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January 2019, 18(1): 15-32. doi: 10.3934/cpaa.2019002

Global stability and uniform persistence for an infection load-structured SI model with exponential growth velocity

UMR CNRS 6249 Chrono-environnement, 16 route de Gray, F-25030 Besançon cedex FRANCE, University Bourgogne Franche-Comté

The author would like to thank the reviewers whose remarks and suggestions greatly improved the manuscript

Received  June 2017 Revised  January 2018 Published  August 2018

In this article is perfomed a global stability analysis of an infection load-structured epidemic model using tools of dynamical systems theory. An explicit Duhamel formulation of the semiflow allows us to prove the existence of a compact attractor for the trajectories of the system. Then, according to the sharp threshold $\mathcal R_0$, the basic reproduction number of the disease, we make explicit the basins of attractions of the equilibria of the system and prove their global stability with respect to these basins, the attractivness property being obtained using infinite dimensional Lyapunov functions.

Citation: 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
References:
[1]

R. M. Anderson and R. M. May, Infectious Diseases of Humans: Dynamics and Control, Oxford University Press, 1991.

[2]

O. ArinoA. BertuzziA. GandolfiE. Sánchez and C. Sinisgalli, The asynchronous exponential growth property in a model for the kinetic heterogeneity of tumour cell populations, Journal of Mathematical Analysis and Applications, 302 (2005), 521-542. doi: 10.1016/j.jmaa.2004.08.024.

[3]

O. ArinoE. Sánchez and G. F. Webb, Necessary and sufficient conditions for asynchronous exponential growth in age structured cell populations with quiescence, Journal of Mathematical Analysis and Applications, 215 (1997), 499-513. doi: 10.1006/jmaa.1997.5654.

[4]

D. BicharaA. Iggidr and G. Sallet, Global analysis of multi-strains SIS, SIR and MSIR epidemic models, J. Appl. Math. Comput., 44 (2014), 273-292. doi: 10.1007/s12190-013-0693-x.

[5]

O. Diekmann and J. A. P. Heesterbeek, Mathematical Epidemiology of Infectious Diseases, Wiley Series in Mathematical and Computational Biology, John Wiley & Sons, 2000.

[6]

X. DuanS. YuanZ. Qiu and J. Ma, Global stability of an SVEIR epidemic model with ages of vaccination and latency, Comput. Math. Appl., 68 (2014), 288-308. doi: 10.1016/j.camwa.2014.06.002.

[7]

Janet DysonRosanna Villella-Bressan and G. F. Webb, Asynchronous exponential growth in an age structured population of proliferating and quiescent cells, Mathematical Biosciences, 177-178 (2002), 73-83. doi: 10.1016/S0025-5564(01)00097-9.

[8]

A. FallA. IggidrG. Sallet and J. J. Tewa, Epidemiological models and Lyapunov functions, Math. Model. Nat. Phenom., 2 (2007), 55-73. doi: 10.1051/mmnp:2008011.

[9]

Jozsef Z. Farkas, Note on asynchronous exponential growth for structured population models, Nonlinear Analysis: Theory, Methods & Applications, 67 (2007), 618-622. doi: 10.1016/j.na.2006.06.016.

[10]

J. K. Hale, Asymptotic Behavior of Dissipative Systems, volume 25 of Mathematical Surveys and Monographs, American Mathematical Society, Providence, RI, 1988.

[11]

J. K. Hale and P. Waltman, Persistence in infinite-dimensional systems, SIAM J. Math. Anal., 20 (1989), 388-395. doi: 10.1137/0520025.

[12]

G. HuangX. Liu and Y. Takeuchi, Lyapunov functions and global stability for age-structured HIV infection model, SIAM J. Appl. Math., 72 (2012), 25-38. doi: 10.1137/110826588.

[13]

W. O. Kermack and A. G. McKendrick, A contribution to the mathematical theory of epidemics, Proc. R. Soc. Lond. Ser. A, 115 (1927), 700-721.

[14]

W. O. Kermack and A. G. McKendrick, Contributions to the mathematical theory of epidemics: 2, Proc. R. Soc. Lond. Ser. B, 138 (1932), 55-83.

[15]

W. O. Kermack and A. G. McKendrick, Contributions to the mathematical theory of epidemics: 3, Proc. R. Soc. Lond. Ser. B, 141 (1933), 94-112.

