Mathematical Modeling, Optimal Control and Cost-Effectiveness Analysis of Diphtheria Transmission Dynamics

Ayodeji Sunday Afolabi, Miswanto Miswanto

Abstract


Diphtheria remains a serious public health concern in regions with low vaccination coverage and limited access to timely treatment, highlighting the urgent need for effective modeling and control strategies to guide intervention efforts. A nonlinear mathematical model is developed to describe the transmission dynamics of diphtheria. The well-posedness of the model is analyzed by investigating the positivity and boundedness of its solutions. The solutions of the disease-free equilibrium points are obtained analytically. The basic reproduction number () is determined using Diekmann-Heesterbeek-Metz Next Generation Matrix approach. The stability of the disease-free and endemic equilibrium points are rigorously analyzed. Sensitivity analysis of the model parameters with respect to  is conducted to assess the relative impact of each parameter on the transmission dynamics of the disease. Based on the results of the sensitivity analysis, the proposed diphtheria model is extended into an optimal control problem by introducing four time-dependent control variables: personal protection, booster vaccine administration, detection/treatment of the asymptomatic infected humans and reduction of bacteria concentration. Four different scenarios with each involving at least three of the control variables are examined. We evaluated the cost-effectiveness of each control strategy using IAR, ACER and ICER methods in order to identify the most economically efficient strategy. The findings demonstrate that Strategy A is the most cost-effective startegy that can significantly reduce diphtheria transmission throught optimal personal protection, detection/treatment of the asymptomatic infected humans and reduction of bacteria concentration.


Keywords


Diphtheria; Non-linear; Stability analysis; Sensitivity analysis; Optimal control problem; Cost-effectiveness analysis

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References


W. H. Organization, “Who african region health emergency situation report-multi-country outbreak of diphtheria, consolidated regional situation report number 006 – as of january 14, 2024,“ 2024, https://reliefweb.int/report/nigeria/who-african-region-health- emergency-situation-report-multi-country-outbreak-diphtheria-consolidated-regional-situation-report-006-january-14-2024, Accesed on 11 March 2025.

L. Blumberg et al., “The preventable tragedy of diphtheria in the 21st century,“ International Journal of Infectious Diseases, vol. 71, pp. 122–123, 2018.

D. D. Gaiya et al., “Diphtheria outbreak in nigeria: what we know now,“ Infection Prevention in Practice, vol. 6, no. 1, p. 100345, 2024. DOI:10.1016/j.infpip.2024.100345

H. Husain, “An sir mathematical model for dipterid disease,” in Journal of Physics: Conference Series, vol. 1280, no. 2, p. 022051, 2019. DOI:10.1088/1742-6596/1280/2/022051

N. Kitamura, “Understanding factors contributing to outbreaks of diphtheria and implications for vaccination policy in vietnam [Dissertation],” London: London School of Hygiene & Tropical Medicine, 2023.

D. Kolibo and S. Romaniuk, “Mathematical model of the infection process in diphtheria for determining the therapeutic dose of antitoxic anti-diphtheria serum,” Ukrains’ kyi Biokhimichnyi Zhurnal, vol. 73, no. 2, pp. 144–151, 2001.

S. Latifah et al., “Mathematical study for an infectious disease with awareness-based sis-m model,” in Journal of Physics: Conference Series, vol. 1747, no. 1, p. 012017, 2021. DOI:10.1088/1742-6596/1747/1/012017

M. Muscat et al., “Diphtheria in the who european region, 2010 to 2019,” Eurosurveillance, vol. 27, no. 8, p. 2100058, 2022. DOI:10.2807/1560-7917.ES.2022.27.8.2100058

O. N. Olulaja et al., “A looming epidemic: combating the recurrent outbreaks of diphtheria in nigeria,” The Pan African Medical Journal, vol. 45, 2023. DOI:10.11604/pamj.2023.45.186.41328

P. O. Omosigho et al., “The re-emergence of diphtheria amidst multiple outbreaks in nigeria,” Infectious Disorders-Drug Targets, vol. 24, no. 4, pp. 20–28, 2024. DOI:10.2174/0118715265251299231117045940

S. Sharma and G. Samanta, “Stability analysis and optimal control of an epidemic model with vaccination,” International Journal of Biomathematics, vol. 8, no. 3, p. 1550030, 2015. DOI:10.1142/S1793524515500308

