APPROXIMATE SOLUTION OF FRACTIONAL ORDER MATHEMATICAL MODEL ON THE CO-TRANSMISSION OF ZIKA AND CHIKUNGUNYA VIRUS USING LAPLACE ADOMIAN DECOMPOSITION METHOD
Keywords:
Laplace Adomians Decomposition Method (LADM) , Zika virus , Chikungunya virus, co-infections, series solutions
Abstract
Gaining insight into the transmission dynamics of the Zika and Chikungunya viruses, as well as their co-infection, is essential for implementing efficient public health interventions. This paper presents a comprehensive fractional order mathematical model consisting of thirteen non-linear compartments to accurately represent the intricate interactions between humans and infected mosquito populations, as well as the challenges associated with their identification. In order to solve this model, we utilize the Laplace Adomians Decomposition Method (LADM), which is a very effective analytical technique for solving nonlinear differential equations. By utilizing LADM, we obtained infinite series solutions for the previously given model that ultimately converged to its precise solutions. The numerical simulations of the model demonstrate the transmission patterns of Zika virus, Chikungunya virus, and their co-infections for different values of . We utilized the fmicon algorithm, a MATLAB optimization tool, to accurately fit into the model, real-life data from Espirito Santos State in Brazil, where two viruses are concurrently spreading. The simulation deduce that, reducing mosquito biting rates and promoting compliance with treated bed net usage can substantially mitigate Zika-Chikungunya co-infection dynamics.
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References
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[23] Chahar, H. S., Bharaj, P., Dar, L., Guleria, R., Kabra, S. K., &Broor, S. (2009). Co-infections with chikungunya virus and dengue virus in Delhi, India. Emerg Infect Dis., 15, 1077–1080.
[24] Villamil-Gómez, W. E., González-Camargo, O., Rodriguez-Ayubi, J., Zapata-Serpa, D., &Rodriguez-Morales, A. J. (2016). Dengue, chikungunya and zika co-infection in a patient from Colombia. Journal of Infection and Public Health, 9(4), 684–686.
[25] Atokolo, W., Aja, R., Stephen, A., Onah, I., & Mbah, G.,. (2022). Approximate Solution of the Fractional Order Sterile Insect Technology Model via the Laplace-Adomian Decomposition Method for the Spread of Zika Virus Disease. International Journal of Mathematics and Mathematical Sciences. 1-24. 10.1155/2022/2297630
[26]Acheneje, G.O., Omale, D., Agbata, B.C., Atokolo, W., Shior, M.M., Bolarinwa, B. (2024) Approximate Solution of the Fractional Order Mathematical Model on the Transmission Dynamics on The Co-Infection of COVID-19 and Monkeypox Using the Laplace-Adomian Decomposition Method, International Journal of Mathematics and Statistics Studies, 12 (3), 17-51
[27] Adejoh, P. and Mbah, G. C. E. (2018). “Application of Fractional Differential Equations in Studying the Dynamics of Cancer Disease with its Control”, University of Nigeria, Nsukka, M. Sc,
[28] Haq, F., Shahzad, M., Muhammad, S., Wahab, H. A. & Rahman, G. (2017). “Numerical analysis of fractional order epidemic model of childhood diseases,” Discrete Dynamics in Nature and Society, vol. 2017, pp. 1–7.
[29] Das¸ B. (2021). “Stability analysis of an incommensurate fractional-order SIR model,” Mathematical Modelling and Numerical Simulation with Applications, vol. 1, no. 1, pp. 44–55.
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[31] Sene, N. (022). “Second-grade fluid with Newtonian heating under Caputo fractional derivative: analytical investigations via Laplace transforms,” Mathematical Modelling and Numerical Simulation with Applications, vol. 2, no. 1, pp. 13–252.
[32] Yavuz, M. and Ozdemir, N. “Numerical inverse Laplace homotopy technique for fractional heat equations,” 9ermal Science, vol. 22, pp. 185–194, 2018.
[33] Yavuz, M. and Sene, N. ( 2020). “Approximate solutions of the model describing fluid flow using generalized ρ-laplace transform method and heat balance integral method,” Axioms, vol. 9, no. 4, p. 123.
