Reminder of the objectives given for the last evaluation

The aim of this project is the development of modern tools for multi-scale modeling in biological phenomena. During the period 2014-2017, the objectives we had fixed were to develop modern tools for multi-scale modeling of biological phenomena, as detailed hereafter:

1. Multi-scale modeling of erythropoiesis, the process of red blood cell production, in order to describe normal, stress, and pathological erythropoiesis, using mathematical and computational models. This led to:
2. The modeling of hemoglobin instability in dialysis patients: Thomas Lepoutre has been progressively taking part in this theme through a collaboration with P. Kim (University of Sydney, Australia);
3. Multi-scale modeling of the CD8 T cell immune response, in order to develop a predictive model of the CD8 T cell response, by confronting the model at different stages to in vivo-acquired experimental data;
4. Population dynamics modeling, with the aim to develop general mathematical tools to study them. The main tools we were using were structured equations, in which the cell population is endowed with relevant structures, or traits. We identified limitations in using these formalisms, this is why we started developing multi-scale approaches;
5. Modeling of Chronic Myeloid Leukemia (CML) treatment, using ordinary differential equations models. Our team had already developed a first model of mutant leukemic cells being resistant to chemotherapy. A next step would be to identify the parameters using experimental data;
6. Multi-scale modeling carried out on the basis of hybrid discrete-continuous models, where dissipative particle dynamics (DPD) are used in order to describe individual cells and relatively small cell populations, partial differential equations (PDE) are used to describe concentrations of bio-chemical substances in the extracellular matrix, and ordinary differential equations for intracellular regulatory networks (Figure 1). An emphasis would be made on developing codes that are both flexible and powerful enough to implement variants of the model, perform simulations, produce desired outputs, and provide tools for analysis; to do so:
7. We planned to contribute to a recent project named chronos, whose code (written in C++) represents heterogeneous populations of individual cells evolving in time and interacting physically and biochemically, and the objective is to make the code flexible enough to implement different formalisms within the same model, so that different components of the model can be represented in the most appropriate way;
8. Partial differential equations (PDE) analysis, with a focus on reaction-diffusion equations, transport equations (hyperbolic PDEs) in which the structure can be age, maturity, protein concentration, etc., with particular cases where transport equations are reduced to delay differential equations (DDE).

Comments on these objectives over the evaluation period

We have had strong contributions to objectives 1, 3, 4, 5, and consequently to objective 6, as well as to objective 8, as mentioned in previous sections. These contributions represented the core of the team's research activity over the evaluation period, as stressed by our publications. It is however noticeable that multi-scale modeling of the immune response and of pathological hematopoiesis (leukemia) has come to represent a proportionally more important part of our activity.

Objective 2 has been cancelled few months after the previous evaluation, following meetings with clinicians who did not show any particular interest in our approaches. The modeling of chronic myeloid leukemia instead took a bigger part of the team's research activity, both project being at the time coordinated by Thomas Lepoutre.

Objective 7 has been pursued, the project chronos evolved to a better defined project SiMuScale that is currently being developed and aims at structuring the team's activity and providing a simulation platform that could be adapted to various biological questions necessitating multi-scale modeling.

Objectives for the next four years

The main objectives for the next four years are to continue to improve the 3 previous points: 1) Mathematical and computational modeling for cell population dynamics; 2) Multi-scale modeling of hematopoiesis and leukemia; 3) Multi-scale modeling of the immune response. In addition, we will pursue our effort to develop a simulation platform for multi-scale models (SiMuScale) and we intend to develop the use of mixed effect models and other statistical approaches to deal with the challenges offered by modern biology, in particular the generation of single cell data.

5.1.1 CelDyn

• Keywords: Modeling, Bioinformatics, Biology
• Functional Description: Software "Celdyn" is developed in order to model cell population dynamics for biological applications. Cells are represented either as soft spheres or they can have more complex structure. Cells can divide, move, interact with each other or with the surrounding medium. Different cell types can be introduced. When cells divide, the types of daughter cells are specified. A user interface is developed.
• Authors: Nikolai Bessonov, Vitaly Volpert
• Contacts: Nicolas Jourdan, Vitaly Volpert
• Participants: Alen Tosenberger, Laurent Pujo-Menjouet, Nikolai Bessonov, Vitaly Volpert

