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DYNAMICS OF VISCERAL LEISHMANIA EPIDEMIC MODEL WITH NON-SINGULAR KERNEL

School of Mathematics, Hangzhou Normal University, Hangzhou 311121, P. R. China

Department of Mathematics, Faculty of Science, King Mongkuts, University of Technology Thonburi (KMUTT), 126 Pracha-Uthit Road, Bang Mod, Thrung Khru, Bangkok 10140, Thailand

E-mail Address: amirkhan@uswat.edu.pk

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USA WANNASINGHA HUMPHRIES

Department of Mathematics, Faculty of Science, King Mongkuts, University of Technology Thonburi (KMUTT), 126 Pracha-Uthit Road, Bang Mod, Thrung Khru, Bangkok 10140, Thailand

E-mail Address: usa.wan@kmutt.ac.th

Corresponding author.

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RAHAT ZARIN

Department of Basic Sciences, University of Engineering and Technology, Peshawar, Khyber Pakhtunkhwa, Pakistan

E-mail Address: rahat.zarin@uetpeshawar.edu.pk

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MAJID KHAN

Department of Mathematics and Statistics, University of Swat, Khyber Pakhtunkhawa, Pakistan

E-mail Address: lizaseen@hotmail.com

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, and

ABDULLAHI YUSUF

Department of Computer Engineering, Biruni University, Istanbul, Turkey

Department of Mathematics, Near East University, TRNC, Mersin 10, Turkey

E-mail Address: ayusuf@biruni.edu.tr

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https://doi.org/10.1142/S0218348X22401351Cited by:5 (Source: Crossref)

This article is part of the issue:

Special Issue on Fractal AI-Based Analyses and Applications to Complex Systems: Part III
Lead Guest Editors: Yeliz Karaca & Dumitru Baleanu
Guest Editors: Majaz Moonis, Yu-Dong Zhang & Osvaldo Gervasi

Abstract

In this paper, we have taken into consideration the most dangerous form of leishmaniasis known as visceral leishmaniasis, which has the highest fatality rate and is also famous with the name of kala-azar. The analysis is carried out using Atangana–Baleanu–Caputo (ABC) operator with a convex incidence rate. Using fixed point theorems the existence and uniqueness for the solutions of the fractional leishmania epidemic model have been established. All the basic properties of the given model are studied along with stability analysis. Newton polynomial and Adams–Bashforth methods are taken into account to find the numerical solution.

Keywords:

