Existence theory and numerical solution of leptospirosis disease model via exponential decay law

Amir Khan; Abdur Raouf; Rahat Zarin; Abdullahi Yusuf; Usa Wannasingha Humphries; Amir Khan; Abdur Raouf; Rahat Zarin; Abdullahi Yusuf; Usa Wannasingha Humphries

doi:10.3934/math.2022492

AIMS Mathematics

2022, Volume 7, Issue 5: 8822-8846. doi: 10.3934/math.2022492

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Existence theory and numerical solution of leptospirosis disease model via exponential decay law

1.
Department of Mathematics and Statistics, University of Swat, Khyber Pakhtunkhawa, Pakistan
2.
Department of Basic Sciences, University of Engineering and Technology Peshawar, Khyber Pakhtunkhwa, Pakistan
3.
Department of Computer Engineering, Biruni University, Istanbul, Turkey
4.
Department of Mathematics, Near East University TRNC, Mersin 10, Nicosia 99138, Turkey
5.
Department of Mathematics, Faculty of Science, King Mongkut's University of Technology Thonburi (KMUTT), 126 Pracha-Uthit Road, Bang Mod, Thrung Khru, Bangkok 10140, Thailand

Received: 26 November 2021 Revised: 23 January 2022 Accepted: 28 January 2022 Published: 04 March 2022
MSC : 92B05, 37G05, 37G15, 39A28

We investigated the leptospirosis epidemic model by using Caputo and Fabrizio fractional derivatives. Picard's successive iterative method and Sumudu transform are taken into consideration for developing the iterative solutions for the leptospirosis disease. Employing nonlinear functional analysis, the stability and uniqueness of the proposed model are established. Sensitivity analysis is taken into account to highlight the most sensitive parameters corresponding to the basic reproductive number. Various solutions to the proposed system have been interpolated by graphs with the application of Matlab software.
- mathematical model,
- exponential decay law,
- Sumudu transform,
- existence and uniqueness
Citation: Amir Khan, Abdur Raouf, Rahat Zarin, Abdullahi Yusuf, Usa Wannasingha Humphries. Existence theory and numerical solution of leptospirosis disease model via exponential decay law[J]. AIMS Mathematics, 2022, 7(5): 8822-8846. doi: 10.3934/math.2022492

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Abstract

We investigated the leptospirosis epidemic model by using Caputo and Fabrizio fractional derivatives. Picard's successive iterative method and Sumudu transform are taken into consideration for developing the iterative solutions for the leptospirosis disease. Employing nonlinear functional analysis, the stability and uniqueness of the proposed model are established. Sensitivity analysis is taken into account to highlight the most sensitive parameters corresponding to the basic reproductive number. Various solutions to the proposed system have been interpolated by graphs with the application of Matlab software.

