Dynamics analysis of dengue fever model with harmonic mean type under fractal-fractional derivative

Khaled A. Aldwoah; Mohammed A. Almalahi; Kamal Shah; Muath Awadalla; Ria H. Egami; Khaled A. Aldwoah; Mohammed A. Almalahi; Kamal Shah; Muath Awadalla; Ria H. Egami

doi:10.3934/math.2024676

AIMS Mathematics

2024, Volume 9, Issue 6: 13894-13926. doi: 10.3934/math.2024676

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Research article

Dynamics analysis of dengue fever model with harmonic mean type under fractal-fractional derivative

1.
Department of Mathematics, Faculty of Science, Islamic University of Madinah, Al Madinah 42351, Saudi Arabia
2.
Department of Mathematics, Hajjah University, Hajjah, Yemen
3.
Department of Mathematics, University of Malakand, Chakdara Dir (Lower), 18000 Khyber Pakhtunkhawa, Pakistan
4.
Department of Computer Science and Mathematics, Lebanese American University, Lebanon
5.
Department of Mathematics and Statistics, College of Science, King Faisal University, Hafuf 31982, Saudi Arabia
6.
Department of Mathematics, College of Science and Humanities in Sulail, Prince Sattam Bin Abdulaziz University, Saudi Arabia

Received: 09 February 2024 Revised: 23 March 2024 Accepted: 02 April 2024 Published: 16 April 2024
MSC : 92D30, 26A33, 34D20, 34Cxx, 92D30, 65C05

Dengue is a viral illness transmitted by Aedes mosquitoes and is a significant global threat. In this study, we developed a model of the dengue epidemic that incorporates larvicide and adulticide, as well as the harmonic mean incidence rate under fractal-fractional derivatives. We examined various theoretical aspects of the model, including nonnegativity, boundedness, existence, uniqueness, and stability. We computed the basic reproduction number $ \Re _{0} $ using the next-generation matrix. The model has two disease-free equilibriums, a trivial equilibrium, and a biologically realistic, along with one endemic equilibrium point. These findings enhanced our understanding of dengue transmission, providing valuable insights for awareness campaigns, control strategies, intervention approaches, decision support, guiding public health planning, and resource allocation to manage dengue effectively.
- dengue diseases,
- fractal-fractional model,
- stability,
- equilibrium points,
- basic reproduction numbers,
- simulation analysis
Citation: Khaled A. Aldwoah, Mohammed A. Almalahi, Kamal Shah, Muath Awadalla, Ria H. Egami. Dynamics analysis of dengue fever model with harmonic mean type under fractal-fractional derivative[J]. AIMS Mathematics, 2024, 9(6): 13894-13926. doi: 10.3934/math.2024676

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Abstract

Dengue is a viral illness transmitted by Aedes mosquitoes and is a significant global threat. In this study, we developed a model of the dengue epidemic that incorporates larvicide and adulticide, as well as the harmonic mean incidence rate under fractal-fractional derivatives. We examined various theoretical aspects of the model, including nonnegativity, boundedness, existence, uniqueness, and stability. We computed the basic reproduction number $ \Re _{0} $ using the next-generation matrix. The model has two disease-free equilibriums, a trivial equilibrium, and a biologically realistic, along with one endemic equilibrium point. These findings enhanced our understanding of dengue transmission, providing valuable insights for awareness campaigns, control strategies, intervention approaches, decision support, guiding public health planning, and resource allocation to manage dengue effectively.

