Monotonic and nonmonotonic immune responses in viral infection systems

Shaoli Wang; Huixia Li; Fei Xu

doi:10.3934/dcdsb.2021035

Article Contents

2022, Volume 27, Issue 1: 141-165. Doi: 10.3934/dcdsb.2021035

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Monotonic and nonmonotonic immune responses in viral infection systems

1.
School of Mathematics and Statistics, Bioinformatics Center of Henan University, Kaifeng 475001, Henan, China
2.
Department of Mathematics, Nanjing University of Aeronautics and Astronautics, Nanjing, 210016, Jiangsu, China
3.
Department of Mathematics, Wilfrid Laurier University, Waterloo, Ontario, N2L 3C5, Canada

^* Corresponding author: Shaoli Wang
^* Corresponding author: Shaoli Wang

Received: August 31, 2019

Revised: October 31, 2020

Early access: January 2021

Published: January 2022

This work is supported by Science and Foundation of Technology Department of Henan Province (No.192102310089), Foundation of Henan Educational Committee (No.19A110009), Natural Science Foundation of Henan (No. 202300410045) and Grant of Bioinformatics Center of Henan University (No. 2019YLXKJC02)

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Abstract

In this paper, we study two-dimensional, three-dimensional monotonic and nonmonotonic immune responses in viral infection systems. Our results show that the viral infection systems with monotonic immune response has no bistability appear. However, the systems with nonmonotonic immune response has bistability appear under some conditions. For immune intensity, we got two important thresholds, post-treatment control threshold and elite control threshold. When immune intensity is less than post-treatment control threshold, the virus will be rebound. The virus will be under control when immune intensity is larger than elite control threshold. While between the two thresholds is a bistable interval. When immune intensity is in the bistable interval, the system can have bistability appear. Select the rate of immune cells stimulated by the viruses as a bifurcation parameter for nonmonotonic immune responses, we prove that the system exhibits saddle-node bifurcation and transcritical bifurcation.

Keywords:

Monotonic and nonmonotonic immune response,
post-treatment control threshold,
elite control threshold,
saddle-node bifurcation,
transcritical bifurcation.

Mathematics Subject Classification: Primary: 34D20, 37N25; Secondary: 92B05.

Citation:

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Figure 1. Bifurcation diagram of system (3). The solid line represents the stable equilibrium of infected CD4+ T cells and the dashed line represents the unstable equilibrium of infected CD4+ T cells. The post-treatment control threshold is $ c_{2} = 0.2500 $, the elite control threshold is $ c^{**}_1\approx0.6505 $ and the bistable interval is $ (0.2500, 0.6505). $ Here, $ c = 0.37\; \; \mbox{day}^{-1} $ and the values of other parameters are listed in (4)

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Figure 2. Time histories and trajectories of system (3) with different initial conditions. The system has a stable equilibria $ E^{(2)}_{1} $. Here, $ c = 0.2\; \; \mbox{day}^{-1} $ is less than the post-treatment control threshold $ P_I $ and other parameter values are listed in (4)

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Figure 3. Time histories and trajectories of system (3) with different initial conditions. Here, system (3) has two different stable equilibria $ E^{(2)}_{1} $ and $ E_{-}^{2*} $ with $ c = 0.37\; \; \mbox{day}^{-1} $. Other parameter values are listed in (4)

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Figure 4. Time histories and trajectories of system (3) with different initial conditions. System (3) only has the positive equilibrium $ E_{-}^{2*} $, which is stable with $ c = 0.65\; \; \mbox{day}^{-1} $. Other parameter values are listed in (4)

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Figure 5. Bifurcation diagram of system (6). The solid line is the stable equilibrium and the dashed line denotes the unstable equilibrium. The post-treatment control threshold is $ c_{2} = 2.5000 $, the elite control threshold is $ c^{**}_2\approx3.5278 $ and the bistable interval is $ (2.5000, 3.5278). $ Here, $ c = 3\; \; \mbox{day}^{-1} $ and other parameter values are listed in (7)

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Figure 6. Time histories and phase portraits of system (6). System (6) has two different stable equilibria $ E^{(4)}_{1} $ and $ E_{*}^{4-} $. Here, $ c = 3\; \; \mbox{day}^{-1} $ and other parameter values are listed in (7). We choose different initial values

