Caputo-Fabrizio Fractional Order Derivative Mathematical Modeling and Optimal Control, Cost-Effectiveness Analysis of Diphtheria Transmission Dynamics in Nigeria
Article Sidebar
Main Article Content
Diphtheria, a potentially fatal vaccine-preventable disease caused by Corynebacterium diphtheriae, has seen a resurgence in several regions, including sub-Saharan Africa. This study presents a comprehensive mathematical model for diphtheria transmission dynamics that incorporates environmental bacterial persistence, public awareness campaigns, and vaccine hesitancy. The model subdivides the human population into epidemiologically relevant classes, including susceptible, exposed, symptomatic infectious, asymptomatic infectious, vaccinated, and hesitant individuals, with an additional compartment representing environmental bacterial load. Analytical results establish the model’s positivity, boundedness, and stability properties, with the basic reproduction number ( ) derived to determine disease threshold conditions. A global sensitivity analysis using the Partial Rank Correlation Coefficient (PRCC) identified the transmission rate , exposure progression rate , and vaccination rate as the most influential parameters affecting . The optimal control analysis, formulated via Pontryagin’s Maximum Principle, evaluated time-dependent interventions representing vaccination, treatment, and public enlightenment efforts. Numerical simulations using parameterized incidence data from Bauchi State, Northeast Nigeria, revealed that combined control strategies significantly reduce infection prevalence and environmental contamination compared to single interventions. The cost-effectiveness analysis (CEA) demonstrated that the combined vaccination and public enlightenment campaign strategy produced the lowest Incremental Cost-Effectiveness Ratio (ICER) and was classified as “very cost-effective” according to WHO-CHOICE thresholds. The findings underscore that effective control requires a multi-pronged, dynamically-adapted strategy targeting all transmission pathways. High vaccination coverage remains the cornerstone, but its impact is significantly amplified when synergistically combined with public health campaigns and environmental decontamination. The results offer evidence-based, cost-effective guidance for public health policymakers to design efficient diphtheria outbreak response and elimination programs. Moreover, numerical solutions obtained for different fractional orders highlight the Caputo-Fabrizio fractional derivative's capability to simulate memory effects, thereby offering a more nuanced representation of the real-life problem under study.
Downloads
References
Adegboye, O. A., Alele, F. O., Pak, A., Castellanos, M. E., Abdullahi, M. A., Okeke, M. I., ... & McBryde, E. S. (2023). A resurgence and re-emergence of diphtheria disease in Nigeria, 2023. Therapeutic Advances in Infectious Diseases. https://doi.org/10.20499361231161936
Adekunle, A., Meehan, M. T., & McBryde, E. S. (2020). Modeling combined intervention strategies for infectious disease control. Journal of Theoretical Biology, 497, 110282.
Anderson, A. (2022). What to know about diphtheria. WebMD LLC. https://www.webmd.com/a-to-z-guides/what-to-know-diphtheria-causes
Anderson, R. M., & May, R. M. (1991). Infectious diseases of humans: Dynamics and control. Oxford University Press.
Atangana, A., & Gómez-Aguilar, J.F. (2018). Decolonisation of fractional calculus rules: breaking commutativity and associativity to capture more natural phenomena. Eur. Phys. J. Plus, 133(4), 166.
Diethelm K., N. J. Ford, A. D. Freed, (2002). A predictor-corrector approach for the numerical solution of fractional differential equations, Nonlinear Dynam., 29 (2002), 3–22. https://doi.org/10.1023/A:1016592219341
Brigitte, C., Margaret, B., Fred, Z., Jussi, M., Joanne, W., & Lode, S. (2004). Anti-diphtheria antibody seroprotection rates are similar 10 years after vaccination with dTpa or DTPa using a mathematical model. Vaccine, 23(3), 336–342.
Britannica, The Editors of Encyclopaedia. (2023). Diphtheria. Encyclopedia Britannica. https://www.britannica.com/science/diphtheria
Caputo, M., & Fabrizio, M. (2015). A new definition of a fractional derivative without singular kernel. Prog. Fract. Differ. Appl., 1(2), 1–13.
Diethelm K., N. J. Ford, A. D. Freed, (2004). Detailed error analysis for a fractional Adams method, Numer. Algorithms, 36, 31–52. https://doi.org/10.1023/B:NUMA.0000027736.85078.
Centers for Disease Control and Prevention (CDC). (2022). Diphtheria. https://www.cdc.gov/diphtheria/about/causes-transmission.html
Njagarah J. B. H., & C. B. Tabi, (2018). Spatial synchrony in fractional order metapopulation cholera transmission, Chaos Soliton. Fract., 117, 37–49. https://doi.org/10.1016/j.chaos.2018.10.004
Diethelm, K. (2010). The Analysis of Fractional Differential Equations: An Application-Oriented Exposition Using Differential Operators of Caputo Type. Springer.
Drummond, M. F., Sculpher, M. J., Claxton, K., Stoddart, G. L., & Torrance, G. W. (2015). Methods for the Economic Evaluation of Health Care Programmes. Oxford University Press.
Ekere, S. U., Ubong, D. A., Joy, I. U., & Henry, S. T. (2024). Age structured deterministic model of diphtheria infection. Earthline Journal of Mathematical Sciences, 14(3), 391–404.
Hunter, J.K., Nachtergaele, B. (2001): Applied Analysis. World Scientific, Singapore.
Hackbusch, W. (1978). A numerical method for solving parabolic equations with opposite orientations. Computing, 20(3), 229–240.
Iboi, E., Ndinya, C., & Okuonghae, D. (2020). Modeling environmental contribution to disease transmission. Mathematical Biosciences, 329, 108455.