References

[1]	A. R. Bharti, J. E. Nally, J. N. Ricaldi, M. A Matthias, M. M. Diaz, M. A. Lovett, et al., Leptospirosis: A zoonotic disease of global importance, Lancet Infect. Dis., 3 (2003), 757–771. https://doi.org/10.1016/S1473-3099(03)00830-2 doi: 10.1016/S1473-3099(03)00830-2
[2]	G. Zaman, M. A. Khan, S. Islam, M. I. Chohan, I. H. Jung, Modeling dynamical interactions between leptospirosis infected vector and human population, Appl. Math. Sci., 6 (2012), 1287–1302. https://doi.org/10.4236/apm.2010.38086 doi: 10.4236/apm.2010.38086
[3]	A. I. Ko, M. G. Reis, C. M. R. Dourado, W. D. Johnson, L. W. Riley, Urban epidemic of severe leptospirosis in Brazil, Lancet, 354 (1999), 820–825. https://doi.org/10.1016/s0140-6736(99)80012-9 doi: 10.1016/s0140-6736(99)80012-9
[4]	A. F. B. Victoriano, L. D. Smythe, N. Gloriani-Barzaga, L. L. Cavinta, T. Kasai, K. Limpakarnjanarat, et al., Leptospirosis in the Asia Pacific region, BMC Infect. Dis., 9 (2009), 1–9. https://doi.org/10.1186/1471-2334-9-147 doi: 10.1186/1471-2334-9-147
[5]	A. Din, Y. J. Li, A. Yusuf, Delayed hepatitis B epidemic model with stochastic analysis, Chaos Solitons Fract., 146 (2021), 110839. https://doi.org/10.1016/j.chaos.2021.110839 doi: 10.1016/j.chaos.2021.110839
[6]	A. Din, Y. J. Li, T. Khan, G. Zaman, Mathematical analysis of spread and control of the novel corona virus (COVID-19) in China, Chaos Solitons Fract., 141 (2020), 110286. https://doi.org/10.1016/j.chaos.2020.110286 doi: 10.1016/j.chaos.2020.110286
[7]	A. Din, Y. J. Li, M. A. Shah, The complex dynamics of hepatitis B infected individuals with optimal control, J. Syst. Sci. Complex., 34 (2021), 1301–1323. https://doi.org/10.1007/s11424-021-0053-0 doi: 10.1007/s11424-021-0053-0
[8]	A. Din, Y. J. Li, Stationary distribution extinction and optimal control for the stochastic hepatitis B epidemic model with partial immunity, Phys. Scr., 96 (2021), 074005. https://doi.org/10.1088/1402-4896/abfacc doi: 10.1088/1402-4896/abfacc
[9]	R. Zarin, I. Ahmed, P. Kumam, A. Zeb, A. Din, Fractional modeling and optimal control analysis of rabies virus under the convex incidence rate, Results Phys., 28 (2021), 104665. https://doi.org/10.1016/j.rinp.2021.104665 doi: 10.1016/j.rinp.2021.104665
[10]	A. Din, Y. J. Li, F. M. Khan, Z. U. Khan, P. J. Liu, On analysis of fractional order mathematical model of hepatitis B using Atangana-Baleanu Caputo (ABC) derivative, Fractals, 2021, 2240017. https://doi.org/10.1142/S0218348X22400175 doi: 10.1142/S0218348X22400175
[11]	A. Din, Y. J. Li, A. Yusuf, A. I. Ali, Caputo type fractional operator applied to hepatitis B system, Fractals, 2021, 2240023. https://doi.org/10.1142/S0218348X22400230 doi: 10.1142/S0218348X22400230
[12]	W. K. Reisen, Landscape epidemiology of vector-borne diseases, Annu. Rev. Entomol., 55 (2010), 461–483. https://doi.org/10.1146/annurev-ento-112408-085419 doi: 10.1146/annurev-ento-112408-085419
[13]	A. Kilicman, A fractional order SIR epidemic model for dengue transmission, Chaos Solitons Fract., 114 (2018), 55–62. https://doi.org/10.1016/j.chaos.2018.06.031 doi: 10.1016/j.chaos.2018.06.031
[14]	P. Pongsuumpun, T. Miami, R. Kongnuy, Age structural transmission model for leptospirosis, In: Proceedings of the 3rd international symposium on biomedical engineering, 2008,411–416.
[15]	W. Triampo, D. Baowan, I. M. Tang, N. Nuttavut, J. Wong-Ekkabut, G. Doungchawee, A simple deterministic model for the spread of leptospirosis in Thailand, Int. J. Bio. Med. Sci., 2 (2007), 22–26.
[16]	G. Zaman, Dynamical behavior of leptospirosis disease and role of optimal control theory, Int. J. Math. Comput., 7 (2010), 80–92.
[17]	A. Khan, R. Zarin, M. Inc, G. Zaman, B. Almohsen, Stability analysis of leishmania epidemic model with harmonic mean type incidence rate, Eur. Phys. J. Plus., 135 (2020), 1–20. https://doi.org/10.1140/epjp/s13360-020-00535-0 doi: 10.1140/epjp/s13360-020-00535-0
