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GLOBAL STABILITY OF EQUILIBRIA AND LYAPUNOV FUNCTIONS IN A VACCINATING BEHAVIOURAL SIR MODEL WITH LAGGED INFORMATION
Keywords:
SIR models with vaccination, behavioural epidemiology, rational exemption, lagged information, global stability, Lyapunov functionsDOI:
https://doi.org/10.17654/0972087125004Abstract
In this work, we prove two global asymptotic stability (GAS) results for the endemic equilibrium (EE) of an SIR model with information-dependent vaccinating behaviour and information delay, introduced in [7]. Differently from [3], we obtain our results by the direct method based on Lyapunov functions: the first one in a general context, the second one for a piecewise linear coverage function. For this last case, we compare our results and the one in [3] both analytically and numerically, thus enriching the range of exploration of the model parameters.
Received: March 8, 2024
Revised: November 2, 2024
Accepted: November 17, 2024
References
R. M. Anderson and R. M. May, Infectious Diseases of Humans: Dynamics and Control, Oxford University Press, Oxford, 1991.
E. Beretta and V. Capasso, On the general structure of epidemic systems. Global asymptotic stability, Comput. Math. Apps. Ser. A 12(6) (1986), 677-694.
B. Buonomo, A. d’Onofrio and D. Lacitignola, Global stability of an SIR epidemic model with information dependent vaccination, Math. Biosci. 216(1) (2008), 9-16.
N. Cangiotti, M. Capolli, M. Sensi and S. Sottile, A survey on Lyapunov functions for epidemic compartmental models, Boll. Unione Mat. Ital. 17 (2024), 241-257.
V. Capasso, Mathematical Structure of Epidemic Systems, Springer, Berlin, 1993.
F. Centrone and S. Salinelli, Lyapunov functions in SIR model with vaccinating behaviour, Far East J. Math. Sci. (FJMS) 125(2) (2020), 139-150.
A. d’Onofrio, P. Manfredi and E. Salinelli, Vaccinating behaviour, information, and the dynamics of SIR vaccine preventable diseases, Theoretical Population Biology 71 (2007), 301-317.
A. d’Onofrio, P. Manfredi and E. Salinelli, Bifurcation thresholds in an SIR model with information-dependent vaccination, Math. Model. Nat. Phenom. 2(1) (2007), 23-38.
A. d’Onofrio, P. Manfredi and E. Salinelli, Lyapunov stability of an SIRS epidemic model with varying population: ecological VS physical approaches, Far East J. Math. Sci. (FJMS) 117(2) (2019), 147-160.
H. W. Hethcote, The mathematics of infectious diseases, SIAM Rev. 42 (2000), 599-653.
W. O. Kermack and A. G. McKendrick, A contribution to the mathematical theory of epidemics, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 155 (1927), 700-721.
A. Korobeinikov and Graeme C. Wake, Lyapunov functions and global stability for SIR, SIRS, and SIS epidemiological models, Appl. Math. Lett. 15(8) (2002), 955-960.
M. I. Li and J. S. Muldowney, On Bendixon’s criterion, J. Differential Equations 106(1) (1993), 27-39.
M. I. Li and J. S. Muldowney, A geometric approach to global-stability problems, SIAM J. Math. Anal. 27(4) (1996), 1070-1083.
J. Li, Y. Yang, Y. Xiao and S. Liu, A class of Lyapunov functions and the global stability of some epidemic models with nonlinear incidence, J. Appl. Anal. Comput. 6(1) (2016), 38-46.
N. MacDonald, Biological Delay Systems: Linear Stability Theory, Cambridge University Press, 1989.
M. O. Oke, O. M. Ogunmiloro, C. T. Akinwumi and R. A. Raji, Mathematical modeling and stability analysis of a SIRV epidemic model with non-linear force of infection and treatment, Communications in Mathematics and Applications 10(4) (2019), 717-731.
S. Zhisheng and P. van den Driessche, Global stability of infectious disease models using Lyapunov functions, SIAM J. Appl. Math. 73(4) (2013), 1513-1532.
S. Syafruddin and M. S. M. Noorani, Lyapunov function of SIR and SEIR model for transmission of dengue fever disease, Int. J. Simul. Process. Model. 8(2/3) (2013), 177-184.
Q. Tang, Z. Teng and X. Abdurahman, A new Lyapunov function for SIRS epidemic models, Bull. Malays. Math. Sci. Soc. 40 (2017), 237-258.
C. Vargas-De-Leòn, On the global stability of SIS, SIR and SIRS epidemic models with standard incidence, Chaos Solitons Fractals 44 (2011), 1106-1110.
E. Vynnycky and R. White, An Introduction to Infectious Disease Modelling, Oxford University Press, Oxford, 2010.
Z. Wang, C. T. Bauch, S. Bhattacharyya, A. d’Onofrio, P. Manfredi, M. Perc, N. Perra, M. Salathé and D. Zhao, Statistical physics of vaccination, Physics Reports 664 (2016), 1-113.
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