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Figure 7. Phase portraits of system (6). (A) Choosing $ c = 2\; \; \mbox{day}^{-1} $, which is less than the post-treatment control threshold $ c_{2} = 2.5000 $, system (6) only has a stable equilibrium $ E^{(4)}_{1} $; (B) Choosing $ c = 4\; \; \mbox{day}^{-1} $, which is greater than the elite control threshold $ c^{**}_2\approx3.5278 $, system (6) only has the stable equilibria $ E_{*}^{4-} $. Other parameter values are listed in (7)

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Table 1. The stabilities of the equilibria and the behaviors of system (3) in the case $ 1<\mathcal {R}^{(2)}_{0}<\mathcal {R}^{(1)}_{c} $

	$ E^{(2)}_{0} $	$ E^{(2)}_{1} $	$ E_{*}^{2-} $	$ E_{*}^{2+} $	System (3)
$ R^{(2)}_{0}<1 $	GAS	—	—	—	Converges to $ E^{(2)}_{0} $
$ 1<R^{(2)}_{0}<R^{(1)}_{c}, $ $ 0<c<c^{**}_1 $	US	LAS	—	—	Converges to $ E^{(2)}_{1} $
$ 1<R^{(2)}_{0}<R^{(1)}_{c}, $ $ c^{**}_1<c $	US	US	LAS	—	Converges to $ E_{*}^{2-} $

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Table 2. The stabilities of the equilibria and the behaviors of system (3) in the case $ \mathcal {R}^{(2)}_{0}>\mathcal {R}^{(1)}_{c} $

	$ E^{(2)}_{0} $	$ E^{(2)}_{1} $	$ E_{*}^{2-} $	$ E_{*}^{2+} $	System (3)
$ R^{(2)}_{0}<1 $	GAS	—	—	—	Converges to $ E^{(2)}_{0} $
$ R^{(2)}_{0}>1 $, $ 0<c<c_{2} $	US	LAS	—	—	Converges to $ E^{(2)}_{1} $
$ R^{(2)}_{0}>R^{(1)}_{c}>1, $ $ c_{2}<c<c^{**}_1 $	US	LAS	LAS	US	Bistable
$ R^{(2)}_{0}>R^{(1)}_{c}>1, $ $ c>c^{**}_1 $	US	US	LAS	—	Converges to $ E_{*}^{2-} $

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Table 3. The stabilities of the equilibria and the behaviors of system (6) in the case $ 1<\mathcal {R}^{(4)}_{0}<\mathcal {R}^{(2)}_{c} $

	$ E^{(4)}_{0} $	$ E^{(4)}_{1} $	$ E_{*}^{4-} $	$ E_{*}^{4+} $	System (6)
$ R^{(4)}_{0}<1 $	GAS	—	—	—	Converges to $ E^{(4)}_{0} $
$ 1<R^{(4)}_{0}<R^{(2)}_{c}, $ $ 0<c<c^{**}_2 $	US	GAS	—	—	Converges to $ E^{(4)}_{1} $
$ 1<R^{(4)}_{0}<R^{(2)}_{c}, $ $ c^{**}_2<c $	US	US	GAS	—	Converges to $ E_{*}^{4-} $

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Table 4. The stabilities of the equilibria and the behaviors of system (6) in the case $ \mathcal {R}^{(4)}_{0}>\mathcal {R}^{(2)}_{c} $

	$ E^{(4)}_{0} $	$ E^{(4)}_{1} $	$ E_{*}^{4-} $	$ E_{*}^{4+} $	System (6)
$ R^{(4)}_{0}<1 $	GAS	—	—	—	Converges to $ E^{(4)}_{0} $
$ R^{(4)}_{0}>1 $, $ 0<c<c_{2} $	US	GAS	—	—	Converges to $ E^{(4)}_{1} $
$ R^{(4)}_{0}>R^{(2)}_{c}>1, $ $ c_{2}<c<c^{**}_2 $	US	GAS	GAS	US	Bistable
$ R^{(4)}_{0}>R^{(2)}_{c}>1, $ $ c>c^{**}_2 $	US	US	GAS	—	Converges to $ E_{*}^{4-} $

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