En este trabajo se estudia la dinámica de infección por VIH, a través de los procesos estocásticos de nacimiento y muerte y los sistemas de ecuaciones diferenciales que representan un sistema real. Para éste caso en específico, se describe un proceso estocástico que interpreta la dinámica de infección del VIH al interior del organismo de una persona en sus etapas iniciales de infección (post exposición o periodo de ventana); es decir, se considera que el momento mismo en que el virus ingresa en el organismo corresponde al tiempo inicial para el modelo, y a partir de entonces se tiene en cuenta el proceso de replicación y las incidencias que el virus genera cuando ataca las células T CD4+, las cuales, son pieza fundamental en el sistema inmunológico del paciente. El proceso estocástico permite deducir a partir de primeros principios, un modelo básico para la infección por VIH, similar a los estudiados en la literatura; es decir, un sistema basado en ecuaciones diferenciales ordinarias de variable estocástica, donde las variables de estado corresponden a valores esperados (promedios) y en ese sentido se encuentran también ecuaciones diferenciales para la varianza de esas variables de estado, lo que proporcionará información adicional sobre el sistema. Finalmente se presenta el estudio analítico local del modelo completo y un estudio numérico de las soluciones del sistema usando valores de los parámetros obtenidos de fuentes secundarias, con el fin de ilustrar los resultados analíticos.

ObjectivesMathematical models can be helpful to understand the complex dynamics of human immunodeficiency virus infection within a host. Most of work has studied the interactions of host responses and virus in the presence of active cytotoxic immune cells, which decay to zero when there is no virus. However, recent research highlights that cytotoxic immune cells can be inactive but never be depleted.MethodsWe propose a mathematical model to investigate the human immunodeficiency virus dynamics in the presence of both active and inactive cytotoxic immune cells within a host. We explore the impact of the immune responses on the dynamics of human immunodeficiency virus infection under different disease stages.ResultsStandard mathematical and numerical analyses are presented for this new model. Specifically, the basic reproduction number is computed and local and global stability analyses are discussed.ConclusionOur results can give helpful insights when designing more effective drug schedules in the presence of active and inactive immune responses.

HIV interaction with the immune response is modeled mathematically. Initially, a detailed model is proposed that consists of a system of differential equations including immune cells (antigen presenting cells, T latent infected cells, T actively infected cells, resting T cells, helper T cells, inactive cytotoxic cells and active cytotoxic cells) and viral particles. Then, stability conditions are given from the basic reproduction number and numerical simulations are performed. From this it is possible to conclude what are the most influential parameters to reduce infection. From the initial model, a control problem is formulated in order to determine the most appropriate type of intervention to ensure high levels of activated T cells and immune response. Five different control strategies based on antiretroviral are evaluated to conclude that a strategy of constant control, obtained as the average value of optimal control, provides satisfactory results.

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A mathematical model for the transmission dynamics of human immunodeficiency virus (HIV) within a host is developed. Our model focuses on the roles of immune response cells or cytotoxic lymphocytes (CTLs). The model includes active and inactive cytotoxic immune cells. The basic reproduction number and the global stability of the virus free equilibrium is carried out. The model is modified to include anti-retroviral treatment interventions and the controlled reproduction number is explored. Their effects on the HIV infection dynamics are investigated. Two different disease stage scenarios are assessed: early-stage and advanced-stage of the disease. Furthermore, optimal control theory is employed to enhance healthy CD4 + T cells, active cytotoxic immune cells and minimize the total cost of anti-retroviral treatment interventions. Two different anti-retroviral treatment interventions (RTI and PI) are incorporated. The results highlight the key roles of cytotoxic immune response in the HIV infection dynamics and corresponding optimal treatment strategies. It turns out that the combined control (both RTI and PI) and stronger immune response is the best intervention to maximize healthy CD4 + T cells at a minimal cost of treatments.