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ATTAINABLE SETS OF INTEGRAL CONSTRAINED SEIR CONTROL SYSTEM WITH NONLINEAR INCIDENCE

Year 2023, , 322 - 337, 30.09.2023
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.

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.
Year 2023, , 322 - 337, 30.09.2023
https://doi.org/10.59313/jsr-a.1312173

Abstract

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.
There are 20 citations in total.

Details

Primary Language English
Subjects Biological Mathematics, Dynamical Systems in Applications, Calculus of Variations, Mathematical Aspects of Systems Theory and Control Theory
Journal Section Research Articles
Authors

Ali Serdar Nazlıpınar 0000-0002-5114-208X

Farıdeh Mohammadımehr 0000-0003-0122-7920

Publication Date September 30, 2023
Submission Date June 9, 2023
Published in Issue Year 2023

Cite

IEEE A. S. Nazlıpınar and F. Mohammadımehr, “ATTAINABLE SETS OF INTEGRAL CONSTRAINED SEIR CONTROL SYSTEM WITH NONLINEAR INCIDENCE”, JSR-A, no. 054, pp. 322–337, September 2023, doi: 10.59313/jsr-a.1312173.