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doi: 10.3934/dcdsb.2022048
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## Mathematical analysis of a triple age dependent epidemiological model with including a protection strategy

 1 Laboratoire d'Analyse Non linéaire et Mathématiques Appliquées, Département de Mathématiques, Université Aboubekr Belkaïd Tlemcen, 13000 Tlemcen, Algeria 2 Laboratoire d'Analyse Non Linéaire et Mathématiques Appliquées, University of Tlemcen, Algeria 3 Faculty of Exact sciences and informatics, Mathematic Department, Hassiba Benbouali university, Chlef, Algeria

* Corresponding author

Received  April 2021 Revised  January 2022 Early access March 2022

In this research, we consider the influence of protection measures on the spread of infectious diseases in an age-structured population. Protection strategy can take different forms as isolation, treatment, or renewable vaccine; to mathematically represent it, we include a new compartment p standing for protected individuals, in a classical age structured si model. Global analysis of the proposed model is made by the introduction of total trajectories and a suitable Lyapunov functional. We give a particular importance to the protection strategy and many numerical simulations are provided to illustrate our theoretical results.

Citation: Fatima Zohra Hathout, Tarik Mohammed Touaoula, Salih Djilali. Mathematical analysis of a triple age dependent epidemiological model with including a protection strategy. Discrete and Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2022048
##### References:
 [1] M. Adimy, A. Chekroun and C. P. Ferreira, Global dynamics of a differential-difference system: A case of Kermack-McKendrick SIR model with age-structured protection phase, Math. Biosci. Eng., 17 (2020), 1329-1354.  doi: 10.3934/mbe.2020067. [2] 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. [3] S. Bentout, A. Tridane, S. Djilali and T. M. Touaoula, Age-structured modeling of COVID-19 epidemic in the USA, UAE and algeria, Alexandria Engin. J., 60 (2020), 401-411.  doi: 10.1016/j.aej.2020.08.053. [4] I. Boudjema and T. M. Touaoula, Global stability of an infection and vaccination age-structured model with general nonlinear incidence, J. Nonlinear Functional Analysis, (2018), 1–21. doi: 10.2395/jnfa2018.33.. [5] C. J. Browne and S. S. Pilyugin, Global analysis of age-structured within-host virus model, Discrete Contin. Dyn. Syst. Ser. B, 18 (2013), 1999-2017.  doi: 10.3934/dcdsb.2013.18.1999. [6] L. M. Cai, M. Martcheva and X. Z. Li, Epidemic models with age of infection, indirect transmission and incomplete treatment, Discrete Contin. Dyn. Syst. Ser. B, 18 (2013), 2239-2265.  doi: 10.3934/dcdsb.2013.18.2239. [7] L.