[34] Jan, A., Jan, R., Khan, H., Zobaer, M. S. & Shah, R. (2020). “Fractional-order dynamics of rift valley fever in ruminant host with vaccination” Commun. Math. Biol. Neurosci. Article-ID.Google Scholar
[35] Jan, R. et al. (2023). “Optimization of the fractional-order parameter with the error analysis for human immunodeficiency virus under Caputo operator”. Discrete Contin. Dyn. Syst. S 16, 2118–2140 .Article MathSciNet MATHGoogle Schola.
[36] Bonyah, E., Khan, M. A., Okosun, K. O., & Gómez-Aguilar, J. F. (2019). On the co-infection of dengue fever and Zika virus. Optimization: A Journal of Mathematical Programming and Operations Research, 40(3), 394–421.
[37] Omame, A., Isah, M. E., & Abbas, M. (2022). An optimal control model for COVID-19, Zikadengue, and chikungunya co-dynamics with reinfection. Optimization: A Journal of Mathematical Programming and Operations Research,1-35. doi:10.1002/oca.2936.
[38] Goswami,N.K., &Shanmukha,B. (2019). Stability and optimal control analysis of Zika virus with saturated incident rate. Modern Applied Science ,13(3),78-85.
[39] Jose,S.A., Raja,R., Omede,B.I., Agarwal, R.P., Alzabut, J., Cao,J., &Balas,V.E. (2022). Mathematical modelling on coinfection: transmission dynamics of Zika virus and Dengue fever.NonlinearDyn.https://doi.org/10.1007/S11071-022-08063-5
[40] Van-Driessche,D., &Watmough,J., (2002). Reproduction numbers and sub-threshold endemic equillibria for compartmental models of disease transmission. Math.BioSci. 180(1),2948.
serologicalspecificity.Trans. R. Soc. Trop. Med. Hyg.,46(2), 509–520.https://doi.org/10.1016/0035-9203(52)900424
[2] Macnamara, F. N. (1954). Zika virus: A report on three cases of human infection during an
epidemic of jaundice in Nigeria. Transactions of the Royal Society of Tropical Medicine
and Hygiene, 48(1), 139–145. https://doi.org/10.1016/0035-9203(54)90006-1
[3] Olson, J. G., Ksiazek, T. G., Suhandiman, T., Triwibowo, T. (1981). Zika virus, a cause of
fever in Central Java, Indonesia. Transactions of the Royal Society of Tropical Medicine and Hygiene, 75(3), 389–393. https://doi.org/10.1016/0035-9203(81)90100-0
[4] Duffy, M. R., Chen, T. H., Hancock, W. T., Powers, A. M., Kool, J. L., Lanciotti, R. S.,
(2009). Zika virus outbreak on Yap Island, Federated States of Micronesia. New England Journal of Medicine, 360(25), 2536–2543. https://doi:10.1056/NEJMoa0805715
[5] Zanluca, C., Melo, V. C. A. D., Mosimann, A. L. P., Santos, C. N. D. D., Santos, G. I. V., &Luz, K. (2015). First report of autochthonous transmission of Zika virus in Brazil. Memorias do Instituto Oswaldo Cruz, 110(3), 569–572. https://doi:10.1590/0074- 02760150192
[6] Rocklöv, J., Quam, M. B., Sudre, B., German, M., Kraemer, M. U. G., Brady, O., ... (2016). Assessing seasonal risks for the introduction and mosquito-borne spread of Zika virus in Europe. EBioMedicine, 9, 250–256. https://doi:10.1016/j.ebiom.2016.06.009
[7] Watson-Brown, P., Viennet, E., Mincham, G., Williams, C. R., Jansen, C. C., Montgomery, B. L.,(2019). Epidemic potential of Zika virus in Australia: implications for blood transfusion safety. Transfusion, 59(5), 648–658. https://doi:10.1111/trf.15095
[8] Centers for Disease Control and Prevention. (2018). Zika virus. Retrieved on 28/10/2023 from https://www.cdc.gov/zika/
[9] Tataryn, J., Vrbova, L., Drebot, M., Wood, H., Payne, E., Connors, S., ... (2018). Travel-related Zika virus cases in Canada: October 2015-June 2017. Canadian Communicable Disease Report, 44(ACS-2), 18–26. https://doi:10.14745/ccdr.v44i01a05
[10] Hashimoto, T., Kutsuna, S., Tajima, S., Nakayama, E., Maeki, T., Taniguchi, S., ... (2017). Importation of Zika Virus from Vietnam to Japan, November 2016. Emerging Infectious Diseases, 23(7), 1223–1225. https://doi:10.3201/eid2307.170519
[11] Jia, H., Zhang, M., Chen, M., Yang, Z., Li, J., Huang, G., ... (2018). Zika virus infection in travelers returning from countries with local transmission, Guangdong, China, 2016. Travel Medicine and Infectious Disease, 21, 56–61. https://doi:10.1016/j.tmaid.2017.11.012
[12] Singh, H., Singh, O. P., Akhtar, N., Sharma, G., Sindhania, A., Gupta, N., ... (2019). First report on the transmission of Zika virus by Aedes (Stegomyia) aegypti (L.) (Diptera: Culicidae) during the 2018 Zika outbreak in India. ActaTropica, 199, 1–6.
