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Saade M, Ghosh S, Banerjee M, Volpert V. Delay epidemic models determined by latency, infection, and immunity duration. Math Biosci 2024; 370:109155. [PMID: 38316373 DOI: 10.1016/j.mbs.2024.109155] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/12/2023] [Revised: 01/17/2024] [Accepted: 01/27/2024] [Indexed: 02/07/2024]
Abstract
We propose new single and two-strain epidemic models represented by systems of delay differential equations and based on the number of newly exposed individuals. Transitions between exposed, infectious, recovered, and back to susceptible compartments are determined by the corresponding time delays. Existence and positiveness of solutions are proved. Reduction of delay differential equations to integral equations allows the analysis of stationary solutions and their stability. In the case of two strains, they compete with each other, and the strain with a larger individual basic reproduction number dominates the other one. However, if the basic reproduction number exceeds some critical values, stationary solution loses its stability resulting in periodic time oscillations. In this case, both strains are present and their dynamics is not completely determined by the basic reproduction numbers but also by other parameters. The results of the work are illustrated by comparison with data on seasonal influenza.
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Affiliation(s)
- Masoud Saade
- Peoples Friendship University of Russia (RUDN University), 6 Miklukho-Maklaya St, 117198 Moscow, Russia.
| | - Samiran Ghosh
- Department of Mathematics and Statistics, IIT Kanpur, Kanpur, , 208016, India.
| | - Malay Banerjee
- Department of Mathematics and Statistics, IIT Kanpur, Kanpur, , 208016, India.
| | - Vitaly Volpert
- Peoples Friendship University of Russia (RUDN University), 6 Miklukho-Maklaya St, 117198 Moscow, Russia; Institut Camille Jordan, UMR 5208 CNRS, University Lyon 1, 69622 Villeurbanne, France.
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2
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Bouchnita A, Volpert V. Phenotype-structured model of intra-clonal heterogeneity and drug resistance in multiple myeloma. J Theor Biol 2024; 576:111652. [PMID: 37952610 DOI: 10.1016/j.jtbi.2023.111652] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/24/2023] [Revised: 09/26/2023] [Accepted: 10/22/2023] [Indexed: 11/14/2023]
Abstract
Multiple myeloma (MM) is a genetically complex hematological cancer characterized by the abnormal proliferation of malignant plasma cells in the bone marrow. This disease progresses from a premalignant condition known as monoclonal gammopathy of unknown significance (MGUS) through sequential genetic alterations involving various genes. These genetic changes contribute to the uncontrolled growth of multiple clones of plasma cells. In this study, we present a phenotype-structured model that captures the intra-clonal heterogeneity and drug resistance in multiple myeloma (MM). The model accurately reproduces the branching evolutionary pattern observed in MM progression, aligning with a previously developed multiscale model. Numerical simulations reveal that higher mutation rates enhance tumor phenotype diversity, while access to growth factors accelerates tumor evolution and increases its final size. Interestingly, the model suggests that further increasing growth factor access primarily amplifies tumor size rather than altering clonal dynamics. Additionally, the model emphasizes that higher mutation frequencies and growth factor availability elevate the chances of drug resistance and relapse. It indicates that the timing of the treatment could trajectory of tumor evolution and clonal emergence in the case of branching evolutionary pattern. Given its low computational cost, our model is well-suited for quantitative studies on MM clonal heterogeneity and its interaction with chemotherapeutic treatments.
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Affiliation(s)
- Anass Bouchnita
- Department of Mathematical Sciences, The University of Texas at El Paso, El Paso, 79968, TX, United States.
| | - Vitaly Volpert
- Institut Camille Jordan, UMR 5208 CNRS, University Lyon 1, 69622 Villeurbanne, France; Peoples Friendship University of Russia (RUDN University), 6 Miklukho-Maklaya St, 117198 Moscow, Russian Federation
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3
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Meziane M, Moussaoui A, Volpert V. On a two-strain epidemic model involving delay equations. Math Biosci Eng 2023; 20:20683-20711. [PMID: 38124571 DOI: 10.3934/mbe.2023915] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 12/23/2023]
Abstract
We propose an epidemiological model for the interaction of either two viruses or viral strains with cross-immunity, where the individuals infected by the first virus cannot be infected by the second one, and without cross-immunity, where a secondary infection can occur. The model incorporates distributed recovery and death rates and consists of integro-differential equations governing the dynamics of susceptible, infectious, recovered, and dead compartments. Assuming that the recovery and death rates are uniformly distributed in time throughout the duration of the diseases, we can simplify the model to a conventional ordinary differential equation (ODE) model. Another limiting case arises if the recovery and death rates are approximated by the delta-function, thereby resulting in a new point-wise delay model that incorporates two time delays corresponding to the durations of the diseases. We establish the positiveness of solutions for the distributed delay models and determine the basic reproduction number and an estimate for the final size of the epidemic for the delay model. According to the results of the numerical simulations, both strains can coexist in the population if the disease transmission rates for them are close to each other. If the difference between them is sufficiently large, then one of the strains dominates and eliminates the other one.
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Affiliation(s)
- Mohammed Meziane
- Laboratoire d'Analyse Non linéaire et Mathématiques Appliquées, Department of Mathematics, Faculty of Sciences, University of Tlemcen, Algeria
| | - Ali Moussaoui
- Laboratoire d'Analyse Non linéaire et Mathématiques Appliquées, Department of Mathematics, Faculty of Sciences, University of Tlemcen, Algeria
| | - Vitaly Volpert
- Institut Camille Jordan, UMR 5208 CNRS, University Lyon 1, 69622 Villeurbanne, France
- Peoples' Friendship University of Russia (RUDN University), 6 Miklukho-Maklaya St, Moscow, 117198, Russian Federation
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4
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Bessonov N, Volpert V. Airway obstruction in respiratory viral infections due to impaired mucociliary clearance. Int J Numer Method Biomed Eng 2023; 39:e3707. [PMID: 37073098 DOI: 10.1002/cnm.3707] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/25/2022] [Revised: 03/14/2023] [Accepted: 04/02/2023] [Indexed: 05/03/2023]
Abstract
Respiratory viral infections, such as SARS-CoV-2 or influenza, can lead to impaired mucociliary clearance in the bronchial tree due to increased mucus viscosity and its hyper-secretion. We develop in this work a mathematical model to study the interplay between viral infection and mucus motion. The results of numerical simulations show that infection progression can be characterized by three main stages. At the first stage, infection spreads through the most part of mucus producing airways (about 90% of the length) without significant changes in mucus velocity and thickness layer. During the second stage, when it passes through the remaining generations, mucus viscosity increases, its velocity drops down, and it forms a plug. At the last stage, the thickness of the mucus layer gradually increases because mucus is still produced but not removed by the flow. After some time, the thickness of the mucus layer in the small airways becomes comparable with their diameter leading to their complete obstruction.
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Affiliation(s)
- N Bessonov
- Institute of Problems of Mechanical Engineering, Russian Academy of Sciences, Saint Petersburg, Russian Federation
| | - V Volpert
- Institut Camille Jordan, UMR 5208 CNRS, University Lyon 1, Villeurbanne, France
- S.M. Nikolskii Mathematical Institute, Peoples Friendship University of Russia (RUDN University), Moscow, Russian Federation
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El Hajj W, El Khatib N, Volpert V. Inflammation propagation modeled as a reaction-diffusion wave. Math Biosci 2023; 365:109074. [PMID: 37689347 DOI: 10.1016/j.mbs.2023.109074] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [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: 04/07/2023] [Revised: 08/28/2023] [Accepted: 08/29/2023] [Indexed: 09/11/2023]
Abstract
Inflammation is a physiological process aimed to protect the organism in various diseases and injuries. This work presents a generic inflammation model based on the reaction-diffusion equations for the concentrations of uninflamed cells, inflamed cells, immune cells and the inflammatory cytokines. The analysis of the model shows the existence of three different regimes of inflammation progression depending on the value of a parameter R called the inflammation number. If R>1, then inflammation propagates in cell culture or tissue as a reaction-diffusion wave due to diffusion of inflammatory cytokines produced by inflamed cells. If 0
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Affiliation(s)
- W El Hajj
- Institut Camille Jordan, UMR 5208 CNRS, University Lyon 1, 69622 Villeurbanne, France
| | - N El Khatib
- Department of Computer Science and Mathematics, Lebanese American University, P.O. Box 36, Byblos, Lebanon.
| | - V Volpert
- Institut Camille Jordan, UMR 5208 CNRS, University Lyon 1, 69622 Villeurbanne, France; Peoples' Friendship University of Russia (RUDN University), 6 Miklukho-Maklaya St, Moscow 117198, Russian Federation
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Moussaoui A, Volpert V. The influence of immune cells on the existence of virus quasi-species. Math Biosci Eng 2023; 20:15942-15961. [PMID: 37919996 DOI: 10.3934/mbe.2023710] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 11/04/2023]
Abstract
This article investigate a nonlocal reaction-diffusion system of equations modeling virus distribution with respect to their genotypes in the interaction with the immune response. This study demonstrates the existence of pulse solutions corresponding to virus quasi-species. The proof is based on the Leray-Schauder method, which relies on the topological degree for elliptic operators in unbounded domains and a priori estimates of solutions. Furthermore, linear stability analysis of a spatially homogeneous stationary solution identifies the critical conditions for the emergence of spatial and spatiotemporal structures. Finally, numerical simulations are used to illustrate nonlinear dynamics and pattern formation in the nonlocal model.
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Affiliation(s)
- Ali Moussaoui
- Laboratoire d'Analyse Non linéaire et Mathématiques Appliquées, Department of Mathematics, Faculty of Sciences, University of Tlemcen, Algeria
| | - Vitaly Volpert
- Institut Camille Jordan, UMR 5208 CNRS, University Lyon 1, Villeurbanne 69622, France
- Peoples' Friendship University of Russia (RUDN University), 6 Miklukho-Maklaya St, Moscow 117198, Russian Federation
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Saade M, Ghosh S, Banerjee M, Volpert V. An epidemic model with time delays determined by the infectivity and disease durations. Math Biosci Eng 2023; 20:12864-12888. [PMID: 37501470 DOI: 10.3934/mbe.2023574] [Citation(s) in RCA: 1] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 07/29/2023]
Abstract
We propose an epidemiological model with distributed recovery and death rates. It represents an integrodifferential system of equations for susceptible, exposed, infectious, recovered and dead compartments. This model can be reduced to the conventional ODE model under the assumption that recovery and death rates are uniformly distributed in time during disease duration. Another limiting case, where recovery and death rates are given by the delta-function, leads to a new point-wise delay model with two time delays corresponding to the infectivity period and disease duration. Existence and positiveness of solutions for the distributed delay model and point-wise delay model are proved. The basic reproduction number and the final size of the epidemic are determined. Both, the ODE model and the delay models are used to describe COVID-19 epidemic progression. The delay model gives a better approximation of the Omicron data than the conventional ODE model from the point of view of parameter estimation.
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Affiliation(s)
- Masoud Saade
- Peoples Friendship University of Russia (RUDN University) 6 Miklukho-Maklaya St, Moscow 117198, Russia
| | - Samiran Ghosh
- Department of Mathematics and Statistics, IIT Kanpur, Kanpur 208016, India
| | - Malay Banerjee
- Department of Mathematics and Statistics, IIT Kanpur, Kanpur 208016, India
| | - Vitaly Volpert
- Peoples Friendship University of Russia (RUDN University) 6 Miklukho-Maklaya St, Moscow 117198, Russia
- Institut Camille Jordan, UMR 5208 CNRS, University Lyon 1, Villeurbanne 69622, France
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8
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Volpert V, Xu B, Tchechmedjiev A, Harispe S, Aksenov A, Mesnildrey Q, Beuter A. Characterization of spatiotemporal dynamics in EEG data during picture naming with optical flow patterns. Math Biosci Eng 2023; 20:11429-11463. [PMID: 37322989 DOI: 10.3934/mbe.2023507] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/17/2023]
Abstract
In this study, we investigate the spatiotemporal dynamics of the neural oscillations by analyzing the electric potential that arises from neural activity. We identify two types of dynamics based on the frequency and phase of oscillations: standing waves or as out-of-phase and modulated waves, which represent a combination of standing and moving waves. To characterize these dynamics, we use optical flow patterns such as sources, sinks, spirals and saddles. We compare analytical and numerical solutions with real EEG data acquired during a picture-naming task. Analytical approximation of standing waves helps us to establish some properties of pattern location and number. Specifically, sources and sinks are mainly located in the same location, while saddles are positioned between them. The number of saddles correlates with the sum of all the other patterns. These properties are confirmed in both the simulated and real EEG data. In particular, source and sink clusters in the EEG data overlap with each other with median percentages around 60%, and hence have high spatial correlation, while source/sink clusters overlap with saddle clusters in less than 1%, and have different locations. Our statistical analysis showed that saddles account for about 45% of all patterns, while the remaining patterns are present in similar proportions.
