Controlling IL-7 Injections in HIV-Infected Patients

Immune interventions consisting in repeated injections are broadly used as they are thought to improve the quantity and the quality of the immune response. However, they also raise several questions that remain unanswered, in particular the number of injections to make or the delay to respect between different injections to achieve this goal. Practical and financial considerations add constraints to these questions, especially in the framework of human studies. We specifically focus here on the use of interleukin-7 (IL-7) injections in HIV-infected patients under antiretroviral treatment, but still unable to restore normal levels of \(\hbox {CD}4^{+}\) T lymphocytes. Clinical trials have already shown that repeated cycles of injections of IL-7 could help maintaining \(\hbox {CD}4^{+}\) T lymphocytes levels over the limit of 500 cells/\(\upmu \)L, by affecting proliferation and survival of \(\hbox {CD}4^{+}\) T cells. We then aim at answering the question: how to maintain a patients level of \(\hbox {CD}4^{+}\) T lymphocytes by using a minimum number of injections (i.e., optimizing the strategy of injections)? Based on mechanistic models that were previously developed for the dynamics of \(\hbox {CD}4^{+}\) T lymphocytes in this context, we model the process by a piecewise deterministic Markov model. We then address the question by using some recently established theory on impulse control problem in order to develop a numerical tool determining the optimal strategy. Results are obtained on a reduced model, as a proof of concept: the method allows to define an optimal strategy for a given patient. This method could be applied to optimize injections schedules in clinical trials.