Amazon molly (Poecilia formosa) females reproduce asexually, but theyneed sperm to initiate the process. Such gynogenetic reproduction can be called spermparasitism since the DNA in the sperm is not used. Since all offspring of asexuallyreproducing females are females, they can locally outcompete sexually reproducing ones,but their persistence is threatened by the lack of males. Therefore, the existence ofAmazon mollies is puzzling. A metapopulation structure has been suggested to enable thecoexistence of gynogenetic and sexual species. Previously only Levins-type metapopulationmodels have been used to investigate this question, but they are not defined on theindividual level. Therefore we investigate the evolution of sperm parasitism in astructured metapopulation model, which incorporates both realistic local populationdynamics and individual-level dispersal. If the reproduction strategy is freely evolvingin a large well-mixed population or in the structured metapopulation model, strongdiscrimination of asexually reproducing females by males results in evolution to fullsexuality, whereas mild discrimination leads to too small probability of sexualreproduction, so that the lack of males causes the extinction of the evolving population,resulting in evolutionary suicide. This classification remains the same also when bothsexual reproduction and dispersal are freely evolving. Sexual and asexual behaviour can beobserved at the same time in this model in the presence of a trade-off between thereproduction and dispersal traits. However, we do not observe disruptive selectionresulting in the evolutionarily stable coexistence of fully sexual and fully asexualfemales. Instead, the presence of sexual and asexual behaviour is due to females with amixed reproduction trait.

The stochastic dynamics of chemical reactions can be accurately described by chemicalmaster equations. An approximated time-evolution equation of the Langevin type has beenproposed by Gillespie based on two explicit dynamical conditions. However, whennumerically solve these chemical Langevin equations, we often have a small stopping time–atime point of having an unphysical solution–in the case of low molecular numbers. Thispaper proposes an approach to simulate stochasticities in chemical reactions withdeterministic delay differential equations. We introduce a deterministic Brownian motiondescribed by delay differential equations, and replace the Gaussian noise in the chemicalLangevin equations by the solutions of these deterministic equations. This modificationcan largely increase the stopping time in simulations and regain the accuracy as in thechemical Langevin equations. The novel aspect of the present study is to apply thedeterministic Brownian motion to chemical reactions. It suggests a possible direction ofdeveloping a hybrid method of simulating dynamic behaviours of complex gene regulationnetworks.

A reaction–diffusion replicator equation is studied. A novel method to apply theprinciple of global regulation is used to write down a model with explicit spatialstructure. Properties of stationary solutions together with their stability are analyzedanalytically, and relationships between stability of the rest points of thenon-distributed replicator equation and the distributed system are shown. In particular,we present the conditions on the diffusion coefficients under which the non-distributedreplicator equation can be used to describe the number and stability of the stationarysolutions to the distributed system. A numerical example is given, which shows that thesuggested modeling framework promotes the system’s persistence, i.e., a scenario ispossible when in the spatially explicit system all the interacting species survive whereassome of them go extinct in the non-distributed one.

We analyse a reduced version of the Grodins et al. control model [14] of respiration involving only CO2, andshow that it can be dramatically simplified by the use of judicious approximations. Inparticular, we show that the conceptual basis of the popular model of Mackey and Glass[20] is at odds with the important transportprocesses of the Grodins model. Despite this, a realistic approximation of the Grodinsmodel yields a Mackey-Glass type model with almost the same criterion for the onset ofCheyne-Stokes breathing.

While the reduced Grodins model does apparently provide a realistic mechanism forinstability, consideration of the buffering effect of the blood-brain barrier appears tomake it unlikely. We conclude that a realistic physiological model of Grodins type toexplain Cheyne–Stokes breathing is not yet in place, and raise the question whether thebicarbonate buffering system has a rôle to play.

We extrapolate from the exact master equations of epidemic dynamics on fully connectedgraphs to non-fully connected by keeping the size of the state space N + 1, whereN is thenumber of nodes in the graph. This gives rise to a system of approximate ODEs (ordinarydifferential equations) where the challenge is to compute/approximate analytically thetransmission rates. We show that this is possible for graphs with arbitrary degreedistributions built according to the configuration model. Numerical tests confirm that:(a) the agreement of the approximate ODEs system with simulation is excellent and (b) thatthe approach remains valid for clustered graphs with the analytical calculations of thetransmission rates still pending. The marked reduction in state space gives good results,and where the transmission rates can be analytically approximated, the model provides astrong alternative approximate model that agrees well with simulation. Given that thetransmission rates encompass information both about the dynamics and graph properties, thespecific shape of the curve, defined by the transmission rate versus the number ofinfected nodes, can provide a new and different measure of network structure, and themodel could serve as a link between inferring network structure from prevalence orincidence data.

