Mathematical analysis of carbon nanotube model

In this paper, we present a recently developed mathematical model for a short double-wall carbon nanotube. The model is governed by a system of two coupled hyperbolic equations and is reduced to an evolution equation. This equation defines a dissipative semi-group. We show that the semi-group generator is an unbounded nonselfadjoint operator with compact resolvent. Moreover, this operator is a relatively compact perturbation of a certain selfadjoint operator.     

Mathematical analysis of a three-dimensional eutrophication model

In this paper we present and analyze a nutrient-phytoplankton-zooplankton-organic detritus-dissolved oxygen mathematical model simulating eutrophication processes into aquatic media. As a main result, we obtain existence and uniqueness results for the solution of the system, under realistic hypotheses of non-smooth coefficients (in particular, a non-regular water velocity). This lack or regularity prevent us from using the standard semigroup approach, forcing us towards the utilization of more refined techniques.     

Mathematical analysis of a three-strain tuberculosis transmission model

In recent years, bacteria have become resistant to antibiotics, leading to a decline in the effectiveness of antibiotics in treating infectious diseases. A mathematical model for multi-strain tuberculosis transmission dynamics to assess the burden of drug-sensitive, multidrug-resistant and extensively drug-resistant tuberculosis is formulated and analyzed. Each single strain submodel is shown to exhibit backward bifurcation when the threshold parameter is less than unity. Both analytical and numerical results show that resistance to drugs increase with increase in drug use, that is, active tuberculosis treatment results in a reduction of drug sensitive and increase in multidrug-resistant tuberculosis. Furthermore, use of second line drugs results in a decrease of the multidrug-resistant and increase of extensively drug resistant tuberculosis as most cases of multidrug resistant tuberculosis occur as a result of inappropriate, misuse or mismanaged treatment. Both the analytic results and numerical simulations suggest that quarantine of extensively drug resistant TB cases in addition to treatment of other forms of TB may be able to reduce the spread of the epidemic in poor resource-settings.     

Mathematical analysis of a wave energy converter model

In a context where sustainable development should be a priority, Orazov et al. have proposed in 2010, an excitation scheme for buoy-type ocean wave energy converters. The simplest model for this scheme is a non autonomous piecewise linear second order differential equation. The goal of the present paper is to give a mathematical framework for this model and to highlight some properties of its solutions. In particular, we will look at bounded and periodic solutions, and compare the energy-harvesting capabilities of this novel WEC with respect to that of a wave energy converter without mass modulation.     

Mathematical analysis of autonomous and non-autonomous age-structured reaction-diffusion-advection population model

In this paper, we study an age-structured reaction-diffusion-advection population model. First, we use a non-densely defined operator to the linear age-structured reaction-diffusion-advection population model in a patchy environment. By spectral analysis, we obtain the asynchronous exponential growth of the population model. Then we consider nonlinear death rate and birth rate, which all depend on the function related to the generalized total population, and we prove the existence of a steady state of the system. Finally, we study the age-structured reaction-diffusion-advection population model in non-autonomous situations. We give the comparison principle and prove the eventual compactness of semiflow by using integrated semigroup. We also prove the existence of compact attractors under the periodic situation.     

A mathematical analysis of a fish school model

A model proposed in the literature for fish schools of relatively large size is studied for mathematical and qualitative properties. Existence, uniqueness and positivity of solutions are established and bifurcation properties relative to diffusion and alignment parameters are studied.     

Dynamics of populations in fish farm: Analysis of stability and direction of Hopf-bifurcating periodic oscillation

The paper deals with the dynamical behavior of fish and mussel population in a fish farm where external food is supplied. The ecosystem of the fish farm is represented by a set of nonlinear differential equations involving nutrient (food), fish and mussel. We have listed some results already obtained. We have analyzed for the direction of Hopf-bifurcation, stability of the Hopf-bifurcating periodic orbits, and the period of the periodic orbits by using Poincare’ normal form and center manifold theory. We have performed numerical simulation to support the analytical results.     

Mathematical analysis and stability of a chemotaxis model with logistic term

In this paper we study a non‐linear system of differential equations arising in chemotaxis. The system consists of a PDE that describes the evolution of a population and an ODE which models the concentration of a chemical substance. We study the number of steady states under suitable assumptions, the existence of one global solution to the evolution problem in terms of weak solutions and the stability of the steady states. Copyright © 2004 John Wiley & Sons, Ltd.     

