Stability Analysis of a Population Growth Model of the Aedes aegypti via Hurwitz Polynomials

El presente trabajo busca criterios generales que garanticen la estabilidad asintótica alrededor de los puntos fijos de un modelo de crecimiento poblacional para el mosquito Aedes aegypti, estudiado previamente en otro trabajo, con el fin de obtener nuevas perspectivas sobre el modelo biológico y probar la utilidad de un procedimiento computacional para este análisis. El modelo había sido estudiado asumiendo valores estrictamente positivos para las tasas con las que tres importantes tipos de criaderos de mosquitos son introducidos en el ambiente. En este trabajo, esas suposiciones son relajadas y se derivan condiciones más generales para la estabilidad asintótica a partir del sistema de cinco ecuaciones resultante, en términos de parámetros no especificados, y mediante la aplicación de un resultado relacionado a los polinomios de Hurwitz (el Teorema de Liénard-Chipart). Se procura una implementación algorítmica de este resultado y se usa para el análisis previamente mencionado. Esta implementación se muestra como una herramienta adicional que ayuda con el proceso de analizar cuándo se tiene estabilidad asintótica en puntos fijos de sistemas autónomos en general.This work presents the search for general criteria to guarantee asymptotic stability around the fixed points of a population growth model of the Aedes aegypti mosquito previously studied in another work, with the goal of obtaining new insights about the biological model and to test the usefulness of a computational procedure in this analysis. The model had been studied assuming exclusively-positive values for the rates at which three different important types of mosquito breeding sites are introduced into the environment. In this work, these assumptions are dropped and more general conditions for asymptotic stability are derived for the resulting five-equation system, in terms or unspecified parameters, through the application of a result related to Hurwitz polynomials, namely the Liénard-Chipart Theorem. An algorithmic implementation of this result is sought and used for the aforementioned analysis and is seen to provide an additional tool to assist in the process of analyzing fixed points for asymptotic stability in general autonomous systems.1 Introduction 6 1.1 The Aedes aegypti mosquito . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.2 Control of mosquito population . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.3 Mathematical modeling of the Aedes aegypti mosquito . . . . . . . . . . . . 8 2 Stability theory 10 2.1 Basic concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.2 Linear systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.2.1 Definition and solution . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.2.2 Change of variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.3 Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.3.1 Stability of trajectories . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.3.2 Fixed points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.3.3 Nonlinear systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 2.3.4 Lyapunov stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 2.3.5 Limit points and limit cycles . . . . . . . . . . . . . . . . . . . . . . . 19 2.3.6 Hurwitz polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 2.3.6.1 Routh-Hurwitz criterion . . . . . . . . . . . . . . . . . . . . 20 2.3.6.2 Liénard-Chipart conditions . . . . . . . . . . . . . . . . . . 21 2.3.6.3 Routh’s algorithm . . . . . . . . . . . . . . . . . . . . . . . 21 2.3.6.4 Implications on stability . . . . . . . . . . . . . . . . . . . . 23 3 Analysis of models 24 3.1 Model with a logistic carrying capacity . . . . . . . . . . . . . . . . . . . . . 24 3.1.1 Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 3.1.2 Proposed analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 3.2 Complete model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 3.2.1 Analysis using factored characteristic polynomial . . . . . . . . . . . 28 3.2.1.1 Summary of analysis . . . . . . . . . . . . . . . . . . . . . . 29 3.2.2 Analysis using expanded characteristic polynomial . . . . . . . . . . . 30 3.2.2.1 Algorithm overview . . . . . . . . . . . . . . . . . . . . . . . 30 3.2.2.2 Tests with the algorithm . . . . . . . . . . . . . . . . . . . . 31 Conclusions 38 Appendix 40MaestríaMagíster en Matemátic

A feedback control perspective on biological control of dengue vectors by Wolbachia infection

International audienceControlling diseases such as dengue fever, chikungunya and zika fever by introduction of the intracellular parasitic bacterium $Wolbachia$ in mosquito populations which are their vectors, is presently quite a promising tool to reduce their spread. While description of the conditions of such experiments has received ample attention from biologists, entomologists and applied mathematicians, the issue of effective scheduling of the releases remains an interesting problem. Having in mind the important uncertainties present in the dynamics of the two populations in interaction, we attempt here to identify general ideas for building feedback-based release strategies, enforceable to a variety of models and situations. These principles are exemplified by several feedback control laws whose stabilizing properties are demonstrated, illustrated numerically and compared, when applied to a model retrieved from [P.-A. Bliman et al., Ensuring successful introduction of $Wolbachia$ in natural populations of $Aedes aegypti$ by means of feedback control. $J. of Math. Bio.$ 76(5):1269-1300, 2018]. The contribution is believed to be also of potential interest to tackle other important issues related to the biological control of vectors and pests. A crucial use of the theory of monotone dynamical systems is made in the derivations

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Aerial robots with perception, navigation, and manipulation capabilities are extending the range of applications of drones, allowing the integration of different sensor devices and robotic manipulators to perform inspection and maintenance operations on infrastructures such as power lines, bridges, viaducts, or walls, involving typically physical interactions on flight. New research and technological challenges arise from applications demanding the benefits of aerial robots, particularly in outdoor environments. This book collects eleven papers from different research groups from Spain, Croatia, Italy, Japan, the USA, the Netherlands, and Denmark, focused on the design, development, and experimental validation of methods and technologies for inspection and maintenance using aerial robots

Critical Thinking Skills Profile of High School Students In Learning Science-Physics

This study aims to describe Critical Thinking Skills high school students in the city of Makassar. To achieve this goal, the researchers conducted an analysis of student test results of 200 people scattered in six schools in the city of Makassar. The results of the quantitative descriptive analysis of the data found that the average value of students doing the interpretation, analysis, and inference in a row by 1.53, 1.15, and 1.52. This value is still very low when compared with the maximum value that may be obtained by students, that is equal to 10.00. This shows that the critical thinking skills of high school students are still very low. One fact Competency Standards science subjects-Physics is demonstrating the ability to think logically, critically, and creatively with the guidance of teachers and demonstrate the ability to solve simple problems in daily life. In fact, according to Michael Scriven stated that the main task of education is to train students and or students to think critically because of the demands of work in the global economy, the survival of a democratic and personal decisions and decisions in an increasingly complex society needs people who can think well and make judgments good. Therefore, the need for teachers in the learning device scenario such as: driving question or problem, authentic Investigation: Science Processes

Actes du CARI 2016 (Colloque africain sur la recherche en informatique et mathématiques appliquées)

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