Analysis of a model to control the co-dynamics of Chlamydia and Gonorrhea using Caputo fractional derivative
Year 2023,
Volume: 3 Issue: 2, 111 - 140, 30.06.2023
Udoka Benedict ODİONYENMA
Nometa IKENNA
Bolarinwa BOLAJİ
Abstract
This paper investigates a fractional derivative model of Chlamydia-Gonorrhea co-infection using Caputo derivative definition. The positivity boundedness of the model is established using Laplace transform. Additionally, we investigated the existence and uniqueness of the model using methods established by some fixed point theorems. We concluded that the model is Ulam-Hyers-Rassias stable. Furthermore, we obtained plots of the model at different fractional derivative orders, which show the significant role played by the fractional order on various classes of the model as it varies. We observe distinct results for each class in different orders, highlighting the importance of considering the fractional order in modeling Chlamydia-Gonorrhea co-infection. Moreover, the fractional model presented in this paper can be used to study the dynamics of Chlamydia-Gonorrhea co-infection in a more accurate and realistic way compared to traditional integer-order models.
References
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- Omame, A., Nnanna, C.U. and Inyama, S.C. Optimal control and cost-effectiveness analysis of an HPV–Chlamydia trachomatis co-infection model. Acta Biotheoretica, 69, 185-223, (2021).
- Omame, A., Okuonghae, D., Nwafor, U.E. and Odionyenma, B.U. A co-infection model for HPV and syphilis with optimal control and cost-effectiveness analysis. International Journal of Biomathematics, 14(07), 2150050, (2021).
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- Omame, A., Abbas, M. and Onyenegecha, C.P. A fractional order model for the co-interaction of COVID-19 and Hepatitis B virus. Results in Physics, 37, 105498, (2022).
- Ullah, S., Khan, M.A., Farooq, M., Hammouch, Z. and Baleanu, D. A fractional model for the dynamics of tuberculosis infection using Caputo-Fabrizio derivative. Discrete and Continuous Dynamical Systems Series S, 13(3), 975-993, (2020).
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- Yildiz, T.A., Jajarmi, A., Yildiz, B. and Baleanu, D. New aspects of time fractional optimal control problems within operators with nonsingular kernel. Discrete and Continuous Dynamical Systems Series S, 13(3), 407-428, (2020).
- Shah, K., Jarad, F. and Abdeljawad, T. On a nonlinear fractional order model of dengue fever disease under Caputo-Fabrizio derivative. Alexandria Engineering Journal, 59(4), 2305-2313, (2020).
- Dokuyucu, M.A., Celik, E., Bulut, H. and Baskonus, H.M. Cancer treatment model with the Caputo-Fabrizio fractional derivative. The European Physical Journal Plus, 133, 92, (2018).
- Atede, A.O., Omame, A. and Inyama, S.C. A fractional order vaccination model for COVID- 19 incorporating environmental transmission: a case study using Nigerian data. Bulletin of Biomathematics, 1(1), 78-110, (2023).
- Omame, A., Nwajeri, U.K., Abbas, M. and Onyenegecha, C.P. AA fractional order control model for Diabetes and COVID-19 co-dynamics with Mittag-Leffler function. Alexandria Engineering Journal, 61(10), 7619-7635, (2022).
- Omame, A., Isah, M.E., Abbas, M., Abdel-Aty, A. and Onyenegecha, C.P. A fractional order model for Dual Variants of COVID-19 and HIV co-infection via Atangana-Baleanu derivative. Alexandria Engineering Journal, 61(12), 9715-9731, (2022).
- Evirgen, F., Uçar, E., Uçar, S. and Özdemir, N. Modelling influenza a disease dynamics under Caputo-Fabrizio fractional derivative with distinct contact rates. Mathematical Modelling and Numerical Simulation with Applications, 3(1), 58-72, (2023).
- Ogunrinde, R.B., Nwajeri, U.K., Fadugba, S.E., Ogunrinde, R.R. and Oshinubi, K.I. Dynamic model of COVID-19 and citizens reaction using fractional derivative. Alexandria Engineering Journal, 60(2), 2001-2012, (2021).
- Khan, M.A, Hammouch, Z. and Baleanu, D. Modeling the dynamics of hepatitis E via the Caputo-Fabrizio derivative. Mathematical Modelling of Natural Phenomena, 14(3), 311, (2019).
- Ucar, E., Ozdemir, N. and Altun, E. Fractional order model of immune cells influenced by cancer cells. Mathematical Modelling of Natural Phenomena, 14(3), 308, (2019).
