Vitalii Volpert
Doctor of Physical and Mathematical Sciences
Director of the interdisciplinary research center “Mathematical modeling in Biomedicine” of S.M. Nikol’skii Mathematical Institute of RUDN, s.M. Nikol’skii Mathematical Institute

Mathematics and modeling for the study of natural phenomena.


Graduated with honors from the Faculty of Mechanics and Mathematics of the Southern Federal University (SFU, Rostov–on-Don).


PhD. thesis “Autowave processes in chemically active media”.

1980 - 1991

Worked at the Institute of Chemical Physics of the USSR Academy of Sciences (Chernogolovka), worked his way up from engineer to head of the laboratory of Macrokinetics of polymerization processes.


Schelkin prize of the Soviet Academy of Sciences.


Visiting researcher at the Courant Institute Mathematical Sciences, New York University, USA. 

1991 - 1992

Research Fellow, Department of Materials Science, Northwestern University, USA.

1992 - present

Works at the French National Center for Scientific Research (Directeur de recherche, Centre National de la Recherche Scientifique) and at University Lyon 1, France. 


Habilitation, “Mathematical theory of reaction-diffusion equations and their application in chemical physics”, University Lyon 1.

2004 - 2006

Deputy Director of Camille Jordan Institute, Lyon, France.

2010 - 2012

Member of the board of Directors of the Institute for Systems Biology and Medicine, Lyon, France.

2012 - 2018

Member of the Council of the European Society for Theoretical and Mathematical Biology.

2018 - present

Director of the interdisciplinary research center “Mathematical modeling in Biomedicine” of S.M. Nikol’skii Mathematical Institute of RUDN.

Member of editorial boards of the following journals:

  • Mathematical modelling of natural phenomena (founder and editor-in-chief, 2006),
  • Complex variables and elliptic equations (2017),
  • Computation (2018),
  • Computer research and modelling (2018),
  • Pure and Applied Functional Analysis (2016),
  • Mathematics (2019).


  1. Gives the course of lectures “Reaction-diffusion equations and applications” for postgraduate students of the direction “Differential equations, dynamical systems and optimal control”.
  2. 2014 - 2018 gave courses on partial differential equations and mathematical modeling to students, postgraduate students at the Faculty of Mathematics at the Higher school of Kouba. (Ecole Normale de Kouba, Algeria) and other educational institutions in Algeria.
  3. 2017 gave courses on partial differential equations and mathematical modeling to students, postgraduate students at the Faculty of Mathematics of the National Institute of Technology Patna, India.
  4. 2012 - 2013 gave courses on partial differential equations and mathematical modeling to students at the Engineering Faculty of Ecole Centrale de Lyon France.
  5. Repeatedly was invited for scientific visits and lectures to England, Israel, India, Poland, USA, Chile and other countries. Last three years:
    • 2018 - University of Talca (Chili), University of Warsaw (Poland), University of Tlemcen (Algeria)
    • 2017 - King’s college London (UK), University of Tlemcen (Algeria), University of Talca (Chili), Baumann University Moscow (Russia), Indian Institute of Technology Patna (India)
    • 2016 - University of Leiden (Netherlands), University of Talca (Chili), Institute of numerical mathematics, Moscow (Russia)