[16]

A. Korobeinikov, Lyapunov functions and global stability for SIR and SIRS epidemiological models with non-linear transmission, Bull. Math. Biol., 68 (2006), 615-626. doi: 10.1007/s11538-005-9037-9.

[17]

A. Korobeinikov and G. C. Wake, Lyapunov functions and global stability for SIR, SIRS, and SIS epidemiological models, Appl. Math. Lett., 15 (2002), 955-960. doi: 10.1016/S0893-9659(02)00069-1.

[18]

B. Laroche and A. Perasso, Threshold behaviour of a SI epidemiological model with two structuring variables, J. Evol. Equ., 16 (2016), 293-315. doi: 10.1007/s00028-015-0303-5.

[19]

P. Magal and C. C. McCluskey, Two group infection age model: an application to nosocomial infection, SIAM J. Appl. Math., 73 (2013), 1058-1095. doi: 10.1137/120882056.

[20]

P. MagalC. C. McCluskey and G. F. Webb, Lyapunov functional and global asymptotic stability for an infection-age model, Appl. Anal., 89 (2010), 1109-1140. doi: 10.1080/00036810903208122.

[21]

A. V. Melnik and A. Korobeinikov, Lyapunov functions and global stability for SIR and SEIR models with age-dependent susceptibility, Math. Biosci. Eng., 10 (2013), 369-378.

[22]

K. MischaikowH. Smith and H. Thieme, Asymptotically autonomous semiflows: chain recurrence and Lyapunov functions, Trans. Amer. Math. Soc., 347 (1995), 1669-1685. doi: 10.2307/2154964.

[23]

R. Peralta, C. Vargas-De-León and P. Miramontes, Global stability results in a SVIR epidemic model with immunity loss rate depending on the vaccine-age, Abstr. Appl. Anal., pages Art. ID 341854, 8, 2015. doi: 10.1155/2015/341854.

[24]

A. Perasso and B. Laroche, Well-posedness of an epidemiological problem described by an evolution PDE, ESAIM: Proc., 25 (2008), 29-43. doi: 10.1051/proc:082503.

[25]

A. PerassoB. LarocheY. Chitour and S. Touzeau, Identifiability analysis of an epidemiological model in a structured population, J. Math. Anal. Appl., 374 (2011), 154-165. doi: 10.1016/j.jmaa.2010.08.072.

[26]

A. Perasso and U. Razafison, Infection load structured si model with exponential velocity and external source of contamination, In Proceedings of the World Congress on Engineering (WCE), volume 1, pages 263-267, 2013.

[27]

A. Perasso and U. Razafison, Asymptotic behavior and numerical simulations for an infection load-structured epidemiological models; application to the transmission of prion pathologies, SIAM J. Appl. Math., 74 (2014), 1571-1597. doi: 10.1137/130946058.

[28]

H. L. Smith, Monotone Dynamical Systems, volume 41 of Mathematical Surveys and Monographs, American Mathematical Society, Providence, RI, 1995, An introduction to the theory of competitive and cooperative systems.

[29]

H. L. Smith and H. R. Thieme, Dynamical Systems and Population Persistence, Graduate Studies in Mathematics 118, American Mathematical Society, 2011.

[30]

H. R. Thieme and C. Castillo-Chavez, How may infection-age-dependent infectivity affect the dynamics of hiv/aids? SIAM J. Appl. Math., 53 (1993), 1447-1479. doi: 10.1137/0153068.

[31]

C. Vargas-De-LeónE. Lourdes and A. Korobeinikov, Age-dependency in host-vector models: the global analysis, Appl. Math. Comput., 243 (2014), 969-981. doi: 10.1016/j.amc.2014.06.042.

[32]

J. A. Walker, Dynamical Systems and Evolution Equations, volume 20 of Mathematical Concepts and Methods in Science and Engineering, Plenum Press, New York-London, 1980, Theory and applications.

[33]

G. F. Webb, Theory of Nonlinear Age-Dependent Population Dynamics, Marcel Dekker, New York, 1985.

[34]

G. F. Webb, An operator-theoretic formulation of asynchronous exponential growth, Trans. Amer. Math. Soc., 303 (1987), 751-763. doi: 10.2307/2000695.