Mathematical analysis of a model of chemotaxis arising from morphogenesis

We consider non‐negative solution couples (u,v) of with positive parameters χ and λ, where the spatial domain is the interval (0,1). This system appears as a limit case of a model for morphogenesis proposed by Bollenbach et al. (Phys. Rev. E. 75 , 2007). Under suitable boundary conditions, modeling the presence of a morphogen source at x = 0, we prove the existence of a global and bounded weak solution using an approximation by problems where diffusion is introduced in the ordinary differential equation. Moreover, we prove the convergence of the solution to the unique steady state provided that χ is small and λ is large enough. Numerical simulations both illustrate these results and give rise to further conjectures on the solution behavior that go beyond the rigorously proved statements. Copyright © 2012 John Wiley & Sons, Ltd.     

Mathematical Modeling and Analysis of Spatial Neuron Dynamics: Dendritic Integration and Beyond

Neurons compute by integrating spatiotemporal excitatory (E) and inhibitory (I) synaptic inputs received from the dendrites. The investigation of dendritic integration is crucial for understanding neuronal information processing. Yet quantitative rules of dendritic integration and their mathematical modeling remain to be fully elucidated. Here neuronal dendritic integration is investigated by using theoretical and computational approaches. Based on the passive cable theory, a PDE-based cable neuron model with spatially branched dendritic structure is introduced to describe the neuronal subthreshold membrane potential dynamics, and the analytical solutions in response to conductance-based synaptic inputs are derived. Using the analytical solutions, a bilinear dendritic integration rule is identified, and it characterizes the change of somatic membrane potential when receiving multiple spatiotemporal synaptic inputs from the dendrites. In addition, the PDE-based cable neuron model is reduced to an ODE-based point-neuron model with the feature of bilinear dendritic integration inherited, thus providing an efficient computational framework of neuronal simulation incorporating certain important dendritic functions. The above results are further extended to active dendrites by numerical verification in realistic neuron simulations. Our work provides a comprehensive and systematic theoretical and computational framework for the study of spatial neuron dynamics. © 2021 The Authors. Communications on Pure and Applied Mathematics published by Wiley Periodicals LLC.     

Mathematical analysis of a simple model for the stripping reaction

Zusammenfassung Die Integralgleichung (1.6), die aus einem sehr vereinfachten Modell der stripping reaction hergeleitet ist, wird diskutiert. Es ist bemerkenswert, dass die bekannte Methode vonN. Wiener undE. Hopf auf diese komplizierte Integralgleichung angewendet werden kann. Die entstehende Funktionalgleichung (1.12) wird im § 2 auf die lösbare Differenzengleichung (2.20) reduziert. § 3 behandelt die Eindeutigkeit der Lösung und § 4 die explizite Lösung in einem Spezialfall. Die explizite Lösung des allgemeinen Falls wird nicht angegeben.

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Most of the content of this paper grew out of discussions, which the author had withJ. H. D. Jensen, J. M. Luttinger, andT. H. Berlin during their respective stays at the Institute for Advanced Study. The author's only merit consists in having written down the results.

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Institute for Advanced Study.     

Mathematical analysis for an epidemic model with spatial and age structure

A system of parabolic–hyperbolic equations with a non-local boundary condition, arising in mathematical theory of epidemics, is analyzed. For such a system, well–posedness as well as Sobolev regularity with respect to the space variables is proved. Asymptotic behavior of the solutions is also investigated.     

A simple computational model of digestion in the marine mussel

A simplified model of a marine mussel is proposed. We model the digestive system as a cellular analogue, physically partitioned into a lysosome and endosome, with the flow of particulate carbon to represent energy. The parameters in the rate equations that govern the interactions between compartments are assumed to be a simple linear form but are consistent with experiment and observation. The model is tested first by using continuous feeding and then by using a periodic feeding representative of tidal input. The responses in both cases produce interesting results compatible with observations of the biology of the animal. We define a “health function” which indicates whether or not the animal survives. It is found that survival is critically dependent on lysosomal efficiency.     

Mathematical model and optimization in production investment

We study a kind of nonlinear programming models derived from various investment problems, like industrial production investment, educational investment, farming investment, etc. We will analyze the properties of solutions of the models and obtain a polynomial algorithm. We will also apply our theory to a concrete example to demonstrate the algorithm complexity.     

Mathematical analysis and application of a cholera transmission model with waning vaccine-induced immunity

In this paper, a cholera infection model with vaccination is investigated, in which hyperinfectious and hypoinfectious vibrios, both human-to-human and environment-to-human transmission pathways, and waning vaccine-induced immunity are considered. The basic reproduction number is calculated and verified to be a threshold determining the global dynamics of the model. In addition, an application is demonstrated by investigating the cholera outbreak in Somalia, and the relevant control measures in the short term are given by elasticity and sensitivity analysis.