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October  2020, 19(10): 4955-4972. doi: 10.3934/cpaa.2020220

## Periodic solutions of an age-structured epidemic model with periodic infection rate

 1 Department of Mathematics, University of Miami, Coral Gables, FL 33146, USA 2 Department of Mathematics, Applied Mathematics and Statistics, Case Western Reserve University, Cleveland, OH 44106, USA

* Corresponding author

Received  January 2020 Revised  May 2020 Published  July 2020

Fund Project: Research was partially supported by National Science Foundation (DMS-1853622)

In this paper we consider an age-structured epidemic model of the susceptible-exposed-infectious-recovered (SEIR) type. To characterize the seasonality of some infectious diseases such as measles, it is assumed that the infection rate is time periodic. After establishing the well-posedness of the initial-boundary value problem, we study existence of time periodic solutions of the model by using a fixed point theorem. Some numerical simulations are presented to illustrate the obtained results.

Citation: Hao Kang, Qimin Huang, Shigui Ruan. Periodic solutions of an age-structured epidemic model with periodic infection rate. Communications on Pure and Applied Analysis, 2020, 19 (10) : 4955-4972. doi: 10.3934/cpaa.2020220
##### References:
 [1] R. Anderson and R. May, Age-related changes in the rate of disease transmission: implications for the design of vaccination programmes, Epidemiol. Infect., 94 (1985), 365-436. [2] V. Andreasen, Disease regulation of age-structured host populations, Theor. Popul. Biol., 36 (1989), 214-239.  doi: 10.1016/0040-5809(89)90031-2. [3] N. Bacaër, Approximation of the basic reproduction number $R_0$ for vector-borne diseases with a periodic vector population, Bull. Math. Biol., 69 (2007), 1067-1091.  doi: 10.1007/s11538-006-9166-9. [4] N. Bacaër and S. Guernaoui, The epidemic threshold of vector-borne diseases with seasonality, J. Math. Biol., 53 (2006), 421-436.  doi: 10.1007/s00285-006-0015-0. [5] D. Bentil and J. Murray, Modelling bovine tuberculosis in badgers, J. Anim. Ecol., 239–250. [6] H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Springer Science and Business Media, New York, 2010. [7] S. Busenberg, M. Iannelli and H. Thieme, Dynamics of an age-structured epidemic model, in Dynamical Systems, Proceedings of the Special Program at Nankai Institute of Mathematics, World Scientific Pub., Singapore, (1993), 1–19. doi: 10.1007/978-3-642-75301-5_1. [8] S. N. Busenberg, M. Iannelli and H. R. Thieme, Global behavior of an age-structured epidemic model, SIAM J. Math. Anal., 22 (1991), 1065-1080.  doi: 10.1137/0522069. [9] Y. Cha, M. Iannelli and F. A. Milner, Existence and uniqueness of endemic states for the age-structured S-I-R epidemic model, Math. Biosci., 150 (1998), 177-190.  doi: 10.1016/S0025-5564(98)10006-8. [10] D. J. Earn, P. Rohani, B. M. Bolker and B. T. Grenfell, A simple model for complex dynamical transitions in epidemics, Science, 287 (2000), 667-670. [11] Z. Feng, W. Huang and C. Castillo-Chavez, Global behavior of a multi-group sis epidemic model with age structure, J. Differ. Equ., 218 (2005), 292-324.  doi: 10.1016/j.jde.2004.10.009. [12] D. Greenhalgh, Analytical results on the stability of age-structured recurrent epidemic models, Math. Med. Biol. JIMA, 4 (1987), 109-144. [13] H. W. Hethcote, Optimal ages of vaccination for measles, Math. Biosci., 89 (1988), 29-52.  doi: 10.1016/0025-5564(88)90111-3. [14] H. W. Hethcote and S. A. Levin, Periodicity in epidemiological models, in Applied Mathematical Ecology (eds. S. A. Levin, T. G. Hallam and L. J. Gross), Biomathematics Vol. 18, Springer, Berlin, (1989), 193–211. doi: 10.1007/978-3-642-61317-3_8. [15] F. Hoppensteadt, An age dependent epidemic model, J. Franklin Inst., 297 (1974), 325-333. [16] J. Huang, S. Ruan, X. Wu and X. Zhou, Seasonal transmission dynamics of measles in China, Theory Biosci., 137 (2018), 185-195.  