References

[1]	A. Abidemi, N. A. B. Aziz, Optimal control strategies for dengue fever spread in Johor, Malaysia, Comput. Meth. Programs Biomed., 196 (2020), 105585. https://doi.org/10.1016/j.cmpb.2020.105585 doi: 10.1016/j.cmpb.2020.105585
[2]	A. Abidemi, N. A. B. Aziz, Analysis of deterministic models for dengue disease transmission dynamics with vaccination perspective in Johor, Malaysia, Int. J. Appl. Comput. Math., 8 (2022), 45. https://doi.org/10.1007/s40819-022-01250-3 doi: 10.1007/s40819-022-01250-3
[3]	A. Abidemi, M. I. Abd Aziz, R. Ahmad, Vaccination and vector control effect on dengue virus transmission dynamics: modelling and simulation, Chaos Solitons Fract., 133 (2020), 109648. https://doi.org/10.1016/j.chaos.2020.109648 doi: 10.1016/j.chaos.2020.109648
[4]	F. B. Agusto, M. A. Khan, Optimal control strategies for dengue transmission in Pakistan, Math. Biosci., 305 (2018), 102–121. https://doi.org/10.1016/j.mbs.2018.09.007 doi: 10.1016/j.mbs.2018.09.007
[5]	S. Ahmad, S. Javeed, H. Ahmad, J. Khushi, S. K. Elagan, A. Khames, Analysis and numerical solution of novel fractional model for dengue, Results Phys., 28 (2021), 104669. https://doi.org/10.1016/j.rinp.2021.104669 doi: 10.1016/j.rinp.2021.104669
[6]	K. A. Aldwoah, M. A. Almalahi, M. A. Abdulwasaa, K. Shah, S. V. Kawale, M. Awadalla, et al., Mathematical analysis and numerical simulations of the piecewise dynamics model of Malaria transmission: a case study in Yemen, AIMS Math., 9 (2024), 4376–4408. https://doi.org/10.3934/math.2024216 doi: 10.3934/math.2024216
[7]	K. A. Aldwoah, M. A. Almalahi, K. Shah, Theoretical and numerical simulations on the hepatitis B virus model through a piecewise fractional order, Fractal Fract., 7 (2023), 844. https://doi.org/10.3390/fractalfract7120844 doi: 10.3390/fractalfract7120844
[8]	M. A. Almalahi, A. B. Ibrahim, A. Almutairi, O. Bazighifan, T. A. Aljaaidi, J. Awrejcewicz, A qualitative study on second-order nonlinear fractional differential evolution equations with generalized ABC operator, Symmetry, 14 (2022), 207. https://doi.org/10.3390/sym14020207 doi: 10.3390/sym14020207
[9]	M. A. Almalahi, S. K. Panchal, W. Shatanawi, M. S. Abdo, K. Shah, K. Abodayeh, Analytical study of transmission dynamics of 2019-nCoV pandemic via fractal fractional operator, Results Phys., 24 (2021), 104045. https://doi.org/10.1016/j.rinp.2021.104045 doi: 10.1016/j.rinp.2021.104045
[10]	J. K. K. Asamoah, E. Yankson, E. Okyere, G. Q. Sun, Z. Jin, R. Jan, Optimal control and cost-effectiveness analysis for dengue fever model with asymptomatic and partial immune individuals, Results Phys., 31 (2021), 104919. https://doi.org/10.1016/j.rinp.2021.104919 doi: 10.1016/j.rinp.2021.104919
[11]	A. Atangana, Fractal-fractional differentiation and integration: connecting fractal calculus and fractional calculus to predict complex system, Chaos Solitons Fract., 102 (2017), 396–406. https://doi.org/10.1016/j.chaos.2017.04.027 doi: 10.1016/j.chaos.2017.04.027
[12]	A. Atangana, S. Araz, New Numerical Scheme with Newton Polynomial: Theory, Methods, and Applications, New York: Academic Press, 2021.
[13]	A. Atangana, D. Baleanu, New fractional derivatives with nonlocal and non-singular kernel: theory and application to heat transfer model, preprint paper, 2016. https://doi.org/10.48550/arXiv.1602.03408
[14]	D. Baleanu, F. Akhavan Ghassabzade, J. J. Nieto, A. Jajarmi, On a new and generalized fractional model for a real cholera outbreak, Alex. Eng. J., 61 (2022), 9175–9186. https://doi.org/10.1016/j.aej.2022.02.054 doi: 10.1016/j.aej.2022.02.054
[15]	L. R. Bowman, S. Donegan, P. J. Mc Call, Is dengue vector control deficient in effectiveness or evidence?: Systematic review and meta-analysis, PLoS Neglect. Trop. Dis., 10 (2016), e0004551. https://doi.org/10.1371/journal.pntd.0004551 doi: 10.1371/journal.pntd.0004551