[18]	K. Khan, R. Zarin, A. Khan, A. Yusuf, M. Al-Shomrani, A. Ullah, Stability analysis of five-grade Leishmania epidemic model with harmonic mean-type incidence rate, Adv. Differ. Equ., 2021 (2021), 1–27. https://doi.org/10.1186/s13662-021-03249-4 doi: 10.1186/s13662-021-03249-4
[19]	A. Khan, R. Zarin, G. Hussain, A. H. Usman, U. W. Humphries, J. F. Gomez-Aguilar, Modeling and sensitivity analysis of HBV epidemic model with convex incidence rate, Results Phys., 22 (2021), 103836. https://doi.org/10.1016/j.rinp.2021.103836 doi: 10.1016/j.rinp.2021.103836
[20]	A. Khan, R. Zarin, G. Hussain, N. A. Ahmad, M. H. Mohd, A. Yusuf, Stability analysis and optimal control of COVID-19 with convex incidence rate in Khyber Pakhtunkhawa (Pakistan), Results Phys., 20 (2021), 103703. https://doi.org/10.1016/j.rinp.2020.103703 doi: 10.1016/j.rinp.2020.103703
[21]	D. Baleanu, B. Ghanbari, J. H. Asad, A. Jajarmi, H. M. Pirouz, Planar system-masses in an equilateral triangle: Numerical study within fractional calculus, Comput. Model. Eng. Sci., 124 (2020), 953–968. https://doi.org/10.32604/cmes.2020.010236 doi: 10.32604/cmes.2020.010236
[22]	A. Jajarmi, D. Baleanu, A new iterative method for the numerical solution of high-order non-linear fractional boundary value problems, Front. Phys., 8 (2020), 220. https://doi.org/10.3389/fphy.2020.00220 doi: 10.3389/fphy.2020.00220
[23]	R. Zarin, A. Khan, M. Inc, U. W. Humphries, T. Karite, Dynamics of five grade leishmania epidemic model using fractional operator with Mittag-Leffler kernel, Chaos Solitons Fract., 147 (2021), 110985. https://doi.org/10.1016/j.chaos.2021.110985 doi: 10.1016/j.chaos.2021.110985
[24]	F. Mohammadi, L. Moradi, D. Baleanu, A. Jajarmi, A hybrid functions numerical scheme for fractional optimal control problems: Application to nonanalytic dynamical systems, J. Vib. Control, 24 (2018), 5030–5043. https://doi.org/10.1177/1077546317741769 doi: 10.1177/1077546317741769
[25]	H. M. Srivastava, V. P. Dubey, R. Kumar, J. Singh, D. Kumar, D. Baleanu, An efficient computational approach for a fractional-order biological population model with carrying capacity, Chaos Solitons Fract., 138 (2020), 109880. https://doi.org/10.1016/j.chaos.2020.109880 doi: 10.1016/j.chaos.2020.109880
[26]	J. Singh, Analysis of fractional blood alcohol model with composite fractional derivative, Chaos Solitons Fract., 140 (2020), 110127. https://doi.org/10.1016/j.chaos.2020.110127 doi: 10.1016/j.chaos.2020.110127
[27]	A. Khan, G. Hussain, M. Inc, G. Zaman, Existence, uniqueness, and stability of fractional hepatitis B epidemic model, Chaos, 30 (2020), 103104. https://doi.org/10.1063/5.0013066 doi: 10.1063/5.0013066
[28]	R. Zarin, A. Khan, A. Yusuf, S. Abdel-Khalek, M. Inc, Analysis of fractional COVID-19 epidemic model under Caputo operator, Math. Meth. Appl. Sci., 2021, 1–21. https://doi.org/10.1002/mma.7294 doi: 10.1002/mma.7294
[29]	E. Bonyah, R. Zarin, Fatmawati, Mathematical modeling of cancer and hepatitis co-dynamics with non-local and non-singular kernal, Commun. Math. Biol. Neurosci., 2020 (2020), 91. https://doi.org/10.28919/cmbn/5029 doi: 10.28919/cmbn/5029
[30]	A. Imitaz, A. Aamina, F. Ali, I. Khan, K. S. Nisar, Two-phase flow of blood with magnetic dusty particles in cylindrical region: A Caputo Fabrizio fractional model, Comput. Mater. Con., 66 (2021), 2253–2264. https://doi.org/10.32604/cmc.2021.012470 doi: 10.32604/cmc.2021.012470
[31]	Z. A. Khan, S. U. Haq, T. S. Khan, I. Khan, K. S. Nisar, Fractional Brinkman type fluid in channel under the effect of MHD with Caputo-Fabrizio fractional derivative, Alex. Eng. J., 59 (2020), 2901–2910. https://doi.org/10.1016/j.aej.2020.01.056 doi: 10.1016/j.aej.2020.01.056