https://doi:10.1016/j.actatropica.2019.105114
[13] International Travel Health Advisory Network. (2022). ITHC: Evidence-based travel health &tropical medicine portal. Retrieved on 22/10/2023from https://www.ithc.cn/article/460057.html
[14] World Health Organization. (2018).Zika virus.Retrieved ON 05/11/2023 from https://www.who.int/mediacentre/factsheets/zika/en/
[15] Russell, K., Hills, S. L., Oster, A. M., Porse, C. C., Danyluk, G., Cone, M., ... (2017). Male to-female sexual transmission of Zika virus-United States, January-April 2016. Clinical Infectious Diseases, 64(2), 211–213. https://doi:10.1093/cid/ciw692
[16] Deckard, D. T., Chung, W. M., Brooks, J. T., Smith, J. C., Woldai, S., Hennessey, M., (2016). Male-to-male sexual transmission of Zika virus-Texas, January 2016. Morbidity and Mortality Weekly Report, 65(13), 371–374.http://dx.doi.org/10.15585/mmwr.mm6514a3
[17] Thangamani, S., Huang, J., Hart, C. E., Guzman, H.,&Tesh, R. B. (2016). Vertical transmission of Zika virus in Aedesaegypti mosquitoes. American Journal of Tropical Medicine and Hygiene, 95(6), 1169–1173. doi:10.4269/ajtmh.16-0448.
[18] Du, S., Liu, Y., Liu, J., Zhao, J., Champagne, C., Tong, L., (2019). Aedes mosquitoes acquire and transmit Zika virus by breeding in contaminated aquatic environments. Nature Communications, 10, 1–11. doi:10.1038/s41467-019-09256-0.
[19] Microcephaly Epidemic Research Group. (2016). Microcephaly in infants, Pernambuco state, Brazil,2015.Emerging Infectious Diseases, 22(6), 1090–1093. doi:10.3201/eid2206.160062.
[20] Munoz, L. S., Barreras, P., & Pardo, C. A. (2016). Zika virus-associated neurological disease in the adult: Guillain-Barré syndrome, encephalitis, and myelitis. Seminars in Reproductive Medicine, 34(5), 273–279. doi:10.1055/s-0036-1592066.
[21]History of Chikungunya Disease. (2017, July 15). Retrieved from http://www.chikungunya.in/
[22] Dupont-Rouzeyrol, M., O'Connor, O., Calvez, E., Daurès, M., John, M., &Grangeon, J. P. (2015). Co-infection with Zika and dengue viruses in 2 patients, New Caledonia, 2014. Emerging Infectious Diseases, 21(2), 381–382.
[23] Chahar, H. S., Bharaj, P., Dar, L., Guleria, R., Kabra, S. K., &Broor, S. (2009). Co-infections with chikungunya virus and dengue virus in Delhi, India. Emerg Infect Dis., 15, 1077–1080.
[24] Villamil-Gómez, W. E., González-Camargo, O., Rodriguez-Ayubi, J., Zapata-Serpa, D., &Rodriguez-Morales, A. J. (2016). Dengue, chikungunya and zika co-infection in a patient from Colombia. Journal of Infection and Public Health, 9(4), 684–686.