S. A. Truelove et al., “Clinical and epidemiological aspects of diphtheria: a systematic review and pooled analysis,” Clinical Infectious Diseases, vol. 71, no. 1, pp. 89–97, 2020. DOI:10.1093/cid/ciz808

S. S. Voss et al., “Underreporting of the 5-year tetanus, diphtheria, pertussis and polio booster vaccination in the danish vaccination register,” BMC Public Health, vol. 20, no. 1, pp. 1–6, 2020. DOI:10.1186/s12889-020-09816-w

C. E. Madubueze, K. A. Tijani, and Fatmawati, “A deterministic mathematical model for optimal control of diphtheria disease with booster vaccination,” Healthcare Analytics, vol. 4, p. 100281, 2023. DOI:10.1016/j.health.2023.100281

N. Rahmi and M. I. Pratama, “Model analysis of diphtheria disease transmission with vaccination, quarantine, and hand-washing behavior,” JTAM (Jurnal Teori dan Aplikasi Matematika), vol. 7, no. 2, pp. 462–474, 2023. DOI:10.31764/jtam.v7i2.13466

N. Medugu et al., “A review of the current diphtheria outbreaks,” African Journal of Clinical and Experimental Microbiology, vol. 24, no. 2, pp. 120–129, 2023. DOI:10.4314/ajcem.v24i2.2

E. S. Udofia et al., “Age structured deterministic model of diphtheria infection,” Earthline Journal of Mathematical Sciences, vol. 14, no. 3, pp. 391–404, 2024. DOI:10.34198/ejms.14324.391404

F. Finger et al., “Real-time analysis of the diphtheria outbreak in forcibly displaced myanmar nationals in bangladesh,” BMC Medicine, vol. 17, pp. 1–11, 2019. DOI:10.1186/s12916-019-1288-7

Z. Islam et al., “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, vol. 2022, pp. 1–13, 2022. DOI:10.1155/2022/6545179

N. Izzati and A. Andriani, “Dynamical analysis of diphtheria epidemic model with natural immunity rate on exposed individuals,” in Journal of Physics: Conference Series, vol. 1869, no. 1, p. 012117, 2021. DOI:10.1088/1742-6596/1869/1/012117

N. Izzati, A. Andriani, and R. Robi’Aqolbi, “Optimal control of diphtheria epidemic model with prevention and treatment,” in Journal of Physics: Conference Series, vol. 1663, no. 1, p. 012042. DOI:10.1088/1742-6596/1663/1/012042

M. Grasse et al., “Booster vaccination against tetanus and diphtheria: insufficient protection against diphtheria in young and elderly adults,” Immunity & Ageing, vol. 13, pp. 1–9, 2016. DOI:10.1186/s12979-016-0081-0

N. Abdulrasheed et al., “Recurrent diphtheria outbreaks in nigeria: A review of the underlying factors and remedies,” Immunity, Inflammation and Disease, vol. 11, no. 11, p. e1096, 2023. DOI:10.1002/iid3.1096

F. Ilahi and A. Widiana, “The effectiveness of vaccine in the outbreak of diphtheria: Mathematical model and simulation,” in IOP Conference Series: Materials Science and Engineering, vol. 434, no. 1, p. 012006, 2018. DOI:10.1088/1757-899X/434/1/012006

S. Kanchanarat, S. Chinviriyasit, and W. Chinviriyasit, “Mathematical assessment of the impact of the imperfect vaccination on diphtheria transmission dynamics,” Symmetry, vol. 14, no. 10, p. 2000, 2022. DOI:10.3390/sym14102000

I. S. Fauzi et al., “Assessing the impact of booster vaccination on diphtheria transmission: Mathematical modeling and risk zone mapping,” Infectious Disease Modelling, vol. 9, no. 1, pp. 245–262, 2024. DOI:10.1016/j.idm.2024.01.004

S. Adewale et al., “Mathematical analysis of quarantine on the dynamical transmission of diphtheria disease,” International Journal of Science and Engineering Investigations, vol. 6, no. 5, pp. 8–17, 2017.

W. L. Conklin, “Clinical versus bacteriological diagnosis and quarantine of diphtheria,” Buffalo Medical Journal, vol. 41, no. 9, p. 660, 1902.

