Mathematics Letters

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Global Stability Analysis of the SEIR Deterministic Model in the Presence of Treatment at the Latent Period

Received: 05 November 2018    Accepted: 19 November 2018    Published: 03 January 2019
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Abstract

In this paper, the objective was to find out the influence of introducing treatment at the latent period of disease (infectious) transmission as a result the global dynamics of the SEIR epidemic model with the introduction of treatment at the latent period is explored. The basic reproduction number is estimated. Whenever the basic reproduction number is not larger than unity (R0≤1) then the disease – free equilibrium is globally stable and the disease dies out. But when the basic reproduction number is larger than unity (R0>1), then there exist the endemic equilibrium point which is stable and hence the disease will persist. This was demonstrated with a tuberculosis data obtained from Amansie west district health directorate in the Ashanti region of Ghana.In this instance the endemic equilibrium point was found to be stable. The sensitivity analysis also revealed that increasing the treatment rate introduced at the latent period will reduce the value of the basic reproduction number.

DOI 10.11648/j.ml.20180404.12
Published in Mathematics Letters (Volume 4, Issue 4, December 2018)
Page(s) 67-73
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This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.

Copyright

Copyright © The Author(s), 2024. Published by Science Publishing Group

Keywords

Basic Reproduction Number, Disease – Free Equilibrium, Endemic Equilibrium Point

References
[1] D. Bernoulli, Reflexionssur les avantages de l’inoculation, Mercure de France (1760) 173. (Reprinted in L.P. Bouckaert, B.L. van der Waerden (Eds), Die Werke von Daniel Bernoulli, Bd. 2 Analysis und Wahrscheinlich- keitsrechnung, Birkhauser, Basel, 1982, p. 268).
[2] R. M. Anderson and R. M. May (1991). Infectious diseases of humans. Oxford, UK:Oxford university press.
[3] N. T. J. Bailey (1975). The mathematical theory of infectious diseases and its applications. London: Charles Griffin and company.
[4] H. Hethcote (2000). The mathematics of infectious diseases. SIAM review, 42, 599-653.
[5] J. M. Hyman and J. Li (1998). Modeling the effectiveness of isolation strategies in preventing STD epidemics. SIAM journal of applied mathematics, 58, 912.
[6] S. Olaniyi, M. A. Lawal, and O. S. Obabiyi, “Stability and sensitivity analysis of a deterministic epidemiological model with pseudo-recovery,” IAENG International Journal of AppliedMathematics, vol. 46, no.2, pp. 1–8, 2016.
[7] O. Diekmann, J. A. Heesterbeek, and J. A.Metz, “On the definition and the computation of the basic reproduction ratio R0 in models for infectious diseases in heterogeneous populations,” Journal of Mathematical Biology, vol. 28, no. 4, pp. 365–382, 1990.
[8] P. van den Driessche and J. Watmough (2002). Reproduction numbers and sub thresholdEndemic equilibria for compartmental models of disease transmission. Mathematical biosciences, 180, 29-48.
[9] Tom et al. SEIRS epidemics with disease fatalities in growing populations, Mathematical Biosciences (2017), doi: 10.1016/j.mbs.2017.11.006.
[10] F. Brauer, P. van den Driessche and J. Wu (2008). Mathematical epidemiology,Springer.
[11] X. Zhou and J.Cui (2011). Analysis of stability and bifurcation for an SEIR epidemic model with saturated recovery rate. Communications in nonlinear science and numerical simulation, 16, 4438-4450.
[12] Sarah A. Al – Sheikh (2012). Modeling and Analysis of an SEIR Epidemic Model with a Limited Resource for Treatment. Global Journals Inc. (USA).
[13] J. Zhang, J. Li, and Z. Ma (2006). Global dynamics of an SEIR epidemic model withimmigration of different compartments. ActaMathematicaScientia, 26, 551-567.
[14] N. Yi, Q. Zhang, K. Mao, D. Yang, and Q. Li (2009). Analysis and control of an SEIRepidemic system with nonlinear transmission rate. Mathematical and computer modeling, 50, 1498-1513.
[15] C. Sun, Y. Hsieh (2010).Global analysis of an SEIR model with varying populationsize and vaccination. Applied mathematical modeling, 34, 2685-2697.
[16] H. Shu, D. Fan, and J. Wei Global (2012). Stability of multi-group SEIR epidemic models with distributed delays and nonlinear transmission. Nonlinear Analysis: Real World Applications, 13, 1581-1592.
[17] J. M. Hyman and J. Li (1998). Modeling the effectiveness of isolation strategies in preventing STD epidemics. SIAM journal of applied mathematics, 58, 912.
[18] Z. Fang and H. R. Thieme (1995). Recurrent outbreak of childhood diseases revisited:The impact of solution. Mathematical biosciences, 128, 93.
[19] L. Wu and Z. Feng (2000). Homoclinic bifurcation in an SIQR model for childhooddiseases. Journal of differential equations, 168, 150.
[20] J. Wang, S. Liu, B. Zheng and Y. Takeuchi (2012). Qualitative and bifurcation analysis using an SIR model with a saturated treatment function. Mathematical and computer modeling, 55, 710-722.
[21] Index mundi, (2012). http://www.indexmundi.com/ghana/death_rate.html.
[22] Wikipedia. http://www.wikipedia.com/Transmission risks and rates, 2018.
[23] Jones, J. H., (2007). Notes on Reproductive Number.
[24] Sarkodie Eric (2014). Modeling the spread of tuberculosis in central region using the susceptible-exposed-infected-susceptible (seis) mathematical model.
[25] Saltelli, A., Ratto, M., Andres, T., Campolongo, F., Cariboni, J., Gatelli, D. Saisana, M., and Tarantola, S. (2008). Global Sensitivity Analysis. The Primer, John Wiley & Sons.
Author Information
  • Department of Mathematics and Statistics, University of Ghana, Legon, Ghana

