Analysis and treatment impact to control the transmission dynamics of skin sores disease with novel hybrid fractional operators
DOI:
https://doi.org/10.48165/jmmfc.2025.2105Keywords:
Constant Proportional(CP) operator, Impetigo (skin sores), Strength number, Hilfer Gen eralized ProportionalAbstract
Skin sores (impetigo) are a frequent and contagious skin infection in young children, caus ing blisters and ulcers. It is usually not hazardous and resolves within a week of treatment or a few weeks if no treatment is used. Treatment is commonly recommended since it can shorten the length of the condition and reduce the risk of the virus spreading to others. We developed and tested a non linear hybrid fractional order model to investigate skin sores infection and the transmission dynamics. The study examines the mathematical properties of the suggested model, such as the feasible region, equilibrium points, basic reproduction number, and existence of the system’s unique solutions using Banach fixed-point theory. It also uses the appropriate lyapunov function to examine the stability of equilibrium states. This work gives a thorough examination of several hybrid fractional operators and numerically simulates the suggested skin sores system using the Laplace-Adomian decomposition approach, demonstrating its efficacy in simulating theoretical scenarios. This research advances our knowledge of the mechanisms underlying disease transmission by taking into account fractional-order dynamics and a variety of routes, offering suggestions for better disease management and control.
References
A. C. Bowen, S. Y. Tong, M. D. Chatfield, J. R. Carapetis. The microbiology of impetigo in indigenous children: associations between Streptococcus pyogenes, Staphylococcus aureus, scabies, and nasal carriage. BMC Infect. Dis. 14 (2014) 1–6.
T. Millard. Impetigo: recognition and management. Practice Nurs. 19(9) (2008) 432–434.
A. K. Fantaye, M. D. Goshu, B. B. Zeleke, A. A. Gessesse, M. F. Endalew, Z. K. Birhanu. Mathematical model and stability analysis on the transmission dynamics of skin sores. Epidemiol. Infect. 150 (2022) e207.
R. J. Tanaka, M. Ono. Skin disease modeling from a mathematical perspective. J. Invest. Dermatol. 133(6) (2013) 1472–1478.
Y. G. Anissimov. Mathematical models for skin toxicology. Expert Opin. Drug Metab. Toxicol. 10(4) (2014) 551–560.
S. Nakaoka, S. Kuwahara, C. H. Lee, H. Jeon, J. Lee, Y. Takeuchi, Y. Kim. Chronic inflammation in the epidermis: a mathematical model. Appl. Sci. 6(9) (2016) 252.
M. J. Lydeamore, P. T. Campbell, D. J. Price, Y. Wu, A. J. Marcato, W. Cuningham, J. R. Carapetis, R. M. Andrews, M. T. McDonald, J. McVernon, S. Y. Tong. Estimation of the force of infection and infectious period of skin sores in remote Australian communities using interval censored data. PLoS Comput. Biol. 16(10) (2020) e1007838.
M. Saidalieva, M. Hidirova, A. Shakarov. Mathematical model of skin epidermis regulatory mechanisms in squamous cell cancers development. In 2020 International Conference on Information Science and Communications Technologies (ICISCT), IEEE (2020) 1–4.
J. Q. Zhao, E. Bonyah, B. Yan, M. A. Khan, K. O. Okosun, M. Y. Alshahrani, T. Muhammad. A mathematical model for the coinfection of buruli ulcer and cholera. Results Phys. 29 (2021) 104746.
E. T. Greugny, G. N. Stamatas, F. Fages. Stability versus meta-stability in a skin microbiome model. In International Conference on Computational Methods in Systems Biology, Springer (2022) 179–197.
T. Parvin, M. H. Biswas, B. K. Datta. Mathematical analysis of the transmission dynamics of skin cancer caused by UV radiation. J. Appl. Math. 2022(1) (2022) 5445281.
C. Xu, M. Farman, A. Shehzad. Analysis and chaotic behavior of a fish farming model with singular and non-singular kernel. Int. J. Biomath. 18(3) (2025) 2350105.
M. Farman, A. Shehzad, K. S. Nisar, E. Hincal, A. Akgul, A. M. Hassan. Generalized Ulam-Hyers-Rassias stability and novel sustainable techniques for dynamical analysis of global warming impact on ecosystem. Sci. Rep. 13(1) (2023) 22441.
C. Xu, M. Liao, M. Farman, A. Shehzade. Hydrogenolysis of glycerol by heterogeneous catalysis: a fractional order kinetic model with analysis. MATCH Commun. Math. Comput. Chem. 91(3) (2024) 635–664.
M. Farman, C. Xu, A. Shehzad, A. Akgul. Modeling and dynamics of measles via fractional differential operator of singular and non-singular kernels. Math. Comput. Simul. 221 (2024) 461–488.
