Hybrid multi-step fractional numerical schemes for human-wildlife zoonotic disease dynamics

Muflih Alhazmi; Safa M. Mirgani; Abdullah Alahmari; Sayed Saber; Muflih Alhazmi; Safa M. Mirgani; Abdullah Alahmari; Sayed Saber

doi:10.3934/math.2025944

AIMS Mathematics

2025, Volume 10, Issue 9: 21126-21158. doi: 10.3934/math.2025944

Previous Article Next Article

Research article Special Issues

Hybrid multi-step fractional numerical schemes for human-wildlife zoonotic disease dynamics

1.
Mathematics Department, Faculty of Science, Northern Border University, Arar, Saudi Arabia
2.
Imam Mohammad Ibn Saud Islamic University (IMSIU), College of Science Department of Mathematics and Statistics, Riyadh, Saudi Arabia
3.
Department of Mathematics, Faculty of Sciences, Umm Al-Qura University, Saudi Arabia
4.
Department of Mathematics, Faculty of Science, Al-Baha University, Al-Baha, Saudi Arabia
5.
Department of Mathematics and Computer Science, Faculty of Science, Beni-Suef University, Egypt

Received: 12 June 2025 Revised: 21 July 2025 Accepted: 24 July 2025 Published: 15 September 2025
MSC : 34A08, 34L99, 92D30

In this study, the transmission dynamics of zoonotic diseases between baboons and humans were explored by examining increased interactions between humans and wild animals. We established the model's well-posedness through proofs of existence, uniqueness, non-negativity, and boundedness of solutions. Stability and sensitivity analyses identified key parameters affecting disease dynamics, particularly the baboon-to-human transmission rate $ (\beta_h) $, the human recovery rate $ (\gamma_h) $, and the human-side contact control parameter $ (H_i) $. The basic reproduction number $ (R_0) $ governed disease outcomes: If $ R_0 < 1 $, the disease died out and the infection-free equilibrium was globally asymptotically stable; if $ R_0 > 1 $, a unique endemic equilibrium emerged and was locally asymptotically stable, indicating the potential for disease persistence. Numerical simulations were conducted using the Multi-Step Generalized Differential Transform Method and the Adams-Bashforth-Moulton scheme, confirming the model's biological relevance. Our results indicated that sterilization reduced infected baboons by up to 40%, while food access restrictions lowered human infections by approximately 25%. By leveraging fractional calculus and advanced numerical methods, this study provides a robust framework for modeling zoonotic diseases and offers actionable insights for public health and wildlife management.
- fractional derivatives,
- nonlinear equations,
- simulation,
- numerical results,
- iterative method,
- zoonotic disease
Citation: Muflih Alhazmi, Safa M. Mirgani, Abdullah Alahmari, Sayed Saber. Hybrid multi-step fractional numerical schemes for human-wildlife zoonotic disease dynamics[J]. AIMS Mathematics, 2025, 10(9): 21126-21158. doi: 10.3934/math.2025944

Related Papers:

Abstract

In this study, the transmission dynamics of zoonotic diseases between baboons and humans were explored by examining increased interactions between humans and wild animals. We established the model's well-posedness through proofs of existence, uniqueness, non-negativity, and boundedness of solutions. Stability and sensitivity analyses identified key parameters affecting disease dynamics, particularly the baboon-to-human transmission rate $ (\beta_h) $, the human recovery rate $ (\gamma_h) $, and the human-side contact control parameter $ (H_i) $. The basic reproduction number $ (R_0) $ governed disease outcomes: If $ R_0 < 1 $, the disease died out and the infection-free equilibrium was globally asymptotically stable; if $ R_0 > 1 $, a unique endemic equilibrium emerged and was locally asymptotically stable, indicating the potential for disease persistence. Numerical simulations were conducted using the Multi-Step Generalized Differential Transform Method and the Adams-Bashforth-Moulton scheme, confirming the model's biological relevance. Our results indicated that sterilization reduced infected baboons by up to 40%, while food access restrictions lowered human infections by approximately 25%. By leveraging fractional calculus and advanced numerical methods, this study provides a robust framework for modeling zoonotic diseases and offers actionable insights for public health and wildlife management.

