# American Institute of Mathematical Sciences

October  2014, 19(8): 2383-2399. doi: 10.3934/dcdsb.2014.19.2383

## A singularly perturbed age structured SIRS model with fast recovery

 1 School of Mathematical Sciences, University of KwaZulu-Natal, Durban 2 School of Mathematical Sciences, University of KwaZulu-Natal, Durban 4041, South Africa

Received  November 2013 Revised  March 2014 Published  August 2014

Age structure of a population often plays a significant role in the spreading of a disease among its members. For instance, childhood diseases mostly affect the juvenile part of the population, while sexually transmitted diseases spread mostly among the adults. Thus, it is important to build epidemiological models which incorporate the demography of the affected populations. Doing this we must be careful as many diseases act on a time scale different from that of the vital processes. For many diseases, e.g. measles, influenza, the typical time unit is one day or one week, whereas the proper time unit for the vital processes is the average lifespan in the population; that is, 10-100 years. In such a case, the epidemiological model with vital dynamics becomes a multiple time scale model and thus it often can be significantly simplified by various asymptotic methods. The presented paper is concerned with an SIRS type disease spreading in a population with a continuous age structure modelled by the McKendrick-von Foerster equation. We consider a disease with a quick recovery rate in a large population. Though it is not too surprising that in such a model the introduced disease quickly vanishes, the result is mathematically interesting as the error estimates are uniform on the whole infinite time interval, in contrast to the typical results based on the Tikhonov theorem and classical asymptotic expansions.
Citation: Jacek Banasiak, Rodrigue Yves M'pika Massoukou. A singularly perturbed age structured SIRS model with fast recovery. Discrete and Continuous Dynamical Systems - B, 2014, 19 (8) : 2383-2399. doi: 10.3934/dcdsb.2014.19.2383
##### References:
 [1] J. Banasiak, Mathematical Modelling in One Dimension, Cambridge University Press, Cambridge, 2013. [2] J. Banasiak, Introduction to Mathematical Methods in Population Dynamics,, in preparation., (). [3] J. Banasiak and M. Lachowicz, On a macroscopic limit of a kinetic model of alignment, Math. Models Methods Appl. Sci., 23 (2013), 2647-2670. doi: 10.1142/S0218202513500425. [4] J. Banasiak and M. Lachowicz, Methods of Small Parameter in Mathematical Biology, Birkhäuser/Springer Cham, Heidelberg/New York, 2014. doi: 10.1007/978-3-319-05140-6. [5] J. Banasiak and W. Lamb, Coagulation, fragmentation and growth processes in a size structured population, Discrete Contin. Dyn. Syst., Ser. B, 17 (2012), 445-472 doi: 10.3934/dcdsb.2012.17.445. [6] J. Banasiak, A. Goswami and S. Shindin, Aggregation in age and space structured population models: an asymptotic analysis approach, J. Evol. Equ., 11 (2011), 121-154. doi: 10.1007/s00028-010-0086-7. [7] S. N. Busenberg, M. Iannelli and H. R. Thieme, Global behaviour of an age-structured epidemic model, SIAM J. Math. Anal., 22 (1991), 1065-1080. doi: 10.1137/0522069. [8] D. J. D. Earn, A Light Introduction to Modelling Recurrent Epidemics, in Mathematical Epidemiology (eds. F. Brauer, P. van den Driessche and J. Wu), LNM 1945, Springer, Berlin, 2008, 3-18. doi: 10.1007/978-3-540-78911-6_1. [9] M. Iannelli, Mathematical Theory of Age-Structured Population Dynamics, Applied Mathematics Monographs, 7, Consiglio Nazionale delle Ricerche, Giardini, Pisa, 1995. [10] H. Inaba, A semigroup approach to the strong ergodic theorem of the multi state stable population process, Mathematical Population Studies, 1 (1988), 49-77. doi: 10.1080/08898488809525260. [11] R. M'pika Massoukou, Age Structured Models of Mathematical Epidemiology, Ph.D thesis, UKZN, 2013. [12] J. Prüss, Equilibrium Solutions of Age-Specific Population Dynamics of Several Species, J. Math. Biol., 11 (1981), 65-84. doi: 10.1007/BF00275825. [13] J. Prüss, Stability analysis for equilibria in age-specific population dynamics, Nonlinear Anal., 7 (1983), 1291-1313. doi: 10.1016/0362-546X(83)90002-0. [14] H. R. Thieme, Mathematics in Population Biology, Princeton University Press, Princeton, 2003. [15] G. F. Webb, Theory of Non-linear Age Dependent Population Dynamics, Marcel Dekker, New York, 1985.

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##### References:
 [1] J. Banasiak, Mathematical Modelling in One Dimension, Cambridge University Press, Cambridge, 2013. [2] J. Banasiak, Introduction to Mathematical Methods in Population Dynamics,, in preparation., (). [3] J. Banasiak and M. Lachowicz, On a macroscopic limit of a kinetic model of alignment, Math. Models Methods Appl. Sci., 23 (2013), 2647-2670. doi: 10.1142/S0218202513500425. [4] J. Banasiak and M. Lachowicz, Methods of Small Parameter in Mathematical Biology, Birkhäuser/Springer Cham, Heidelberg/New York, 2014. doi: 10.1007/978-3-319-05140-6. [5] J. Banasiak and W. Lamb, Coagulation, fragmentation and growth processes in a size structured population, Discrete Contin. Dyn. Syst., Ser. B, 17 (2012), 445-472 doi: 10.3934/dcdsb.2012.17.445. [6] J. Banasiak, A. Goswami and S. Shindin, Aggregation in age and space structured population models: an asymptotic analysis approach, J. Evol. Equ., 11 (2011), 121-154. doi: 10.1007/s00028-010-0086-7. [7] S. N. Busenberg, M. Iannelli and H. R. Thieme, Global behaviour of an age-structured epidemic model, SIAM J. Math. Anal., 22 (1991), 1065-1080. doi: 10.1137/0522069. [8] D. J. D. Earn, A Light Introduction to Modelling Recurrent Epidemics, in Mathematical Epidemiology (eds. F. Brauer, P. van den Driessche and J. Wu), LNM 1945, Springer, Berlin, 2008, 3-18. doi: 10.1007/978-3-540-78911-6_1. [9] M. Iannelli, Mathematical Theory of Age-Structured Population Dynamics, Applied Mathematics Monographs, 7, Consiglio Nazionale delle Ricerche, Giardini, Pisa, 1995. [10] H. Inaba, A semigroup approach to the strong ergodic theorem of the multi state stable population process, Mathematical Population Studies, 1 (1988), 49-77. doi: 10.1080/08898488809525260. [11] R. M'pika Massoukou, Age Structured Models of Mathematical Epidemiology, Ph.D thesis, UKZN, 2013. [12] J. Prüss, Equilibrium Solutions of Age-Specific Population Dynamics of Several Species, J. Math. Biol., 11 (1981), 65-84. doi: 10.1007/BF00275825. [13] J. Prüss, Stability analysis for equilibria in age-specific population dynamics, Nonlinear Anal., 7 (1983), 1291-1313. doi: 10.1016/0362-546X(83)90002-0. [14] H. R. Thieme, Mathematics in Population Biology, Princeton University Press, Princeton, 2003. [15] G. F. Webb, Theory of Non-linear Age Dependent Population Dynamics, Marcel Dekker, New York, 1985.
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