Caputo fractional order derivative model of Zika virus transmission dynamics

Volume 28, Issue 2, pp 145--157 http://dx.doi.org/10.22436/jmcs.028.02.03

Publication Date: May 14, 2022 Submission Date: January 08, 2022 Revision Date: March 06, 2022 Accteptance Date: March 21, 2022

Download PDF

Download XML

• Export citations
  • CITATIONS & REFERENCES
  • EndNote (.ENW)
  • Mendeley, JabRef (.BIB)
  • Papers, Zotero, Reference Manager, RefWorks (.RIS)
  • ARTICLE CITATION
  • EndNote (.ENW)
  • Mendeley, JabRef (.BIB)
  • Papers, Zotero, Reference Manager, RefWorks (.RIS)
  • REFERENCES
  • EndNote (.ENW)
  • Mendeley, JabRef (.BIB)
  • Papers, Zotero, Reference Manager, RefWorks (.RIS)

• 1301 Downloads
• 2872 Views

Authors

R. Prasad - Department of Mathematics, Gargi College (University of Delhi), New Delhi, Delhi 110049, India. K. Kumar - Department of Mathematics, Atma Ram Sanatan Dharma College (University of Delhi), New Delhi, Delhi 110021, India. R. Dohare - Centre for Interdisciplinary Research in Basic Sciences, Jamia Millia Islamia, New Delhi, Delhi 110025, India.

Abstract

The Zika Virus (ZIKV) is a highly contagious disease, and several outbreaks have occurred since it emerged. It is transmitted from one to another human via a mosquito Aedes aegypti. There is no vaccine or established medicine available for ZIKV to date. There is an urgent need to enhance an understanding of the progression mechanism of the disease when drugs or vaccines are not available. Mathematical modeling is a tool that might be helpful to understand the progression dynamics of ZIKV which can enable us to make control strategies for invading the progression dynamics of disease. SEIR-SEI is a famous compartmental deterministic modeling based on integer-order derivative calculus. Nowadays, conversion from integer to fractional order-based derivative modeling is in trend, and it is a very effective and high degree of accuracy. In this paper, we proposed a Caputo fractional-order based susceptible-exposed-infected-recovered (SEIR) structure for hosts and a susceptible-exposed-infected (SEI) structure for mosquitoes for transmission dynamics of ZIKV. For this purpose, we modified the classical compartmental model used in the study of progression dynamics of the Zika fever outbreak in El-Salvador during 2015-16. The modified model involves nonlinear differential equations of fractional (non-integer) order which has an advantage over the classical model due to its memory effect property. Our study includes eight regions across the globe where the Zika outbreak has occurred during the year 2013-2016 including six major archipelagos of French Polynesia, i.e., Tahiti, Sous-le-vent, Moorea, Tuamotu, Marquises, and Australes. The other two regions included Costa Rica and Colombia. The outbreak in selected regions was studied first using a classical model and then compared by a fractional-order model. The data of outbreaks are best fitted with the fractional-order model which enables us to estimate the best parameters values for the outbreaks. Using this modeling, the epidemic threshold parameter $R_0$ was computed which is more accurate than the classical one. Hence, the fractional-order model for ZIKV transmission dynamics is much better prediction, analysis, and disease parameters estimation than the classical model. This modeling enhances the knowledge in the field of fractional order and understanding the ZIKV transmission accurately.

Share and Cite

Keywords

• Fractional-order
• transmission dynamics
• basics reproduction number
• ZIKV

MSC

•  92B05
•  92D25
•  92D30
•  26A33

References

• [1] F. B. Agusto, S. Bewick, W. F. Fagan, Mathematical model of Zika virus with vertical transmission, Infectious Disease Modelling, 2 (2017), 244--267
  • View Article
  • Google Scholar


• [2] A. Ali, S. Islam, M. R. Khan, S. Rasheed, F. M. Allehiany, J. Baili, M. A. Khan, H. Ahmad, Dynamics of a fractional order Zika virus model with mutant, Alexandria Eng. J., 61 (2022), 4821--4836
  • View Article
  • Google Scholar