References

[1]	K. E. Jones, N. G. Patel, M. A. Levy, A. Storeygard, D. Balk, J. L. Gittleman, et al., Global trends in emerging infectious diseases, Nature, 451 (2008), 990–993. https://doi.org/10.1038/nature06536 doi: 10.1038/nature06536
[2]	M. E. J. Woolhouse, S. Gowtage-Sequeria, Host range and emerging and reemerging pathogens, Emerg. Infect. Dis., 11 (2005), 1842–1847. https://doi.org/10.3201/eid1112.050997 doi: 10.3201/eid1112.050997
[3]	A. Al-Aklabi, A. W. Al-Khulaidi, A. Hussain, N. Al-Sagheer, Main vegetation types and plant species diversity along an altitudinal gradient of Al Baha region, Saudi Arabia, Saudi J. Biol. Sci., 23 (2016), 687–697. https://doi.org/10.1016/j.sjbs.2016.02.007 doi: 10.1016/j.sjbs.2016.02.007
[4]	P. Daszak, A. A. Cunningham, A. D. Hyatt, Emerging infectious diseases of wildlife-Threats to biodiversity and human health, Science, 287 (2000), 443–449. https://doi.org/10.1126/science.287.5452.443 doi: 10.1126/science.287.5452.443
[5]	N. D. Wolfe, C. P. Dunavan, J. Diamond, Origins of major human infectious diseases, Nature, 447 (2007), 279–283. https://doi.org/10.1038/nature05775 doi: 10.1038/nature05775
[6]	M. Kazimírová, B. Mangová, M. Chvostáč, Y. M. Didyk, P. de Alba, A. Mira, et al., The role of wildlife in the epidemiology of tick-borne diseases in Slovakia, Current Res. Parasitol. V., 6 (2024), 100195. https://doi.org/10.1016/j.crpvbd.2024.100195 doi: 10.1016/j.crpvbd.2024.100195
[7]	K. Mapagha-Boundoukou, M. H. Mohamed-Djawad, N. M. Longo-Pendy, P. Makouloutou-Nzassi, F. Bangueboussa, M. B. Said, et al., Gastrointestinal parasitic infections in non-human primates at Gabon's primatology center: Implications for zoonotic diseases, J. Zool. Bot. Gard., 5 (2024), 733–744. https://doi.org/10.3390/jzbg5040048 doi: 10.3390/jzbg5040048
[8]	B. Tulu, A. Zewede, M. Belay, M. Zeleke, M. Girma, M. Tegegn, et al., Epidemiology of bovine tuberculosis and its zoonotic implication in Addis Ababa milkshed, central Ethiopia, Front Vet Sci., 17 (2021), 595511. https://doi.org/10.3389/fvets.2021.595511 doi: 10.3389/fvets.2021.595511
[9]	A. A. Al-Huqail, Z. Islam, Ecological stress assessment on vegetation in the Al-Baha highlands, Saudi Arabia (1991-2023), Sustainability, 17 (2025), 2854. https://doi.org/10.3390/su17072854 doi: 10.3390/su17072854
[10]	P. van den Driessche, J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Math. Biosci., 180 (2002), 29–48. https://doi.org/10.1016/S0025-5564(02)00108-6 doi: 10.1016/S0025-5564(02)00108-6
[11]	M. Marsudi, T. Trisilowati, R. R. Musafir, Bifurcation and optimal control analysis of an HIV/AIDS model with saturated incidence rate, Mathematics, 13 (2025), 2149. https://doi.org/10.3390/math13132149 doi: 10.3390/math13132149
[12]	I. Podlubny, Fractional differential equations, Academic Press, 1998.
[13]	C. A. Monje, Y. Q. Chen, B. M. Vinagre, D. Xue, V. Feliu, Fractional-order systems and controls: Fundamentals and applications, London: Springer, 2010. https://doi.org/10.1007/978-1-84996-335-0
[14]	L. J. S. Allen, V. L. Brown, C. B. Jonsson, S. L. Klein, S. M. Laverty, K. Magwedere, et al., Mathematical modeling of viral zoonoses in wildlife, Nat. Resour. Model., 25 (2012), 5–51. https://doi.org/10.1111/j.1939-7445.2011.00104.x doi: 10.1111/j.1939-7445.2011.00104.x
[15]	E. Babaie, A. A. Alesheikh, M. Tabasi, Spatial modeling of zoonotic cutaneous leishmaniasis with regard to potential environmental factors using ANFIS and PCA-ANFIS methods, Acta Trop., 228 (2022), 106296. https://doi.org/10.1016/j.actatropica.2021.106296 doi: 10.1016/j.actatropica.2021.106296
