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

Mamadou L. Diagne; Ousmane Seydi; Aissata A. B. Sy

doi:10.3934/dcdsb.2019202

Previous Article
The operating diagram for a model of competition in a chemostat with an external lethal inhibitor
DCDS-B Home
This Issue
Next Article
Takens–Bogdanov singularity for age structured models

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

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

Mamadou L. Diagne ^1, , Ousmane Seydi ^2,, and Aissata A. B. Sy ^1,

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.

Keywords: Age of infection, periodic solution, uniform persistence, global stability, coexistence.

Mathematics Subject Classification: Primary: 92D30, 34K20; Secondary: 34K13.

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]	Carlota Rebelo, Alessandro Margheri, Nicolas Bacaër. Persistence in some periodic epidemic models with infection age or constant periods of infection. Discrete & Continuous Dynamical Systems - B, 2014, 19 (4) : 1155-1170. doi: 10.3934/dcdsb.2014.19.1155
[2]	Antoine Perasso. Global stability and uniform persistence for an infection load-structured SI model with exponential growth velocity. Communications on Pure & Applied Analysis, 2019, 18 (1) : 15-32. doi: 10.3934/cpaa.2019002
[3]	Yuming Chen, Junyuan Yang, Fengqin Zhang. The global stability of an SIRS model with infection age. Mathematical Biosciences & Engineering, 2014, 11 (3) : 449-469. doi: 10.3934/mbe.2014.11.449
[4]	Kaifa Wang, Aijun Fan. Uniform persistence and periodic solution of chemostat-type model with antibiotic. Discrete & Continuous Dynamical Systems - B, 2004, 4 (3) : 789-795. doi: 10.3934/dcdsb.2004.4.789
[5]	Kazuo Yamazaki, Xueying Wang. Global stability and uniform persistence of the reaction-convection-diffusion cholera epidemic model. Mathematical Biosciences & Engineering, 2017, 14 (2) : 559-579. doi: 10.3934/mbe.2017033
[6]	Yu Yang, Shigui Ruan, Dongmei Xiao. Global stability of an age-structured virus dynamics model with Beddington-DeAngelis infection function. Mathematical Biosciences & Engineering, 2015, 12 (4) : 859-877. doi: 10.3934/mbe.2015.12.859
[7]	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
[8]	Yan-Xia Dang, Zhi-Peng Qiu, Xue-Zhi Li, Maia Martcheva. Global dynamics of a vector-host epidemic model with age of infection. Mathematical Biosciences & Engineering, 2017, 14 (5&6) : 1159-1186. doi: 10.3934/mbe.2017060
[9]	Jianxin Yang, Zhipeng Qiu, Xue-Zhi Li. Global stability of an age-structured cholera model. Mathematical Biosciences & Engineering, 2014, 11 (3) : 641-665. doi: 10.3934/mbe.2014.11.641
[10]	Liang Kong, Tung Nguyen, Wenxian Shen. Effects of localized spatial variations on the uniform persistence and spreading speeds of time periodic two species competition systems. Communications on Pure & Applied Analysis, 2019, 18 (4) : 1613-1636. doi: 10.3934/cpaa.2019077
[11]	Hal L. Smith, Horst R. Thieme. Persistence and global stability for a class of discrete time structured population models. Discrete & Continuous Dynamical Systems - A, 2013, 33 (10) : 4627-4646. doi: 10.3934/dcds.2013.33.4627
[12]	Jinliang Wang, Jiying Lang, Yuming Chen. Global dynamics of an age-structured HIV infection model incorporating latency and cell-to-cell transmission. Discrete & Continuous Dynamical Systems - B, 2017, 22 (10) : 3721-3747. doi: 10.3934/dcdsb.2017186
[13]	Yicang Zhou, Zhien Ma. Global stability of a class of discrete age-structured SIS models with immigration. Mathematical Biosciences & Engineering, 2009, 6 (2) : 409-425. doi: 10.3934/mbe.2009.6.409
[14]	Geni Gupur, Xue-Zhi Li. Global stability of an age-structured SIRS epidemic model with vaccination. Discrete & Continuous Dynamical Systems - B, 2004, 4 (3) : 643-652. doi: 10.3934/dcdsb.2004.4.643
[15]	C. Connell McCluskey. Global stability for an $SEI$ model of infectious disease with age structure and immigration of infecteds. Mathematical Biosciences & Engineering, 2016, 13 (2) : 381-400. doi: 10.3934/mbe.2015008
[16]	Jinliang Wang, Xiu Dong. Analysis of an HIV infection model incorporating latency age and infection age. Mathematical Biosciences & Engineering, 2018, 15 (3) : 569-594. doi: 10.3934/mbe.2018026
[17]	Bao-Zhu Guo, Li-Ming Cai. A note for the global stability of a delay differential equation of hepatitis B virus infection. Mathematical Biosciences & Engineering, 2011, 8 (3) : 689-694. doi: 10.3934/mbe.2011.8.689
[18]	Yu Ji. Global stability of a multiple delayed viral infection model with general incidence rate and an application to HIV infection. Mathematical Biosciences & Engineering, 2015, 12 (3) : 525-536. doi: 10.3934/mbe.2015.12.525
[19]	Chunqing Wu, Patricia J.Y. Wong. Global asymptotical stability of the coexistence fixed point of a Ricker-type competitive model. Discrete & Continuous Dynamical Systems - B, 2015, 20 (9) : 3255-3266. doi: 10.3934/dcdsb.2015.20.3255
[20]	Rafael Luís, Sandra Mendonça. A note on global stability in the periodic logistic map. Discrete & Continuous Dynamical Systems - B, 2020, 25 (11) : 4211-4220. doi: 10.3934/dcdsb.2020094

American Institute of Mathematical Sciences

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

References:

References:

Tools

Metrics

Other articles
by authors

Tools

Tools

Metrics

Other articles
by authors

American Institute of Mathematical Sciences

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

References:

References:

Tools

Metrics

Other articlesby authors

Tools

Tools

Metrics

Other articlesby authors

Other articles
by authors

Other articles
by authors