• [3] B. S. T. Alkahtani, A. Atangana, I. Koca, Novel analysis of the fractional Zika model using the Adams type predictorcorrector rule for non-singular and non-local fractional operators, J. Nonlinear Sci. Appl., 10 (2017), 3191--3200
  • View Article
  • MathSciNet


• [4] R. M. Anderson, Directly transmitted viral and bacterial infections of man, in: The Population Dynamics of Infectious Diseases, 1982 (1982), 1--37
  • View Article
  • Google Scholar


• [5] M. Andraud, N. Hens, P. Beutels, A simple periodic-forced model for dengue fitted to incidence data in Singapore, Math. Biosci., 244 (2013), 22--28
  • View Article
  • MathSciNet


• [6] A. Atangana, E. F. Doungmo Goufo, Computational analysis of the model describing HIV infection of CD4+ T cells, BioMed Res. Int., 2014 (2014), 14 pages
  • View Article
  • Google Scholar


• [7] S. Binder, A. M. Levitt, J. J. Sacks, J. M. Hughes, Emerging infectious diseases: public health issues for the 21st century, Science, 284 (1999), 1311--1313
  • View Article
  • Google Scholar


• [8] C. Caminade, J. Turner, S. Metelmann, J. C. Hesson, M. S. C. Blagrove, T. Solomon, A. P. Morse, M. Baylis, Global risk model for vector-borne transmission of Zika virus reveals the role of El Niño 2015, Proc. Nat. Acad. Sci., 114 (2017), 119--124
  • View Article
  • Google Scholar


• [9] ] G. S. Campos, A. C. Bandeira, S. I. Sardi, Zika virus outbreak, Bahia, Brazil, Emerging Infectious Diseases, 21 (2015), 1885--1886
  • View Article
  • Google Scholar


• [10] V. M. Cao-Lormeau, C. Roche, A. Teissier,E. Robin, A.-L. Berry, H.-P. Mallet, A. A. Sall, D. Musso, Zika virus, French polynesia, South pacific, Emerging Infectious Diseases, 20 (2014), 16 pages
  • View Article
  • Google Scholar


• [11] C. Chiyaka, W. Garira, S. Dube, Modelling immune response and drug therapy in human malaria infection, Comput. Math. Methods Med., 9 (2008), 143--163
  • View Article
  • MathSciNet


• [12] L. C. C. Medeiros, C. A. R. Castilho, C. Braga, W. V. de Souza, L. Regis, A. M. V. Monteiro, Modeling the dynamic transmission of dengue fever: investigating disease persistence, PLOS Neglected Tropical Diseases, 5 (2011), 13 pages
  • View Article
  • Google Scholar


• [13] M. Derouich, A. Boutayeb, E. H. Twizell, A model of dengue fever, BioMed. Eng. OnLine, 2 (2003), 10 pages
  • View Article
  • Google Scholar


• [14] O. Diekmann, J. A. P. Heesterbeek, Mathematical epidemiology of infectious diseases: model building, analysis and interpretation, John Wiley & Sons, New York (2000)
  • View Article
  • Google Scholar


• [15] O. Diekmann, J. A. P. Heesterbeek, J. A. J. Metz, On the definition and the computation of the basic reproduction ratio R 0 in models for infectious diseases in heterogeneous populations, J. Math. Biology, 28 (1990), 365--382
  • View Article
  • Google Scholar


• [16] K. Diethelm, The analysis of fractional differential equations: An application-oriented exposition using differential operators of Caputo type, Springer Science & Business Media, Berlin (2010)
  • View Article
  • Google Scholar


• [17] K. Diethelm, A fractional calculus based model for the simulation of an outbreak of dengue fever, Nonlinear Dynam., 71 (2013), 613--619
  • View Article
  • MathSciNet