The projection method is a simple way of constructing functions and filters byintegrating multidimensional functions and filters along parallel superplanes in the spacedomain. Equivalently expressed in the frequency domain, the projection method constructs anew function by simply taking a cross-section of the Fourier transform of amultidimensional function. The projection method is linked to several areas such as boxsplines in approximation theory and the projection-slice theorem in image processing. Inthis paper, we shall systematically study and discuss the projection method in the area ofmultidimensional framelet and wavelet analysis. We shall see that the projection methodnot only provides a painless way for constructing new wavelets and framelets but also is auseful analysis tool for studying various optimal properties of multidimensional refinablefunctions and filters. Using the projection method, we shall explicitly and easilyconstruct a tight framelet filter bank from every box spline filter having at least orderone sum rule. As we shall see in this paper, the projection method is particularlysuitable to be applied to frequency-based nonhomogeneous framelets and wavelets in anydimensions, and the periodization technique is a special case of the projection method forobtaining periodic wavelets and framelets from wavelets and framelets on Euclidean spaces.

A two dimensional two-delays differential system modeling the dynamics of stem-like cellsand white-blood cells in Chronic Myelogenous Leukemia is considered. All three types ofstem cell division (asymmetric division, symmetric renewal and symmetric differentiation)are present in the model. Stability of equilibria is investigated and emergence ofperiodic solutions of limit cycle type, as a result of a Hopf bifurcation, is eventuallyshown. The stability of these limit cycles is studied using the first Lyapunovcoefficient.

The reduced basis method is a model reduction technique yielding substantial savings ofcomputational time when a solution to a parametrized equation has to be computed for manyvalues of the parameter. Certification of the approximation is possible by means of ana posteriori error bound. Under appropriate assumptions, this errorbound is computed with an algorithm of complexity independent of the size of the fullproblem. In practice, the evaluation of the error bound can become very sensitive toround-off errors. We propose herein an explanation of this fact. A first remedy has beenproposed in [F. Casenave, Accurate a posteriori error evaluation in thereduced basis method. C. R. Math. Acad. Sci. Paris 350(2012) 539–542.]. Herein, we improve this remedy by proposing a new approximationof the error bound using the empirical interpolation method (EIM). This method achieveshigher levels of accuracy and requires potentially less precomputations than the usualformula. A version of the EIM stabilized with respect to round-off errors is also derived.The method is illustrated on a simple one-dimensional diffusion problem and athree-dimensional acoustic scattering problem solved by a boundary element method.

We consider the spread of an infectious disease on a heterogeneous metapopulation definedby any (correlated or uncorrelated) network. The infection evolves under transmission,recovery and migration mechanisms. We study some spectral properties of a connectivitymatrix arising from the continuous-time equations of the model. In particular we show thatthe classical sufficient condition of instability for the disease-free equilibrium, wellknown for the particular case of uncorrelated networks, works also for the general case.We give also an alternative condition that yields a more accurate estimation of theepidemic threshold for correlated (either assortative or dissortative) networks.

Increasing experimental evidence suggests that epigenetic and microenvironmental factors play a key role in cancer progression. In this respect, it is now generally recognized that the immune system can act as an additional selective pressure, which modulates tumor development and leads, through cancer immunoediting, to the selection for resistance to immune effector mechanisms. This may have serious implications for the design of effective anti-cancer protocols. Motivated by these considerations, we present a mathematical model for the dynamics of cancer and immune cells under the effects of chemotherapy and immunity-boosters. Tumor cells are modeled as a population structured by a continuous phenotypic trait, that is related to the level of resistance to receptor-induced cell death triggered by effector lymphocytes. The level of resistance can vary over time due to the effects of epigenetic modifications. In the asymptotic regime of small epimutations, we highlight the ability of the model to reproduce cancer immunoediting. In an optimal control framework, we tackle the problem of designing effective anti-cancer protocols. The results obtained suggest that chemotherapeutic drugs characterized by high cytotoxic effects can be useful for treating tumors of large size. On the other hand, less cytotoxic chemotherapy in combination with immunity-boosters can be effective against tumors of smaller size. Taken together, these results support the development of therapeutic protocols relying on combinations of less cytotoxic agents and immune-boosters to fight cancer in the early stages.

This paper presents numerical results based on a macroscopic blood coagulation modelcoupled with a non-linear viscoelastic model for blood flow. The system of governingequations is solved using a central finite-volume scheme for space discretization and anexplicit Runge-Kutta time-integration. An artificial compressibility method is used toresolve pressure and a non-linear TVD filter is applied for stabilization. A simple testcase of flowing blood over a clotting surface in a straight 3D vessel is solved. This workpresents a significant extension of the previous studies [10] and [9].

In mammalian cells, the p53 pathway regulates the response to a variety of stresses,including oncogene activation, heat and cold shock, and DNA damage. Here we explore amathematical model of this pathway, composed of a system of partial differentialequations. In our model, the p53 pathway is activated by a DNA-compromising event of shortduration. As is typical for mathematical models of the p53 pathway, our model contains anegative feedback loop representing interactions between the p53 and Mdm2 proteins. Anovel feature of our model is that we combine a spatio-temporal approach with theappearance and repair of DNA damage. We investigate the behaviour of our model throughnumerical simulations. By ignoring the possibility of DNA repair, we first explore thescenario in which the cell has a very inefficient DNA repair mechanism. We find thatspatio-temporal oscillations in p53 and Mdm2 may occur, consistent with experimental data.We then allow p53 to be directly involved in repairing DNA damage, since experimentalevidence suggests this can happen. We find that oscillations in p53 and Mdm2 can stilloccur, but their amplitude damps down quickly as the DNA damage is repaired. Finally, wefind that a minor change to the location of the DNA damage can notably change the spatialdistribution of p53 within the nucleus. We discuss the biological implications of ourresults.