doi: 10.1007/s11538-020-00747-6. [17] M. Iannelli, Mathematical Theory of Age-Structured Population Dynamics, Giardini editori e stampatori, Pisa, 1995. [18] M. Iannelli, M. Y. Kim and E. J. Park, Asymptotic behavior for an sis epidemic model and its approximation, Nonlinear Anal. Theory Meth. Appl., 35 (1999), 797-814.  doi: 10.1016/S0362-546X(97)00597-X. [19] H. Inaba, Threshold and stability results for an age-structured epidemic model, J. Math. Biol., 28 (1990), 411-434.  doi: 10.1007/BF00178326. [20] H. Inaba, Kermack and McKendrick revisited: the variable susceptibility model for infectious diseases, Japan J. Indust. Appl. Math., 18 (2001), 273-292.  doi: 10.1007/BF03168575. [21] H. Inaba, The Malthusian parameter and is $R_0$ for heterogeneous populations in periodic environments, Math. Biosci. Eng., 9 (2012), 313-346.  doi: 10.3934/mbe.2012.9.313. [22] H. Inaba, On a new perspective of the basic reproduction number in heterogeneous environments, J. Math. Biol., 65 (2012), 309-348.  doi: 10.1007/s00285-011-0463-z. [23] H. Inaba, Age-structured Population Dynamics in Demography and Epidemiology, Springer, New York, 2017. doi: 10.1007/978-981-10-0188-8. [24] W. O. Kermack and A. G. McKendrick, A contribution to the mathematical theory of epidemics, Proc. R. Soc. Lond. A, 115 (1927), 700-721. [25] W. O. Kermack and A. G. McKendrick, Contributions to the mathematical theory of epidemics. Ⅱ. The problem of endemicity, Proc. R. Soc. Lond. A, 138 (1932), 55-83. [26] W. O. Kermack and A. G. McKendrick, Contributions to the mathematical theory of epidemics. Ⅲ. Further studies of the problem of endemicity, Proc. R. Soc. Lond. A, 141 (1933), 94-122. [27] M. A. Krasnoselskii, Positive Solutions of Operator Equations, Noordhoff, Groningen, 1964. [28] M. Kubo and M. Langlais, Periodic solutions for nonlinear population dynamics models with age-dependence and spatial structure, J. Differ. Equ., 109 (1994), 274-294.  doi: 10.1006/jdeq.1994.1050. [29] T. Kuniya, Existence of a nontrivial periodic solution in an age-structured SIR epidemic model with time periodic coefficients, Appl. Math. Lett., 27 (2014), 15-20.  doi: 10.1016/j.aml.2013.08.008. [30] T. Kuniya and M. Iannelli, $R_0$ and the global behavior of an age-structured SIS epidemic model with periodicity and vertical transmission, Math. Biosci. Eng., 11 (2014), 929-945.  doi: 10.3934/mbe.2014.11.929. [31] T. Kuniya and H. Inaba, Endemic threshold results for an age-structured SIS epidemic model with periodic parameters, J. Math. Anal. Appl., 402 (2013), 477-492.  doi: 10.1016/j.jmaa.2013.01.044. [32] M. Langlais and S. Busenberg, Global behaviour in age structured SIS models with seasonal periodicities and vertical transmission, J. Math. Anal. Appl., 213 (1997), 511-533.  doi: 10.1006/jmaa.1997.5554. [33] X.-Z. Li, G. Gupur and G.-T. Zhu, Threshold and stability results for an age-structured SEIR epidemic model, Comput. Math. Appl., 42 (2001), 883-907.  doi: 10.1016/S0898-1221(01)00206-1. [34] I. Marek, Frobenius theory of positive operators: Comparison theorems and applications, SIAM J. Appl. Math., 19 (1970), 607-628.  doi: 10.1137/0119060. [35] I. Sawashima, On spectral properties of some positive operators, Natural Sci. Rep. Ochanomizu Univ., 15 (1964), 53-64. [36] D. Schenzle, An age-structured model of pre-and post-vaccination measles transmission, Math. Med. Biol. JIMA, 1 (1984), 169-191. [37] H. R. Thieme, Renewal theorems for linear periodic Volterra integral equations, J. Integral Equ., 7 (1984), 253-277. [38] D. W. Tudor, An age-dependent epidemic model with application to measles, Math. Biosci., 73 (1985), 131-147.  doi: 10.1016/0025-5564(85)90081-1. [39] W. Wang and X.-Q. Zhao, Threshold dynamics for compartmental epidemic models in periodic environments, J. Dyn. Differ. Equ., 20 (2008), 699-717.  doi: 10.1007/s10884-008-9111-8.

