A striking correspondence between the dynamics generated by the vector fields and by the scalar parabolic equations

The purpose of this paper is to enhance a correspondence between the dynamics of the differential equations $\dot y(t)=g(y(t))$ on $\mathbb{R}^d$ and those of the parabolic equations $\dot u=\Delta u +f(x,u,\nabla u)$ on a bounded domain $\Omega$. We give details on the similarities of these dynamics in the cases $d=1$, $d=2$ and $d\geq 3$ and in the corresponding cases $\Omega=(0,1)$, $\Omega=\mathbb{T}^1$ and dim($\Omega$)$\geq 2$ respectively. In addition to the beauty of such a correspondence, this could serve as a guideline for future research on the dynamics of parabolic equations

Nonautonomous Conley index: theory and applications

Conley index theory associates isolated invariant sets with an index e.g, a topological space. This theory is generalized to nonautonomous dynamical systems. A corresponding theory of Morse-decompositions is developed. Applications to dynamical systems obtained from ordinary and partial differential equations are discussed

Global dynamics of periodic infectious disease models with time-dependent delays

Many infectious diseases have seasonal trends and exhibit variable periods of peak seasonality. Understanding the population dynamics due to seasonal changes becomes very important for predicting and controlling disease transmission risks. For some directly transmitted and vector-borne diseases, the length of the incubation period strongly depends on the temperature. This thesis is devoted to the study of the global dynamics of some periodic epidemic models with periodic incubation periods. We start with a classical SEIRS epidemic model with a time-dependent latent period in Chapter 2. Moreover, vector-borne diseases, such as West Nile virus, bluetongue, and malaria, are always highly dependent on seasonal change, especially the temperature. To investigate the seasonal effects and temperature-dependent delays on West Nile virus, we present a periodic functional differential equations model with the vertical transmission, the periodic maturation delay, and the periodic extrinsic incubation period in Chapter 3. In Chapter 4, we propose a bluetongue model with seasonality and temperature-dependent incubation period, which describes the dynamics of bluetongue transmission via cattle and sheep as hosts and midges as vectors. To explore the effects of the spatial and temporal heterogeneity in hosts and vectors, and only vector movements on the spread of bluetongue, we develop a nonlocal periodic reaction-diffusion model of bluetongue disease with periodic time delays in Chapter 5. Based on the theory of the basic reproduction ratio, we derive and numerically compute the basic reproduction ratio for our models. By the theory of dynamical systems, we show that the basic reproduction ratio acts as a threshold parameter for the global dynamics for each model. Numerical simulations or case studies are carried out to illustrate the analytic results and help us provide some new findings. At the end of this thesis, we present a brief summary and some interesting future works

Systems biology approaches to the dynamics of gene expression and chemical reactions

