often assume that both the tree topology and edge lengths are known. In some scenarios the tree topology might be reasonably well known for the taxa under study. When sequence length is much smaller than the number of species, however, edge lengths are not likely to be accurately estimated. We study the consistency of the maximum likelihood and empirical Bayes estimators of the ancestral state of discrete traits in such settings under a star tree. We prove that the likelihood-based reconstruction is consistent under symmetric models but can be inconsistent under non-symmetric models. We show, however, that a simple consistent estimator for the ancestral states is available under non-symmetric models. The results illustrate that likelihood methods can unexpectedly have undesirable properties as the number of sequences considered gets very large. Broader implications of the results are discussed.

During their lifetimes, individuals in populations pass through different states, and the notion of an occupancy time describes the amount of time an individual spends in a given set of states. Questions related to this idea were studied in a recent paper by Roth and Caswell for cases where the environmental conditions are constant. However, it is truly important to consider the case where environments are changing randomly or in directional way through time, so the transition probabilities between different states change over time, motivating the use of time-dependent stage-structured models. Using absorbing inhomogenous Markov chains and the discrete-time McKendrick-von Foerster equation, we derive explicit formulas for the occupancy time, its expectation, and its higher-order moments for stage-structured models with time-dependent transition rates. The results provide insights into the dynamics of long lived plant or animal populations where individuals transition in both directions between reproductive and non reproductive stages. We apply our approach to study a specific time-dependent model of the Southern Fulmar, and obtain insights into how the number of breeding attempts depends on external conditions that vary through time.

We explore the interaction between two genetic incompatibilities (underdominant loci in diploid organisms) in a population occupying a one-dimensional space. We derive a system of partial differential equations describing the dynamics of allele frequencies and linkage disequilibrium between the two loci, and use a quasi-linkage equilibrium approximation in order to reduce the number of variables. We investigate the solutions of this system and demonstrate the existence of a solution in which the two clines in allele frequency remain stacked together. In the case of asymmetric incompatibilities (i.e. when one homozygote is favored over the other at each locus), these stacked clines propagate in the form of a traveling wave. We obtain an approximation for the speed of this wave which, in particular, is decreased by recombination between the two loci but is always larger than the speed of "one cline alone".

How do environmental heterogeneity influence propagation dynamics of the age-structured invasive species? We investigate this problem by considering a yearly generation invasive species in time-space periodic habitat. Starting from an age-structured population growth law, we formulate a reaction-diffusion model with time-space periodic dispersal, mortality and recruitment. Thanks to the fundamental solution for linear part of the model, we reduce to study the dynamics of a time-space periodic semiflow which is defined by the solution map. By the recent developed dynamical theory in Fang et al. (J Funct Anal 272:4222-4262, 2017), we obtained the spreading speed and its coincidence with the minimal wave speed of time-space periodic traveling waves, as well as the variational characterization of spreading speed in terms of a principal eigenvalue problem. Such results are also proved back to the reaction-diffusion model.

In this paper, we propose a diffusive competition model with habitat degradation and homogeneous Neumann boundary conditions in a bounded domain that is partitioned into the healthy region (undisturbed habitat) and the degraded region (due to anthropogenic habitat disturbance). Species follow the Lotka-Volterra competition in the healthy region while in the degraded region species experience only exponential decay (not necessarily at the same rate). This setup is novel in that it requires no positivity assumption on the environmental heterogeneity, either absolute or on average, which would be far too restrictive for the study of the effects of habitat degradation. We rigorously show competitive exclusion and coexistence via global stability analysis. A remarkable finding is that the quality heterogeneity of landscapes can lead to the competitive exclusion of the slower species by the faster species. This result is robust as long as the degraded region has positive area, and moreover is at odds with classical results predicting the deterministic extinction of the stronger species. On the other hand, if the degraded region has intermediate negative effect on the faster competitor, species can coexist. Differing from comparable existing results, coexistence does not rely on a limit as the diffusion coefficients tend to zero or infinity. Together, these results imply that coexistence is always a possibility under this basic, yet general, configuration, providing insights into the varying impacts found through empirical study of habitat loss and fragmentation on species.

