# American Institute of Mathematical Sciences

June  2020, 25(6): 2057-2092. doi: 10.3934/dcdsb.2019202

## A two-group age of infection epidemic model with periodic behavioral changes

 1 University of Thies, Senegal 2 Polytechnic School of Thies, Senegal

* Corresponding author: Ousmane Seydi

Received  January 2019 Published  September 2019

Fund Project: All the authors are supported by the CEA-MITIC (Senegal)

In this paper we propose a two-group SIR age of infection epidemic model by incorporating periodical behavioral changes for both susceptible and infected individuals. Our model allows different incubation periods for the two groups. It is proved in this paper that the persistence and extinction of the disease are determined by a threshold condition given in term of the basic reproductive number $R_0$. That is, the disease is uniformly persistent if $R_0 >1$ with the existence of a positive periodic solution, while the disease goes to extinction if $R_0< 1$ with the global asymptotic stability of the disease free periodic solution. The model we have proposed is general and can be applied to a wide class of diseases.

Citation: Mamadou L. Diagne, Ousmane Seydi, Aissata A. B. Sy. A two-group age of infection epidemic model with periodic behavioral changes. Discrete & Continuous Dynamical Systems - B, 2020, 25 (6) : 2057-2092. doi: 10.3934/dcdsb.2019202
##### References:
 [1] P. Auger, P. Magal and S. Ruan, Structured Population Models in Biology and Epidemiology, Lecture Notes in Mathematics, 1936. Mathematical Biosciences Subseries. Springer-Verlag, Berlin, 2008. doi: 10.1007/978-3-540-78273-5.  Google Scholar [2] N. Bacaër and S. Guernaoui, The epidemic threshold of vector-borne diseases with seasonality: The case of cutaneous leishmaniasis in Chichaoua, Journal of Mathematical Biology, 53 (2006), 421-436.  doi: 10.1007/s00285-006-0015-0.  Google Scholar [3] Z. G. Bai, Threshold dynamics of a time-delayed SEIRS model with pulse vaccination, Mathematical biosciences, 269 (2015), 178-185.  doi: 10.1016/j.mbs.2015.09.005.  Google Scholar [4] K. Deimling, Nonlinear Functional Analysis, Springer-Verlag, Berlin, 1985. doi: 10.1007/978-3-662-00547-7.  Google Scholar [5] S. Funk, M. Salathé and V. A. A. Jansen, Modelling the influence of human behaviour on the spread of infectious diseases: A review, Journal of the Royal Society Interface, 7 (2010), 1247-1256.  doi: 10.1098/rsif.2010.0142.  Google Scholar [6] J. K. Hale, Asymptotic Behavior of Dissipative Systems, Mathematical Surveys and Monographs, 25. American Mathematical Society, Providence, RI, 1988.  Google Scholar [7] M. Iannelli, Mathematical Theory of Age-Structured Population Dynamics, Applied Mathematics Monograph, Giardini editori e stampatori, 1995. Google Scholar [8] H. Inaba, Age-Structured Population Dynamics in Demography and Epidemiology, Springer-Verlag, Singapore, 2017. doi: 10.1007/978-981-10-0188-8.  Google Scholar [9] H. Inaba, The Malthusian parameter and $R_0$ for heterogeneous populations in periodic environments, Mathematical Biosciences and Engineering, 9 (2012), 313-346.  doi: 10.3934/mbe.2012.9.313.  Google Scholar [10] H. Inaba, Weak ergodicity of population evolution processes, Mathematical Biosciences, 96 (1989), 195-219.  doi: 10.1016/0025-5564(89)90059-X.  Google Scholar [11] K. H. Liu, Y. j. Lou and J. H. Wu, Analysis of an age structured model for tick populations subject to seasonal effects, Journal of Differential Equations, 263 (2017), 2078-2112.  doi: 10.1016/j.jde.2017.03.038.  Google Scholar [12] P. Magal, Perturbation of a globally stable steady state and uniform persistence, Journal of Dynamics and Differential Equations, 21 (2009), 1-20.  doi: 10.1007/s10884-008-9127-0.  Google Scholar [13] P. Magal and O. Arino, Existence of periodic solutions for a state dependent delay differential equation, Journal of Differential Equations, 165 (2000), 61-95.  doi: 10.1006/jdeq.1999.3759.  Google Scholar [14] P. Magal and C. McCluskey, Two group infection age model: An application to nosocomial infection, SIAM Journal on Applied Mathematics, 73 (2013), 1058-1095.  doi: 10.1137/120882056.  Google Scholar [15] P. Magal and S. G. Ruan, On semilinear Cauchy problems with non-dense domain, Advances in Differential Equations, 14 (2009), 1041-1084.   Google Scholar [16] P. Magal and S. G. Ruan, Theory and Applications of Abstract Semilinear Cauchy Problems, Applied Mathematical Sciences, 201. Springer, Cham, 2018. doi: 10.1007/978-3-030-01506-0.  Google Scholar [17] P. Magal, O. Seydi and F.