6.1.1 Modeling and characterization of inter-individual variability in CD8 T cell responses

To develop vaccines, it is mandatory yet challenging to account for inter-individual variability during immune responses. Even in laboratory mice, T cell responses of single individuals exhibit a high heterogeneity that may come from genetic backgrounds, intra-specific processes (e.g. antigen-processing and presentation) and immunization protocols. To account for inter-individual variability in CD8 T cell responses in mice, we propose in 4, a dynamical model coupled to a statistical, nonlinear mixed effects model. Average and individual dynamics during a CD8 T cell response are characterized in different immunization contexts (vaccinia virus and tumor). On one hand, we identify biological processes that generate inter-individual variability (activation rate of naive cells, the mortality rate of effector cells, and dynamics of the immunogen). On the other hand, introducing categorical covariates to analyze two different immunization regimens, we highlight the steps of the response impacted by immunogens (priming, differentiation of naive cells, expansion of effector cells and generation of memory cells). The robustness of the model is assessed by confrontation to new experimental data. Our approach allows to investigate immune responses in various immunization contexts, when measurements are scarce or missing, and contributes to a better understanding of inter-individual variability in CD8 T cell immune responses.

6.1.2 Multiscale modeling of germinal center recapitulates the temporal transition from memory B cells to plasma cells

Germinal centers play a key role in the adaptive immune system since they are able to produce memory B cells and plasma cells that produce high affinity antibodies for an effective immune protection. The mechanisms underlying cell-fate decisions are not well understood but asymmetric division of antigen, B-cell receptor affinity, interactions between B-cells and T follicular helper cells (triggering CD40 signaling), and regulatory interactions of transcription factors have all been proposed to play a role. In addition, a temporal switch from memory B-cell to plasma cell differentiation during the germinal center reaction has been shown. To investigate if antigen affinity-based Tfh cell help recapitulates the temporal switch we implemented (see 20) a multiscale model that integrates cellular interactions with a core gene regulatory network comprising BCL6, IRF4, and BLIMP1. Using this model we show that affinity-based CD40 signaling in combination with asymmetric division of B-cells result in switch from memory B-cell to plasma cell generation during the course of the germinal center reaction. We also show that cell fate division is unlikely to be (solely) based on asymmetric division of Ag but that BLIMP1 is a more important factor. Altogether, our model enables to test the influence of molecular modulations of the CD40 signaling pathway on the production of germinal center output cells.

6.1.3 Maternal passive immunity and dengue hemorrhagic fever in infants

Dengue hemorrhagic fever (DHF) can occur in primary dengue virus infection of infants less than one year of age. To understand the presumed role of maternal dengue-specific antibodies received until birth in the development of this primary DHF in infants, we investigated in 2 a mathematical model based on a system of nonlinear ordinary differential equations that mimics cells, virus and antibodies interactions. The neutralization and enhancement activities of maternal antibodies against the virus are represented by a function derived from experimental data and knowledge from the medical literature. The analytic study of the model shows the existence of two equilibriums, a disease-free equilibrium and an endemic one. We performed the asymptotic stability analysis for these two equilibriums. The local asymptotic stability of the endemic equilibrium (DHF equilibrium) corresponds to the occurrence of DHF. Numerical results are also presented in order to illustrate the mathematical analysis performed, highlighting the most important parameters that drive model dynamics. We defined the age at which DHF occurs as the time when the infection takes off that means at the inflection point of the curve of infected cell population. We showed that this age corresponds to the one at which maximum enhancing activity for dengue infection appears. This critical time for the occurrence of DHF is calculated from the model to be approximately two months after the time for maternal dengue neutralizing antibodies to degrade below a protective level, which corresponds to what is observed in the experimental data from the literature.

6.1.4 Viral infection model with CTL immunity

A mathematical model describing viral dynamics in the presence of the latently infected cells and the cytotoxic T-lymphocytes cells (CTL), taking into consideration the spatial mobility of free viruses, is presented and studied in 16. The model includes five nonlinear differential equations describing the interaction among the uninfected cells, the latently infected cells, the actively infected cells, the free viruses, and the cellular immune response. First, we establish the existence, positivity, and boundedness for the suggested diffusion model. Moreover, we prove the global stability of each steady state by constructing some suitable Lyapunov functionals. Finally, we validated our theoretical results by numerical simulations for each case. A review on mathematical modeling of the immune system in homeostasis, infection and disease is presented in 17.