- Sene, N. SIR epidemic model with Mittag–Leffler fractional derivative. Chaos, Solitons & Fractals, 137, 109833, (2020).
- Omame, A., Abbas, M. and Abdel-Aty, A.H. Assessing the impact of SARS-CoV-2 infection on the dynamics of dengue and HIV via fractional derivatives. Chaos, Solitons & Fractals, 162, 112427, (2022).
- Nwajeri, U.K., Omame, A. and Onyenegecha, C.P. Analysis of a fractional order model for HPV and CT co-infection. Results in Physics, 28, 104643, (2021).
- Nwajeri, U.K., Panle, A.B., Omame, A., Obi, M.C. and Onyenegecha, C.P. On the fractional order model for HPV and Syphilis using non–singular kernel. Results in Physics, 37, 10546, (2022).
- Omame, A., Abbas, M. and Onyenegecha, C.P. A fractional-order model for COVID-19 and tuberculosis co-infection using Atangana–Baleanu derivative. Chaos, Solitons & Fractals, 153, 111486, (2021).
- Sene, N. Theory and applications of new fractional-order chaotic system under Caputo operator. An International Journal of Optimization and Control: Theories & Applications (IJOCTA), 12(1), 20–38, (2022).
- Veeresha, P., Yavuz, M. and Baishya, C. A computational approach for shallow water forced Korteweg–De Vries equation on critical flow over a hole with three fractional operators. An International Journal of Optimization and Control: Theories & Applications (IJOCTA), 11(3), 52–67, (2021).
- Uçar, E., Uçar, S., Evirgen, F. and Özdemir, N. A fractional SAIDR model in the frame of Atangana–Baleanu derivative. Fractal and Fractional, 5(2), 32, (2021).
- Diouf, M. and Sene, N. Analysis of the financial chaotic model with the fractional derivative operator. Complexity, 2020, 9845031, (2020).
- Carpinteri, A. and Mainardi, F. Fractals and Fractional Calulus in Continum Mechanics, (Vol. 378). Springer-Verlag Wien GmbH: New York, (1997).
- Liu, K., Feckan, M. and Wang, J. Hyers–Ulam stability and existence of solutions to the generalized Liouville–Caputo fractional differential equations. Symmetry, 12(6), 955, (2020).
- Van den Driessche, P. and Watmough, J. Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission. Mathematical Biosciences, 180(1- 2), 29-48, (2002).
- Castillo-Chavez, C., Feng, Z. and Huang, W. On the computation of R0 and its role on global stability. In Mathematical Approaches for Emerging and Reemerging Infectious Diseases: An Introduction, (vol. 125), pp. 229–250. (Minneapolis, MN, 1999), Springer: New York, (2002).
- Diethelm, K. and Freed, A.D. The FracPECE subroutine for the numerical solution of differential equations of fractional order. Forschung und Wissenschaftliches Rechnen, 1999, 57-71, (1998).
Year 2023,
Volume: 3 Issue: 2, 111 - 140, 30.06.2023
Udoka Benedict ODİONYENMA
Nometa IKENNA
Bolarinwa BOLAJİ
References
- Centers for Disease Control and Prevention (CDC). (2021). Chlamydia - CDC Fact Sheet (Detailed Version). Retrieved from https://www.cdc.gov/std/chlamydia/stdfact-chlamydiadetailed.htm (Accessed Date: 20.04.2023).
- Centers for Disease Control and Prevention (CDC). (2021). Gonorrhea - CDC Fact Sheet (Detailed Version). Retrieved from https://www.cdc.gov/std/gonorrhea/stdfact-gonorrheadetailed.htm (Accessed Date: 20.04.2023).
- Centers for Disease Control and Prevention (CDC). (2016). STDs and HIV – CDC Fact Sheet. Retrieved from https://www.cdc.gov/std/hiv/stdfact-std-hiv-detailed.htm (Accessed Date: 20.04.2023).
- Odionyenma, U.B., Omame, A., Ukanwoke, N.O. and Nometa, I. Optimal control of Chlamydia model with vaccination. International Journal of Dynamics and Control, 10, 332-348, (2022).
- Omame, A., Okuonghae, D., Umana, R.A. and Inyama, S.C. Analysis of a co-infection model for HPV-TB. Applied Mathematical Modelling, 77, 881-901, (2020).
- Omame, A., Okuonghae, D. and Inyama, S.C. A mathematical study of a model for HPVwith two high-risk strains. In Mathematical Modelling in Health, Social and Applied Sciences (pp. 107-149). Springer, Singapore, Forum for Interdisciplinary Mathematics, (2020).