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Affiliation(s)
- V Volpert
- Institut Camille Jordan, UMR 5208 CNRS, University Lyon 1, 69622 Villeurbanne, France
| | - B Xu
- EuroMov Digital Health in Motion, Univ Montpellier, IMT Mines Ales, Ales, France
| | - A Tchechmedjiev
- EuroMov Digital Health in Motion, Univ Montpellier, IMT Mines Ales, Ales, France
| | - S Harispe
- EuroMov Digital Health in Motion, Univ Montpellier, IMT Mines Ales, Ales, France
| | | | | | - A Beuter
- CorStim SAS, Montpellier, France
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9
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Eymard N, Bessonov N, Volpert V, Kurbatova P, Gueyffier F, Nony P. Pharmacokinetic/pharmacodynamic model of a methionine starvation based anti-cancer drug. Med Biol Eng Comput 2023:10.1007/s11517-023-02786-2. [PMID: 36882575 DOI: 10.1007/s11517-023-02786-2] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [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: 12/28/2020] [Accepted: 01/17/2023] [Indexed: 03/09/2023]
Abstract
A new therapeutic approach against cancer is developed by the firm Erytech. This approach is based on starved cancer cells of an amino acid essential to their growth (the L-methionine). The depletion of plasma methionine level can be induced by an enzyme, the methionine-γ-lyase. The new therapeutic formulation is a suspension of erythrocytes encapsulating the activated enzyme. Our work reproduces a preclinical trial of a new anti-cancer drug with a mathematical model and numerical simulations in order to replace animal experiments and to have a deeper insight on the underlying processes. With a combination of a pharmacokinetic/pharmacodynamic model for the enzyme, substrate, and co-factor with a hybrid model for tumor, we develop a "global model" that can be calibrated to simulate different human cancer cell lines. The hybrid model includes a system of ordinary differential equations for the intracellular concentrations, partial differential equations for the concentrations of nutrients and drugs in the extracellular matrix, and individual based model for cancer cells. This model describes cell motion, division, differentiation, and death determined by the intracellular concentrations. The models are developed on the basis of experiments in mice carried out by Erytech. Parameters of the pharmacokinetics model were determined by fitting a part of experimental data on the concentration of methionine in blood. Remaining experimental protocols effectuated by Erytech were used to validate the model. The validated PK model allowed the investigation of pharmacodynamics of cell populations. Numerical simulations with the global model show cell synchronization and proliferation arrest due to treatment similar to the available experiments. Thus, computer modeling confirms a possible effect of treatment based on the decrease of methionine concentration. The main goal of the study is the development of an integrated pharmacokinetic/pharmacodynamic model for encapsulated methioninase and of a mathematical model of tumor growth/regression in order to determine the kinetics of L-methionine depletion after co-administration of Erymet product and Pyridoxine.
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Affiliation(s)
- N Eymard
- Institut Camille Jordan, UMR 5208 CNRS, University Lyon 1, 69622, Villeurbanne, France.
| | - N Bessonov
- Institute of Mechanical Engineering Problems, 199178, Saint Petersburg, Russia
| | - V Volpert
- Institut Camille Jordan, UMR 5208 CNRS, University Lyon 1, 69622, Villeurbanne, France.,Peoples Friendship University of Russia (RUDN University), 6 Miklukho-Maklaya St, Moscow, 117198, Russia
| | - P Kurbatova
- Institut Camille Jordan, UMR 5208 CNRS, University Lyon 1, 69622, Villeurbanne, France
| | - F Gueyffier
- CNRS, Laboratoire de Biométrie et Biologie Evolutive UMR 5558, Université Lyon 1, F-69622, Villeurbanne, France
| | - P Nony
- Institut Camille Jordan, UMR 5208 CNRS, University Lyon 1, 69622, Villeurbanne, France.,CNRS, Laboratoire de Biométrie et Biologie Evolutive UMR 5558, Université Lyon 1, F-69622, Villeurbanne, France
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Volpert V. Functional properties of morphological networks. Comment on "Networks behind the morphology and structural design of living systems" by Marko Gosak et al. Phys Life Rev 2023; 44:58-60. [PMID: 36502607 DOI: 10.1016/j.plrev.2022.12.001] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/02/2022] [Accepted: 12/02/2022] [Indexed: 12/12/2022]
Affiliation(s)
- V Volpert
- Institut Camille Jordan, UMR 5208 CNRS, University Lyon 1, 69622 Villeurbanne, France; Peoples' Friendship University of Russia, 6 Miklukho-Maklaya St, Moscow, 117198, Russia.
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Bessonov N, Neverova D, Popov V, Volpert V. Emergence and competition of virus variants in respiratory viral infections. Front Immunol 2023; 13:945228. [PMID: 37168105 PMCID: PMC10165551 DOI: 10.3389/fimmu.2022.945228] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/16/2022] [Accepted: 11/22/2022] [Indexed: 02/16/2023] Open
Abstract
The emergence of new variants of concern (VOCs) of the SARS-CoV-2 infection is one of the main factors of epidemic progression. Their development can be characterized by three critical stages: virus mutation leading to the appearance of new viable variants; the competition of different variants leading to the production of a sufficiently large number of copies; and infection transmission between individuals and its spreading in the population. The first two stages take place at the individual level (infected individual), while the third one takes place at the population level with possible competition between different variants. This work is devoted to the mathematical modeling of the first two stages of this process: the emergence of new variants and their progression in the epithelial tissue with a possible competition between them. The emergence of new virus variants is modeled with non-local reaction–diffusion equations describing virus evolution and immune escape in the space of genotypes. The conditions of the emergence of new virus variants are determined by the mutation rate, the cross-reactivity of the immune response, and the rates of virus replication and death. Once different variants emerge, they spread in the infected tissue with a certain speed and viral load that can be determined through the parameters of the model. The competition of different variants for uninfected cells leads to the emergence of a single dominant variant and the elimination of the others due to competitive exclusion. The dominant variant is the one with the maximal individual spreading speed. Thus, the emergence of new variants at the individual level is determined by the immune escape and by the virus spreading speed in the infected tissue.
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Leon C, Tokarev A, Bouchnita A, Volpert V. Modelling of the Innate and Adaptive Immune Response to SARS Viral Infection, Cytokine Storm and Vaccination. Vaccines (Basel) 2023; 11:vaccines11010127. [PMID: 36679972 PMCID: PMC9861811 DOI: 10.3390/vaccines11010127] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [Track Full Text] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/09/2022] [Revised: 12/20/2022] [Accepted: 12/24/2022] [Indexed: 01/06/2023] Open
Abstract
In this work, we develop mathematical models of the immune response to respiratory viral infection, taking into account some particular properties of the SARS-CoV infections, cytokine storm and vaccination. Each model consists of a system of ordinary differential equations that describe the interactions of the virus, epithelial cells, immune cells, cytokines, and antibodies. Conventional analysis of the existence and stability of stationary points is completed by numerical simulations in order to study the dynamics of solutions. The behavior of the solutions is characterized by large peaks of virus concentration specific to acute respiratory viral infections. At the first stage, we study the innate immune response based on the protective properties of interferon secreted by virus-infected cells. Viral infection down-regulates interferon production. This competition can lead to the bistability of the system with different regimes of infection progression with high or low intensity. After that, we introduce the adaptive immune response with antigen-specific T- and B-lymphocytes. The resulting model shows how the incubation period and the maximal viral load depend on the initial viral load and the parameters of the immune response. In particular, an increase in the initial viral load leads to a shorter incubation period and higher maximal viral load. The model shows that a deficient production of antibodies leads to an increase in the incubation period and even higher maximum viral loads. In order to study the emergence and dynamics of cytokine storm, we consider proinflammatory cytokines produced by cells of the innate immune response. Depending on the parameters of the model, the system can remain in the normal inflammatory state specific for viral infections or, due to positive feedback between inflammation and immune cells, pass to cytokine storm characterized by the excessive production of proinflammatory cytokines. Finally, we study the production of antibodies due to vaccination. We determine the dose-response dependence and the optimal interval of vaccine dose. Assumptions of the model and obtained results correspond to the experimental and clinical data.
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Affiliation(s)
- Cristina Leon
- Interdisciplinary Center for Mathematical Modelling in Biomedicine, S.M. Nikol’skii Mathematical Institute, Peoples Friendship University of Russia (RUDN University), 6 Miklukho-Maklaya St., 117198 Moscow, Russia
- M&S Decisions, 5 Naryshkinskaya Alley, 125167 Moscow, Russia
- Department of Foreign Languages No. 2, Plekhanov Russian University of Economics, 36 Stremyanny Lane, 115093 Moscow, Russia
- Correspondence:
| | - Alexey Tokarev
- Interdisciplinary Center for Mathematical Modelling in Biomedicine, S.M. Nikol’skii Mathematical Institute, Peoples Friendship University of Russia (RUDN University), 6 Miklukho-Maklaya St., 117198 Moscow, Russia
- Semenov Institute of Chemical Physics, 4 Kosygin St., 119991 Moscow, Russia
- Bukhara Engineering Technological Institute, 15 Murtazoyeva Street, Bukhara 200100, Uzbekistan
| | - Anass Bouchnita
- Department of Mathematical Sciences, The University of Texas at El Paso, El Paso, TX 79902, USA
| | - Vitaly Volpert
- Interdisciplinary Center for Mathematical Modelling in Biomedicine, S.M. Nikol’skii Mathematical Institute, Peoples Friendship University of Russia (RUDN University), 6 Miklukho-Maklaya St., 117198 Moscow, Russia
- Institut Camille Jordan, UMR 5208 CNRS, University Lyon 1, 69622 Villeurbanne, France
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Ghosh S, Volpert V, Banerjee M. An age-dependent immuno-epidemiological model with distributed recovery and death rates. J Math Biol 2023; 86:21. [PMID: 36625974 PMCID: PMC9838470 DOI: 10.1007/s00285-022-01855-8] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [Track Full Text] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/24/2022] [Revised: 12/06/2022] [Accepted: 12/09/2022] [Indexed: 01/11/2023]
Abstract
The work is devoted to a new immuno-epidemiological model with distributed recovery and death rates considered as functions of time after the infection onset. Disease transmission rate depends on the intra-subject viral load determined from the immunological submodel. The age-dependent model includes the viral load, recovery and death rates as functions of age considered as a continuous variable. Equations for susceptible, infected, recovered and dead compartments are expressed in terms of the number of newly infected cases. The analysis of the model includes the proof of the existence and uniqueness of solution. Furthermore, it is shown how the model can be reduced to age-dependent SIR or delay model under certain assumptions on recovery and death distributions. Basic reproduction number and final size of epidemic are determined for the reduced models. The model is validated with a COVID-19 case data. Modelling results show that proportion of young age groups can influence the epidemic progression since disease transmission rate for them is higher than for other age groups.
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Affiliation(s)
- Samiran Ghosh
- grid.417965.80000 0000 8702 0100Department of Mathematics and Statistics, Indian Institute of Technology Kanpur, Kanpur, 208016 Uttar Pradesh India
| | - Vitaly Volpert
- grid.493248.00000 0004 0383 2988Institut Camille Jordan, UMR 5208 CNRS, University Lyon 1, 69622 Villeurbanne, France ,grid.77642.300000 0004 0645 517XPeoples Friendship University of Russia (RUDN University), 6 Miklukho-Maklaya St, 117198 Moscow, Russian Federation
| | - Malay Banerjee
- Department of Mathematics and Statistics, Indian Institute of Technology Kanpur, Kanpur, 208016, Uttar Pradesh, India.