6.1.5 Genotype-dependent virus distribution and competition of virus strains

Virus density distribution as a function of genotype considered as a continuous variable and of time is studied with a nonlocal reaction-diffusion equation taking into account virus competition for the host cells and its elimination by the immune response and by the genotype-dependent mortality, 9. The existence of virus strains, that is, of positive stable stationary solutions decaying at infinity, is determined by the admissible intervals in the genotype space where the genotype-dependent mortality is less than the virus reproduction rate, and by the immune response under some appropriate assumptions on the immune response function characterizing virus elimination by immune cells. The competition of virus strains is studied, first, without immune response and then with the immune response. In the absence of immune response, the strain dynamics is different in a short time scale where they converge to some intermediate slowly evolving solutions depending on the initial conditions, and in a long time scale where their distribution converges to a stationary solution. Immune response can essentially influence the strain dynamics either stabilizing them or eliminating one of the strains. An antiviral treatment can also influence the competition of virus strains, and it can lead to the emergence of resistant strains, which were absent before the treatment because of the competition with susceptible strains.

6.1.6 Reaction–diffusion waves in a model of immune response

In 27, existence of travelling waves is studied for a reaction-diffusion system of equations describing the distribution of viruses and immune cells in the tissue. The proof uses the Leray-Schauder method based on the topological degree for elliptic operators in unbounded domains and on a priori estimates of solutions in weighted spaces.

6.1.7 Nonlocal reaction–diffusion model of viral evolution: emergence of virus strains

The paper 10 is devoted to the investigation of virus quasi-species evolution and diversification due to mutations, competition for host cells, and cross-reactive immune responses. The model consists of a nonlocal reaction-diffusion equation for the virus density depending on the genotype considered to be a continuous variable and on time. This equation contains two integral terms corresponding to the nonlocal effects of virus interaction with host cells and with immune cells. In the model, a virus strain is represented by a localized solution concentrated around some given genotype. Emergence of new strains corresponds to a periodic wave propagating in the space of genotypes. The conditions of appearance of such waves and their dynamics are described.

6.4.1 A mathematical model for treatment of papillary thyroid cancer using the allee effect

The incidence of thyroid cancer is rising all over the world, and the papillary subtype (PTC) is the primary factor for this increase. The presence of thyroid tumors is commonly associated with increased levels of cytokines, such as interleukin 6 (IL-6). Considering PTC patients treated with thyroidectomy and radioactive iodine (RAI), we propose in 15 an ordinary differential system using four variables: the RAI activity, the number of cancer cells and the serum concentrations of IL-6 and thyroglobulin (Tg). Our objective is to study the efficacy of different therapeutic doses of RAI in the treatment of thyroid cancer. The Allee effect is taken into account when modeling cancer cells growth under the influence of IL-6. From the results obtained, the main factors and conditions correlated with successful treatment, such as the RAI activity used and the tumor response are addressed. The detection of minimal doses of RAI that can cause tumor extinction is performed, though this has also meant longer periods for tumor cell elimination. The critical number of tumor cells due to the Allee effect is analyzed and linked to the immune system or biological factors that can slow the progression of the tumor but are insufficient after thyroid resection surgery.

6.4.2 A multiscale model to design therapeutic strategies that overcome drug resistance to tyrosine kinase inhibitors in multiple myeloma

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 13, 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. Predictions of numerical simulations using such a model can be considered as testable hypotheses in future pre-clinical experiments and clinical studies.

6.5.1 Coronavirus - Scientific insights and societal aspects

In December 2019, the first case of infection with a new virus COVID-19 (SARS-CoV-2), named coronavirus, was reported in the city of Wuhan, China. At that time, almost nobody paid any attention to it. The new pathogen, however, fast proved to be extremely infectious and dangerous. Over the few months that followed, coronavirus has spread over entire world. At the end of March, the total number of infections is fast approaching the psychological threshold of one million, resulting so far in tens of thousands of deaths. Due to the high number of lives already lost and the virus high potential for further spread, and due to its huge overall impact on the economies and societies, it is widely admitted that coronavirus poses the biggest challenge to the humanity after the second World war. The COVID-19 epidemic is provoking numerous questions at all levels. It also shows that modern society is extremely vulnerable and unprepared to such events. A wide scientific and public discussion becomes urgent. Some possible directions of this discussion are suggested in 28 and 26.