References

1. A. Stauch, H. Duerr, J. Dujardin, M. Vanaerschot, S. Sundar and M. Eichner, Treatment of visceral leishmaniasis: Model-based analyses on the spread of antimony-resistant L. donovani in Bihar, India, PLoS Negl. Trop. Dis. 6 (2012) 1973. Crossref, Web of Science, Google Scholar
2. WHO, Leishmaniasis fact sheet (2014), http://www.who.int/mediacentre/factsheets/fs375/en/. Google Scholar
3. R. W. Ashford, The leishmaniases as emerging and reemerging zoonoses, Int. J. Parasitol. 30 (2000) 1269–1281. Crossref, Web of Science, Google Scholar
4. U. Brahmachari, A new form of cutaneous leishmaniasis-dermal leishmanoid, Ind. Med. Gaz. 57 (1992) 125–127. Google Scholar
5. A. Khan, R. Zarin, G. Hussain, N. A. Ahmad, M. H. Mohd and A. Yusuf, Stability analysis and optimal control of COVID-19 with convex incidence rate in Khyber Pakhtunkhawa (Pakistan), Results Phys. 20 (2021) 103703. Crossref, Web of Science, Google Scholar
6. K. Khan, R. Zarin, A. Khan, A. Yusuf, M. Al-Shomrani and A. Ulla, Stability analysis of five-grade Leishmania epidemic model with harmonic mean-type incidence rate, Adv. Diff. Equ. 86 (2021) 1–7, https://doi.org/10.1186/s13662-021-03249-4. Google Scholar
7. A. Khan, G. Hussain, A. Yusuf, A. H. Usman and U. W. Humphrie, A hepatitis stochastic epidemic model with acute and chronic stages, Adv. Diff. Equ. 181 (2021) 1–10, https://doi.org/10.1186/s13662-021-03335-7. Google Scholar
8. I. Ahmed, E. F. D. Goufo, A. Yusuf, P. Kumam, P. Chaipanya and K. Nonlaopon, An epidemic prediction from analysis of a combined HIV-COVID-19 co-infection model via ABC fractional operator, Alex. Eng. J. 60 (2021) 2979–2995. Crossref, Web of Science, Google Scholar
9. R. Zarin, A. Khan, A. Yusuf, S. Abdel-Khalek and M. Inc, Analysis of fractional COVID-19 epidemic model under Caputo operator, Math. Methods Appl. 2021 (2021) 234–241, https://doi.org/10.1002/mma.7294. Google Scholar
10. S. Lisa and R. Newton, A review of preventative methods against human leishmaniasis infection, PLoS Negl. Trop. Dis. 7(6) (2011) 2278. Web of Science, Google Scholar
11. Z. Wang, Y. Moreno, S. Boccaletti and M. Perc, Vaccination and epidemics in networked populations an introduction, Chaos Solitons Fractals 103 (2017) 177–183. Crossref, Web of Science, Google Scholar
12. Z. Rahat, I. Ahmed, P. Kumam, A. Zeb and A. Din, Fractional modeling and optimal control analysis of rabies virus under the convex incidence rate, Results Phys. 28 (2021) 104665. Crossref, Web of Science, Google Scholar
13. S. Gui-Quan, M. Jusup, Z. Jin, Y. Wang and Z. Wang, Pattern transitions in spatial epidemics: Mechanisms and emergent properties, Phys. Life Rev. 19 (2016) 43–73. Crossref, Web of Science, Google Scholar
14. K. Amir, R. Zarin, I. Ahmed, A. Yusuf and U. W. Humphries, Numerical and theoretical analysis of Rabies model under the harmonic mean type incidence rate, Results Phys. 29 (2021) 104652. Crossref, Web of Science, Google Scholar
15. S. Gui-Quan, Mathematical modeling of population dynamics with Allee effect, Nonlinear Dyn. 85(1) (2016) 1–12. Crossref, Web of Science, Google Scholar
16. K. Amir, R. Zarin, S. Khan, A. Saeed, T. Gul and U. W. Humphries, Fractional dynamics and stability analysis of COVID-19 pandemic model under the harmonic mean type incidence rate, Comput. Methods Biomech. Biomed. Eng. 25 (2021) 1–22. Web of Science, Google Scholar
17. X. Yi, L. Song, G. Sun, Z. Jin and J. Zhang, Assessing reappearance factors of H7N9 avian influenza in China, Appl. Math. Comput. 309 (2017) 192–204. Web of Science, Google Scholar
18. D. Christopher and D. M. Wolpert, Earthquakes, influenza and cycles of Indian kala-azar, Trans. R. Soc. 82(6) (1988) 843–850. Google Scholar
19. Z. Rahat, A. Khan, M. Inc, U. W. Humphries and T. Karite, Dynamics of five grade leishmania epidemic model using fractional operator with Mittag-Leffler kernel, Chaos Solitons Fractals 147 (2021) 110985. Crossref, Web of Science, Google Scholar
20. D. Bora, Epidemiology of visceral leishmaniasis in India, Natl. Med. J. India 12 (1999) 62–68. Web of Science, Google Scholar
21. A. Ephraim, D. S. Ross and K. Bathena, A model of the transmission dynamics of leishmaniasis, J. Biol. Syst. 19(2) (2011) 237–250. Link, Web of Science, Google Scholar
22. B. Santanu, A. Subramanian, I. M. Elmojtaba, J. Chattopadhyay and R. R. Sarkar, Optimal combinations of control strategies and cost-effective analysis for visceral leishmaniasis disease transmission, PLoS One 12(2) (2017) 0172465. Web of Science, Google Scholar
23. I. M. ELmojtaba, J. Y. T. Mugisha and M. H. A. Hashim, Mathematical analysis of the dynamics of visceral leishmaniasis in the Sudan, Appl. Math. Comput. 217 (2010) 2567–2578. Crossref, Web of Science, Google Scholar
24. M. Anuj, C. Castillo-Chavez, G. Chowell, C. Kribs-Zaleta, N. A. Siddiqui, N. Kumar and P. Das, Transmission dynamics and under reporting of kala-azarin the Indian state of Bihar, J. Theor. Biol. 262 (2010) 177–185. Crossref, Web of Science, Google Scholar
25. L. Wayne and J. A. Yorke, Recurrent outbreaks of measles, chicken pox and mumps, Amer. J. Epidemiol. 98 (1973) 469–482. Crossref, Web of Science, Google Scholar