We study optimal sustainable policies in a benchmark logistic world (where both population and technological progress follow logistic laws of motion) subject to a pollution ceiling. The main policy in the hands of the benevolent planner is pollution abatement, ultimately leading to the control of a dirtiness index as in the early literature of the limits to growth literature. Besides inclusion of demographic dynamics, we also hypothesize that population size affects negatively the natural regeneration or assimilation rate, as a side product of human activities (like increasing pollution, deforestation,...). We first characterize optimal sustainable policies. Under certain conditions, the planner goes to the pollution ceiling value and stays on, involving a more stringent environmental policy and a sacrifice in terms of consumption per capita. Second, we study how the sustainable problem is altered when we depart from the logistic world by considering exponential technical progress (keeping population growth logistic). It’s shown that, as expected, introducing such an asymmetry widens the margins of optimal policies as sustainable environmental policies are clearly less stringent under exponential technical progress. Third, we connect our model to the data, using in particular UN population projections.

We define two notions of discrete dimension based on the Minkowski and Hausdorffdimensions in the continuous setting. After proving some basic results illustrating thesedefinitions, we apply this machinery to the study of connections between the Erdős andFalconer distance problems in geometric combinatorics and geometric measure theory,respectively.

The problem of transformation of quasimonochromatic wavetrains of surface gravity waveswith narrow spatial-temporal spectra on the bottom shelf is considered in the linearapproximation. By means of numerical modeling, the transmission and reflectioncoefficients are determined as functions of the depth ratio and wave number (frequency) ofan incident wave. The approximation formulae are proposed for the coefficients of wavetransformation. The characteristic features of these formulae are analyzed. It is shownthat the numerical results agree quite satisfactorily with the proposed approximationformulae.

We perform a sensitivity analysis for a thus far unstudied mathematical model for theformation, growth and lysis of clots in vitro. The sensitivity analysis procedure uses anensemble standard deviation for species concentrations, and is equivalent to a variancedecomposition procedure also available in the literature. Our analysis shows that fibrinproduction is most sensitive to the rate constant governing activation of prothrombin tothrombin. Further, the time-averaged sum of all species’ concentrations is most sensitiveto the rate constants governing the inactivation of VIIIa (intrinsic as well as by APC).We therefore conclude that the rate constants for VIIIa inactivation affect the model thegreatest: this conclusion must be experimentally verified to determine if such is indeedthe case for hemostasis.

We propose two new algorithms to improve greedy sampling of high-dimensional functions. While the techniques have a substantial degree of generality, we frame the discussion in the context of methods for empirical interpolation and the development of reduced basis techniques for high-dimensional parametrized functions. The first algorithm, based on a saturation assumption of the error in the greedy algorithm, is shown to result in a significant reduction of the workload over the standard greedy algorithm. In a further improved approach, this is combined with an algorithm in which the train set for the greedy approach is adaptively sparsified and enriched. A safety check step is added at the end of the algorithm to certify the quality of the sampling. Both these techniques are applicable to high-dimensional problems and we shall demonstrate their performance on a number of numerical examples.

Thrombus formation in flowing blood is a complex time- and space-dependent process ofcell adhesion and fibrin gel formation controlled by huge intricate networks ofbiochemical reactions. This combination of complex biochemistry, non-Newtonianhydrodynamics, and transport processes makes thrombosis difficult to understand. That iswhy numerous attempts to use mathematical modeling for this purpose were undertaken duringthe last decade. In particular, recent years witnessed something of a transition from the“systems biology” to the “systems pharmacology/systems medicine” stage: computationalmodeling is being increasingly applied to practical problems such as drug development,investigation of particular events underlying disease, analysis of the mechanism(s) ofdrug’s action, determining an optimal dosing protocols, etc. Here we review recentadvances and challenges in our understanding of thrombus formation.

The paper investigates an age-structured infinite-horizon optimal control model ofharvesting a biological resource, interpreted as fish. Time and age are considered ascontinuum variables. The main result shows that in case of selective fishing, where onlyfish of prescribed sizes is harvested, it may be advantageous in the log run to implementa periodic fishing effort, rather than constant (the latter suggested by single-fishmodels that disregard the age-heterogeneity). Thus taking into account the age-structureof the fish may qualitatively change the theoretically optimal fishing mode. This resultis obtained by developing a technique for reliable numerical verification of second ordernecessary optimality conditions for the considered problem. This technique could be usefulfor other optimal control problems of periodic age-structured systems.