6.5.2 On a quarantine model of coronavirus infection and data analysis

Attempts to curb the spread of coronavirus by introducing strict quarantine measures apparently have different effect in different countries: while the number of new cases has reportedly decreased in China and South Korea, it still exhibit significant growth in Italy and other countries across Europe. In the brief note 25, we endeavour to assess the efficiency of quarantine measures by means of mathematical modelling. Instead of the classical SIR model, we introduce a new model of infection progression under the assumption that all infected individual are isolated after the incubation period in such a way that they cannot infect other people. Disease progression in this model is determined by the basic reproduction number R0 (the number of newly infected individuals during the incubation period), which is different compared to that for the standard SIR model. If R0>1, then the number of latently infected individuals exponentially grows. However, if R0<1 (e.g. due to quarantine measures and contact restrictions imposed by public authorities), then the number of infected decays exponentially. We then consider the available data on the disease development in different countries to show that there are three possible patterns: growth dynamics, growth-decays dynamics, and patchy dynamics (growth-decay-growth). Analysis of the data in China and Korea shows that the peak of infection (maximum of daily cases) is reached about 10 days after the restricting measures are introduced. During this period of time, the growth rate of the total number of infected was gradually decreasing. However, the growth rate remains exponential in Italy. Arguably, it suggests that the introduced quarantine is not sufficient and stricter measures are needed.

6.5.3 Global dynamics of a differential-difference system: a case of Kermack-McKendrick SIR model with age-structured protection phase

In 1, we are concerned with an epidemic model of susceptible, infected and recovered (SIR) population dynamic by considering an age-structured phase of protection with limited duration, for instance due to vaccination or drugs with temporary immunity. The model is reduced to a delay differential-difference system, where the delay is the duration of the protection phase. We investigate the local asymptotic stability of the two steady states: disease-free and endemic. We also establish when the endemic steady state exists, the uniform persistence of the disease. We construct quadratic and logarithmic Lyapunov functions to establish the global asymptotic stability of the two steady states. We prove that the global stability is completely determined by the basic reproduction number.

6.8.1 Prey-predator model with nonlocal and global consumption in the prey dynamics

A prey-predator model with a nonlocal or global consumption of resources by prey is studied in 5. Linear stability analysis about the homogeneous in space stationary solution is carried out to determine the conditions of the bifurcation of stationary and moving pulses in the case of global consumption. Their existence is confirmed in numerical simulations. Periodic travelling waves and multiple pulses are observed for the nonlocal consumption.

6.8.2 Modeling the dynamics of Wolbachia-infected and uninfected A edes aegypti populations by delay differential equations

In the paper 7, starting from an age structured partial differential model, constructed taking into account the mosquito life cycle and the main features of the Wolbachia-infection, we derived a delay differential model using the method of characteristics, to study the colonization and persistence of the Wolbachia-transinfected Aedes aegypti mosquito in an environment where the uninfected wild mosquito population is already established. Under some conditions, the model can be reduced to a Nicholson-type delay differential system; here, the delay represents the duration of mosquito immature phase that comprises egg, larva and pupa. In addition to mortality and oviposition rates characteristic of the life cycle of the mosquito, other biological features such as cytoplasmic incompatibility, bacterial inheritance, and deviation on sex ratio are considered in the model. The model presents three equilibriums: the extinction of both populations, the extinction of Wolbachia-infected population and persistence of uninfected one, and the coexistence. The conditions of existence for each equilibrium are obtained analytically and have been interpreted biologically. It is shown that the increase of the delay can promote, through Hopf bifurcation, stability switch towards instability for the nonzero equilibriums. Overall, when the delay increases and crosses predetermined thresholds, the populations go to extinction.

6.8.3 Spatio-temporal Bazykin’s model for prey-predator interaction

The paper 22 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.

6.8.4 Dynamics of a diffusive two-prey-one-predator model with nonlocal intra-specific competition for both the prey species

