Persistence in some periodic epidemic models with infection age or constant periods of infection

Previous Article
Multistability and localized attractors in a dissipative discrete NLS equation
DCDS-B Home
This Issue
Next Article
Asymptotic pattern of a migratory and nonmonotone population model

2014, 19(4): 1155-1170. doi: 10.3934/dcdsb.2014.19.1155

Persistence in some periodic epidemic models with infection age or constant periods of infection

Carlota Rebelo ^1, , Alessandro Margheri ^1, and Nicolas Bacaër ^2,

1.	Centro de Matemática e Aplicações Fundamentais, Faculdade de Ciências, Universidade de Lisboa, 1749-016 Lisboa, Portugal, Portugal
2.	IRD and University Paris 6, Research group UMMISCO, Bondy, France

Received June 2013 Revised February 2014 Published April 2014

Abstract
Related Papers

Much recent work has focused on persistence for epidemic models with periodic coefficients. But the case where the infected compartments satisfy a delay differential equation or a partial differential equation does not seem to have been considered so far. The purpose of this paper is to provide a framework for proving persistence in such a case. Some examples are presented, such as a periodic SIR model structured by time since infection and a periodic SIS delay model.

Keywords: seasonality, epidemiological models, delay differential equation., Persistence.

Mathematics Subject Classification: Primary: 92D30, 45J05; Secondary: 54H2.

Citation: 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

References:

[1]	S. Anita, Analysis and control of age-dependent population dynamics,, Kluwer, (2000).
[2]	N. Bacaër, Approximation of the basic reproduction number $R_0$ for vector-borne diseases with a periodic vector population,, Bull. Math. Biol., 69 (2007), 1067. doi: 10.1007/s11538-006-9166-9.
[3]	N. Bacaër and E. Ait Dads, Genealogy with seasonality, the basic reproduction number, and the influenza pandemic,, J. Math. Biol., 62 (2011), 741. doi: 10.1007/s00285-010-0354-8.
[4]	N. Bacaër and S. Guernaoui, The epidemic threshold of vector-borne diseases with seasonality,, J. Math. Biol., 53 (2006), 421. doi: 10.1007/s00285-006-0015-0.
[5]	N. Bacaër and R. Ouifki, Growth rate and basic reproduction number for population models with a simple periodic factor,, Math. Biosci., 210 (2007), 647. doi: 10.1016/j.mbs.2007.07.005.
[6]	C. Castillo-Chavez and H.R. Thieme, How may infection-age dependent infectivity affect the dynamics of HIV/AIDS?,, SIAM J. Appl. Math., 53 (1993), 1447. doi: 10.1137/0153068.
[7]	K. Cooke and J. L. Kaplan, A periodicity threshold theorem for epidemics and population growth,, Math. Biosci., 31 (1976), 87. doi: 10.1016/0025-5564(76)90042-0.
[8]	G. Degla, An overview of semi-continuity results on the spectral radius and positivity,, J. Math. Anal. Appl., 338 (2008), 101. doi: 10.1016/j.jmaa.2007.05.011.
