An SEIR model with infectious latent and a periodic vaccination strategy
Abstract
An SEIR epidemic model with a nonconstant vaccination strategy is studied. This SEIR model has two disease transmission rates β1 and β2 which imitate the fact that, for some infectious diseases, a latent person can pass the disease into a susceptible one. Here we study the spread of some childhood infectious diseases as good examples of diseases with infectious latent. We found that our SEIR model has a unique disease free solution (DFS). A lower bound 
and an upper bound 
of the basic reproductive number, R0 are estimated. We show that, the DFS is globally asymptotically stable when 
and unstable if 
Computer simulations have been conducted to show that non trivial periodic solutions are possible. Moreover the impact of the contact rate between the latent and the susceptibles is simulated. Different periodic solutions with different periods including one, two and three years, are obtained. These results give a clearer view for the decision makers to know how and when they should take action against a possible new wave of these infectious diseases. This action is mainly, applying a suitable dose of vaccination just before a severe peak of infection occurs.
Keyword : modelling, simulation, disease free solution, two contact rates, global stability, periodic vaccination, R0
How to Cite
Moneim, I. A. (2021). An SEIR model with infectious latent and a periodic vaccination strategy . Mathematical Modelling and Analysis, 26(2), 236-252. https://doi.org/10.3846/mma.2021.12945
This work is licensed under a Creative Commons Attribution 4.0 International License.
References
E. Avila, E. Rivero and G. Emilio. Golobal dynamics of a periodic SEIRS model with general incidence rate. International Journal of Differential Equations, 2017:1–14, 2017. https://doi.org/10.1155/2017/5796958
M.F. Boni, B.H. Manh, P.Q. Thai and et al. Modelling the progression of pandemic influenza a (H1N1) in Vietnam and the opportunities for reassortment with other influenza viruses. BMC Medicene, 7(43), 2009. https://doi.org/10.1186/1741-7015-7-43
T.A. Burton. Stability and periodic solutions of ordinary and functional differential equations. Academic Press, New York, 1985.
D. Greenhalgh. Hopf bifurcation in epidemic models with a latent period and nonpermanent immunity. Math. Comput. Model, 25:85–93, 1997.
D. Greenhalgh and I.A. Moneim. SIRS epidemic model and simulations using different types of seasonal contact rate. Systems Analysis Modelling Simulation, 43(5):573–600, 2003. https://doi.org/10.1080/023929021000008813
D. Grenhalgh. Some results for an SEIR epidemic model with density dependence in the death rate. IMA J. Math. Appl. Med. Biol., 9(2):67–106, 1992. https://doi.org/10.1093/imammb/9.2.67
R. Jan and Y. Xiao. Effect of pulse vaccination on dynamics of dengue with periodic transmission functions. Advances in Difference Equations, 219(1):368, 2019. https://doi.org/10.1186/s13662-019-2314-y
T. Kitano. Dynamic transmission model of routine mumps vaccination in Japan. Epidemiology and Infection, 147:1–8, 2018. https://doi.org/10.1017/S0950268818003230
M. De la Sen, A. Ibeas and S. Alonso-Quesada. On vaccination controls for the SEIR epidemic model. Communications in Nonlinear Science and Numerical Simulation, 17(6):3888–3904, 2012. https://doi.org/10.1016/j.cnsns.2011.10.012
M. De la Sen and S.Alonso-Quesada. Vaccination strategies based on feedback control techniques for a general SEIR epidemic model. Applied Mathematics and Computation, 218(7):2637–2658, 2011. https://doi.org/10.1016/j.amc.2011.09.036
G. Li and Z. Jin. Global stability of a SEIR epidemic model with infectious force in latent, infected and immune period. Chaos, Solitons and Fractals, 25(2):1177–1184, 2005. https://doi.org/10.1016/j.chaos.2004.11.062
M.Y. Li, J.S. Muldoweney, L.C. Wang and J. Karsai. Global dynamics of an SEIR epidemic model with a varying total population size. Math. Biosci., 160(2):191– 213, 1999. https://doi.org/10.1016/S0025-5564(99)00030-9
I.A. Moneim. Seasonally varying epidemics with and without latent period: A comparative simulation study. IMA J. of Mathematical Medicine and Biology, 24(1):1–15, 2007. https://doi.org/10.1093/imammb/dql023
I.A. Moneim. Different vaccination strategies for measles diseases: A simulation study. Journal of Informatics and Mathematical Sciences, 3:227–236, 2011.
