Analysis of an age structured SIR epidemic model with fractional Caputo derivative

Document Type : Research Paper


Laboratory LMACS, FST of Beni-Mellal, Sultan Moulay Slimane University, Morocco


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.


[1] L.J.S. Allen, Discrete and continuous models of populations and epidemics, J. Math. Syst. Estim. Control 1 (1991), no. 3, 335–369.
[2] C. Barril, A`. Calsina and J. Rippol, A practical approach to R0 in continuous time ecological models, Math. Meth. Appl. Sci. 41 (2017), 8432–8445.
[3] S. Busenberg, M. Iannelli and H. Thieme, Global behavior of an age-structured epidemic model, SIAM J. Math. Anal. 22 (1991), no. 4, 1065–1080.
[4] Y. Cha, M. Iannelli, and E. Milner, Existence and uniqueness of endemic states for the age-structured SIR epidemic model, Math. Biosci. 150 (1998), 177–190.
[5] O. Diekmann, J.A.P. Heesterbeek, Mathematical Epidemiology of Infectious Diseases: Model Building, Analysis and Interpretation, Wiley, Chichester, 2000.
[6] O. Diekmann, J.A.P. Heesterbeek and J.A.J. Metz, On the definition and the computation of the basic reproduction ratio R in models for infectious diseases in heterogeneous populations, J. Math. Biol. 28 (1990), 365–382.
[7] K. Dietz and D. Schenzle, Proportionate mixingmodels for age-dependent infection transmission, J. Math. Biol. 22 (1985), 117–120.
[8] N. Dunford and J. T. Schwartz, Linear Operators Part I: General Theory, Interscience publishers, New York, 1958.
[9] M. El-Doma, Analysis of an age-dependent SIS epidemic model with vertical transmission and proportionate mixing assumption, Math. Comput. Model. 29 (1999), 31–43.
[10] D. Greenhalgh, Analytical threshold and stability results on age-structured epidemic models with vaccination, Theor. Popul. Bio. 33 (1988), 266–290.
[11] D. Greenhalgh and K. Dietz, Some bounds on estimates for reproductive ratios derived from the age-specific force of infection, Math. Biosci. 124 (1994), 9–57.
[12] M.E. Gurtin and R.C. MacCamy, Product Solutions and Asymptotic Behavior for Age-Dependent, Dispersing Populations, 1981.
[13] H.J. Heijmans, The dynamical behaviour of the age-size-distribution of a cell population, Springer Berlin Heidelberg, 1986, pp. 185–202.
[14] M. Iannelli, Mathematical Theory of Age-Structured Population Dynamics, Giardini Editori e Stampatori in Pisa, 1995.
[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
[21] A.A. Kilbas, H.H. Srivastava and J.J. Trujillo, Theory and Applications of Fractional Differential Equations, . Vol. 204, Elsevier, 2006.
[22] M.A. Krasnoselskii, Positive Solutions of Operator Equations, Groningen, Noordhoff, 1964.
[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.
Volume 14, Issue 5
May 2023
Pages 79-93
  • Receive Date: 10 February 2023
  • Accept Date: 28 March 2023