Investigation of interacting populations is an active area of research, and various modeling approaches have been adopted to describe their dynamics. Mathematical models of such interactions using differential equations are capable to mimic the stationary and oscillating (regular or irregular) population distributions. Recently, some researchers have paid their attention to explain the consequences of transient dynamics of population density (especially the long transients) and able to capture such behaviors with simple models. Existence of multiple stationary patches and settlement to a stable distribution after a long quasi-stable transient dynamics can be explained by spatiotemporal models with nonlocal interaction terms. However, the studies of such interesting phenomena for three interacting species are not abundant in literature. Motivated by these facts here we have considered in 19 a three species prey-predator model where the predator is generalist in nature as it survives on two prey species. Nonlocalities are introduced in the intra-specific competition terms for the two prey species in order to model the accessibility of nearby resources. Using linear analysis, we have derived the Turing instability conditions for both the spatiotemporal models with and without nonlocal interactions. Validation of such conditions indicates the possibility of existence of stationary spatially heterogeneous distributions for all the three species. Existence of long transient dynamics has been presented under certain parametric domain. Exhaustive numerical simulations reveal various scenarios of stabilization of population distribution due to the presence of nonlocal intra-specific competition for the two prey species. Chaotic oscillation exhibited by the temporal model is significantly suppressed when the populations are allowed to move over their habitat and prey species can access the nearby resources.

6.9.1 Dynamics of periodic waves in a neural field model

Periodic traveling waves are observed in various brain activities, including visual, motor, language, sleep, and so on. There are several neural field models describing periodic waves assuming nonlocal interaction, and possibly, inhibition, time delay or some other properties. In 8, we study the influences of asymmetric connectivity functions and of time delay for symmetric connectivity functions on the emergence of periodic waves and their properties. Nonlinear wave dynamics are studied, including modulated and aperiodic waves. Multiplicity of waves for the same values of parameters is observed. External stimulation in order to restore wave propagation in a damaged tissue is discussed.

6.9.2 Nonlinear analysis of periodic waves in a neural field model

In the paper 14, 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.

6.9.3 Cortical stimulation in aphasia following ischemic stroke

The aim of the paper 11 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 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 integrodifferential equation describing spatially distributed and synaptically coupled neural networks producing traveling wave solutions. The model is used to calculate wave parameters (speed, amplitude and 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.

6.11.1 Dynamics of solutions of a reaction-diffusion equation with delayed inhibition

Reaction-diffusion equation with a logistic production term and a delayed inhibition term is studied in 21. Global stability of the homogeneous in space equilibrium is proved under some conditions on the delay term. In the case where these conditions are not satisfied, this solution can become unstable resulting in the emergence of spatiotemporal pattern formation studied in numerical simulations.

6.11.2 Existence of solutions for some non-Fredholm integro-differential equations with the bi-Laplacian

In the paper 29, we prove the existence of solutions for some semilinear elliptic equations in the appropriate spaces using the fixed‐point technique where the elliptic equation contains fourth-order differential operators with and without Fredholm property, generalizing the previous results.

6.11.3 Dynamics of convective thermal explosion in porous media

In the paper 3, we study complex dynamics of the interaction between natural convection and thermal explosion in porous media. This process is modeled with the nonlinear heat equation coupled with the nonstationary Darcy equation under the Boussinesq approximation for a fluid-saturated porous medium in a rectangular domain. Numerical simulations with the Radial Basis Functions Method (RBFM) reveal complex dynamics of solutions and transitions to chaos after a sequence of period doubling bifurcations. Several periodic windows alternate with chaotic regimes due to intermittence or crisis. After the last chaotic regime, a final periodic solution precedes transition to thermal explosion.

7.1.1 Inria associate team not involved in an IIL

7.2.1 FP7 & H2020 Projects

7.3.1 ANR

• ANR SinCity “Single cell transcriptomics on genealogically identified differentiating cells” (https://anr.fr/Projet-ANR-17-CE12-0031), 2017-2020.

  Participants: Olivier Gandrillon.

• Olivier Gandrillon participates in the ANR MEMOIRE (head Jacqueline Marvel) dedicated to “MultiscalE MOdeling of CD8 T cell Immune REsponses”, 2018-2021.

7.3.2 Other projects

• Thomas Lepoutre is a member of the ERC MESOPROBIO (head Vincent Calvez) dedicated to "Mesoscopic models for propagation in biology", 2015-2020: (http://vcalvez.perso.math.cnrs.fr/mesoprobio.html).

