Mathematical Modeling and Analysis of Epidemiological and Chemical Systems

This dissertation focuses on three interdisciplinary areas of applied mathematics, mathematical biology/epidemiology, economic epidemiology and mathematical physics, interconnected by the concepts and applications of dynamical systems.;In mathematical biology/epidemiology, a new deterministic SIS modeling framework for the dynamics of malaria transmission in which the malaria vector population is accounted for at each of its developmental stages is proposed. Rigorous qualitative and quantitative techniques are applied to acquire insights into the dynamics of the model and to identify and study two epidemiological threshold parameters reals* and R0 that characterize disease transmission and prevalence, and that can be used for disease control. It is shown that nontrivial disease-free and endemic equilibrium solutions, which can become unstable via a Hopf bifurcation exist. By incorporating vector demography; that is, by interpreting an aspect of the life cycle of the malaria vector, natural fluctuations known to exist in malaria prevalence are captured without recourse to external seasonal forcing and delays. Hence, an understanding of vector demography is necessary to explain the observed patterns in malaria prevalence. Additionally, the model exhibits a backward bifurcation. This implies that simply reducing R0 below unity may not be enough to eradicate the malaria disease. Since, only the female adult mosquitoes involved in disease transmission are identified and fully accounted for, the basic reproduction number (R0) for this model is smaller than that for previous SIS models for malaria. This, and the occurrence of both oscillatory dynamics and a backward bifurcation provide a novel and plausible framework for developing and implementing optimal malaria control strategies, especially those strategies that are associated with vector control.;In economic epidemiology, a deterministic and a stochastic model are used to investigate the effects of determinism, stochasticity, and safety nets on disease-driven poverty traps; that is, traps of low per capita income and high infectious disease prevalence. It is shown that economic development in deterministic models require significant external changes to the initial economic and health care conditions or a change in the parametric structure of the system. Therefore, poverty traps arising from deterministic models lead to more limited policy options. In contrast, there is always some probability that a population will escape or fall into a poverty trap in stochastic models. It is demonstrated that in stochastic models, a safety net can guarantee ultimate escape from the poverty trap, even when it is set within the basin of attraction of the poverty trap or when it is implemented only as an economic or health care intervention. It is also shown that the benefits of safety nets for populations that are close to the poverty trap equilibrium are highest for the stochastic model and lowest for the deterministic model. Based on the analysis of the stochastic model, the following optimal economic development and public health intervention questions are answered: (i) Is it preferable to provide health care, income/income generating resources, or both health care and income/income generating resources to enable populations to break cycles of poverty and disease; that is, escape from poverty traps? (ii) How long will it take a population that is caught in a poverty trap to attain economic development when the initial health and economic conditions are reinforced by safety nets?;In mathematical physics, an unusual form of multistability involving the coexistence of an infinite number of attractors that is exhibited by specially coupled chaotic systems is explored. It is shown that this behavior is associated with generalized synchronization and the emergence of a conserved quantity. The robustness of the phenomenon in relation to a mismatch of parameters of the coupled systems is studied, and it is shown that the special coupling scheme yields a new class of dynamical systems that manifests characteristics of dissipative and conservative systems

Entropy analysis and image encryption application based on a new chaotic system crossing a cylinder

Designing chaotic systems with specific features is a hot topic in nonlinear dynamics. In this study, a novel chaotic system is presented with a unique feature of crossing inside and outside of a cylinder repeatedly. This new system is thoroughly analyzed by the help of the bifurcation diagram, Lyapunov exponents' spectrum, and entropy measurement. Bifurcation analysis of the proposed system with two initiation methods reveals its multistability. As an engineering application, the system's efficiency is tested in image encryption. The complexity of the chaotic attractor of the proposed system makes it a proper choice for encryption. States of the chaotic attractor are used to shue the rows and columns of the image, and then the shued image is XORed with the states of chaotic attractor. The unpredictability of the chaotic attractor makes the encryption method very safe. The performance of the encryption method is analyzed using the histogram, correlation coefficient, Shannon entropy, and encryption quality. The results show that the encryption method using the proposed chaotic system has reliable performance. - 2019 by the authors.Scopu

Coherent multi-mode dynamics in a Quantum Cascade Laser: Amplitude and Frequency-modulated Optical Frequency Combs

We cast a theoretical model based on Effective Semiconductor Maxwell-Bloch Equations and study the dynamics of a multi-mode mid-Infrared Quantum Cascade Laser in Fabry Perot with the aim to investigate the spontaneous generation of optical frequency combs. This model encompasses the key features of a semiconductor active medium such as asymmetric,frequency-dependent gain and refractive index as well as the phase-amplitude coupling of the field dynamics provided by the linewidth enhancement factor. Our numerical simulations are in excellent agreement with recent experimental results, showing broad ranges of comb formationin locked regimes, separated by chaotic dynamics when the field modes unlock. In the former case, we identify self-confined structures travelling along the cavity, while the instantaneous frequency is characterized by a linear chirp behaviour. In such regimes we show that OFC are characterized by concomitant and relevant amplitude and frequency modulation

