Global stability of an age-structured epidemic model with general Lyapunov functional

Abdennasser Chekroun; Mohammed Nor Frioui; Toshikazu Kuniya; Tarik Mohammed Touaoula; Abdennasser Chekroun; Mohammed Nor Frioui; Toshikazu Kuniya; Tarik Mohammed Touaoula

doi:10.3934/mbe.2019073

Mathematical Biosciences and Engineering

2019, Volume 16, Issue 3: 1525-1553. doi: 10.3934/mbe.2019073

Previous Article Next Article

Research article Special Issues

Global stability of an age-structured epidemic model with general Lyapunov functional

1.
Laboratoire d’Analyse Nonlinéaire et Mathématiques Appliquées, University of Tlemcen, Tlemcen 13000, Algeria
2.
Graduate School of System Informatics, Kobe University, 1-1 Rokkodai-cho, Nada-ku, Kobe 657-8501, Japan

Received: 19 December 2018 Accepted: 27 January 2019 Published: 26 February 2019

In this paper, we focus on the study of the dynamics of a certain age structured epidemic model. Our aim is to investigate the proposed model, which is based on the classical SIR epidemic model, with a general class of nonlinear incidence rate with some other generalization. We are interested to the asymptotic behavior of the system. For this, we have introduced the basic reproduction number ${\cal R}_0$ of model and we prove that this threshold shows completely the stability of each steady state. Our approach is the use of general constructed Lyapunov functional with some results on the persistence theory. The conclusion is that the system has a trivial disease-free equilibrium which is globally asymptotically stable for ${\cal R}_0 \lt 1$ and that the system has only a unique positive endemic equilibrium which is globally asymptotically stable whenever ${\cal R}_0 \gt 1$. Several numerical simulations are given to illustrate our results.
- SIR epidemic model,
- infection age,
- nonlinear incidence,
- persistence,
- Lyapunov function,
- global stability
Citation: Abdennasser Chekroun, Mohammed Nor Frioui, Toshikazu Kuniya, Tarik Mohammed Touaoula. Global stability of an age-structured epidemic model with general Lyapunov functional[J]. Mathematical Biosciences and Engineering, 2019, 16(3): 1525-1553. doi: 10.3934/mbe.2019073

Related Papers:

Abstract

In this paper, we focus on the study of the dynamics of a certain age structured epidemic model. Our aim is to investigate the proposed model, which is based on the classical SIR epidemic model, with a general class of nonlinear incidence rate with some other generalization. We are interested to the asymptotic behavior of the system. For this, we have introduced the basic reproduction number ${\cal R}_0$ of model and we prove that this threshold shows completely the stability of each steady state. Our approach is the use of general constructed Lyapunov functional with some results on the persistence theory. The conclusion is that the system has a trivial disease-free equilibrium which is globally asymptotically stable for ${\cal R}_0 \lt 1$ and that the system has only a unique positive endemic equilibrium which is globally asymptotically stable whenever ${\cal R}_0 \gt 1$. Several numerical simulations are given to illustrate our results.

