Mathematical and computational neuroscience have contributed to the brain sciences by thestudy of the dynamics of individual neurons and more recently the study of the dynamics ofelectrophysiological networks. Often these studies treat individual neurons as points orthe nodes in networks and the biochemistry of the brain appears, if at all, as someintermediate variables by which the neurons communicate with each other. In fact, manyneurons change brain function not by communicating in one-to-one fashion with otherneurons, but instead by projecting changes in biochemistry over long distances. Thisbiochemical network is of crucial importance for brain function and it influences and isinfluenced by the more traditional electrophysiological networks. Understanding howbiochemical networks interact with electrophysiological networks to produce brain functionboth in health and disease poses new challenges for mathematical neuroscience.

Dirac structures are used as the underlying structure to mathematically formalizeport-Hamiltonian systems. This note approaches the Dirac structures forinfinite-dimensional systems using the theory of linear relations on Hilbert spaces.First, a kernel representation for a Dirac structure is proposed. The one-to-onecorrespondence between Dirac structures and unitary operators is revisited. Further, theproposed kernel representation and a scattering representation are constructively related.Several illustrative examples are also presented in the paper.

The paper combines analytic and numeric tools to investigate a nonlinear optimal control problem relevant to the economics of climate change. The problem describes optimal investments into pollution mitigation and environmental adaptation at a macroeconomic level. The steady-state analysis of this problem focuses on the optimal ratio between adaptation and mitigation. In particular, we analytically prove that the long-term investments into adaptation are profitable only for economies above certain efficiency threshold. Numerical simulation is provided to estimate how the economic efficiency and capital deterioration affect the optimal policy.

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 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 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.

An overview of a general approach for mathematical modeling of evolving heterogeneouspopulations using a wide class of selection systems and replicator equations (RE) ispresented. The method allows visualizing evolutionary trajectories of evolvingheterogeneous populations over time, while still enabling use of analytical tools ofbifurcation theory. The developed theory involves introducing escort systems of auxiliary“keystone" variables, which reduce complex multi-dimensional inhomogeneous models tolow dimensional systems of ODEs that in many cases can be investigated analytically. Inaddition to a comprehensive theoretical framework, a set of examples of the method’sapplicability to questions ranging from preventing the tragedy of the commons to cancertherapy is presented.

We are concerned with the proof of a generalized solution to an ill-posed variational inequality. This is determined as a solution to an appropriate minimization problem involving a nonconvex functional, treated by an optimal control technique.

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.

In epidemic modeling, the Susceptible-Alert-Infected-Susceptible (SAIS) model extends theSIS (Susceptible-Infected-Susceptible) model. In the SAIS model, “alert” individualsobserve the health status of neighbors in their contact network, and as a result, they mayadopt a set of cautious behaviors to reduce their infection rate. This alertness, whenincorporated in the mathematical model, increases the range of effective/relativeinfection rates for which initial infections die out. Built upon the SAIS model, this workinvestigates how information dissemination further increases this range. Informationdissemination is realized through an additional network (e.g., an online social network)sharing the contact network nodes (individuals) with different links. These “informationlinks” provide the health status of one individual to all the individuals she is connectedto in the information dissemination network. We propose an optimal informationdissemination strategy with an index in quadratic form relative to the informationdissemination network adjacency matrix and the dominant eigenvector of the contactnetwork. Numerical tools to exactly solve steady state infection probabilities andinfluential thresholds are developed, providing an evaluative baseline for our informationdissemination strategy. We show that monitoring the health status of a small but “central”subgroup of individuals and circulating their incidence information optimally enhances theresilience of the society against infectious diseases. Extensive numerical simulations ona survey–based contact network for a rural community in Kansas support these findings.

