Modeling Rift Valley fever transmission: insights from fractal-fractional dynamics with the Caputo derivative

Rashid Jan; Normy Norfiza Abdul Razak; Sania Qureshi; Imtiaz Ahmad; Salma Bahramand; Rashid Jan; Normy Norfiza Abdul Razak; Sania Qureshi; Imtiaz Ahmad; Salma Bahramand

doi:10.3934/mmc.2024015

Mathematical Modelling and Control

2024, Volume 4, Issue 2: 163-177. doi: 10.3934/mmc.2024015

Previous Article Next Article

Research article

Modeling Rift Valley fever transmission: insights from fractal-fractional dynamics with the Caputo derivative

1.
Institute of Energy Infrastructure, Department of Civil Engineering, College of Engineering, Universiti Tenaga Nasional, Putrajaya Campus, Jalan Ikram-Uniten, Kajang 43000, Selangor, Malaysia
2.
Mathematics Research Center, Near East University North Cyprus, Mersin 99138, Turkey
3.
Department of Basic Sciences and Related Studies, Mehran University of Engineering and Technology, Jamshoro 76062, Pakistan
4.
Department of Computer Science and Mathematics, Lebanese American University, Beirut, P. O. Box 135053, Lebanon
5.
Department of Mathematics, Near East University, Mersin 99138, Turkey
6.
Institute of Informatics and Computing in Energy, Universiti Tenaga Nasional, Kajang, Selangor, Malaysia
7.
Department of Political Science, Bacha Khan, University Charsadda, Charsadda 24420, Pakistan

Received: 07 September 2023 Revised: 17 December 2023 Accepted: 29 December 2023 Published: 15 May 2024

The infection caused by Rift Valley fever (RVF) virus is a dangerous vector-borne disease found in humans, domestic, and wild animals. It is transferred through insect vectors to ruminant host and then spread through direct contact of infected animals with their body fluid or organs. In this paper, a fractal-fractional model for the transmission of RVF in the Caputo's sense was presented. We analyzed the model and determined the basic reproduction number through the next-generation matrix technique, indicated by $ \mathcal{R}_0 $. The global sensitivity technique is used for the sensitivity test of $ \mathcal{R}_0 $ to find out the most sensitive input-factors to the reproduction parameter $ \mathcal{R}_0 $. The existence and uniqueness results of the proposed fractal-fractional model were established. Then, we presented the fractal-fractional dynamics of the proposed RVF model through a novel numerical scheme under the fractal-fractional Caputo operator. In the end, the recommended model of RVF was highlighted numerically with the variation of different input parameters of the system. The key factors of the system were highlighted to the policymakers for the control and prevention of the infection.
- Rift Valley fever,
- fractal-fractional operator,
- mathematical model,
- sensitivity analysis,
- numerical analysis
Citation: Rashid Jan, Normy Norfiza Abdul Razak, Sania Qureshi, Imtiaz Ahmad, Salma Bahramand. Modeling Rift Valley fever transmission: insights from fractal-fractional dynamics with the Caputo derivative[J]. Mathematical Modelling and Control, 2024, 4(2): 163-177. doi: 10.3934/mmc.2024015

Related Papers:

Abstract

The infection caused by Rift Valley fever (RVF) virus is a dangerous vector-borne disease found in humans, domestic, and wild animals. It is transferred through insect vectors to ruminant host and then spread through direct contact of infected animals with their body fluid or organs. In this paper, a fractal-fractional model for the transmission of RVF in the Caputo's sense was presented. We analyzed the model and determined the basic reproduction number through the next-generation matrix technique, indicated by $ \mathcal{R}_0 $. The global sensitivity technique is used for the sensitivity test of $ \mathcal{R}_0 $ to find out the most sensitive input-factors to the reproduction parameter $ \mathcal{R}_0 $. The existence and uniqueness results of the proposed fractal-fractional model were established. Then, we presented the fractal-fractional dynamics of the proposed RVF model through a novel numerical scheme under the fractal-fractional Caputo operator. In the end, the recommended model of RVF was highlighted numerically with the variation of different input parameters of the system. The key factors of the system were highlighted to the policymakers for the control and prevention of the infection.

