Maximum response perturbation-based control of virus infection model with time-delays

Abstract

A new method for constructing the multi-modal impacts on the immune systemin the chronic phase of viral infection, based on mathematical models formulated with delay-differential equations is proposed. The so called, optimal disturbances, widely used in the aerodynamic stability theory for mathematical models without delays are constructed for perturbing the steady states of the dynamical system for maximizing the perturbation-induced response. The concept of optimal disturbances is generalized on the systems with delayed argument. An algorithm for computing the optimal disturbances is developed for such systems. The elaborated computational technology is tested on a system of four nonlinear delay-differential equations which represents the model of experimental infection in mice caused by lymphocytic choriomeningitis virus. The steady-state perturbations resulting in a maximum responsewere computed with the proposed algorithm for two types of steady states characterized by a low and a high levels of viral load. The possibility of correction of the infection dynamics and the restoration of virus-specific lymphocyte functioning of the immune system by perturbing the steady states is demonstrated.

Bistability analysis of virus infection models with delayed arguments

“…При ее решении наиболее важным является предсказание бистабильности в зависимости от параметров математических моделей, поскольку это, в конечном счете, определяет стратегию оптимального воздействия на рассматриваемую систему. Для бистабильной динамической системы перевод в благоприятное стационарное состояние может быть осуществлен на основе оптимальных возмущений [8,9,10]. В случае же моностабильности перевод из области нежелательного стационарного состояния предполагает применение стратегий стабилизации динамики системы по принципу обратной связи или же программного управления [11].…”

“…Эта технология включает в себя методы вычисления всех стационарных состояний при фиксированных значениях параметров, их трассировки по параметрам модели и анализа их устойчивости. Методы построения оптимальных возмущений описаны в работах [8,9,10,12].…”

Nonlinear effects in the dynamics of HIV-1 infection predicted by mathematical model with multiple delays

“…Finally, in order to eradicate most efficiently the infection the powerful mathematical methods of optimal control of the system dynamics and the clinical data-based parameter estimation are required to be a part of the modelling-guided studies. Some progress made in this direction can be found in [1], [4], [3], [11], [10].…”

Bistability analysis of virus infection models with time delays

“…Understanding it is most important for elaboration of an optimal treatment strategy of the offset system dynamics. Indeed, a bistable dynamical system can be transferred to a favourable steady state using the optimal disturbance approach as outlined in [5,6,7]. In the situation of monostability, the transfer of the system away from unwanted steady state requires the implementation of control taking the system to a neighborhood of a given trajectory (e.g., feedback stabilization, extremal shift or the optimal programme (open loop) control [8]).…”

“…The technology includes methods which allow one to compute all steady states for fixed parameter values, to trace them along parameters, and to analyze their stability. Methods for constructing the optimal disturbances were proposed and justified in [5,6,7,25].…”