S. Withers, J. R. Ranson, and E. D. Humphrys, “Shortening the quarantine period for diphtheria convalescents and carriers,” Journal of the American Medical Association, vol. 87, no. 16, pp. 1266–1269, 1926. DOI:10.1001/jama.1926.02680160014004

R. Kurniati, S. Sugiarto, and S. Nurwijaya, “Dynamical system for tuberculosis outbreak with vaccination treatment and different interventions on the burden of drug resistance,” Jambura Journal of Biomathematics (JJBM), vol. 5, no. 1, pp. 10–18, 2024. DOI:10.37905/jjbm.v5i1.21903

M. M. Ojo and E. F. Doungmo Goufo, “Assessing the impact of control interventions and awareness on malaria: a mathematical modeling approach,” Communications in Mathematical Biology and Neuroscience, vol. 2021, pp. 1–31, 2021. DOI:10.28919/cmbn/6632

Statista, “Population growth in nigeria from 2012 to 2022,” 2022, https://www.statista.com/statistics/382235/population-growth-in-nigeria/, Accesed on 11 March 2025.

Statista, “Population of nigeria in selected years between 1950 and 2023,” 2023, https://www.statista.com/statistics/1122838/population-of-nigeria/, Accesed on 11 March 2025.

Statista, “Life expectancy at birth in nigeria in 2023, by gender,” 2023, https://www.statista.com/statistics/1122851/life-expectancy-in-nigeria-by-gender/ Accessed on 29 April 2025.

N. KidsHealth, “Diphtheria,” 2024, https://kidshealth.org/en/parents/, Accesed on 29 April 2025.

J. Hallare and V. Gerriets, “Half life,” StatPearls, 2020, Accessed on April 29, 2025.

N. C. Marshall et al., “Ten years of diphtheria toxin testing and toxigenic cutaneous diphtheria investigations in alberta, canada: A highly vaccinated population,” in Open Forum Infectious Diseases, vol. 9, no. 1, p. ofab414, 2022. DOI:10.1093/ofid/ofab414

A. M. Acosta et al., “Diphtheria,” Epidemiology and Prevention of Vaccine-Preventable Diseases, 2021.

A. Abidemi, J. Akanni, and O. Makinde, “A non-linear mathematical model for analysing the impact of covid-19 disease on higher education in developing countries,” Healthcare Analytics, vol. 3, p. 100193, 2023. DOI:10.1016/j.health.2023.100193

M. O. Akinade and A. S. Afolabi, “Sensitivity and stability analyses of a lassa fever disease model with control strategies,” IOSR Journal of Mathematics (IOSR-JM), vol. 16, no. 1, pp. 29–42, 2020. DOI: 10.9790/5728-1601022942

F. O. Akinpelu and R. Akinwande, “Mathematical model for lassa fever and sensitivity analysis,” Journal of Science and Engineering Research, vol. 5, no. 6, pp. 1–9, 2018.

E. Bakare and C. Nwozo, “Bifurcation and sensitivity analysis of malaria–schistosomiasis co-infection model,” International Journal of Applied and Computational Mathematics, vol. 3, pp. 971–1000, 2017. DOI:10.1007/s40819-017-0394-5

C. M. Veronica et al., “Mathematical modeling and stability analyses on the transmission dynamics of bacterial meningitis,” Journal of Mathematics and Computer Science, vol. 11, no. 6, pp. 7384–7413, 2021. DOI:10.28919/jmcs/6513

E. Kanyi, A. S. Afolabi, and N. O. Onyango, “Optimal control analysis of schistosomiasis dynamics,” Journal of Mathematics and Computer Science, vol. 11, no. 4, pp. 4599–4630, 2021. DOI:10.28919/jmcs/5847

A. Abidemi, Fatmawati, and O. J. Peter, “An optimal control model for dengue dynamics with asymptomatic, isolation, and vigilant compartments,” Decision Analytics Journal, p. 100413, 2024. DOI:10.1016/j.dajour.2024.100413

I. Kour, L. Singhal, and V. Gupta, “Diphtheria: A paradigmatic vaccine-preventable toxigenic disease with changing epidemiology.” in Recent Advances in Pharmaceutical Innovation and Research, pp. 749–759. Singapore: Springer, 2023. DOI:10.1007/978-981-99-2302-1_30