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    Prince Osei Affi. (2019). Global Stability Analysis of the SEIR Deterministic Model in the Presence of Treatment at the Latent Period. Mathematics Letters, 4(4), 67-73. https://doi.org/10.11648/j.ml.20180404.12

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    Prince Osei Affi. Global Stability Analysis of the SEIR Deterministic Model in the Presence of Treatment at the Latent Period. Math. Lett. 2019, 4(4), 67-73. doi: 10.11648/j.ml.20180404.12

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    AMA Style

    Prince Osei Affi. Global Stability Analysis of the SEIR Deterministic Model in the Presence of Treatment at the Latent Period. Math Lett. 2019;4(4):67-73. doi: 10.11648/j.ml.20180404.12

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  • @article{10.11648/j.ml.20180404.12,
      author = {Prince Osei Affi},
      title = {Global Stability Analysis of the SEIR Deterministic Model in the Presence of Treatment at the Latent Period},
      journal = {Mathematics Letters},
      volume = {4},
      number = {4},
      pages = {67-73},
      doi = {10.11648/j.ml.20180404.12},
      url = {https://doi.org/10.11648/j.ml.20180404.12},
      eprint = {https://download.sciencepg.com/pdf/10.11648.j.ml.20180404.12},
      abstract = {In this paper, the objective was to find out the influence of introducing treatment at the latent period of disease (infectious) transmission as a result the global dynamics of the SEIR epidemic model with the introduction of treatment at the latent period is explored. The basic reproduction number is estimated. Whenever the basic reproduction number is not larger than unity (R0≤1) then the disease – free equilibrium is globally stable and the disease dies out. But when the basic reproduction number is larger than unity (R0>1), then there exist the endemic equilibrium point which is stable and hence the disease will persist. This was demonstrated with a tuberculosis data obtained from Amansie west district health directorate in the Ashanti region of Ghana.In this instance the endemic equilibrium point was found to be stable. The sensitivity analysis also revealed that increasing the treatment rate introduced at the latent period will reduce the value of the basic reproduction number.},
     year = {2019}
    }
    

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