[9]	R. Drnovšek, Bounds for the spectral radius of positive operators,, Comment. Math. Univ. Carolinae, 41 (2000), 459.
[10]	A. Fonda, Uniformly persistent semidynamical systems,, Proc. Amer. Math. Soc., 104 (1988), 111. doi: 10.1090/S0002-9939-1988-0958053-2.
[11]	J. Hale, Dissipation and compact attractors,, J. Dynam. Differ. Equat., 18 (2006), 485. doi: 10.1007/s10884-006-9021-6.
[12]	J. Hale, Asymptotic behavior of dissipative systems,, American Mathematical Society, (1988).
[13]	J. Hofbauer, A unified approach to persistence,, Acta Applicandae Math., 14 (1989), 11. doi: 10.1007/BF00046670.
[14]	J. Hofbauer and J. W. H. So, Uniform persistence and repellors for maps,, Proc. Amer. Math. Soc., 107 (1989), 1137. doi: 10.1090/S0002-9939-1989-0984816-4.
[15]	T. Kato, Perturbation theory for linear operators,, Springer, (1995).
[16]	P. Magal, Compact attractors for time-periodic age-structured population models,, Electron. J. Differ. Equat., 65 (2001), 1.
[17]	P. Magal, C. C. McCluskey and G. F. Webb, Lyapunov functional and global asymptotic stability for an infectious-age model,, Applic. Anal., 89 (2010), 1109. doi: 10.1080/00036810903208122.
[18]	R. D. Nussbaum, Periodic solutions of some integral equations from the theory of epidemics, in Nonlinear systems and applications* (ed. V. Lakshmikantham)*, (1977), 235.
[19]	R. D. Nussbaum, A periodicity threshold theorem for some nonlinear integral equations,, SIAM J. Math. Anal., 9 (1978), 356. doi: 10.1137/0509024.
[20]	C. Rebelo, A. Margheri and N. Bacaër, Persistence in seasonally forced epidemiological models,, J. Math. Biol., 64 (2012), 933. doi: 10.1007/s00285-011-0440-6.
[21]	H. L. Smith, On periodic solutions of a delay integral equation modelling epidemics,, J. Math. Biol., 4 (1977), 69. doi: 10.1007/BF00276353.
[22]	H. L. Smith, An introduction to delay differential equations with applications to the life sciences,, Springer, (2011). doi: 10.1007/978-1-4419-7646-8.
[23]	H. L. Smith and H. R. Thieme, Dynamical systems and population persistence,, American Mathematical Society, (2011).
[24]	H. R. Thieme, Spectral bound and reproduction number for infinite-dimensional population structure and time heterogeneity,, SIAM J. Appl. Math., 70 (2009), 188. doi: 10.1137/080732870.
[25]	H. R. Thieme, Renewal theorems for linear periodic Volterra integral equations,, J. Integral Equat., 7 (1984), 253.
[26]	G. F. Webb, Theory of nonlinear age-dependent population dynamics,, Marcel Dekker, (1985).
[27]	G. F. Webb, E. D'Agata, P. Magal and S. Ruan, A model of antibiotic resistant bacterial epidemics in hospitals,, Proc. Nat. Acad. Sci., 102 (2005), 13343. doi: 10.1073/pnas.0504053102.
[28]	F. Zhang and X. Q. Zhao, A periodic epidemic model in a patchy environment,, J. Math. Anal. Appl., 325 (2007), 496. doi: 10.1016/j.jmaa.2006.01.085.