[35]

G. F. Webb, Population models structured by age, size, and spatial position, In Structured Population Models in Biology and Epidemiology, Lecture Notes in Math. 1936, pages 1-49. Springer, Berlin, 2008. doi: 10.1007/978-3-540-78273-5_1.

show all references

References:
[1]

R. M. Anderson and R. M. May, Infectious Diseases of Humans: Dynamics and Control, Oxford University Press, 1991.

[2]

O. ArinoA. BertuzziA. GandolfiE. Sánchez and C. Sinisgalli, The asynchronous exponential growth property in a model for the kinetic heterogeneity of tumour cell populations, Journal of Mathematical Analysis and Applications, 302 (2005), 521-542. doi: 10.1016/j.jmaa.2004.08.024.

[3]

O. ArinoE. Sánchez and G. F. Webb, Necessary and sufficient conditions for asynchronous exponential growth in age structured cell populations with quiescence, Journal of Mathematical Analysis and Applications, 215 (1997), 499-513. doi: 10.1006/jmaa.1997.5654.

[4]

D. BicharaA. Iggidr and G. Sallet, Global analysis of multi-strains SIS, SIR and MSIR epidemic models, J. Appl. Math. Comput., 44 (2014), 273-292. doi: 10.1007/s12190-013-0693-x.

[5]

O. Diekmann and J. A. P. Heesterbeek, Mathematical Epidemiology of Infectious Diseases, Wiley Series in Mathematical and Computational Biology, John Wiley & Sons, 2000.

[6]

X. DuanS. YuanZ. Qiu and J. Ma, Global stability of an SVEIR epidemic model with ages of vaccination and latency, Comput. Math. Appl., 68 (2014), 288-308. doi: 10.1016/j.camwa.2014.06.002.

[7]

Janet DysonRosanna Villella-Bressan and G. F. Webb, Asynchronous exponential growth in an age structured population of proliferating and quiescent cells, Mathematical Biosciences, 177-178 (2002), 73-83. doi: 10.1016/S0025-5564(01)00097-9.

[8]

A. FallA. IggidrG. Sallet and J. J. Tewa, Epidemiological models and Lyapunov functions, Math. Model. Nat. Phenom., 2 (2007), 55-73. doi: 10.1051/mmnp:2008011.

[9]

Jozsef Z. Farkas, Note on asynchronous exponential growth for structured population models, Nonlinear Analysis: Theory, Methods & Applications, 67 (2007), 618-622. doi: 10.1016/j.na.2006.06.016.

[10]

J. K. Hale, Asymptotic Behavior of Dissipative Systems, volume 25 of Mathematical Surveys and Monographs, American Mathematical Society, Providence, RI, 1988.

[11]

J. K. Hale and P. Waltman, Persistence in infinite-dimensional systems, SIAM J. Math. Anal., 20 (1989), 388-395. doi: 10.1137/0520025.

[12]

G. HuangX. Liu and Y. Takeuchi, Lyapunov functions and global stability for age-structured HIV infection model, SIAM J. Appl. Math., 72 (2012), 25-38. doi: 10.1137/110826588.

[13]

W. O. Kermack and A. G. McKendrick, A contribution to the mathematical theory of epidemics, Proc. R. Soc. Lond. Ser. A, 115 (1927), 700-721.

[14]

W. O. Kermack and A. G. McKendrick, Contributions to the mathematical theory of epidemics: 2, Proc. R. Soc. Lond. Ser. B, 138 (1932), 55-83.

[15]

W. O. Kermack and A. G. McKendrick, Contributions to the mathematical theory of epidemics: 3, Proc. R. Soc. Lond. Ser. B, 141 (1933), 94-112.

[16]

A. Korobeinikov, Lyapunov functions and global stability for SIR and SIRS epidemiological models with non-linear transmission, Bull. Math. Biol., 68 (2006), 615-626. doi: 10.1007/s11538-005-9037-9.

[17]

A. Korobeinikov and G. C. Wake, Lyapunov functions and global stability for SIR, SIRS, and SIS epidemiological models, Appl. Math. Lett., 15 (2002), 955-960. doi: 10.1016/S0893-9659(02)00069-1.

[18]

B. Laroche and A. Perasso, Threshold behaviour of a SI epidemiological model with two structuring variables, J. Evol. Equ., 16 (2016), 293-315. doi: 10.1007/s00028-015-0303-5.

[19]

P. Magal and C. C. McCluskey, Two group infection age model: an application to nosocomial infection, SIAM J. Appl. Math., 73 (2013), 1058-1095. doi: 10.1137/120882056.