In this study, the cardiac electro-mechanical model in a deforming domain is taken with the addition of mechanical feedback and stretch-activated channel current coupled with the ten Tusscher human ventricular cell level model that results in a coupled PDE-ODE system. The existence and uniqueness of such a coupled system in a deforming domain is proved. At first, the existence of a solution is proved in the deformed domain. The local existence of the solution is proved using the regularization and the Faedo-Galerkin technique. Then, the global existence is proved using the energy estimates in appropriate Banach spaces, Gronwall lemma, and the compactness procedure. The existence of the solution in an undeformed domain is proved using the lower semi-continuity of the norms. Uniqueness is proved using Young's inequality, Gronwall lemma, and the Cauchy-Schwartz inequality. For the application purpose, this model is applied to understand the electro-mechanical activity in ischemic cardiac tissue. It also takes care of the development of active tension, conductive, convective, and ionic feedback. The Second Piola-Kirchoff stress tensor arising in Lagrangian mapping between reference and moving frames is taken as a combination of active, passive, and volumetric components. We investigated the effect of varying strength of hyperkalemia and hypoxia, in the ischemic subregions of human cardiac tissue with local multiple ischemic subregions, on the electro-mechanical activity of healthy and ischemic zones. This system is solved numerically using the [Formula: see text] finite element method in space and the implicit-explicit Euler method in time. Discontinuities arising with the modeled multiple ischemic regions are treated to the desired order of accuracy by a simple regularization technique using the interpolating polynomials. We examined the cardiac electro-mechanical activity for several cases in multiple hyperkalemic and hypoxic human cardiac tissue. We concluded that local multiple ischemic subregions severely affect the cardiac electro-mechanical activity more, in terms of action potential (v) and mechanical parameters, intracellular calcium ion concentration [Formula: see text], active tension ([Formula: see text]), stretch ([Formula: see text]) and stretch rate ([Formula: see text]), of a healthy cell in its vicinity, compared to a single Hyperkalemic or Hypoxic subregion. The four moderate hypoxically generated ischemic subregions affect the waveform of the stretch along the fiber and the stretch rate more than a single severe ischemic subregion.

When studying the dynamics of trait distribution of populations in a heterogeneous environment, classical models from quantitative genetics choose to look at its system of moments, specifically the first two ones. Additionally, in order to close the resulting system of equations, they often assume the local trait distributions are Gaussian [see for instance Ronce and Kirkpatrick (Evolution 55(8):1520-1531, 2001. https://doi.org/10.1111/j.0014-3820.2001.tb00672.x.37 )]. The aim of this paper is to introduce a mathematical framework that follows the whole trait distribution (without prior assumption) to study evolutionary dynamics of sexually reproducing populations. Specifically, it focuses on complex traits, whose inheritance can be encoded by the infinitesimal model of segregation (Fisher in Trans R Soc Edinb 52(2):399-433, 1919. https://doi.org/10.1017/S0080456800012163 ). We show that it allows us to derive a regime in which our model gives the same dynamics as when assuming Gaussian local trait distributions. To support that, we compare the stationary problems of the system of moments derived from our model with the one given in Ronce and Kirkpatrick (Evolution 55(8):1520-1531, 2001. https://doi.org/10.1111/j.0014-3820.2001.tb00672.x.37 ) and show that they are equivalent under this regime and do not need to be otherwise. Moreover, under this regime of equivalence, we show that a separation bewteen ecological and evolutionary time scales arises. A fast relaxation toward monomorphism allows us to reduce the complexity of the system of moments, using a slow-fast analysis. This reduction leads us to complete, still in this regime, the analytical description of the bistable asymmetrical equilibria numerically found in Ronce and Kirkpatrick (Evolution 55(8):1520-1531, 2001. https://doi.org/10.1111/j.0014-3820.2001.tb00672.x.37 ). More globally, we provide explicit modelling hypotheses that allow for such local adaptation patterns to occur.