Fractal-fractional order dynamics and numerical simulations of a Zika epidemic model with insecticide-treated nets. (English) Zbl 07569870

Summary: Fractional order and fractal order are mathematical tools that can be used to model real-world problems. In order to demonstrate the usefulness of these operators, we develop a new fractal-fractional model for the propagation of the Zika virus. This model includes insecticide-treated nets and the generalized fractal-fractional Mittag-Leffler kernel. The existence, uniqueness, and Ulam-Hyres stability conditions for the given system are determined. Using the Newton polynomial, the numerical scheme is described. From the numerical simulations, we notice that a change in the fractal-fractional order directly affects the dynamics of the Zika virus. We also notice that the use of fractal order only converges to faster than the use of fractional order only. Testing the inherent potency of insecticide-treated nets when the fractal-fractional order is 0.99 indicates that increased use of insecticide-treated nets increases the number of healthy humans. The fractal-fractional analysis captures the geometric pattern of the Zika virus that is repeated at every scale, which cannot be captured by classical geometry. This backs up the idea that the best way to control the disease is to know enough about how it spread in the past.


82-XX Statistical mechanics, structure of matter
Full Text: DOI


[1] Nishiura, H.; Kinoshita, R.; Mizumoto, K.; Yasuda, Y.; Nah, K., Transmission potential of Zika virus infection in the South Pacific, Int. J. Infect. Dis., 45, 95-97 (2016)
[2] WHO, Zika Virus, https://www.who.int/news-room/fact-sheets/detail/zika-virus.
[3] CDC, Zika Virus, how the Zika virus is transmitted, https://www.cdc.gov/zika/about/overview.html.
[4] González-Parra, G.; Benincasa, T., Mathematical modeling and numerical simulations of Zika in Colombia considering mutation, Math. Comput. Simulation, 163, 1-18 (2019) · Zbl 07316703
[5] Asamoah, J. K.K.; Owusu, M. A.; Jin, Z.; Oduro, F. T.; Abidemi, A.; Gyasi, E. O., Global stability and cost-effectiveness analysis of COVID-19 considering the impact of the environment: using data from Ghana, Chaos Solitons Fractals, 140, Article 110103 pp. (2020) · Zbl 07508183
[6] Sene, N., Analysis of the stochastic model for predicting the novel coronavirus disease, Adv. Difference Equ., 2020, 1, 1-19 (2020)
[7] Asamoah, J. K.K.; Oduro, F. T.; Bonyah, E.; Seidu, B., Modelling of rabies transmission dynamics using optimal control analysis, J. Appl. Math., 2017 (2017) · Zbl 1437.92109
[8] Asamoah, J. K.K.; Jin, Z.; Sun, G. Q., Non-seasonal and seasonal relapse model for q fever disease with comprehensive cost-effectiveness analysis, Results Phys., 22, Article 103889 pp. (2021)
[9] Omame, A.; Sene, N.; Nometa, I.; Nwakanma, C. I.; Nwafor, E. U.; Iheonu, N. O.; Okuonghae, D., Analysis of COVID-19 and comorbidity co-infection model with optimal control, Optim. Control Appl. Methods, 42, 6, 1568-1590 (2021) · Zbl 1486.92266
[10] Abidemi, A.; Fatoyinbo, H. O.; Asamoah, J. K.K., Analysis of dengue fever transmission dynamics with multiple controls: a mathematical approach, (2020 International Conference on Decision Aid Sciences and Application (DASA) (2020), IEEE), 971-978
[11] Omame, A.; Rwezaura, H.; Diagne, M. L.; Inyama, S. C.; Tchuenche, J. M., COVID-19 And dengue co-infection in Brazil: optimal control and cost-effectiveness analysis, Eur. Phys. J. Plus, 136, 10, 1-33 (2021)
[12] Acheampong, E.; Okyere, E.; Iddi, S.; Bonney, J. H.; Asamoah, J. K.K.; Wattis, J. A.; Gomes, R. L., Mathematical modelling of earlier stages of COVID-19 transmission dynamics in Ghana, Results Phys., 34, Article 105193 pp. (2022)
