We explicitly give all stationary solutions to the focusing cubic NLS on the line, in thepresence of a defect of the type Dirac’s delta or delta prime. The models proveinteresting for two features: first, they are exactly solvable and all quantities can beexpressed in terms of elementary functions. Second, the associated dynamics is far frombeing trivial. In particular, the NLS with a delta prime potential shows two symmetrybreaking bifurcations: the first concerns the ground state and was already known. Thesecond emerges on the first excited state, and up to now had not been revealed. Wehighlight such bifurcations by computing the nonlinear and the no-defect limits of thestationary solutions.

We are interested in optimizing the co-administration of two drugs for some acute myeloidleukemias (AML), and we are looking for in vitro protocols as a first step. This issue canbe formulated as an optimal control problem. The dynamics of leukemic cell populations inculture is given by age-structured partial differential equations, which can be reduced toa system of delay differential equations, and where the controls represent the action ofthe drugs. The objective function relies on eigenelements of the uncontrolled model and ongeneral relative entropy, with the idea to maximize the efficiency of the protocols. Theconstraints take into account the toxicity of the drugs. We present in this paper themodeling aspects, as well as theoretical and numerical results on the optimal controlproblem that we get.

We consider the spread of infectious disease through contact networks of ConfigurationModel type. We assume that the disease spreads through contacts and infected individualsrecover into an immune state. We discuss a number of existing mathematical models used toinvestigate this system, and show relations between the underlying assumptions of themodels. In the process we offer simplifications of some of the existing models. Thedistinctions between the underlying assumptions are subtle, and in many if not most casesthis subtlety is irrelevant. Indeed, under appropriate conditions the models areequivalent. We compare the benefits and disadvantages of the different models, and discusstheir application to other populations (e.g., clustered networks).Finally we discuss ongoing challenges for network-based epidemic modeling.

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 aim of this paper is to introduce and study multilinear pseudo-differential operatorson Zn and Tn =(Rn/2πZn) then-torus.More precisely, we give sufficient conditions and sometimes necessary conditions forLp-boundedness of theseclasses of operators. L2-boundedness results for multilinearpseudo-differential operators on Zn and Tn withL2-symbols are stated. The proofs of theseresults are based on elementary estimates on the multilinear Rihaczek transforms forfunctions in L2(Zn)respectively L2(Tn)which are also introduced.

We study the weak continuity of multilinear operators on the m-fold product of Lebesguespaces Lpj(Zn),j =1,...,m and thelink with the continuity of multilinear pseudo-differential operators on Zn.

Necessary and sufficient conditions for multilinear pseudo-differential operators onZn or Tn to be aHilbert-Schmidt operators are also given. We give a necessary condition for a multilinearpseudo-differential operators on Zn to be compact. A sufficientcondition for compactness is also given.

Darwin illustrated his theory about emergence and evolution of biological species with adiagram. It shows how species exist, evolve, appear and disappear. The goal of this workis to give a mathematical interpretation of this diagram and to show how it can bereproduced in mathematical models. It appears that conventional models in populationdynamics are not sufficient, and we introduce a number of new models which take intoaccount local, nonlocal and global consumption of resources, and models with space andtime dependent coefficients.

We discuss the zero-controllability and the zero-stabilizability for the nonnegative solutions to some Fisher-like models with nonlocal terms describing the dynamics of biological populations with diffusion, logistic term and migration. A necessary and sufficient condition for the nonnegative zero-stabilizabiity for a linear integro-partial differential equation is obtained in terms of the sign of the principal eigenvalue to a certain non-selfadjoint operator. For a related semilinear problem a necessary condition and a sufficient condition for the local nonnegative zero-stabilizability are also derived in terms of the magnitude of the above mentioned principal eigenvalue. The rate of stabilization corresponding to a simple feedback stabilizing control is dictated by the principal eigenvalue. A large principal eigenvalue leads to a fast stabilization to zero. A necessary condition and a sufficient condition for the stabilization to zero of the predator population in a predator-prey system is also investigated. Finally, a method to approximate the above mentioned principal eigenvalues is indicated.

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 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.

In this paper we present a fluid-structure interaction model of neuron’s membranedeformation. The membrane-actin is considered as an elastic solid layer, while thecytoplasm is considered as a viscous fluid one. The membrane-actin layer is governed byelasticity equations while the cytoplasm is described by the Navier-Stokes equations. Atthe interface between the cytoplasm and the membrane we consider a match between the solidvelocity displacement and the fluid velocity as well as the mechanical equilibrium. Themembrane, which faces the extracellular medium, is free to move. This will change thegeometry in time. To take into account the deformation of the initial configuration, weuse the Arbitrary Lagrangian Eulerian method in order to take into account the meshdisplacement. The numerical simulations, show the emergence of a filopodium, a typicalstructure in cells undergoing deformation.

In this paper we explore the eco-evolutionary dynamics of a predator-prey model, wherethe prey population is structured according to a certain life history trait. The traitdistribution within the prey population is the result of interplay between geneticinheritance and mutation, as well as selectivity in the consumption of prey by thepredator. The evolutionary processes are considered to take place on the same time scaleas ecological dynamics, i.e. we consider the evolution to be rapid. Previously publishedresults show that population structuring and rapid evolution in such predator-prey systemcan stabilize an otherwise globally unstable dynamics even with an unlimited carryingcapacity of prey. However, those findings were only based on direct numerical simulationof equations and obtained for particular parameterizations of model functions, whichobviously calls into question the correctness and generality of the previous results. Themain objective of the current study is to treat the model analytically and considervarious parameterizations of predator selectivity and inheritance kernel. We investigatethe existence of a coexistence stationary state in the model and carry out stabilityanalysis of this state. We derive expressions for the Hopf bifurcation curve which can beused for constructing bifurcation diagrams in the parameter space without the need for adirect numerical simulation of the underlying integro-differential equations. Weanalytically show the possibility of stabilization of a globally unstable predator-preysystem with prey structuring. We prove that the coexistence stationary state is stablewhen the saturation in the predation term is low. Finally, for a class of kernelsdescribing genetic inheritance and mutation we show that stability of the predator-preyinteraction will require a selectivity of predation according to the life trait.

We prove that the minimal operator and the maximal operator of the Hermite operator arethe same on Lp(ℝn), 4 / 3<p< 4. Thedomain and the spectrum of the minimal operator (=maximal operator) of the Hermiteoperator on Lp(ℝn),4/3 <p<4, are computed. In addition, we can give anestimate for the Lp-norm of thesolution to the initial value problem for the heat equation governed by the minimal(maximal) operator for 4/3<p<4.

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.

In this paper we explore the potential of the pairwise-type modelling approach to beextended to weighted networks where nodal degree and weights are not independent. As abaseline or null model for weighted networks, we consider undirected, heterogenousnetworks where edge weights are randomly distributed. We show that the pairwise modelsuccessfully captures the extra complexity of the network, but does this at the cost oflimited analytical tractability due the high number of equations. To circumvent thisproblem, we employ the edge-based modelling approach to derive models corresponding to twodifferent cases, namely for degree-dependent and randomly distributed weights. Thesemodels are more amenable to compute important epidemic descriptors, such as early growthrate and final epidemic size, and produce similarly excellent agreement with simulation.Using a branching process approach we compute the basic reproductive ratio for both modelsand discuss the implication of random and correlated weight distributions on this as wellas on the time evolution and final outcome of epidemics. Finally, we illustrate that thetwo seemingly different modelling approaches, pairwise and edge-based, operate on similarassumptions and it is possible to formally link the two.

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.