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Ait Mahiout L, Mozokhina A, Tokarev A, Volpert V. Virus replication and competition in a cell culture: Application to the SARS-CoV-2 variants. Appl Math Lett 2022; 133:108217. [PMID: 35669092 PMCID: PMC9155187 DOI: 10.1016/j.aml.2022.108217] [Citation(s) in RCA: 5] [Impact Index Per Article: 2.5] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/12/2022] [Revised: 05/23/2022] [Accepted: 05/23/2022] [Indexed: 05/29/2023]
Abstract
Viral replication in a cell culture is described by a delay reaction-diffusion system. It is shown that infection spreads in cell culture as a reaction-diffusion wave, for which the speed of propagation and viral load can be determined both analytically and numerically. Competition of two virus variants in the same cell culture is studied, and it is shown that the variant with larger individual wave speed out-competes another one, and eliminates it. This approach is applied to the Delta and Omicron variants of the SARS-CoV-2 infection in the cultures of human epithelial and lung cells, allowing characterization of infectivity and virulence of each variant, and their comparison.
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Affiliation(s)
- L Ait Mahiout
- Laboratoire d'équations aux dérivées partielles non linéaires et histoire des mathématiques Ecole Normale Supérieure, B.P. 92, Vieux Kouba, 16050 Algiers, Algeria
| | - A Mozokhina
- Peoples Friendship University of Russia (RUDN University), 6 Miklukho-Maklaya St, Moscow, 117198, Russia
| | - A Tokarev
- Peoples Friendship University of Russia (RUDN University), 6 Miklukho-Maklaya St, Moscow, 117198, Russia
- Semenov Institute of Chemical Physics RAS, Kosygin St., 4, 119991 Moscow, Russia
| | - V Volpert
- Institut Camille Jordan, UMR 5208 CNRS, University Lyon 1, 69622 Villeurbanne, France
- Peoples Friendship University of Russia (RUDN University), 6 Miklukho-Maklaya St, Moscow, 117198, Russia
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15
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Kuznetsov M, Clairambault J, Volpert V. Perspectives in cancer treatment. Phys Life Rev 2022; 42:15-18. [DOI: 10.1016/j.plrev.2022.05.003] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/06/2022] [Accepted: 05/07/2022] [Indexed: 11/17/2022]
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16
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Ait Mahiout L, Bessonov N, Kazmierczak B, Volpert V. Mathematical modeling of respiratory viral infection and applications to SARS-CoV-2 progression. Math Methods Appl Sci 2022; 46:MMA8606. [PMID: 36247228 PMCID: PMC9538414 DOI: 10.1002/mma.8606] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [Grants] [Track Full Text] [Figures] [Subscribe] [Scholar Register] [Received: 02/26/2022] [Revised: 07/19/2022] [Accepted: 07/20/2022] [Indexed: 06/16/2023]
Abstract
Viral infection in cell culture and tissue is modeled with delay reaction-diffusion equations. It is shown that progression of viral infection can be characterized by the viral replication number, time-dependent viral load, and the speed of infection spreading. These three characteristics are determined through the original model parameters including the rates of cell infection and of virus production in the infected cells. The clinical manifestations of viral infection, depending on tissue damage, correlate with the speed of infection spreading, while the infectivity of a respiratory infection depends on the viral load in the upper respiratory tract. Parameter determination from the experiments on Delta and Omicron variants allows the estimation of the infection spreading speed and viral load. Different variants of the SARS-CoV-2 infection are compared confirming that Omicron is more infectious and has less severe symptoms than Delta variant. Within the same variant, spreading speed (symptoms) correlates with viral load allowing prognosis of disease progression.
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Affiliation(s)
- Latifa Ait Mahiout
- Laboratoire d'équations aux dérivées partielles non linéaires et histoire des mathématiquesEcole Normale SupérieureAlgiersAlgeria
| | - Nikolai Bessonov
- Institute of Problems of Mechanical EngineeringRussian Academy of SciencesSaint PetersburgRussia
| | - Bogdan Kazmierczak
- Institute of Fundamental Technological ResearchPolish Academy of SciencesWarsawPoland
| | - Vitaly Volpert
- Institut Camille Jordan, UMR 5208 CNRSUniversity Lyon 1VilleurbanneFrance
- Peoples' Friendship University of Russia6 Miklukho‐Maklaya StMoscowRussia
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17
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Affiliation(s)
- Samiran Ghosh
- Department of Mathematics and Statistics, Indian Institute of Technology Kanpur, Kanpur, 208016 Uttar Pradesh India
| | - Vitaly Volpert
- Institut Camille Jordan, UMR 5208 CNRS, University Lyon 1, 69622 Villeurbanne, France
- Peoples Friendship University of Russia (RUDN University), 6 Miklukho-Maklaya St, Moscow, 117198 Russian Federation
| | - Malay Banerjee
- Department of Mathematics and Statistics, Indian Institute of Technology Kanpur, Kanpur, 208016 Uttar Pradesh India
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18
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Banerjee M, Kuznetsov M, Udovenko O, Volpert V. Nonlocal Reaction-Diffusion Equations in Biomedical Applications. Acta Biotheor 2022; 70:12. [PMID: 35298702 DOI: 10.1007/s10441-022-09436-4] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/19/2021] [Accepted: 02/28/2022] [Indexed: 11/01/2022]
Abstract
Nonlocal reaction-diffusion equations describe various biological and biomedical applications. Their mathematical properties are essentially different in comparison with the local equations, and this difference can lead to important biological implications. This review will present the state of the art in the investigation of nonlocal reaction-diffusion models in biomedical applications. We will consider various models arising in mathematical immunology, neuroscience, cancer modelling, and we will discuss their mathematical properties, nonlinear dynamics, resulting spatiotemporal patterns and biological significance.
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19
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Ait Mahiout L, Mozokhina A, Tokarev A, Volpert V. The Influence of Immune Response on Spreading of Viral Infection. Lobachevskii J Math 2022; 43:2699-2713. [PMCID: PMC9907882 DOI: 10.1134/s1995080222130285] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/14/2022] [Revised: 08/24/2022] [Accepted: 09/06/2022] [Indexed: 05/19/2023]
Abstract
In this work we develop a model of viral infection in host tissues in order to study the influence of the immune response on the infection spreading speed and on the viral load characterizing, respectively, severity of symptoms and infection transmission rate. Dynamics of the interaction between viral infection and the immune response is studied with nonlocal reaction-diffusion equations for the concentrations of virus, interferon, immune cells and antibodies. Analytical results for infection spreading speed and viral load are completed by numerical simulations. At the first stage, progression of viral infection is confronted by the innate immune response mostly determined by the local interferon production. The modeling results show in this case that infection spreading speed does not depend on interferon concentration, while the total viral load decreases with the increase of its concentration. Next, we consider the influence of globally circulating interferon and show that, in contrast to local interferon diffusion, infection spreading speed decreases with increasing of global interferon level, and the total viral load also decreases. At the next stage, adaptive immune response mediated by antibodies and cytotoxic T cells (CTL) further influences infection progression. In this case, the infection propagation speed and the total viral load are decreased by the immune response. The humoral adaptive response (antibodies) increases the global interferon concentration through the viral load, while the cellular adaptive response (CTL) decreases it.
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Affiliation(s)
- L. Ait Mahiout
- Laboratoire d’Équations aux Dérivées Partielles Non Linéaires et Histoire des Mathématiques, Ecole Normale Supérieure, 16050 Algiers, Algeria
| | - A. Mozokhina
- Peoples’ Friendship University of Russia (RUDN University), 117198 Moscow, Russia
| | - A. Tokarev
- Peoples’ Friendship University of Russia (RUDN University), 117198 Moscow, Russia
- Semenov Federal Research Center for Chemical Physics of Russian Academy of Sciences, 119991 Moscow, Russia
| | - V. Volpert
- Peoples’ Friendship University of Russia (RUDN University), 117198 Moscow, Russia
- Institut Camille Jordan, UMR 5208 CNRS, University Lyon 1, Villeurbanne, 69622 France
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20
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Masalceva AA, Kaneva VN, Panteleev MA, Ataullakhanov F, Volpert V, Afanasyev I, Nechipurenko DY. Analysis of microvascular thrombus mechanobiology with a novel particle-based model. J Biomech 2021; 130:110801. [PMID: 34768079 DOI: 10.1016/j.jbiomech.2021.110801] [Citation(s) in RCA: 4] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/11/2021] [Revised: 08/20/2021] [Accepted: 09/16/2021] [Indexed: 11/18/2022]
Abstract
Platelet accumulation at the site of a vascular injury is regulated by soluble platelet agonists, which induce various types of platelet responses, including integrin activation and granule secretion. The interplay between local biochemical cues, mechanical interactions between platelets and macroscopic thrombus dynamics is poorly understood. Here we describe a novel computational model of microvascular clot formation for the detailed analysis of thrombus mechanics. We adopt a previously developed two-dimensional particle-based model focused on the thrombus shell formation and revise it to introduce the platelet agonists. Blood flow is simulated via a computational fluid dynamics approach. In order to model soluble platelet activators, we apply Langevin dynamics to a large number of non-dimensional virtual particles. Taking advantage of the available data on platelet dense granule secretion kinetics, we model platelet degranulation as a stochastic agonist-dependent process. The new model qualitatively reproduces the enhanced thrombus formation due to dense granule secretion, in line with in vivo findings, and provides a mechanism for the thrombin confinement at the early stages of clot formation. Our calculations also predict that the release of platelet dense granules results in the additional mechanical stabilization of the inner layers of thrombus. Distribution of the inter-platelet forces throughout the aggregate reveals multiple weak spots in the outer regions of a thrombus, which are expected to result in the mechanical disruptions at the later stages of clot formation.
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Affiliation(s)
- Anastasia A Masalceva
- Faculty of Physics, Lomonosov Moscow State University, Moscow, Russia; Center for Theoretical Problems of Physico-chemical Pharmacology, Russian Academy of Science, Moscow, Russia
| | - Valeriia N Kaneva
- Faculty of Physics, Lomonosov Moscow State University, Moscow, Russia; Center for Theoretical Problems of Physico-chemical Pharmacology, Russian Academy of Science, Moscow, Russia; Dmitry Rogachev National Research Center of Pediatric Hematology, Oncology and Immunology, Moscow, Russia
| | - Mikhail A Panteleev
- Faculty of Physics, Lomonosov Moscow State University, Moscow, Russia; Center for Theoretical Problems of Physico-chemical Pharmacology, Russian Academy of Science, Moscow, Russia; Dmitry Rogachev National Research Center of Pediatric Hematology, Oncology and Immunology, Moscow, Russia; Faculty of Biological and Medical Physics, Moscow Institute of Physics and Technology, Dolgoprudnyi, Russia
| | - Fazoil Ataullakhanov
- Faculty of Physics, Lomonosov Moscow State University, Moscow, Russia; Center for Theoretical Problems of Physico-chemical Pharmacology, Russian Academy of Science, Moscow, Russia; Dmitry Rogachev National Research Center of Pediatric Hematology, Oncology and Immunology, Moscow, Russia; Faculty of Biological and Medical Physics, Moscow Institute of Physics and Technology, Dolgoprudnyi, Russia
| | - Vitaly Volpert
- Institut Camille Jordan, UMR 5208 CNRS, University Lyon 1, 69622 Villeurbanne, France; INRIA Team Dracula, INRIA Lyon La Doua, 69603 Villeurbanne, France; Peoples' Friendship University of Russia (RUDN University), 6 Miklukho-Maklaya St, Moscow 117198, Russia
| | - Ilya Afanasyev
- Moscow Center of Fundamental and Applied Mathematics, Moscow, Russia; Research Computing Center of Moscow State University, Moscow, Russia
| | - Dmitry Yu Nechipurenko
- Faculty of Physics, Lomonosov Moscow State University, Moscow, Russia; Center for Theoretical Problems of Physico-chemical Pharmacology, Russian Academy of Science, Moscow, Russia; Dmitry Rogachev National Research Center of Pediatric Hematology, Oncology and Immunology, Moscow, Russia.