-B. Wang, Monotone abstract non-densely defined Cauchy problems applied to age structured population dynamic models, J. Math. Anal. Appl., 479 (2019), 450–481, arXiv: math.AP/1901.01231. doi: 10.1016/j.jmaa.2019.06.034.  Google Scholar [18] P. Magal and H. R. Thieme, Eventual compactness for a semiflow generated by an age-structured models, Communications on Pure and Applied Analysis, 3 (2004), 695-727.  doi: 10.3934/cpaa.2004.3.695.  Google Scholar [19] P. Magal and X.-Q. Zhao, Global attractors and steady states for uniformly persistent dynamical systems, SIAM Journal on Mathematical Analysis, 37 (2005), 251-275.  doi: 10.1137/S0036141003439173.  Google Scholar [20] P. Manfredi and A. D'Onofrio, Modeling the Interplay Between Human Behavior and the Spread of Infectious Diseases, Springer-Verlag, New York, 2013. doi: 10.1007/978-1-4614-5474-8.  Google Scholar [21] R. H. Martin, Nonlinear Operators and Differential Equations in Banach Spaces, Pure and Applied Mathematics, Wiley-Interscience, New York-London-Sydney, 1976.  Google Scholar [22] C. McCluskey, Global stability for an SEI model of infectious disease with age structure and immigration of infecteds, Mathematical Biosciences Engineering, 13 (2016), 381-400.  doi: 10.3934/mbe.2015008.  Google Scholar [23] J. A. J. Metz and O. Diekmann, The Dynamics of Physiologically Structured Populations, Lecture Notes in Biomathematics, 68. Springer-Verlag, Berlin, 1986. doi: 10.1007/978-3-662-13159-6.  Google Scholar [24] C. Rebelo, A. Margheri and N. Bacaër, Persistence in some periodic epidemic models with infection age or constant periods of infection, Discrete and Continuous Dynamical Systems-Series B, 19 (2014), 1155-1170.  doi: 10.3934/dcdsb.2014.19.1155.  Google Scholar [25] G. R. Sell and Y. C. You, Dynamics of Evolutionary Equations, Applied Mathematical Sciences, 143. Springer-Verlag, New York, 2002. doi: 10.1007/978-1-4757-5037-9.  Google Scholar [26] H. L. Smith and H. R. Thieme, Dynamical Systems and Population Persistence, Graduate Studies in Mathematics, 118. American Mathematical Society, RI, 2011.  Google Scholar [27] H. L. Smith and P. Waltman, Perturbation of a globally stable steady state, Proceedings of the American Mathematical Society, 127 (1999), 447-453.  doi: 10.1090/S0002-9939-99-04768-1.  Google Scholar [28] H. R. Thieme, Mathematics in Population Biology, Princeton Series in Theoretical and Computational Biology, Princeton University Press, Princeton, NJ, 2003.   Google Scholar [29] H. R. Thieme, Renewal theorems for linear periodic Volterra integral equations, Journal of Integral Equations, 7 (1984), 253-277.   Google Scholar [30] H. R. Thieme, Semiflows generated by Lipschitz perturbations of non-densely defined operators, Differential Integral Equations, 3 (1990), 1035-1066.   Google Scholar [31] H. R. Thieme and I. I. Vrabie, Relatively compact orbits and compact attractors for a class of nonlinear evolution equations, Journal of Dynamics and Differential Equations, 15 (2003), 731-750.  doi: 10.1023/B:JODY.0000010063.69213.7c.  Google Scholar [32] H.-O. Walther, A periodic solution of a differential equation with state-dependent delay, Journal of Differential Equations, 244 (2008), 1910-1945.  doi: 10.1016/j.jde.2008.02.001.  Google Scholar [33] W. D. Wang and X.-Q. Zhao, Threshold dynamics for compartmental epidemic models in periodic environments, Journal of Dynamics and Differential Equations, 20 (2008), 699-717.  doi: 10.1007/s10884-008-9111-8.  Google Scholar [34] G. F. Webb, Theory of Nonlinear Age-Dependent Population Dynamics, Monographs and Textbooks in Pure and Applied Mathematics, 89. Marcel Dekker, Inc., New York, 1985.  Google Scholar [35] S. X. Zhang and H. B. Guo, Global analysis of age-structured multi-stage epidemic models for infectious diseases, Applied Mathematics and Computation, 337 (2018), 214-233.  doi: 10.1016/j.amc.2018.05.020.  Google Scholar [36] L. Zhao, Z.-C. Wang and L. Zhang, Threshold dynamics of a time periodic and two-group epidemic model with distributed delay, Mathematical Biosciences and Engineering, 14 (2017), 1535-1563.  doi: 10.3934/mbe.2017080.  Google Scholar [37] X.-Q. Zhao, Basic reproduction ratios for periodic compartmental models with time delay, Journal of Dynamics and Differential Equations, 29 (2017), 67-82.  doi: 10.1007/s10884-015-9425-2.  Google Scholar [38] X.-Q. Zhao, Dynamical Systems in Population Biology, CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC, 16. Springer-Verlag, New York, 2003. doi: 10.1007/978-0-387-21761-1.  Google Scholar

