A new model of dengue fever in terms of fractional derivative

Fatmawati; Rashid Jan; Muhammad Altaf Khan; Yasir Khan; Saif ullah; Fatmawati; Rashid Jan; Muhammad Altaf Khan; Yasir Khan; Saif ullah

doi:10.3934/mbe.2020285

Mathematical Biosciences and Engineering

2020, Volume 17, Issue 5: 5267-5287. doi: 10.3934/mbe.2020285

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A new model of dengue fever in terms of fractional derivative

1.
Department of Mathematics, Faculty of Science and Technology,Universitas Airlangga, Surabaya 60115, Indonesia
2.
School of Mathematics and Statistics, Xi’an Jiaotong University, Xi’an 710049, PR China
3.
Informetrics Research Group, Ton Duc Thang University, Ho Chi Minh City, Vietnam
4.
Faculty of Mathematics and Statistics, Ton Duc Thang University, Ho Chi Minh City, Vietnam
5.
Department of Mathematics,University of Hafr Al-Batin, Hafr Al-Batin 31991, Saudi Arabia
6.
Department of Mathematics,University of Peshawar, Pakistan

Received: 08 June 2020 Accepted: 26 July 2020 Published: 12 August 2020

It is eminent that the epidemiological patterns of dengue are threatening for both the global economy and human health. The experts in the field are always in search to have better mathematician models in order to understand the transmission dynamics of epidemics models and to suggest possible control or the minimization of the infection from the community. In this research, we construct a new fractional-order system for dengue infection with carrier and partially immune classes to visualize the intricate dynamics of dengue. By using the basics of fractional theory, we determine the fundamental results of the proposed fractional-order dengue model. We obtain the basic reproduction number $R_0$ by next generation method and present the results based on it. The stability results are established for the infection-free state of the system. Moreover, sensitivity of $R_0$ is analyzed through partial rank correlation coefficient(PRCC) method to show the importance of different parameters in $R_0$. The influence of different input factors is shown on the output of basic reproduction number $R_0$ numerically. Our result showed that the threshold parameter $R_0$ can be decreased by increasing vaccination and treatment in the system. Finally, we illustrate the solution of the suggested dengue system through a numerical scheme to notice the influence of the fractional-order $\vartheta$ on the system. We observed that the fractional-order dynamics can explain the complex system of dengue infection more precisely and accurately rather than the integer-order dynamics. In addition, we noticed that the index of memory and biting rate of vector can play a significant part in the prevention of the infection.
- fractional derivative,
- dengue disease,
- mathematical model,
- sensitivity analysis,
- numerical simulations
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

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Abstract

It is eminent that the epidemiological patterns of dengue are threatening for both the global economy and human health. The experts in the field are always in search to have better mathematician models in order to understand the transmission dynamics of epidemics models and to suggest possible control or the minimization of the infection from the community. In this research, we construct a new fractional-order system for dengue infection with carrier and partially immune classes to visualize the intricate dynamics of dengue. By using the basics of fractional theory, we determine the fundamental results of the proposed fractional-order dengue model. We obtain the basic reproduction number $R_0$ by next generation method and present the results based on it. The stability results are established for the infection-free state of the system. Moreover, sensitivity of $R_0$ is analyzed through partial rank correlation coefficient(PRCC) method to show the importance of different parameters in $R_0$. The influence of different input factors is shown on the output of basic reproduction number $R_0$ numerically. Our result showed that the threshold parameter $R_0$ can be decreased by increasing vaccination and treatment in the system. Finally, we illustrate the solution of the suggested dengue system through a numerical scheme to notice the influence of the fractional-order $\vartheta$ on the system. We observed that the fractional-order dynamics can explain the complex system of dengue infection more precisely and accurately rather than the integer-order dynamics. In addition, we noticed that the index of memory and biting rate of vector can play a significant part in the prevention of the infection.

References

[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.

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