Oscillation patterns in negative feedback loops

Organisms are equipped with regulatory systems that display a variety of dynamical behaviours ranging from simple stable steady states, to switching and multistability, to oscillations. Earlier work has shown that oscillations in protein concentrations or gene expression levels are related to the presence of at least one negative feedback loop in the regulatory network. Here we study the dynamics of a very general class of negative feedback loops. Our main result is that in these systems the sequence of maxima and minima of the concentrations is uniquely determined by the topology of the loop and the activating/repressing nature of the interaction between pairs of variables. This allows us to devise an algorithm to reconstruct the topology of oscillating negative feedback loops from their time series; this method applies even when some variables are missing from the data set, or if the time series shows transients, like damped oscillations. We illustrate the relevance and the limits of validity of our method with three examples: p53-Mdm2 oscillations, circadian gene expression in cyanobacteria, and cyclic binding of cofactors at the estrogen-sensitive pS2 promoter.Comment: 10 pages, 8 figure

Dynamical principles in neuroscience

Dynamical modeling of neural systems and brain functions has a history of success over the last half century. This includes, for example, the explanation and prediction of some features of neural rhythmic behaviors. Many interesting dynamical models of learning and memory based on physiological experiments have been suggested over the last two decades. Dynamical models even of consciousness now exist. Usually these models and results are based on traditional approaches and paradigms of nonlinear dynamics including dynamical chaos. Neural systems are, however, an unusual subject for nonlinear dynamics for several reasons: (i) Even the simplest neural network, with only a few neurons and synaptic connections, has an enormous number of variables and control parameters. These make neural systems adaptive and flexible, and are critical to their biological function. (ii) In contrast to traditional physical systems described by well-known basic principles, first principles governing the dynamics of neural systems are unknown. (iii) Many different neural systems exhibit similar dynamics despite having different architectures and different levels of complexity. (iv) The network architecture and connection strengths are usually not known in detail and therefore the dynamical analysis must, in some sense, be probabilistic. (v) Since nervous systems are able to organize behavior based on sensory inputs, the dynamical modeling of these systems has to explain the transformation of temporal information into combinatorial or combinatorial-temporal codes, and vice versa, for memory and recognition. In this review these problems are discussed in the context of addressing the stimulating questions: What can neuroscience learn from nonlinear dynamics, and what can nonlinear dynamics learn from neuroscience?This work was supported by NSF Grant No. NSF/EIA-0130708, and Grant No. PHY 0414174; NIH Grant No. 1 R01 NS50945 and Grant No. NS40110; MEC BFI2003-07276, and Fundación BBVA

Dynamics of microresonator frequency comb generation: models and stability

Microresonator frequency combs hold promise for enabling a new class of light sources that are simultaneously both broadband and coherent, and that could allow for a profusion of potential applications. In this article, we review various theoretical models for describing the temporal dynamics and formation of optical frequency combs. These models form the basis for performing numerical simulations that can be used in order to better understand the comb generation process, for example helping to identify the universal combcharacteristics and their different associated physical phenomena. Moreover, models allow for the study, design and optimization of comb properties prior to the fabrication of actual devices. We consider and derive theoretical formalisms based on the Ikeda map, the modal expansion approach, and the Lugiato-Lefever equation. We further discuss the generation of frequency combs in silicon resonators featuring multiphoton absorption and free-carrier effects. Additionally, we review comb stability properties and consider the role of modulational instability as well as of parametric instabilities due to the boundary conditions of the cavity. These instability mechanisms are the basis for comprehending the process of frequency comb formation, for identifying the different dynamical regimes and the associated dependence on the comb parameters. Finally, we also discuss the phenomena of continuous wave bi- and multistability and its relation to the observation of mode-locked cavity solitons

A New Chaotic System with Multiple Attractors: Dynamic Analysis, Circuit Realization and S-Box Design

This paper reports about a novel three-dimensional chaotic system with three nonlinearities. The system has one stable equilibrium, two stable equilibria and one saddle node, two saddle foci and one saddle node for different parameters. One salient feature of this novel system is its multiple attractors caused by different initial values. With the change of parameters, the system experiences mono-stability, bi-stability, mono-periodicity, bi-periodicity, one strange attractor, and two coexisting strange attractors. The complex dynamic behaviors of the system are revealed by analyzing the corresponding equilibria and using the numerical simulation method. In addition, an electronic circuit is given for implementing the chaotic attractors of the system. Using the new chaotic system, an S-Box is developed for cryptographic operations. Moreover, we test the performance of this produced S-Box and compare it to the existing S-Box studies

Non-classical behavior in multimode and disordered transverse structures in OPO. Use of the Q representation

We employ the Q representation to study the non-classical correlations that are present from below to above-threshold in the degenerate optical parametric oscillator. Our study shows that such correlations are present just above threshold, in the regime in which stripe patterns are formed, but that they also persist further above threshold in the presence of spatially disordered structures