We present and characterize a multi-host epidemic model of Rift Valley fever (RVF) virusin East Africa with geographic spread on a network, rule-based mitigation measures, andmosquito infection and population dynamics. Susceptible populations are depleted bydisease and vaccination and are replenished with the birth of new animals. We observe thatthe severity of the epidemics is strongly correlated with the duration of the rainy seasonand that even severe epidemics are abruptly terminated when the rain stops. Becausenaturally acquired herd immunity is established, total mortality across 25 years isrelatively insensitive to many mitigation approaches. Strong reductions in cattlemortality are expected, however, with sufficient reduction in population densities ofeither vectors or susceptible (ie. unvaccinated) hosts. A better understanding of RVFepidemiology would result from serology surveys to quantify the importance of herdimmunity in epidemic control, and sequencing of virus from representative animals toquantify the realative importance of transportation and local reservoirs in nucleatingyearly epidemics. Our results suggest that an effective multi-layered mitigation strategywould include vector control, movement control, and vaccination of young animals yearly,even in the absence of expected rainfall.

We formulate an immuno-epidemiological model of coupled “within-host” model of ODEs and“between-host” model of ODE and PDE, using the Human Immunodeficiency Virus (HIV) forillustration. Existence and uniqueness of solution to the “between-host” model isestablished, and an explicit expression for the basic reproduction number of the“between-host” model derived. Stability of disease-free and endemic equilibria isinvestigated. An optimal control problem with drug-treatment control on the within-hostsystem is formulated and analyzed; these results are novel for optimal control of ODEslinked with such first order PDEs. Numerical simulations based on the forward-backwardsweep method are obtained.

Recent advances in high-resolution fluorescence microscopy have enabled the systematicstudy of morphological changes in large populations of cells induced by chemical andgenetic perturbations, facilitating the discovery of signaling pathways underlyingdiseases and the development of new pharmacological treatments. In these studies, though,due to the complexity of the data, quantification and analysis of morphological featuresare for the vast majority handled manually, slowing significantly data processing andlimiting often the information gained to a descriptive level. Thus, there is an urgentneed for developing highly efficient automated analysis and processing tools forfluorescent images. In this paper, we present the application of a method based on theshearlet representation for confocal image analysis of neurons. The shearletrepresentation is a newly emerged method designed to combine multiscale data analysis withsuperior directional sensitivity, making this approach particularly effective for therepresentation of objects defined over a wide range of scales and with highly anisotropicfeatures. Here, we apply the shearlet representation to problems of soma detection ofneurons in culture and extraction of geometrical features of neuronal processes in braintissue, and propose it as a new framework for large-scale fluorescent image analysis ofbiomedical data.

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.

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.

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.

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.

We consider a family of discrete Jacobi operators on the one-dimensional integer latticewith Laplacian and potential terms modulated by a primitive invertible two-lettersubstitution. We investigate the spectrum and the spectral type, the fractal structure andfractal dimensions of the spectrum, exact dimensionality of the integrated density ofstates, and the gap structure. We present a review of previous results, some applications,and open problems. Our investigation is based largely on the dynamics of trace maps. Thiswork is an extension of similar results on Schrödinger operators, although some of theresults that we obtain differ qualitatively and quantitatively from those for theSchrödinger operators. The nontrivialities of this extension lie in the dynamics of theassociated trace map as one attempts to extend the trace map formalism from theSchrödinger cocycle to the Jacobi one. In fact, the Jacobi operators considered here are,in a sense, a test item, as many other models can be attacked via the same techniques, andwe present an extensive discussion on this.

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

Consider the Schrödinger operator −∇2 + q with a smooth compactly supportedpotential q,q =q(x),x ∈R3.

Let A(β,α,k)be the corresponding scattering amplitude, k2 be the energy, α ∈S2 be the incident direction,β ∈S2 be the direction of scattered wave,S2 be the unit sphere in R3. Assume thatk =k0> 0 is fixed, andα =α0 is fixed. Then the scattering data areA(β) =A(β,α0,k0)= Aq(β)is a function on S2. The following inverse scatteringproblem is studied: IP: Given an arbitrary f ∈L2(S2)and an arbitrary small number ϵ> 0, can one find q ∈ C0∞(D) , where D ∈R3 is an arbitrary fixed domain, suchthat ||Aq(β) −f(β)||L2(S2)<ϵ? A positive answer to this question is given. A method for constructing such aq isproposed. There are infinitely many such q, not necessarily real-valued.

Scattering of electromagnetic (EM) waves by small (ka ≪ 1) impedance particleD of anarbitrary shape, embedded in a homogeneous medium, is studied. Analytic, closed form,formula for the scattered field is derived. The scattered field is of the orderO(a2 −κ), where κ ∈ [ 0,1)is a number. This field is much larger than in the case of Rayleigh-type scattering. Thenumerical results demonstrate a wide range of applicability of the analytic formula forthe scattered field. Comparison with Mie-type solution is carried out for various boundaryimpedances and radii of the particle.

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.

We investigate several equivalent notions of the Jost solution associated with a unitaryCMV matrix and provide a necessary and sufficient conditions for the Jost solution toconsist of entire functions of finite growth order in terms of super exponential decay ofVerblunsky coefficients. We also establish several one-to-one correspondences between CMVmatrices with super-exponentially decaying Verblunsky coefficients and spectral dataassociated with the first component of the Jost solution.

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.