Periodic trajectories for an "SEIR" epidemic model in a seasonal environment with general incidence rate
Periodic trajectories for an "SEIR" epidemic model in a seasonal environment
Resumen
Infectious diseases take different forms. They are caused by bacteria, viruses, parasitesor fungi that enter the body to weaken it. These diseases are transmitted from person to person
or can be carried by animals. The list of infectious diseases is very long: gastroenteritis,
bronchiolitis, tuberculosis, hepatitis, measles, seasonal infection, influenza, malaria, chikungunya, etc.
Many epidemic diseases exhibit seasonal peak periods. Studying the population behaviours due to seasonal
environment becomes a necessity for predicting the risk of disease transmission and trying to control it.
In this work, we considered a four-dimensional system for a fatal disease, in a seasonal environment.
We establish the existence, uniqueness, positivity and boundedness of the solution then we prove that it is a periodic orbit. We show that the global dynamics is determined using the basic reproduction number denoted by R_0 and calculated
using the spectral radius of an integral operator. We show the global stability of the disease-free periodic solution
if R_0<1 and we also show the persistence of the disease if R_0>1. Finally, we display some numerical investigations supporting the theoretical findings where the trajectories
converge to a limit cycle if R_0>1.
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Citas
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anaerobic digestion process,” Math. Biosci. Eng., vol. 20, no. 4, pp. 6591–6611, 2023.
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the presence of a virus,” Mathematics, vol. 11, no. 4, p. 883, 2023.
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vol. 5, pp. 98–112, 2017.
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seasonal succession,” Mathematical Biosciences and Engineering, vol. 14, no. 5&6, pp. 1407–1424, 2017.
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3 Conference on Computational and Mathematical Population Dynamics, (Bordeaux, France), May 2010.
[15] H. Wei, Y. Jiang, X. Song, G. Su, and S. Qiu, “Global attractivity and permanence of a SVEIR epidemic
model with pulse vaccination and time delay,” Journal of Computational and Applied Mathematics, vol. 229,
no. 1, pp. 302–312, 2009.
[16] A. B. Gumel, C. C. McCluskey, and J. Watmough, “An SVEIR model for assessing potential impact of an
imperfect anti-sars vaccine,” Mathematical Biosciences and Engineering, vol. 3, no. 3, pp. 485–512, 2006.
[17] M. El Hajji, A. Zaghdani, and S. Sayari, “Mathematical analysis and optimal control for Chikungunya virus
with two routes of infection with nonlinear incidence rate,” Int. J. Biomath., vol. 15(1), p. 2150088, 2022.
[18] M. El Hajji, S. Sayari, and A. Zaghdani, “Mathematical analysis of an SIR epidemic model in a continuous
reactor - deterministic and probabilistic approaches,” J. Korean Math. Soc., vol. 58(1), pp. 45–67, 2021.
[19] D. Xiao, “Dynamics and bifurcations on a class of population model with seasonal constant-yield harvesting,”
Discrete Contin. Dynam. Syst. -B, vol. 21, pp. 699–719, 2016.
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pp. 1954–1966, 2009.
[21] F. Kermack and D. McKendrick, “A contribution to the mathematical theory of epidemics,” In Proceedings
of the Royal Society of London A : Mathematical, Physical and Engineering Sciences, vol. 115, pp. 700–721,
1927.
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and Engineering, vol. 3, no. 1, pp. 161–172, 2006.
[23] S. Guerrero-Flores, O. Osuna, and C. V. de Leon, “Periodic solutions for seasonal SIQRS models with
nonlinear infection terms,” Electron. J. Differ. Equations, vol. 2019 (92), pp. 1–13, 2019.
[24] Z. T. Tailei Zhang, “On a nonautonomous SEIRS model in epidemiology,” Bull. Math. Biol., vol. 69 (8),
pp. 2537–2559, 2007.
