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Numerical Analysis of Stationary Solutions of Systems with Delayed Argument in Mathematical Immunology

Authors:
M. Yu. Khristichenko
M. Yu. Khristichenko
M. Yu. Khristichenko
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Yuri Mikhailovich Nechepurenko at Institute of Numerical Mathematics, Russian Academy of Sciences
Yuri Mikhailovich Nechepurenko
D. S. Grebennikov
D. S. Grebennikov
D. S. Grebennikov
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G. A. Bocharov
G. A. Bocharov
G. A. Bocharov
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Abstract

This work is devoted to the technology developed by the authors that allows one for fixed values of parameters and tracing by parameters to calculate stationary solutions of systems with delay and analyze their stability. We discuss the results of applying this technology to the Marchuk–Petrov antiviral immune response model with parameter values corresponding to hepatitis B infection. The presence of bistability and hysteresis properties in this model is shown for the first time.
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Journal of Mathematical Sciences, Vol. 283, No. 1, July, 2024
NUMERICAL ANALYSIS OF STATIONARY SOLUTIONS OF SYSTEMS
WITH DELAYED ARGUMENT IN MATHEMATICAL IMMUNOLOGY
M. Yu. Khristichenko, Yu. M. Nechepurenko, D. S. Grebennikov,
andG.A.Bocharov UDC 517.929, 517.958
Abstract. This work is devoted to the technology developed by the authors that allows one for fixed
values of parameters and tracing by parameters to calculate stationary solutions of systems with delay
and analyze their stability. We discuss the results of applying this technology to the Marchuk–Petrov
antiviral immune response model with parameter values corresponding to hepatitis B infection. The
presence of bistability and hysteresis properties in this model is shown for the first time.
Keywords: Marchuk–Petrov antiviral immune response model, delayed argument, stationary solu-
tions, tracing by parameters, numerical experiment, hepatitis B infection, bistability, hysteresis.
1. Introduction
Mathematical modeling is the most important analytical tool of modern mathematical immunol-
ogy [6, 12]. In recent years intensive research in the field of contagion of SARS-CoV-2 viruses demon-
strates the capabilities of mathematical models and emerging challenges in the development of numer-
ical analysis and technologies for such a research [9]. The fundamental problem of modern immunol-
ogy is the investigation of the mechanisms of development of unfavorable forms of viral infections,
in particular, chronic viral hepatitis B virus. It is a well known fact that chronic viral hepatitis is a
multifactorial disease [7]. Previously, by analyzing the sensitivity of the solutions of the mathematical
model of the antiviral immune response, we established that a change in the parameters of activation
of the antigen-presenting function of macrophages can serve as one of the factors in the transition from
an acute form of infection with recovery to chronic persistence of viruses [3]. However, a systematic
study of the characteristics of chronic viral hepatitis B remains a relatively uninvestigated problem.
In particular, detecting regions in the parameter space of mathematical models that correspond to
bistable or multistable modes of interaction between the infection and the body’s immune system,
and investigating the possibilities of transitions between them are open problems.The key element of
the analysis of such properties is the existence of efficient methods for calculating and numerically
analyzing of the stability of stationary solutions of nonlinear systems of differential equations with a
delay argument used to model viral infections.
Bistability, as the ability of the “virus – human body” system to coexist in two stable equilibrium
states, is a very important property, since it allows to solve the problem of functional treatment of
viral infection by moving from a chronic state with a higher viral load to a more favorable state
with a low viral load due to the activation of components of the immune system. The verification of
such a concept from biological perspective was established in paper [2]. Note that the transfer of the
infectious process to a state with a lower viral load is a nontrivial problem, whose solution requires the
activation of both humoral link and T-cell link of immunity. At the same time, the multiplier effect of
the antibodies and cytotoxic T-lymphocytes contribution to the reduction of the level of the infectious
process allows for reduction of the thresholds for the size of these populations required to control the
infection at a lower level. It is possible to investigate the fundamental conditions of bistability (or)
multistability of the infectious process and the cooperation of immunity links in ensuring the control
Translated from Sovremennaya Matematika. Fundamental’nye Napravleniya (Contemporary Mathematics.
Fundamental Directions), Vol. 68, No. 4,Differential and Functional Differential Equations, 2022.
1072–3374/24/2831–0125 ©2024 Springer Nature Switzerland AG 125
DOI 10.1007/s10958-024-07243-5
Content courtesy of Springer Nature, terms of use apply. Rights reserved.
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