Russian version English version
Volume 13   Issue 1   Year 2018
Nikolay V. Pertsev, Boris Yu. Pichugin, Anna N. Pichugina

Application of M-Matrices for the Study of Mathematical Models of Living Systems

Mathematical Biology & Bioinformatics. 2018;13(1):208-237.

doi: 10.17537/2018.13.208.



  1. Voevodin V.V., Kuznetsov Yu.A. Matrices and Calculations. Moscow: Nauka; 1984. 320 p. (in Russ.).
  2. Bellman R. Introduction to Matrix Theory. Moscow: Nauka; 1976. 352 p. (in Russ.).
  3. Sevastyanov B.A. Branching Processes. Moscow: Nauka; 1971. 436 p. (in Russ.).
  4. Ortega J.M., Rheinboldt W.C. Iterative Solution of Nonlinear Equations in Several Variables. New York and London: Academic Press; 1970.
  5. Berman A., Plemmons R.J. Nonnegative matrices in the mathematical sciences. New York: Academic Press; 1979. 340 p.
  6. Hale J. Theory of Functional Differential Equations. New York-Heidelberg-Berlin: Springer-Verlag, 1977. 365 p. doi: 10.1007/978-1-4612-9892-2
  7. Kolmanovsky V.B., Nosov V.R. Stability and Periodical Regiments with After-Effect. Moscow: Nauka; 1981. 448 p. (in Russ.).
  8. Obolenskii A.Y. Stability of Solutions of Autonomous Wazewski Systems with Delayed Action. Ukranian Mathematical Journal. 1984;35(5):486-492. doi: 10.1007/BF01061640
  9. Volz R. Stability conditions for systems of linear nonautonomous delay differential equations. J. Math. Anal. Appl. 1986;120(2):584-595. doi: 10.1016/0022-247X(86)90181-2
  10. Gyori I., Pertsev N.V. On the stability of Equilibrium States of Functional-Differential Equations of Retarded Type Possessing a Mixed Monotone Property. Doklady Akademii Nauk SSSR. 1987;297(1):23-25 (in Russ.).
  11. Demidovich B.P. Lectures on Mathematical Stability Theory. Moscow: Nauka, 1967. 472 p. (in Russ.).
  12. Marchuk G.I. Mathematical Models in Immunology. Moscow: Nauka, 1985. 240 p. (in Russ.).
  13. Marchuk G.I. Mathematical Models in Immunology. Computational Methods and Experiments. Moscow: Nauka; 1991. 304 p. (in Russ.).
  14. Els’golts L.I., Norkin S.B. Introduction to the Theory and Application of Differential Equations with Deviating Arguments. Academic Press, Mathematics in Science and Engineering; 1973.
  15. Nelson P.W., Perelson A.S. Mathematical analysis of delay differential equation models of HIV-1 infection. Math. Biosci. 2002;179:73-94. doi: 10.1016/S0025-5564(02)00099-8
  16. Bocharov G., Chereshnev V., Gainova I., Bazhan S., Bachmetyev B., Argilaguet J., Martinez J., Meyerhans A. Human Immunodeficiency Virus Infection: from Biological Observations to Mechanistic Mathematical Modelling. Math. Model. Nat. Phenom. 2012;7(5):78-104. doi: 10.1051/mmnp/20127507
  17. Pawelek K.A., Liu S., Pahlevani F., Rong L. A model of HIV-1 infection with two time delays: mathematical analysis and comparison with patient data. Math. Biosci. 2012;235(1):98-109. doi: 10.1016/j.mbs.2011.11.002
  18. Pitchaimani M., Monica C. Global stability analysis of HIV-1 infection model with three time delays. J. Appl. Math. Comput. 2015;48:293-319. doi: 10.1007/s12190-014-0803-4
  19. Wang J., Lang J., Zou X. Analysis of an age structured HIV infection model with virus-to-cell infection and cell-to-cell transmission. Nonlinear Analysis: Real World Applications. 2017;34:75-96. doi: 10.1016/j.nonrwa.2016.08.001
