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Abstract
This paper represents a literature review on traveling waves described by delayed reactiondiffusion (RD, for short) equations. It begins with the presentation of different types of equations arising in applications. The main results on wave existence and stability are presented for the equations satisfying the monotonicity condition that provides the applicability of the maximum and comparison principles. Other methods and results are described for the case where the monotonicity condition is not satisfied. The last two sections deal with delayed RD equations in mathematical immunology and in neuroscience. Existence, stability, and dynamics of wavefronts and of periodic waves are discussed.
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Affiliation(s)
- Sergei Trofimchuk
- Instituto de Matemáticas, Universidad de Talca, Casilla 747, Talca, Chile
| | - Vitaly Volpert
- Institut Camille Jordan, UMR 5208 CNRS, University Lyon 1, Villeurbanne 69622, France
- INRIA Team Dracula, INRIA Lyon La Doua, Villeurbanne 69603, France
- Peoples' Friendship University of Russia (RUDN University), Miklukho-Maklaya St, Moscow 117198, Russia
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Beuter A, Balossier A, Trofimchuk S, Volpert V. Modeling of post-stroke stimulation of cortical tissue. Math Biosci 2018; 305:146-159. [DOI: 10.1016/j.mbs.2018.08.014] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [What about the content of this article? (0)] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/16/2018] [Revised: 08/23/2018] [Accepted: 08/29/2018] [Indexed: 11/28/2022]
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Bocharov G, Meyerhans A, Bessonov N, Trofimchuk S, Volpert V. Interplay between reaction and diffusion processes in governing the dynamics of virus infections. J Theor Biol 2018; 457:221-236. [PMID: 30170043 DOI: 10.1016/j.jtbi.2018.08.036] [Citation(s) in RCA: 10] [Impact Index Per Article: 1.7] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/26/2017] [Revised: 08/22/2018] [Accepted: 08/28/2018] [Indexed: 02/02/2023]
Abstract
Spreading of viral infection in the tissues such as lymph nodes or spleen depends on virus multiplication in the host cells, their transport and on the immune response. Reaction-diffusion systems of equations with delays in cell proliferation and death by apoptosis represent an appropriate model to study this process. The properties of the cells of the immune system and the initial viral load determine the spatiotemporal regimes of infection spreading. Infection can be completely eliminated or it can persist at some level together with a certain chronic immune response in a spatially uniform or oscillatory mode. Finally, the immune cells can be completely exhausted leading to a high viral load persistence in the tissue. It has been found experimentally, that virus proteins can affect the immune cell migration. Our study shows that both the motility of immune cells and the virus infection spreading represented by the diffusion rate coefficients are relevant control parameters determining the fate of virus-host interaction.
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Affiliation(s)
- G Bocharov
- Marchuk Institute of Numerical Mathematics, Russian Academy of Sciences Gubkina Street 8, 119333 Moscow, Russian Federation; Peoples Friendship University of Russia (RUDN University) 6 Miklukho-Maklaya St, Moscow, 117198, Russian Federation; Gamaleya Center of Epidemiology and Microbiology, Moscow, Russian Federation.
| | - A Meyerhans
- Infection Biology Laboratory, Department of Experimental and Health Sciences Universitat Pompeu Fabra, Barcelona, Spain; ICREA, Pg. Llus Companys 23, 08010 Barcelona, Spain
| | - N Bessonov
- Institute of Problems of Mechanical Engineering, Russian Academy of Sciences 199178 Saint Petersburg, Russia
| | - S Trofimchuk
- Instituto de Matematica y Fisica, Universidad de Talca, Casilla 747, Talca, Chile
| | - V Volpert
- Institut Camille Jordan, UMR 5208 CNRS, University Lyon 1, 69622 Villeurbanne, France; INRIA, Université de Lyon, Université Lyon 1, Institut Camille Jordan 43 Bd. du 11 Novembre 1918, 69200 Villeurbanne Cedex, France; Poncelet Center, UMI 2615 CNRS, 11 Bolshoy Vlasyevskiy, 119002 Moscow, Russian Federation; Peoples Friendship University of Russia (RUDN University) 6 Miklukho-Maklaya St, Moscow, 117198, Russian Federation
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Bocharov G, Meyerhans A, Bessonov N, Trofimchuk S, Volpert V. Spatiotemporal Dynamics of Virus Infection Spreading in Tissues. PLoS One 2016; 11:e0168576. [PMID: 27997613 PMCID: PMC5173377 DOI: 10.1371/journal.pone.0168576] [Citation(s) in RCA: 28] [Impact Index Per Article: 3.5] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/22/2016] [Accepted: 12/03/2016] [Indexed: 12/21/2022] Open
Abstract
Virus spreading in tissues is determined by virus transport, virus multiplication in host cells and the virus-induced immune response. Cytotoxic T cells remove infected cells with a rate determined by the infection level. The intensity of the immune response has a bell-shaped dependence on the concentration of virus, i.e., it increases at low and decays at high infection levels. A combination of these effects and a time delay in the immune response determine the development of virus infection in tissues like spleen or lymph nodes. The mathematical model described in this work consists of reaction-diffusion equations with a delay. It shows that the different regimes of infection spreading like the establishment of a low level infection, a high level infection or a transition between both are determined by the initial virus load and by the intensity of the immune response. The dynamics of the model solutions include simple and composed waves, and periodic and aperiodic oscillations. The results of analytical and numerical studies of the model provide a systematic basis for a quantitative understanding and interpretation of the determinants of the infection process in target organs and tissues from the image-derived data as well as of the spatiotemporal mechanisms of viral disease pathogenesis, and have direct implications for a biopsy-based medical testing of the chronic infection processes caused by viruses, e.g. HIV, HCV and HBV.