References

[1]	R. Daubney, J. R. Hudson, Enzootic hepatitis or Rift Valley fever, an un-described virus disease of sheep, cattle and man from East Africa, J. Pathol. Bacteriol., 34 (1931), 545–579. https://doi.org/10.1002/path.1700340418 doi: 10.1002/path.1700340418
[2]	S. Abdo-Salem, A. Tran, V. Grosbois, G. Gerbier, M. Al-Qadasi, K. Saeed, et al., Can environmental and socioeconomic factors explain the recent emergence of Rift Valley fever in Yemen, 2000-2001? Vector Borne Zoonot. Dis., 11 (2011), 773–779. https://doi.org/10.1089/vbz.2010.0084
[3]	C. A. Mebus, Rift Valley fever, Access Sci., 2014. https://doi.org/10.1036/1097-8542.757367
[4]	A. Anyamba, J. P. Chretien, J. Small, C. J. Tucker, P. B. Formenty, J. H. Richardson, et al., Prediction of a Rift Valley fever outbreak, Proc. Natl. Acad. Sci., 106 (2009), 955–959. https://doi.org/10.1073/pnas.0806490106 doi: 10.1073/pnas.0806490106
[5]	N. Chitnis, J. M. Hyman, C. A. Manore, Modelling vertical transmission in vector-borne diseases with applications to Rift Valley fever, J. Biol. Dyn., 7 (2013), 11–40. https://doi.org/10.1080/17513758.2012.733427 doi: 10.1080/17513758.2012.733427
[6]	S. Sankhe, C. Talla, M. S. Thiam, M. Faye, M. A. Barry, M. Diarra, et al., Seroprevalence of Crimean-Congo Hemorrhagic Fever virus and Rift Valley fever virus in human population in Senegal from October to November 2020, IJID Regions, 7 (2023), 216–221. https://doi.org/10.1016/j.ijregi.2023.03.016 doi: 10.1016/j.ijregi.2023.03.016
[7]	M. K.Trabelsi, A. Hachid, F. Derrar, N. E. Messahel, T. Bia, Y. Mockbel, et al., Serological evidence of Rift Valley fever viral infection among camels imported into Southern Algeria, Comp. Immunol. Microbiol. Infect. Dis., 100 (2023), 102035. https://doi.org/10.1016/j.cimid.2023.102035 doi: 10.1016/j.cimid.2023.102035
[8]	S. C. Mpeshe, H. Haario, J. M. Tchuenche, A mathematical model of Rift Valley fever with human host, Acta Biotheor., 59 (2011), 231–250. https://doi.org/10.1007/s10441-011-9132-2 doi: 10.1007/s10441-011-9132-2
[9]	J. K. Gitau, R. W. Macharia, K. W. Mwangi, N. Ongeso, E. Murungi, Gene co-expression network identifies critical genes, pathways and regulatory motifs mediating the progression of rift valley fever in Bos taurus, Heliyon, 9 (2023), e18175. https://doi.org/10.1016/j.heliyon.2023.e18175 doi: 10.1016/j.heliyon.2023.e18175
[10]	L. Xue, H. M. Scott, L. W. Cohnstaedt, C. Scoglio, A network-based meta-population approach to model Rift Valley fever epidemics, J. Theor. Biol., 306 (2012), 129–144. https://doi.org/ 10.1016/j.jtbi.2012.04.029 doi: 10.1016/j.jtbi.2012.04.029
[11]	D. Gao, C. Cosner, R. S. Cantrell, J. C. Beier, S. Ruan, Modeling the spatial spread of Rift Valley fever in Egypt, Bullt. Math. Biol., 75 (2013), 523–542. https://doi.org/10.1007/s11538-013-9818-5 doi: 10.1007/s11538-013-9818-5
[12]	C. Catre-Sossah, C. Lebon, P. Rabarison, E. Cardinale, P. Mavingui, C. Atyame, Evidence of Eretmapodites subsimplicipes and Aedes albopictus as competent vectors for Rift Valley fever virus transmission in Mayotte, Acta Trop., 239 (2023), 106835. https://doi.org/10.1016/j.actatropica.2023.106835 doi: 10.1016/j.actatropica.2023.106835
[13]	R. Jan, Y. Xiao, Effect of partial immunity on transmission dynamics of dengue disease with optimal control, Math. Methods Appl. Sci., 42 (2019), 1967–1983. https://doi.org/10.1002/mma.5491 doi: 10.1002/mma.5491