Systems biology is an emergent interdisciplinary field of study whose main goal is to understand the global properties and functions of a biological system by investigating its structure and dynamics [74]. This high-level knowledge can be reached only with a coordinated approach involving researchers with different backgrounds in molecular biology, the various omics (like genomics, proteomics, metabolomics), computer science and dynamical systems theory. The history of systems biology as a distinct discipline began in the 1960s, and saw an impressive growth since year 2000, originated by the increased accumulation of biological information, the development of high-throughput experimental techniques, the use of powerful computer systems for calculations and database hosting, and the spread of Internet as the standard medium for information diffusion [77]. In the last few years, our research group tried to tackle a set of systems biology problems which look quite diverse, but share some topics like biological networks and system dynamics, which are of our interest and clearly fundamental for this field. In fact, the first issue we studied (covered in Part I) was the reverse engineering of large-scale gene regulatory networks. Inferring a gene network is the process of identifying interactions among genes from experimental data (tipically microarray expression profiles) using computational methods [6]. Our aim was to compare some of the most popular association network algorithms (the only ones applicable at a genome-wide level) in different conditions. In particular we verified the predictive power of similarity measures both of direct type (like correlations and mutual information) and of conditional type (partial correlations and conditional mutual information) applied on different kinds of experiments (like data taken at equilibrium or time courses) and on both synthetic and real microarray data (for E. coli and S. cerevisiae). In our simulations we saw that all network inference algorithms obtain better performances from data produced with \u201cstructural\u201d perturbations (like gene knockouts at steady state) than with just dynamical perturbations (like time course measurements or changes of the initial expression levels). Moreover, our analysis showed differences in the performances of the algorithms: direct methods are more robust in detecting stable relationships (like belonging to the same protein complex), while conditional methods are better at causal interactions (e.g. transcription factor\u2013binding site interactions), especially in presence of combinatorial transcriptional regulation. Even if time course microarray experiments are not particularly useful for inferring gene networks, they can instead give a great amount of information about the dynamical evolution of a biological process, provided that the measurements have a good time resolution. Recently, such a dataset has been published [119] for the yeast metabolic cycle, a well-known process where yeast cells synchronize with respect to oxidative and reductive functions. In that paper, the long-period respiratory oscillations were shown to be reflected in genome-wide periodic patterns in gene expression. As explained in Part II, we analyzed these time series in order to elucidate the dynamical role of post-transcriptional regulation (in particular mRNA stability) in the coordination of the cycle. We found that for periodic genes, arranged in classes according either to expression profile or to function, the pulses of mRNA abundance have phase and width which are directly proportional to the corresponding turnover rates. Moreover, the cascade of events which occurs during the yeast metabolic cycle (and their correlation with mRNA turnover) reflects to a large extent the gene expression program observable in other dynamical contexts such as the response to stresses or stimuli. The concepts of network and of systems dynamics return also as major arguments of Part III. In fact, there we present a study of some dynamical properties of the so-called chemical reaction networks, which are sets of chemical species among which a certain number of reactions can occur. These networks can be modeled as systems of ordinary differential equations for the species concentrations, and the dynamical evolution of these systems has been theoretically studied since the 1970s [47, 65]. Over time, several independent conditions have been proved concerning the capacity of a reaction network, regardless of the (often poorly known) reaction parameters, to exhibit multiple equilibria. This is a particularly interesting characteristic for biological systems, since it is required for the switch-like behavior observed during processes like intracellular signaling and cell differentiation. Inspired by those works, we developed a new open source software package for MATLAB, called ERNEST, which, by checking these various criteria on the structure of a chemical reaction network, can exclude the multistationarity of the corresponding reaction system. The results of this analysis can be used, for example, for model discrimination: if for a multistable biological process there are multiple candidate reaction models, it is possible to eliminate some of them by proving that they are always monostationary. Finally, we considered the related property of monotonicity for a reaction network. Monotone dynamical systems have the tendency to converge to an equilibrium and do not present chaotic behaviors. Most biological systems have the same features, and are therefore considered to be monotone or near-monotone [85, 116]. Using the notion of fundamental cycles from graph theory, we proved some theoretical results in order to determine how distant is a given biological network from being monotone. In particular, we showed that the distance to monotonicity of a network is equal to the minimal number of negative fundamental cycles of the corresponding J-graph, a signed multigraph which can be univocally associated to a dynamical system

Modeling the impact of climate change on tick population dynamics and tick-borne disease spread

There is increasing evidence that Lyme disease is an emerging threat for public health in Canada. In this dissertation, we study the impact of climate change on establishment/extinction of the Lyme disease tick vector Ixodes scapularis and Lyme pathogen Borrelia burgdorferi in the Canadian landscape by utilizing mathematical techniques and computer simulations. We develop principal mathematical frameworks using ordinary differential equations with periodic coefficients, partial differential equations with seasonality and a periodic system of delay differential equations with periodic delay. We develop analysis and tools to predict the long-term status of Lyme disease transmission dynamics in the vector population. We determine factors, which are of interest to public health policy makers, for Lyme disease prevention and control in Canada in varying environmental conditions. We provide the theoretical foundation of a risk map of Lyme disease in Canada

Advances in Robotics, Automation and Control

The book presents an excellent overview of the recent developments in the different areas of Robotics, Automation and Control. Through its 24 chapters, this book presents topics related to control and robot design; it also introduces new mathematical tools and techniques devoted to improve the system modeling and control. An important point is the use of rational agents and heuristic techniques to cope with the computational complexity required for controlling complex systems. Through this book, we also find navigation and vision algorithms, automatic handwritten comprehension and speech recognition systems that will be included in the next generation of productive systems developed by man