[25] W.Wang and X. Zhao, “Threshold dynamics for compartmental epidemic models in periodic environments,”
J. Dynam. Differential Equations, vol. 20 (3), pp. 699–717, 2008.
[26] M. El Hajji, D. M. Alshaikh, and N. A. Almuallem, “Periodic behaviour of an epidemic in a seasonal environment
with vaccination,” Mathematics, vol. 11, no. 10, p. 2350, 2023.
[27] O. Diekmann and J. Heesterbeek, “On the definition and the computation of the basic reproduction ratio
R0 in models for infectious diseases in heterogeneous populations,” J. Math. Bio., vol. 28(4), pp. 365–382,
1990.
[28] P. V. den Driessche and J. Watmough, “Reproduction numbers and sub-threshold endemic equilibria for
compartmental models of disease transmission,” Math. Biosci., vol. 180, pp. 29–48, 2002.
[29] J. LaSalle, The Stability of Dynamical Systems. SIAM, 1976.
[30] F. Zhang and X. Zhao, “A periodic epidemic model in a patchy environment,” J. Math. Anal. Appl., vol. 325
(1), pp. 496–516, 2007.
[31] X. Zhao, “Dynamical systems in population biology,” CMS Books Math./Ouvrages Math. SMC Springer-
Verlag, New York, vol. 16, 2003.
(NCIRD), “Types of influenza viruses,” Available from: https://www.cdc.gov/flu/about/index.html, 2023.
[2] WHO, “Influenza (seasonal),” Available from: https://www.emro.who.int/health-topics/influenza/influenzaseasonal.
html, 2023.
[3] Y. Nakata and T. Kuniya, “Global dynamics of a class of SEIRS epidemic models in a periodic environment,”
J. Math. Anal. Appl., vol. 363, pp. 230–237, 2010.
[4] N. Bacaër and R. Ouifki, “Growth rate and basic reproduction number for population models with a simple
periodic factor,” Mathematical Biosciences, vol. 210, no. 2, pp. 647–658, 2007.
[5] N. Bacaër, “Approximation of the basic reproduction number r0 for vector-borne diseases with a periodic
vector population,” Bull. Math. Biol., vol. 69 (3), pp. 1067–1091, 2007.
[6] N. Bacaër and S. Guernaoui, “The epidemic threshold of vector-borne diseases with seasonality,” J. Math.
Biol., vol. 53, pp. 421–436, 2006.
[7] M. El Hajji and A. H. Albargi, “A mathematical investigation of an "SVEIR" epidemic model for the measles
transmission,” Math. Biosci. Eng., vol. 19(3), pp. 2853–2875, 2022.
[8] A. Alshehri and M. El Hajji, “Mathematical study for zika virus transmission with general incidence rate,”
AIMS Mathematics, vol. 7(4), pp. 7117–7142, 2022.
[9] M. El Hajji, “Modelling and optimal control for Chikungunya disease,” Theory Biosci., vol. 140(1), pp. 27–
44, 2021.
[10] A. H. Albargi and M. El Hajji, “Mathematical analysis of a two-tiered microbial food-web model for the
anaerobic digestion process,” Math. Biosci. Eng., vol. 20, no. 4, pp. 6591–6611, 2023.
[11] A. A. Alsolami and M. El Hajji, “Mathematical analysis of a bacterial competition in a continuous reactor in
the presence of a virus,” Mathematics, vol. 11, no. 4, p. 883, 2023.
[12] L. Nkamba, J. Ntaganda, H. Abboubakar, J. Kamgang, and L. Castelli, “Global stability of a SVEIR epidemic
model: Application to poliomyelitis transmission dynamics,” Open Journal of Modelling and Simulation,
vol. 5, pp. 98–112, 2017.
[13] Y. Tang, D. Xiao, W. Zhang, and D. Zhu, “Dynamics of epidemic models with asymptomatic infection and
seasonal succession,” Mathematical Biosciences and Engineering, vol. 14, no. 5&6, pp. 1407–1424, 2017.