show all references

References:

[1]	S. Anita, Analysis and control of age-dependent population dynamics,, Kluwer, (2000).
[2]	N. Bacaër, Approximation of the basic reproduction number $R_0$ for vector-borne diseases with a periodic vector population,, Bull. Math. Biol., 69 (2007), 1067. doi: 10.1007/s11538-006-9166-9.
[3]	N. Bacaër and E. Ait Dads, Genealogy with seasonality, the basic reproduction number, and the influenza pandemic,, J. Math. Biol., 62 (2011), 741. doi: 10.1007/s00285-010-0354-8.
[4]	N. Bacaër and S. Guernaoui, The epidemic threshold of vector-borne diseases with seasonality,, J. Math. Biol., 53 (2006), 421. doi: 10.1007/s00285-006-0015-0.
[5]	N. Bacaër and R. Ouifki, Growth rate and basic reproduction number for population models with a simple periodic factor,, Math. Biosci., 210 (2007), 647. doi: 10.1016/j.mbs.2007.07.005.
[6]	C. Castillo-Chavez and H.R. Thieme, How may infection-age dependent infectivity affect the dynamics of HIV/AIDS?,, SIAM J. Appl. Math., 53 (1993), 1447. doi: 10.1137/0153068.
[7]	K. Cooke and J. L. Kaplan, A periodicity threshold theorem for epidemics and population growth,, Math. Biosci., 31 (1976), 87. doi: 10.1016/0025-5564(76)90042-0.
[8]	G. Degla, An overview of semi-continuity results on the spectral radius and positivity,, J. Math. Anal. Appl., 338 (2008), 101. doi: 10.1016/j.jmaa.2007.05.011.
[9]	R. Drnovšek, Bounds for the spectral radius of positive operators,, Comment. Math. Univ. Carolinae, 41 (2000), 459.
[10]	A. Fonda, Uniformly persistent semidynamical systems,, Proc. Amer. Math. Soc., 104 (1988), 111. doi: 10.1090/S0002-9939-1988-0958053-2.
[11]	J. Hale, Dissipation and compact attractors,, J. Dynam. Differ. Equat., 18 (2006), 485. doi: 10.1007/s10884-006-9021-6.
[12]	J. Hale, Asymptotic behavior of dissipative systems,, American Mathematical Society, (1988).
[13]	J. Hofbauer, A unified approach to persistence,, Acta Applicandae Math., 14 (1989), 11. doi: 10.1007/BF00046670.
[14]	J. Hofbauer and J. W. H. So, Uniform persistence and repellors for maps,, Proc. Amer. Math. Soc., 107 (1989), 1137. doi: 10.1090/S0002-9939-1989-0984816-4.
[15]	T. Kato, Perturbation theory for linear operators,, Springer, (1995).
[16]	P. Magal, Compact attractors for time-periodic age-structured population models,, Electron. J. Differ. Equat., 65 (2001), 1.
[17]	P. Magal, C. C. McCluskey and G. F. Webb, Lyapunov functional and global asymptotic stability for an infectious-age model,, Applic. Anal., 89 (2010), 1109. doi: 10.1080/00036810903208122.
[18]	R. D. Nussbaum, Periodic solutions of some integral equations from the theory of epidemics, in Nonlinear systems and applications* (ed. V. Lakshmikantham)*, (1977), 235.
[19]	R. D. Nussbaum, A periodicity threshold theorem for some nonlinear integral equations,, SIAM J. Math. Anal., 9 (1978), 356. doi: 10.1137/0509024.
[20]	C. Rebelo, A. Margheri and N. Bacaër, Persistence in seasonally forced epidemiological models,, J. Math. Biol., 64 (2012), 933. doi: 10.1007/s00285-011-0440-6.
[21]	H. L. Smith, On periodic solutions of a delay integral equation modelling epidemics,, J. Math. Biol., 4 (1977), 69. doi: 10.1007/BF00276353.
[22]	H. L. Smith, An introduction to delay differential equations with applications to the life sciences,, Springer, (2011). doi: 10.1007/978-1-4419-7646-8.
[23]	H. L. Smith and H. R. Thieme, Dynamical systems and population persistence,, American Mathematical Society, (2011).
[24]	H. R. Thieme, Spectral bound and reproduction number for infinite-dimensional population structure and time heterogeneity,, SIAM J. Appl. Math., 70 (2009), 188. doi: 10.1137/080732870.
[25]	H. R. Thieme, Renewal theorems for linear periodic Volterra integral equations,, J. Integral Equat., 7 (1984), 253.
[26]	G. F. Webb, Theory of nonlinear age-dependent population dynamics,, Marcel Dekker, (1985).
[27]	G. F. Webb, E. D'Agata, P. Magal and S. Ruan, A model of antibiotic resistant bacterial epidemics in hospitals,, Proc. Nat. Acad. Sci., 102 (2005), 13343. doi: 10.1073/pnas.0504053102.
[28]	F. Zhang and X. Q. Zhao, A periodic epidemic model in a patchy environment,, J. Math. Anal. Appl., 325 (2007), 496. doi: 10.1016/j.jmaa.2006.01.085.