M. Petráš et al., “Factors influencing persistence of diphtheria immunity and immune response to a booster dose in healthy slovak adults,” Vaccines, vol. 7, no. 4, p. 139, 2019. DOI:10.3390/vaccines7040139

V. D. Bampoe et al., “A review of adverse events from the use of diphtheria antitoxin (dat) in the united states, 2004–2019,” Clinical Infectious Diseases, vol. 74, no. 11, pp. 2082–2083, 2022. DOI:10.1093/cid/ciab899

B. L. Wiedermann, “Diphtheria in the 21st century: new insights and a wake-up call,” Clinical Infectious Diseases, vol. 71, no. 1, pp. 98–99, 2020. DOI:10.1093/cid/ciz813

A. Adikari and Y. Jayathunga, “Optimal control for resource allocation in a multi-patch epidemic model with human dispersal behavior,” Communication in Biomathematical Sciences, vol. 8, no. 1, pp. 1–18, 2025. DOI:10.5614/cbms.2025.8.1.1

A. B. Gumel and S. Lenhart, Modeling Paradigms and Analysis of Disease Transmission Models. in Providence, vol. 75. USA: American Mathematical Society, 2010.

S. Lenhart and J. T. Workman, Optimal control applied to biological models. in Boca Raton. USA: Chapman and Hall/CRC, 2007, ISBN 978-1-58488-640-2.

F. A. Oguntolu et al., “Mathematical modeling on the transmission dynamics of diphtheria with optimal control strategies,” Jambura Journal of Biomathematics (JJBM), vol. 6, no. 1, pp. 1–22, 2025. DOI:10.37905/jjbm.v6i1.29716

H. Alrabaiah et al., “Optimal control analysis of hepatitis b virus with treatment and vaccination,” Results in Physics, vol. 19, p. 103599, 2020. DOI:10.1016/j.rinp.2020.103599

A. Altamirano-Fernández, A. Rojas-Palma, and S. Espinoza-Meza, “Existence of solutions for an optimal control problem in forestry management,” in Journal of Physics: Conference Series, vol. 2515, no. 1, p. 012001, 2023. DOI:10.1088/1742-6596/2515/1/012001

W. H. Fleming and R. W. Rishel, Deterministic and Stochastic Optimal Control. New York: Springer New York, 2012, ISBN:978-1-4612-6382-1. DOI:10.1007/978-1-4612-6380-7

O. D. Falowo, S. Olaniyi, and A. T. Oladipo, “Optimal control assessment of rift valley fever model with vaccination and environmental sanitation in the presence of treatment delay,” Modeling Earth Systems and Environment, vol. 9, no. 1, pp. 457–471, 2023. DOI:10.1007/s40808-022-01508-1

E. E. Joshua, E. T. Akpan, and U. G. Inyang, “Computational nonlinear dynamics: Analysis and assessment in optimal control of covid-19 in akwa ibom state, nigeria,” Journal of Advances in Mathematics and Computer Science, vol. 39, no. 1, pp. 1–19, 2024. DOI:10.9734/jamcs/2024/v39i11858

H. R. Joshi, “Optimal control problems in PDE and ODE systems [Dissertation],“ in Knoxville. USA: The University of Tennessee, 2002.

F. Agusto and M. C. A. Leite, “Optimal control and cost-effective analysis of the 2017 meningitis outbreak in nigeria,” Infectious Disease Modelling, vol. 4, pp. 161–187, 2019. DOI:10.1016/j.idm.2019.05.003

L. J. Allen et al., “Mathematical Epidemiology.“ in Berlin. Germany: Springer, 2008.

F. Brauer et al., “Mathematical Models in Epidemiology,“ Cham. Switzerland: Springer, 2019.

J. K. K. Asamoa et al., “Optimal control and comprehensive cost-effectiveness analysis for covid-19,” Results in Physics, vol. 33, p. 105177, 2022. DOI:10.1016/j.rinp.2022.105177

F. S. García, “Mathematical modeling approaches in epidemiology: within-host dynamics, control strategies and cost-effectiveness analysis [Dissertation],” in Centro de Investigación en Matemáticas. Mexico: Guanajuato, 2020.