[14] P. Adda, L. Nkague Nkamba, G. Sallet, and L. Castelli, “A SVEIR model with Imperfect Vaccine,” in CMPD
3 Conference on Computational and Mathematical Population Dynamics, (Bordeaux, France), May 2010.
[15] H. Wei, Y. Jiang, X. Song, G. Su, and S. Qiu, “Global attractivity and permanence of a SVEIR epidemic
model with pulse vaccination and time delay,” Journal of Computational and Applied Mathematics, vol. 229,
no. 1, pp. 302–312, 2009.
[16] A. B. Gumel, C. C. McCluskey, and J. Watmough, “An SVEIR model for assessing potential impact of an
imperfect anti-sars vaccine,” Mathematical Biosciences and Engineering, vol. 3, no. 3, pp. 485–512, 2006.
[17] M. El Hajji, A. Zaghdani, and S. Sayari, “Mathematical analysis and optimal control for Chikungunya virus
with two routes of infection with nonlinear incidence rate,” Int. J. Biomath., vol. 15(1), p. 2150088, 2022.
[18] M. El Hajji, S. Sayari, and A. Zaghdani, “Mathematical analysis of an SIR epidemic model in a continuous
reactor - deterministic and probabilistic approaches,” J. Korean Math. Soc., vol. 58(1), pp. 45–67, 2021.
[19] D. Xiao, “Dynamics and bifurcations on a class of population model with seasonal constant-yield harvesting,”
Discrete Contin. Dynam. Syst. -B, vol. 21, pp. 699–719, 2016.
[20] N. Bacaër and M. Gomes, “On the final size of epidemics with seasonality,” Bull. Math. Biol., vol. 71,
pp. 1954–1966, 2009.
[21] F. Kermack and D. McKendrick, “A contribution to the mathematical theory of epidemics,” In Proceedings
of the Royal Society of London A : Mathematical, Physical and Engineering Sciences, vol. 115, pp. 700–721,
1927.
[22] J. Ma and Z. Ma, “Epidemic threshold conditions for seasonally forced SEIR models,” Mathematical Biosciences
and Engineering, vol. 3, no. 1, pp. 161–172, 2006.
[23] S. Guerrero-Flores, O. Osuna, and C. V. de Leon, “Periodic solutions for seasonal SIQRS models with
nonlinear infection terms,” Electron. J. Differ. Equations, vol. 2019 (92), pp. 1–13, 2019.
[24] Z. T. Tailei Zhang, “On a nonautonomous SEIRS model in epidemiology,” Bull. Math. Biol., vol. 69 (8),
pp. 2537–2559, 2007.
[25] W.Wang and X. Zhao, “Threshold dynamics for compartmental epidemic models in periodic environments,”
J. Dynam. Differential Equations, vol. 20 (3), pp. 699–717, 2008.
[26] M. El Hajji, D. M. Alshaikh, and N. A. Almuallem, “Periodic behaviour of an epidemic in a seasonal environment
with vaccination,” Mathematics, vol. 11, no. 10, p. 2350, 2023.
[27] O. Diekmann and J. Heesterbeek, “On the definition and the computation of the basic reproduction ratio
R0 in models for infectious diseases in heterogeneous populations,” J. Math. Bio., vol. 28(4), pp. 365–382,
1990.
[28] P. V. den Driessche and J. Watmough, “Reproduction numbers and sub-threshold endemic equilibria for
compartmental models of disease transmission,” Math. Biosci., vol. 180, pp. 29–48, 2002.
[29] J. LaSalle, The Stability of Dynamical Systems. SIAM, 1976.
[30] F. Zhang and X. Zhao, “A periodic epidemic model in a patchy environment,” J. Math. Anal. Appl., vol. 325
(1), pp. 496–516, 2007.
[31] X. Zhao, “Dynamical systems in population biology,” CMS Books Math./Ouvrages Math. SMC Springer-
Verlag, New York, vol. 16, 2003.
Publicado
2025-12-05
Número
Sección
Research Articles
Derechos de autor 2025 Boletim da Sociedade Paranaense de Matemática
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