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.

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.

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.

Cancer treatment using the antiangiogenic agents targets the evolution of the tumor vasculature. The aim is to significantly reduce supplies of oxygen and nutrients, and thus starve the tumor and induce its regression. In the paper we consider well established family of tumor angiogenesis models together with their recently proposed modification, that increases accuracy in the case of treatment using VEGF antibodies. We consider the optimal control problem of minimizing the tumor volume when the maximal admissible drug dose (the total amount of used drug) and the final level of vascularization are also taken into account. We investigate the solution of that problem for a fixed therapy duration. We show that the optimal strategy consists of the drug-free, full-dose and singular (with intermediate values of the control variable) intervals. Moreover, no bang-bang switch of the control is possible, that is the change from the no-dose to full-dose protocol (or in opposite direction) occurs on the interval with the singular control. For two particular models, proposed by Hahnfeldt et al. and Ergun et al., we provide additional theorems about the optimal control structure. We investigate the optimal controls numerically using the customized software written in MATLAB®, which we make freely available for download. Utilized numerical scheme is based on the composition of the well known gradient and shooting methods.

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.

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 prove that localization operators associated to ridgelet transforms withLp symbols are boundedlinear operators on L2(Rn).Operators closely related to these localization operators are shown to be in the traceclass and a trace formula for them is given.

Properties of blood cells and their interaction determine their distribution in flow. Itis observed experimentally that erythrocytes migrate to the flow axis, platelets to thevessel wall, and leucocytes roll along the vessel wall. In this work, a three-dimensionalmodel based on Dissipative Particle Dynamics method and a new hybrid (discrete-continuous)model for blood cells is used to study the interaction of erythrocytes with platelets andleucocytes in flow. Erythrocytes are modelled as elastic highly deformable membranes,while platelets and leucocytes as elastic membranes with their shape close to a sphere.Separation of erythrocytes and platelets in flow is shown for different values ofhematocrit. Erythrocyte and platelet distributions are in a good qualitative agreementwith the existing experimental results. Migration of leucocyte to the vessel wall and itsrolling along the wall is observed.

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.

In this paper we deal with a semilinear hyperbolic chemotaxis model in one space dimension evolving on a network, with suitable transmission conditions at nodes. This framework is motivated by tissue-engineering scaffolds used for improving wound healing. We introduce a numerical scheme, which guarantees global mass densities conservation. Moreover our scheme is able to yield a correct approximation of the effects of the source term at equilibrium. Several numerical tests are presented to show the behavior of solutions and to discuss the stability and the accuracy of our approximation.

We present sufficient conditions of local controllability for a class of models of treatment response to combined anticancer therapies which include delays in control strategies. The combined therapy is understood as combination of direct anticancer strategy e.g. chemotherapy and indirect modality (in this case antiangiogenic therapy). Controllability of the models in the form of semilinear second order dynamic systems with delays in control enables to answer the questions of realizability of different objectives of multimodal therapy in the presence of PK/PD effects. We compare results for the models without delays and conditions for relative local controllability of models with delays.

In a recent paper [E. Chacón Vera and D. Franco Coronil, J. Numer. Math. 20 (2012) 161–182.] a non standard mortar method for incompressible Stokes problem was introduced where the use of the trace spaces H1 / 2and H1/200and a direct computation of the pairing of the trace spaces with their duals are the main ingredients. The importance of the reduction of the number of degrees of freedom leads naturally to consider the stabilized version and this is the results we present in this work. We prove that the standard Brezzi–Pitkaranta stabilization technique is available and that it works well with this mortar method. Finally, we present some numerical tests to illustrate this behaviour.

We give a necessary and sufficient condition for pseudo-differential operators with SGsymbols to be compact from L2(ℝn)into L2(ℝn).

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.

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.