International journals

• 1 articleMostafaM. Adimy, AbdennasserA. Chekroun and ClaudiaC. Pio Ferreira. Global dynamics of a differential-difference system: a case of Kermack-McKendrick SIR model with age-structured protection phaseMathematical Biosciences and Engineering1722020, 1329-1354
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• 2 article MostafaM. Adimy, PauloP. Mancera, Diego SD. Rodrigues, Fernando L PF. Santos and Cláudia PC. Ferreira. Maternal Passive Immunity and Dengue Hemorrhagic Fever in Infants Bulletin of Mathematical Biology 82 2 February 2020
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• 3 articleKaramK. Allali, YoussefY. Joundy, AhmedA. Taik and VitalyV. Volpert. Dynamics of Convective Thermal Explosion in Porous MediaInternational journal of bifurcation and chaos in applied sciences and engineering3006May 2020, 2050081
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• 4 articleChloeC. Audebert, DaphnéD. Laubreton, ChristopheC. Arpin, OlivierO. Gandrillon, JacquelineJ. Marvel and FabienF. Crauste. Modeling and characterization of inter-individual variability in CD8 T cell responses in miceIn Silico BiologyFebruary 2021, 1-27
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• 5 articleMalayM. Banerjee, NayanaN. Mukherjee and VitalyV. Volpert. Prey-predator model with nonlocal and global consumption in the prey dynamicsDiscrete & Continuous Dynamical Systems - S1382020, 2109-2120
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• 6 article MM. Banerjee and VV. Volpert. Stochastic intracellular regulation can remove oscillations in a model of tissue growth Mathematical Medicine and Biology July 2020
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• 7 articleA.S. SA. Benedito, C.P. PC. Ferreira and MostafaM. Adimy. Modeling the dynamics of Wolbachia -infected and uninfected A edes aegypti populations by delay differential equationsMathematical Modelling of Natural Phenomena152020, 76
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• 8 articleNikolaiN. Bessonov, AnneA. Beuter, SergeiS. Trofimchuk and VitalyV. Volpert. Dynamics of Periodic Waves in a Neural Field ModelMathematics87July 2020, 1076
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• 9 articleNikolaiN. Bessonov, GennadyG. Bocharov, CristinaC. Leon, VladimirV. Popov and VitalyV. Volpert. Genotype-dependent virus distribution and competition of virus strainsMathematics and Mechanics of Complex Systems822020, 101-126
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• 10 articleNikolaiN. Bessonov, GennadyG. Bocharov, AndreasA. Meyerhans, VladimirV. Popov and VitalyV. Volpert. Nonlocal Reaction–Diffusion Model of Viral Evolution: Emergence of Virus StrainsMathematics81January 2020, 117
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• 11 article AnneA. Beuter, AnneA. Balossier, FrançoisF. Vassal, SimoneS. Hemm and VitalyV. Volpert. Cortical stimulation in aphasia following ischemic stroke: toward model-guided electrical neuromodulation Biological Cybernetics (Modeling) November 2020
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• 12 articleAnassA. Bouchnita, KirillK. Terekhov, PatriceP. Nony, YuriY. Vassilevski and VitalyV. Volpert. A mathematical model to quantify the effects of platelet count, shear rate, and injury size on the initiation of blood coagulation under venous flow conditionsPLoS ONE157July 2020, e0235392
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• 13 articleAnassA. Bouchnita, VitalyV. Volpert, MarkM. Koury and AndreasA. Hellander. A multiscale model to design therapeutic strategies that overcome drug resistance to tyrosine kinase inhibitors in multiple myelomaMathematical Biosciences319January 2020, 108293
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• 14 article SS. Budzinskiy, AA. Beuter and VV. Volpert. Nonlinear analysis of periodic waves in a neural field model Chaos: An Interdisciplinary Journal of Nonlinear Science November 2020
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• 15 articleJairo GomesJ. Da Silva, Rafael MartinsR. De Morais, Izabel Cristina RodriguesI. Da Silva, MostafaM. Adimy and Paulo FernandoP. De Arruda Mancera. A Mathematical model for treatment of papillary thyroid cancer using the allee effectJournal of Biological Systems2803September 2020, 701-718
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• 16 articleJaouadJ. Danane, KaramK. Allali, LeonL. Tine and VitalyV. Volpert. Nonlinear Spatiotemporal Viral Infection Model with CTL Immunity: Mathematical AnalysisMathematics81January 2020, 52
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• 17 article Vitaly VV. Ganusov, VitalyV. Volpert, BurkhardB. Ludewig and AndreasA. Meyerhans. Editorial: Mathematical Modeling of the Immune System in Homeostasis, Infection and Disease Frontiers in Immunology 2020
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• 18 articlePaulP. Lemarre, LaurentL. Pujo-Menjouet and Suzanne SS. Sindi. A unifying model for the propagation of prion proteins in yeast brings insight into the [PSI+] prionPLoS Computational Biology165May 2020, e1007647
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• 19 articleKalyanK. Manna, VitalyV. Volpert and MalayM. Banerjee. Dynamics of a Diffusive Two-Prey-One-Predator Model with Nonlocal Intra-Specific Competition for Both the Prey SpeciesMathematics81January 2020, 101
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• 20 article ElenaE. Merino Tejero, DanialD. Lashgari, RodrigoR. García-Valiente, XuefengX. Gao, FabienF. Crauste, PhilippeP. Robert, MichaelM. Meyer-Hermann, María RodríguezM. Martínez, S. MariekeS. van Ham, JeroenJ. Guikema, HuubH. Hoefsloot and AntoineA. Van Kampen. Multiscale Modeling of Germinal Center Recapitulates the Temporal Transition From Memory B Cells to Plasma Cells Differentiation as Regulated by Antigen Affinity-Based Tfh Cell Help Frontiers in Immunology 11 February 2021
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• 21 articleTarikT. Mohammed Touaoula, MohammedM. Nor Frioui, NikolayN. Bessonov and VitalyV. Volpert. Dynamics of solutions of a reaction-diffusion equation with delayed inhibitionDiscrete & Continuous Dynamical Systems - S1392020, 2425-2442
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• 22 articleSwadeshS. Pal, MalayM. Banerjee and VitalyV. Volpert. Spatio-temporal Bazykin’s model with space-time nonlocalityMathematical Biosciences and Engineering1752020, 4801-4824
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• 23 article CharlesC. Rocabert, GuillaumeG. Beslon, CaroleC. Knibbe and SamuelS. Bernard. Phenotypic noise and the cost of complexity Evolution - International Journal of Organic Evolution August 2020
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• 24 articleTimT. Squire, A. Ryan and SamuelS. Bernard. Radioprotective effects of induced astronaut torpor and advanced propulsion systems during deep space travelLife Sciences in Space Research26August 2020, 105-113
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• 25 articleVitalyV. Volpert, MalayM. Banerjee and SergeiS. Petrovskii. On a quarantine model of coronavirus infection and data analysisMathematical Modelling of Natural Phenomena152020, 24
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• 26 articleVitalyV. Volpert, MalayM. Banerjee, AlbertoA. d'Onofrio, TomaszT. Lipniacki, SergeiS. Petrovskii and Viet ChiV. Tran. Coronavirus – Scientific insights and societal aspectsMathematical Modelling of Natural Phenomena152020, E2
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• 27 article VitalyV. Volpert. Existence of Reaction–Diffusion Waves in a Model of Immune Response Mediterranean Journal of Mathematics 17 2 April 2020
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• 28 articleVitalyV. Volpert. Mathematical Modelling in the era of coronavirus (Six month in a new reality)Mathematical Modelling of Natural Phenomena152020, E3
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• 29 article VitaliV. Vougalter and VitalyV. Volpert. Existence of solutions for some non‐Fredholm integro‐differential equations with the bi‐Laplacian Mathematical Methods in the Applied Sciences November 2020
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Reports & preprints

