Generalized Models of the Seir Epidemiological Model and Their Discrete Analogues
Generalized Models of the Seir Epidemiological Model and Their Discrete Analogues
DOI:
https://doi.org/10.56143/ujmcs.v1i1.2Ключевые слова:
The SEIR model; Lotka-Volterra mapping; simplex; graph; trajectory; skew-symmetric matrix; viral diseases.Аннотация
The paper considers a discrete analogue of the $SEIR$ model, which, unlike SIR, includes a group of infected individuals in the incubation (latent) period. This model is based on Lotka-Volterra mappings operating in a three-dimensional simplex with degenerate skew-symmetric matrices, which correspond to mixed graphs. The models considered in this paper are intended to study the course of viral diseases transmitted by the air-capillary route without a repeated effect.
Библиографические ссылки
[1] https://www.hmong.press/wiki/ Mathematical modelling in epidemiology.
[2] Daly, D.J., Ghani, J.: Epidemic modeling: an introduction. New York: Cambridge University Press.(2005).
[3] Murray, J.D.: Mathematical biology. Third Edition. Springer.p.776. (2009).
[4] Hammer, W.: Epidemiology is old and new . London: Kegan Paul. (1928)
[5] Ross, R.: The prevention of malaria. New York: Dutton. (2010)
[6] Ganikhodzhaev, R.N.: Quadratic stochastic operators, Lyapunov function and tournaments, Acad. Sci. Sb. Math., 76(2), pp.489-506, (1993)
[7] Ganikhodzhaev, R.N., Tadzieva, M.A., Eshmamatova, D.B.: Dynamical Proporties of Quadratic Homeomorphisms of a Finite-Dimensional Simplex. Journal of Mathematical Sciences — 245 — (3). — P. — 398-402, (2024)
[8] Ganikhodzhaev, R.N., Eshmamatova, D.B.: Quadratic automorphisms of a simplex and the asymptotic behavior of their trajectories. Vladikavkaz. Mat. Zh., Volume 8, Number 2, 12-28, (2006)
[9] Harary, F., Palmer, E.M.: Graphical enumeration. Academic Press New York and London. (1973)
[10] Kermack, W.O. and McKendrick, A.G.: A contribution to the mathematical theory of epidemics, Proc. R. Soc. Lond, 115, 700-721. (1927)
[11] Eshmamatova, D.B., Ganikhodzhaev, R.N.: Tournaments of Volterra type transversal operators acting in the simplex Sm-1
. AIP Conference
Proceedings 2365, 060009, https:doi.org/10.1063/5.0057303. (2021)
[12] Ganikhodzhaev, R.N., Tadzhieva, M.A.: Stability of fixed points of discrete dynamic systems of Volterra type. AIP Conference Proceedings, V. 2365. P. 060005-1 − 060005-7. https://doi.org/10.1063/5.0057979. (Scopus. IF=0.7). (2021)
[13] Ganikhodzhaev, R.N.: A chart of fixed points and Lyapunov functions for a class of discrete dynamical systems, Math. Notes, 56 (5-6), pp.1125-1131, (1994)
[14] Kuznetsov, Y.A.: Elements of applied bifurcation theory. Second edition. NewYork 10027, p 592, (1998)
[15] Murray, Y. D.: On a necessary condition for the ergodicity of quadratic operators defined on a two-dimensional simplex, Russ. Marh. Surv 59. 13. p.571-573, (2004)
[16] Ulam, S.: A collection of mathematical problems.Interscience. New-York, 150 p.Graph Theory. Addison-Wesley.1969. p.274. (1960)
[17] Volterra, V.: Theorie mathematique de la lutte pour la vie. Paris, p.290, (1931)