The Susceptible-Infected-Recovered (SIR) model for the spread of an infectious disease ina population of constant size is considered. In order to control the spread of infection,we propose the model with four bounded controls which describe vaccination of newborns,vaccination of the susceptible, treatment of infected, and indirect strategies aimed at areduction of the incidence rate (e. g. quarantine). The optimal control problem ofminimizing the total number of the infected individuals on a given time interval is statedand solved. The optimal solutions are obtained with the use of the Pontryagin MaximumPrinciple and investigated analytically. Numerical results are presented to illustrate theoptimal solutions.

In this paper, we shall obtain the symmetries of the mathematical model describingspontaneous relaxation of eastward jets into a meandering state and use these symmetriesfor constructing the conservation laws. The basic eastward jet is a spectral parameter ofthe model, which is in geostrophic equilibrium with the basic density structure and whichguarantees the existence of nontrivial conservation laws.

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.

We first investigate linear size-structured population models with spacial diffusion. Existence of a unique mild solution is established. Then we consider a harvesting problem for linear size-structured models with diffusion and show the existence of an optimal harvesting effort to maximize the total price or total harvest.

We present a fully automatic approach to recover boundary conditions and locations of thevessel wall, given a crude initial guess and some velocity cross-sections, which can becorrupted by noise. This paper contributes to the body of work regarding patient-specificnumerical simulations of blood flow, where the computational domain and boundaryconditions have an implicit uncertainty and error, that derives from acquiring andprocessing clinical data in the form of medical images. The tools described in this paperfit well in the current approach of performing patient-specific simulations, where areasonable segmentation of the medical images is used to form the computational domain,and boundary conditions are obtained as velocity cross-sections from phase-contrastmagnetic resonance imaging. The only additional requirement in the proposed methods is toobtain additional velocity cross-section measurements throughout the domain. The toolsdeveloped around optimal control theory, would then minimize a user defined cost functionto fit the observations, while solving the incompressible Navier-Stokes equations.Examples include two-dimensional idealized geometries and an anatomically realisticsaccular geometry description.

Efficiency of mixing, resulting from the reflection of an internal wave field imposed onthe oscillatory background flow with a three-dimensional bottom topography, isinvestigated using a linear approximation. The radiating wave field is associated with thespectrum of the linear model, which consists of those mode numbers n and slope valuesα, forwhich the solution represents the internal waves of frequencies ω =nω0 radiating upwrad ofthe topography, where ω0 is the fundamental frequency at whichinternal waves are generated at the topography. The effects of the bottom topography andthe earth’s rotation on the spectrum is analyzed analytically and numerically in thevicinity of the critical slope

αn,θc = arcsin (n 2ω02-f 2 / N 2-f 2) 1/2

which is a slope with the same angle to the horizontal as the internal wavecharacteristic. In this notation, θ is latitude, f is the Coriolis parameterand N is thebuoyancy frequency, which is assumed to be a constant, which corresponds to the uniformstratification.

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.

A mathematical model, coupling the dynamics of short-term stem-like cells and matureleukocytes in leukemia with that of the immune system, is investigated. The model isdescribed by a system of seven delay differential equations with seven delays. Threeequilibrium points E0, E1,E2 are highlighted. The stability and the existence of theHopf bifurcation for the equilibrium points are investigated. In the analysis of themodel, the rate of asymmetric division and the rate of symmetric division are veryimportant.

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.

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.

We introduce a game theoretical model of stealing interactions. We model the situation asan extensive form game when one individual may attempt to steal a valuable item fromanother who may in turn defend it. The population is not homogeneous, but rather eachindividual has a different Resource Holding Potential (RHP). We assume that RHP not onlyinfluences the outcome of the potential aggressive contest (the individual with the largerRHP is more likely to win), but that it also influences how an individual values aparticular resource. We investigate several valuation scenarios and study the prevalenceof aggressive behaviour. We conclude that the relationship between RHP and resource valueis crucial, where some cases lead to fights predominantly between pairs of strongindividuals, and some between pairs of weak individuals. Other cases lead to no fightswith one individual conceding, and the order of strategy selection is crucial, where theindividual which picks its strategy first often has an advantage.