  • Elliptic problems in unbounded domains were studied. Fredholm conditions for general elliptic operators in unbounded domains were obtained. Solvability conditions of linear problems, index, solvability conditions for non-Fredholm operators were studied. Property conditions of general nonlinear elliptic operators in unbounded domains were obtained and a topological degree was constructed.
  • Reaction-diffusion waves, in particular, existence and stability of waves for monotonic and locally monotonic systems, minimax representation of wave velocity were studied. Application to various problems of chemical kinetics, population dynamics, biomedicine. Generalized travelling waves.
  • Nonlocal reaction-diffusion equations and equations with delay were studied. Local and global stability of equilibrium states, existence of travelling waves and generalized travelling waves, bifurcations and nonlinear dynamics were studied. Applications to various problems of population dynamics and biomedicine.
  • Waves in combustion and chemical kinetics, the existence, stability, bifurcations, nonlinear dynamics were studied. Thermal explosion with convection, thermal explosion conditions, convection influence, space-time structures, vibrational thermal explosion, thermal explosion in porous medium were studied.
  • Existence, stability, propagation velocity, nonlinear dynamics of front polymerization waves were studied. Production technology of poly-epsilon-caprolactam based on frontal polymerization. Influence of convection on the propagation of frontal polymerization waves was studied. Low-temperature waves accompanied by the destruction of a solid.
  • Interphase and capillary phenomena in miscible liquids were studied, and experiments on the international space station were prepared.
  • Mathematical modeling of atherosclerosis. Models of atherosclerosis as a chronic inflammation of the artery walls were developed. Development of atherosclerosis depending on the cholesterol content was studied. Development of atherosclerosis as a reaction-diffusion wave. Existence and stability of waves in one-dimensional formulation and in two-dimensional formulation with nonlinear boundary conditions were investigated. Interaction of an atherosclerotic plaque with the flow.
  • Blood clotting and comorbidities. Different modes of blood clotting and conditions of their implementation. Blood clotting as a reaction-diffusion wave. Existence, stability and speed of propagation. Initial conditions for initiation of coagulation and existence of a solution in the form of a stationary pulse. Influence of various factors on clotting: blood flow, platelets, inflammation. Thrombosis and bleeding. Identification of patients with hemophilia.
  • Modeling of cancer: leukemia and multiple myeloma. Development of mathematical models, analysis and numerical modeling, characterization of myeloblastic leukemia by flow cytometry. Modeling of the development of multiple myeloma. Selection of optimal treatment protocols.
  • Mathematical modeling of erythropoiesis. Development of hybrid models of erythropoiesis taking into account the presence of cells of different types, intracellular and intercellular regulation. Functioning of erythroblastic islets and production of erythrocytes, reaction to hypoxia, regulation by erythropoietin.
  • Mathematical immunology. Development of viral infection as a reaction-diffusion phenomenon. Existence, resistance and modes of spread; influence of delayed immune response; virus mutations; antiviral therapy and emergence of resistant strains.
  • Electrical stimulation of the cerebral cortex for rehabilitation of patients after stroke. Modeling of electric potential waves in the cerebral cortex on the basis of integro-differential equations of the mean field theory. Different wave propagation modes, stability, bifurcations, nonlinear dynamics. Selection of stimulation modes to restore waves characteristics in damaged areas of the brain.
  • Development of hybrid models in biomedicine based on the combination of discrete cell models and continuous models for intracellular regulation (ODE) and intercellular regulation (PDE). Application to various problems of biomedicine (leukemia, lymphoma, myeloma, erythropoiesis). Hybrid models with dissipative particle dynamics to study blood flow clotting.
  • Evolution of biological species. Development of models of evolution based on nonlocal reaction-diffusion equations, taking into account competition for resources. Conditions of emergence of new species, description of different modes of evolution of species. Interrelation of different definitions of species (by Darwin and Mair). Nonlocal predator-prey models.
  • Morphogenesis and modeling of plant growth. Development of plant growth models, how to explain plant diversity, branching; vegetative hormones and nutrients. Two-dimensional models based on cell division, plant growth as self-similar structures. Different models of morphogenesis and wound healing.
  • Other issues: economic and demographic models, phase transitions in metal oxides, propagation of calcium waves, etc.