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21
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Kaneva VN, Dunster JL, Volpert V, Ataullahanov F, Panteleev MA, Nechipurenko DY. Modeling Thrombus Shell: Linking Adhesion Receptor Properties and Macroscopic Dynamics. Biophys J 2021; 120:334-351. [PMID: 33472026 DOI: 10.1016/j.bpj.2020.10.049] [Citation(s) in RCA: 10] [Impact Index Per Article: 3.3] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/02/2020] [Revised: 09/10/2020] [Accepted: 10/15/2020] [Indexed: 02/06/2023] Open
Abstract
Damage to arterial vessel walls leads to the formation of platelet aggregate, which acts as a physical obstacle for bleeding. An arterial thrombus is heterogeneous; it has a dense inner part (core) and an unstable outer part (shell). The thrombus shell is very dynamic, being composed of loosely connected discoid platelets. The mechanisms underlying the observed mobility of the shell and its (patho)physiological implications are unclear. To investigate arterial thrombus mechanics, we developed a novel, to our knowledge, two-dimensional particle-based computational model of microvessel thrombosis. The model considers two types of interplatelet interactions: primary reversible (glycoprotein Ib (GPIb)-mediated) and stronger integrin-mediated interaction, which intensifies with platelet activation. At high shear rates, the former interaction leads to adhesion, and the latter is primarily responsible for stable platelet aggregation. Using a stochastic model of GPIb-mediated interaction, we initially reproduced experimental curves that characterize individual platelet interactions with a von Willebrand factor-coated surface. The addition of the second stabilizing interaction results in thrombus formation. The comparison of thrombus dynamics with experimental data allowed us to estimate the magnitude of critical interplatelet forces in the thrombus shell and the characteristic time of platelet activation. The model predicts moderate dependence of maximal thrombus height on the injury size in the absence of thrombin activity. We demonstrate that the developed stochastic model reproduces the observed highly dynamic behavior of the thrombus shell. The presence of primary stochastic interaction between platelets leads to the properties of thrombus consistent with in vivo findings; it does not grow upstream of the injury site and covers the whole injury from the first seconds of the formation. А simplified model, in which GPIb-mediated interaction is deterministic, does not reproduce these features. Thus, the stochasticity of platelet interactions is critical for thrombus plasticity, suggesting that interaction via a small number of bonds drives the dynamics of arterial thrombus shell.
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Affiliation(s)
- Valeriia N Kaneva
- Center for Theoretical Problems of Physico-chemical Pharmacology, Russian Academy of Sciences, Moscow, Russia
| | - Joanne L Dunster
- Institute for Cardiovascular and Metabolic Research, School of Biological Sciences, University of Reading, Whiteknights, Reading, United Kingdom
| | - Vitaly Volpert
- Institut Camille Jordan, UMR 5208 CNRS, University Lyon 1, Villeurbanne, France; INRIA Team Dracula, INRIA Lyon La Doua, Villeurbanne, France; Peoples Friendship University of Russia (RUDN University), Moscow, Russian Federation
| | - Fazoil Ataullahanov
- Center for Theoretical Problems of Physico-chemical Pharmacology, Russian Academy of Sciences, Moscow, Russia; Dmitry Rogachev National Medical Research Centre of Pediatric Hematology, Oncology and Immunology, Moscow, Russia; Faculty of Physics, Lomonosov Moscow State University, Moscow, Russia; Faculty of Biological and Medical Physics, Moscow Institute of Physics and Technology, Dolgoprudnyi, Russia
| | - Mikhail A Panteleev
- Center for Theoretical Problems of Physico-chemical Pharmacology, Russian Academy of Sciences, Moscow, Russia; Dmitry Rogachev National Medical Research Centre of Pediatric Hematology, Oncology and Immunology, Moscow, Russia; Faculty of Physics, Lomonosov Moscow State University, Moscow, Russia; Faculty of Biological and Medical Physics, Moscow Institute of Physics and Technology, Dolgoprudnyi, Russia
| | - Dmitry Yu Nechipurenko
- Center for Theoretical Problems of Physico-chemical Pharmacology, Russian Academy of Sciences, Moscow, Russia; Dmitry Rogachev National Medical Research Centre of Pediatric Hematology, Oncology and Immunology, Moscow, Russia; Faculty of Physics, Lomonosov Moscow State University, Moscow, Russia.
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22
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Zincenko A, Petrovskii S, Volpert V, Banerjee M. Turing instability in an economic-demographic dynamical system may lead to pattern formation on a geographical scale. J R Soc Interface 2021; 18:20210034. [PMID: 33906386 DOI: 10.1098/rsif.2021.0034] [Citation(s) in RCA: 6] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/12/2022] Open
Abstract
Spatial distribution of the human population is distinctly heterogeneous, e.g. showing significant difference in the population density between urban and rural areas. In the historical perspective, i.e. on the timescale of centuries, the emergence of densely populated areas at their present locations is widely believed to be linked to more favourable environmental and climatic conditions. In this paper, we challenge this point of view. We first identify a few areas at different parts of the world where the environmental conditions (quantified by the temperature, precipitation and elevation) show a relatively small variation in space on the scale of thousands of kilometres. We then examine the population distribution across those areas to show that, in spite of the approximate homogeneity of the environment, it exhibits a significant variation revealing a nearly periodic spatial pattern. Based on this apparent disagreement, we hypothesize that there may exist an inherent mechanism that may lead to pattern formation even in a uniform environment. We consider a mathematical model of the coupled demographic-economic dynamics and show that its spatially uniform, locally stable steady state can give rise to a periodic spatial pattern due to the Turing instability, the spatial scale of the emerging pattern being consistent with observations. Using numerical simulations, we show that, interestingly, the emergence of the Turing patterns may eventually lead to the system collapse.
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Affiliation(s)
- Anna Zincenko
- School of Mathematics and Actuarial Science, University of Leicester, Leicester LE1 7RH, UK.,School of Mathematics, University of Birmingham, Birmingham B15 2TT, UK
| | - Sergei Petrovskii
- School of Mathematics and Actuarial Science, University of Leicester, Leicester LE1 7RH, UK
| | - Vitaly Volpert
- Institut Camille Jordan, UMR, 5208, CNRS, University Lyon 1, 69622 Villeurbanne, France.,INRIA Team Dracula, INRIA Lyon La Doua, 69603 Villeurbanne, France.,Peoples Friendship University of Russia (RUDN University), 6 Miklukho-Maklaya St, Moscow 117198, Russian Federation
| | - Malay Banerjee
- Department of Mathematics and Statistics, IIT Kanpur, Kanpur 208016, India
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23
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Abstract
In nature, different species compete among themselves for common resources and favorable habitat. Therefore, it becomes really important to determine the key factors in maintaining the bio-diversity. Also, some competing species follow cyclic competition in real world where the competitive dominance is characterized by a cyclic ordering. In this paper, we study the formation of a wide variety of spatiotemporal patterns including stationary, periodic, quasi-periodic and chaotic population distributions for a diffusive Lotka-Volterra type three-species cyclic competition model with two different types of cyclic ordering. For both types of cyclic ordering, the temporal dynamics of the corresponding non-spatial system show the extinction of two species through global bifurcations such as homoclinic and heteroclinic bifurcations. For the spatial system, we show that the existence of Turing patterns is possible for a particular cyclic ordering, while it is not the case for the other cyclic ordering through both the analytical and numerical methods. Further, we illustrate an interesting scenario of short-range invasion as opposed to the usual invasion phenomenon over the entire habitat. Also, our study reveals that both the stationary and dynamic population distributions can coexist in different parts of a habitat. Finally, we extend the spatial system by incorporating nonlocal intra-specific competition terms for all the three competing species. Our study shows that the introduction of nonlocality in intra-specific competitions stabilizes the system dynamics by transforming a dynamic population distribution to stationary. Surprisingly, this nonlocality-induced stationary pattern formation leads to the extinction of one species and hence, gives rise to the loss of bio-diversity for intermediate ranges of nonlocality. However, the bio-diversity can be restored for sufficiently large extent of nonlocality.
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Affiliation(s)
- Kalyan Manna
- Department of Mathematics and Statistics, Indian Institute of Technology Kanpur, Kanpur, 208016, Uttar Pradesh, India
| | - Vitaly Volpert
- Institut Camille Jordan, UMR 5208 CNRS, University Lyon 1, 69622, Villeurbanne, France.,INRIA, Team Dracula, INRIA Lyon La Doua, 69603, Villeurbanne, France.,Peoples' Friendship University of Russia (RUDN University), 6 Miklukho-Maklaya St, Moscow, 117198, Russia
| | - Malay Banerjee
- Department of Mathematics and Statistics, Indian Institute of Technology Kanpur, Kanpur, 208016, Uttar Pradesh, India.
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24
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Ratto N, Bouchnita A, Chelle P, Marion M, Panteleev M, Nechipurenko D, Tardy-Poncet B, Volpert V. Patient-Specific Modelling of Blood Coagulation. Bull Math Biol 2021; 83:50. [PMID: 33772645 PMCID: PMC7998098 DOI: 10.1007/s11538-021-00890-8] [Citation(s) in RCA: 3] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/09/2020] [Accepted: 03/12/2021] [Indexed: 10/24/2022]
Abstract
Blood coagulation represents one of the most studied processes in biomedical modelling. However, clinical applications of this modelling remain limited because of the complexity of this process and because of large inter-patient variation of the concentrations of blood factors, kinetic constants and physiological conditions. Determination of some of these patients-specific parameters is experimentally possible, but it would be related to excessive time and material costs impossible in clinical practice. We propose in this work a methodological approach to patient-specific modelling of blood coagulation. It begins with conventional thrombin generation tests allowing the determination of parameters of a reduced kinetic model. Next, this model is used to study spatial distributions of blood factors and blood coagulation in flow, and to evaluate the results of medical treatment of blood coagulation disorders.
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Affiliation(s)
- N Ratto
- UMR 5208 CNRS, Institute Camille Jordan, Ecole Centrale de Lyon, Ecully, France
| | - A Bouchnita
- University of Texas at Austin, Austin, TX, 78712, USA
| | - P Chelle
- Center for Health Engineering, UMR 5307, Ecole Nationale Superieure des Mines de Saint-Etienne, 2023, Saint-Étienne, France.,EA3065, University Jean Monnet, 42023, Saint-Étienne, France
| | - M Marion
- UMR 5208 CNRS, Institute Camille Jordan, Ecole Centrale de Lyon, Ecully, France
| | - M Panteleev
- Faculty of Physics, Lomonosov Moscow State University, Moscow, Russia.,Center for Theoretical Problems of Physicochemical Pharmacology of the Russian Academy of Sciences, Moscow, Russia.,National Medical Research Center of Pediatric Hematology, Oncology and Immunology named after Dmitry Rogachev, Moscow, Russia
| | - D Nechipurenko
- Faculty of Physics, Lomonosov Moscow State University, Moscow, Russia.,Center for Theoretical Problems of Physicochemical Pharmacology of the Russian Academy of Sciences, Moscow, Russia.,National Medical Research Center of Pediatric Hematology, Oncology and Immunology named after Dmitry Rogachev, Moscow, Russia
| | - B Tardy-Poncet
- EA3065, University Jean Monnet, 42023, Saint-Étienne, France.,Inserm CIC1408, CHU de Saint-Etienne, 42023, Saint-Étienne, France
| | - V Volpert
- UMR 5208 CNRS, Institut Camille Jordan, University Lyon 1, 69622, Villeurbanne, France. .,INRIA Team Dracula, INRIA Lyon La Doua, 69603, Villeurbanne, France. .,Peoples' Friendship University of Russia (RUDN University), 6 Miklukho-Maklaya St, Moscow, Russia, 117198.
