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

• Autores: Hernán Darío Toro Zapata, Carlos Andrés Trujillo Salazar
• Localización: Revista de Matemática: Teoría y Aplicaciones, ISSN 2215-3373, ISSN-e 2215-3373, Vol. 25, Nº. 2, 2018, págs. 261-294
• Idioma: español
• DOI: 10.15517/rmta.v25i2.33625
• Enlaces
  • Texto completo (pdf)
• Resumen
  • español

    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 transcriptasainversa 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 Principio del Máximo de Pontryagin. Se presentan simulaciones numéricas del modelo sin tratamiento y de los diferentes escenarios con tratamiento.

  • English

    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 reproductionnumber, 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 Maximum Principle of Pontryagin. Numerical simulations of the model without treatment and of the different scenarios with treatment are presented.

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