[16]	R. L. Bagley, R. A. Calico, Fractional order state equations for the control of viscoelastically damped structures, J. Guid. Control Dynam., 14 (1991), 304–311. https://doi.org/10.2514/3.20641 doi: 10.2514/3.20641
[17]	D. Kusnezov, A. Bulgac, G. D. Dang, Quantum Lévy processes and fractional kinetics, Phys. Rev. Lett., 82 (1999), 1136. https://doi.org/10.1103/PhysRevLett.82.1136 doi: 10.1103/PhysRevLett.82.1136
[18]	H. A. Hammad, M. Qasymeh, M. Abdel-Aty, Existence and stability results for a Langevin system with Caputo-Hadamard fractional operators, Int. J. Geom. Methods M., 21 (2024), 2450218. https://doi.org/10.1142/S0219887824502189 doi: 10.1142/S0219887824502189
[19]	S. Saber, Control of chaos in the Burke-Shaw system of fractal-fractional order in the sense of Caputo-Fabrizio, J. Appl. Math. Comput. Mech., 23 (2024), 83–96. https://doi.org/10.17512/jamcm.2024.1.07 doi: 10.17512/jamcm.2024.1.07
[20]	M. Alhazmi, S. Saber, Glucose-insulin regulatory system: Chaos control and stability analysis via Atangana-Baleanu fractal-fractional derivatives, Alex. Eng. J., 122 (2025), 77–90. https://doi.org/10.1016/j.aej.2025.02.066 doi: 10.1016/j.aej.2025.02.066
[21]	S. Saber, E. Solouma, R. A. Alharb, A. Alalyani, Chaos in fractional-order glucose-insulin models with variable derivatives: Insights from the Laplace-Adomian decomposition method and generalized Euler techniques, Fractal Fract., 9 (2025), 149. https://doi.org/10.3390/fractalfract9030149 doi: 10.3390/fractalfract9030149
[22]	S. Saber, S. M. Mirgani, Numerical analysis and stability of a system (2) using the Laplace residual power series method incorporating the Atangana-Baleanu derivative, Int. J. Model. Simul. Sc., 16 (2025), 2550030. https://doi.org/10.1142/S1793962325500308 doi: 10.1142/S1793962325500308
[23]	S. Saber, S. M. Mirgani, Numerical solutions, stability, and chaos control of Atangana-Baleanu variable-order derivatives in glucose-insulin dynamics, J. Appl. Math. Comput. Mech., 24 (2025), 44–55. https://doi.org/10.17512/jamcm.2025.1.04 doi: 10.17512/jamcm.2025.1.04
[24]	S. Saber, S. Mirgani, Analyzing fractional glucose-insulin dynamics using Laplace residual power series methods via the Caputo operator: Stability and chaotic behavior, Beni-Suef Univ. J. Basic Appl. Sci., 14 (2025), 28. https://doi.org/10.1186/s43088-025-00608-y doi: 10.1186/s43088-025-00608-y
[25]	M. Alhazmi, A. F. Aljohani, N. E. Taha, S. Abdel-Khalek, M. Bayram, S. Saber, Application of a fractal fractional operator to nonlinear glucose-insulin systems: Adomian decomposition solutions, Comput. Biol. Med., 196 (2025), 110453. https://doi.org/10.1016/j.compbiomed.2025.110453 doi: 10.1016/j.compbiomed.2025.110453
[26]	M. Alhazmi, S. Saber, Application of a fractal fractional derivative with a power-law kernel to the glucose-insulin interaction system based on Newton's interpolation polynomials, Fractals, 2025. https://doi.org/10.1142/S0218348X25402017
[27]	S. Saber, B. Dridi, A. Alahmari, M. Messaoudi, Application of Jumarie-Stancu collocation series method and multi-step generalized differential transform method to fractional glucose-insulin, Int. J. Optimiz. Contro., 25 (2025), 464–482. https://doi.org/10.36922/IJOCTA025120054 doi: 10.36922/IJOCTA025120054
[28]	S. Saber, B. Dridi, A. Alahmari, M. Messaoudi, Hyers-Ulam stability and control of fractional glucose-insulin systems, Eur. J. Pure Appl. Math., 18 (2025), 6152. https://doi.org/10.29020/nybg.ejpam.v18i2.6152 doi: 10.29020/nybg.ejpam.v18i2.6152
[29]	S. Saber, A. Alahmari, Impact of fractal-fractional dynamics on pneumonia transmission modeling, Eur. J. Pure Appl. Math., 18 (2025), 5901. https://doi.org/10.29020/nybg.ejpam.v18i2.5901 doi: 10.29020/nybg.ejpam.v18i2.5901
