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Estimation of Parameters in the SIR Epidemic Model Using Particle Swarm Optimization

Received: 30 September 2019     Accepted: 25 October 2019     Published: 30 October 2019
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Abstract

Susceptible, Infected and Resistant (SIR) models are used to observe the spread of infection from infected populations into healthy populations. Stability analysis of the model is done using the Routh-Hurwitz criteria, basic reproduction number or the Lyapunov Stability. For stability analysis, parameters value are needed and these values are usually assumed. Given data cannot be used to determine the parameter values of SIR model because analytic solution of system of nonlinear differential equation cannot be determined. In this article, we determine the parameters of the exponential growth model, logistic model and SIR models using the Particle Swarm Optimization (PSO) algorithm. The SIR model is solved numerically using the Euler method based on the parameter values determined by PSO. The simulation results show that the PSO algorithm is good enough in determining the parameters of the three models compared to analytical methods and the Gauss-Newton’s method. Based on the average hypothesis test the relative error obtained from the PSO algorithm to determine the parameters is less than 3% with a significance level of 1%.

Published in American Journal of Mathematical and Computer Modelling (Volume 4, Issue 4)
DOI 10.11648/j.ajmcm.20190404.11
Page(s) 83-93
Creative Commons

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), 2019. Published by Science Publishing Group

Keywords

Growth Mathematical Model, SIR Model, Curve Fitting, PSO Algorithm, Estimation of Parameters

References
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[2] Bonyah, E., Okosun, K. O., Mathematical modeling of Zika virus, Asian Pasific Journal of Tropical Disease, 6 (9): 673-679, 2016.
[3] Gebremeskel, A. A., Krogstad, H. E., Mathematical modeling of Endemic Malaria Transmission, American Journal of Applied Mathematics, 3 (2): 36-46, 2015.
[4] Sandhya, Kumar, D., Mathematical model for glucose-insulin regulatory system of diabetes mellitus, Advances in Applied Mathematical Biosciences, Vol 2 No 1, 2011.
[5] Mu'tamar, K., Optimal control strategy for alcoholism model with two infected compartments, IOSR Journal of Mathematics, Vol. 14 Issue 3 Ver. I, 58-67, 2018.
[6] Shukla, J. B., Singh, G., Shukla, P., Tripathi, A., Modeling and analysis of the effects of antivirus software on an infected computer network, Applied Mathematic and Computation, 227 (2014): 11-18, 2014.
[7] Kennedy, J. and Eberhart, R. Particle Swarm Optimization. Proceedings of the IEEE International Conference on Neural Networks, 4, 1942-1948, 1995.
[8] Bratton, D., Kennedy, J., Defining a standard for particle swarm optimization. Proceeding of the 2007 IEEE Swarm Intelligence Symposium, (1-4244-0708-7/07), 2007.
[9] Naiborhu, J., Firman, Mu'tamar, K., Particle swarm optimization in the exact linearization technic for output tracking of non-minimum phase nonlinear systems, Applied Mathematical Sciences, Vol. 7 No 109, 5427-5442, 2013.
[10] Mu'tamar, K., Naiborhu, J., Penentuan matriks pembobot pada kontrol optimal menggunakan adaptive particle swarm optimization, Jurnal Aplikasi Teknologi Universitas Pasir Pengaraian, Vol 8 No 1, 2016.
[11] Hasni, A. Taibi, R., Draoui, B., Boulard, T., Optimization of greenhouse climate model parameters using particle swarm optimization and genetic algorithms, ScienceDirect: Energy Procidia, 6, 371-380, 2011.
[12] Jalilvand, A., Kimiyaghalam, A., Ashouri, A., Kord, H., Optimal tunning of PID controller parameters on a DC motor based on advanced particle swarm optimization algorithm, International Journal on technical and physical problems of Engineering, Vol 3 No 4 Issue 9, 10-17, 2011.
[13] Chiu, C. C., Cheng, Y. T., Chang, C. W., Comparison of particle swarm optimization and genetic algorithm for the path loss reduction in an urban area, Journal of applied science and engineering, Vol 15 No 4, pp. 371-380, 2012.
[14] Solihin, M. I., Akmeliawati, R., Particle swarm optimization for stabilizing controller of self-erecting linear inverted pendulum, International Journal of Electrical and Electronic Systems Research, Vol. 3, 13-23, 2010.
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  • APA Style