[14]	N. N. H. Shah, R. Jan, H. Ahmad, N. N. A. Razak, I. Ahmad, H. Ahmad, Enhancing public health strategies for tungiasis: a mathematical approach with fractional derivative, AIMS Bioeng., 10 (2023), 384–405. https://doi.org/10.3934/bioeng.2023023 doi: 10.3934/bioeng.2023023
[15]	R. Jan, Y. Xiao, Effect of pulse vaccination on dynamics of dengue with periodic transmission functions, Adv. Differ. Equations, 2019 (2019), 368. https://doi.org/10.1186/s13662-019-2314-y doi: 10.1186/s13662-019-2314-y
[16]	T. Ikegami, S. Makino, Rift valley fever vaccines, Vaccine, 27 (2009), 69–72. https://doi.org/10.1016/j.vaccine.2009.07.046
[17]	G. F. Ronchi, L. Testa, M. Iorio, C. Pinoni, G. Bortone, A. C. Dondona, et al., Immunogenicity and safety studies of an inactivated vaccine against Rift Valley fever, Acta Trop., 232 (2022), 106498. https://doi.org/10.1016/j.actatropica.2022.106498 doi: 10.1016/j.actatropica.2022.106498
[18]	J. C. Morrill, C. J. Peters, G. E. Bettinger, P. M. Palermo, D. R. Smith, D. M. Watts, Rift Valley fever MP-12 vaccine elicits an early protective immune response in mice, Vaccine, 40 (2022), 7255–7261. https://doi.org/10.1016/j.vaccine.2022.10.062 doi: 10.1016/j.vaccine.2022.10.062
[19]	F. Chamchod, R. S. Cantrell, C. Cosner, A. N. Hassan, J. C. Beier, S. Ruan, A modeling approach to investigate epizootic outbreaks and enzootic maintenance of Rift Valley fever virus, Bull. Math. Biol., 76 (2014), 2052–2072. https://doi.org/10.1007/s11538-014-9998-7 doi: 10.1007/s11538-014-9998-7
[20]	I. Ahmad, M. Ahsan, I. Hussain, P. Kumam, W. Kumam, Numerical simulation of PDEs by local meshless differential quadrature collocation method, Symmetry, 11 (2019), 394. https://doi.org/10.3390/sym11030394 doi: 10.3390/sym11030394
[21]	M. N. Khan, I. Ahmad, M. Shakeel, R. Jan, Fractional calculus analysis: investigating Drinfeld-Sokolov-Wilson system and Harry Dym equations via meshless procedures, Math. Model. Control, 4 (2024), 86–100. https://doi.org/10.3934/mmc.2024008 doi: 10.3934/mmc.2024008
[22]	M. Shakeel, M. N. Khan, I. Ahmad, H. Ahmad, N. Jarasthitikulchai, W. Sudsutad, Local meshless collocation scheme for numerical simulation of space fractional PDE, Therm. Sci., 27 (2023), 101–109. https://doi.org/10.2298/TSCI23S1101S doi: 10.2298/TSCI23S1101S
[23]	F. Wang, J. Zhang, I. Ahmad, A. Farooq, H. Ahmad, A novel meshfree strategy for a viscous wave equation with variable coefficients, Front. Phys., 9 (2021), 701512. https://doi.org/10.3389/fphy.2021.701512 doi: 10.3389/fphy.2021.701512
[24]	I. Ahmad, I. Ali, R. Jan, S. A. Idris, M. Mousa, Solutions of a three-dimensional multi-term fractional anomalous solute transport model for contamination in groundwater, Plos One, 18 (2023), e0294348. https://doi.org/10.1371/journal.pone.0294348 doi: 10.1371/journal.pone.0294348
[25]	F. Wang, I. Ahmad, H. Ahmad, M. D. Alsulami, K. S. Alimgeer, C. Cesarano, et al., Meshless method based on RBFs for solving three-dimensional multi-term time fractional PDEs arising in engineering phenomenons, J. King Saud Univ., 33 (2021), 101604. https://doi.org/10.1016/j.jksus.2021.101604 doi: 10.1016/j.jksus.2021.101604
[26]	I. Ahmad, H. Ahmad, A. E. Abouelregal, P. Thounthong, M. Abdel-Aty, Numerical study of integer-order hyperbolic telegraph model arising in physical and related sciences, Eur. Phys. J. Plus, 135 (2020), 759. https://doi.org/10.1140/epjp/s13360-020-00784-z doi: 10.1140/epjp/s13360-020-00784-z
[27]	Z. U. Rehman, S. Boulaaras, R. Jan, I. Ahmad, S. Bahramand, Computational analysis of financial system through non-integer derivative, J. Comput. Sci., 75 (2024), 102204. https://doi.org/10.1016/j.jocs.2023.102204 doi: 10.1016/j.jocs.2023.102204
