Abstract.
Dengue fever is a vector-borne disease and is still epidemic in most countries of the world by providing so many outbreaks. The present paper investigates the dengue dynamics for the real cases reported in Pakistan in the period 2003–2015. The model is formulated and the associated properties are presented. We show, for the given period, a basic reproduction, \( {R}_0 = 3.8\). The parameters are parameterized for model simulation by using the leaset square curve fitting in MATLAB. We use the Caputo derivative and formulate the fractional dengue model. The stability analysis for the fractional dengue model in both disease-free and endemic cases is presented. We show that, in the disease-free case, the fractional dengue model is locally and globally stable when \( {R}_0 < 1\). Then, we prove the model stability in the endemic case and present the results for \( {R}_0 > 1\) . We provide some graphical illustrations and show that the dengue model with fractional derivative is more useful than that of the integer order model.
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G. Chowell, P. Diaz-Duenas, J.C. Miller, A. Alcazar-Velazco, J.M. Hyman, P.W. Fenimore, C. Castillo-Chavez, Math. Biosci. 208, 571 (2007)
Chandra Shekhar, Chem. Biol. 14, 871 (2007)
Samir Bhatt, Peter W. Gething, Oliver J. Brady, Jane P. Messina, Andrew W. Farlow, Catherine L. Moyes, John M. Drake, John S. Brownstein, Anne G. Hoen, Osman Sankoh et al., Nature 496, 504 (2013)
Oliver J. Brady, Peter W. Gething, Samir Bhatt, Jane P. Messina, John S. Brownstein, Anne G. Hoen, Catherine L. Moyes, Andrew W. Farlow, Thomas W. Scott, Simon I. Hay, PLOS Negl. Trop. Dis. 6, e1760 (2012)
Thomas L. Bancroft, Austral. Med. Gaz. 25, 17 (1906)
Helena Sofia Rodrigues, M. Teresa, T. Monteiro, Delfim F.M. Torres, Math. Biosci. 247, 1 (2014)
Scott B. Halstead, World Health Stat. Q. 45, 292 (1992)
Gustavo P. Kouri, María G. Guzmán, José R. Bravo, Bull. Pan. Am. Health Organ. 20, 24 (1986)
World Health Organization (WHO), Dengue Vaccine Research: Immunization, Vaccines and Biologicals, https://www.who.int/immunization/research/development/dengue_vaccines/en/ (2017)
Joseph E. Blaney, Jennifer M. Matro, Brian R. Murphy, Stephen S. Whitehead, J. Virol. 79, 5516 (2005)
Joseph E. Blaney, Neeraj S. Sathe, Christopher T. Hanson, Cai Yen Firestone, Brian R. Murphy, Stephen S. Whitehead, Virol. J. 4, 23 (2007)
Matthieu Lesnoff, Géraud Laval, Pascal Bonnet, Karine Chalvet-Monfray, Renaud Lancelot, Francois Thiaucourt, Prev. Vet. Med. 62, 101 (2004)
Eunha Shim, Am. J. Trop. Med. Hyg. 95, 1137 (2016)
F.B. Agusto, M.A. Khan, Math. Biosci. 305, 102 (2018)
Stefan G. Samko, Anatoly A. Kilbas, Oleg I. Marichev, Fractional Integrals and Derivatives: Theory and Applications (CRC Press, 1993)
Michele Caputo, Mauro Fabrizio, Progr. Fract. Differ. Appl. 1, 1 (2015)
Abdon Atangana, Dumitru Baleanu, arXiv:1602.03408 (2016)
