ATTAINABLE SETS OF INTEGRAL CONSTRAINED SEIR CONTROL SYSTEM WITH NONLINEAR INCIDENCE

Year 2023, Issue: 054, 322 - 337, 30.09.2023

Ali Serdar Nazlıpınar , Farıdeh Mohammadımehr

https://doi.org/10.59313/jsr-a.1312173

Abstract

In this survey, we consider the dynamics of a contagious disease spread by employing a nonlinear dynamical control system of differential equations. It considers treatment and vaccination as key control parameters to discern their influence on disease control. The study, approximate the attainable sets of a given control system and presents visual results, while also discussing potential biological applications of their findings.

Keywords

Attainable set, nonlinear incidence, SEIR model

References

• [1] Kermack W.O., Mckendric A.G. (1927). Contributions to the mathematical theory of epidemics, part i, Proceedings of the Royal Society of Edinburgh. Section A Mathematics, 115 (772), 700-721.
• [2] Hethcote H.W. (2000). The mathematics of infectious diseases, SIAM Review, 42(4), 599–653.
• [3] Hoppensteadt F.C. (1982). Mathematical methods in population biology, Cambridge University Press, Cambridge.
• [4] Anderson R.M. (1982). Population dynamics of infectious diseases: Theory and applications, Chapman and Hall, London.
• [5] Grassly N.C., Fraser C. (2008). Mathematical models of infectious disease transmission, Nature Reviews Microbiology 6, 477-487. doi:10.1038/nrmicro1845.
• [6] Keeling M.J., Danon L. (2009). Mathematical modelling of infectious diseases, Br Med Bull, 92(1), 33-42. doi: 10.1093/bmb/ldp038.
• [7] Biswas M.H.A., Paiva L.T., Pinho M. (2014). A SEIR model for control of infectious diseases with constraints, Mathematical Biosciences and Engineering, 11(4), 761-784. doi:10.3934/mbe.2014.11.761.
• [8] Neilan R.M., Lenhart S. (2010). An Introduction to Optimal Control with an Application in Disease Modeling, Modeling Paradigms and Analysis of Disease Trasmission Models, 49, 67-82.
• [9] Gaff H., Schaffer E. (2009). Optimal control applied to vaccination and treatment strategies for various epidemiological models, Math. Bio. Sci. Eng. (MBE), 6, 469-492.
• [10] Guseinov Kh. G., Ozer O., Akyar E. (2004). On the continuity properties of the attainable sets of control systems with integral constraints on control, Nonl. Anal.: Theo., Meth. App. 56, 433–449.
• [11] Guseinov Kh. G., Ozer O., Akyar E., Ushakov V.N. (2007). The approximation of reachable sets of control systems with integral constraint on controls, Non. Diff. Equat. Appl. 14, 57–73.
• [12] Guseinov Kh.G., Nazlipinar A.S. (2007). On the continuity property of Lp balls and an application, J.Math. Anal. Appl., 335, 1347-1359.
• [13] Guseinov Kh.G., Nazlipinar A.S. (2008). On the continuity properties of attainable sets of nonlinear control systems with integral constraint on controls, Abstr. Appl. Anal., p.14.
• [14] Guseinov KH.G. (2009). Approximation of the attainable sets of the nonlinear control systems with integral constraints on control, Nonlinear Analysis, TMA, 71, 622-645.
• [15] Guseinov Kh.G., Nazlipinar A.S. (2011). An algorithm for approximate calculation of the attainable sets of the nonlinear control systems with integral constraint on controls, Comp. Math. Appl., 62(4), 1887-1895.
• [16] Krasovskii N.N., Subbotin A.I. (1988). Game-theoretical control problems, Springer, NewYork.
• [17] Krasovskii N.N. (1968). Theory of control of motion: Linear systems, Nauka, Moscow.
• [18] Nazlipinar A.S., Basturk B. (2020). Attainable set of a SIR epidemiological model with constraints on vaccination and treatment stocks, Tbilisi Mathematical Journal 13(1), pp. 11-22.
• [19] Hethcote H.W. (1989). Three Basic Epidemiological Models, In Levin SA, Hallam TG, Gross LJ (eds.). Applied Mathematical Ecology. Biomathematics. Vol. 18. Berlin: Springer. pp. 119–144. doi:10.1007/978-3-642-61317-3_5. ISBN 3-540-19465-7.
• [20] Padua RN, Tulang A.B. (2010). A Density–Dependent Epidemiological Model for the Spread of Infectious Diseases, Liceo Journal of Higher Education Research. 6 (2). doi:10.7828/ljher.v6i2.62.