-M. Cai, C. Modnak and J. Wang, An age-structured model for cholera control with vaccination, Appl. Math. Comput., 299 (2017), 127-140.  doi: 10.1016/j.amc.2016.11.013. [8] C. Castillo-Chavez, H. W. Hethecote, V. Andreasen, S. A. Levin and W. M. Liu, Epidemiological models with age structure, proportionate mixing and cross-immunity, J. Math. Biol., 27 (1989), 233-258.  doi: 10.1007/BF00275810. [9] O. Diekmann and J. A. P. Heesterbeek, Mathematical Epidemiology of Infectious Diseases: Model Building, Analysis and Interpretation, Wiley, Chichester, UK, 2000. [10] S. Djilali, T. M. Touaoula and S. E. Miri, A heroin epidemic model: Very general non linear incidence, treat-age, and global stability, Acta Appl. Math., 152 (2017), 171-194.  doi: 10.1007/s10440-017-0117-2. [11] Z. Feng, Y. Feng and J. W. Glasser, Influence of demographically-realistic mortality schedules on vaccination strategies in age-structured models, Theoretical Population Biology, 132 (2020), 24-32. [12] M. N. Frioui, T. M. Touaoula and B. Ainseba, Global dynamics of an age-structured model with relapse, Disc. Cont. Dyn. Syst B, 25 (2020), 2245-2270.  doi: 10.3934/dcdsb.2019226. [13] J. K. Hale, Asymptotic Behavior of Dissipative Systems, Mathematical Surveys and Monographsvol. 25, American Mathematical Society, Providence, RI, 1988. doi: 10.1090/surv/025. [14] 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. [15] P. Magal, C. 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. [16] P. Magal and H. R. Thieme, Eventual compactness for a semiflow generated by an age- structured models, Commun Pure Appl. Anal., 3 (2004), 695-727.  doi: 10.3934/cpaa.2004.3.695. [17] 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. [18] 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. [19] C. C. McCluskey, Complete global stability for a SIR epidemic model with delay-distributed or discrete, Nonlinear Anal. Real World Appl., 11 (2010), 55-59.  doi: 10.1016/j.nonrwa.2008.10.014. [20] C. C. McCluskey, Global stability for an SEI epidemiological model with continuous age-structure in the exposed and infectious classes, Math. Biosci. Eng., 9 (2012), 819-841.  doi: 10.3934/mbe.2012.9.819. [21] H. L. Smith and H. R. Thieme, Dynamical Systems and Population Persistence, Grad. Stud. Math., vol. 118 AMS, 2011. doi: 10.1090/gsm/118. [22] H. L. Smith and X.-Q. Zhao, Robust persistence for semidynamical systems, Nonlinear Anal., 47 (2001), 6169-6179.  doi: 10.1016/S0362-546X(01)00678-2. [23] Y. Takeuchi, W. Ma and E. Beretta, Global asymptotic properties of a delay SIR epidemic model with finite incubation times, Nonlinear Anal., 42 (2000), 931-947.  doi: 10.1016/S0362-546X(99)00138-8. [24] G. F. Webb, Theory of Nonlinear Age-Dependent Population Dynamics, Marcel Dekker, New York, 1985.

