Novel parameter estimation techniques for a multi-term fractional dynamical epidemic model of dengue fever

Abstract

An inverse problem to identify parameters for the single-term (and multi-term) fractional-order system of an outbreak of dengue fever is considered. Firstly, we propose a numerical method for the fractional-order dengue fever system based on the Gorenflo-Mainardi-Moretti-Paradisi (GMMP) scheme and the Newton method. Secondly, two methods, the modified grid approximation method (MGAM) and the modified hybrid Nelder-Mead simplex search and particle swarm optimization (MH-NMSS-PSO) algorithm are expanded to estimate the fractional orders and coefficients for fractional differential equations. Then, we use GMMP and MH-NMSS-PSO to estimate the parameters of the fractional-order dengue fever system. With the new fractional orders and parameters, our fractional-order dengue fever system is capable of providing numerical results that agree very well with the real data. Furthermore, for searching a better dengue fever system, a multi-term fractional-order epidemic system of dengue fever is proposed. We also use the MGAM and MH-NMSS-PSO to estimate the fractional orders and coefficients of the multi-term fractional-order system. To verify the efficiency and accuracy of the proposed methods in dealing with the fractional inverse problem, a numerical example with real data is investigated. Using the statistics from the 2009 outbreak of the disease in the Cape Verde islands, we are able to predict the fractional orders and parameters of the fractional dengue fever system. With the new fractional orders and parameters, our multi-term fractional-order dengue fever system is capable of providing numerical results that agree better with the real data than other integer-order models.

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Acknowledgements

The authors would like to thank the editor and the anonymous reviewers for their constructive comments and suggestions to improve the quality of the paper.

Funding

This work is partly supported by Australian Research Council (ARC) via the Discovery Projects DP180103858 and DP190101889; Natural Science Foundation of China (Grant Nos. 11771364, 11701397, 61573010), Sichuan Youth Science and Technology Foundation (Grant No.2016JQ0046), Found of Sichuan University of Science and Engineering (Grant No. 2016RCL33), and the State Scholarship Fund from China Scholarship Council.

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Li, T., Wang, Y., Liu, F. et al. Novel parameter estimation techniques for a multi-term fractional dynamical epidemic model of dengue fever. Numer Algor 82, 1467–1495 (2019). https://doi.org/10.1007/s11075-019-00665-2

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Keywords

  • Fractional dynamical epidemic model
  • Parameter estimation
  • Simplex search method
  • Grid approximation method
  • Inverse problem

Mathematics Subject Classification (2010)

  • 26A33
  • 34A24
  • 37M05