[1]	P. Dormayer, A. F. Ivanov. Symmetric periodic solutions of a delay differential equation. Conference Publications, 1998, 1998 (Special) : 220-230. doi: 10.3934/proc.1998.1998.220
[2]	Gergely Röst, Jianhong Wu. SEIR epidemiological model with varying infectivity and infinite delay. Mathematical Biosciences & Engineering, 2008, 5 (2) : 389-402. doi: 10.3934/mbe.2008.5.389
[3]	Guy V. Norton, Robert D. Purrington. The Westervelt equation with a causal propagation operator coupled to the bioheat equation.. Evolution Equations & Control Theory, 2016, 5 (3) : 449-461. doi: 10.3934/eect.2016013
[4]	Dan Stanescu, Benito Chen-Charpentier. Random coefficient differential equation models for Monod kinetics. Conference Publications, 2009, 2009 (Special) : 719-728. doi: 10.3934/proc.2009.2009.719
[5]	Vincent Giovangigli. Persistence of Boltzmann entropy in fluid models. Discrete & Continuous Dynamical Systems - A, 2009, 24 (1) : 95-114. doi: 10.3934/dcds.2009.24.95
[6]	Jacek Banasiak, Eddy Kimba Phongi. Canard-type solutions in epidemiological models. Conference Publications, 2015, 2015 (special) : 85-93. doi: 10.3934/proc.2015.0085
[7]	Ábel Garab. Unique periodic orbits of a delay differential equation with piecewise linear feedback function. Discrete & Continuous Dynamical Systems - A, 2013, 33 (6) : 2369-2387. doi: 10.3934/dcds.2013.33.2369
[8]	Michael Scheutzow. Exponential growth rate for a singular linear stochastic delay differential equation. Discrete & Continuous Dynamical Systems - B, 2013, 18 (6) : 1683-1696. doi: 10.3934/dcdsb.2013.18.1683
[9]	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
[10]	Arne Ogrowsky, Björn Schmalfuss. Unstable invariant manifolds for a nonautonomous differential equation with nonautonomous unbounded delay. Discrete & Continuous Dynamical Systems - B, 2013, 18 (6) : 1663-1681. doi: 10.3934/dcdsb.2013.18.1663
[11]	A. R. Humphries, O. A. DeMasi, F. M. G. Magpantay, F. Upham. Dynamics of a delay differential equation with multiple state-dependent delays. Discrete & Continuous Dynamical Systems - A, 2012, 32 (8) : 2701-2727. doi: 10.3934/dcds.2012.32.2701
[12]	C. Connell McCluskey. Global stability for an SEIR epidemiological model with varying infectivity and infinite delay. Mathematical Biosciences & Engineering, 2009, 6 (3) : 603-610. doi: 10.3934/mbe.2009.6.603
[13]	Guihong Fan, Yijun Lou, Horst R. Thieme, Jianhong Wu. Stability and persistence in ODE models for populations with many stages. Mathematical Biosciences & Engineering, 2015, 12 (4) : 661-686. doi: 10.3934/mbe.2015.12.661
[14]	Pasquale Palumbo, Simona Panunzi, Andrea De Gaetano. Qualitative behavior of a family of delay-differential models of the Glucose-Insulin system. Discrete & Continuous Dynamical Systems - B, 2007, 7 (2) : 399-424. doi: 10.3934/dcdsb.2007.7.399
[15]	Yu Yang, Dongmei Xiao. Influence of latent period and nonlinear incidence rate on the dynamics of SIRS epidemiological models. Discrete & Continuous Dynamical Systems - B, 2010, 13 (1) : 195-211. doi: 10.3934/dcdsb.2010.13.195
[16]	Andrei Korobeinikov, Philip K. Maini. A Lyapunov function and global properties for SIR and SEIR epidemiological models with nonlinear incidence. Mathematical Biosciences & Engineering, 2004, 1 (1) : 57-60. doi: 10.3934/mbe.2004.1.57
[17]	Holly Gaff, Elsa Schaefer. Optimal control applied to vaccination and treatment strategies for various epidemiological models. Mathematical Biosciences & Engineering, 2009, 6 (3) : 469-492. doi: 10.3934/mbe.2009.6.469
[18]	Mustapha Ait Rami, Vahid S. Bokharaie, Oliver Mason, Fabian R. Wirth. Stability criteria for SIS epidemiological models under switching policies. Discrete & Continuous Dynamical Systems - B, 2014, 19 (9) : 2865-2887. doi: 10.3934/dcdsb.2014.19.2865
[19]	Glenn Ledder, Donna Sylvester, Rachelle R. Bouchat, Johann A. Thiel. Continuous and pulsed epidemiological models for onchocerciasis with implications for eradication strategy. Mathematical Biosciences & Engineering, 2018, 15 (4) : 841-862. doi: 10.3934/mbe.2018038
[20]	Sebastián Ferrer, Martin Lara. Families of canonical transformations by Hamilton-Jacobi-Poincaré equation. Application to rotational and orbital motion. Journal of Geometric Mechanics, 2010, 2 (3) : 223-241. doi: 10.3934/jgm.2010.2.223

2016 Impact Factor: 0.994

American Institute of Mathematical Sciences