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##### References:
 [1] P. Auger, P. Magal and S. Ruan, Structured Population Models in Biology and Epidemiology, Lecture Notes in Mathematics, 1936. Mathematical Biosciences Subseries. Springer-Verlag, Berlin, 2008. doi: 10.1007/978-3-540-78273-5.  Google Scholar [2] N. Bacaër and S. Guernaoui, The epidemic threshold of vector-borne diseases with seasonality: The case of cutaneous leishmaniasis in Chichaoua, Journal of Mathematical Biology, 53 (2006), 421-436.  doi: 10.1007/s00285-006-0015-0.  Google Scholar [3] Z. G. Bai, Threshold dynamics of a time-delayed SEIRS model with pulse vaccination, Mathematical biosciences, 269 (2015), 178-185.  doi: 10.1016/j.mbs.2015.09.005.  Google Scholar [4] K. Deimling, Nonlinear Functional Analysis, Springer-Verlag, Berlin, 1985. doi: 10.1007/978-3-662-00547-7.  Google Scholar [5] S. Funk, M. Salathé and V. A. A. Jansen, Modelling the influence of human behaviour on the spread of infectious diseases: A review, Journal of the Royal Society Interface, 7 (2010), 1247-1256.  doi: 10.1098/rsif.2010.0142.  Google Scholar [6] J. K. Hale, Asymptotic Behavior of Dissipative Systems, Mathematical Surveys and Monographs, 25. American Mathematical Society, Providence, RI, 1988.  Google Scholar [7] M. Iannelli, Mathematical Theory of Age-Structured Population Dynamics, Applied Mathematics Monograph, Giardini editori e stampatori, 1995. Google Scholar [8] H. Inaba, Age-Structured Population Dynamics in Demography and Epidemiology, Springer-Verlag, Singapore, 2017. doi: 10.1007/978-981-10-0188-8.  Google Scholar [9] H. Inaba, The Malthusian parameter and $R_0$ for heterogeneous populations in periodic environments, Mathematical Biosciences and Engineering, 9 (2012), 313-346.  doi: 10.3934/mbe.2012.9.313.  Google Scholar [10] H. Inaba, Weak ergodicity of population evolution processes, Mathematical Biosciences, 96 (1989), 195-219.  doi: 10.1016/0025-5564(89)90059-X.  Google Scholar [11] K. H. Liu, Y. j. Lou and J. H. Wu, Analysis of an age structured model for tick populations subject to seasonal effects, Journal of Differential Equations, 263 (2017), 2078-2112.  doi: 10.1016/j.jde.2017.03.038.  Google Scholar [12] P. Magal, Perturbation of a globally stable steady state and uniform persistence, Journal of Dynamics and Differential Equations, 21 (2009), 1-20.  doi: 10.1007/s10884-008-9127-0.  Google Scholar [13] P. Magal and O. Arino, Existence of periodic solutions for a state dependent delay differential equation, Journal of Differential Equations, 165 (2000), 61-95.  doi: 10.1006/jdeq.1999.3759.  Google Scholar [14] P. Magal and C. McCluskey, Two group infection age model: An application to nosocomial infection, SIAM Journal on Applied Mathematics, 73 (2013), 1058-1095.  doi: 10.1137/120882056.  Google Scholar [15] P. Magal and S. G. Ruan, On semilinear Cauchy problems with non-dense domain, Advances in Differential Equations, 14 (2009), 1041-1084.   Google Scholar [16] P. Magal and S. G. Ruan, Theory and Applications of Abstract Semilinear Cauchy Problems, Applied Mathematical Sciences, 201. Springer, Cham, 2018. doi: 10.1007/978-3-030-01506-0.  Google Scholar [17] P. Magal, O. Seydi and F.-B. Wang, Monotone abstract non-densely defined Cauchy problems applied to age structured population dynamic models, J. Math. Anal. Appl., 479 (2019), 450–481, arXiv: math.AP/1901.01231. doi: 10.1016/j.jmaa.2019.06.034.  Google Scholar [18] P. Magal and H. R. Thieme, Eventual compactness for a semiflow generated by an age-structured models, Communications on Pure and Applied Analysis, 3 (2004), 695-727.  doi: 10.3934/cpaa.2004.3.695.  Google Scholar [19] P. Magal and X.-Q. Zhao, Global attractors and steady states for uniformly persistent dynamical systems, SIAM Journal on Mathematical Analysis, 37 (2005), 251-275.  doi: 10.1137/S0036141003439173.  