[20]

P. MagalC. C. McCluskey and G. F. Webb, Lyapunov functional and global asymptotic stability for an infection-age model, Appl. Anal., 89 (2010), 1109-1140. doi: 10.1080/00036810903208122.

[21]

A. V. Melnik and A. Korobeinikov, Lyapunov functions and global stability for SIR and SEIR models with age-dependent susceptibility, Math. Biosci. Eng., 10 (2013), 369-378.

[22]

K. MischaikowH. Smith and H. Thieme, Asymptotically autonomous semiflows: chain recurrence and Lyapunov functions, Trans. Amer. Math. Soc., 347 (1995), 1669-1685. doi: 10.2307/2154964.

[23]

R. Peralta, C. Vargas-De-León and P. Miramontes, Global stability results in a SVIR epidemic model with immunity loss rate depending on the vaccine-age, Abstr. Appl. Anal., pages Art. ID 341854, 8, 2015. doi: 10.1155/2015/341854.

[24]

A. Perasso and B. Laroche, Well-posedness of an epidemiological problem described by an evolution PDE, ESAIM: Proc., 25 (2008), 29-43. doi: 10.1051/proc:082503.

[25]

A. PerassoB. LarocheY. Chitour and S. Touzeau, Identifiability analysis of an epidemiological model in a structured population, J. Math. Anal. Appl., 374 (2011), 154-165. doi: 10.1016/j.jmaa.2010.08.072.

[26]

A. Perasso and U. Razafison, Infection load structured si model with exponential velocity and external source of contamination, In Proceedings of the World Congress on Engineering (WCE), volume 1, pages 263-267, 2013.

[27]

A. Perasso and U. Razafison, Asymptotic behavior and numerical simulations for an infection load-structured epidemiological models; application to the transmission of prion pathologies, SIAM J. Appl. Math., 74 (2014), 1571-1597. doi: 10.1137/130946058.

[28]

H. L. Smith, Monotone Dynamical Systems, volume 41 of Mathematical Surveys and Monographs, American Mathematical Society, Providence, RI, 1995, An introduction to the theory of competitive and cooperative systems.

[29]

H. L. Smith and H. R. Thieme, Dynamical Systems and Population Persistence, Graduate Studies in Mathematics 118, American Mathematical Society, 2011.

[30]

H. R. Thieme and C. Castillo-Chavez, How may infection-age-dependent infectivity affect the dynamics of hiv/aids? SIAM J. Appl. Math., 53 (1993), 1447-1479. doi: 10.1137/0153068.

[31]

C. Vargas-De-LeónE. Lourdes and A. Korobeinikov, Age-dependency in host-vector models: the global analysis, Appl. Math. Comput., 243 (2014), 969-981. doi: 10.1016/j.amc.2014.06.042.

[32]

J. A. Walker, Dynamical Systems and Evolution Equations, volume 20 of Mathematical Concepts and Methods in Science and Engineering, Plenum Press, New York-London, 1980, Theory and applications.

[33]

G. F. Webb, Theory of Nonlinear Age-Dependent Population Dynamics, Marcel Dekker, New York, 1985.

[34]

G. F. Webb, An operator-theoretic formulation of asynchronous exponential growth, Trans. Amer. Math. Soc., 303 (1987), 751-763. doi: 10.2307/2000695.

[35]

G. F. Webb, Population models structured by age, size, and spatial position, In Structured Population Models in Biology and Epidemiology, Lecture Notes in Math. 1936, pages 1-49. Springer, Berlin, 2008. doi: 10.1007/978-3-540-78273-5_1.

Figure 1.  Three examples of shapes of function $\beta$ - left : $i_0 < \underline i $ and ${\bar i} = +\infty$; right : $i_0 < \underline i $ and ${\bar i} < +\infty$; down : $i_0 = \underline i $ and ${\bar i} < +\infty$
Table 1.  Parameters involved in the model
Parameter definition symbol
recruitment flux $ \gamma $
minimal infection load $i_0$
basic mortality rate $\mu _0$
disease mortality rate $\mu$
horizontal transmission rate $\beta$
infection load velocity $\nu$
infection load distribution at contamination $ \Phi$
Parameter definition symbol
recruitment flux $ \gamma $
minimal infection load $i_0$
basic mortality rate $\mu _0$
disease mortality rate $\mu$
horizontal transmission rate $\beta$
infection load velocity $\nu$
infection load distribution at contamination $ \Phi$
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