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Affiliation(s)
- Gennady Bocharov
- Institute of Numerical Mathematics, Russian Academy of Sciences, Moscow, Russian Federation
- Gamaleya Center of Epidemiology and Microbiology, Moscow, Russian Federation
- RUDN University, Moscow, Russian Federation
| | - Andreas Meyerhans
- Institute of Numerical Mathematics, Russian Academy of Sciences, Moscow, Russian Federation
- Infection Biology Laboratory, Department of Experimental and Health Sciences, Universitat Pompeu Fabra, Barcelona, Spain
- ICREA, Pg. Lluís Companys 23, Barcelona, Spain
| | - Nickolai Bessonov
- Institute of Problems of Mechanical Engineering, Russian Academy of Sciences, Saint Petersburg, Russian Federation
| | - Sergei Trofimchuk
- Instituto de Matemática y Fisica, Universidad de Talca, Talca, Chile
| | - Vitaly Volpert
- Institute of Numerical Mathematics, Russian Academy of Sciences, Moscow, Russian Federation
- Institut Camille Jordan, UMR 5208 CNRS, University Lyon 1, Villeurbanne, France
- INRIA Team Dracula, INRIA Lyon La Doua, Villeurbanne, France
- Laboratoire Poncelet, UMI 2615 CNRS, Moscow, Russian Federation
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Abstract
We consider positive travelling fronts,
u
(
t
,
x
)=
ϕ
(
ν
.
x
+
ct
),
ϕ
(−∞)=0,
ϕ
(∞)=
κ
, of the equation
u
t
(
t
,
x
)=Δ
u
(
t
,
x
)−
u
(
t
,
x
)+
g
(
u
(
t
−
h
,
x
)),
x
∈
m
. This equation is assumed to have exactly two non-negative equilibria:
u
1
≡0 and
u
2
≡
κ
>0, but the birth function
g
∈
C
2
(
,
) may be non-monotone on [0,
κ
]. We are therefore interested in the so-called monostable case of the time-delayed reaction–diffusion equation. Our main result shows that for every fixed and sufficiently large velocity
c
, the positive travelling front
ϕ
(
ν
.
x
+
ct
) is unique (modulo translations). Note that
ϕ
may be non-monotone. To prove uniqueness, we introduce a small parameter
ϵ
=1/
c
and realize a Lyapunov–Schmidt reduction in a scale of Banach spaces.
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Affiliation(s)
- Maitere Aguerrea
- Instituto de Matemática y Fisica, Universidad de TalcaCasilla 747 Talca, Chile
| | - Sergei Trofimchuk
- Instituto de Matemática y Fisica, Universidad de TalcaCasilla 747 Talca, Chile
| | - Gabriel Valenzuela
- Instituto de Matemática y Fisica, Universidad de TalcaCasilla 747 Talca, Chile
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Liz E, Tkachenko V, Trofimchuk S. Global stability in discrete population models with delayed-density dependence. Math Biosci 2005; 199:26-37. [PMID: 16337248 DOI: 10.1016/j.mbs.2005.03.016] [Citation(s) in RCA: 25] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/07/2004] [Revised: 03/11/2005] [Accepted: 03/14/2005] [Indexed: 11/30/2022]
Abstract
We address the global stability issue for some discrete population models with delayed-density dependence. Applying a new approach based on the concept of the generalized Yorke conditions, we establish several criteria for the convergence of all solutions to the unique positive steady state. Our results support the conjecture stated by Levin and May in 1976 affirming that the local asymptotic stability of the equilibrium of some delay difference equations (including Ricker's and Pielou's equations) implies its global stability. We also discuss the robustness of the obtained results with respect to perturbations of the model.
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Affiliation(s)
- Eduardo Liz
- Departamento de Matemática Aplicada II, E.T.S.I. Telecomunicación, Campus Marcosende, Universidad de Vigo, 36280 Vigo, Spain.
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