References

[1]	N. Bacaër, A Short History of Mathematical Population Dynamics, Springer-Verlag, London, UK, 2011.
[2]	W. O. Kermack and A. G. McKendrick, Contributions to the mathematical theory of epidemics - I, Proc. R. Soc., 115 (1927), 700–721.
[3]	O. Diekmann and J. A. P. Heesterbeek, Mathematical epidemiology of infectious diseases: Model building, analysis and interpretation, Wiley, Chichester, UK, 2000.
[4]	H. Inaba, Age-Structured Population Dynamics in Demography and Epidemiology, Springer, Singapore, 2017.
[5]	P. Magal, C. C. McCluskey and G. F. Webb, Lyapunov functional and global asymptotic stability for an infection-age model, Appl. Anal., 89 (2010), 1109–1140.
[6]	Y. Chen, S. Zou and J. Yang, Global analysis of an SIR epidemic model with infection age and saturated incidence, Nonlinear Anal. RWA, 30 (2016), 16–31.
[7]	L. Liu, J. Wang and X. Liu, Global stability of an SEIR epidemic model with age-dependent latency and relapse, Nonlinear Anal. RWA, 24 (2015), 18–35.
[8]	C. C. McCluskey, Global stability for an SEI epidemiological model with continuous age-structure in the exposed and infectious classes, Math. Biosci. Eng., 9 (2012), 819–841.
[9]	C. Vargas-De-Leon, Global stability properties of age-dependent epidemic models with varying rates of recurrence, Math. Meth. Appl. Sci., 39 (2016), 2057–2064.
[10]	J. Yang, Y. Chen and T. Kuniya, Threshold dynamics of an age-structured epidemic model with relapse and nonlinear incidence, IMA J. Appl. Math., 82 (2017), 629–655.
[11]	A. Chekroun and T. Kuniya, Stability and existence results for a time-delayed nonlocal model of hematopoietic stem cells dynamics, J. Math. Anal. Appl., 463 (2018), 1147–1168.
[12]	K. Hattaf and Y. Yang, Global dynamics of an age-structured viral infection model with general incidence function and absorption, Int. J. Biomath., 11 (2018). 1850065
[13]	G. ur Rahman, R. P. Agarwal, L. Liu and A. Khan, Threshold dynamics and optimal control of an age-structured giving up smoking model, Nonlinear Anal. RWA, 43 (2018), 96–120.
[14]	V. Capasso and G. Serio, A generalization of the Kermack-McKendrick deterministic epidemic model, Math. Biosci., 42 (1978), 43–61.
[15]	S. Djilali, T. M. Touaoula and S. E. H. Miri, A heroin epidemic model: very general non linear incidence, treat-age, and global stability, Acta Applicandae Mathematicae, 152 (2017), 171–194.
[16]	Z. Feng and H. R. Thieme, Endemic models with arbitrarily distributed periods of infection I: fundamental properties of the model, SIAM J. Appl. Math., 61 (2000), 803–833.
[17]	Z. Feng and H. R. Thieme, Endemic models with arbitrarily distributed periods of infection II: fast disease dynamics and permanent recovery, SIAM J. Appl. Math., 61 (2000), 983–1012.
[18]	M. N. Frioui, S. E. Miri and T. M. Touaoula, Unified Lyapunov functional for an age-structured virus model with very general nonlinear infection response, J. Appl. Math. Comput., 58 (2018), 47–73.
[19]	K. Hattaf, A. A. Lashari, Y. Louartassi and N. Yousfi, A delayed SIR epidemic model with general incidence rate, Electron. J. Qual. Theo. Diff. Equ., 3 (2013), 1–9.
[20]	G. Huang and Y. Takeuchi, Global analysis on delay epidemiological dynamic models with nonlinear incidence, J. Math. Biol., 63 (2011), 125–139.
[21]	G. Huang, Y. Takeuchi, W. Ma and D. Wei, Global stability for delay SIR and SEIR epidemic models with nonlinear incidence rate, Bull. Math. Biol., 72 (2010), 1192–1207.
[22]	A. Korobeinikov, Lyapunov functions and global stability for SIR and SIRS epidemiological models with non-linear transmission, Bull. Math. Biol., 68 (2006), 615–626.
[23]	A. Korobeinikov, Global properties of infectious disease models with nonlinear incidence, Bull. Math. Biol., 69 (2007), 1871-1886.
[24]	A. Korobeinikov and P. K. Maini, Nonlinear incidence and stability of infectious disease models, Math. Med. Biol., 22 (2005), 113–128.
[25]	S. Liu, S.Wang and L.Wang, Global dynamics of delay epidemic models with nonlinear incidence rate and relapse, Nonlinear Anal. RWA, 12 (2011), 119–127.
[26]	J.Wang, J. Pang and X. Liu, Modelling diseases with relapse and nonlinear incidence of infection: a multi-group epidemic model, J. Biol. Dyn., 8 (2014), 99–116.
[27]	S. Bentout and T. M. Touaoula, Global analysis of an infection age model with a class of nonlinear incidence rates, J. Math. Anal. Appl., 434 (2016), 1211–1239.
[28]	P. van den Driessche and X. Zou, Modeling relapse in infectious diseases, Math. Biosci., 207 (2007), 89–103.
[29]	B. Fang, X. Z. Li, M. Martcheva and L. M. Cai, Global asymptotic properties of a heroin epidemic model with treat-age, Appl. Math. Comput., 263 (2015), 315–331.
[30]	G. Mulone and B. Straughan, A note on heroin epidemics, Math. Biosci., 218 (2009), 138–141.
[31]	Y. Muroya, H. Li and T. Kuniya, Complete global analysis of an SIRS epidemic model with grated cure and incomplete recovery rates, J. Math. Anal. Appl., 410 (2014), 719–732.
[32]	E. White and C. Comiskey, Heroin epidemics, treatment and ODE modelling, Math. Biosci. 208 (2007), 312–324.
[33]	I. M. Wangari and L. Stone, Analysis of a heroin epidemic model with saturated treatment function, J. Appl. Math., 2017 (2017), Article ID 1953036.
[34]	P. Munz, I. Hudea, J. Imad and R. J. Smith, When zombies attack!: mathematical modelling of an outbreak of zombie infection, in Infectious Disease Modelling Research Progress (eds. J.M. Tchuenche and C. Chiyaka), New York, Nova Science Publishers, (2009), 133–150.
[35]	J. K. Hale and S. M. Verduyn Lunel, Introduction to Functional Differential Equations, V. 99, AMS, 1993.
[36]	B. Balachandran, T. Kalmar-Nagy and D. E. Gilsin, Delay Differential Equations: Recent Advances and New Direction, Springer-verlag, 2009.
[37]	H. L. Smith and H. R. Thieme, Dynamical Systems and Population Persistence, Graduate Studies in Mathematics V. 118, AMS, 2011.
[38]	H. R. Thieme, Mathematics in Population Biology, Princeton University Press, Princeton 2003.
[39]	B. Perthame, Transport Equations in Biology, Birkhäuser, Berlin, 2007.

Reader Comments

Your name:*

Email:*
© 2019 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)