International Journal of Nonlinear Analysis and Applications
Analysis of an age structured SIR epidemic model with fractional Caputo derivative
Document Type : Research Paper
Authors
Laboratory LMACS, FST of Beni-Mellal, Sultan Moulay Slimane University, Morocco
Abstract
In this paper, we consider a mathematical model with fractional derivatives in the sense of Caputo with respect to time. It describes the spread of an infectious disease that is directly transmitted in an age-structured population and whose transmission coefficient varies with age. We formulate the basic model as an abstract fractional Cauchy problem on a Banach space to prove the existence, and uniqueness of a local mild solution and ensure the global existence of a solution. Moreover, the results for the existence and uniqueness of non-trivial steady states are also demonstrated under the appropriate conditions.
Keywords
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[15] H. Inaba, On a new perspective of the basic reproduction number in heterogeneous environments, J. Math.Bio. 65 (2012), 309–348.
[16] H. Inaba, Threshold and stability results for an age-structured epidemic model, J. Math. Bio. 28 (1990), 149–175.
[17] H. Inaba, A semigroup approach to the strong ergodic theorem of the multistate stable population process, Math. Popul. Stud. 1 (1988), 49–77.
[18] H. Inaba, Age-Structured Population Dynamics in Demography and Epidemiology, Springer, Singapore, 2017.
[19] W.O. Kermack and A.G. McKendrick, Contributions to the mathematical theory of epidemics I, Proc. R. Soc. 115 (1927), 700–721.
[20] S.G. Kilbas, A.A. Kilbas, and O.I. Marichev, Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach, Yverdon, 1993
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[23] M. Langlais, Large time behaviour in a nonlinear age-dependent population dynamics problem with spatial diffusion, J. Math. Bio. 26 (1988), no. 3, 319–346. Math., 19(3): 607-628 (1970).
[24] R.M. May and R.M. Anderson, Endemic infections in growing populations, Math. Biosci. 77 (1985), 141–156.
[25] A. Mckendrick, Applications of mathematics to medical problems, Proc. Edinburgh Math. Soc. 44 (1926), 98–130.
[26] K.S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, Wiley, 1993.
[27] H.L. Smith and H.R. Thieme, Dynamical Systems and Population Persistence, Graduate Studies in Mathematics 118, Amer. Math. Soc. Providence, Rhode Island, 2011.
[28] D.W. Tudor, An age-dependent epidemic model with applications to measles, Math. Biosci. 73 (1985), 131–147.
[29] G.B. Webb, Theory of Nonlinear Age-dependent Population Dynamics, Marcel Dekker, New York and Basel, 1985.
[30] H.M. Yang, Directly transmitted infections modeling considering an age-structured contact rate, Math. Comput. Model. 29 (1999), 39–48.
[31] H. Ye, J. Gao, and Y. Ding, A generalized Gronwall inequality and its application to a fractional differential equation J. Math. Anal. Appl. 328 (2007), no. 2, 1075–1081.
[32] M. Zbair, A.Qaffou, F. Cherkaoui and K. Hilal, Bayesian Inference of a Discrete Fractional SEIRD Model, Recent Advances in Fuzzy Sets Theory, Fractional Calculus, Dynamic Systems and Optimization, Cham: Springer International Publishing, 2022, pp. 138–146.
[33] Y. Zhou and F. Jiao, Existence of mild solutions for fractional neutral evolution equations, Comput. Math. Appl. 59 (2010), no. 3, 1063–1077.
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Volume 14, Issue 5
May 2023Pages 79-93
- Receive Date: 10 February 2023
- Accept Date: 28 March 2023