S. Olaniyi et al., “Efficiency and economic analysis of intervention strategies for recurrent malaria transmission,” Quality & Quantity, vol. 58, no. 1, pp. 627–645, 2024. DOI:10.1007/s11135-023-01664-1

B. E. Nichols et al., “Cost-effectiveness analysis of pre-exposure prophylaxis for hiv-1 prevention in the netherlands: a mathematical modelling study,” The Lancet Infectious Diseases, vol. 16, no. 12, pp. 1423–1429, 2016. DOI:10.1016/S1473-3099(16)30311-5

D. Aldila et al., “On the role of early case detection and treatment failure in controlling tuberculosis transmission: A mathematical modeling study,” Communication in Biomathematical Sciences, vol. 7, no. 1, pp. 61–86, 2024. DOI:10.5614/cbms.2024.7.1.4

H. A. Fatahillah and D. Aldila, “Forward and backward bifurcation analysis from an imperfect vaccine efficacy model with saturated treatment and saturated infection,” Jambura Journal of Biomathematics (JJBM), vol. 5, no. 2, pp. 132–143, 2024. DOI:10.37905/jjbm.v5i2.28810

A. Abidemi and O. J. Peter, “Deterministic double dose vaccination model of covid-19 transmission dynamics–optimal control strategies with cost-effectiveness analysis,” Communication in Biomathematical Sciences, vol. 7, no. 1, pp. 1–33, 2024. DOI:10.5614/cbms.2024.7.1.1

F. B. Agusto and I. M. ELmojtaba, “Optimal control and cost-effective analysis of malaria/visceral leishmaniasis co-infection,” PLOS ONE, vol. 12, no. 2, p. e0171102, 2017. DOI:10.1371/journal.pone.0171102

R. Boucekkine and T. Loch-Temzelides, “Introduction to the special issue on mathematical economic epidemiology models,” Economic Theory, vol. 77, no. 1–2, pp. 1–7, 2024. DOI:10.1007/s00199-023-01541-w

E. J. Dasbach, E. H. Elbasha, and R. P. Insinga, “Mathematical models for predicting the epidemiologic and economic impact of vaccination against human papillomavirus infection and disease,” Epidemiologic Reviews, vol. 28, no. 1, pp. 88–100, 2006. DOI:10.1093/epirev/mxj006

P. J. White, “nfectious Diseases (Fourth Edition),” in 5 - Mathematical models in infectious disease epidemiology, pp. 49–53.e1. Elsevier, 2017. DOI:10.1016/B978-0-7020-6285-8.00005-8

Y. A. Adi, N. Irsalinda, and M. Z. Ndii, “Optimal control and cost-effectiveness analysis in an epidemic model with viral mutation and vaccine intervention,” CAUCHY: Jurnal Matematika Murni dan Aplikasi, vol. 7, no. 2, pp. 173–185, 2022. DOI:10.18860/ca.v7i2.13184

D. Angulo et al., “Fine-grained mathematical modeling for cost-effectiveness evaluation of public health policies for cervical cancer, with application to a colombian case study,” BMC Public Health, vol. 23, no. 1, p. 1470, 2023. DOI:10.1186/s12889-023-16022-x

P. Asplin et al., “Epidemiological and health economic implications of symptom propagation in respiratory pathogens: A mathematical modelling investigation,” PLOS Computational Biology, vol. 20, no. 5, p. e1012096, 2024. DOI:10.1371/journal.pcbi.1012096

H. Bang and H. Zhao, “Average cost-effectiveness ratio with censored data,” Journal of Biopharmaceutical Statistics, vol. 22, no. 2, pp. 401–415, 2012. DOI:10.1080/10543406.2010.544437

S. Kim et al., “The epidemiologic and economic impact of varicella and herpes zoster vaccination in south korea: A mathematical modelling study,” Vaccine, vol. 42, no. 19, pp. 4046–4055, 2024. DOI:10.1016/j.vaccine.2024.05.016

K. N. Wanis et al., “Health and economic impact of intensive surveillance for distant recurrence after curative treatment of colon cancer: A mathematical modeling study,” Diseases of the Colon & Rectum, vol. 62, no. 7, pp. 872–881, 2019. DOI:10.1097/DCR.0000000000001364




DOI: https://doi.org/10.37905/jjbm.v6i2.30851

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