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25
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Banerjee M, Volpert V. Stochastic intracellular regulation can remove oscillations in a model of tissue growth. Math Med Biol 2020; 37:551-568. [PMID: 32735317 DOI: 10.1093/imammb/dqaa009] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/17/2019] [Revised: 03/28/2020] [Accepted: 06/19/2020] [Indexed: 01/06/2023]
Abstract
The work is devoted to the analysis of cell population dynamics where cells make a choice between differentiation and apoptosis. This choice is based on the values of intracellular proteins whose concentrations are described by a system of ordinary differential equations with bistable dynamics. Intracellular regulation and cell fate are controlled by the extracellular regulation through the number of differentiated cells. It is shown that the total cell number necessarily oscillates if the initial condition in the intracellular regulation is fixed. These oscillations can be suppressed if the initial condition is a random variable with a sufficiently large variation. Thus, the result of the work suggests a possible answer to the question about the role of stochasticity in the intracellular regulation.
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Affiliation(s)
- M Banerjee
- Department of Mathematics and Statistics, Indian Institute of Technology Kanpur, Kanpur 208016, India
| | - 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.,Peoples' Friendship University of Russia (RUDN University), 6 Miklukho-Maklaya St, Moscow 117198, Russian Federation
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26
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Abstract
An extended SEIQR type model is considered in order to model the COVID-19 epidemic. It contains the classes of susceptible individuals, exposed, infected symptomatic and asymptomatic, quarantined, hospitalized and recovered. The basic reproduction number and the final size of epidemic are determined. The model is used to fit available data for some European countries. A more detailed model with two different subclasses of susceptible individuals is introduced in order to study the influence of social interaction on the disease progression. The coefficient of social interaction K characterizes the level of social contacts in comparison with complete lockdown (K=0) and the absence of lockdown (K=1). The fitting of data shows that the actual level of this coefficient in some European countries is about 0.1, characterizing a slow disease progression. A slight increase of this value in the autumn can lead to a strong epidemic burst.
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Affiliation(s)
- Swarnali Sharma
- Department of Mathematics, Vijaygarh Jyotish Ray College, Kolkata - 700032, India
| | - Vitaly Volpert
- Institut Camille Jordan, UMR 5208 CNRS, University Lyon 1, 69622 Villeurbanne, France
- INRIA Team Dracula, INRIA Lyon La Doua, 69603 Villeurbanne, France
- Peoples Friendship University of Russia (RUDN University), 6 Miklukho-Maklaya St, Moscow, 117198, Russian Federation
| | - Malay Banerjee
- Department of Mathematics & Statistics, IIT Kanpur, Kanpur - 208016, India
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27
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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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28
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Budzinskiy S, Beuter A, Volpert V. Nonlinear analysis of periodic waves in a neural field model. Chaos 2020; 30:083144. [PMID: 32872829 DOI: 10.1063/5.0012010] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/30/2020] [Accepted: 08/04/2020] [Indexed: 06/11/2023]
Abstract
Various types of brain activity, including motor, visual, and language, are accompanied by the propagation of periodic waves of electric potential in the cortex, possibly providing the synchronization of the epicenters involved in these activities. One example is cortical electrical activity propagating during sleep and described as traveling waves [Massimini et al., J. Neurosci. 24, 6862-6870 (2004)]. These waves modulate cortical excitability as they progress. Clinically related examples include cortical spreading depression in which a wave of depolarization propagates not only in migraine but also in stroke, hemorrhage, or traumatic brain injury [Whalen et al., Sci. Rep. 8, 1-9 (2018)]. Here, we consider the possible role of epicenters and explore a neural field model with two nonlinear integrodifferential equations for the distributions of activating and inhibiting signals. It is studied with symmetric connectivity functions characterizing signal exchange between two populations of neurons, excitatory and inhibitory. Bifurcation analysis is used to investigate the emergence of periodic traveling waves and of standing oscillations from the stationary, spatially homogeneous solutions, and the stability of these solutions. Both types of solutions can be started by local oscillations indicating a possible role of epicenters in the initiation of wave propagation.
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Affiliation(s)
- S Budzinskiy
- Faculty of Computational Mathematics and Cybernetics, Lomonosov Moscow State University, Leninskie Gory 1, 119991 Moscow, Russia
| | | | - V Volpert
- Peoples' Friendship University of Russia (RUDN University), Miklukho-Maklaya St. 6, 117198 Moscow, Russia
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29
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Bouchnita A, Terekhov K, Nony P, Vassilevski Y, Volpert V. A mathematical model to quantify the effects of platelet count, shear rate, and injury size on the initiation of blood coagulation under venous flow conditions. PLoS One 2020; 15:e0235392. [PMID: 32726315 PMCID: PMC7390270 DOI: 10.1371/journal.pone.0235392] [Citation(s) in RCA: 14] [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: 10/16/2019] [Accepted: 06/16/2020] [Indexed: 11/18/2022] Open
Abstract
Platelets upregulate the generation of thrombin and reinforce the fibrin clot which increases the incidence risk of venous thromboembolism (VTE). However, the role of platelets in the pathogenesis of venous cardiovascular diseases remains hard to quantify. An experimentally validated model of thrombin generation dynamics is formulated. The model predicts that a high platelet count increases the peak value of generated thrombin as well as the endogenous thrombin potential (ETP) as reported in experimental data. To investigate the effects of platelets density, shear rate, and wound size on the initiation of blood coagulation, we calibrate a previously developed model of venous thrombus formation and implement it in 3D using a novel cell-centered finite-volume solver. We conduct numerical simulations to reproduce in vitro experiments of blood coagulation in microfluidic capillaries. Then, we derive a reduced one-equation model of thrombin distribution from the previous model under simplifying hypotheses and we use it to determine the conditions of clotting initiation on the platelet count, the shear rate, and the plasma composition. The initiation of clotting also exhibits a threshold response to the size of the wounded region in good agreement with the reported experimental findings.
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Affiliation(s)
| | - Kirill Terekhov
- Marchuk Institute of Numerical Mathematics, Russian Academy of Sciences, Moscow, Russia
| | - Patrice Nony
- Services de Pharmacologie Clinique, Hospices Civils de Lyon, Lyon, France
| | - Yuri Vassilevski
- Marchuk Institute of Numerical Mathematics, Russian Academy of Sciences, Moscow, Russia
- Sechenov University, Moscow, Russia
- Moscow Institute of Physics and Technology, Dolgoprudny, Russia
| | - Vitaly Volpert
- Marchuk Institute of Numerical Mathematics, Russian Academy of Sciences, Moscow, Russia
- Institut Camille Jordan, Université Lyon 1, Villeurbanne, France
- INRIA team Dracula, INRIA Lyon La Doua, Villeurbanne, France
- Peoples’ Friendship University of Russia (RUDN University), Moscow, Russia
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30
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Abstract
This work deals with a reaction-diffusion model for prey-predator interaction with Bazykin's reaction kinetics and a nonlocal interaction term in prey growth. The kernel of the integral characterizes nonlocal consumption of resources and depends on space and time. Linear stability analysis determines the conditions of the emergence of Turing patterns without and with nonlocal term, while weakly nonlinear analysis allows the derivation of amplitude equations. The bifurcation analysis and numerical simulation carried out in this work reveal the existence of stationary and dynamic patterns appearing due to the loss of stability of the coexistence homogeneous steady-state.
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Affiliation(s)
- Swadesh Pal
- Department of Mathematics & Statistics, IIT Kanpur, Kanpur, 208016, India
| | - Malay Banerjee
- Department of Mathematics & Statistics, IIT Kanpur, Kanpur, 208016, India
| | - Vitaly Volpert
- Institut Camille Jordan, UMR 5208 CNRS, University Lyon 1, 69622 Villeurbanne, France
- INRIA, Team Dracula, INRIA Lyon La Doua, 69603 Villeurbanne, France
- Peoples' Friendship University of Russia (RUDN University), 6 Miklukho-Maklaya St, Moscow, 117198, Russia
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31
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Ratto N, Tokarev A, Chelle P, Tardy-Poncet B, Volpert V. Clustering of Thrombin Generation Test Data Using a Reduced Mathematical Model of Blood Coagulation. Acta Biotheor 2020; 68:21-43. [PMID: 31853681 DOI: 10.1007/s10441-019-09372-w] [Citation(s) in RCA: 5] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/31/2019] [Accepted: 12/04/2019] [Indexed: 11/28/2022]
Abstract
Correct interpretation of the data from integral laboratory tests, including Thrombin Generation Test (TGT), requires biochemistry-based mathematical models of blood coagulation. The purpose of this study is to describe the experimental TGT data from healthy donors and hemophilia A (HA) and B (HB) patients. We derive a simplified ODE model and apply it to analyze the TGT data from healthy donors and HA/HB patients with in vitro added tissue factor pathway inhibitor (TFPI) antibody. This model allows the characterization of hemophilia patients in the space of three most important model parameters. The proposed approach may provide a new quantitative tool for the analysis of experimental TGT. Also, it gives a reduced model of coagulation verified against clinical data to be used in future theoretical large-scale modeling of thrombosis in flow.
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Affiliation(s)
- N Ratto
- Institute Camille Jordan, UMR 5208 CNRS, University Lyon 1, 69622, Villeurbanne, France.
| | - A Tokarev
- Peoples' Friendship University of Russia (RUDN University), Moscow, 117198, Russian Federation
| | - P Chelle
- Center for Health Engineering, UMR 5307, Ecole Nationale Supérieure des Mines de Saint-Etienne, 42023, Saint-Etienne, France
- Unite Inserm Sainbiose U1059, Université Jean Monnet, 42023, Saint-Etienne, France
| | - B Tardy-Poncet
- Unite Inserm Sainbiose U1059, Université Jean Monnet, 42023, Saint-Etienne, France
- Inserm CIC1408, CHU de Saint-Etienne, 42023, Saint-Etienne, France
- CRC Hémophilie CHU St Etienne, 42023, Saint-Etienne, France
| | - V Volpert
- Institute Camille Jordan, UMR 5208 CNRS, University Lyon 1, 69622, Villeurbanne, France
- Institute Camille Jordan, INRIA, Université de Lyon, Université Lyon 1, 69200, Villeurbanne, France
- Peoples' Friendship University of Russia (RUDN University), Moscow, 117198, Russian Federation
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32
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Beuter A, Balossier A, Vassal F, Hemm S, Volpert V. Cortical stimulation in aphasia following ischemic stroke: toward model-guided electrical neuromodulation. Biol Cybern 2020; 114:5-21. [PMID: 32020368 DOI: 10.1007/s00422-020-00818-w] [Citation(s) in RCA: 12] [Impact Index Per Article: 3.0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/24/2019] [Accepted: 01/28/2020] [Indexed: 06/10/2023]
Abstract
The aim of this paper is to integrate different bodies of research including brain traveling waves, brain neuromodulation, neural field modeling and post-stroke language disorders in order to explore the opportunity of implementing model-guided, cortical neuromodulation for the treatment of post-stroke aphasia. Worldwide according to WHO, strokes are the second leading cause of death and the third leading cause of disability. In ischemic stroke, there is not enough blood supply to provide enough oxygen and nutrients to parts of the brain, while in hemorrhagic stroke, there is bleeding within the enclosed cranial cavity. The present paper focuses on ischemic stroke. We first review accumulating observations of traveling waves occurring spontaneously or triggered by external stimuli in healthy subjects as well as in patients with brain disorders. We examine the putative functions of these waves and focus on post-stroke aphasia observed when brain language networks become fragmented and/or partly silent, thus perturbing the progression of traveling waves across perilesional areas. Secondly, we focus on a simplified model based on the current literature in the field and describe cortical traveling wave dynamics and their modulation. This model uses a biophysically realistic integro-differential equation describing spatially distributed and synaptically coupled neural networks producing traveling wave solutions. The model is used to calculate wave parameters (speed, amplitude and/or frequency) and to guide the reconstruction of the perturbed wave. A stimulation term is included in the model to restore wave propagation to a reasonably good level. Thirdly, we examine various issues related to the implementation model-guided neuromodulation in the treatment of post-stroke aphasia given that closed-loop invasive brain stimulation studies have recently produced encouraging results. Finally, we suggest that modulating traveling waves by acting selectively and dynamically across space and time to facilitate wave propagation is a promising therapeutic strategy especially at a time when a new generation of closed-loop cortical stimulation systems is about to arrive on the market.
