Examinando por Materia "Ordinary differential equations"
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Ítem Descripción del Modelo de Lorenz con aplicaciones(Universidad EAFIT, 2007-11-13) Calderón Saavedra, Pablo Emilio; Chaux M., Víctor Humberto; Montealegre Cárdenas, MauroÍtem Los sistemas dinámicos relacionados con el Efecto Josephson(Universidad EAFIT, 2008-04) Montealegre Cárdenas, Edgar; Montealegre Cárdenas, MauroÍtem Solving stochastic epidemiological models using computer algebra(SPIE-INT SOC OPTICAL ENGINEERING, 2011-01-01) Hincapie, D.; Ospina, J.; Hincapie, D.; Ospina, J.; Universidad EAFIT. Departamento de Ciencias; Lógica y ComputaciónMathematical modeling in Epidemiology is an important tool to understand the ways under which the diseases are transmitted and controlled. The mathematical modeling can be implemented via deterministic or stochastic models. Deterministic models are based on short systems of non-linear ordinary differential equations and the stochastic models are based on very large systems of linear differential equations. Deterministic models admit complete, rigorous and automatic analysis of stability both local and global from which is possible to derive the algebraic expressions for the basic reproductive number and the corresponding epidemic thresholds using computer algebra software. Stochastic models are more difficult to treat and the analysis of their properties requires complicated considerations in statistical mathematics. In this work we propose to use computer algebra software with the aim to solve epidemic stochastic models such as the SIR model and the carrier-borne model. Specifically we use Maple to solve these stochastic models in the case of small groups and we obtain results that do not appear in standard textbooks or in the books updated on stochastic models in epidemiology. From our results we derive expressions which coincide with those obtained in the classical texts using advanced procedures in mathematical statistics. Our algorithms can be extended for other stochastic models in epidemiology and this shows the power of computer algebra software not only for analysis of deterministic models but also for the analysis of stochastic models. We also perform numerical simulations with our algebraic results and we made estimations for the basic parameters as the basic reproductive rate and the stochastic threshold theorem. We claim that our algorithms and results are important tools to control the diseases in a globalized world. © 2011 Copyright Society of Photo-Optical Instrumentation Engineers (SPIE).Ítem Understanding epidemics from mathematical models: Details of the 2010 dengue epidemic in Bello (Antioquia, Colombia)(Elsevier Inc., 2017-03-01) Lizarralde-Bejarano, D.P.; Arboleda-Sánchez, S.; Puerta-Yepes, M.E.; Lizarralde-Bejarano, D.P.; Arboleda-Sánchez, S.; Puerta-Yepes, M.E.; Universidad EAFIT. Departamento de Ciencias; Matemáticas y AplicacionesDengue is the most threatening vector-borne viral disease in Colombia. At the moment, there is no treatment or vaccine available for its control or prevention; therefore, the main measure is to exert control over mosquito population. To reduce the economic impact of control measures, it is important to focus on specific characteristics related to local dengue epidemiology at the local level, and know the main factors involved in an epidemic. To this end, we used a mathematical model based on ordinary differential equations and experimental data regarding mosquito populations from Bello (Antioquia, Colombia) to simulate the epidemic occurred in 2010. The results showed that the parameters to which the incidence of dengue cases are most sensitive are the biting and mortality rates of adult mosquitoes as well as the virus transmission probabilities. Finally, we found that the Basic Reproductive Number (R0) of this epidemic was between 1.5 and 2.7, with an infection force (?) of 0.061, meaning that R0 values slightly above one are sufficient to result in a significant dengue outbreak in this region. © 2016 Elsevier Inc.