show all references

##### References:
 [1] R. Anderson and R. May, Age-related changes in the rate of disease transmission: implications for the design of vaccination programmes, Epidemiol. Infect., 94 (1985), 365-436. [2] V. Andreasen, Disease regulation of age-structured host populations, Theor. Popul. Biol., 36 (1989), 214-239.  doi: 10.1016/0040-5809(89)90031-2. [3] N. Bacaër, Approximation of the basic reproduction number $R_0$ for vector-borne diseases with a periodic vector population, Bull. Math. Biol., 69 (2007), 1067-1091.  doi: 10.1007/s11538-006-9166-9. [4] N. Bacaër and S. Guernaoui, The epidemic threshold of vector-borne diseases with seasonality, J. Math. Biol., 53 (2006), 421-436.  doi: 10.1007/s00285-006-0015-0. [5] D. Bentil and J. Murray, Modelling bovine tuberculosis in badgers, J. Anim. Ecol., 239–250. [6] H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Springer Science and Business Media, New York, 2010. [7] S. Busenberg, M. Iannelli and H. Thieme, Dynamics of an age-structured epidemic model, in Dynamical Systems, Proceedings of the Special Program at Nankai Institute of Mathematics, World Scientific Pub., Singapore, (1993), 1–19. doi: 10.1007/978-3-642-75301-5_1. [8] S. N. Busenberg, M. Iannelli and H. R. Thieme, Global behavior of an age-structured epidemic model, SIAM J. Math. Anal., 22 (1991), 1065-1080.  doi: 10.1137/0522069. [9] Y. Cha, M. Iannelli and F. A. Milner, Existence and uniqueness of endemic states for the age-structured S-I-R epidemic model, Math. Biosci., 150 (1998), 177-190.  doi: 10.1016/S0025-5564(98)10006-8. [10] D. J. Earn, P. Rohani, B. M. Bolker and B. T. Grenfell, A simple model for complex dynamical transitions in epidemics, Science, 287 (2000), 667-670. [11] Z. Feng, W. Huang and C. Castillo-Chavez, Global behavior of a multi-group sis epidemic model with age structure, J. Differ. Equ., 218 (2005), 292-324.  doi: 10.1016/j.jde.2004.10.009. [12] D. Greenhalgh, Analytical results on the stability of age-structured recurrent epidemic models, Math. Med. Biol. JIMA, 4 (1987), 109-144. [13] H. W. Hethcote, Optimal ages of vaccination for measles, Math. Biosci., 89 (1988), 29-52.  doi: 10.1016/0025-5564(88)90111-3. [14] H. W. Hethcote and S. A. Levin, Periodicity in epidemiological models, in Applied Mathematical Ecology (eds. S. A. Levin, T. G. Hallam and L. J. Gross), Biomathematics Vol. 18, Springer, Berlin, (1989), 193–211. doi: 10.1007/978-3-642-61317-3_8. [15] F. Hoppensteadt, An age dependent epidemic model, J. Franklin Inst., 297 (1974), 325-333. [16] J. Huang, S. Ruan, X. Wu and X. Zhou, Seasonal transmission dynamics of measles in China, Theory Biosci., 137 (2018), 185-195.  doi: 10.1007/s11538-020-00747-6. [17] M. Iannelli, Mathematical Theory of Age-Structured Population Dynamics, Giardini editori e stampatori, Pisa, 1995. [18] M. Iannelli, M. Y. Kim and E. J. Park, Asymptotic behavior for an sis epidemic model and its approximation, Nonlinear Anal. Theory Meth. Appl., 35 (1999), 797-814.  doi: 10.1016/S0362-546X(97)00597-X. [19] H. Inaba, Threshold and stability results for an age-structured epidemic model, J. Math. Biol., 28 (1990), 411-434.  doi: 10.1007/BF00178326. [20] H. Inaba, Kermack and McKendrick revisited: the variable susceptibility model for infectious diseases, Japan J. Indust. Appl. Math., 18 (2001), 273-292.  doi: 10.1007/BF03168575. [21] H. Inaba, The Malthusian parameter and is $R_0$ for heterogeneous populations in periodic environments, Math. Biosci. Eng., 9 (2012), 313-346.  doi: 10.3934/mbe.2012.9.313. [22] H. Inaba, On a new perspective of the basic reproduction number in heterogeneous environments, J. Math. Biol., 65 (2012), 309-348.  