We review mathematical results about the qualitative structure of chemotherapy protocols that were obtained with the methods of optimal control. As increasingly more complex features are incorporated into the mathematical model—progressing from models for homogeneous, chemotherapeutically sensitive tumor cell populations to models for heterogeneous agglomerations of subpopulations of various sensitivities to models that include tumor immune-system interactions—the structures of optimal controls change from bang-bang solutions (which correspond to maximum dose rate chemotherapy with restperiods) to solutions that favor singular controls (representing reduced dose rates). Medically, this corresponds to a transition from standard MTD (maximum tolerated dose) type protocols to chemo-switch strategies towards metronomic dosing.

A traceless variant of the Johnson-Segalman viscoelastic model is presented and appliedto blood flow simulations. The viscoelastic extra stress tensor is decomposed into itstraceless (deviatoric) and spherical parts, leading to a reformulation of the classicalJohnson-Segalman model. The equivalence of the two models is established comparing modelpredictions for simple test cases. The new model is validated using several 2D benchmarkproblems, designed to reproduce difficulties that arise in the simulation of blood flow inblood vessels or medical devices. The structure and behaviour of the new model arediscussed and the future use of the new model in envisioned, both on the theoretical andnumerical perspectives.

In the present paper we study the reconstruction of a structured quadratic pencil fromeigenvalues distributed on ellipses or parabolas. A quadratic pencil is a square matrixpolynomial

QP(λ) = M λ2+Cλ +K,

where M,C, andK are realsquare matrices. The approach developed in the paper is based on the theory of orthogonalpolynomials on the real line. The results can be applied to more general distribution ofeigenvalues. The problem with added single eigenvector is also briefly discussed. As anillustration of the reconstruction method, the eigenvalue problem on linearized stabilityof certain class of stationary exact solution of the Navier-Stokes equations describingatmospheric flows on a spherical surface is reformulated as a simple mass-spring system bymeans of this method.

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.

A feature often observed in epidemiological networks is significant heterogeneity indegree. A popular modelling approach to this has been to consider large populations withhighly heterogeneous discrete contact rates. This paper defines an individual-levelnon-Markovian stochastic process that converges on standard ODE models of such populationsin the appropriate asymptotic limit. A generalised Sellke construction is derived for thismodel, and this is then used to consider final outcomes in the case where heterogeneityfollows a truncated Zipf distribution.

We show that simple diffusion processes are weak limits of piecewise continuous processesconstructed within a totally deterministic framework. The proofs are based on thecontinuous mapping theorem and the functional central limit theorem.

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.

The majority of species are under predatory risk in their natural habitat and targeted bypredators as part of the food web. During the evolution of ecosystems, manifold mechanismshave emerged to avoid predation. So called secondary defences, which are used after apredator has initiated prey-catching behaviour, commonly involve the expression of toxinsor deterrent substances which are not observable by the predator. Hence, the possession ofsuch secondary defence in many prey species comes with a specific signal of that defence(aposematism). This paper builds on the ideas of existing models of such signallingbehaviour, using a model of co-evolution and generalisation of aversive information andintroduces a new methodology of numerical analysis for finite populations. This newmethodology significantly improves the accessibility of previous models.

In finite populations, investigating the co-evolution of defence and signalling requiresan understanding of natural selection as well as an assessment of the effects of drift asan additional force acting on stability. The new methodology is able to reproduce thepredicted solutions of preceding models and finds additional solutions involving negativecorrelation between signal strength and the extent of secondary defence. In addition,genetic drift extends the range of stable aposematic solutions through the introduction ofa new pseudo-stability and gives new insights into the diversification of aposematicdisplays.

We consider Schrödinger operators Hα given by equation(1.1) below. We study the asymptotic behavior of the spectral density E(Hα,λ)for λ → 0 andthe L1 →L∞ dispersive estimates associated to theevolution operator e−itHα.In particular we prove that for positive values of α, the spectral densityE(Hα,λ)tends to zero as λ →0 with higher speed compared to the spectral density of Schrödingeroperators with a short-range potential V. We then show how the long time behavior ofe−itHαdepends on α.More precisely we show that the decay rate of e−itHαfor t → ∞ canbe made arbitrarily large provided we choose α large enough and consider a suitable operatornorm.