- Samanta, G.P. Mathematical analysis of a Chlamydia epidemic model with pulse vaccination strategy. Acta Biotheoretica, 63, 1-21, (2015).
- Sharomi, O. and Gumel, A.B. Mathematical study of a risk-structured two-group model for Chlamydia transmission dynamics. Applied Mathematical Modelling, 35(8), 3653-3673, (2011).
- Sharomi, O. and Gumel, A.B. Re-infection-induced backward bifurcation in the transmission dynamics of Chlamydia trachomatis. Journal of Mathematical Analysis and Applications, 356(1), 96-118, (2009).
- Hussen, S., Wachamo, D., Yohannes, Z. and Tadesse, E. Prevalence of Chlamydia trachomatis infection among reproductive age women in sub Saharan Africa: a systematic review and meta-analysis. BMC Infectious Diseases, 18, 596, (2018).
- Zhu, H., Shen, Z., Luo ,H., Zhang, W. and Zhu, X. Chlamydia trachomatis infection-associated risk of cervical cancer: a meta-analysis. Medicine, 95(13), 1-10, (2016).
- Paba, P., Bonifacio, D., Bonito, L.D., Ombres, D., Favalli, C., Syrjänen, K. and Ciotti, M. Co-expression of HSV2 and Chlamydia trachomatis in HPV-positive cervical cancer and cervical intraepithelial neoplasia lesions is associated with aberrations in key intracellular pathways. Intervirology, 51(4), 230-234, (2008).
- Omede, B.I., Odionyenma, U.B., Ibrahim, A.A. and Bolaji B. Third wave of COVID-19: mathematical model with optimal control strategy for reducing the disease burden in Nigeria. International Journal of Dynamics and Control, 11, 411–427, (2023).
- Gopalkrishna, V., Aggarwal, N., Malhotra, V.L., Koranne, R.V., Mohan, V.P., Mittal, A. and Das, B.C. Chlamydia trachomatis and human papillomavirus infection in Indian women with sexually transmitted diseases and cervical precancerous and cancerous lesions. Clinical Microbiology and Infection, 6(2), 88-93, (2000).
- Omame, A., Nnanna, C.U. and Inyama, S.C. Optimal control and cost-effectiveness analysis of an HPV–Chlamydia trachomatis co-infection model. Acta Biotheoretica, 69, 185-223, (2021).
- Omame, A., Okuonghae, D., Nwafor, U.E. and Odionyenma, B.U. A co-infection model for HPV and syphilis with optimal control and cost-effectiveness analysis. International Journal of Biomathematics, 14(07), 2150050, (2021).
- Chukukere, E.C, Omame, A., Onyenegecha, C.P. and Inyama, S.C. Mathematical analysis of a model for Chlamydia and Gonorrhea co-dynamics with optimal control. Results in Physics, 27, 104566, (2021).
- Omame, A., Abbas, M. and Onyenegecha, C.P. A fractional order model for the co-interaction of COVID-19 and Hepatitis B virus. Results in Physics, 37, 105498, (2022).
- Ullah, S., Khan, M.A., Farooq, M., Hammouch, Z. and Baleanu, D. A fractional model for the dynamics of tuberculosis infection using Caputo-Fabrizio derivative. Discrete and Continuous Dynamical Systems Series S, 13(3), 975-993, (2020).
- Thabet, S.T.M., Abdo, M.S. and Shah, K. Theoretical and numerical analysis for transmission dynamics of COVID-19 mathematical model involving Caputo–Fabrizio derivative. Advances in Difference Equations, 2021, 184, (2021).
- Joshi, H., Yavuz, M. and Stamova, I. Analysis of the disturbance effect in intracellular calcium dynamic on fibroblast cells with an exponential kernel law. Bulletin of Biomathematics, 1(1), 24-39, (2023).
- Yildiz, T.A., Jajarmi, A., Yildiz, B. and Baleanu, D. New aspects of time fractional optimal control problems within operators with nonsingular kernel. Discrete and Continuous Dynamical Systems Series S, 13(3), 407-428, (2020).
- Shah, K., Jarad, F. and Abdeljawad, T. On a nonlinear fractional order model of dengue fever disease under Caputo-Fabrizio derivative. Alexandria Engineering Journal, 59(4), 2305-2313, (2020).
- Dokuyucu, M.A., Celik, E., Bulut, H. and Baskonus, H.M. Cancer treatment model with the Caputo-Fabrizio fractional derivative. The European Physical Journal Plus, 133, 92, (2018).