I.A. Moneim. Efficiency of different vaccination strategies for childhood diseases: A simulation study. Advances in Bioscience and Biotechnology, 4(2):193–205, 2013. https://doi.org/10.4236/abb.2013.42028
I.A. Moneim. Modeling and simulation of the spread of H1N1 flu with periodic vaccination. IJB., 9(1), 2016. https://doi.org/10.1142/S1793524516500030
I.A. Moneim and D. Greenhalgh. Threshold and stability results for an SIRS epidemic model with a general periodic vaccination strategy. Journal of biological systems, 13(2):131–150, 2005. https://doi.org/10.1142/S0218339005001446
I.A. Moneim and D. Greenhalgh. Use of a periodic vaccination strategy to control the spread of epidemics with seasonally varying contact rate. Mathematical Bioscience and Engineering, 13(2):591–611, 2005. https://doi.org/10.3934/mbe.2005.2.591
I.A. Moneim and H.A. Khalil. Modelling and simulation of the spread of HBV disease with infectious latent. A M., 6(5):745–753, 2015. https://doi.org/10.4236/am.2015.65070
M.Y.Li and J.S. Muldoweney. Global dynamics of a SEIR epidemic model with vertical transmission. SIAM J. Appl.Math., 62(1):58–69, 2001. https://doi.org/10.1137/S0036139999359860
H. Nishiur and H. Inaba. Estimation of the incubation period of influenza a (H1N1) among imported cases: Addressing censoring using outbreak data origin of importation. J Theor Biol., 272(1):123–130, 2011. https://doi.org/10.1016/j.jtbi.2010.12.017
O.J. Peter, O.A. Afolabi, A.A. Victor and et al. Mathematical model for the control of measles. J. Appl. Sci Environ. Manage, 22(4):571–576, 2018. https://doi.org/10.4314/jasem.v22i4.24
S. Rausanu and C. Grosan. A hierarchical network model for epidemic spreading. Analysis of a H1N1 virus spreading in Romania. Appl. Math Comput, 233:39–54, 2014. https://doi.org/10.1016/j.amc.2013.12.176
S. Saha and G.P. Samanta. Modelling and optimal control of HIV/AIDS prevention through PrEP and limited treatment. Physica A., 516:280–307, 2019. https://doi.org/10.1016/j.physa.2018.10.033
G.P. Samanta. Analysis of a nonautonomous HIV/AIDS epidemic model with distributed time delay. Mathematical Modelling and Analysis, 15(3):327–347, 2010. https://doi.org/10.3846/1392-6292.2010.15.327-347
G.P. Samanta, P. Sen and A. Maiti. A delayed epidemic model of diseases through droplet infection and direct contact with saturation incidence and pulse vaccination. Systems Science & Control Engenering: An Open Access Journal, 4(1):320–333, 2016. https://doi.org/10.1080/21642583.2016.1246982
M. Sasuzzoha, M. Singh and D. Lucy. Parameter estimation of influenza epidemic model. Appl. Math Comput, 220:616–629, 2013. https://doi.org/10.1016/j.amc.2013.07.040
S. Sharma and G.P. Samanta. Dynamical behaviour of an HIV/AIDS epidemic model. Differ Equ Dyn Syst., 22:369–395, 2014. https://doi.org/10.1007/s12591-013-0173-7
B. Shulgin, L. Stone and Z. Agur. Pulse vaccination strategy in the SIR epidemic model. Bull. Math. Biol., 60:1123–1148, 1998.
L. Stone, B. Shulgin and Z. Agur. Theoretical examination of the pulse vaccination policy in the SIR epidemic model. Math. Comput. Model., 31(4–5):207–215, 2000.
X.Li and B. Fang. Stability of an age-structured SEIR epidemic model with infectivity in latent period. Appl. Appl. Math, 4(1):218–236, 2009.
J. Zhang and Z. Ma. Global stability of SEIR model with saturating contact rate. Math. Biosci., 185(1):15–32, 2003. https://doi.org/10.1016/S0025-5564(03)00087-7
M.F. Boni, B.H. Manh, P.Q. Thai and et al. Modelling the progression of pandemic influenza a (H1N1) in Vietnam and the opportunities for reassortment with other influenza viruses. BMC Medicene, 7(43), 2009. https://doi.org/10.1186/1741-7015-7-43
T.A. Burton. Stability and periodic solutions of ordinary and functional differential equations. Academic Press, New York, 1985.
D. Greenhalgh. Hopf bifurcation in epidemic models with a latent period and nonpermanent immunity. Math. Comput. Model, 25:85–93, 1997.
D. Greenhalgh and I.A. Moneim. SIRS epidemic model and simulations using different types of seasonal contact rate. Systems Analysis Modelling Simulation, 43(5):573–600, 2003. https://doi.org/10.1080/023929021000008813
D. Grenhalgh. Some results for an SEIR epidemic model with density dependence in the death rate. IMA J. Math. Appl. Med. Biol., 9(2):67–106, 1992. https://doi.org/10.1093/imammb/9.2.67
R. Jan and Y. Xiao. Effect of pulse vaccination on dynamics of dengue with periodic transmission functions. Advances in Difference Equations, 219(1):368, 2019. https://doi.org/10.1186/s13662-019-2314-y
T. Kitano. Dynamic transmission model of routine mumps vaccination in Japan. Epidemiology and Infection, 147:1–8, 2018. https://doi.org/10.1017/S0950268818003230
M. De la Sen, A. Ibeas and S. Alonso-Quesada. On vaccination controls for the SEIR epidemic model. Communications in Nonlinear Science and Numerical Simulation, 17(6):3888–3904, 2012. https://doi.org/10.1016/j.cnsns.2011.10.012
M. De la Sen and S.Alonso-Quesada. Vaccination strategies based on feedback control techniques for a general SEIR epidemic model. Applied Mathematics and Computation, 218(7):2637–2658, 2011. https://doi.org/10.1016/j.amc.2011.09.036
G. Li and Z. Jin. Global stability of a SEIR epidemic model with infectious force in latent, infected and immune period. Chaos, Solitons and Fractals, 25(2):1177–1184, 2005. https://doi.org/10.1016/j.chaos.2004.11.062
M.Y. Li, J.S. Muldoweney, L.C. Wang and J. Karsai. Global dynamics of an SEIR epidemic model with a varying total population size. Math. Biosci., 160(2):191– 213, 1999. https://doi.org/10.1016/S0025-5564(99)00030-9
I.A. Moneim. Seasonally varying epidemics with and without latent period: A comparative simulation study. IMA J. of Mathematical Medicine and Biology, 24(1):1–15, 2007. https://doi.org/10.1093/imammb/dql023
I.A. Moneim. Different vaccination strategies for measles diseases: A simulation study. Journal of Informatics and Mathematical Sciences, 3:227–236, 2011.
