AIMS Mathematics, 2020, 5(3): 2758-2779. doi: 10.3934/math.2020178.

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Parameter estimation and fractional derivatives of dengue transmission model

1 Department of Mathematics, Faculty of Science and Technology, Universitas Airlangga, Surabaya 60115, Indonesia
2 Faculty of Natural and Agricultural Sciences, University of the Free state, South Africa

Special Issues: Recent Advances in Fractional Calculus with Real World Applications

In this paper, we propose a parameter estimation of dengue fever transmission model using a particle swarm optimization method. This method is applied to estimate the parameters of the host-vector and SIR type dengue transmission models by using cumulative data of dengue patient in East Java province, Indonesia. Based on the parameter values, the basic reproduction number of both models are greater than one and obtained their value for SIR is $\mathcal{R}_0=1.4159$ and for vector host is $\mathcal{R}_0=1.1474$. We then formulate the models in fractional Atangana-Baleanu derivative that possess the property of nonlocal and nonsingular kernel that has been remained effective to many real-life problems. A numerical procedure for the solution of the model SIR model is shown. Some specific numerical values are considered to obtain the graphical results for both the SIR and Vector Host model. We show that the model vector host provide good results for data fitting than that of the SIR model.
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Keywords dengue model; parameter estimation; particle swarm optimization method; Atangana-Baleanu derivative

Citation: Windarto, Muhammad Altaf Khan, Fatmawati. Parameter estimation and fractional derivatives of dengue transmission model. AIMS Mathematics, 2020, 5(3): 2758-2779. doi: 10.3934/math.2020178

References

  • 1. World Health Organization, Fact sheet on the Dengue and severe dengue, WHO, 2017. Available from: http://www.who.int/mediacentre/factsheets/fs117/en/.
  • 2. World Health Organization, Global strategy for dengue prevention and control 2012-2020, WHO, 2012. Available from: http://www.who.int/denguecontrol/9789241504034/en/.
  • 3. Health Office (Dinas Kesehatan) of the East Java, Dinas Kesehatan Provinsi Jawa Timur, Surabaya, Indonesia, 2009.
  • 4. Ministry of Health Republic of Indonesia, Kementerian Kesehatan Republik Indonesia Jakarta, 2014.
  • 5. D. Aldila, T. Gotz, E. Soewono, An optimal control problem arising from a dengue disease transmission model, Math. Biosci., 242 (2013), 9-16.    
  • 6. H. Tasman, A. K. Supriatna, N. Nuraini, et al. A dengue vaccination model for immigrants in a two-age-class population, Int. J. Math., 2012 (2012), 1-15.
  • 7. J. P. Chavez, T. Gotz, S. Siegmund, et al. An SIR-Dengue transmission model with seasonal effects and impulsive control, Math. Biosci., 289 (2017), 29-39.    
  • 8. A. Pandey, A. Mubayi, J. Medlock, Comparing vector-host and SIR models for dengue transmission, Math. Biosci., 246 (2013), 252-259.    
  • 9. T. Gotz, N. Altmeier, W. Bock, et al. Modeling dengue data from Semarang, Indonesia, Ecol. Complex., 30 (2017), 57-62.    
  • 10. F. B. Agusto, M. A. Khan, Optimal control strategies for dengue transmission in pakistan, Math. Biosci., 305 (2018), 102-121.    
  • 11. N. Tutkun, Parameter estimation in mathematical models using the real coded genetic algorithms, Expert Syst. Appl., 36 (2009), 3342-3345.    
  • 12. R. L. Haupt, S. E. Haupt, Practical Genetic Algorithms, Second Edition, John Wiley & Sons, 2004.
  • 13. W. B. Roush, S. L. Branton, A Comparison of fitting growth models with a genetic algorithm and nonlinear regression, Poultry Sci., 84 (2005), 494-502.    
  • 14. W. S. W. Indratno, N. Nuraini, E. Soewono, A comparison of binary and continuous genetic algorithm in parameter estimation of a logistic growth model, American Institute of Physics Conference Series, 1587 (2014), 139-142.
  • 15. Windarto, An Implementation of continuous genetic algorithm in parameter estimation of predator-prey model, AIP Conference Proceedings, 2016.
  • 16. D. Akman, O. Akman, E. Schaefer, Parameter Estimation in Ordinary Differential Equations Modeling via Particle Swarm Optimization, J. Appl. Math., 2018 (2018), 1-9.
  • 17. Windarto, Eridani, U. D. Purwati, A comparison of continuous genetic algorithm and particle swarm optimization in parameter estimation of Gompertz growth model, AIP Conference Proceedings, 2084 (2019), 020017.
  • 18. R. Eberhart, J. Kennedy, A new optimizer using particle swarm theory, Proceedings of the Sixth International Symposium on Micro Machine and Human Science, 1995.
  • 19. K. Diethelm, A fractional calculus based model for the simulation of an outbreak of dengue fever, Nonlinear Dyn., 71 (2013), 613-619.    
  • 20. T. Sardar, S. Rana, J. Chattopadhyay, A mathematical model of dengue transmission with memory, Commun. Nonlinear Sci., 22 (2015), 511-525.    
  • 21. M. A. Khan, S. Ullah, M. Farooq, A new fractional model for tuberculosis with relapse via Atangana-Baleanu derivative, Chaos, Solitons & Fractals, 116 (2018), 227-238.
  • 22. S. Ullah, M. A. Khan, M. Farooq, Modeling and analysis of the fractional HBV model with Atangana-Baleanu derivative, The European Physical Journal Plus, 133 (2018), 313.
  • 23. E. O. Alzahrani, M. A. Khan, Modeling the dynamics of Hepatitis E with optimal control, Chaos, Solitons & Fractals, 116 (2018), 287-301.
  • 24. Fatmawati, E. M. Shaiful, M. I. Utoyo, A fractional order model for HIV dynamics in a two-sex population, Int. J. Math., 2018 (2018), 1-11.
  • 25. Fatmawati, M. A. Khan, M. Azizah, et al. A fractional model for the dynamics of competition between commercial and rural banks in Indonesia, Chaos, Solitons & Fractals, 122 (2019), 32-46.
  • 26. A. Atangana, D. Baleanu, New fractional derivatives with nonlocal and non-singular kernel: theory and application to heat transfer model, Therm. Sci., 20 (2016), 763-769.    
  • 27. S. Qureshi, A. Atangana, Mathematical analysis of dengue fever outbreak by novel fractional operators with field data, Physica A, 526 (2019), 121127.
  • 28. O. Diekmann, J. A. P. Heesterbeek, J. A. J. Metz, On the definition and the computation of the basic reproduction ratio R0 in models for infectious diseases in heterogenous populations, J. Math. Biol., 28 (1990), 362-382.
  • 29. O. Diekmann, J. A. P. Heesterbeek, Mathematical Epidemiology of Infectious Diseases, Model Building, Analysis and Interpretation, John Wiley & Son, 2000.
  • 30. P. van den Driessche, J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmition, Math. Biosci., 180 (2002), 29-48.    
  • 31. Wikipedia contributors: East Java, Wikipedia. Available from: https://en.wikipedia.org/wiki/East-Java.
  • 32. M. Toufik, A. Atangana, New numerical approximation of fractional derivative with non-local and non-singular kernel: application to chaotic models, Eur. Phys. J. Plus, 132 (2017), 444.

 

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