[28]	H. Ahmad, M. N. Khan, I. Ahmad, M. Omri, M. F. Alotaibi, A meshless method for numerical solutions of linear and nonlinear time-fractional Black-Scholes models, AIMS Math., 8 (2023), 19677–19698. https://doi.org/ 10.3934/math.20231003 doi: 10.3934/math.20231003
[29]	I. Ahmad, A. A. Bakar, I. Ali, S. Haq, S. Yussof, A. H. Ali, Computational analysis of time-fractional models in energy infrastructure applications, Alex. Eng. J., 82 (2023), 426–436. https://doi.org/10.1016/j.aej.2023.09.057 doi: 10.1016/j.aej.2023.09.057
[30]	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
[31]	W. Deebani, R. Jan, Z. Shah, N. Vrinceanu, M. Racheriu, Modeling the transmission phenomena of water-borne disease with non-singular and non-local kernel, Comput. Methods Biomech. Biomed. Eng., 26 (2022), 1294–1307. https://doi.org/10.1080/10255842.2022.2114793 doi: 10.1080/10255842.2022.2114793
[32]	M. Farman, H. Besbes, K. S. Nisar, M. Omri, Analysis and dynamical transmission of Covid-19 model by using Caputo-Fabrizio derivative, Alex. Eng. J., 66 (2023), 597–606. https://doi.org/10.1016/j.aej.2022.12.026 doi: 10.1016/j.aej.2022.12.026
[33]	J. Li, I. Ahmad, H. Ahmad, D. Shah, Y. M. Chu, P. Thounthong, et al., Numerical solution of two-term time-fractional PDE models arising in mathematical physics using local meshless method, Open Phys., 18 (2020), 1063–1072. https://doi.org/10.1515/phys-2020-0222 doi: 10.1515/phys-2020-0222
[34]	I. Ahmad, M. Riaz, M. Ayaz, M. Arif, S. Islam, P. Kumam, Numerical simulation of partial differential equations via local meshless method, Symmetry, 11 (2019), 257. https://doi.org/10.3390/sym11020257 doi: 10.3390/sym11020257
[35]	K. S. Nisar, M. Farman, E. Hincal, A. Shehzad, Modelling and analysis of bad impact of smoking in society with Constant Proportional-Caputo Fabrizio operator, Chaos Solitons Fract., 172 (2023), 113549. https://doi.org/10.1016/j.chaos.2023.113549 doi: 10.1016/j.chaos.2023.113549
[36]	A. Atangana, J. J. Nieto, Numerical solution for the model of RLC circuit via the fractional derivative without singular kernel, Adv. Mech. Eng., 7 (2015), 1687814015613758. https://doi.org/10.1177/1687814015613758 doi: 10.1177/1687814015613758
[37]	H. M. Srivastava, R. Jan, A. Jan, W. Deebani, M. Shutaywi, Fractional-calculus analysis of the transmission dynamics of the dengue infection, Chaos, 31 (2021), 053130. https://doi.org/10.1063/5.0050452 doi: 10.1063/5.0050452
[38]	S. Qureshi, A. Yusuf, Fractional derivatives applied to MSEIR problems: comparative study with real world data, Eur. Phys. J. Plus, 134 (2019), 171. https://doi.org/10.1140/epjp/i2019-12661-7 doi: 10.1140/epjp/i2019-12661-7
[39]	T. Q. Tang, Z. Shah, R. Jan, W. Deebani, M. Shutaywi, A robust study to conceptualize the interactions of $CD4^+$ T-cells and human immunodeficiency virus via fractional-calculus, Phys. Scr., 96 (2021), 125231. https://doi.org/10.1088/1402-4896/ac2d7b doi: 10.1088/1402-4896/ac2d7b
[40]	O. M. Ogunmiloro, A fractional order mathematical model of teenage pregnancy problems and rehabilitation in Nigeria, Math. Modell. Control, 2 (2022), 139–152. https://doi.org/10.3934/mmc.2022015 doi: 10.3934/mmc.2022015
[41]	M. Caputo, Linear models of dissipation whose Q is almost frequency independent-II, Geophys. J. Int., 13 (1967), 529–539. https://doi.org/10.1111/j.1365-246X.1967.tb02303.x doi: 10.1111/j.1365-246X.1967.tb02303.x
