Bistability analysis of virus infection models with time delays

Yuri Nechepurenko; Michael Khristichenko; Dmitry Grebennikov; Gennady Bocharov

doi:10.3934/dcdss.2020166

Article Contents

2020, Volume 13, Issue 9: 2385-2401. Doi: 10.3934/dcdss.2020166

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Bistability analysis of virus infection models with time delays

1.
Marchuk Institute of Numerical Mathematics, Russian Academy of Sciences, Moscow, 119333, Russian Federation
2.
Keldysh Institute of Applied Mathematics, Russian Academy of Sciences, Moscow, 125047, Russia

^* Corresponding author: Yuri Nechepurenko
^* Corresponding author: Yuri Nechepurenko

Received: December 31, 2018

Revised: March 31, 2019

Early access: December 2019

Published: June 2020

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Abstract

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.

Keywords:

Mathematics Subject Classification: Primary: 97M60, 34K10, 65L07; Secondary: 37N25, 34K28, 34L16.

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Figure 1. The first step of constructing elementary components

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Figure 2. Steady states (Ⅰ — black, Ⅱ — red, Ⅲ — green, Ⅳ — blue) and their stability (solid line) and instability (dashed line) as functions of $ b_p $

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Figure 3. Leading eigenvalues in steady states Ⅰ, Ⅱ, Ⅲ and Ⅳ at $ b_p = 10^{-5} $ ("x"), $ 3.5\cdot10^{-4} $ ("+"), $ 6.7\cdot10^{-4} $ ("o")

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Figure 4. Steady states (Ⅰ — black, Ⅱ — red, Ⅲ — green, Ⅳ — blue) and their stability (solid line) and instability (dashed line) as functions of $ \beta $

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Figure 5. Steady state values of $ V $ as a function of parameter $ b_p $ for the following values of $ \beta $: $ 1.72 $ (A), $ 1.69 $ (B), $ 1.671 $ (C), $ 1.67 $ (D)

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Table 1. Biological meaning of the model (1) parameters

Parameter	Biological meaning
$ \beta $	Viruses replication rate constant
$ \gamma_{VE} $	Rate constant of virus clearance due to effector CTLs
$ V_{mvc} $	Maximum possible virus concentration in spleen
$ \tau $	Typical duration of CTL division cycle
$ b_p $	Rate constant of CTL stimulation
$ b_d $	Rate constant of CTL differentiation
$ \theta_p $	Cumulative viral load threshold for anergy induction
$ \theta_p $	in precursor CTLs
$ \theta_E $	Cumulative viral load threshold for anergy induction
$ \theta_E $	in effector CTLs
$ \alpha_{E_{p}} $	Precursor CTL natural death rate constant
$ \alpha_{E_{e}} $	Effector CTL natural death rate constant
$ E_p^0 $	Concentration of precursor CTLs
$ E_p^0 $	in spleen of unprimed mouse
$ \tau_A $	Typical duration of CTLs commitment for apoptosis
$ \alpha_{AP} $	Precursor CTL apoptosis rate constant
$ \alpha_{AE} $	Effector CTL apoptosis rate constant
$ b_{W} $	Rate constant of cumulative viral load increase
$ \alpha_{W} $	Rate constant of restoration from
$ \alpha_{W} $	the inhibitory effect of cumulative viral load

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