[32]	S. Jitsinchayakul, R. Zarin, A. Khan, A. Yusuf, G. Zaman, U. W. Humphries, T. A. Sulaiman, Fractional modeling of COVID-19 epidemic model with harmonic mean type incidence rate, Open Phys., 19 (2021), 693–709. https://doi.org/10.1515/phys-2021-0062 doi: 10.1515/phys-2021-0062
[33]	A. S. Shaikh, K. S. Nisar, Transmission dynamics of fractional order Typhoid fever model using Caputo-Fabrizio operator, Chaos Solitons Fract., 128 (2019), 355–365. https://doi.org/10.1016/j.chaos.2019.08.012 doi: 10.1016/j.chaos.2019.08.012
[34]	M. Arif, F. Ali, N. A. Sheikh, I. Khan, K. S. Nisar, Fractional model of couple stress fluid for generalized Couette flow: A comparative analysis of Atangana-Baleanu and Caputo-Fabrizio fractional derivatives, IEEE Access, 7 (2019), 88643–88655. https://doi.org/10.1109/ACCESS.2019.2925699 doi: 10.1109/ACCESS.2019.2925699
[35]	R. Zarin, H. Khaliq, A. Khan, D. Khan, A. Akgül, U. W. Humphries, Deterministic and fractional modeling of a computer virus propagation, Results Phys., 33 (2022), 105130. https://doi.org/10.1016/j.rinp.2021.105130 doi: 10.1016/j.rinp.2021.105130
[36]	A. M. S. Mahdy, Numerical solutions for solving model time-fractional Fokker-Planck equation, Numer. Methods Partial Differ. Equ., 37 (2021), 1120–1135. https://doi.org/10.1002/num.22570 doi: 10.1002/num.22570
[37]	K. A. Gepreel, M. Higazy, A. M. S. Mahdy, Optimal control, signal flow graph, and system electronic circuit realization for nonlinear Anopheles mosquito model, Int. J. Mod. Phys. C, 31 (2020), 2050130. https://doi.org/10.1142/S0129183120501302 doi: 10.1142/S0129183120501302
[38]	A. M. S. Mahdy, Y. A. E. Amer, M. S. Mohamed, E. Sobhy, General fractional financial models of awareness with Caputo-Fabrizio derivative, Adv. Mech. Eng., 12 (2020), 1–9. https://doi.org/10.1177/1687814020975525 doi: 10.1177/1687814020975525
[39]	A. M. S. Mahdy, K. A. Gepreel, K. Lotfy, A. A. El-Bary, A numerical method for solving the Rubella ailment disease model, Int. J. Mod. Phys. C, 32 (2021), 1–15. https://doi.org/10.1142/S0129183121500972 doi: 10.1142/S0129183121500972
[40]	A. M. S. Mahdy, M. S. Mohamed, K. Lotfy, M. Alhazmi, A. A. El-Bary, M. H. Raddadi, Numerical solution and dynamical behaviors for solving fractional nonlinear Rubella ailment disease model, Results Phys., 24 (2021), 104091. https://doi.org/10.1016/j.rinp.2021.104091 doi: 10.1016/j.rinp.2021.104091
[41]	Y. Zhao, A. Khan, U. W. Humphries, R. Zarin, M. Khan, A. Yusuf, Dynamics of visceral leishmania epidemic model with non-singular kernel, Fractals, 2022, 1–15. https://doi.org/10.1142/S0218348X22401351 doi: 10.1142/S0218348X22401351
[42]	A. M. S. Mahdy, M. Higazy, K. A. Gepreel, A. A. A. El-Dahdouh, Optimal control and bifurcation diagram for a model nonlinear fractional SIRC, Alex. Eng. J., 59 (2020), 3481–3501. https://doi.org/10.1016/j.aej.2020.05.028 doi: 10.1016/j.aej.2020.05.028
[43]	A. M. S. Mahdy, M. S. Mohamed, K. A. Gepreel, A. Al-Amiri, M. Higazy, Dynamical characteristics and signal flow graph of nonlinear fractional smoking mathematical model, Chaos Solitons Fract., 141 (2020), 110308. https://doi.org/10.1016/j.chaos.2020.110308 doi: 10.1016/j.chaos.2020.110308
[44]	M. Caputo, M. Fabrizio, A new definition of fractional derivative without singular kernel, Progr. Fract. Differ. Appl., 1 (2015), 73–85.
[45]	M. Caputo, M. Fabrizio, Applications of new time and spatial fractional derivatives with exponential kernels, Progr. Fract. Differ. Appl. Sci., 2 (2016), 1–11. https://doi.org/10.18576/pfda/020101 doi: 10.18576/pfda/020101
[46]	J. Losada, J. J. Nieto, Properties of a new fractional derivative without singular kernel, Progr. Fract. Differ. Appl., 1 (2015), 87–92.
[47]	Z. L. Wang, D. S. Yang, T. D. Ma, N. Sun, Stability analysis for nonlinear fractional-order systems based on comparison principle, Nonlinear Dyn., 75 (2014), 387–402. https://doi.org/10.1007/s11071-013-1073-7 doi: 10.1007/s11071-013-1073-7

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