[13] Asamoah, J. K.K.; Yankson, E.; Okyere, E.; Sun, G. Q.; Jin, Z.; Jan, R., Optimal control and cost-effectiveness analysis for dengue fever model with asymptomatic and partial immune individuals, Results Phys., 31, Article 104919 pp. (2021)
[14] Asamoah, J. K.K.; Jin, Z.; Sun, G. Q.; Seidu, B.; Yankson, E.; Abidemi, A.; Oduro, F. T.; Moore, S. E.; Okyere, E., Sensitivity assessment and optimal economic evaluation of a new COVID-19 compartmental epidemic model with control interventions, Chaos Solitons Fractals, 146, Article 110885 pp. (2021) · Zbl 07526777
[15] Seidu, B.; Makinde, O. D., Optimal control of HIV/AIDS in the workplace in the presence of careless individuals, Comput. Math. Methods Med., 2014 (2014) · Zbl 1307.92354
[16] Asamoah, J. K.K.; Nyabadza, F.; Jin, Z.; Bonyah, E.; Khan, M. A.; Li, M. Y.; Hayat, T., Backward bifurcation and sensitivity analysis for bacterial meningitis transmission dynamics with a nonlinear recovery rate, Chaos Solitons Fractals, 140, Article 110237 pp. (2020) · Zbl 1495.92069
[17] Seidu, B., Optimal strategies for control of COVID-19: A mathematical perspective, Scientifica, 2020 (2020)
[18] Asamoah, J. K.K.; Jin, Z.; Seidu, B.; Sun, G. Q.; Oduro, F. T.; Alzahrani, F., A mathematical model and sensitivity assessment of COVID-19 Outbreak for Ghana and Egypt (2020), Available at SSRN 3612877
[19] Abidemi, A.; Abd Aziz, M. I.; Ahmad, R., Vaccination and vector control effect on dengue virus transmission dynamics: Modelling and simulation, Chaos Solitons Fractals, 133, Article 109648 pp. (2020) · Zbl 1483.92118
[20] Irish, S. R.; Al-Amin, H. M.; Paulin, H. N.; Mahmood, A. S.; Khan, R. K.; Muraduzzaman, A. K.M.; Worrell, C. M.; Flora, M. S.; Karim, M. J.; Shirin, T.; Shamsuzzaman, A. K.M., Molecular xenomonitoring for Wuchereria bancrofti in culex quinquefasciatus in two districts in Bangladesh supports transmission assessment survey findings, PLoS Negl. Trop. Dis., 12, 7, Article e0006574 pp. (2018)
[21] Dinh, L.; Chowell, G.; Mizumoto, K.; Nishiura, H., Estimating the subcritical transmissibility of the Zika outbreak in the State of Florida, USA, Theor. Biol. Med. Model., 13, 1, 1-7 (2016)
[22] Ali, A.; Iqbal, Q.; Asamoah, J. K.K.; Islam, S., Mathematical modeling for the transmission potential of Zika virus with optimal control strategies, Eur. Phys. J. Plus, 137, 1, 1-30 (2022)
[23] Alzahrani, E. O.; Ahmad, W.; Khan, M. A.; Malebary, S. J., Optimal control strategies of Zika virus model with mutant, Commun. Nonlinear Sci. Numer. Simul., 93, Article 105532 pp. (2021) · Zbl 1453.92285
[24] Agusto, F. B.; Bewick, S.; Fagan, W. F., Mathematical model for Zika virus dynamics with sexual transmission route, Ecol. Complex., 29, 61-81 (2017)
[25] Mpeshe, S. C.; Nyerere, N.; Sanga, S., Modeling approach to investigate the dynamics of Zika virus fever: A neglected disease in Africa, Int. J. Adv. Appl. Math. Mech., 4, 3, 14-21 (2017) · Zbl 1382.92244
[26] Bonyah, E.; Khan, M. A.; Okosun, K. O.; Islam, S., A theoretical model for Zika virus transmission, PLoS One, 12, 10, Article e0185540 pp. (2017)
[27] Tesla, B.; Demakovsky, L. R.; Mordecai, E. A.; Ryan, S. J.; Bonds, M. H.; Ngonghala, C. N.; Brindley, M. A.; Murdock, C. C., Temperature drives Zika virus transmission: evidence from empirical and mathematical models, Proc. R. Soc. B, 285, 1884, Article 20180795 pp. (2018)
[28] Suparit Wiratsudakul, A.; Modchang, C., A mathematical model for Zika virus transmission dynamics with a time-dependent mosquito biting rate, Theor. Biol. Med. Model., 15, 1, 1-11 (2018)
[29] Cai, Y.; Wang, K.; Wang, W., Global transmission dynamics of a Zika virus model, Appl. Math. Lett., 92, 190-195 (2019) · Zbl 1414.35232
