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

## Global dynamics of an age-structured model with relapse

 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.

Citation: Mohammed Nor Frioui, Tarik Mohammed Touaoula, Bedreddine Ainseba. Global dynamics of an age-structured model with relapse. Discrete and 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. [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. [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. [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. [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. [6] O. Diekmann and J. A. P. Heesterbeek, Mathematical Epidemiology of Infectious Diseases: Model Building, Analysis and Interpretation, Wiley, Chichester, UK, 2000. [7] N. Dunford and J. T. Schwartz, Linear Operators, Interscience Publishers, New York, 1971. [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. [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. [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. [11] J. Hale and P. Waltman, Persistence in infinite dimensional systems, SIAM J. Math. Anal., 20 (1989), 388-395.  doi: 10.1137/0520025. [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. [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. [14] A. Korobeinikov and P. K. Maini, Nonlinear incidence and stability of infectious disease models, Math. Med. Biol., 22 (2005), 113-128. [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. [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. [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. [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. [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. [20] H. L. Smith and H. R. Thieme, Dynamical Systems and Population Persistence, Graduate Studies in Mathematics, 118. American Mathematical Society, Providence, RI, 2011. [21] H. R. Thieme, Mathematics in Population Biology, Princeton University Press, Princeton 2003. [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. [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. [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.

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##### 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. [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. [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. [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. [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. [6] O. Diekmann and J. A. P. Heesterbeek, Mathematical Epidemiology of Infectious Diseases: Model Building, Analysis and Interpretation, Wiley, Chichester, UK, 2000. [7] N. Dunford and J. T. Schwartz, Linear Operators, Interscience Publishers, New York, 1971. [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. [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. [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. [11] J. Hale and P. Waltman, Persistence in infinite dimensional systems, SIAM J. Math. Anal., 20 (1989), 388-395.  doi: 10.1137/0520025. [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. [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. [14] A. Korobeinikov and P. K. Maini, Nonlinear incidence and stability of infectious disease models, Math. Med. Biol., 22 (2005), 113-128. [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. [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. [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. [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. [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. [20] H. L. Smith and H. R. Thieme, Dynamical Systems and Population Persistence, Graduate Studies in Mathematics, 118. American Mathematical Society, Providence, RI, 2011. [21] H. R. Thieme, Mathematics in Population Biology, Princeton University Press, Princeton 2003. [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. [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. [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.
A schematic diagram of the epidemic model with quarantine
The functions $\beta,\theta$ and $\delta$ with respect to age $a$
The evolution of solution $S$ with respect to time $t$
The evolution of solutions $i$ and $q$ with respect to time $t$ and age $a$
The functions $\beta, \theta$ and $\delta$ with respect to age $a$
The evolution of solution $S$ with respect to time $t$
The evolution of solutions $i$ and $q$ with respect to time $t$ and age $a$
The functions $\beta,\theta$ and $\delta$ with respect to age $a$
The evolution of solution $S$ with respect to time $t$
The evolution of solutions $i$ and $q$ with respect to time $t$ and age $a$
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 $R_0 > 1$
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
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
The evolution of solution q with respect to time t and age a : δ ≡ 0 such that R0 < 1
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