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Affiliation(s)
- Anne Beuter
- Bordeaux INP, University of Bordeaux, Bordeaux, France.
| | - Anne Balossier
- Service de neurochirurgie fonctionnelle et stéréotaxique, AP-HM La Timone, Aix-Marseille University, Marseille, France
| | - François Vassal
- INSERM U1028 Neuropain, UMR 5292, Centre de Recherche en Neurosciences, Universités Lyon 1 et Saint-Etienne, Saint-Étienne, France
- Service de Neurochirurgie, Hôpital Nord, Centre Hospitalier Universitaire de Saint-Etienne, Saint-Étienne, France
| | - Simone Hemm
- School of Life Sciences, Institute for Medical Engineering and Medical Informatics, University of Applied Sciences and Arts Northwestern Switzerland, 4132, Muttenz, Switzerland
| | - Vitaly Volpert
- Institut Camille Jordan, UMR 5208 CNRS, University Lyon 1, 69622, Villeurbanne, France
- INRIA Team Dracula, INRIA Lyon La Doua, 69603, Villeurbanne, France
- People's Friendship University of Russia (RUDN University), Miklukho-Maklaya St, Moscow, Russian Federation, 117198
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33
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Bocharov G, Volpert V, Ludewig B, Meyerhans A. Editorial: Mathematical Modeling of the Immune System in Homeostasis, Infection and Disease. Front Immunol 2020; 10:2944. [PMID: 31998286 PMCID: PMC6961534 DOI: 10.3389/fimmu.2019.02944] [Citation(s) in RCA: 7] [Impact Index Per Article: 1.8] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Key Words] [Track Full Text] [Download PDF] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/05/2019] [Accepted: 12/02/2019] [Indexed: 11/17/2022] Open
Affiliation(s)
- Gennady Bocharov
- Marchuk Institute of Numerical Mathematics of the Russian Academy of Sciences, Moscow, Russia.,Institute for Personalized Medicine, Sechenov First Moscow State Medical University, Moscow, Russia
| | - Vitaly Volpert
- Institut Camille Jordan, Université Lyon 1, Villeurbanne, France.,INRIA Team Dracula, INRIA Lyon La Doua, Villeurbanne, France.,Peoples' Friendship University of Russia (RUDN University), Moscow, Russia
| | - Burkhard Ludewig
- Institute of Immunobiology, Kantonsspital St. Gallen, St. Gallen, Switzerland
| | - Andreas Meyerhans
- Infection Biology Laboratory, Department of Experimental and Health Sciences, Universitat Pompeu Fabra, Barcelona, Spain.,Institució Catalana de Recerca i Estudis Avançats (ICREA), Barcelona, Spain
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34
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Bouchnita A, Volpert V, Koury MJ, Hellander A. A multiscale model to design therapeutic strategies that overcome drug resistance to tyrosine kinase inhibitors in multiple myeloma. Math Biosci 2019; 319:108293. [PMID: 31809782 DOI: 10.1016/j.mbs.2019.108293] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.6] [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: 05/19/2019] [Revised: 11/10/2019] [Accepted: 11/26/2019] [Indexed: 02/06/2023]
Abstract
Drug resistance (DR) is a phenomenon characterized by the tolerance of a disease to pharmaceutical treatment. In cancer patients, DR is one of the main challenges that limit the therapeutic potential of the existing treatments. Therefore, overcoming DR by restoring the sensitivity of cancer cells would be greatly beneficial. In this context, mathematical modeling can be used to provide novel therapeutic strategies that maximize the efficiency of anti-cancer agents and potentially overcome DR. In this paper, we present a new multiscale model devoted to the interaction of potential treatments with multiple myeloma (MM) development. In this model, MM cells are represented as individual objects that move, divide, and die by apoptosis. The fate of each cell depends on intracellular and extracellular regulation, as well as the administered treatment. The model is used to explore the combined effects of a tyrosine-kinase inhibitor (TKI) with a pentose phosphate pathway (PPP) inhibitor. We use numerical simulations to tailor effective and safe treatment regimens that may eradicate the MM tumors. The model suggests that an interval for the daily dose of the PPP inhibitor can maximize the responsiveness of MM cells to the treatment with TKIs. Then, it demonstrates that the combination of high-dose pulsatile TKI treatment with high-dose daily PPP inhibitor therapy can potentially eradicate the tumor.The predictions of numerical simulations using such a model can be considered as testable hypotheses in future pre-clinical experiments and clinical studies.
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Affiliation(s)
- Anass Bouchnita
- Division of Scientific Computing, Department of Information Technology, Uppsala University, Uppsala 75105, Sweden; Ecole Centrale Casablanca, Ville Verte, Bouskoura, 20000 Casablanca, Morocco.
| | - Vitaly Volpert
- Institut Camille Jordan, Université Lyon 1, Villeurbanne 69622, France; INRIA Team Dracula, INRIA Lyon La Doua, Villeurbanne 69603, France; Peoples Friendship University of Russia (RUDN University), 6 Miklukho-Maklaya St, Moscow 117198, Russian Federation
| | - Mark J Koury
- Vanderbilt University Medical Center, Nashville, TN 37232-6307, USA
| | - Andreas Hellander
- Division of Scientific Computing, Department of Information Technology, Uppsala University, Uppsala 75105, Sweden
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35
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Lupica A, Manfredi P, Volpert V, Palumbo A, d'Onofrio A. Spatio-temporal games of voluntary vaccination in the absence of the infection: the interplay of local versus non-local information about vaccine adverse events. Math Biosci Eng 2019; 17:1090-1131. [PMID: 32233572 DOI: 10.3934/mbe.2020058] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/11/2023]
Abstract
Under voluntary vaccination, a critical role in shaping the level and trends of vaccine uptake is played by the type and structure of information that is received and used by parents of children eligible for vaccination. In this article we investigate the feedbacks of spatial mobility and the spatial structure of information on vaccination dynamics, by extending to a continuous spatially structured setting existing behavioral epidemiology models for the impact of vaccine adverse events (VAEs) on vaccination choices. We considered the simplest spatial setting, namely classical 'Fickian' diffusion, and focused on the noteworthy case where the infection is absent. This scenario mimics the important case of a population where a previously endemic vaccine preventable infection was successfully eliminated, but the re-emergence of the disease must be prevented. This is, for example, the case of poliomyelitis in most countries worldwide. In such a situation, the dynamics of VAEs and of the related information arguably become the key determinant of vaccination decision and of collective coverage. In relation to this 'information issue', we compared the effects of three main cases: (i) purely local information, where agents react only to locally occurred events; (ii) a mix of purely local and global, country-wide, information due e.g., to country-wide media and the internet; (iii) a mix of local and non-local information. By representing these different information options through a range of different spatial information kernels, we investigated: the presence and stability of space-homogeneous, nontrivial, behavior-induced equilibria; the existence of bifurcations; the existence of classical and generalized traveling waves; and the effects of awareness campaigns enacted by the Public Health System to sustain vaccine uptake. Finally, we analyzed some analogies and differences between our models and those of the Theory of Innovation Diffusion.
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Affiliation(s)
- Antonella Lupica
- Department of Mathematics and Computer Sciences, University of Catania, V.le A. Doria 6, 95125 Catania, Italy
- Department of Mathematical and Computer Sciences, Physical Sciences and Earth Sciences, University of Messina, V.le F. D'Alcontres 31, 98166 Messina, Italy
| | - Piero Manfredi
- Department of Economics and Management, University of Pisa, Via Ridolfi 10, 5612 Pisa, Italy
| | - Vitaly Volpert
- Institut Camille Jordan, UMR 5208 CNRS, University Lyon 1, 69622 Villeurbanne, France
- INRIA Team Dracula, INRIA Lyon La Doua, 69603 Villeurbanne, France
- Peoples Friendship University of Russia (RUDN University), 6 Miklukho-Maklaya St, Moscow, 117198, Russia
| | - Annunziata Palumbo
- Department of Mathematical and Computer Sciences, Physical Sciences and Earth Sciences, University of Messina, V.le F. D'Alcontres 31, 98166 Messina, Italy
| | - Alberto d'Onofrio
- International Prevention Research Institute, 95 Cours Lafayette 69006 Lyon, France
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36
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Grebennikov D, Bouchnita A, Volpert V, Bessonov N, Meyerhans A, Bocharov G. Corrigendum: Spatial Lymphocyte Dynamics in Lymph Nodes Predicts the Cytotoxic T Cell Frequency Needed for HIV Infection Control. Front Immunol 2019; 10:1538. [PMID: 31333669 PMCID: PMC6616112 DOI: 10.3389/fimmu.2019.01538] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/18/2019] [Accepted: 06/19/2019] [Indexed: 12/03/2022] Open
Affiliation(s)
- Dmitry Grebennikov
- Moscow Institute of Physics and Technology, National Research University, Dolgoprudny, Russia.,Marchuk Institute of Numerical Mathematics, Russian Academy of Sciences, Moscow, Russia.,Peoples' Friendship University of Russia (RUDN University), Moscow, Russia
| | - Anass Bouchnita
- Division of Scientific Computing, Department of Information Technology, Uppsala University, Uppsala, Sweden
| | - Vitaly Volpert
- Peoples' Friendship University of Russia (RUDN University), Moscow, Russia.,Institut Camille Jordan, UMR 5208 CNRS, University Lyon 1, Villeurbanne, France.,INRIA Team Dracula, INRIA Lyon La Doua, Villeurbanne, France
| | - Nikolay Bessonov
- Institute of Problems of Mechanical Engineering, Russian Academy of Sciences, Saint Petersburg, Russia
| | - Andreas Meyerhans
- Infection Biology Laboratory, Department of Experimental and Health Sciences, Universitat Pompeu Fabra, Barcelona, Spain.,Institució Catalana de Recerca i Estudis Avançats (ICREA), Barcelona, Spain
| | - Gennady Bocharov
- Marchuk Institute of Numerical Mathematics, Russian Academy of Sciences, Moscow, Russia.,Sechenov First Moscow State Medical University, Moscow, Russia
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37
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Grebennikov D, Bouchnita A, Volpert V, Bessonov N, Meyerhans A, Bocharov G. Spatial Lymphocyte Dynamics in Lymph Nodes Predicts the Cytotoxic T Cell Frequency Needed for HIV Infection Control. Front Immunol 2019; 10:1213. [PMID: 31244829 PMCID: PMC6579925 DOI: 10.3389/fimmu.2019.01213] [Citation(s) in RCA: 5] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/08/2019] [Accepted: 05/13/2019] [Indexed: 11/29/2022] Open
Abstract
The surveillance of host body tissues by immune cells is central for mediating their defense function. In vivo imaging technologies have been used to quantitatively characterize target cell scanning and migration of lymphocytes within lymph nodes (LNs). The translation of these quantitative insights into a predictive understanding of immune system functioning in response to various perturbations critically depends on computational tools linking the individual immune cell properties with the emergent behavior of the immune system. By choosing the Newtonian second law for the governing equations, we developed a broadly applicable mathematical model linking individual and coordinated T-cell behaviors. The spatial cell dynamics is described by a superposition of autonomous locomotion, intercellular interaction, and viscous damping processes. The model is calibrated using in vivo data on T-cell motility metrics in LNs such as the translational speeds, turning angle speeds, and meandering indices. The model is applied to predict the impact of T-cell motility on protection against HIV infection, i.e., to estimate the threshold frequency of HIV-specific cytotoxic T cells (CTLs) that is required to detect productively infected cells before the release of viral particles starts. With this, it provides guidance for HIV vaccine studies allowing for the migration of cells in fibrotic LNs.
