# 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

show all references

##### 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
 [1] Zhihua Liu, Yayun Wu, Xiangming Zhang. Existence of periodic wave trains for an age-structured model with diffusion. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2021009 [2] Hao Wang. Uniform stability estimate for the Vlasov-Poisson-Boltzmann system. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 657-680. doi: 10.3934/dcds.2020292 [3] A. M. Elaiw, N. H. AlShamrani, A. Abdel-Aty, H. Dutta. Stability analysis of a general HIV dynamics model with multi-stages of infected cells and two routes of infection. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020441 [4] Kerioui Nadjah, Abdelouahab Mohammed Salah. Stability and Hopf bifurcation of the coexistence equilibrium for a differential-algebraic biological economic system with predator harvesting. Electronic Research Archive, 2021, 29 (1) : 1641-1660. doi: 10.3934/era.2020084 [5] Yi Guan, Michal Fečkan, Jinrong Wang. Periodic solutions and Hyers-Ulam stability of atmospheric Ekman flows. Discrete & Continuous Dynamical Systems - A, 2021, 41 (3) : 1157-1176. doi: 10.3934/dcds.2020313 [6] Izumi Takagi, Conghui Zhang. Existence and stability of patterns in a reaction-diffusion-ODE system with hysteresis in non-uniform media. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020400 [7] Rong Chen, Shihang Pan, Baoshuai Zhang. Global conservative solutions for a modified periodic coupled Camassa-Holm system. Electronic Research Archive, 2021, 29 (1) : 1691-1708. doi: 10.3934/era.2020087 [8] Vandana Sharma. Global existence and uniform estimates of solutions to reaction diffusion systems with mass transport type boundary conditions. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021001 [9] Ting Liu, Guo-Bao Zhang. Global stability of traveling waves for a spatially discrete diffusion system with time delay. Electronic Research Archive, , () : -. doi: 10.3934/era.2021003 [10] Jiangtao Yang. Permanence, extinction and periodic solution of a stochastic single-species model with Lévy noises. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020371 [11] Laurent Di Menza, Virginie Joanne-Fabre. An age group model for the study of a population of trees. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020464 [12] Yunfeng Geng, Xiaoying Wang, Frithjof Lutscher. Coexistence of competing consumers on a single resource in a hybrid model. Discrete & Continuous Dynamical Systems - B, 2021, 26 (1) : 269-297. doi: 10.3934/dcdsb.2020140 [13] Hirofumi Izuhara, Shunsuke Kobayashi. Spatio-temporal coexistence in the cross-diffusion competition system. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 919-933. doi: 10.3934/dcdss.2020228 [14] Qiang Fu, Xin Guo, Sun Young Jeon, Eric N. Reither, Emma Zang, Kenneth C. Land. The uses and abuses of an age-period-cohort method: On the linear algebra and statistical properties of intrinsic and related estimators. Mathematical Foundations of Computing, 2020  doi: 10.3934/mfc.2021001 [15] Mengyu Cheng, Zhenxin Liu. Periodic, almost periodic and almost automorphic solutions for SPDEs with monotone coefficients. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021026 [16] Julian Tugaut. Captivity of the solution to the granular media equation. Kinetic & Related Models, , () : -. doi: 10.3934/krm.2021002 [17] Yanan Li, Zhijian Yang, Na Feng. Uniform attractors and their continuity for the non-autonomous Kirchhoff wave models. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021018 [18] Yolanda Guerrero–Sánchez, Muhammad Umar, Zulqurnain Sabir, Juan L. G. Guirao, Muhammad Asif Zahoor Raja. Solving a class of biological HIV infection model of latently infected cells using heuristic approach. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020431 [19] Alex P. Farrell, Horst R. Thieme. Predator – Prey/Host – Parasite: A fragile ecoepidemic system under homogeneous infection incidence. Discrete & Continuous Dynamical Systems - B, 2021, 26 (1) : 217-267. doi: 10.3934/dcdsb.2020328 [20] Tin Phan, Bruce Pell, Amy E. Kendig, Elizabeth T. Borer, Yang Kuang. Rich dynamics of a simple delay host-pathogen model of cell-to-cell infection for plant virus. Discrete & Continuous Dynamical Systems - B, 2021, 26 (1) : 515-539. doi: 10.3934/dcdsb.2020261

2019 Impact Factor: 1.27