I.A. Moneim. Efficiency of different vaccination strategies for childhood diseases: A simulation study. Advances in Bioscience and Biotechnology, 4(2):193–205, 2013. https://doi.org/10.4236/abb.2013.42028
I.A. Moneim. Modeling and simulation of the spread of H1N1 flu with periodic vaccination. IJB., 9(1), 2016. https://doi.org/10.1142/S1793524516500030
I.A. Moneim and D. Greenhalgh. Threshold and stability results for an SIRS epidemic model with a general periodic vaccination strategy. Journal of biological systems, 13(2):131–150, 2005. https://doi.org/10.1142/S0218339005001446
I.A. Moneim and D. Greenhalgh. Use of a periodic vaccination strategy to control the spread of epidemics with seasonally varying contact rate. Mathematical Bioscience and Engineering, 13(2):591–611, 2005. https://doi.org/10.3934/mbe.2005.2.591
I.A. Moneim and H.A. Khalil. Modelling and simulation of the spread of HBV disease with infectious latent. A M., 6(5):745–753, 2015. https://doi.org/10.4236/am.2015.65070
M.Y.Li and J.S. Muldoweney. Global dynamics of a SEIR epidemic model with vertical transmission. SIAM J. Appl.Math., 62(1):58–69, 2001. https://doi.org/10.1137/S0036139999359860
H. Nishiur and H. Inaba. Estimation of the incubation period of influenza a (H1N1) among imported cases: Addressing censoring using outbreak data origin of importation. J Theor Biol., 272(1):123–130, 2011. https://doi.org/10.1016/j.jtbi.2010.12.017
O.J. Peter, O.A. Afolabi, A.A. Victor and et al. Mathematical model for the control of measles. J. Appl. Sci Environ. Manage, 22(4):571–576, 2018. https://doi.org/10.4314/jasem.v22i4.24
S. Rausanu and C. Grosan. A hierarchical network model for epidemic spreading. Analysis of a H1N1 virus spreading in Romania. Appl. Math Comput, 233:39–54, 2014. https://doi.org/10.1016/j.amc.2013.12.176
S. Saha and G.P. Samanta. Modelling and optimal control of HIV/AIDS prevention through PrEP and limited treatment. Physica A., 516:280–307, 2019. https://doi.org/10.1016/j.physa.2018.10.033
G.P. Samanta. Analysis of a nonautonomous HIV/AIDS epidemic model with distributed time delay. Mathematical Modelling and Analysis, 15(3):327–347, 2010. https://doi.org/10.3846/1392-6292.2010.15.327-347
G.P. Samanta, P. Sen and A. Maiti. A delayed epidemic model of diseases through droplet infection and direct contact with saturation incidence and pulse vaccination. Systems Science & Control Engenering: An Open Access Journal, 4(1):320–333, 2016. https://doi.org/10.1080/21642583.2016.1246982
M. Sasuzzoha, M. Singh and D. Lucy. Parameter estimation of influenza epidemic model. Appl. Math Comput, 220:616–629, 2013. https://doi.org/10.1016/j.amc.2013.07.040
S. Sharma and G.P. Samanta. Dynamical behaviour of an HIV/AIDS epidemic model. Differ Equ Dyn Syst., 22:369–395, 2014. https://doi.org/10.1007/s12591-013-0173-7
B. Shulgin, L. Stone and Z. Agur. Pulse vaccination strategy in the SIR epidemic model. Bull. Math. Biol., 60:1123–1148, 1998.
L. Stone, B. Shulgin and Z. Agur. Theoretical examination of the pulse vaccination policy in the SIR epidemic model. Math. Comput. Model., 31(4–5):207–215, 2000.
X.Li and B. Fang. Stability of an age-structured SEIR epidemic model with infectivity in latent period. Appl. Appl. Math, 4(1):218–236, 2009.
J. Zhang and Z. Ma. Global stability of SEIR model with saturating contact rate. Math. Biosci., 185(1):15–32, 2003. https://doi.org/10.1016/S0025-5564(03)00087-7