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June  2020, 25(6): 2245-2270. doi: 10.3934/dcdsb.2019226

Global dynamics of an age-structured model with relapse

Mohammed Nor Frioui 1, , Tarik Mohammed Touaoula 1, and  Bedreddine Ainseba 2,,

1. 

Laboratoire d'Analyse Non linéaire et Mathématiques Appliquées, Département de Mathématiques, Université Aboubekr Belkaïd Tlemcen, 13000 Tlemcen, Algeria
2. 

Institut de Mathématiques de Bordeaux, Université de Bordeaux, 33000, Bordeaux, France

* Corresponding author

Revised  February 2019 Published  June 2020 Early access  September 2019

The aim of this paper is to study a general class of $ SIRI $ age infection structured model where infectivity depends on the age since infection and where some individuals from the $ R $ class, also called quarantaine class in this work, can return to the infectiousness class after a while. Using classical technics we compute a basic reproductive number $ R_0 $ and show that the disease dies out when $ R_0 < 1 $ and persists if $ R_0 > 1 $. Some Lyapunov suitable functions are derived to prove global stability for the disease free equilibrium (DFE) when $ R_0 < 1 $ and for the endemic equilibrium (EE) when $ R_0 > 1 $. Using numerical results we show that the non homogeneous infectivity combined with the feedback to the infectiousness class of a part of the quarantaine population modifies drastically the behavior of the epidemic.

Keywords: Epidemic model, infection age, nonlinear incidence, persistence, Lyapunov function, global stability.
Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C35.
Citation: Mohammed Nor Frioui, Tarik Mohammed Touaoula, Bedreddine Ainseba. Global dynamics of an age-structured model with relapse. Discrete & Continuous Dynamical Systems - B, 2020, 25 (6) : 2245-2270. doi: 10.3934/dcdsb.2019226
References:
[1]

B. Ainseba, Z. Feng, M. Iannelli and F. A. Milner, Control Strategies for TB Epidemics, SIAM Journal on Applied Mathematics, 77 (2017), 82-107.  doi: 10.1137/15M1048719.  Google Scholar

[2]

B. E. Ainseba and M. Iannelli, Optimal screening in structured SIR epidemics, Math. Modelling of Natural Phenomena, 7 (2012), 12-27.  doi: 10.1051/mmnp/20127302.  Google Scholar

[3]

S. Bentout and T. M. Touaoula, Global analysis of an infection age model with a class of nonlinear incidence rates, J. Math. Anal. Appl, 434 (2016), 1211-1239.  doi: 10.1016/j.jmaa.2015.09.066.  Google Scholar

[4]

F. Brauer and C. Castillo-Chavez, Mathematical Models in Population Biology and Epidemiology, Springer, New York, 2012. doi: 10.1007/978-1-4614-1686-9.  Google Scholar

[5]

A. Chekroun, M. N. Frioui, T. Kuniya and T. M. Touaoula, Global stability of an age-structured epidemic model with general Lyapunov functional, Math. Biosci. Eng., 16 (2019), 1525-1553.  doi: 10.3934/mbe.2019073.  Google Scholar

[6]

O. Diekmann and J. A. P. Heesterbeek, Mathematical Epidemiology of Infectious Diseases: Model Building, Analysis and Interpretation, Wiley, Chichester, UK, 2000.  Google Scholar

[7]

N. Dunford and J. T. Schwartz, Linear Operators, Interscience Publishers, New York, 1971. Google Scholar

[8]

Z. Feng and H. R. Thieme, Endemic Models With Arbitrarily Distributed Periods of Infection I: Fundamental Properties of The Model, SIAM J. APPL. Math., 61 (2000), 803-833.  doi: 10.1137/S0036139998347834.  Google Scholar

[9]

Z. Feng and H. R. Thieme, Endemic models with arbitrarily distributed periods of infection II: Fast disease dynamics and permanent recovery, SIAM J. Appl. Math., 61 (2000), 983-1012.  doi: 10.1137/S0036139998347846.  Google Scholar

[10]

M. N. Frioui, T. M. Touaoula and S. E. Miri, Unified Lyapunov functional for an age-structured virus model with very general nonlinear infection response, J. Appl. Math. Comput, 58 (2018), 47-73.  doi: 10.1007/s12190-017-1133-0.  Google Scholar