Google Scholar [20] P. Manfredi and A. D'Onofrio, Modeling the Interplay Between Human Behavior and the Spread of Infectious Diseases, Springer-Verlag, New York, 2013. doi: 10.1007/978-1-4614-5474-8.  Google Scholar [21] R. H. Martin, Nonlinear Operators and Differential Equations in Banach Spaces, Pure and Applied Mathematics, Wiley-Interscience, New York-London-Sydney, 1976.  Google Scholar [22] C. McCluskey, Global stability for an SEI model of infectious disease with age structure and immigration of infecteds, Mathematical Biosciences Engineering, 13 (2016), 381-400.  doi: 10.3934/mbe.2015008.  Google Scholar [23] J. A. J. Metz and O. Diekmann, The Dynamics of Physiologically Structured Populations, Lecture Notes in Biomathematics, 68. Springer-Verlag, Berlin, 1986. doi: 10.1007/978-3-662-13159-6.  Google Scholar [24] C. Rebelo, A. Margheri and N. Bacaër, Persistence in some periodic epidemic models with infection age or constant periods of infection, Discrete and Continuous Dynamical Systems-Series B, 19 (2014), 1155-1170.  doi: 10.3934/dcdsb.2014.19.1155.  Google Scholar [25] G. R. Sell and Y. C. You, Dynamics of Evolutionary Equations, Applied Mathematical Sciences, 143. Springer-Verlag, New York, 2002. doi: 10.1007/978-1-4757-5037-9.  Google Scholar [26] H. L. Smith and H. R. Thieme, Dynamical Systems and Population Persistence, Graduate Studies in Mathematics, 118. American Mathematical Society, RI, 2011.  Google Scholar [27] H. L. Smith and P. Waltman, Perturbation of a globally stable steady state, Proceedings of the American Mathematical Society, 127 (1999), 447-453.  doi: 10.1090/S0002-9939-99-04768-1.  Google Scholar [28] H. R. Thieme, Mathematics in Population Biology, Princeton Series in Theoretical and Computational Biology, Princeton University Press, Princeton, NJ, 2003.   Google Scholar [29] H. R. Thieme, Renewal theorems for linear periodic Volterra integral equations, Journal of Integral Equations, 7 (1984), 253-277.   Google Scholar [30] H. R. Thieme, Semiflows generated by Lipschitz perturbations of non-densely defined operators, Differential Integral Equations, 3 (1990), 1035-1066.   Google Scholar [31] H. R. Thieme and I. I. Vrabie, Relatively compact orbits and compact attractors for a class of nonlinear evolution equations, Journal of Dynamics and Differential Equations, 15 (2003), 731-750.  doi: 10.1023/B:JODY.0000010063.69213.7c.  Google Scholar [32] H.-O. Walther, A periodic solution of a differential equation with state-dependent delay, Journal of Differential Equations, 244 (2008), 1910-1945.  doi: 10.1016/j.jde.2008.02.001.  Google Scholar [33] W. D. Wang and X.-Q. Zhao, Threshold dynamics for compartmental epidemic models in periodic environments, Journal of Dynamics and Differential Equations, 20 (2008), 699-717.  doi: 10.1007/s10884-008-9111-8.  Google Scholar [34] G. F. Webb, Theory of Nonlinear Age-Dependent Population Dynamics, Monographs and Textbooks in Pure and Applied Mathematics, 89. Marcel Dekker, Inc., New York, 1985.  Google Scholar [35] S. X. Zhang and H. B. Guo, Global analysis of age-structured multi-stage epidemic models for infectious diseases, Applied Mathematics and Computation, 337 (2018), 214-233.  doi: 10.1016/j.amc.2018.05.020.  Google Scholar [36] L. Zhao, Z.-C. Wang and L. Zhang, Threshold dynamics of a time periodic and two-group epidemic model with distributed delay, Mathematical Biosciences and Engineering, 14 (2017), 1535-1563.  doi: 10.3934/mbe.2017080.  Google Scholar [37] X.-Q. Zhao, Basic reproduction ratios for periodic compartmental models with time delay, Journal of Dynamics and Differential Equations, 29 (2017), 67-82.  doi: 10.1007/s10884-015-9425-2.  Google Scholar [38] X.-Q. Zhao, Dynamical Systems in Population Biology, CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC, 16. Springer-Verlag, New York, 2003. doi: 10.1007/978-0-387-21761-1.  Google Scholar
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