Comptes Rendus
no. 11-12
Systèmes dynamiques
On the global attractors in one mathematical model of antiviral immunity
[Sur les attracteurs globaux dans un modèle mathématique de l’immunité antivirale]
Alexei Tsygvintsev1 
1 ENS de Lyon, UMPA, 46 allée d’Italie, 69364 Lyon Cedex 07, FRANCE
Comptes Rendus. Mathématique, Volume 358 (2020) no. 11-12, pp. 1199-1205.

Nous considérons le modèle mathématique introduit par Batholdy et al. [1] décrivant l’interaction entre les agents pathogènes viraux et le système immunitaire. Nous prouvons la stabilité globale asymptotique de l’état d’équilibre de l’infection si le taux de reproduction de base R 0 est supérieur à 1. Cela résout la conjecture annoncée dans [7].

We consider the mathematical model introduced by Batholdy et al. [1] describing the interaction between viral pathogens and immune system. We prove the global asymptotic stability of the infection steady-state if the basic reproductive ratio R 0 is greater than unity. That solves the conjecture announced in [7].

Reçu le :
Révisé le :
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Publié le :
DOI : 10.5802/crmath.140
Classification : 92B05
Alexei Tsygvintsev 1

1 ENS de Lyon, UMPA, 46 allée d’Italie, 69364 Lyon Cedex 07, FRANCE
Licence : CC-BY 4.0
Droits d'auteur : Les auteurs conservent leurs droits 
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     author = {Alexei Tsygvintsev},
     title = {On the global attractors in one mathematical model of antiviral immunity},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {1199--1205},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {358},
     number = {11-12},
     year = {2020},
     doi = {10.5802/crmath.140},
     language = {en},
}
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Alexei Tsygvintsev. On the global attractors in one mathematical model of antiviral immunity. Comptes Rendus. Mathématique, Volume 358 (2020) no. 11-12, pp. 1199-1205. doi : 10.5802/crmath.140. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.140/

[1] Christina Bartholdy; Jan Pravsgaard Christensen; Dominik Wodarz; Allan Randrup Thomsen Persistent Virus Infection despite Chronic Cytotoxic T-Lymphocyte Activation in Gamma Interferon-Deficient Mice Infected with Lymphocytic Choriomeningitis Virus, Journal of Virology, Volume 74 (2000) no. 22, pp. 10304-10311 | DOI

[2] Philip Hartman; Czesław Olech On Global Asymptotic Stability of Solutions of Differential Equations, Trans. Am. Math. Soc., Volume 104 (1962), pp. 154-178 | MR | Zbl

[3] Joseph P. LaSalle Some Extensions of Liapunov’s Second Method, IRE Trans. Inf. Theory, Volume 7 (1960) no. 4, pp. 520-527 | DOI | MR

[4] Joseph P. LaSalle; Solomon Lefschetz Stability by Liapunov’s Direct Method. With Applications., Mathematics in Science and Engineering, 4, Academic Press Inc., 1961 | MR | Zbl

[5] Kenneth Murphy; Paul Travers; Mark Walport; Charles Janeway Janeway’s immunobiology, Garland Science, 2012

[6] Kaifa Wang; Yu Jin; Aijun Fan The effect of immune responses in viral infections: A mathematical model view, Discrete Contin. Dyn. Syst., Volume 19 (2014) no. 10, pp. 3379-3396 | MR | Zbl

[7] Kaifa Wang; Wendi Wang; Xianning Liu Global stability in a viral infection model with lytic and non-lytic immune responses, Comput. Math. Appl., Volume 51 (2006) no. 9-10, pp. 1593-1610 | DOI | Zbl

[8] Robert F. Williams The structure of Lorenz attractors, Publ. Math., Inst. Hautes Étud. Sci., Volume 50 (1979), pp. 73-99 | DOI | Numdam | Zbl

[9] Dominik Wodarz; Jan Pravsgaard Christensen; Allan Randrup Thomsen The importance of lytic and non-lytic immune responses in viral infections, Trends Immunol., Volume 23 (2002) no. 4, pp. 194-200 | DOI

[10] Carolin Zitzmann; Lars Kaderali Mathematical Analysis of Viral Replication Dynamics and Antiviral Treatment Strategies: From Basic Models to Age-Based Multi-Scale Modeling, Front. Microbiol., Volume 9 (2018), 1546 | DOI

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