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

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

We consider the spread of infectious disease through contact networks of Configuration Model type. We assume that the disease spreads through contacts and infected individuals recover into an immune state. We discuss a number of existing mathematical models used to investigate this system, and show relations between the underlying assumptions of the models. In the process we offer simplifications of some of the existing models. The distinctions between the underlying assumptions are subtle, and in many if not most cases this subtlety is irrelevant. Indeed, under appropriate conditions the models are equivalent. We compare the benefits and disadvantages of the different models, and discuss their 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 of cell adhesion and fibrin gel formation controlled by huge intricate networks of biochemical reactions. This combination of complex biochemistry, non-Newtonian hydrodynamics, and transport processes makes thrombosis difficult to understand. That is why numerous attempts to use mathematical modeling for this purpose were undertaken during the last decade. In particular, recent years witnessed something of a transition from the “systems biology” to the “systems pharmacology/systems medicine” stage: computational modeling is being increasingly applied to practical problems such as drug development, investigation of particular events underlying disease, analysis of the mechanism(s) of drug’s action, determining an optimal dosing protocols, etc. Here we review recent advances and challenges in our understanding of thrombus formation.

The aim of this paper is to introduce and study multilinear pseudo-differential operators on Zn and Tn = (Rn/ 2πZn) the n-torus. More precisely, we give sufficient conditions and sometimes necessary conditions for Lp-boundedness of these classes of operators. L2-boundedness results for multilinear pseudo-differential operators on Zn and Tn with L2-symbols are stated. The proofs of these results are based on elementary estimates on the multilinear Rihaczek transforms for functions in L2(Zn) respectively L2(Tn) which are also introduced.

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

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

Darwin illustrated his theory about emergence and evolution of biological species with a diagram. It shows how species exist, evolve, appear and disappear. The goal of this work is to give a mathematical interpretation of this diagram and to show how it can be reproduced in mathematical models. It appears that conventional models in population dynamics are not sufficient, and we introduce a number of new models which take into account local, nonlocal and global consumption of resources, and models with space and time 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.

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

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

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

In this paper we explore the eco-evolutionary dynamics of a predator-prey model, where the prey population is structured according to a certain life history trait. The trait distribution within the prey population is the result of interplay between genetic inheritance and mutation, as well as selectivity in the consumption of prey by the predator. The evolutionary processes are considered to take place on the same time scale as ecological dynamics, i.e. we consider the evolution to be rapid. Previously published results show that population structuring and rapid evolution in such predator-prey system can stabilize an otherwise globally unstable dynamics even with an unlimited carrying capacity of prey. However, those findings were only based on direct numerical simulation of equations and obtained for particular parameterizations of model functions, which obviously calls into question the correctness and generality of the previous results. The main objective of the current study is to treat the model analytically and consider various parameterizations of predator selectivity and inheritance kernel. We investigate the existence of a coexistence stationary state in the model and carry out stability analysis of this state. We derive expressions for the Hopf bifurcation curve which can be used for constructing bifurcation diagrams in the parameter space without the need for a direct numerical simulation of the underlying integro-differential equations. We analytically show the possibility of stabilization of a globally unstable predator-prey system with prey structuring. We prove that the coexistence stationary state is stable when the saturation in the predation term is low. Finally, for a class of kernels describing genetic inheritance and mutation we show that stability of the predator-prey interaction 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 are the same on Lp(ℝn) , 4 / 3 <p< 4. The domain and the spectrum of the minimal operator (=maximal operator) of the Hermite operator on Lp(ℝn), 4/3 <p< 4, are computed. In addition, we can give an estimate for the Lp-norm of the solution 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 the formation, growth and lysis of clots in vitro. The sensitivity analysis procedure uses an ensemble standard deviation for species concentrations, and is equivalent to a variance decomposition procedure also available in the literature. Our analysis shows that fibrin production is most sensitive to the rate constant governing activation of prothrombin to thrombin. Further, the time-averaged sum of all species’ concentrations is most sensitive to 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 the greatest: this conclusion must be experimentally verified to determine if such is indeed the case for hemostasis.

In this paper we explore the potential of the pairwise-type modelling approach to be extended to weighted networks where nodal degree and weights are not independent. As a baseline or null model for weighted networks, we consider undirected, heterogenous networks where edge weights are randomly distributed. We show that the pairwise model successfully captures the extra complexity of the network, but does this at the cost of limited analytical tractability due the high number of equations. To circumvent this problem, we employ the edge-based modelling approach to derive models corresponding to two different cases, namely for degree-dependent and randomly distributed weights. These models are more amenable to compute important epidemic descriptors, such as early growth rate and final epidemic size, and produce similarly excellent agreement with simulation. Using a branching process approach we compute the basic reproductive ratio for both models and discuss the implication of random and correlated weight distributions on this as well as on the time evolution and final outcome of epidemics. Finally, we illustrate that the two seemingly different modelling approaches, pairwise and edge-based, operate on similar assumptions 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, and show that it can be dramatically simplified by the use of judicious approximations. In particular, we show that the conceptual basis of the popular model of Mackey and Glass [20] is at odds with the important transport processes of the Grodins model. Despite this, a realistic approximation of the Grodins model yields a Mackey-Glass type model with almost the same criterion for the onset of Cheyne-Stokes breathing.

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