[42]	K. S. Nisar, M. Farman, M. Abdel-Aty, J. Cao, A review on epidemic models in sight of fractional calculus, Alex. Eng. J., 75 (2023), 81–113. https://doi.org/10.1016/j.aej.2023.05.071 doi: 10.1016/j.aej.2023.05.071
[43]	Z. Li, Z. Liu, M. A. Khan, Fractional investigation of bank data with fractal-fractional Caputo derivative, Chaos Solitons Fract., 131 (2020), 109528. https://doi.org/10.1016/j.chaos.2019.109528 doi: 10.1016/j.chaos.2019.109528
[44]	M. Farman, S. Jamil, K. S. Nisar, A. Akgul, Mathematical study of fractal-fractional leptospirosis disease in human and rodent populations dynamical transmission, Ain Shams Eng. J., 15 (2023), 102452. https://doi.org/10.1016/j.asej.2023.102452 doi: 10.1016/j.asej.2023.102452
[45]	J. A. P. Heesterbeek, A brief history of R0 and a recipe for its calculation, Acta Biotheor., 50 (2002), 189–204. https://doi.org/10.1023/a:1016599411804 doi: 10.1023/a:1016599411804
[46]	J. Yangla, H. Abboubakar, E. Dangbe, R. Yankoulo, A. A. A. Ari, I. Damakoa, et al., Fractional dynamics of a Chikungunya transmission model, Sci. Afr., 21 (2023), e01812. https://doi.org/10.1016/j.sciaf.2023.e01812 doi: 10.1016/j.sciaf.2023.e01812
[47]	K. N. Nabi, H. Abboubakar, P. Kumar, Forecasting of Covid-19 pandemic: from integer derivatives to fractional derivatives, Chaos Solitons Fract., 141 (2020), 110283. https://doi.org/10.1016/j.chaos.2020.110283 doi: 10.1016/j.chaos.2020.110283
[48]	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. https://doi.org/10.1016/j.jtbi.2008.04.011 doi: 10.1016/j.jtbi.2008.04.011
[49]	A. A. Majok, K. H. Zessin, M. P. O. Baumann, T. B. Farver, Analyses of baseline survey data on rinderpest in Bahr el Ghazal Province, with proposal of an improved vaccination strategy against rinderpest for southern Sudan, Trop. Anim. Health Prod., 23 (1991), 186–196. https://doi.org/10.1007/BF02357004 doi: 10.1007/BF02357004
[50]	D. V. Canyon, J. L. K Hii, R. Muller, The frequency of host biting and its effect on oviposition and survival in Aedes aegypti (Diptera: Culicidae), Bull. Entomol. Res., 89 (1999), 35–39. https://doi.org/10.1017/S000748539900005X doi: 10.1017/S000748539900005X
[51]	M. J. Turell, K. J. Linthicum, L. A. Patrican, F. G. Davies, A. Kairo, C. L. Bailey, Vector competence of selected African mosquito (Diptera: Culicidae) species for Rift Valley fever virus, J. Med. Entomol., 45 (2008), 102–108. https://doi.org/10.1093/jmedent/45.1.102 doi: 10.1093/jmedent/45.1.102
[52]	M. H. Reiskind, C. J. Westbrook, L. P. Lounibos, Exposure to chikungunya virus and adult longevity in Aedes aegypti (L.) and Aedes albopictus (Skuse), J. Vector Ecol., 35 (2010), 61–68. https://doi.org/10.1111/j.1948-7134.2010.00059.x doi: 10.1111/j.1948-7134.2010.00059.x
[53]	A. Atangana, Q. Sania, Modeling attractors of chaotic dynamical systems with fractal-fractional operators, Chaos Solitons Fract., 123 (2019), 320–337. https://doi.org/10.1016/j.chaos.2019.04.020 doi: 10.1016/j.chaos.2019.04.020
[54]	M. L. Danzetta, R. Bruno, F. Sauro, L. Savini, P. Calistri, Rift Valley fever transmission dynamics described by compartmental models, Prev. Vet. Med., 134 (2016), 197–210. https://doi.org/10.1016/j.prevetmed.2016.09.007 doi: 10.1016/j.prevetmed.2016.09.007
[55]	C. Yang, L. Nie, Modelling the use of impulsive vaccination to control Rift Valley fever virus transmission, Adv. Differ. Equations, 2016 (2016), 134. https://doi.org/10.1186/s13662-016-0835-1 doi: 10.1186/s13662-016-0835-1

Reader Comments

Your name:*

Email:*
© 2024 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)