• 30 misc KyriakiK. Dariva and ThomasT. Lepoutre. Influence of the age structure on the stability in a tumor-immune model for chronic myeloid leukemia February 2021
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• 31 misc LéonardL. Dekens. Evolutionary dynamics of complex traits in sexual populations in a strongly heterogeneous environment: how normal? December 2020
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• 32 report StéphaneS. Derrode, RomainR. Gauchon, NicolasN. Ponthus, ChristopheC. Rigotti, CatherineC. Pothier, VitalyV. Volpert, StéphaneS. Loisel, Jean-PierreJ.-P. Bertoglio and PascalP. Roy. Piecewise estimation of R0 by a simple SEIR model. Application to COVID-19 in French regions and departments until June 30, 2020 LIRIS UMR CNRS 5205; Ecole centrale de lyon; INSA LYON; Université Lyon 1 - Claude Bernard September 2020
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• 33 misc Nan PapiliN. Gao, OlivierO. Gandrillon, AndrásA. Páldi, UlysseU. Herbach and RudiyantoR. Gunawan. Universality of cell differentiation trajectories revealed by a reconstruction of transcriptional uncertainty landscapes from single-cell transcriptomic data February 2021
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• 34 misc PaulP. Lemarre, Suzanne SS. Sindi and LaurentL. Pujo-Menjouet. A unifying model for the propagation of prion proteins in yeast brings insight into the [PSI+] prion March 2020
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