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Affiliation(s)
- Dmitry Grebennikov
- Moscow Institute of Physics and Technology, National Research University, Dolgoprudny, Russia.,Marchuk Institute of Numerical Mathematics, Russian Academy of Sciences, Moscow, Russia.,Peoples' Friendship University of Russia (RUDN University), Moscow, Russia
| | - Anass Bouchnita
- Division of Scientific Computing, Department of Information Technology, Uppsala University, Uppsala, Sweden
| | - Vitaly Volpert
- Peoples' Friendship University of Russia (RUDN University), Moscow, Russia.,Institut Camille Jordan, UMR 5208 CNRS, University Lyon 1, Villeurbanne, France.,INRIA Team Dracula, INRIA Lyon La Doua, Villeurbanne, France
| | - Nikolay Bessonov
- Institute of Problems of Mechanical Engineering, Russian Academy of Sciences, Saint Petersburg, Russia
| | - Andreas Meyerhans
- Infection Biology Laboratory, Department of Experimental and Health Sciences, Universitat Pompeu Fabra, Barcelona, Spain.,Institució Catalana de Recerca i Estudis Avançats (ICREA), Barcelona, Spain
| | - Gennady Bocharov
- Marchuk Institute of Numerical Mathematics, Russian Academy of Sciences, Moscow, Russia.,Sechenov First Moscow State Medical University, Moscow, Russia
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38
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Benchaib MA, Bouchnita A, Volpert V, Makhoute A. Mathematical Modeling Reveals That the Administration of EGF Can Promote the Elimination of Lymph Node Metastases by PD-1/PD-L1 Blockade. Front Bioeng Biotechnol 2019; 7:104. [PMID: 31157216 PMCID: PMC6528618 DOI: 10.3389/fbioe.2019.00104] [Citation(s) in RCA: 13] [Impact Index Per Article: 2.6] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/08/2019] [Accepted: 04/24/2019] [Indexed: 01/29/2023] Open
Abstract
In the advanced stages of cancers like melanoma, some of the malignant cells leave the primary tumor and infiltrate the neighboring lymph nodes (LNs). The interaction between secondary cancer and the immune response in the lymph node represents a complex process that needs to be fully understood in order to develop more effective immunotherapeutic strategies. In this process, antigen-presenting cells (APCs) approach the tumor and initiate the adaptive immune response for the corresponding antigen. They stimulate the naive CD4+ and CD8+ T lymphocytes which subsequently generate a population of helper and effector cells. On one hand, immune cells can eliminate tumor cells using cell-cell contact and by secreting apoptosis inducing cytokines. They are also able to induce their dormancy. On the other hand, the tumor cells are able to escape the immune surveillance using their immunosuppressive abilities. To study the interplay between tumor progression and the immune response, we develop two new models describing the interaction between cancer and immune cells in the lymph node. The first model consists of partial differential equations (PDEs) describing the populations of the different types of cells. The second one is a hybrid discrete-continuous model integrating the mechanical and biochemical mechanisms that define the tumor-immune interplay in the lymph node. We use the continuous model to determine the conditions of the regimes of tumor-immune interaction in the lymph node. While we use the hybrid model to elucidate the mechanisms that contribute to the development of each regime at the cellular and tissue levels. We study the dynamics of tumor growth in the absence of immune cells. Then, we consider the immune response and we quantify the effects of immunosuppression and local EGF concentration on the fate of the tumor. Numerical simulations of the two models show the existence of three possible outcomes of the tumor-immune interactions in the lymph node that coincide with the main phases of the immunoediting process: tumor elimination, equilibrium, and tumor evasion. Both models predict that the administration of EGF can promote the elimination of the secondary tumor by PD-1/PD-L1 blockade.
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Affiliation(s)
| | - Anass Bouchnita
- Division of Scientific Computing, Department of Information Technology, Uppsala University, Uppsala, Sweden
| | - Vitaly Volpert
- Institut Camille Jordan, Université Lyon 1, Villeurbanne, France
- INRIA Team Dracula, INRIA Lyon La Doua, Villeurbanne, France
- Peoples Friendship University of Russia (RUDN University), Moscow, Russia
- Marchuk Institute of Numerical Mathematics of the RAS, Moscow, Russia
| | - Abdelkader Makhoute
- Faculté des Sciences, Université Moulay Ismail, Meknes, Morocco
- Faculty of Sciences, Université Libre de Bruxelles (ULB), Brussels, Belgium
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39
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Ratto N, Marion M, Volpert V. Existence of pulses for a reaction-diffusion system of blood coagulation in flow. Math Biosci Eng 2019; 16:4196-4212. [PMID: 31499658 DOI: 10.3934/mbe.2019209] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/10/2023]
Abstract
A reaction-diffusion system describing blood coagulation in flow is studied. We prove the existence of stationary solutions provided that the speed of the travelling wave problem for the limiting value of the velocity is positive. The implications to the problem of clot growth are discussed.
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Affiliation(s)
- Nicolas Ratto
- Institut Camille Jordan, UMR 5585 CNRS, Ecole Centrale de Lyon, 69134 Ecully, France
| | - Martine Marion
- Institut Camille Jordan, UMR 5585 CNRS, Ecole Centrale de Lyon, 69134 Ecully, France
| | - Vitaly Volpert
- Institut Camille Jordan, UMR 5585 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
- Peoples' Friendship University of Russia (RUDN University) 6 Miklukho-Maklaya St, Moscow, 117198, Russian Federation
- Marchuk Institute of Numerical Mathematics of the RAS, ul. Gubkina 8, 119333 Moscow, Russian Federation
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Abstract
Darwin described biological species as groups of morphologically similar individuals. These groups of individuals can split into several subgroups due to natural selection, resulting in the emergence of new species. Some species can stay stable without the appearance of a new species, some others can disappear or evolve. Some of these evolutionary patterns were described in our previous works independently of each other. In this work we have developed a single model which allows us to reproduce the principal patterns in Darwin's diagram. Some more complex evolutionary patterns are also observed. The relation between Darwin's definition of species, stated above, and Mayr's definition of species (group of individuals that can reproduce) is also discussed.
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Affiliation(s)
- Nikolai Bessonov
- Institute of Problems of Mechanical Engineering, Russian Academy of Sciences, Saint Petersburg, Russian Federation, 199178
| | | | - Malay Banerjee
- Department of Mathematics and Statistics, IIT Kanpur, Kanpur, India
| | - Vitaly Volpert
- Institut Camille Jordan, UMR 5208 CNRS, University Lyon 1, 69622, Villeurbanne, France.
- INRIA, Universite de Lyon, Universite Lyon 1, Institut Camille Jordan, 43 Bd. du 11 Novembre 1918, 69200, Villeurbanne Cedex, France.
- Peoples Friendship University of Russia (RUDN University), 6 Miklukho-Maklaya St, Moscow, Russian Federation, 117198.
- Poncelet Center, UMI 2615 CNRS, 11 Bolshoy Vlasyevskiy, Moscow, Russian Federation, 119002.
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41
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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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42
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Eymard N, Volpert V, Kurbatova P, Volpert V, Bessonov N, Ogungbenro K, Aarons L, Janiaud P, Nony P, Bajard A, Chabaud S, Bertrand Y, Kassaï B, Cornu C, Nony P. Mathematical model of T-cell lymphoblastic lymphoma: disease, treatment, cure or relapse of a virtual cohort of patients. Math Med Biol 2018; 35:25-47. [PMID: 28082512 DOI: 10.1093/imammb/dqw019] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/27/2015] [Accepted: 10/12/2016] [Indexed: 12/19/2022]
Abstract
T lymphoblastic lymphoma (T-LBL) is a rare type of lymphoma with a good prognosis with a remission rate of 85%. Patients can be completely cured or can relapse during or after a 2-year treatment. Relapses usually occur early after the remission of the acute phase. The median time of relapse is equal to 1 year, after the occurrence of complete remission (range 0.2-5.9 years) (Uyttebroeck et al., 2008). It can be assumed that patients may be treated longer than necessary with undue toxicity.The aim of our model was to investigate whether the duration of the maintenance therapy could be reduced without increasing the risk of relapses and to determine the minimum treatment duration that could be tested in a future clinical trial.We developed a mathematical model of virtual patients with T-LBL in order to obtain a proportion of virtual relapses close to the one observed in the real population of patients from the EuroLB database. Our simulations reproduced a 2-year follow-up required to study the onset of the disease, the treatment of the acute phase and the maintenance treatment phase.
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Affiliation(s)
- N Eymard
- Institut Camille Jordan, UMR, CNRS, University Lyon 1, Villeurbanne, France
| | - V Volpert
- Institut Camille Jordan, UMR, CNRS, University Lyon 1, Villeurbanne, France
| | - P Kurbatova
- Institut Camille Jordan, UMR, CNRS, University Lyon 1, Villeurbanne, France
| | - V Volpert
- INRIA Team Dracula, INRIA Antenne Lyon la Doua 69603 Villeurbanne, France
| | - N Bessonov
- Institute of Mechanical Engineering Problems, Saint Petersburg, Russia
| | - K Ogungbenro
- Centre for Applied Pharmacokinetic Research, Manchester Pharmacy School The University of Manchester, Manchester, UK
| | - L Aarons
- Centre for Applied Pharmacokinetic Research, Manchester Pharmacy School The University of Manchester, Manchester, UK
| | - P Janiaud
- University of Lyon 1, UMR, CNRS, Lyon, France
| | - P Nony
- University of Lyon 1, UMR, CNRS, Lyon, France
| | - A Bajard
- Unité de Biostatistique et d'Evaluation des Thérapeutiques Centre Léon Bérard, Lyon, France
| | - S Chabaud
- Unité de Biostatistique et d'Evaluation des Thérapeutiques Centre Léon Bérard, Lyon, France
| | - Y Bertrand
- Institute of Hematology and Oncology Paediatrics, Hospices Civils de Lyon, University Claude Bernard Lyon I, Lyon, France
| | - B Kassaï
- Hospices Civils de Lyon, Centre d'Investigation Clinique, INSERM CIC1407, Lyon, France
| | - C Cornu
- Hospices Civils de Lyon, Centre d'Investigation Clinique, INSERM CIC1407, Lyon, France
| | - P Nony
- CHU Lyon, Service de Pharmacologie Clinique et Essais Thérapeutiques, Lyon, France
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Belyaev AV, Dunster JL, Gibbins JM, Panteleev MA, Volpert V. Advancing research on blood coagulation and thrombosis: Reply to the comments on "Modeling thrombosis in silico: Frontiers, challenges, unresolved problems and milestones". Phys Life Rev 2018; 26-27:116-119. [PMID: 30297158 DOI: 10.1016/j.plrev.2018.09.006] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/25/2018] [Accepted: 09/25/2018] [Indexed: 11/17/2022]
Affiliation(s)
- A V Belyaev
- M.V. Lomonosov Moscow State University, 119991 Moscow, Russia; RUDN University, ul. Miklukho-Maklaya 6, Moscow, 117198, Russia
| | - J L Dunster
- Institute for Cardiovascular and Metabolic Research, School of Biological Sciences, University of Reading, Berkshire, RG6 6AX, United Kingdom
| | - J M Gibbins
- Institute for Cardiovascular and Metabolic Research, School of Biological Sciences, University of Reading, Berkshire, RG6 6AX, United Kingdom
| | - M A Panteleev
- M.V. Lomonosov Moscow State University, 119991 Moscow, Russia; National Medical Research Center of Pediatric Hematology, Oncology and Immunology named after Dmitry Rogachev, 117997 Moscow, Russia; Center for Theoretical Problems of Physico-Chemical Pharmacology, Russian Academy of Sciences, 119991, Moscow, Russia; Phystech School of Biological and Medical Physics, Moscow Institute of Physics and Technology, Dolgoprudny 141701, Russia
| | - V Volpert
- Institut Camille Jordan, UMR 5208 CNRS, University Lyon 1, 69622 Villeurbanne, France; INRIA Team Dracula, INRIA Lyon La Doua, 69603 Villeurbanne, France; RUDN University, ul. Miklukho-Maklaya 6, Moscow, 117198, Russia.
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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, Volpert V, Ludewig B, Meyerhans A. Modelling of Experimental Infections. Mathematical Immunology of Virus Infections 2018. [PMCID: PMC7123718 DOI: 10.1007/978-3-319-72317-4_4] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Download PDF] [Figures] [Subscribe] [Scholar Register] [Indexed: 12/01/2022]
Abstract
This chapter aims to give a clear idea of how mathematical analysis for experimental systems could help in the process of data assimilation, parameter estimation and hypothesis testing.