A stenosis is the narrowing of the artery, this narrowing is usually the result of theformation of an atheromatous plaque infiltrating gradually the artery wall, forming a bumpin the ductus arteriosus. This arterial lesion falls within the general context ofatherosclerotic arterial disease that can affect the carotid arteries, but also thearteries of the heart (coronary), arteries of the legs (PAD), the renal arteries... It cancause a stroke (hemiplegia, transient paralysis of a limb, speech disorder, sailing beforethe eye). In this paper we study the blood-plaque and blood-wall interactions using afluid-structure interaction model. We first propose a 2D analytical study of thegeneralized Navier-Stokes equations to prove the existence of a weak solution forincompressible non-Newtonian fluids with non standard boundary conditions. Then, coupled,based on the results of the theoretical study approach is given. And to form a realisticmodel, with high accuracy, additional conditions due to fluid-structure coupling areproposed on the border undergoing inetraction. This coupled model includes (a) a fluidmodel, where blood is modeled as an incompressible non-Newtonian viscous fluid, (b) asolid model, where the arterial wall and atherosclerotic plaque will be treated as nonlinear hyperelastic solids, and (c) a fluid-structure interaction (FSI) model whereinteractions between the fluid (blood) and structures (the arterial wall and atheromatousplaque) are conducted by an Arbitrary Lagrangian Eulerian (ALE) method that allowsaccurate fluid-structure coupling.

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.

When modeling the spread of infectious diseases, it is important to incorporate riskbehavior of individuals in a considered population. Not only risk behavior, but also thenetwork structure created by the relationships among these individuals as well as thedynamical rules that convey the spread of the disease are the key elements in predictingand better understanding the spread. In this work we propose the weighted randomconnection model, where each individual of the population is characterized by twoparameters: its position and risk behavior. A goal is to model the effect that theprobability of transmissions among individuals increases in the individual risk factors,and decays in their Euclidean distance. Moreover, the model incorporates a combined riskbehavior function for every pair of the individuals, through which the spread can bedirectly modeled or controlled. We derive conditions for the existence of an outbreak ofinfectious diseases in this model. Our main result is the almost sure existence of aninfinite component in the weighted random connection model. We use results on the randomconnection model and site percolation in Z2.

A general framework for age-structured predator-prey systems is introduced. Individualsare distinguished into two classes, juveniles and adults, and several possibleinteractions are considered. The initial system of partial differential equations isreduced to a system of (neutral) delay differential equations with one or two delays.Thanks to this approach, physically correct models for predator-prey with delay areprovided. Previous models are considered and analysed in view of the above results. ARosenzweig-MacArthur model with delay is presented as an example.

In this paper, an SIS (susceptible-infected-susceptible)-type epidemic propagation isstudied on a special class of 3-regular graphs, called modified cycle graphs. The modifiedcycle graph is constructed from a cycle graph with N nodes by connecting nodei to thenode i +d in a way that every node has exactly three links.Monte-Carlo simulations show that the propagation process depends on the value ofd in anon-monotone way. A new theoretical model is developed to explain this phenomenon. Thisreveals a new relation between the spreading process and the average path length in thegraph.

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.

The CMV matrices are unitary analogues of the discrete one-dimensional Schrödingeroperators. We review spectral properties of a few classes of CMV matrices and describefamilies of random and deterministic CMV matrices which exhibit a transition in thedistribution of their eigenvalues.

The basic reproduction number, R0, is often defined as the averagenumber of infections generated by a newly infected individual in a fully susceptiblepopulation. The interpretation, meaning, and derivation of R0 arecontroversial. However, in the context of mean field models, R0 demarcatesthe epidemic threshold below which the infected population approaches zero in the limit oftime. In this manner, R0 has been proposed as a method forunderstanding the relative impact of public health interventions with respect to diseaseeliminations from a theoretical perspective. The use of R0 is made morecomplex by both the strong dependency of R0 on the model form and the stochasticnature of transmission. A common assumption in models of HIV transmission that have closedform expressions for R0 is that a single individual’sbehavior is constant over time. In this paper we derive expressions for bothR0 and probability of an epidemic in afinite population under the assumption that people periodically change their sexualbehavior over time. We illustrate the use of generating functions as a general frameworkto model the effects of potentially complex assumptions on the number of transmissionsgenerated by a newly infected person in a susceptible population. We find that therelationship between the probability of an epidemic and R0 is notstraightforward, but, that as the rate of change in sexual behavior increases bothR0 and the probability of an epidemic alsodecrease.