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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 & 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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5]

D. Bentil and J. Murray, Modelling bovine tuberculosis in badgers, J. Anim. Ecol., 239–250. Google Scholar

[6]

H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Springer Science and Business Media, New York, 2010.  Google Scholar

[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.  Google Scholar

[8]

S. N. BusenbergM. 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.  Google Scholar

[9]

Y. ChaM. 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.  Google Scholar

[10]

D. J. EarnP. RohaniB. M. Bolker and B. T. Grenfell, A simple model for complex dynamical transitions in epidemics, Science, 287 (2000), 667-670.   Google Scholar

[11]

Z. FengW. 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.  Google Scholar

[12]

D. Greenhalgh, Analytical results on the stability of age-structured recurrent epidemic models, Math. Med. Biol. JIMA, 4 (1987), 109-144.   Google Scholar

[13]

H. W. Hethcote, Optimal ages of vaccination for measles, Math. Biosci., 89 (1988), 29-52.  doi: 10.1016/0025-5564(88)90111-3.  Google Scholar

[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.  Google Scholar

[15]

F. Hoppensteadt, An age dependent epidemic model, J. Franklin Inst., 297 (1974), 325-333.   Google Scholar

[16]

J. HuangS. RuanX. Wu and X. Zhou, Seasonal transmission dynamics of measles in China, Theory Biosci., 137 (2018), 185-195.  doi: 10.1007/s11538-020-00747-6.  Google Scholar

[17]

M. Iannelli, Mathematical Theory of Age-Structured Population Dynamics, Giardini editori e stampatori, Pisa, 1995. Google Scholar

[18]

M. IannelliM. 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.  Google Scholar

[19]

H. Inaba, Threshold and stability results for an age-structured epidemic model, J. Math. Biol., 28 (1990), 411-434.  doi: 10.1007/BF00178326.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[23]

H. Inaba, Age-structured Population Dynamics in Demography and Epidemiology, Springer, New York, 2017. doi: 10.1007/978-981-10-0188-8.  Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[27]

M. A. Krasnoselskii, Positive Solutions of Operator Equations, Noordhoff, Groningen, 1964.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[33]

X.-Z. LiG. 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.  Google Scholar

[34]

I. Marek, Frobenius theory of positive operators: Comparison theorems and applications, SIAM J. Appl. Math., 19 (1970), 607-628.  doi: 10.1137/0119060.  Google Scholar

[35]

I. Sawashima, On spectral properties of some positive operators, Natural Sci. Rep. Ochanomizu Univ., 15 (1964), 53-64.   Google Scholar

[36]

D. Schenzle, An age-structured model of pre-and post-vaccination measles transmission, Math. Med. Biol. JIMA, 1 (1984), 169-191.   Google Scholar

[37]

H. R. Thieme, Renewal theorems for linear periodic Volterra integral equations, J. Integral Equ., 7 (1984), 253-277.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5]

D. Bentil and J. Murray, Modelling bovine tuberculosis in badgers, J. Anim. Ecol., 239–250. Google Scholar

[6]

H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Springer Science and Business Media, New York, 2010.  Google Scholar

[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.  Google Scholar

[8]

S. N. BusenbergM. 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.  Google Scholar

[9]

Y. ChaM. 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.  Google Scholar

[10]

D. J. EarnP. RohaniB. M. Bolker and B. T. Grenfell, A simple model for complex dynamical transitions in epidemics, Science, 287 (2000), 667-670.   Google Scholar

[11]

Z. FengW. 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.  Google Scholar

[12]

D. Greenhalgh, Analytical results on the stability of age-structured recurrent epidemic models, Math. Med. Biol. JIMA, 4 (1987), 109-144.   Google Scholar

[13]

H. W. Hethcote, Optimal ages of vaccination for measles, Math. Biosci., 89 (1988), 29-52.  doi: 10.1016/0025-5564(88)90111-3.  Google Scholar

[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.  Google Scholar

[15]

F. Hoppensteadt, An age dependent epidemic model, J. Franklin Inst., 297 (1974), 325-333.   Google Scholar

[16]

J. HuangS. RuanX. Wu and X. Zhou, Seasonal transmission dynamics of measles in China, Theory Biosci., 137 (2018), 185-195.  doi: 10.1007/s11538-020-00747-6.  Google Scholar

[17]

M. Iannelli, Mathematical Theory of Age-Structured Population Dynamics, Giardini editori e stampatori, Pisa, 1995. Google Scholar

[18]

M. IannelliM. 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.  Google Scholar

[19]

H. Inaba, Threshold and stability results for an age-structured epidemic model, J. Math. Biol., 28 (1990), 411-434.  doi: 10.1007/BF00178326.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[23]

H. Inaba, Age-structured Population Dynamics in Demography and Epidemiology, Springer, New York, 2017. doi: 10.1007/978-981-10-0188-8.  Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[27]

M. A. Krasnoselskii, Positive Solutions of Operator Equations, Noordhoff, Groningen, 1964.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[33]

X.-Z. LiG. 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.  Google Scholar

[34]

I. Marek, Frobenius theory of positive operators: Comparison theorems and applications, SIAM J. Appl. Math., 19 (1970), 607-628.  doi: 10.1137/0119060.  Google Scholar

[35]

I. Sawashima, On spectral properties of some positive operators, Natural Sci. Rep. Ochanomizu Univ., 15 (1964), 53-64.   Google Scholar

[36]

D. Schenzle, An age-structured model of pre-and post-vaccination measles transmission, Math. Med. Biol. JIMA, 1 (1984), 169-191.   Google Scholar

[37]

H. R. Thieme, Renewal theorems for linear periodic Volterra integral equations, J. Integral Equ., 7 (1984), 253-277.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

Figure 1.  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. $
Figure 2.  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. $
Figure 3.  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
Figure 4.  Plots of the infected population $ i(t, a) $ and exposed population $ e(t, a) $ versus age $ a $ and time $ t $ (in three periods)
Figure 5.  Age distribution of the infected population at the peak of a periodic solution ($ t = 60.3 $)
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