Abstract
This chapter is concerned with mathematical analysis of complex ODE models and their global stability analysis. The chapter introduces Lyapunov function and LaSalle Theorem as tools and illustrates their use on the SEIR model. Hopf bifurcation theorem in higher dimensions is included and its use is illustrated on the SIR model with isolation. The concept of backward bifurcation is introduced and illustrated on an SEI model with standard incidence. Castillo-Chavez and Song Theorem for detecting backward bifurcation in higher dimensions is introduced and illustrated on the SEI example.
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References
F. Brauer and J. A. Noel, The Qualitative Theory of Ordinary Differential Equations: An Introduction, Dover Publications, Inc., New York, 1989.
C. Castillo-Chavez and B. Song, Dynamical models of tuberculosis and their applications, Math. Biosci. Eng., 1 (2004), pp. 361–404.
C. Chicone, Ordinary differential equations with applications, vol. 34 of Texts in Applied Mathematics, Springer, New York, second ed., 2006.
C. V. de León, Constructions of Lyapunov functions for classics SIS, SIR and SIRS epidemic model with variable population size, Foro-Red-Mat: Revista electrónica de contenido matemático, 26 (2009).
C. V. de León and J. A. C. Hernández, Local and global stability of host–vector disease models, Foro-Red-Mat: Revista electrónica de contenido matemático, 25 (2008).
M. El Kahoui and A. Weber, Deciding Hopf bifurcations by quantifier elimination in a software-component architecture, J. Symbolic Comput., 30 (2000), pp. 161–179.
Z. Feng, C. Castillo-Chavez, and A. Capurro, A model for tuberculosis with exogenous reinfection, Theor. Pop, Biol., 57 (2000), pp. 235–247.
Z. Feng and H. R. Thieme, Recurrent outbreaks of childhood diseases revisited: the impact of isolation, Math. Biosci., 128 (1995), pp. 93–130.
A. B. Gumel, Causes of backward bifurcations in some epidemiological models, J. Math. Anal. Appl., 395 (2012), pp. 355–365.
K. P. Hadeler and P. van den Driessche, Backward bifurcation in epidemic control, Math. Biosci., 146 (1997), pp. 15–35.
A. Korobeinikov, Lyapunov functions and global properties for SEIR and SEIS epidemic models, Math Med Biol., 21 (2004), pp. 75–83.
A. Korobeinikov and P. K. Maini, A Lyapunov function and global properties for SIR and SEIR epidemiological models with nonlinear incidence, Math. Biosci. Eng., 1 (2004), pp. 57–60.
N. N. Krasovskiĭ, Nekotorye zadachi teorii ustoichivosti dvizheniya, Gosudarstv. Izdat. Fiz.-Mat. Lit., Moscow, 1959.
J. P. LaSalle, Some extensions of Liapunov’s second method, IRE Trans., CT-7 (1960), pp. 520–527.
M. Y. Li and J. S. Muldowney, Global stability for the SEIR model in epidemiology, Math. Biosci., 125 (1995), pp. 155–164.
A. M. Liapunov, Stability of motion, With a contribution by V. A. Pliss and an introduction by V. P. Basov. Translated from the Russian by Flavian Abramovici and Michael Shimshoni. Mathematics in Science and Engineering, Vol. 30, Academic Press, New York, 1966.
M. Lipsitch and M. B. Murray, Multiple equilibria: tuberculosis transmission require unrealistic assumptions, Theor. Popul. Biol., 63 (2003), pp. 169–170.
G. A. Ngwa, On the population dynamics of the malaria vector, Bull. Math. Biol., 68 (2006), pp. 2161–2189.
L. F. Olsen, G. L. Truty, and W. M. Schaffer, Oscillations and chaos in epidemics: a nonlinear dynamic study of six childhood diseases in Copenhagen, Denmark, Theoret. Population Biol., 33 (1988), pp. 344–370.
Z. Shuai and P. van den Driessche, Global stability of infectious disease models using Lyapunov functions, SIAM J. Appl. Math., 73 (2013), pp. 1513–1532.
P. N. V. Tu, Dynamical systems, Springer-Verlag, Berlin, 1992. An introduction with applications in economics and biology.
L. Zhou and M. Fan, Dynamics of an SIR epidemic model with limited medical resources revisited, Nonlinear Anal. Real World Appl., 13 (2012), pp. 312–324.
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Martcheva, M. (2015). Analysis of Complex ODE Epidemic Models: Global Stability. In: An Introduction to Mathematical Epidemiology. Texts in Applied Mathematics, vol 61. Springer, Boston, MA. https://doi.org/10.1007/978-1-4899-7612-3_7
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