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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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

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

[11]

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

[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.  Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[29]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[33]

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

[34]

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

[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.  Google Scholar

show all references

References:
[1]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

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

[11]

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

[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.  Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[29]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[33]

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

[34]

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

[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.  Google Scholar

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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