[11]

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

[12]

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

[13]

A. Korobeinikov, Global properties of infectious disease models with nonlinear incidence, Bull Math. Biol., 69 (2007), 1871-1886.  doi: 10.1007/s11538-007-9196-y.  Google Scholar

[14]

A. Korobeinikov and P. K. Maini, Nonlinear incidence and stability of infectious disease models, Math. Med. Biol., 22 (2005), 113-128.   Google Scholar

[15]

P. Magal and X.-Q. Zhao, Global attractors and steady states for uniformly persistent dynamical systems, SIAM J. Math. Anal., 37 (2005), 251-275.  doi: 10.1137/S0036141003439173.  Google Scholar

[16]

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

[17]

C. C. McCluskey, Complete global stability for a SIR epidemic model with delay-distributed or discrete, Nonlinear Anal, 11 (2010), 55-59.  doi: 10.1016/j.nonrwa.2008.10.014.  Google Scholar

[18]

C. C. McCluskey, Global stability for an SEIR epidemiological model with varying infectivity and infinite delay, Math. Biosci. Eng., 6 (2009), 603-610.  doi: 10.3934/mbe.2009.6.603.  Google Scholar

[19]

C. C. McCluskey, Global stability for an SIR epidemic model with delay and general nonlinear incidence, Math. Biosci. Eng., 7 (2010), 837-850.  doi: 10.3934/mbe.2010.7.837.  Google Scholar

[20]

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

[21]

H. R. Thieme, Mathematics in Population Biology, Princeton University Press, Princeton 2003.  Google Scholar

[22]

H. R. Thieme, Uniform persistence and permanence for nonautonomus semiflows in population biology, Math. Biosci., 166 (2000), 173-201.  doi: 10.1016/S0025-5564(00)00018-3.  Google Scholar

[23]

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

[24]

H. R. Thieme, Global stability of the endemic equilibrium in infinite dimension: Lyapunov functions and positive operators, J. Differential Equations, 250 (2011), 3772-3801.  doi: 10.1016/j.jde.2011.01.007.  Google Scholar

show all references

References:
[1]

B. Ainseba, Z. Feng, M. Iannelli and F. A. Milner, Control Strategies for TB Epidemics, SIAM Journal on Applied Mathematics, 77 (2017), 82-107.  doi: 10.1137/15M1048719.  Google Scholar

[2]

B. E. Ainseba and M. Iannelli, Optimal screening in structured SIR epidemics, Math. Modelling of Natural Phenomena, 7 (2012), 12-27.  doi: 10.1051/mmnp/20127302.  Google Scholar

[3]

S. Bentout and T. M. Touaoula, Global analysis of an infection age model with a class of nonlinear incidence rates, J. Math. Anal. Appl, 434 (2016), 1211-1239.  doi: 10.1016/j.jmaa.2015.09.066.  Google Scholar

[4]

F. Brauer and C. Castillo-Chavez, Mathematical Models in Population Biology and Epidemiology, Springer, New York, 2012. doi: 10.1007/978-1-4614-1686-9.  Google Scholar

[5]

A. Chekroun, M. N. Frioui, T. Kuniya and T. M. Touaoula, Global stability of an age-structured epidemic model with general Lyapunov functional, Math. Biosci. Eng., 16 (2019), 1525-1553.  doi: 10.3934/mbe.2019073.  Google Scholar

[6]

O. Diekmann and J. A. P. Heesterbeek, Mathematical Epidemiology of Infectious Diseases: Model Building, Analysis and Interpretation, Wiley, Chichester, UK, 2000.  Google Scholar

[7]

N. Dunford and J. T. Schwartz, Linear Operators, Interscience Publishers, New York, 1971. Google Scholar

[8]

Z. Feng and H. R. Thieme, Endemic Models With Arbitrarily Distributed Periods of Infection I: Fundamental Properties of The Model, SIAM J. APPL. Math., 61 (2000), 803-833.  doi: 10.1137/S0036139998347834.  Google Scholar

[9]

Z. Feng and H. R. Thieme, Endemic models with arbitrarily distributed periods of infection II: Fast disease dynamics and permanent recovery, SIAM J. Appl. Math., 61 (2000), 983-1012.  doi: 10.1137/S0036139998347846.  Google Scholar