[30]	M. Althubyani, S. Saber, Hyers-Ulam stability of fractal-fractional computer virus models with the Atangana-Baleanu operator, Fractal Fract., 9 (2025), 158. https://doi.org/10.3390/fractalfract9030158 doi: 10.3390/fractalfract9030158
[31]	M. Althubyani, H. D. S. Adam, A. Alalyani, N. E. Taha, K. O. Taha, R. A. Alharbi, et al., Understanding zoonotic disease spread with a fractional order epidemic model, Sci. Rep., 15 (2025), 13921. https://doi.org/10.1038/s41598-025-95943-6 doi: 10.1038/s41598-025-95943-6
[32]	H. D. S. Adam, M. Althubyani, S. M. Mirgani, S. Saber, An application of Newton's interpolation polynomials to the zoonotic disease transmission between humans and baboons system based on a time-fractal fractional derivative with a power-law kernel, AIP Adv., 15 (2025), 045217. https://doi.org/10.1063/5.0253869 doi: 10.1063/5.0253869
[33]	S. Saber, E. Solouma, The generalized Euler method for analyzing zoonotic disease dynamics in baboon-human populations, Symmetry, 17 (2025), 541. https://doi.org/10.3390/sym17040541 doi: 10.3390/sym17040541
[34]	S. Saber, E. Solouma, M. Althubyani, M. Messaoudi, Statistical insights into zoonotic disease dynamics: Simulation and control strategy evaluation, Symmetry, 17 (2025), 733. https://doi.org/10.3390/sym17050733 doi: 10.3390/sym17050733
[35]	S. Saber, A. Alahmari, Mathematical insights into zoonotic disease spread: Application of the Milstein method, Eur. J. Pure Appl. Math., 18 (2025), 58–81. https://doi.org/10.29020/nybg.ejpam.v18i2.5881 doi: 10.29020/nybg.ejpam.v18i2.5881
[36]	H. Khan, J. Alzabut, A. Shah, S. Etemad, S. Rezapour, C. Park, A study on the fractal-fractional tobacco smoking model, AIMS Math., 7 (2022), 13887–13909. https://doi.org/10.3934/math.2022767 doi: 10.3934/math.2022767
[37]	S. Saber, A. M. Alghamdi, G. A. Ahmed, K. M. Alshehri, Mathematical modelling and optimal control of pneumonia disease in sheep and goats in Al-Baha region with cost-effective strategies, AIMS Math., 7 (2022), 12011–12049. https://doi.org/10.3934/math.2022669 doi: 10.3934/math.2022669
[38]	M. Althubyani, N. E. Taha, K. O. Taha, R. A. Alharb, S. Saber, Epidemiological modeling of pneumococcal pneumonia: Insights from ABC fractal-fractional derivatives, CMES-Comp. Model. Eng., 143 (2025), 3491–3521. https://doi.org/10.32604/cmes.2025.061640 doi: 10.32604/cmes.2025.061640
[39]	F. Evirgen, E. Uçar, S. Uçar, N. Özdemir, Modelling influenza a disease dynamics under Caputo-Fabrizio fractional derivative with distinct contact rates, Math. Model. Numer. Simul. Appl., 3 (2023), 58–73. https://doi.org/10.53391/mmnsa.1274004 doi: 10.53391/mmnsa.1274004
[40]	N. Özdemir, E. Uçar, D. Avcı, Dynamic analysis of a fractional SVIR system modeling an infectious disease, Facta Univ. Ser. Math., 37 (2022), 605–619. https://doi.org/10.22190/FUMI211020042O doi: 10.22190/FUMI211020042O
[41]	X. P. Li, S. Ullah, H. Zahir, A. Alshehri, M. B. Riaz, B. A. Alwan, Modeling the dynamics of coronavirus with super-spreader class: A fractal-fractional approach, Results Phys., 34 (2022), 105179. https://doi.org/10.1016/j.rinp.2022.105179 doi: 10.1016/j.rinp.2022.105179
[42]	A. M. Alzubaidi, H. A. Othman, S. Ullah, N. Ahmad, M. M. Alam, Analysis of Monkeypox viral infection with human to animal transmission via a fractional and fractal-fractional operators with power law kernel, Math. Biosci. Eng., 20 (2023), 6666–6690. https://doi.org/10.3934/mbe.2023287 doi: 10.3934/mbe.2023287
[43]	A. Atangana, S. I. Araz, Mathematical model of COVID-19 spread in Turkey and South Africa: Theory, methods, and applications, Adv. Differ. Equ., 2020 (2020), 659. https://doi.org/10.1186/s13662-020-03095-w doi: 10.1186/s13662-020-03095-w