    Supriadi Putra, Khozin Mu'tamar, Zulkarnain. (2019). Estimation of Parameters in the SIR Epidemic Model Using Particle Swarm Optimization. American Journal of Mathematical and Computer Modelling, 4(4), 83-93. https://doi.org/10.11648/j.ajmcm.20190404.11

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    ACS Style

    Supriadi Putra; Khozin Mu'tamar; Zulkarnain. Estimation of Parameters in the SIR Epidemic Model Using Particle Swarm Optimization. Am. J. Math. Comput. Model. 2019, 4(4), 83-93. doi: 10.11648/j.ajmcm.20190404.11

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    AMA Style

    Supriadi Putra, Khozin Mu'tamar, Zulkarnain. Estimation of Parameters in the SIR Epidemic Model Using Particle Swarm Optimization. Am J Math Comput Model. 2019;4(4):83-93. doi: 10.11648/j.ajmcm.20190404.11

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  • @article{10.11648/j.ajmcm.20190404.11,
      author = {Supriadi Putra and Khozin Mu'tamar and Zulkarnain},
      title = {Estimation of Parameters in the SIR Epidemic Model Using Particle Swarm Optimization},
      journal = {American Journal of Mathematical and Computer Modelling},
      volume = {4},
      number = {4},
      pages = {83-93},
      doi = {10.11648/j.ajmcm.20190404.11},
      url = {https://doi.org/10.11648/j.ajmcm.20190404.11},
      eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajmcm.20190404.11},
      abstract = {Susceptible, Infected and Resistant (SIR) models are used to observe the spread of infection from infected populations into healthy populations. Stability analysis of the model is done using the Routh-Hurwitz criteria, basic reproduction number or the Lyapunov Stability. For stability analysis, parameters value are needed and these values are usually assumed. Given data cannot be used to determine the parameter values of SIR model because analytic solution of system of nonlinear differential equation cannot be determined. In this article, we determine the parameters of the exponential growth model, logistic model and SIR models using the Particle Swarm Optimization (PSO) algorithm. The SIR model is solved numerically using the Euler method based on the parameter values determined by PSO. The simulation results show that the PSO algorithm is good enough in determining the parameters of the three models compared to analytical methods and the Gauss-Newton’s method. Based on the average hypothesis test the relative error obtained from the PSO algorithm to determine the parameters is less than 3% with a significance level of 1%.},
     year = {2019}
    }
    

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  • TY  - JOUR
    T1  - Estimation of Parameters in the SIR Epidemic Model Using Particle Swarm Optimization
    AU  - Supriadi Putra
    AU  - Khozin Mu'tamar
    AU  - Zulkarnain
    Y1  - 2019/10/30
    PY  - 2019
    N1  - https://doi.org/10.11648/j.ajmcm.20190404.11
    DO  - 10.11648/j.ajmcm.20190404.11
    T2  - American Journal of Mathematical and Computer Modelling
    JF  - American Journal of Mathematical and Computer Modelling
    JO  - American Journal of Mathematical and Computer Modelling
    SP  - 83
    EP  - 93
    PB  - Science Publishing Group
    SN  - 2578-8280
    UR  - https://doi.org/10.11648/j.ajmcm.20190404.11
    AB  - Susceptible, Infected and Resistant (SIR) models are used to observe the spread of infection from infected populations into healthy populations. Stability analysis of the model is done using the Routh-Hurwitz criteria, basic reproduction number or the Lyapunov Stability. For stability analysis, parameters value are needed and these values are usually assumed. Given data cannot be used to determine the parameter values of SIR model because analytic solution of system of nonlinear differential equation cannot be determined. In this article, we determine the parameters of the exponential growth model, logistic model and SIR models using the Particle Swarm Optimization (PSO) algorithm. The SIR model is solved numerically using the Euler method based on the parameter values determined by PSO. The simulation results show that the PSO algorithm is good enough in determining the parameters of the three models compared to analytical methods and the Gauss-Newton’s method. Based on the average hypothesis test the relative error obtained from the PSO algorithm to determine the parameters is less than 3% with a significance level of 1%.
    VL  - 4
    IS  - 4
    ER  - 

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Author Information
  • Department of Mathematics, University of Riau, Pekanbaru, Indonesia

  • Department of Mathematics, University of Riau, Pekanbaru, Indonesia

  • Department of Mathematics, University of Riau, Pekanbaru, Indonesia

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