[10]

M. N. Frioui, T. M. Touaoula and S. E. Miri, Unified Lyapunov functional for an age-structured virus model with very general nonlinear infection response, J. Appl. Math. Comput, 58 (2018), 47-73.  doi: 10.1007/s12190-017-1133-0.  Google Scholar

[11]

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

[12]

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

[13]

A. Korobeinikov, Global properties of infectious disease models with nonlinear incidence, Bull Math. Biol., 69 (2007), 1871-1886.  doi: 10.1007/s11538-007-9196-y.  Google Scholar

[14]

A. Korobeinikov and P. K. Maini, Nonlinear incidence and stability of infectious disease models, Math. Med. Biol., 22 (2005), 113-128.   Google Scholar

[15]

P. Magal and X.-Q. Zhao, Global attractors and steady states for uniformly persistent dynamical systems, SIAM J. Math. Anal., 37 (2005), 251-275.  doi: 10.1137/S0036141003439173.  Google Scholar

[16]

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

[17]

C. C. McCluskey, Complete global stability for a SIR epidemic model with delay-distributed or discrete, Nonlinear Anal, 11 (2010), 55-59.  doi: 10.1016/j.nonrwa.2008.10.014.  Google Scholar

[18]

C. C. McCluskey, Global stability for an SEIR epidemiological model with varying infectivity and infinite delay, Math. Biosci. Eng., 6 (2009), 603-610.  doi: 10.3934/mbe.2009.6.603.  Google Scholar

[19]

C. C. McCluskey, Global stability for an SIR epidemic model with delay and general nonlinear incidence, Math. Biosci. Eng., 7 (2010), 837-850.  doi: 10.3934/mbe.2010.7.837.  Google Scholar

[20]

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

[21]

H. R. Thieme, Mathematics in Population Biology, Princeton University Press, Princeton 2003.  Google Scholar

[22]

H. R. Thieme, Uniform persistence and permanence for nonautonomus semiflows in population biology, Math. Biosci., 166 (2000), 173-201.  doi: 10.1016/S0025-5564(00)00018-3.  Google Scholar

[23]

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

[24]

H. R. Thieme, Global stability of the endemic equilibrium in infinite dimension: Lyapunov functions and positive operators, J. Differential Equations, 250 (2011), 3772-3801.  doi: 10.1016/j.jde.2011.01.007.  Google Scholar

Figure 1.  A schematic diagram of the epidemic model with quarantine
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Figure 2.  The functions $ \beta,\theta $ and $ \delta $ with respect to age $ a $
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Figure 3.  The evolution of solution $ S $ with respect to time $ t $
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Figure 4.  The evolution of solutions $ i $ and $ q $ with respect to time $ t $ and age $ a $
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Figure 5.  The functions $ \beta, \theta $ and $ \delta $ with respect to age $ a $
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Figure 6.  The evolution of solution $ S $ with respect to time $ t $
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Figure 7.  The evolution of solutions $ i $ and $ q $ with respect to time $ t $ and age $ a $
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Figure 8.  The functions $ \beta,\theta $ and $ \delta $ with respect to age $ a $
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Figure 9.  The evolution of solution $ S $ with respect to time $ t $
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Figure 10.  The evolution of solutions $ i $ and $ q $ with respect to time $ t $ and age $ a $
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Figure 11.  The functions $ \beta,\theta $ and $ \delta $ with respect to age $ a $ : $ \delta \equiv 0 $ such that $ R_0 < 1 $, $ \delta \not\equiv 0 $ such that $ R_0 < 1 $ and $ \delta \not\equiv 0 $ such that $ R_0 > 1 $
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Figure 12.  The evolution of solution S with respect to time t : δ ≡ 0 such that R0 < 1, $\delta \not \equiv 0$ such that $\delta \not \equiv 0$ and $\delta \not \equiv 0$ such that R0 > 1
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Figure 13.  The evolution of solution i with respect to time t and age a : δ ≡ 0 such that R0 < 1, $\delta \not \equiv 0$ such that R0 < 1 and $\delta \not \equiv 0$ such that R0 > 1
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Figure 14.  The evolution of solution q with respect to time t and age a : δ ≡ 0 such that R0 < 1
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