Citation: Fatmawati, Rashid Jan, Muhammad Altaf Khan, Yasir Khan, Saif ullah. A new model of dengue fever in terms of fractional derivative[J]. Mathematical Biosciences and Engineering, 2020, 17(5): 5267-5287. doi: 10.3934/mbe.2020285
[1] | Center for Disease Control and Prevention. Dengue, Accessed January 14, 2013. URL http://www.cdc.gov/dengue/. |
[2] | D. J. Gubler, Dengue and dengue hemorrhagic fever, Clin. Microbiol. Rev., 11(1998), 480-496. |
[3] | M. G. Guzman, G. Kouri, J. Bravo, L. Valdes, S. Vasquez, S. B. Halstead, Effect of age on outcome of secondary dengue 2 infections, Int. J. Infect. Dis., 6(2002), 118-124. |
[4] | East, Susie (6 April 2016). World's first dengue fever vaccine launched in the Philippines. CNN. Archived from the original on 18 October 2016. Retrieved 17 October 2016. |
[5] | Epidemiology, https://www.cdc.gov/dengue/epidemiology/index.html. Retrieved 7 March 2018. |
[6] | Y. Xiao, T. Zhao, S. Tang, Dynamics of an infectious disease with media/ psychology induced non-smooth incidence, Math. Biosci. Eng., 10(2013), 445-461. |
[7] | A. Wang, Y. Xiao, A Filippov system describing media effects on the spread of infectious diseases, Nonlinear Analysis, Hybrid Syst., 11(2014), 84-97. |
[8] | L. Esteva, C. Vargas, Analysis of a dengue disease transmission model, Math. Biosci., 150(1998), 131-151. |
[9] | L. Esteva, C. Vargas, A model for dengue disease with variable human population, J. Math. Biol., 38(1999), 220-240. |
[10] | M. Derouich, A. Boutayeb, E. H. Twizell, A model of Dengue fever, BioMed. Eng. Online, 2(2003). |
[11] | J. J. Tewa, J. L. Dimi, S. Bowang, Lypaunov functions for a dengue disease transmission model, Chaos Solit. Fract., 39(2009), 936-941. |
[12] | H. S. Rodrigues, M. T. T. Monteiro, D. F. M. Torres, Vaccination models and optimal control strategies to Dengue, Math. Biosci., 5(2014), 1-12. |
[13] | R. Jan, Y. Xiao, Effect of partial immunity on transmission dynamics of dengue disease with optimal control, Math. Method Appl. Sci., 42(2019), 1967-1983. |
[14] | D. S. Burke, A. Nisalak, D. E. Johnson, R. M. Scott, A prospective study of dengue infections in Bangkok, Am. J. Trop. Med. Hyg., 38(1988), 172-180. |
[15] | T. P. Endy, S. Chunsuttiwat, A. Nisalak, D. H. Libraty, S. Green, A. L. Rothman, et al., Epidemiology of inapparent and symptomatic acute dengue virus infection: A prospective study of primary school children in Kamphaeng Phet, Thailand Am. J. Epidemiol., 156(2002), 40-51. |
[16] | L. F. Chaves, L. C. Harrington, C. L. Keogh, A. M. Nguyen, U. D. Kitron, Blood feeding patterns of mosquitoes: Random or structured?, Front. Zool., 7(2010), 1-11. |
[17] | C. Vinauger, L. Buratti, C. R. Lazzari, Learning the way to blood: First evidence of dual olfactory conditioning in a blood sucking insect, Rhodnius prolixus. I. Appetitive learning, J. Exp. Biol., 214(2011), 3032-3038. |
[18] | Y. Carrera, G. A. Rosa, E. J. Vernon-Carter, A. Ramirez, A fractional-order Maxwell model for non-Newtonian fluids, Phys. A, 482(2017), 276-285. |
[19] | R. Jan, M. A. Khan, P. Kumam, P. Thounthong, Modeling the transmission of dengue infection through fractioal derivatives, Chaos Solit. Fract., 127(2019), 189-216. |
[20] | T. Kosztolowicz, K. D. Lewandowska, Application of fractional differential equations in modelling the subdiffusion-reaction process, Math. Model Nat. Pheno., 8(2013), 44-54. |
[21] | C. N. Angstmann, B. I. Henry, A. V. McGann, A fractional-order infectivity SIR model, Phys. A, 452(2016), 86-93. |
[22] | M. Alquran, I. Jaradat, Delay-asymptotic solutions for the time fractional delay-type wave equation, Phys. A, 527(2019), 121-275. |
[23] | M. A. Khan, S. Ullah, M. Farooq, A new fractional model for tuberculosis with relapse via Atangana-Baleanu derivative, Chaos Solit. Fract., 116(2018), 227-238. |
[24] | S. Qureshi, A. Atangana, Mathematical analysis of dengue fever outbreak by novel fractional operators with field data, Phys. A, 526(2019), 121-127. |
[25] | A. Atangana, Application of fractional calculus to epidemiology, Fractional Dynamic, (eds. C. Cattani, H. M. Srivastava, X. Yang), De Gruyter Open, (2015), 174-190. |
[26] | A. Atangana, Modelling the spread of COVID-19 with new fractal-fractional operators: Can the lockdown save mankind before vaccination?. Chaos Solit. Fract., 136(2020), 109860. |