Scientific interests

  • General theory of elliptic equations in unbounded domains;
  • Mathematical theory of reaction-diffusion waves with applications in chemical kinetics and combustion;
  • Nonlocal reaction-diffusion equations and equations with delay;
  • Vibrational thermal explosion;
  • Capillary phenomena in miscible fluids;
  • Mathematical theory of the origin and evolution of species;
  • Mathematical theory and modeling of biomedical processes: atherosclerosis and other chronic inflammations, blood clotting and thrombosis, cancer;
  • New methods of mathematical modeling in Biomedicine;
  • The study of mathematical models of various biological and ecological issues: plant growth, morphogenesis, etc.
This monograph concisely but thoroughly introduces the reader to the field of mathematical immunology. The book covers first basic principles of formulating a mathematical model, and an outline on data-driven parameter estimation and model selection. The authors then introduce the modeling of experimental and human infections and provide the reader with helpful exercises. The target audience primarily comprises researchers and graduate students in the field of mathematical biology who wish to be concisely introduced into mathematical immunology.
If we had to formulate in one sentence what this book is about, it might be "How partial differential equations can help to understand heat explosion, tumor growth or evolution of biological species". These and many other applications are described by reaction-diffusion equations. The theory of reaction-diffusion equations appeared in the first half of the last century. In the present time, it is widely used in population dynamics, chemical physics, biomedical modelling. The purpose of this book is to present the mathematical theory of reaction-diffusion equations in the context of their numerous applications. We will go from the general mathematical theory to specific equations and then to their applications. Existence, stability and bifurcations of solutions will be studied for bounded domains and in the case of travelling waves. The classical theory of reaction-diffusion equations and new topics such as nonlocal equations and multi-scale models in biology will be considered.
The theory of elliptic partial differential equations has undergone an important development over the last two centuries. Together with electrostatics, heat and mass diffusion, hydrodynamics and many other applications, it has become one of the most richly enhanced fields of mathematics. This monograph undertakes a systematic presentation of the theory of general elliptic operators. The author discusses a priori estimates, normal solvability, the Fredholm property, the index of an elliptic operator, operators with a parameter, and nonlinear Fredholm operators. Particular attention is paid to elliptic problems in unbounded domains which have not yet been sufficiently treated in the literature and which require some special approaches. The book also contains an analysis of non-Fredholm operators and discrete operators as well as extensive historical and bibliographical comments. The selected topics and the author's level of discourse will make this book a most useful resource for researchers and graduate students working in the broad field of partial differential equations and applications.
Traveling wave solutions of parabolic systems describe a wide class of phenomena in combustion physics, chemical kinetics, biology, and other natural sciences. The book is devoted to the general mathematical theory of such solutions. The authors describe in detail such questions as existence and stability of solutions, properties of the spectrum, bifurcations of solutions, approach of solutions of the Cauchy problem to waves and systems of waves. The final part of the book is devoted to applications to combustion theory and chemical kinetics. The book can be used by graduate students and researchers specializing in nonlinear differential equations, as well as by specialists in other areas (engineering, chemical physics, biology), where the theory of wave solutions of parabolic systems can be applied.
The paper is devoted to a reaction-diffusion equation with delay arising in modeling the immune response. We prove the existence of traveling waves in the bistable case using the Leray–Schauder method. Differently from the previous works, we do not assume here quasi-monotonicity of the delayed reaction term.
The paper is devoted to mathematical modelling of clot growth in blood flow. Great complexity of the hemostatic system dictates the need of usage of the mathematical models to understand its functioning in the normal and especially in pathological situations. In this work we investigate the interaction of blood flow, platelet aggregation and plasma coagulation. We develop a hybrid DPD–PDE model where dissipative particle dynamics (DPD) is used to model plasma flow and platelets, while the regulatory network of plasma coagulation is described by a system of partial differential equations. Modelling results confirm the potency of the scenario of clot growth where at the first stage of clot formation platelets form an aggregate due to weak inter-platelet connections and then due to their activation. This enables the formation of the fibrin net in the centre of the platelet aggregate where the flow velocity is significantly reduced. The fibrin net reinforces the clot and allows its further growth. When the clot becomes sufficiently large, it stops growing due to the narrowed vessel and the increase of flow shear rate at the surface of the clot. Its outer part is detached by the flow revealing the inner part covered by fibrin. This fibrin cap does not allow new platelets to attach at the high shear rate, and the clot stops growing. Dependence of the final clot size on wall shear rate and on other parameters is studied.
The theory of reaction-diffusion waves begins in the 1930s with the works in population dynamics, combustion theory and chemical kinetics. At the present time, it is a well developed area of research which includes qualitative properties of travelling waves for the scalar reaction-diffusion equation and for system of equations, complex nonlinear dynamics, numerous applications in physics, chemistry, biology, medicine. This paper reviews biological applications of reaction-diffusion waves.
We study a reaction-diffusion equation with an integral term describing nonlocal consumption of resources in population dynamics. We show that a homogeneous equilibrium can lose its stability resulting in appearance of stationary spatial structures. They can be related to the emergence of biological species due to the intra-specific competition and random mutations. Various types of travelling waves are observed.
The paper is devoted to general elliptic operators in Holder spaces in bounded or unbounded domains. We discuss the Fredholm property of linear operators and properness of nonlinear operators. We construct a topological degree for Fredholm and proper operators of index zero.