Ilahi, F., & Widiana, A. (2018). The effectiveness of vaccine in the outbreak of diphtheria: Mathematical model and simulation. IOP Conference Series: Materials Science and Engineering, 434(1), 012006.
Isobeye, G. (2023). Diphtheria transmission dynamics: A sensitivity analysis. Faculty of Natural and Applied Sciences Journal of Scientific Innovations, 5(1), 130–140.
Izzati, N., Andriani, A., & Robi’Aqolbi, R. (2020). Optimal control of diphtheria epidemic model with prevention and treatment. Journal of Physics: Conference Series, 1663(1), 012042.
Khalil, H. K. (2002). Nonlinear systems (3rd ed.). Prentice Hall.
Kreyszig, E. (1978): Introductory Functional Analysis with Applications. Wiley, New York .
Lamichhane, A., & Radhakrishnan, S. (2022). Diphtheria. In StatPearls [Internet]. StatPearls Publishing. https://www.ncbi.nlm.nih.gov/books/NBK560911/
LaSalle, J. P. (1976). The stability of dynamical systems. Society for Industrial and Applied Mathematics.
Lenhart, S., & Workman, J. T. (2007). Optimal control applied to biological models. Chapman and Hall/CRC.
Losada, J., & Nieto, J.J. (2015). Properties of a new fractional derivative without singular kernel. Prog. Fract. Differ. Appl., 1(2), 87–92.
Marino, S., Hogue, I. B., Ray, C. J., & Kirschner, D. E. (2008). A methodology for performing global uncertainty and sensitivity analysis in systems biology. Journal of Theoretical Biology, 254(1), 178–196.
Morufu, O. O., & Adedapo, I. A. (2024). Mathematical modelling of diphtheria transmission and vaccine efficacy in Nigeria. Modeling Earth Systems and Environment, 1–27.
Musa, S. M., Abdullahi, A., & Aliyu, M. (2023). Behavioral dynamics and vaccine hesitancy in infectious disease modeling. African Journal of Applied Mathematics, 12(3), 45–59.
NHS. (2022). Diphtheria. https://www.nhs.uk/conditions/diphtheria/
Nigeria Centre for Disease Control (NCDC). (2023). Weekly epidemiological report on diphtheria outbreaks in Nigeria. Abuja: NCDC.
Okeke, C. A., Iboi, E. A., & Musa, S. M. (2023). Cost-effectiveness analysis of vaccination and public health campaigns in epidemic control. African Journal of Epidemiological Modelling, 15(2), 22–36.
Oli, M. K., Venkataraman, M., Klein, P. A., Wendl, L. D., & Brown, M. B. (2006). Population dynamics of infectious disease: A discrete time model. Ecological Modelling, 198(1–2), 183–194. https://doi.org/10.1016/j.ecolmodel.2006.04.007
Pontryagin, L. S., Boltyanskii, V. G., Gamkrelidze, R. V., & Mishchenko, E. F. (1962). The mathematical theory of optimal processes. Interscience Publishers.
Saltelli, A., Ratto, M., Tarantola, S., & Campolongo, F. (2008). Sensitivity analysis for chemical models. Chemical Reviews, 108(7), 2812–2832.
Sanders, G. D., Neumann, P. J., Basu, A., Brock, D. W., Feeny, D., Krahn, M., ... & Ganiats, T. G. (2016). Recommendations for conduct, methodological practices, and reporting of cost-effectiveness analyses: Second panel on cost-effectiveness in health and medicine. JAMA, 316(10), 1093–1103.
Sanusi, O. A., Ojo, O. D., & Onikola, I. O. (2023). Modeling the transmission and control strategies of diphtheria (A case study of Isin Local Government Area of Kwara State). International Journal of Advances in Engineering and Management, 5(9), 904–910.
Van den Driessche, P., & Watmough, J. (2002). Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission. Mathematical Biosciences, 180(1–2), 29–48.
Verguet, S., Laxminarayan, R., & Jamison, D. T. (2020). Economic evaluation of infectious disease interventions in low- and middle-income countries. Health Economics Review, 10(1), 5–14. https://doi.org/10.1186/s13561-020-00271-1
Woods, B., Revill, P., Sculpher, M., & Claxton, K. (2016). Country-level cost-effectiveness thresholds: Initial estimates and the need for further research. Value in Health, 19(8), 929–935.
World Health Organization (WHO). (2013). WHO guide for standardization of economic evaluations of immunization programmes. WHO Press.
World Health Organization (WHO). (2019). Making choices in health: Cost-effectiveness thresholds for health interventions. Geneva: WHO Press.
World Health Organization (WHO). (2020). Global Health Expenditure Database and Economic Evaluation Guidelines. Geneva: WHO.
World Health Organization (WHO). (2023, July). Multi-country diphtheria outbreak report. https://www.who.int/news-room/questions-and-answers/items/diphtheria
Zahurul, I., Shohel, A., Rahman, M. M., Karim, M. F., & Amin, M. R. (2022). Global stability analysis and parameter estimation for a diphtheria model: A case study of an epidemic in Rohingya refugee camp in Bangladesh. Computational and Mathematical Methods in Medicine, 2022(1), 6545179.
Zahurul, I., Kabir, K. M. A., Rahman, M. M., & Rahman, M. D. A. (2024). A cost-effective epidemiological exposition of diphtheria outbreak by the optimal control model: A case study of Rohingya refugee camp in Bangladesh. Complexity, 2024(1), 6654346.
This work is licensed under a Creative Commons Attribution 4.0 International License.
All articles published in our journal are licensed under CC-BY 4.0, which permits authors to retain copyright of their work. This license allows for unrestricted use, sharing, and reproduction of the articles, provided that proper credit is given to the original authors and the source.