[30] Okyere, E.; Olaniyi, S.; Bonyah, E., Analysis of Zika virus dynamics with sexual transmission route using multiple optimal controls, Sci. Afr., 9, Article e00532 pp. (2020)
[31] Chand, M.; Hammouch, Z.; Asamoah, J. K.K.; Baleanu, D., Certain fractional integrals and solutions of fractional kinetic equations involving the product of S-function, (Mathematical Methods in Engineering (2019), Springer: Springer Cham), 213-244
[32] Alhejaili, W.; Alhazmi, S. E.; Nawaz, R.; Ali, A.; Asamoah, J. K.K.; Zada, L., Numerical investigation of Fractional-Order Kawahara and modified Kawahara equations by a semianalytical method, J. Nanomater., 2022 (2022)
[33] Ali, N.; Nawaz, R.; Zada, L.; Mouldi, A.; Bouzgarrou, S. M.; Sene, N., Analytical approximate solution of the fractional order biological population model by using natural transform, J. Nanomater., 2022 (2022)
[34] Baleanu, D.; Jajarmi, A.; Mohammadi, H.; Rezapour, S., A new study on the mathematical modelling of human liver with Caputo-Fabrizio fractional derivative, Chaos Solitons Fractals, 134, Article 109705 pp. (2020) · Zbl 1483.92041
[35] Khan, S. A.; Shah, K.; Zaman, G.; Jarad, F., Existence theory and numerical solutions to smoking model under Caputo-Fabrizio fractional derivative, Chaos, 29, 1, Article 013128 pp. (2019) · Zbl 1406.92309
[36] Asamoah, J. K.K.; Okyere, E.; Yankson, E.; Opoku, A. A.; Adom-Konadu, A.; Acheampong, E.; Arthur, Y. D., Non-fractional and fractional mathematical analysis and simulations for Q fever, Chaos Solitons Fractals, 156, Article 111821 pp. (2022) · Zbl 1506.92083
[37] Owolabi, K. M.; Pindza, E., Modeling and simulation of nonlinear dynamical system in the frame of nonlocal and non-singular derivatives, Chaos Solitons Fractals, 127, 146-157 (2019) · Zbl 1448.35303
[38] Owolabi, K. M.; Gómez-Aguilar, J. F.; Karaagac, B., Modelling, analysis and simulations of some chaotic systems using derivative with Mittag-Leffler kernel, Chaos Solitons Fractals, 125, 54-63 (2019) · Zbl 1448.34023
[39] Aslam, M.; Murtaza, R.; Abdeljawad, T.; Khan, A.; Khan, H.; Gulzar, H., A fractional order HIV/AIDS epidemic model with Mittag-Leffler kernel, Adv. Difference Equ., 2021, 1, 1-15 (2021) · Zbl 1494.92123
[40] Sher, M.; Shah, K.; Khan, Z. A.; Khan, H.; Khan, A., Computational and theoretical modeling of the transmission dynamics of novel COVID-19 under Mittag-Leffler power law, Alexandria Eng. J., 59, 5, 3133-3147 (2020)
[41] Morales-Delgadoa, V. F.; Gomez-Aguilar, J. F.; Taneco-Hernandez, M. A.; Escobar-Jimenezc, R. F.; Olivares-Peregrino, V. H., Mathematical modeling of the smoking dynamics using fractional differential equations with local and nonlocal kernel, J. Nonlinear Sci. Appl., 11, 8, 994-1014 (2018) · Zbl 1438.92034
[42] Atangana, A.; Baleanu, D., New fractional derivatives with nonlocal and non-singular kernel: theory and application to heat transfer model (2016), arXiv preprint arXiv:1602.03408
[43] Atangana, A.; Gómez-Aguilar, J. F., A new derivative with normal distribution kernel: Theory, methods and applications, Physica A, 476, 1-14 (2017) · Zbl 1495.35182
[44] Sene, N., SIR Epidemic model with Mittag-Leffler fractional derivative, Chaos Solitons Fractals, 137, Article 109833 pp. (2020) · Zbl 1489.92176
[45] Karaagac, B.; Owolabi, K. M.; Nisar, K. S., Analysis and dynamics of illicit drug use described by fractional derivative with Mittag-Leffler kernel, CMC-Comput. Mater. Cont., 65, 3, 1905-1924 (2020)
[46] Owolabi, K. M., Analysis and simulation of herd behaviour dynamics based on derivative with nonlocal and nonsingular kernel, Results Phys., 22, Article 103941 pp. (2021)
