Adaptive Step-Size Differential Transformation Method for Solving a Chikungunya Virus System
DOI:
https://doi.org/10.5269/bspm.82711Resumo
An often studied challenge in epidemiology is Chikungunya. To study the dynamics of the differential equation systemmodeling the spread of this Chikungunya virus within the human population, Chamekh et al. (2023) proposed a multi-step
semi-analytical solutions and obtained good results compared to classical numerical methods. However, the use of a fixed
step size in the method can compromise the efficiency and accuracy of calculations, especially in regions where the
system has rapid changes and a small number of subdivisions. To overcome this limitation, this paper proposes an
adaptive multi-step differential transformation method in which the step size is automatically adjusted based on local
error estimation. This adaptive approach improves the accuracy and robustness of the solution while preserving the
semi-analytical nature of the method. Numerical simulations indicate that the proposed method significantly outperforms
the fixed-step version in terms of accuracy and computational efficiency, articulately over long time intervals and
in the presence of stiff dynamics. These results extend and strengthen the scope of semi-analytical techniques in
epidemiological modeling.
Referências
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14. Radid, A. and Rhofir, K., Partitioning differential transformation method to a SIR epidemic model under vaccination strategy. Int. J. of Ad. Applied. Math and Mech, 7, 9-19, (2019).
15. Radid, A. and Rhofir, K. SOR-Like New Iterative Method for Solving the Epidemic Model and the Prey and Predator Problem. Discrete Dynamics in Nature and Society, ID 2020, 1-13, (2020).
16. Reiner, R.C., et al., A systematic review of mathematical models of mosquito-borne pathogen transmission. J R Soc Interface, 10(81), 20120921. (2013).
17. Shampine, L.F., Reichelt, M.W., The MATLAB ODE Suite. SIAM J Sci Comput, 18(1), 1–22, (1997).
18. Staples, J.E., Breiman, R.F., Powers, A.M., Chikungunya fever: an epidemiological review. Clin Infect Dis, 49(6), 942–948.(2009).
19. Tchuenche, J. M., Bauch, C. T., Modelling the impact of vector control on Chikungunya outbreaks. Mathematical Biosciences, 230(1), 20–29. (2011).
20. Weaver, S.C., Lecuit, M., Chikungunya virus and the global spread of a mosquito-borne disease. The New England journal of medicine vol. 372,13, 1231-9, (2015).
21. Zhou, J.K., Differential Transformation and Its Applications. Huazhong University Press. (1986).
2. Anderson, R. M., May, R. M. Infectious Diseases of Humans: Dynamics and Control. Oxford University Press. (1992).
3. Ayaz Fatma, Solutions of the system of differential equations by differential transform method, Appl. Math. Comput., 147, 547-567, (2004).
4. Brauer, F., Castillo-Chavez, C., Feng, Z., Mathematical Models in Epidemiology. Springer. (2019).
5. Burt, F.J., et al., Chikungunya: a re-emerging virus. Lancet, 379(9816), 662–671.(2012).
6. Butcher, J.C., Numerical Methods for Ordinary Differential Equations. Wiley. (2016).
7. Chamekh, M., Latrach, M.A., Jday, F., Multi-step semi-analytical solutions for a Chikungunya virus system. J Umm Al-Qura Univ Appl Sci, 9, 123–131. (2023).
8. Gökdoǧan, A., Merdan, M., Yildirim, A., A multistage differential transformation method for approximate solution of Hantavirus infection model. Commun Nonlinear Sci Numer Simul 17(1):1–8 (2012)
9. Hethcote, H.W., The mathematics of infectious diseases. SIAM Review, 42(4), 599–653. (2000).
10. Keeling, M.J., Rohani, P., Modeling Infectious Diseases in Humans and Animals. Princeton University Press. (2008).
11. Medlock, J. M., Hansford, K. M., Schaffner, F., Versteirt, V., Hendrickx, G., Zeller, H., Bortel, W. V. A review of the invasive mosquitoes in Europe: ecology, public health risks, and control options. Vector-borne and zoonotic diseases, 12(6), 435-447, (2012).
12. Morrison, T.E., Reemergence of chikungunya virus. J Virol, 88(20), 11644–11647.(2014).
13. Powers, A.M., Logue, C.H., Changing patterns of chikungunya virus: re-emergence of a zoonotic arbovirus. J Gen Virol, 88(9), 2363–2377.(2007).
14. Radid, A. and Rhofir, K., Partitioning differential transformation method to a SIR epidemic model under vaccination strategy. Int. J. of Ad. Applied. Math and Mech, 7, 9-19, (2019).
15. Radid, A. and Rhofir, K. SOR-Like New Iterative Method for Solving the Epidemic Model and the Prey and Predator Problem. Discrete Dynamics in Nature and Society, ID 2020, 1-13, (2020).
16. Reiner, R.C., et al., A systematic review of mathematical models of mosquito-borne pathogen transmission. J R Soc Interface, 10(81), 20120921. (2013).
17. Shampine, L.F., Reichelt, M.W., The MATLAB ODE Suite. SIAM J Sci Comput, 18(1), 1–22, (1997).
18. Staples, J.E., Breiman, R.F., Powers, A.M., Chikungunya fever: an epidemiological review. Clin Infect Dis, 49(6), 942–948.(2009).
19. Tchuenche, J. M., Bauch, C. T., Modelling the impact of vector control on Chikungunya outbreaks. Mathematical Biosciences, 230(1), 20–29. (2011).
20. Weaver, S.C., Lecuit, M., Chikungunya virus and the global spread of a mosquito-borne disease. The New England journal of medicine vol. 372,13, 1231-9, (2015).
21. Zhou, J.K., Differential Transformation and Its Applications. Huazhong University Press. (1986).
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Publicado
2026-07-01
Edição
Seção
Conf. Issue: Recent Advances in Applied Mathematics, Modeling, and Engineering
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Copyright (c) 2026 Boletim da Sociedade Paranaense de Matemática
Este trabalho está licenciado sob uma licença Creative Commons Attribution 4.0 International License.
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Como Citar
RADID, A., Rhofir, K., & LAARAJ, M. (2026). Adaptive Step-Size Differential Transformation Method for Solving a Chikungunya Virus System. Boletim Da Sociedade Paranaense De Matemática, 44(18), 1-12. https://doi.org/10.5269/bspm.82711