  20. Pertsev N.V. Global solvability and estimates for solutions to the Cauchy problem for the retarded functional differential equations that are used to the model living systems. Siberian Mathematical Journal. 2018;59(1):113-125. doi: 10.1134/S0037446618010135
  21. Pertsev N.V. Discrete-continuous Model of Tuberculosis Spread and Control. Siberian Journal of Industrial Mathematics. 2014;17(3):86-97 (in Russ.).
  22. Pertsev N.V. Analysis of Solutions to Mathematical Models of Epidemic Processes with Common Structural properties. Siberian Journal of Industrial Mathematics. 2015;18(2):85-98 (in Russ.). doi: 10.17377/sibjim.2015.18.209
  23. Romanyukha A.A., Nosova E.A. Modeling Spread of HIV as Result of Social Maladjustment in Population. UBS. 2011;34:227-253 (in Russ.).
  24. Nosova E.A. Models of Control and Spread of HIV-infection. Mat. Biolog. Bioinform. 2012;7(2):632-675 (in Russ.). doi: 10.17537/2012.7.632
  25. Pertsev N.V., Pichugin B.Yu. , Pichugina A.N. Analysis of the Asymptotic Behavior Solutions of Some Models of Epidemic Processes. Mat. Biol. Bioinform. 2013;8(1):21-48 (in Russ.). doi: 10.17537/2013.8.21
  26. Pichugina A.N. An Integrodifferential Model of a Population under the Effects of Pollutants. Siberian Journal of Industrial Mathematics. 2004;7(4):130-140 (in Russ.).
  27. Pertsev N.V., Tsaregorodtseva G.E. Modeling population dynamics under the influence of harmful substances on the individual reproduction process. Automation and Remote Control. 2011;72(1):129-140. doi: 10.1134/S0005117911010115
  28. Romanovskii Yu.M., Stepanova N.V., Chernavsky D.S. Mathematical Biophysics. Moscow: Nauka, 1984. 304 p. (in Russ.).
  29. Alexandrov A.Yu., Zhabko A.P. On the Asymptotic Stability of Solutions of Nonlinear Systems with Delay. Siberian Mathematical Journal. 2012;53(3):495-508 (in Russ.). doi: 10.1134/S0037446612020218
  30. Balandin A.S., Sabatulina T.L. The Local Stability of a Population Dynamics Model in Conditions of Deleterious Effects. Sib. Èlektron. Mat. Izv. 2015;12:610-624 (in Russ.). doi: 10.17377/semi.2015.12.049
  31. Malygina V.V., Mulyukov M.V. On Local Stability of a Population Dynamics Model with Three Development Stages. Russ Math. 2017;4:35-42 (in Russ.). doi: 10.3103/S1066369X17040053
  32. Golubyatnikov V.P., Kirillova N.E. On Cycles in Models of Functioning of Circular Gene Networks. Sib. J. Pure and Appl. Math. 2018;18(1):54-63 (in Russ.). doi: 10.17377/PAM.2018.18.5
  33. Bocharov G.A, Marchuk G.I. Applied problems of mathematical modeling in immunology. Comput. Math. Math. Phys. 2000;40(12):1830-1844.
  34. Luzyanina T., Sieber J., Engelborghs K., Samaey G., Roose D. Numerical bifurcation analysis of mathematical modelswith time delays with the package DDE-BIFTOOL. Mathematical Biology and Bioinformatics. 2017;12(2):496-520. doi: 10.17537/2017.12.496
Table of Contents Original Article
Math. Biol. Bioinf.
doi: 10.17537/2018.13.208
published in Russian

Abstract (rus.)
Abstract (eng.)
Full text (rus., pdf)
References Translation into English
Math. Biol. Bioinf.
doi: 10.17537/2018.13.t104

Full text (eng., pdf)


  Copyright IMPB RAS © 2005-2024