[47] Owolabi, K. M.; Atangana, A., Computational study of multi-species fractional reaction-diffusion system with ABC operator, Chaos Solitons Fractals, 128, 280-289 (2019) · Zbl 1483.35324
[48] Ávalos-Ruiz, L. F.; Gomez-Aguilar, J. F.; Atangana, A.; Owolabi, K. M., On the dynamics of fractional maps with power-law, exponential decay and Mittag-Leffler memory, Chaos Solitons Fractals, 127, 364-388 (2019) · Zbl 1448.34086
[49] Khan, M. A.; Atangana, A., Modeling the dynamics of novel coronavirus (2019-nCov) with fractional derivative, Alexandria Eng. J., 59, 4, 2379-2389 (2020)
[50] Omame, A.; Nwajeri, U. K.; Abbas, M.; Onyenegecha, C. P., A fractional order control model for diabetes and COVID-19 co-dynamics with Mittag-Leffler function, Alexandria Eng. J. (2022)
[51] Khan, M. A.; Ullah, S.; Farhan, M., The dynamics of Zika virus with Caputo fractional derivative, AIMS Math., 4, 1, 134-146 (2019) · Zbl 1425.37054
[52] Rakkiyappan, R.; Latha, V. P.; Rihan, F. A., A fractional-order model for Zika virus infection with multiple delays, Complexity, 2019 (2019) · Zbl 1432.92100
[53] Farman, M.; Ahmad, A.; Akgül, A.; Saleem, M. U.; Rizwan, M.; Ahmad, M. O., A mathematical analysis and simulation for Zika virus model with time fractional derivative, Math. Methods Appl. Sci. (2020)
[54] Khan, F. M.; Ali, A.; Khan, Z. U.; Alharthi, M. R.; Abdel-Aty, A. H., Qualitative and quantitative study of Zika virus epidemic model under Caputo’s fractional differential operator, Phys. Scr., 96, 12, Article 124030 pp. (2021)
[55] Farman, M.; Akgül, A.; Askar, S.; Botmart, T.; Ahmad, A.; Ahmad, H., Modeling and analysis of fractional order Zika model, Virus, 3, 4 (2021)
[56] Akinyemi, L. Veeresha.; Oluwasegun, K.; Senol, M.; Oduro, B., Numerical surfaces of fractional Zika virus model with diffusion effect of mosquito-borne and sexually transmitted disease, Math. Methods Appl. Sci., 45, 5, 2994-3013 (2022)
[57] Ali, A.; Islam, S.; Khan, M. R.; Rasheed, S.; Allehiany, F. M.; Baili, J.; Khan, M. A.; Ahmad, H., Dynamics of a fractional order Zika virus model with mutant, Aej, 2021, 031 (2021)
[58] Thaiprayoon, C.; Kongson, J.; Sudsutad, W.; Alzabut, J.; Etemad, S.; Rezapour, S., Analysis of a nonlinear fractional system for zika virus dynamics with sexual transmission route under generalized Caputo-type derivative, J. Appl. Math. Comput., 1-31 (2022)
[59] Begum, R.; Tunç, O.; Khan, H.; Gulzar, H.; Khan, A., A fractional order Zika virus model with Mittag-Leffler kernel, Chaos Solitons Fractals, 146, Article 110898 pp. (2021) · Zbl 07526790
[60] Atangana, A., Fractal-fractional differentiation and integration: connecting fractal calculus and fractional calculus to predict complex system, Chaos Solitons Fractals, 102, 396-406 (2017) · Zbl 1374.28002
[61] Owolabi, K. M.; Shikongo, A.; Atangana, A., Fractal fractional derivative operator method on MCF-7 cell line dynamics, (Methods of Mathematical Modelling and Computation for Complex Systems (2022), Springer: Springer Cham), 319-339 · Zbl 1471.92102
[62] Atangana, A.; Akgül, A.; Owolabi, K. M., Analysis of fractal fractional differential equations, Alexandria Eng. J., 59, 3, 1117-1134 (2020)
[63] Owolabi, K. M.; Shikongo, A., Fractal fractional operator method on HER2+ breast cancer dynamics, Int. J. Appl. Comput. Math., 7, 3, 1-19 (2021) · Zbl 1499.92019
[64] Owolabi, K. M.; Atangana, A.; Akgul, A., Modelling and analysis of fractal-fractional partial differential equations: application to reaction-diffusion model, Alexandria Eng. J., 59, 4, 2477-2490 (2020)
[65] Owolabi, K. M.; Pindza, E., Numerical simulation of chaotic maps with the new generalized Caputo-type fractional-order operator, Results Phys., Article 105563 pp. (2022)