26. W.-M. Liu, H. W. Hethcote and S. A. Levin, Dynamical behavior of epidemiological models with nonlinear incidence rates, J. Math. Biol. 25(4) (1987) 359–380. Crossref, Web of Science, Google Scholar
27. K. Amir, R. Zarin, M. Inc, G. Zaman and B. Almohsen, Stability analysis of leishmania epidemic model with harmonic mean type incidence rate, Eur. Phys. J. Plus 135(6) (2020) 1–20. Web of Science, Google Scholar
28. K. Karim, R. Zarin, A. Khan, A. Yusuf, M. Al-Shomrani and A. Ullah, Stability analysis of five-grade Leishmania epidemic model with harmonic mean-type incidence rate, Adv. Differ. Equ. 1 (2021) 1–27. Google Scholar
29. K. Amir, R. Zarin, G. Hussain, A. H. Usman, U. W. Humphries and J. F. Gomez-Aguilar, Modeling and sensitivity analysis of HBV epidemic model with convex incidence rate, Results Phys. 22 (2021) 103836. Crossref, Web of Science, Google Scholar
30. K. Amir, R. Zarin, G. Hussain, N. A. Ahmad, M. H. Mohd and A. Yusuf, Stability analysis and optimal control of COVID-19 with convex incidence rate in Khyber Pakhtunkhawa (Pakistan), Results Phys. 20 (2021) 103703. Crossref, Web of Science, Google Scholar
31. B. Buonomo and D. Lacitignola, On the backward bifurcation of a vaccination model with nonlinear incidence, Nonlinear Anal. Model. 16(1) (2011) 30–46. Crossref, Web of Science, Google Scholar
32. O. Sharomi, C. N. Podder, A. B. Gumel, E. H. Elbasha and J. Watmough, Role of incidence function in vaccine-induced backward bifurcation in some HIV models, Math. Biosci. 210 (2007) 436–463. Crossref, Web of Science, Google Scholar
33. K. Amir, G. Hussain, M. Inc and G. Zaman, Existence, uniqueness, and stability of fractional hepatitis B epidemic model, Chaos 30(10) (2020) 103104. Crossref, Web of Science, Google Scholar
34. B. Dumitru, B. Ghanbari, J. H. Asad, A. Jajarmi and H. M. Pirouz, Planar system-masses in an equilateral triangle, Fract. Calc. Appl. Anal. (2020) 010236. Google Scholar
35. J. Amin and D. Baleanu, A new iterative method for the numerical solution of high-order non-linear fractional boundary value problems, Front. Phys. 8 (2020) 220. Crossref, Web of Science, Google Scholar
36. K. Amir, R. Zarin, A. Akgl, A. Saeed and T. Gul, Fractional optimal control of COVID-19 pandemic model with generalized Mittag-Leffler function, Adv. Differ. Equ. 1 (2021) 1–22. Google Scholar
37. J. Amin and D. Baleanu, On the fractional optimal control problems with a general derivative operator, Asian J. Control 23(2) (2021) 1062–1071. Crossref, Web of Science, Google Scholar
38. H. M. Srivastava, V. P. Dubey, R. Kumar, J. Singh, D. Kumar and D. Baleanu, An efficient computational approach for a fractional-order biological population model with carrying capacity, Chaos Solitons Fractals 138 (2020) 109880. Crossref, Web of Science, Google Scholar
39. P. Kumar, V. S. Erturk, A. Yusuf, K. S. Nisar and S. F. Abdelwahab, A study on canine distemper virus (CDV) and rabies epidemics in the red fox population via fractional derivatives, Results Phys. 25 (2021) 104281. Crossref, Web of Science, Google Scholar
40. P. Kumar, V. S. Erturk, A. Yusuf and S. Kumar, Fractional time-delay mathematical modeling of oncolytic virotherapy, Chaos Solitons Fractals 150 (2021) 111123. Crossref, Web of Science, Google Scholar
41. F. M. Khan, Z. U. Khan, Y. P. Lv, A. Yusuf and A. Din, Investigating of fractional order dengue epidemic model with ABC operator, Results Phys. 24 (2021) 104075. Crossref, Web of Science, Google Scholar
42. S. Qureshi and A. Yusuf, Modeling chickenpox disease with fractional derivatives: From Caputo to Atangana–Baleanu, Chaos Solitons Fractals 122 (2019) 111–118. Crossref, Web of Science, Google Scholar
43. A. Jajarmi, A. Yusuf, D. Baleanu and M. Inc, A new fractional HRSV model and its optimal control: A non-singular operator approach, Physica A 547 (2020) 123860. Crossref, Web of Science, Google Scholar
44. B. Santanu, Mathematical modeling of visceral leishmaniasis and control strategies, Chaos Solitons Fractals 104 (2017) 546–556. Crossref, Web of Science, Google Scholar
45. G. Mittag-Leffler, Sur la nouvelle fontion $E_{α}$ , C. R. Acad. Sci. Ser. 137 (1903) 554–558. Google Scholar
46. A. Wiman, Über den fundamental Satz in der Theorie der Funktionen, Acta Math. 29 (1905) 191–201. Crossref, Google Scholar

Vol. 30, No. 05

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History

Received 17 June 2021

Accepted 23 November 2021

Published: June 30, 2022

Information

This is an Open Access article in the “Special Issue Section on Fractal AI-Based Analyses and Applications to Complex Systems: Part III”, edited by Yeliz Karaca (University of Massachusetts Medical School, USA), Dumitru Baleanu (Cankaya University, Turkey), Majaz Moonis (University of Massachusetts Medical School, USA), Yu-Dong Zhang (University of Leicester, UK) & Osvaldo Gervasi (Perugia University, Italy) published by World Scientific Publishing Company. It is distributed under the terms of the Creative Commons Attribution 4.0 (CC-BY) License which permits use, distribution and reproduction in any medium, provided the original work is properly cited.

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DYNAMICS OF VISCERAL LEISHMANIA EPIDEMIC MODEL WITH NON-SINGULAR KERNEL

Abstract

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