[27] | E. F. Doungmo Goufo, Y. Khan, Q. A. Chaudhry, HIV and shifting epicenters for COVID-19, an alert for some countries,Chaos Solit. Fract., 139(2020), 110030. |
[28] | M. A. Khan, A. Atangana, Modeling the dynamics of novel coronavirus (2019-nCov) with fractional derivative, Alex. Eng. J., (2020). |
[29] | W. Gao, P. Veeresha, D. G. Prakasha, H. M. Baskonus, Novel dynamic structures of 2019-nCoV with nonlocal operator via powerful computational technique, Biology, 9(2020), 107. |
[30] | E. Ilhan, I. O. Kiymaz, A generalization of truncated M-fractional derivative and applications to fractional differential equations, Appl. Math. Non. Sci., 5(2020), 171-188. |
[31] | W. Gao, P. Veeresha, D. G. Prakasha, H. M. Baskonus, G. Yel, New numerical results for the time-fractional Phi-four equation using a novel analytical approach, Symmetry, 12(2020), 478. |
[32] | T. A. Sulaiman, H. Bulut, S. S. Atas, Optical solitons to the fractional Schrodinger-Hirota equation, Appl. Math. Non. Sci., 4(2019),535-542. |
[33] | J. Singh, D. Kumar, Z. Hammouch, A. Atangana, A fractional epidemiological model for computer viruses pertaining to a new fractional derivative, Appl. Math. Comput., 316(2018), 504-515. |
[34] | C. Cattani, A review on Harmonic wavelets and their fractional extension, J. Adv. Eng. Comput., 2(2018), 224-238. |
[35] | X. J. Yang, F. Gao, A new technology for solving diffusion and heat equations, Therm. Sci., 21(2017), 133-140. |
[36] | M. A. Khan, N. Iqbal, Y. Khan, E. Alzahrani, A biologoical mathematical model of vector-host disease with saturated treatment function and optimal control strategies, Math. Biosci. Eng., 17 (2020), 3972. |
[37] | S. K. Panda, T. Abdeljawad, C. Ravichandran, A complex valued approach to the solutions of Riemann-Liouville integral, Atangana-Baleanu integral operator and non-linear Telegraph equation via fixed point method, Chaos Solit. Fract., 130(2020), 109439. |
[38] | X. J. Yang, D. Baleanu, M. P. Lazaveric, M. S. Cajic, Fractal boundary value problems for integral and differential equations with local fractional operators, Therm. Sci., 19(2015), 959-966. |
[39] | R. Almeida, Analysis of a fractional SEIR model with treatment, Appl. Math. Lett., 84(2018), 56-62. |
[40] | R. M. A. Carvalho, C. M. A. Pinto, Immune response in HIV epidemics for distinct transmission rates and for saturated CTL response, Math. Model. Nat. Phenom., 14(2019), 13. |
[41] | K. Diethelm, The analysis of fractional differential equations: An application-oriented exposition using operators of Caputo type, Springer,2004. |
[42] | S. Mao, R. Xu, Y. Li, A fractional order SIRS model with standard incidence rate, J. Beihua Univ., 12(2012), 379-382. |
[43] | Z. M. Odibat, N. T. Shawagteh, Genralized Taylor's formula, Appl. Math. Comput., 186(2007), 286-293. |
[44] | A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and application of fractional differential equations, Elsevier, The Netherlands, 2006. |
[45] | C. Castillo-Chavez, Z. Feng, W. Huang, On the computation of $R_0$ and its role on global stability, Mathematical Approach for Emerging and Reemerging Infectious Diseases: An Introduction, Springer, p. 229, 2002. |
[46] | D. Matignon, Stability results for fractional differential equations with applications to control processing, Comput. Eng. Syst. Appl., 2(1996), 963-968. |
[47] | C. Li, Y. Ma. Fractional dynamical system and its linearization theorem, Nonlin. Dyn., 71(2013), 621-633. |
[48] | Z. Wang, D. Yang, T. Ma, N. Sun, Stability analysis for nonlinear fractional-order system based on comparison principle, Nonlin. Dyn., 75(2014), 387-402. |
[49] | H. I. Freedman, M. X. Tang, S. G. Ruan, Uniform persistence and flows near a closed positively invariant set, J. Dyn. Differ. Equ., 6(1994), 583-600. |
[50] | S. Marino, I. B. Hogue, C. J. Ray, D Kirschner, A methodolojgy for performing global uncertainty and sensitivity analysis in systems biology, J. Theor. Biol., 254(2008), 178-196. |
[51] | A. Jajarmi, D. Baleanu, A new fractional analysis on the interaction of HIV with CD4+ T-cells, Chaos Solit. Fract., 113(2018), 221-229. |
[52] | C. Li, F. Zeng, Numerical methods for fractional calculus, Chapman and Hall/CRC, (2015). |
[53] | KKM. 2007, Health Facts 2007. |
[54] | KKRI, Dengue fever is still high in south sulawesi. Tribun Timur Makassar, Newspaper fact, 2009. |