doi: 10.1007/s00285-011-0463-z. [23] H. Inaba, Age-structured Population Dynamics in Demography and Epidemiology, Springer, New York, 2017. doi: 10.1007/978-981-10-0188-8. [24] W. O. Kermack and A. G. McKendrick, A contribution to the mathematical theory of epidemics, Proc. R. Soc. Lond. A, 115 (1927), 700-721. [25] W. O. Kermack and A. G. McKendrick, Contributions to the mathematical theory of epidemics. Ⅱ. The problem of endemicity, Proc. R. Soc. Lond. A, 138 (1932), 55-83. [26] W. O. Kermack and A. G. McKendrick, Contributions to the mathematical theory of epidemics. Ⅲ. Further studies of the problem of endemicity, Proc. R. Soc. Lond. A, 141 (1933), 94-122. [27] M. A. Krasnoselskii, Positive Solutions of Operator Equations, Noordhoff, Groningen, 1964. [28] M. Kubo and M. Langlais, Periodic solutions for nonlinear population dynamics models with age-dependence and spatial structure, J. Differ. Equ., 109 (1994), 274-294.  doi: 10.1006/jdeq.1994.1050. [29] T. Kuniya, Existence of a nontrivial periodic solution in an age-structured SIR epidemic model with time periodic coefficients, Appl. Math. Lett., 27 (2014), 15-20.  doi: 10.1016/j.aml.2013.08.008. [30] T. Kuniya and M. Iannelli, $R_0$ and the global behavior of an age-structured SIS epidemic model with periodicity and vertical transmission, Math. Biosci. Eng., 11 (2014), 929-945.  doi: 10.3934/mbe.2014.11.929. [31] T. Kuniya and H. Inaba, Endemic threshold results for an age-structured SIS epidemic model with periodic parameters, J. Math. Anal. Appl., 402 (2013), 477-492.  doi: 10.1016/j.jmaa.2013.01.044. [32] M. Langlais and S. Busenberg, Global behaviour in age structured SIS models with seasonal periodicities and vertical transmission, J. Math. Anal. Appl., 213 (1997), 511-533.  doi: 10.1006/jmaa.1997.5554. [33] X.-Z. Li, G. Gupur and G.-T. Zhu, Threshold and stability results for an age-structured SEIR epidemic model, Comput. Math. Appl., 42 (2001), 883-907.  doi: 10.1016/S0898-1221(01)00206-1. [34] I. Marek, Frobenius theory of positive operators: Comparison theorems and applications, SIAM J. Appl. Math., 19 (1970), 607-628.  doi: 10.1137/0119060. [35] I. Sawashima, On spectral properties of some positive operators, Natural Sci. Rep. Ochanomizu Univ., 15 (1964), 53-64. [36] D. Schenzle, An age-structured model of pre-and post-vaccination measles transmission, Math. Med. Biol. JIMA, 1 (1984), 169-191. [37] H. R. Thieme, Renewal theorems for linear periodic Volterra integral equations, J. Integral Equ., 7 (1984), 253-277. [38] D. W. Tudor, An age-dependent epidemic model with application to measles, Math. Biosci., 73 (1985), 131-147.  doi: 10.1016/0025-5564(85)90081-1. [39] W. Wang and X.-Q. Zhao, Threshold dynamics for compartmental epidemic models in periodic environments, J. Dyn. Differ. Equ., 20 (2008), 699-717.  doi: 10.1007/s10884-008-9111-8.
Behavior of the model when $\mathscr{R}_0<1$: (a) Total exposed population $\int_{0}^{80} e(t , a)da$ versus time $t$; (b) Total infected population $\int_{0}^{80} i(t , a)da$ versus time $t.$
Behavior of solutions when $\mathscr{R}_0>1$ and the vaccination rate is zero: (a) Total susceptible population $\int_{0}^{80} s(t , a)da$ versus time t; (b) Total exposed population $\int_{0}^{80} e(t , a)da$ versus time t; (c) Total infected population $\int_{0}^{80} i(t , a)da$ versus time $t.$
Effect of vaccination on the behavior of the solutions (the total exposed population $\int_{0}^{80} e(t , a)da$ and the total infected population $\int_{0}^{80} i(t , a)da$ versus time $t$) and different vaccination rate $\rho$: (a) $\rho = 0$; (b) $\rho = 0.5$; (c) $\rho = 0.7$; (d) $\rho = 0.9$. All other parameters are fixed
Plots of the infected population $i(t, a)$ and exposed population $e(t, a)$ versus age $a$ and time $t$ (in three periods)
Age distribution of the infected population at the peak of a periodic solution ($t = 60.3$)
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