[44]	M. Farman, C. Alfiniyah, A. Shehzad, Modelling and analysis of tuberculosis (TB) model with hybrid fractional operator, Alex. Eng. J., 72 (2023), 463–478. https://doi.org/10.1016/j.aej.2023.04.017 doi: 10.1016/j.aej.2023.04.017
[45]	Z. M. Odibat, C. Bertelle, M. A. Aziz-Alaoui, G. H. E. Duchamp, A multi-step differential transform method and application to non-chaotic or chaotic systems, Comput. Math. Appl., 59 (2010), 1462–1472. https://doi.org/10.1016/j.camwa.2009.11.005 doi: 10.1016/j.camwa.2009.11.005
[46]	H. A. Alkresheh, A. I. Ismail, Multi-step fractional differential transform method for the solution of fractional order stiff systems, Ain Shams Eng. J., 12 (2021), 4223–4231. https://doi.org/10.1016/j.asej.2017.03.017 doi: 10.1016/j.asej.2017.03.017
[47]	K. Diethelm, N. J. Ford, A. D. Freed, A predictor-corrector approach for the numerical solution of fractional differential equations, Nonlinear Dyn., 29 (2002), 3–22. https://doi.org/10.1023/A:1016592219341 doi: 10.1023/A:1016592219341
[48]	L. Sadek, D. Baleanu, M. S. Abdo, W. Shatanawi, Introducing novel $\Theta$-fractional operators: Advances in fractional calculus, J. King Saud Univ. Sci., 36 (2024), 103352. https://doi.org/10.1016/j.jksus.2024.103352 doi: 10.1016/j.jksus.2024.103352
[49]	L. Sadek, A cotangent fractional derivative with the application, Fractal Fract., 7 (2023), 444. https://doi.org/10.3390/fractalfract7060444 doi: 10.3390/fractalfract7060444
[50]	L. Sadek, T. A. Lazǎr, On Hilfer cotangent fractional derivative and a particular class of fractional problems, AIMS Math., 8 (2023), 28334–28352. https://doi.org/10.3934/math.20231450 doi: 10.3934/math.20231450
[51]	L. Sadek, O. Sadek, H. T. Alaoui, M. S. Abdo, K. Shah, T. Abdeljawad, Fractional order modeling of predicting COVID-19 with isolation and vaccination strategies in Morocco, CMES-Comp. Model. Eng., 136 (2023), 1931–1950. https://doi.org/10.32604/cmes.2023.025033 doi: 10.32604/cmes.2023.025033
[52]	O. Sadek, L. Sadek, S. Touhtouh, A. Hajjaji, The mathematical fractional modeling of TiO₂ nanopowder synthesis by sol-gel method at low temperature, Math. Model. Comput., 9 (2022), 616–626. https://doi.org/10.23939/mmc2022.03.616 doi: 10.23939/mmc2022.03.616
[53]	A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and applications of fractional differential equations, Elsevier, 2006.
[54]	Z. Odibat, S. Momani, V. S. Erturk, Generalized differential transform method: Application to differential equations of fractional order, Appl. Math. Comput., 197 (2008), 467–477. https://doi.org/10.1016/j.amc.2007.07.068 doi: 10.1016/j.amc.2007.07.068
[55]	V. S. Ertürk, Z. M. Odibat, S. Momani, The multi-step differential transform method and its application to determine the solutions of non-linear oscillators, Adv. Appl. Math. Mech., 4 (2012), 422–438. https://doi.org/10.1017/S2070073300001727 doi: 10.1017/S2070073300001727
[56]	V. S. Erturk, S. Momani, Z. Odibat, Application of generalized differential transform method to multi-order fractional differential equations, Commun. Nonlinear Sci., 13 (2008), 1642–1654. https://doi.org/10.1016/j.cnsns.2007.02.006 doi: 10.1016/j.cnsns.2007.02.006
[57]	Z. Odibat, V. S. Erturk, P. Kumar, A. B. Makhlouf, V. Govindaraj, An implementation of the generalized differential transform scheme for simulating impulsive fractional differential equations, Math. Probl. Eng., 2022 (2022), 8280203. https://doi.org/10.1155/2022/8280203 doi: 10.1155/2022/8280203

Reader Comments

Your name:*

Email:*
© 2025 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)