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.
Mathematical models with time delays are widely used to analyze the mechanisms of the immune response to virus infections and predict various therapeutic effects. Using the lymphocytic choriomeningitis virus infection model as an example, this work describes an original computational technology for searching the bistable regimes of such models. This technology includes numerical methods for finding all possible steady states at fixed values of parameters, for tracing these states along the parameters and for analyzing their stability.2020 Mathematics Subject Classification. Primary: 97M60, 34K10, 65L07; Secondary: 37N25, 34K28, 34L16.
In this paper, we apply optimal perturbations to control mathematical models of infectious diseases expressed as systems of nonlinear differential equations with delayed argument. We develop the method for calculation of perturbations of the initial state of a dynamical system with delayed argument producing maximal amplification in the given local norm taking into account weights of perturbation components. For the model of experimental virus infection, we construct optimal perturbation for two types of stationary states, with low or high virus load, corresponding to different variants of chronic virus infection flow.
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