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Affiliation(s)
- Gennady Bocharov
- Marchuk Institute of Numerical Mathematics, Russian Academy of Sciences, Moscow, Russia
| | - Vitaly Volpert
- Institut Camille Jordan, UMR 5208 CNRS, Centre National de la Recherche Scientifique (CNRS), Villeurbanne, France
- RUDN University, Moscow, Russia
| | - Burkhard Ludewig
- Institute of Immunobiology, Kantonsspital St. Gallen, St. Gallen, Switzerland
| | - Andreas Meyerhans
- Parc de Recerca Biomedica Barcelona, ICREA and Universitat Pompeu Fabra, Barcelona, Spain
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Bouchnita A, Miossec P, Tosenberger A, Volpert V. Modeling of the effects of IL-17 and TNF-α on endothelial cells and thrombus growth. C R Biol 2017; 340:456-473. [PMID: 29195855 DOI: 10.1016/j.crvi.2017.10.002] [Citation(s) in RCA: 15] [Impact Index Per Article: 2.1] [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: 04/26/2017] [Revised: 08/13/2017] [Accepted: 10/12/2017] [Indexed: 11/24/2022]
Abstract
Rheumatoid and psoriatic arthritis are chronic inflammatory diseases, with massive increase of cardiovascular events (CVE), and contribution of the cytokines TNF-α and IL-17. Chronic inflammation inside the joint membrane or synovium results from the activation of fibroblasts/synoviocytes, and leads to the release of cytokines from monocytes (Tumor Necrosis Factor or TNF) and from T lymphocytes (Interleukin-17 or IL-17). At the systemic level, the very same cytokines affect endothelial cells and vessel wall. We have previously shown [1,2] that IL-17 and TNF-α, specifically when combined, increase procoagulation, decrease anticoagulation and increase platelet aggregation, leading to thrombosis. These results are the basis for the models of interactions between IL-17 and TNF, and genes expressed by activated endothelial cells. This work is devoted to mathematical modeling and numerical simulations of blood coagulation and clot growth under the influence of IL-17 and TNF-α. We show that they can provoke thrombosis, leading to the complete or partial occlusion of blood vessels. The regimes of blood coagulation and conditions of occlusion are investigated in numerical simulations and in approximate analytical models. The results of mathematical modeling allow us to predict thrombosis development for an individual patient.
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Affiliation(s)
- Anass Bouchnita
- Laboratoire de biométrie et biologie évolutive (LBBE), UMR 5558 CNRS, Université Lyon-1, 69376 Lyon, France; Mohammadia School of Engineering (EMI), Université Mohammed-V, 10080 Rabat, Morocco.
| | - Pierre Miossec
- Department of Clinical Immunology and Rheumatology, Immunogenomics and Inflammation, Research Unit EA 4130, Hôpital Édouard-Herriot, Université de Lyon, 69437 Lyon, France
| | - Alen Tosenberger
- Unité de chronobiologie théorique, Faculté des sciences, Université ibre de Bruxelles (ULB), campus Plaine, CP 231, 1050 Bruxelles, Belgium
| | - Vitaly Volpert
- Institut Camille-Jordan (ICJ), UMR 5208 CNRS, Université Lyon-1, 69622 Villeurbanne, France; Intitut national de recherche en informatique et automatique (INRIA), Team Dracula, INRIA Lyon La Doua, 69603 Villeurbanne, France; RUDN University, ul. Miklukho-Maklaya 6, 117198 Moscow, Russia
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Bouchnita A, Galochkina T, Kurbatova P, Nony P, Volpert V. Conditions of microvessel occlusion for blood coagulation in flow. Int J Numer Method Biomed Eng 2017; 33:e2850. [PMID: 27863131 DOI: 10.1002/cnm.2850] [Citation(s) in RCA: 9] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/13/2016] [Revised: 09/26/2016] [Accepted: 10/25/2016] [Indexed: 06/06/2023]
Abstract
Vessel occlusion is a perturbation of blood flow inside a blood vessel because of the fibrin clot formation. As a result, blood circulation in the vessel can be slowed down or even stopped. This can provoke the risk of cardiovascular events. In order to explore this phenomenon, we used a previously developed mathematical model of blood clotting to describe the concentrations of blood factors with a reaction-diffusion system of equations. The Navier-Stokes equations were used to model blood flow, and we treated the clot as a porous medium. We identify the conditions of partial or complete occlusion in a small vessel depending on various physical and physiological parameters. In particular, we were interested in the conditions on blood flow and diameter of the wounded area. The existence of a critical flow velocity separating the regimes of partial and complete occlusion was demonstrated through the mathematical investigation of a simplified model of thrombin wave propagation in Poiseuille flow. We observed different regimes of vessel occlusion depending on the model parameters both for the numerical simulations and in the theoretical study. Then, we compared the rate of clot growth in flow obtained in the simulations with experimental data. Both of them showed the existence of different regimes of clot growth depending on the velocity of blood flow.
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Affiliation(s)
- A Bouchnita
- Institut Camille Jordan, UMR 5208 CNRS, University Lyon 1, Villeurbanne, 69622, France
- Laboratoire de Biométrie et Biologie Evolutive, UMR 5558 CNRS, University Lyon 1, Lyon, 69376, France
- Laboratory of Study and Research in Applied Mathematics, Mohammadia School of Engineers, Mohamed V University, Rabat, Morocco
| | - T Galochkina
- Institut Camille Jordan, UMR 5208 CNRS, University Lyon 1, Villeurbanne, 69622, France
- Department of Biophysics, Faculty of Biology, M.V. Lomonosov Moscow State University, Leninskie gory 1, Moscow, Russia
- Federal Research Clinical Center of Federal Medical & Biological Agency of Russia, Orekhovy boulevard 28, Moscow, Russia
| | - P Kurbatova
- Institut Camille Jordan, UMR 5208 CNRS, University Lyon 1, Villeurbanne, 69622, France
- Laboratoire de Biométrie et Biologie Evolutive, UMR 5558 CNRS, University Lyon 1, Lyon, 69376, France
| | - P Nony
- Laboratoire de Biométrie et Biologie Evolutive, UMR 5558 CNRS, University Lyon 1, Lyon, 69376, France
- Service de Pharmacologie Clinique, Hospices Civils de Lyon, Lyon, France
| | - V Volpert
- Institut Camille Jordan, UMR 5208 CNRS, University Lyon 1, Villeurbanne, 69622, France
- INRIA Team Dracula, INRIA Lyon La Doua, 69603 Villeurbanne, France
- Laboratoire Poncelet, UMI 2615 CNRS, Bolshoy Vlasyevskiy Pereulok 11, 119002 Moscow, Russia
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Abstract
BACKGROUND Moving from the molecular and cellular level to a multi-scale systems understanding of immune responses requires the development of novel approaches to integrate knowledge and data from different biological levels into mechanism-based integrative mathematical models. The aim of our study is to present a methodology for a hybrid modelling of immunological processes in their spatial context. METHODS A two-level hybrid mathematical model of immune cell migration and interaction integrating cellular and organ levels of regulation for a 2D spatial consideration of idealized secondary lymphoid organs is developed. It considers the population dynamics of antigen-presenting cells, CD4 + and CD8 + T lymphocytes in naive-, proliferation- and differentiated states. Cell division is assumed to be asymmetric and regulated by the extracellular concentration of interleukin-2 (IL-2) and type I interferon (IFN), together controlling the balance between proliferation and differentiation. The cytokine dynamics is described by reaction-diffusion PDEs whereas the intracellular regulation is modelled with a system of ODEs. RESULTS The mathematical model has been developed, calibrated and numerically implemented to study various scenarios in the regulation of T cell immune responses to infection, in particular the change in the diffusion coefficient of type I IFN as compared to IL-2. We have shown that a hybrid modelling approach provides an efficient tool to describe and analyze the interplay between spatio-temporal processes in the emergence of abnormal immune response dynamics. DISCUSSION Virus persistence in humans is often associated with an exhaustion of T lymphocytes. Many factors can contribute to the development of exhaustion. One of them is associated with a shift from a normal clonal expansion pathway to an altered one characterized by an early terminal differentiation of T cells. We propose that an altered T cell differentiation and proliferation sequence can naturally result from a spatial separation of the signaling events delivered via TCR, IL-2 and type I IFN receptors. Indeed, the spatial overlap of the concentration fields of extracellular IL-2 and IFN in lymph nodes changes dynamically due to different migration patterns of APCs and CD4 + T cells secreting them. CONCLUSIONS The proposed hybrid mathematical model of the immune response represents a novel analytical tool to examine challenging issues in the spatio-temporal regulation of cell growth and differentiation, in particular the effect of timing and location of activation signals.
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Affiliation(s)
- Anass Bouchnita
- Institut Camille Jordan, UMR 5208 CNRS, University Lyon 1, Villeurbanne, 69622 France
- Laboratoire de Biométrie et Biologie Evolutive (LBBE), UMR 5558 CNRS, University Lyon 1, Villeurbanne, 69622 France
- Mohammadia School of Engineering, Mohamed V University, Rabat, 10080 Morocco
| | - Gennady Bocharov
- Institute of Numerical Mathematics, Russian Academy of Sciences, Gubkina Street 8, Moscow, 119333 Russian Federation
| | - Andreas Meyerhans
- Institute of Numerical Mathematics, Russian Academy of Sciences, Gubkina Street 8, Moscow, 119333 Russian Federation
- Infection Biology Laboratory, Department of Experimental and Health Sciences, Universitat Pompeu Fabra, Doctor Aiguader, 88, Barcelona, 08003 Spain
- ICREA, Pg. Lluís Companys 23, Barcelona, 08010 Spain
| | - Vitaly Volpert
- Institut Camille Jordan, UMR 5208 CNRS, University Lyon 1, Villeurbanne, 69622 France
- Institute of Numerical Mathematics, Russian Academy of Sciences, Gubkina Street 8, Moscow, 119333 Russian Federation
- INRIA Team Dracula, INRIA Lyon La Doua, Villeurbanne, 69603 France
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Banerjee M, Volpert V. Spatio-temporal pattern formation in Rosenzweig–MacArthur model: Effect of nonlocal interactions. Ecological Complexity 2017. [DOI: 10.1016/j.ecocom.2016.12.002] [Citation(s) in RCA: 30] [Impact Index Per Article: 4.3] [Reference Citation Analysis] [What about the content of this article? (0)] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 10/20/2022]
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Galochkina T, Bouchnita A, Kurbatova P, Volpert V. Reaction-diffusion waves of blood coagulation. Math Biosci 2017; 288:130-139. [PMID: 28347652 DOI: 10.1016/j.mbs.2017.03.008] [Citation(s) in RCA: 14] [Impact Index Per Article: 2.0] [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: 09/20/2016] [Revised: 02/08/2017] [Accepted: 03/23/2017] [Indexed: 12/01/2022]
Abstract
One of the main characteristics of blood coagulation is the speed of clot growth. In the current work we consider a mathematical model of the coagulation cascade and study existence, stability and speed of propagation of the reaction-diffusion waves of blood coagulation. We also develop a simplified one-equation model that reflects the main features of the thrombin wave propagation. For this equation we estimate the wave speed analytically. The resulting formulas provide a good approximation for the speed of wave propagation in a more complex model as well as for the experimental data.
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Affiliation(s)
- Tatiana Galochkina
- Camille Jordan Institute, University Lyon 1, Villeurbanne, 69622 France; INRIA Team Dracula, INRIA Antenne Lyon la Doua, Villeurbanne, 69603 France; Department of Biophysics, Faculty of Biology, M.V. Lomonosov Moscow State University, Moscow, 119992 Russia.
| | - Anass Bouchnita
- Camille Jordan Institute, University Lyon 1, Villeurbanne, 69622 France; INRIA Team Dracula, INRIA Antenne Lyon la Doua, Villeurbanne, 69603 France; Laboratoire de Biométrie et Biologie Evolutive, UMR 5558 CNRS, University Lyon 1, Lyon, 69376 France; Laboratory of Study and Research in Applied Mathematics, Mohammadia School of Engineers, Mohamed V university, Rabat, Morocco
| | - Polina Kurbatova
- Laboratoire de Biométrie et Biologie Evolutive, UMR 5558 CNRS, University Lyon 1, Lyon, 69376 France
| | - Vitaly Volpert
- Camille Jordan Institute, University Lyon 1, Villeurbanne, 69622 France; INRIA Team Dracula, INRIA Antenne Lyon la Doua, Villeurbanne, 69603 France; RUDN University, Moscow, 117198 Russia
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