show all references

##### References:
 [1] M. Adimy, A. Chekroun and C. P. Ferreira, Global dynamics of a differential-difference system: A case of Kermack-McKendrick SIR model with age-structured protection phase, Math. Biosci. Eng., 17 (2020), 1329-1354.  doi: 10.3934/mbe.2020067. [2] 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. [3] S. Bentout, A. Tridane, S. Djilali and T. M. Touaoula, Age-structured modeling of COVID-19 epidemic in the USA, UAE and algeria, Alexandria Engin. J., 60 (2020), 401-411.  doi: 10.1016/j.aej.2020.08.053. [4] I. Boudjema and T. M. Touaoula, Global stability of an infection and vaccination age-structured model with general nonlinear incidence, J. Nonlinear Functional Analysis, (2018), 1–21. doi: 10.2395/jnfa2018.33.. [5] C. J. Browne and S. S. Pilyugin, Global analysis of age-structured within-host virus model, Discrete Contin. Dyn. Syst. Ser. B, 18 (2013), 1999-2017.  doi: 10.3934/dcdsb.2013.18.1999. [6] L. M. Cai, M. Martcheva and X. Z. Li, Epidemic models with age of infection, indirect transmission and incomplete treatment, Discrete Contin. Dyn. Syst. Ser. B, 18 (2013), 2239-2265.  doi: 10.3934/dcdsb.2013.18.2239. [7] L.-M. Cai, C. Modnak and J. Wang, An age-structured model for cholera control with vaccination, Appl. Math. Comput., 299 (2017), 127-140.  doi: 10.1016/j.amc.2016.11.013. [8] C. Castillo-Chavez, H. W. Hethecote, V. Andreasen, S. A. Levin and W. M. Liu, Epidemiological models with age structure, proportionate mixing and cross-immunity, J. Math. Biol., 27 (1989), 233-258.  doi: 10.1007/BF00275810. [9] O. Diekmann and J. A. P. Heesterbeek, Mathematical Epidemiology of Infectious Diseases: Model Building, Analysis and Interpretation, Wiley, Chichester, UK, 2000. [10] S. Djilali, T. M. Touaoula and S. E. Miri, A heroin epidemic model: Very general non linear incidence, treat-age, and global stability, Acta Appl. Math., 152 (2017), 171-194.  doi: 10.1007/s10440-017-0117-2. [11] Z. Feng, Y. Feng and J. W. Glasser, Influence of demographically-realistic mortality schedules on vaccination strategies in age-structured models, Theoretical Population Biology, 132 (2020), 24-32. [12] M. N. Frioui, T. M. Touaoula and B. Ainseba, Global dynamics of an age-structured model with relapse, Disc. Cont. Dyn. Syst B, 25 (2020), 2245-2270.  doi: 10.3934/dcdsb.2019226. [13] J. K. Hale, Asymptotic Behavior of Dissipative Systems, Mathematical Surveys and Monographsvol. 25, American Mathematical Society, Providence, RI, 1988. doi: 10.1090/surv/025. [14] 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. [15] P. Magal, C. 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. [16] P. Magal and H. R. Thieme, Eventual compactness for a semiflow generated by an age- structured models, Commun Pure Appl. Anal., 3 (2004), 695-727.  doi: 10.3934/cpaa.2004.3.695. [17] 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. [18] 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. [19] C. C. McCluskey, Complete global stability for a SIR epidemic model with delay-distributed or discrete, Nonlinear Anal. Real World Appl., 11 (2010), 55-59.  doi: 10.1016/j.nonrwa.2008.10.014. [20] C. C. McCluskey, Global stability for an SEI epidemiological model with continuous age-structure in the exposed and infectious classes, Math. Biosci. Eng., 9 (2012), 819-841.  doi: 10.3934/mbe.2012.9.819. [21] H. L. Smith and H. R. Thieme, Dynamical Systems and Population Persistence, Grad. Stud. Math., vol. 118 AMS, 2011. doi: 10.1090/gsm/118. [22] H. L. Smith and X.-Q. Zhao, Robust persistence for semidynamical systems, Nonlinear Anal., 47 (2001), 6169-6179.  doi: 10.1016/S0362-546X(01)00678-2. [23] Y. Takeuchi, W. Ma and E. Beretta, Global asymptotic properties of a delay SIR epidemic model with finite incubation times, Nonlinear Anal., 42 (2000), 931-947.  doi: 10.1016/S0362-546X(99)00138-8. [24] G. F. Webb, Theory of Nonlinear Age-Dependent Population Dynamics, Marcel Dekker, New York, 1985.
Diagram flux of the system (1)
The considered transmission functional in simulating the behavior of the solution in Fig. 4 in the case $\mathcal{R}_0<1$ and the behavior of solution in Fig. 5 for $\mathcal{R}_0>1$
The considered transmission functional in simulating the behavior of the solution in Fig. 4 in the case $\mathcal{R}_0<1$ and the behavior of solution in Fig. 5 in the case of $\mathcal{R}_0>1$
The global stability of the DFE in the case of $\mathcal{R}_0 = 0.4557<1$ for the standard set of parameters mentioned in section 6 and $\theta_1 = 10^{-4}$
The global stability of the EE in the case of $\mathcal{R}_0 = 4.557>1$ for the standard set of parameters mentioned in section 6 and $\theta_1 = 10^{-3}$
The value of the endemic equilibrium state obtained in Fig. 5
The effect of the re-protection rate $\alpha$ on $\mathcal{R}_0$
The the effect of the re-protection rate $\alpha$ on $i(T,a)$ where $T = 1500$
The effect of protection rate $h_0$ on the epidemic peak outbreak
The effect of the protection rate $h_0$ on $\mathcal{R}_0.$
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