[16]	M. Derouich, A. Boutayeb, Dengue fever: mathematical modelling and computer simulation, Appl. Math. Comput., 177 (2006), 528–544. https://doi.org/10.1016/j.amc.2005.11.031 doi: 10.1016/j.amc.2005.11.031
[17]	I. Dorigatti, C. McCormack, G. Nedjati-Gilani, N. M. Ferguson, Using Wolbachia for dengue control: insights from modelling, Trends Parasitol., 34 (2018), 102–113. https://doi.org/10.1016/j.pt.2017.11.002 doi: 10.1016/j.pt.2017.11.002
[18]	L. Esteva, C. Vargas, Influence of vertical and mechanical transmission on the dynamics of dengue disease, Math. Biosci., 167 (2000), 51–64. https://doi.org/10.1016/S0025-5564(00)00024-9 doi: 10.1016/S0025-5564(00)00024-9
[19]	Fatmawati, R. Jan, M. A. Khan, Y. Khan, S. Ullah, A new model of dengue fever in terms of fractional derivative, Math. Biosci. Eng., 17 (2020), 5267–5288.
[20]	Fatmawati, M. A. Khan, C. Alfiniyah, E. Alzahrani, Analysis of dengue model with fractal-fractional Caputo-Fabrizio operator, Adv. Differ. Equ., 2020 (2020), 422. https://doi.org/10.1186/s13662-020-02881-w doi: 10.1186/s13662-020-02881-w
[21]	Y. Gu, M. Khan, R. Zarin, A. Khan, A. Yusuf, U. W. Humphries, Mathematical analysis of a new nonlinear dengue epidemic model via deterministic and fractional approach, Alex. Eng. J., 67 (2023), 1–21. https://doi.org/10.1016/j.aej.2022.10.057 doi: 10.1016/j.aej.2022.10.057
[22]	F. Haq, K. Shah, G. U. Rahman, M. Shahzad, Numerical analysis of fractional order model of HIV-1 infection of CD4+ T-cells, Comput. Meth. Differ. Equ., 5 (2017), 1–11.
[23]	A. Hanif, A. I. K. Butt, T. Ismaeel, Fractional optimal control analysis of COVID-19 and dengue fever co-infection model with Atangana-Baleanu derivative, AIMS Math., 9 (2024), 5171–5203. https://doi.org/10.3934/math.2024251 doi: 10.3934/math.2024251
[24]	R. Jan, S. Boulaaras, Analysis of fractional-order dynamics of dengue infection with non-linear incidence functions, T. Inst. Meas. Control, 44 (2022), 2630–2641. https://doi.org/10.1177/01423312221085049 doi: 10.1177/01423312221085049
[25]	R. Jan, S. Boulaaras, S. Alyobi, K. Rajagopal, M. Jawad, Fractional dynamics of the transmission phenomena of dengue infection with vaccination, Discrete Contin. Dyn. Syst. Ser. S, 2022. https://doi.org/10.3934/dcdss.2022154
[26]	R. Jan, A. Khan, S. Boulaaras, S. A. Zubair, Dynamical behaviour and chaotic phenomena of HIV infection through fractional calculus, Discrete Dyn. Nat. Soc., 2022 (2022), 5937420. https://doi.org/10.1155/2022/5937420 doi: 10.1155/2022/5937420
[27]	R. Jan, M. A. Khan, P. Kumam, P. Thounthong, Modeling the transmission of dengue infection through fractional derivatives, Chaos Solitons Fract., 127 (2019), 189–216. https://doi.org/10.1016/j.chaos.2019.07.002 doi: 10.1016/j.chaos.2019.07.002
[28]	A. Jan, R. Jan, H. Khan, M. S. Zobaer, R. Shah, Fractional-order dynamics of Rift Valley fever in ruminant host with vaccination, Commun. Math. Biol. Neurosci., 2020 (2020), 79. https://doi.org/10.28919/cmbn/5017 doi: 10.28919/cmbn/5017
[29]	R. Jan, Z. Shah, W. Deebani, E. Alzahrani, Analysis and dynamical behavior of a novel dengue model via fractional calculus, Int. J. Biomath., 15 (2022), 2250036. https://doi.org/10.1142/S179352452250036X doi: 10.1142/S179352452250036X
[30]	M. B. Jeelani, A. S. Alnahdi, M. S. Abdo, M. A. Almalahi, N. H. Alharthi, K. Shah, A generalized fractional order model for COV-2 with vaccination effect using real data, Fractals, 31 (2023), 2340042. https://doi.org/10.1142/S0218348X2340042X doi: 10.1142/S0218348X2340042X
[31]	M. A. Khan, The dynamics of dengue infection through fractal-fractional operator with real statistical data, Alex. Eng. J., 60 (2021), 321–336. https://doi.org/10.1016/j.aej.2020.08.018 doi: 10.1016/j.aej.2020.08.018
[32]	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), 528. https://doi.org/10.1140/epjp/s13360-020-00535-0 doi: 10.1140/epjp/s13360-020-00535-0
