On the existence of a nontrivial equilibrium in relation to the basic reproductive number

• Autorzy: Wijaya K. P. , Sutimin , Soewono E. , Götz T.

Identyfikatory

DOI

10.1515/amcs-2017-0044

• Języki publikacji: EN

Abstrakty

EN

Equilibrium analysis in autonomous evolutionary models is of central importance for developing long term treatments. This task typically includes checks on the existence and stability of some equilibria. Prior to touching on the stability, one often attempts to determine the existence where the basic reproductive number R₀ plays a critical role as a threshold parameter. When analyzing a nontrivial equilibrium (e.g., an endemic, boundary, or coexistence equilibrium) where R₀ is explicit, we usually come across a typical result: if R₀ > 1, then a nontrivial equilibrium exists in the biological sense. However, for more sophisticated models, R₀ can be too complicated to be revealed in terms of the involving parameters; the task of relating the formulation of a nontrivial equilibrium to R₀ thus becomes intractable. This paper shows how to mitigate such a problem with the aid of functional analysis, adopting the framework of a nonlinear eigenvalue problem. An equilibrium equation is first to be transformed into a canonical equation in a lower dimension, and then the existence is confirmed under several conditions. Three models are tested showing the applicability of this approach.

Słowa kluczowe

EN

autonomous model; nontrivial equilibrium; basic reproductive number; nonlinear eigenvalue problem

PL

model autonomiczny; podstawowy współczynnik reprodukcji; wartość własna

• Wydawca: Oficyna Wydawnicza Uniwersytetu Zielonogórskiego
• Czasopismo: International Journal of Applied Mathematics and Computer Science
• Rocznik: 2017
• Tom: Vol. 27, no. 3
• Strony: 623--636
• Opis fizyczny: Bibliogr. 16 poz., rys., tab.

Twórcy

autor

Wijaya K. P.

• Mathematical Institute, University of Koblenz, Universität Straße 1, 56070 Koblenz, Germany

autor

Sutimin

• Department of Mathematics, Diponegoro University, Jalan Prof. H. Soedarto, SH, 50275 Semarang, Indonesia

autor

Soewono E.

• Department of Mathematics, Bandung Institute of Technology, Jalan Ganesha 10, 40132 Bandung, Indonesia

autor

Götz T.

• Mathematical Institute, University of Koblenz, Universität Straße 1, 56070 Koblenz, Germany

Bibliografia

• [1] Aguiar, M., Kooi, B.W., Rocha, F., Ghaffari, P. and Stollenwerk, N. (2013). How much complexity is needed to describe the fluctuations observed in dengue hemorrhagic fever incidence data?, Ecological Complexity 16: 31–40.
• [2] Arino, J., Miller, J.M. and van den Driessche, P. (2005). A multi-species epidemic model with spatial dynamics, Mathematical Medicine and Biology: A Journal of the IMA 22(2): 3140.
• [3] Cushing, J.M. (1998). An Introduction to Structured Population Dynamics, CBMS-NSF Regional Conference Series in Applied Mathematics, SIAM, Philadelphia, PA.
• [4] Gelfand, I. (1941). Normierte Ringe, Mathematiceskii Sbornik 9(51)(1): 3–24.
• [5] Golubitsky, M. and Schaeffer, D.G. (1985). Singularities and Groups in Bifurcation Theory, Vol. I, Springer, New York, NY.
• [6] Horn, R.A. and Johnson, C.R. (2013). Matrix Analysis, 2nd Ed., Cambridge University Press, New York, NY.
• [7] Krasnoselskii, M. and Zabreiko, P. (1984). Geometrical Methods of Nonlinear Analysis, Springer, New York, NY.
• [8] Ma, T. and Wang, S. (2005). Bifurcation Theory and Applications, World Scientific Series on Nonlinear Science A, Vol. 53, World Scientific Publishing, Singapore.
• [9] Nirenberg, L. (2001). Topics in Nonlinear Functional Analysis, Courant Lecture Notes in Mathematics 6, New York University Courant Institute of Mathematical Sciences, New York, NY.
• [10] Ortega, J.M. (1932). Numerical Analysis: A Second Course, SIAM, Philadelphia, PA.
• [11] Rabinowitz, P.H. (1971). Some global results for nonlinear eigenvalue problems, Journal of Functional Analysis 7(3): 487–513.
• [12] Rabinowitz, P.H. (1977). A bifurcation theorem for potential operators, Journal of Functional Analysis 25(4): 412–424.
• [13] van den Driessche, P. and Watmough, J. (2002). Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Mathematical Biosciences 180(1): 29–48.
• [14] Wijaya, K.P., Goetz, T. and Soewono, E. (2014). An optimal control model of mosquito reduction management in a dengue endemic region, International Journal of Biomathematics 7(5): 1450056–22.
• [15] Wijaya, K.P., Goetz, T. and Soewono, E. (2016). Advances in mosquito dynamics modeling, Mathematical Methods in the Applied Sciences 39(16): 4750–4763.
• [16] Zadeh, L.A. and Desoer, C.A. (1963). Linear System Theory: The State Approach, McGraw-Hill, New York, NY.

Uwagi

PL

Opracowanie ze środków MNiSW w ramach umowy 812/P-DUN/2016 na działalność upowszechniającą naukę (zadania 2017).