Abdon Atangana, Ilknur Koca, Chaos, Solitons Fractals 89, 447 (2016)
Saif Ullah, Muhammad Altaf Khan, Muhammad Farooq, Eur. Phys. J. Plus 133, 237 (2018)
Abdon Atangana, J.F. Gómez-Aguilar, Eur. Phys. J. Plus 133, 166 (2018)
Zhenhai Liu, Peifen Lu, Adv. Differ. Equ. 2014, 298 (2014)
Emile Franc Doungmo Goufo, J. Theor. Biol. 403, 178 (2016)
Muhammad Altaf Khan, Saif Ullah, Muhammad Farhan, AIMS Math. 4, 134 (2019)
Saif Ullah, Muhammad Altaf Khan, Muhammad Farooq, Chaos, Solitons Fractals 116, 63 (2018)
H. Yépez-Martínez, J.F. Gómez-Aguilar, J. Comput. Appl. Math. 346, 247 (2019)
José Francisco Gómez-Aguilar, Baleanu Dumitru, Z. Naturforsch. A 69, 539 (2014)
J.F. Gómez-Aguilar, H. Yépez-Martínez, R.F. Escobar-Jiménez, Appl. Math. Model. 40, 9079 (2016)
J.F. Gómez-Aguilar, Abdon Atangana, Eur. Phys. J. Plus 132, 13 (2017)
Abdon Atangana, J.F. Gómez-Aguilar, Chaos, Solitons Fractals 102, 285 (2017)
B. Cuahutenango-Barro, M.A. Taneco-Hernández, J.F. Gómez-Aguilar, Chaos, Solitons Fractals 115, 283 (2018)
H. Yépez-Martínez, F. Gómez-Aguilar, I.O. Sosa, J.M. Reyes, J. Torres-Jiménez, Rev. Mex. Fís. 62, 310 (2016)
Abdon Atangana, J.F. Gómez-Aguilar, Chaos, Solitons Fractals 114, 516 (2018)
Sania Qureshi, Abdullahi Yusuf, Eur. Phys. J. Plus 134, 171 (2019)
Sania Qureshi, Abdon Atangana, Physica A 526, 121 (2019)
Abdon Atangana, Sania Qureshi, Chaos, Solitons Fractals 123, 320 (2019)
Sania Qureshi, Abdullahi Yusuf, Chaos, Solitons Fractals 122, 111 (2019)
Igor Podlubny, Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of their Solution and some of Their Applications, Vol. 198 (Elsevier, 1998)
Hadi Delavari, Dumitru Baleanu, Jalil Sadati, Nonlinear Dyn. 67, 2433 (2012)
Cruz Vargas-De-León, Commun. Nonlinear Sci. Numer. Simul. 24, 75 (2015)
Zaid M. Odibat, Nabil T. Shawagfeh, Appl. Math. Comput. 186, 286 (2007)
Wei Lin, J. Math. Anal. Appl. 332, 709 (2007)
H.A. Antosiewicz, Studies in Ordinary Differential Equations, Vol. 14 (Mathematical Association of America, 1977)
Pauline Van den Driessche, James Watmough, Math. Biosci. 180, 29 (2002)
Ana R.M. Carvalho, Carla M.A. Pinto, Dumitru Baleanu, Adv. Differ. Equ. 2018, 2 (2018)
Mohammad Saleh Tavazoei, Mohammad Haeri, Physica D 237, 2628 (2008)
Muhammad Sabir, Yousaf Ali, Noor Muhammad, J. Pakistan Med. Assoc. 68, 1383 (2018)
Who Health Organization (WHO), WHO Country Cooperation Strategies, https://apps.who.int/iris/bitstream/handle/10665/136607/ccsbrief_pak_en.pdf?sequence=1
Carrie A. Manore, Kyle S. Hickmann, Sen Xu, Helen J. Wearing, James M. Hyman, J. Theor. Biol. 356, 174 (2014)
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Khan, M.A., Khan, A., Elsonbaty, A. et al. Modeling and simulation results of a fractional dengue model. Eur. Phys. J. Plus 134, 379 (2019). https://doi.org/10.1140/epjp/i2019-12765-0
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DOI: https://doi.org/10.1140/epjp/i2019-12765-0