D. Baleanu, M. Hasanabadi, A. M. Vaziri, A. Jajarmi. A new intervention strategy for an HIV/AIDS transmission by a general fractional modeling and an optimal control approach. Chaos Solit. 167 (2023) 113078.
A. Ahmad, M. Farman, . A. Naik, K. Faiz, A. Ghaffar, E. Hincal, M. U. Saleem. Analytical analysis and bifurcation of pine wilt dynamical transmission with host vector and nonlinear incidence using sustainable fractional approach. Partial Differ. Equ. Appl. Math. 11 (2024) 100830.
P. A. Naik, M. Farman, K. Jamil, K. S. Nisar, M. A. Hashmi, Z. Huang. Modeling and analysis using piecewise hybrid fractional operator in time scale measure for ebola virus epidemics under Mittag-Leffler kernel. Sci. Rep. 14(1) (2024) 24963.
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(3) (2024) 102452.
S. Qureshi, A. Yusuf. Modeling chickenpox disease with fractional derivatives: From Caputo to Atangana-Baleanu. Chaos Solit. 122 (2019) 111–118.
T. B. Gashirai, S. D. Hove-Musekwa, S. Mushayabasa. Dynamical analysis of a fractional-order foot-and-mouth disease model. Math. Sci. 15(1) (2021) 65–82.
S. Okyere, J. Ackora-Prah. Modeling and analysis of monkeypox disease using fractional derivatives. Results Eng. 17 (2023) 100786.
K. S. Nisar, M. Farman. Analysis of a mathematical model with hybrid fractional derivatives under different kernel for hearing loss due to mumps virus. Int. J. Simul. Model. Simul. (2024) 1–27.
P. Bedi, A. Khan, A. Kumar, T. Abdeljawad. Computational study of fractional-order vector borne diseases model. Fractals 30(5) (2022) 2240149.
K. S. Nisar, A. Ahmad, M. Farman, E. Hincal, A. Zehra. Modeling and mathematical analysis of fractional order eye infection (conjunctivitis) virus model with treatment impact: Prelicence and dynamical transmission. Alex. Eng. J. 107 (2024) 33–46.
M. A. Ullah, N. Raza, T. Nazir. Mathematical simulations and sensitivity visualization of fractional order disease model describing human immunodeficiency. Alex. Eng. J. 87 (2024) 1–6.
S. W. Teklu, B. S. Kotola. Insight into the treatment strategy on pneumonia transmission with asymptotic carrier stage using fractional order modeling approach. Comput. Methods Programs Biomed. Update 5 (2024) 100134.
D. Baleanu, A. Fernandez, A. Akgul. On a fractional operator combining proportional and classical differintegrals. Mathematics 8(3) (2020) 360.
A. Akgul. Some fractional derivatives with different kernels. Int. J. Appl. Comput. Math. 8(4) (2022) 1–10.
M. Farman, A. Shehzad, A. Akgul, D. Baleanu, M. D. Sen. Modelling and analysis of a measles epidemic model with the constant proportional Caputo operator. Symmetry 15(2) (2023) 468.
P. A. Naik, A. Zehra, M. Farman, A. Shehzad, S. Shahzeen, Z. Huang. Forecasting and dynamical modeling of reversible enzymatic reactions with a hybrid proportional fractional derivative. Front. Phys. 11 (2024) 1307307.
A. U. Rehman, S. Hua, M. B. Riaz, J. Awrejcewicz, S. Xiange. A fractional study with Newtonian heating effect on heat absorbing MHD radiative flow of rate type fluid with application of novel hybrid fractional derivative operator. Arab J. Basic Appl. Sci. 30(1) (2023) 482–495.
M. Azeem, M. Farman, A. Shehzad, K. S. Nisar. Hybrid fractional derivative for modeling and analysis of cancer treatment with virotherapy. Stoch. Anal. Appl. 42(6) (2024) 1052–1084.
M. I. Abbas. Existence results and the Ulam stability for fractional differential equations with hybrid proportional-Caputo derivatives. J. Nonlinear Funct. Anal. 2020 (2020) 1–4.
P. Van den Driessche, J. Watmough. Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission. Math. Biosci. 180(1–2) (2002) 29–48.
C. Vargas-De-Leon. Volterra-type Lyapunov functions for fractional-order epidemic systems. Commun. Nonlinear Sci. Numer. 24(1–3) (2015) 75–85.
I. Ahmed, P. Kumam, F. Jarad, P. Borisut, W. Jirakitpuwapat. On Hilfer generalized proportional fractional derivative. Adv. Differ. Equ. 2020 (2020) 1–8.
T. R. Prabhakar. A singular integral equation with a generalized Mittag-Leffler function in the kernel. Yokohama Math. J. 19(1) (1971) 7–15.