• [18] K. Diethelm, N. J. Ford, A. D. Freed, A predictor-corrector approach for the numerical solution of fractional differential equations, Nonlinear Dynam., 29 (2002), 3--22
  • View Article
  • MathSciNet


• [19] K. Diethelm, N. J. Ford, A. D. Freed, Detailed error analysis for a fractional Adams method, Numer. Algorithms, 36 (2004), 31--52
  • View Article
  • MathSciNet


• [20] K. Diethelm, N. J. Ford, A. D. Freed, Y. Luchko, Algorithms for the fractional calculus: a selection of numerical methods, Comput. Methods Appl. Mech. Engrg., 194 (2005), 743--773
  • View Article
  • MathSciNet


• [21] M. R. Duffy, T.-H. Chen, W. T. Hancock, A. M. Powers, J. L. Kool, R. S. Lanciotti, M. Pretrick, Zika virus outbreak on Yap Island, federated states of Micronesia, New England J. Med., 2009 (2009), 2536--2543
  • View Article
  • Google Scholar


• [22] A. Enfissi, J. Codrington, J. Roosblad, M. Kazanji, D. Rousset, Zika virus genome from the Americas, Lancet, 387 (2016), 227--228
  • View Article
  • Google Scholar


• [23] R. A. Erickson, S. M. Presley, L. J. S. Allen, K. R. Long, S. B. Cox, A dengue model with a dynamic Aedes albopictus vector population, Ecolog. Model., 221 (2010), 2899--2908
  • View Article
  • Google Scholar


• [24] M. Farman, A. Akgül, S. Askar, T. Botmart, A. Ahmad, H. Ahmad, Modeling and analysis of fractional order Zika model, Virus, 3 (2021), 15 pages
  • View Article
  • Google Scholar


• [25] D. Gao, Y. Lou, D. He, T. C. Porco, Y. Kuang, G. Chowell, S. G. Ruan, Prevention and control of Zika as a mosquitoborne and sexually transmitted disease: a mathematical modeling analysis, Sci. Rep., 6 (2016), 14 pages
  • View Article
  • Google Scholar


• [26] S. Greenland, C. Poole, Living with P Values: Resurrecting a Bayesian Perspective on Frequentisi Statistics, Epidemiology, 2013 (2013), 62--68
  • View Article
  • Google Scholar


• [27] H. W. Hethcote, The mathematics of infectious diseases, SIAM Rev., 42 (2000), 599--653
  • View Article


• [28] W. Karesh, R. Cook, E. Bennett, J. Newcomb, Wildlife Trade and Global Disease Emergence, Emerging Infectious Diseases, 11 (2005), 1000--1002
  • View Article
  • Google Scholar


• [29] M. Khalil, A. A. M. Arafa, A. Sayed, A variable fractional order network model of zika virus, J. Fract. Calc. Appl., 9 (2018), 204--221
  • View Article
  • MathSciNet


• [30] A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and applications of fractional differential equations, Elsevier Science B.V., Amsterdam (2006)
  • View Article
  • MathSciNet
  • MATH


• [31] A. J. Kucharski, S. Funk, R. M. Eggo, H. P. Mallet, W. Edmunds, E. J. Nilles, Transmission dynamics of Zika virus in island populations: a modelling analysis of the 2013–14 French Polynesia outbreak, PLoS Neglected Tropical Diseases, 10 (2016), 11 pages
  • View Article
  • Google Scholar


• [32] N. Kumar, M. Abdullah, M. I. Faizan, A. Ahmed, H. A. Alsenaidy, R. Dohare, S. Parveen, Progression dynamics of Zika fever outbreak in El Salvador during 2015–2016: a mathematical modeling approach, Future Virology, 12 (2017), 271--281
  • View Article
  • Google Scholar


• [33] C. P. Li, C. X. Tao, On the fractional Adams method, Comput. Math. Appl., 58 (2009), 1573--1588
  • View Article
  • MathSciNet