- Atede, A.O., Omame, A. and Inyama, S.C. A fractional order vaccination model for COVID- 19 incorporating environmental transmission: a case study using Nigerian data. Bulletin of Biomathematics, 1(1), 78-110, (2023).
- Omame, A., Nwajeri, U.K., Abbas, M. and Onyenegecha, C.P. AA fractional order control model for Diabetes and COVID-19 co-dynamics with Mittag-Leffler function. Alexandria Engineering Journal, 61(10), 7619-7635, (2022).
- Omame, A., Isah, M.E., Abbas, M., Abdel-Aty, A. and Onyenegecha, C.P. A fractional order model for Dual Variants of COVID-19 and HIV co-infection via Atangana-Baleanu derivative. Alexandria Engineering Journal, 61(12), 9715-9731, (2022).
- Evirgen, F., Uçar, E., Uçar, S. and Özdemir, N. Modelling influenza a disease dynamics under Caputo-Fabrizio fractional derivative with distinct contact rates. Mathematical Modelling and Numerical Simulation with Applications, 3(1), 58-72, (2023).
- Ogunrinde, R.B., Nwajeri, U.K., Fadugba, S.E., Ogunrinde, R.R. and Oshinubi, K.I. Dynamic model of COVID-19 and citizens reaction using fractional derivative. Alexandria Engineering Journal, 60(2), 2001-2012, (2021).
- Khan, M.A, Hammouch, Z. and Baleanu, D. Modeling the dynamics of hepatitis E via the Caputo-Fabrizio derivative. Mathematical Modelling of Natural Phenomena, 14(3), 311, (2019).
- Ucar, E., Ozdemir, N. and Altun, E. Fractional order model of immune cells influenced by cancer cells. Mathematical Modelling of Natural Phenomena, 14(3), 308, (2019).
- Sene, N. SIR epidemic model with Mittag–Leffler fractional derivative. Chaos, Solitons & Fractals, 137, 109833, (2020).
- Omame, A., Abbas, M. and Abdel-Aty, A.H. Assessing the impact of SARS-CoV-2 infection on the dynamics of dengue and HIV via fractional derivatives. Chaos, Solitons & Fractals, 162, 112427, (2022).
- Nwajeri, U.K., Omame, A. and Onyenegecha, C.P. Analysis of a fractional order model for HPV and CT co-infection. Results in Physics, 28, 104643, (2021).
- Nwajeri, U.K., Panle, A.B., Omame, A., Obi, M.C. and Onyenegecha, C.P. On the fractional order model for HPV and Syphilis using non–singular kernel. Results in Physics, 37, 10546, (2022).
- Omame, A., Abbas, M. and Onyenegecha, C.P. A fractional-order model for COVID-19 and tuberculosis co-infection using Atangana–Baleanu derivative. Chaos, Solitons & Fractals, 153, 111486, (2021).
- Sene, N. Theory and applications of new fractional-order chaotic system under Caputo operator. An International Journal of Optimization and Control: Theories & Applications (IJOCTA), 12(1), 20–38, (2022).
- Veeresha, P., Yavuz, M. and Baishya, C. A computational approach for shallow water forced Korteweg–De Vries equation on critical flow over a hole with three fractional operators. An International Journal of Optimization and Control: Theories & Applications (IJOCTA), 11(3), 52–67, (2021).
- Uçar, E., Uçar, S., Evirgen, F. and Özdemir, N. A fractional SAIDR model in the frame of Atangana–Baleanu derivative. Fractal and Fractional, 5(2), 32, (2021).
- Diouf, M. and Sene, N. Analysis of the financial chaotic model with the fractional derivative operator. Complexity, 2020, 9845031, (2020).
- Carpinteri, A. and Mainardi, F. Fractals and Fractional Calulus in Continum Mechanics, (Vol. 378). Springer-Verlag Wien GmbH: New York, (1997).
- Liu, K., Feckan, M. and Wang, J. Hyers–Ulam stability and existence of solutions to the generalized Liouville–Caputo fractional differential equations. Symmetry, 12(6), 955, (2020).
- Van den Driessche, P. and Watmough, J. Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission. Mathematical Biosciences, 180(1- 2), 29-48, (2002).
- Castillo-Chavez, C., Feng, Z. and Huang, W. On the computation of R0 and its role on global stability. In Mathematical Approaches for Emerging and Reemerging Infectious Diseases: An Introduction, (vol. 125), pp. 229–250. (Minneapolis, MN, 1999), Springer: New York, (2002).
- Diethelm, K. and Freed, A.D. The FracPECE subroutine for the numerical solution of differential equations of fractional order. Forschung und Wissenschaftliches Rechnen, 1999, 57-71, (1998).