Figure 1.
A schematic diagram of the epidemic model with quarantine
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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 |
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. |
| [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. 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. |
| [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. 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. |
| [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. |
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. |
| [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. 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. |
| [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. 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. |
| [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. |
Figure 3.
The evolution of solution $ S $ with respect to time $ t $
Figure 4.
The evolution of solutions $ i $ and $ q $ with respect to time $ t $ and age $ a $
Figure 5.
The functions $ \beta, \theta $ and $ \delta $ with respect to age $ a $
Figure 6.
The evolution of solution $ S $ with respect to time $ t $
Figure 7.
The evolution of solutions $ i $ and $ q $ with respect to time $ t $ and age $ a $
Figure 9.
The evolution of solution $ S $ with respect to time $ t $
Figure 10.
The evolution of solutions $ i $ and $ q $ with respect to time $ t $ and age $ a $
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 $
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
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
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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