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Revista de Matemática Teoría y Aplicaciones

versão impressa ISSN 1409-2433

Rev. Mat vol.25 no.2 San José Jul./Dez. 2018

http://dx.doi.org/10.15517/rmta.v25i2.33625 

Artículos

Modelo para el control óptimo del VIH con tasa de infección dependiente de la densidad del virus

HIV optimal control model with infection rate depending on the virus density

Hernán Darío Toro-Zapata1 

Carlos Andrés Trujillo-Salazar1 

1Licenciatura en Matemáticas, Universidad del Quindío, Quindío, Colombia. E-Mail: hdtoro@uniquindio.edu.co, catrujillo@uniquindio.edu.co

Resumen

[13]

Se propone un modelo en ecuaciones diferenciales ordinarias para describir la dinámica de infección por VIH en una población de células T CD4 susceptibles a la infección y considerando una tasa de infección no lineal densodependiente. Se analiza la estabilidad del modelo con base en el número básico de reproducción, lo que permite determinar resultados de estabilidad y un umbral de control mediante la reducción de la tasa máxima de infección. Luego se formula un problema de control óptimo para establecer funciones óptimas de tratamiento mediante inhibidores de transcriptasa inversa e inhibidores de proteasa, que minimicen la carga viral y los costos directos y/o indirectos de la administración del tratamiento. Se estudian los casos en que la efectividad del tratamiento es nula y plena, y para el caso de efectividad imperfecta del tratamiento se acude al PrincipiodelMáximodePontryagin. Se presentan simulaciones numéricas del modelo sin tratamiento y de los diferentes escenarios con tratamiento.

Palabras clave: sistemas dinámicos; estabilidad; control óptimo; principio del máximo de Pontryagin; VIH; terapia antirretroviral

Abstract

[17]

We propose a model on ordinary differential equations to describe the dynamics of HIV infection in a population of CD4 T cells susceptible to infection and considering a nonlinear infection rate depending on viral density. The stability of the model is analyzed based on the basic reproduction number, which allows us to determine stability results and a control threshold by reducing the rate of maximum infection. An optimal control problem is then formulated to establish optimal treatment functions by reverse transcriptase inhibitors and protease inhibitors that minimize viral load and direct and/or indirect costs of treatment administration. We study the cases in which the effectiveness of the treatment is null and full, and for the case of imperfect effectiveness of the treatment, we refer to the MaximumPrincipleofPontryagin. Numerical simulations of the model without treatment and of the different scenarios with treatment are presented.

Keywords: dynamic system; stability; optimal control; Pontryagin maximum principle; HIV; antirretroviral therapy

Mathematics Subject Classification: 93C15, 49J15, 92B05, 92C50.

[23]

Ver contenido completo en pdf.

Agradecimientos

El presente trabajo es financiado por la Universidad del Quindío, a través de la Vicerrectoría de Investigaciones. Proyecto de investigación No. 573.

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Recibido: 04 de Agosto de 2017; Revisado: 06 de Marzo de 2018; Aprobado: 02 de Mayo de 2018

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