[33]	E. S. Paixão, M. G. Teixeira, L. C. Rodrigues, Zika, chikungunya and dengue: The causes and threats of new and re-emerging arboviral diseases, BMJ Glob. Health, 3 (2018), 000530. https://doi.org/10.1136/bmjgh-2017-000530 doi: 10.1136/bmjgh-2017-000530
[34]	M. Qayyum, E. Ahmad, S. Tauseef Saeed, H. Ahmad, S. Askar, Homotopy perturbation method-based soliton solutions of the time-fractional (2+ 1)-dimensional Wu-Zhang system describing long dispersive gravity water waves in the ocean, Front. Phys., 11 (2023), 1178154. https://doi.org/10.3389/fphy.2023.1178154 doi: 10.3389/fphy.2023.1178154
[35]	S. Qureshi, A. Atangana, Mathematical analysis of dengue fever outbreak by novel fractional operators with field data, Phys. A, 526 (2019), 121127. https://doi.org/10.1016/j.physa.2019.121127 doi: 10.1016/j.physa.2019.121127
[36]	M. Rafiq, M. Kamran, H. Ahmad, A. Saliu, Critical analysis for nonlinear oscillations by least square HPM, Sci. Rep., 14 (2024), 1456. https://doi.org/10.1038/s41598-024-51706-3 doi: 10.1038/s41598-024-51706-3
[37]	H. S. Rodrigues, M. T. T. Monteiro, D. F. Torres, Vaccination models and optimal control strategies to dengue, Math. Biosci., 247 (2014), 1–12. https://doi.org/10.1016/j.mbs.2013.10.006 doi: 10.1016/j.mbs.2013.10.006
[38]	H. S. Rodrigues, M. T. T. Monteiro, D. F. Torres, Dynamics of dengue epidemics when using optimal control, Math. Comput. Model., 52 (2010), 1667–1673. https://doi.org/10.1016/j.mcm.2010.06.034 doi: 10.1016/j.mcm.2010.06.034
[39]	H. S. Rodrigues, M. T. T. Monteiro, D. F. Torres, Bioeconomic perspectives to an optimal control dengue model, Int. J. Comput. Math., 90 (2013), 2126–2136. https://doi.org/10.1080/00207160.2013.790536 doi: 10.1080/00207160.2013.790536
[40]	T. Sardar, S. Rana, J. Chattopadhyay, A mathematical model of dengue transmission with memory, Commun. Nonlinear Sci. Numer. Simul., 22 (2015), 511–525. https://doi.org/10.1016/j.cnsns.2014.08.009 doi: 10.1016/j.cnsns.2014.08.009
[41]	Z. Shah, E. Bonyah, E. Alzahrani, R. Jan, N. A. Alreshidi, Chaotic phenomena and oscillations in dynamical behaviour of financial system via fractional calculus, Complexity, 2022 (2022), 8113760. https://doi.org/10.1155/2022/8113760 doi: 10.1155/2022/8113760
[42]	D. R. Smart, Fixed Point Theorems, Cambridge: Cambridge University Press, 1980.
[43]	J. M. Torres-Flores, A. Reyes-Sandoval, M. I. Salazar, Dengue vaccines: an update, BioDrugs, 36 (2022), 325–336. https://doi.org/10.1007/s40259-022-00531-z
[44]	G. ur Rahman, R. P. Agarwal, Q. Din, Mathematical analysis of giving up smoking model via harmonic mean type incidence rate, Appl. Math. Comput., 354 (2019), 128–148. https://doi.org/10.1016/j.amc.2019.01.053 doi: 10.1016/j.amc.2019.01.053
[45]	H. van den Berg, R. Velayudhan, R. S. Yadav, Management of insecticides for use in disease vector control: Lessons from six countries in Asia and the Middle East, PLoS Neglect. Trop. Dis., 15 (2021), e0009358. https://doi.org/10.1371/journal.pntd.0009358 doi: 10.1371/journal.pntd.0009358
[46]	G. H. Wang, S. Gamez, R. R. Raban, J. M. Marshall, L. Alphey, M. Li, et al., Combating mosquito-borne diseases using genetic control technologies, Nat. Commun., 12 (2021), 4388. https://doi.org/10.1038/s41467-021-24654-z doi: 10.1038/s41467-021-24654-z
[47]	Dengue and Severe Dengue, World Health Organization (WHO), 2019. Available from: https://www.who.int/health-topics/dengue-and-severe-dengue.
[48]	P. S. Yen, A. B. Failloux, A review: Wolbachia-based population replacement for mosquito control shares common points with genetically modified control approaches, Pathogens, 9 (2020), 404. https://doi.org/10.3390/pathogens9050404 doi: 10.3390/pathogens9050404
[49]	R. Zarin, Modeling and numerical analysis of fractional order hepatitis B virus model with harmonic mean type incidence rate, Comput. Meth. Biomec. Biomed. Eng., 26 (2023), 1018–1033. https://doi.org/10.1080/10255842.2022.2103371 doi: 10.1080/10255842.2022.2103371

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