• [34] D. Li, W. B. Ma, Z. C. Jiang, An epidemic model for tick-borne disease with two delays, J. Appl. Math., 2013 (2013), 11 pages
  • View Article
  • MathSciNet


• [35] J. A. Machado, A. M. S. F. Galhano, J. J. Trujillo, On development of fractional calculus during the last fifty years, Scientometrics, 98 (2014), 577--582
  • View Article
  • Google Scholar


• [36] A. N. Maina, C. M. Farris, A. Odhiambo, J. Jiang, J. Laktabai, J. Armstrong, T. Holland, A. L. Richards, W. P. O’Meara, Q fever, scrub typhus, and rickettsial diseases in children, Kenya, 2011–2012, Emerging Infectious Diseases, 22 (2016), 883--886
  • View Article
  • Google Scholar


• [37] D. M. Morens, G. K. Folkers, A. S. Fauci, The challenge of emerging and re-emerging infectious diseases, Nature, 430 (2004), 242--249
  • View Article
  • Google Scholar


• [38] D. Musso, Zika virus transmission from French Polynesia to Brazil, Emerging Infectious Diseases, 21 (2015), 1887--1889
  • View Article
  • Google Scholar


• [39] D. Musso, D. J. Gubler, Zika virus: following the path of dengue and chikungunya?, Lancet, 386 (2015), 243--244
  • View Article
  • Google Scholar


• [40] K. B. Oldham, J. Spanier, The Fractional Calculus, Academic Press, New York (1974)
  • View Article
  • MathSciNet


• [41] T. A. Perkins, A. S. Siraj, C. W. Ruktanonchai, M. U. G. Kraemer, A. J. Tatem, Model-based projections of Zika virus infections in childbearing women in the Americas, Nature Microbiology, 1 (2016), 7 pages
  • View Article
  • Google Scholar


• [42] I. Petráš, R. L. Magin, Simulation of drug uptake in a two compartmental fractional model for a biological system, Commun. Nonlinear Sci. Numer. Simulat., 16 (2011), 4588--4595
  • View Article
  • Google Scholar


• [43] I. Podlubny, Fractional Differential Equations, Academic Press, San Diego (1999)
  • View Article
  • MathSciNet


• [44] S. Pooseh, H. S. Rodrigues, D. F. M. Torres, Fractional derivatives in dengue epidemics, AIP Conference Proceedings, 1389 (2011), 739--742
  • View Article
  • Google Scholar


• [45] R. Rakkiyappan, V. P. Latha, F. A. Rihan, A fractional-order model for Zika virus infection with multiple delays, Complexity, 2019 (2019), 11 pages
  • View Article
  • Google Scholar


• [46] A. Roth, A. Mercier, C. Lepers, D. Hoy, S. Duituturaga, E. Benyon, L. Guillaumot, Y. Souares, Concurrent outbreaks of dengue, chikungunya and Zika virus infections–an unprecedented epidemic wave of mosquito-borne viruses in the Pacific 2012–2014, Eurosurveillance, 19 (2014), 11 pages
  • View Article
  • Google Scholar


• [47] T. Sardar, S. Rana, S. Bhattacharya, K. Al-Khaled, J. Chattopadhyay, A generic model for a single strain mosquitotransmitted disease with memory on the host and the vector, Math. Biosci., 263 (2015), 18--36
  • View Article
  • MathSciNet


• [48] B. H. Song, S. I. Yun, M. Woolley, Y. M. Lee, Zika virus: history, epidemiology, transmission, and clinical presentation, J. Neuroimmunol., 308 (2017), 50--64
  • View Article
  • Google Scholar


• [49] A. K. Srivastav, N. K. Goswami, M. Ghosh, X. Z. Li, Modeling and optimal control analysis of Zika virus with media impact, Int. J. Dyn. Control, 6 (2018), 1673--1689
  • View Article
  • MathSciNet


• [50] P. van den Driessche, J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Math. Biosc., 180 (2002), 29--48
  • View Article
  • Google Scholar