Application of differential equations with variable delay in the compartmental models of living systems

N. V. Pertsev

Sibirskij žurnal industrialʹnoj matematiki, Tome 24 (2021) no. 3, pp. 55-73.

Résumé

An approach to the construction of compartmental models of living systems based on linear nonautonomous differential equations with variable delay is presented. Differential equations describing the dynamics of the number of elements of the living system located in compartments are supplemented by a set of linear integral equations that reflect the dynamics of the number of elements in the process of movement between compartments. The model contains nonnegative initial data that takes into account the prehistory of the dynamics of the number of elements of the living system. The existence, uniqueness, and nonnegativity of solutions of the model on the semi-axis are established. Two-side estimates for the sum of all components of the solution are obtained. The exponential stability of the trivial solution of the system of differential equations in the absence of an external source of influx of elements of living systems is shown. A compartmental model of the dynamics of HIV-1 infection in the body of an infected person is considered. To study the model, the properties of nonsingular M-matrices are used. The conditions for exponential and asymptotic stability of the trivial solution of the model are established. The obtained relations are interpreted as the conditions for eradication of HIV-1 infection due to nonspecific factors of the human body protection.

Détail

Keywords: linear differential equations with variable delay, system of Wazewski equations, positive system, asymptotic stability, nonsingular M-matrix, compartmental model
Mots-clés : HIV-1 infection. .

Bibliographie

[1] R. Bellman, Mathematical Methods in Medicine, World Sci. Publ, Cleveland, 1983 | Zbl

[2] V. N. Dagaev, V. N. Novosel'tsev, “Parametrization of pharmacokinetic models for analysis of the control processes in living organisms”, Avtomat. Telemekh., 1995, no. 4, 130–144 | Zbl

[3] V. I. Sergienko, R. Dzhelliff, I. B. Bondareva, Applied pharmacokinetics: fundamentals and clinical applications, Izd. Ross. Akad. Med. Nauk, M., 2003 (in Russian)

[4] S. Nakaoka, I. Shingo, K. Sato, “Dynamics of HIV infection in lymphoid tissue network”, J. Math. Biol., 72 (2016), 909–938 | DOI | Zbl

[5] A. S. Mozokhina, S. I. Mukhin, G. I. Lobov, “Pump efficiency of lymphatic vessels: numeric estimation”, Russ. J. Numer. Anal. Math. Modelling, 5:34 (2019), 261–268 | DOI | Zbl

[6] R. Savinkov, D. Grebennikov, D. Puchkova, V. Chereshnev, I. Sazonov, G. Bocharov, “Graph theory for modeling and analysis of the human lymphatic system”, Mathematics, 8:12 (2020), 2236 | DOI

[7] V. B. Kolmanovskii, V. R. Nosov, Stability and periodic modes of controlled systems with aftereffect, Nauka, M., 1981 (in Russian)

[8] A. Mazanov, “Stability of multi-pool models with lag”, J. Theor. Biol., 59 (1976), 429–442 | DOI

[9] I. Gyori, J. Eller, “Compartmental systems with pipes”, Math. Bios., 53 (1981), 223–247 | DOI | Zbl

[10] A. Yu. Obolenskii, “Stability of solutions of autonomous wazewski systems with delayed action”, Ukrain. Mat. Zh., 35 (1983), 574–579 | Zbl

[11] N. V. Pertsev, “Two-sided estimates for solutions to the Cauchy problem for wazewski linear differential systems with delay”, Siberian Math. J., 54:6 (2013), 1088–1097 | DOI | Zbl

[12] A. Yu. Aleksandrov, “Construction of the Lyapunov-Krasovskii functionals for some classes of positive delay systems”, Siberian Math. J., 59:5 (2018), 753–762 | DOI | Zbl

[13] J. K. Kheil, Theory of Functional Differential Equations, Springer-Verl., N.Y., 1977

[14] N. V. Pertsev, “Global solvability and estimates of solutions to the Cauchy problem for the retarded functional differential equations that are used to model living systems”, Siberian Math. J., 59:1 (2018), 113–125 | DOI | Zbl

[15] V. A. Chereshnev, G. A. Bocharov, A. V. Kim, S. I. Bazhan, I. A. Gainova, A. N. Krasovskii, N. G. Shmagel', A. V. Ivanov, M. A. Safronov, R. M. Tret'yakova, Introduction to problems of modeling and control of the dynamics of HIV infection, Institute for Computer Research, M.–Izhevsk, 2016

[16] N. Pertsev, K. Loginov, G. Bocharov, “Nonlinear effects in the dynamics of HIV-1 infection predicted by mathematical model with multiple delays”, Discrete and Continuous Dynamical Systems. Series S, 13:9 (2020), 2365–2384 | DOI | Zbl

[17] B. P. Demidovich, Lectures on mathematical theory of stability, Nauka, M., 1967

[18] N. V. Pertsev, “Application of $M$-matrices in construction of exponential estimates for solutions to the Cauchy problem for systems of linear difference and differential equations”, Siberian Adv. Math., 24:2 (2014), 240–2601 | DOI | Zbl

[19] N. V. Pertsev, “Stability of linear delay differential equations arising in models of living systems”, Siberian Adv. Math., 30:2 (2020), 43–54 | DOI

[20] V. V. Voevodin, Yu. A. Kuznetsov, Matrices, Calculations, Nauka, M., 1984 (in Russian)

[21] A. Berman, R. J. Plemmons, Nonnegative Matrices in the Mathematical Sciences, Acad. Press, N.Y., 1979 | Zbl

[22] M. A. Krasnosel'skii, G. M. Vainikko, P. P. Zabreiko, Ya. B. Rutitskii, V. Ya. Stetsenko, Approximate Solution of Operator Equations, Nauka, M., 1969

[23] Z. B. Tsalyuk, “Volterra integral equations”, Results of Science and Technology. Ser. Mat. Analiz, 15, 1977, 131–198 | Zbl

[24] R. P. Agarwal, L. Berezansky, E. Braverman, A. Domoshnitsky, Nonoscillation Theory of Functional Differential Equations with Applications, Springer-Verl, N.Y., 2012 | Zbl

[25] N. V. Pertsev, “Study of solutions of mathematical models of epidemic processes with common structural properties”, Sibir. Zh. Ind. Mat., 18:2 (2015), 85–98 | Zbl

[26] A. S. Mozokhina, S. I. Mukhin, “Some exact solutions to the problem of a liquid flow in a contracting elastic vessel”, Math. Models and Comp. Simulations, 11 (2019), 894–904 | DOI