[66] Asamoah, J. K.K., Fractal-fractional model and numerical scheme based on Newton polynomial for Q fever disease under Atangana-Baleanu derivative, Results Phys., Article 105189 pp. (2022)
[67] Gómez-Aguilar, J. F.; Atangana, A., New chaotic attractors: Application of fractal-fractional differentiation and integration, Math. Methods Appl. Sci., 44, 4, 3036-3065 (2021) · Zbl 1481.34055
[68] Gomez-Aguilar, J. F.; Cordova-Fraga, T.; Abdeljawad, T.; Khan, A.; Khan, H., Analysis of fractal-fractional malaria transmission model, Fractals, 28, 08, Article 2040041 pp. (2020) · Zbl 1482.92097
[69] Zúñiga Aguilar, C. J.; Gómez-Aguilar, J. F.; Romero-Ugalde, H. M.; Jahanshahi, H.; Alsaadi, F. E., Fractal-fractional neuro-adaptive method for system identification, Eng. Comput., 1-24 (2021)
[70] Abro, K. A.; Atangana, A.; Gómez-Aguilar, J. F., Ferromagnetic chaos in thermal convection of fluid through fractal-fractional differentiations, J. Therm. Anal. Calorim., 1-13 (2022)
[71] Najafi, H.; Etemad, S.; Patanarapeelert, N.; Asamoah, J. K.K.; Rezapour, S.; Sitthiwirattham, T., A study on dynamics of CD4+ T-cells under the effect of HIV-1 infection based on a mathematical fractal-fractional model via the Adams-Bashforth Scheme and Newton polynomials, Mathematics, 10, 9, 1366 (2022)
[72] Atangana, A., Modelling the spread of COVID-19 with new fractal-fractional operators: can the lockdown save mankind before vaccination?, Chaos Solitons Fractals, 136, Article 109860 pp. (2020)
[73] Saad, K. M.; Alqhtani, M.; Gómez-Aguilar, J. F., Fractal-fractional study of the hepatitis C virus infection model, Results Phys., 19, Article 103555 pp. (2020)
[74] Zhang, L.; ur Rahman, M.; Haidong, Q.; Arfan, M., Fractal-fractional anthroponotic cutaneous leishmania model study in sense of Caputo derivative, Alexandria Eng. J., 61, 6, 4423-4433 (2022)
[75] Zhou, B.; Zhang, L.; Addai, E.; Zhang, N., Multiple positive solutions for nonlinear high-order Riemann-Liouville fractional differential equations boundary value problems with p-Laplacian operator, Bound. Value Probl., 2020, 1, 1-17 (2020) · Zbl 1495.34048
[76] Zhou, B.; Zhang, L.; Zhang, N.; Addai, E., Existence and monotone iteration of unique solution for tempered fractional differential equations Riemann-Stieltjes integral boundary value problems, Adv. Difference Equ., 2020, 1, 1-19 (2020) · Zbl 1482.34042
[77] Musso, D.; Nhan, T.; Robin, E.; Roche, C.; Bierlaire, D.; Zisou, K.; Shan Yan, A.; Cao-Lormeau, V. M.; Broult, J., Potential for Zika virus transmission through blood transfusion demonstrated during an outbreak in French Polynesia, 2013 to 2014, Euro Surveill., 19, 14 (2014)
[78] M.S. Mojumder, E. Cohn, D. Fish, J.S. Brownstein, Estimating a Feasible Serial Interval Range for Zika Fever, Bull World Health Organization.
[79] Okosun, O. K.; Makinde, O. D., A co-infectionmodel of malaria and cholera diseases with optimal control, Math. Biosci., 258, 19-32 (2014), pmid:25245609 · Zbl 1314.92168
[80] Ahmed, E.; El-Sayed, A. M.A.; El-Saka, H. A., Equilibrium points, stability and numerical solutions of fractional-order predator-prey and rabies models, J. Math. Anal. Appl., 325, 1, 542-553 (2007) · Zbl 1105.65122
[81] Granas, A.; Dugundji, J., Fixed Point Theory (2003), Springer-Verlag: Springer-Verlag New York · Zbl 1025.47002
[82] Hyers, D. H., On the stability of the linear functional equation, Proc. Natl. Acad. Sci. USA, 27, 4, 222 (1941) · JFM 67.0424.01
[83] Rassias, T. M., On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc., 72, 2, 297-300 (1978) · Zbl 0398.47040
[84] Atangana, A.; Araz, S. I., New Numerical Scheme with Newton Polynomial: Theory, Methods, and Applications (2021), Academic Press · Zbl 1462.65001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.