[25] Atokolo, W., Aja, R., Stephen, A., Onah, I., & Mbah, G.,. (2022). Approximate Solution of the Fractional Order Sterile Insect Technology Model via the Laplace-Adomian Decomposition Method for the Spread of Zika Virus Disease. International Journal of Mathematics and Mathematical Sciences. 1-24. 10.1155/2022/2297630
[26]Acheneje, G.O., Omale, D., Agbata, B.C., Atokolo, W., Shior, M.M., Bolarinwa, B. (2024) Approximate Solution of the Fractional Order Mathematical Model on the Transmission Dynamics on The Co-Infection of COVID-19 and Monkeypox Using the Laplace-Adomian Decomposition Method, International Journal of Mathematics and Statistics Studies, 12 (3), 17-51
[27] Adejoh, P. and Mbah, G. C. E. (2018). “Application of Fractional Differential Equations in Studying the Dynamics of Cancer Disease with its Control”, University of Nigeria, Nsukka, M. Sc,
[28] Haq, F., Shahzad, M., Muhammad, S., Wahab, H. A. & Rahman, G. (2017). “Numerical analysis of fractional order epidemic model of childhood diseases,” Discrete Dynamics in Nature and Society, vol. 2017, pp. 1–7.
[29] Das¸ B. (2021). “Stability analysis of an incommensurate fractional-order SIR model,” Mathematical Modelling and Numerical Simulation with Applications, vol. 1, no. 1, pp. 44–55.
[30] Akg¨ul, E. K., Akg¨ul, A. & Yavuz, M. (2021). “New illustrative applications of integral transforms to financial models with different fractional derivatives,” Chaos, Solitons & Fractals, vol. 146, Article ID 110877.
[31] Sene, N. (022). “Second-grade fluid with Newtonian heating under Caputo fractional derivative: analytical investigations via Laplace transforms,” Mathematical Modelling and Numerical Simulation with Applications, vol. 2, no. 1, pp. 13–252.
[32] Yavuz, M. and Ozdemir, N. “Numerical inverse Laplace homotopy technique for fractional heat equations,” 9ermal Science, vol. 22, pp. 185–194, 2018.
[33] Yavuz, M. and Sene, N. ( 2020). “Approximate solutions of the model describing fluid flow using generalized ρ-laplace transform method and heat balance integral method,” Axioms, vol. 9, no. 4, p. 123.
[34] Jan, A., Jan, R., Khan, H., Zobaer, M. S. & Shah, R. (2020). “Fractional-order dynamics of rift valley fever in ruminant host with vaccination” Commun. Math. Biol. Neurosci. Article-ID.Google Scholar
[35] Jan, R. et al. (2023). “Optimization of the fractional-order parameter with the error analysis for human immunodeficiency virus under Caputo operator”. Discrete Contin. Dyn. Syst. S 16, 2118–2140 .Article MathSciNet MATHGoogle Schola.
[36] Bonyah, E., Khan, M. A., Okosun, K. O., & Gómez-Aguilar, J. F. (2019). On the co-infection of dengue fever and Zika virus. Optimization: A Journal of Mathematical Programming and Operations Research, 40(3), 394–421.
[37] Omame, A., Isah, M. E., & Abbas, M. (2022). An optimal control model for COVID-19, Zikadengue, and chikungunya co-dynamics with reinfection. Optimization: A Journal of Mathematical Programming and Operations Research,1-35. doi:10.1002/oca.2936.
[38] Goswami,N.K., &Shanmukha,B. (2019). Stability and optimal control analysis of Zika virus with saturated incident rate. Modern Applied Science ,13(3),78-85.
[39] Jose,S.A., Raja,R., Omede,B.I., Agarwal, R.P., Alzabut, J., Cao,J., &Balas,V.E. (2022). Mathematical modelling on coinfection: transmission dynamics of Zika virus and Dengue fever.NonlinearDyn.https://doi.org/10.1007/S11071-022-08063-5
[40] Van-Driessche,D., &Watmough,J., (2002). Reproduction numbers and sub-threshold endemic equillibria for compartmental models of disease transmission. Math.BioSci. 180(1),2948.
Published
2024-04-19
How to Cite
Agahiu, N., Bolaji, B., Atokolo, W., & Acheneje O., G. (2024). APPROXIMATE SOLUTION OF FRACTIONAL ORDER MATHEMATICAL MODEL ON THE CO-TRANSMISSION OF ZIKA AND CHIKUNGUNYA VIRUS USING LAPLACE ADOMIAN DECOMPOSITION METHOD. GPH - International Journal of Mathematics, 7(03), 47-81. https://doi.org/10.5281/zenodo.10995552
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