Keywords:-
Article Content:-
Abstract
A hybrid deterministic-stochastic (jump-diffusion) model for typhoid fever dynamics, addressing the limitations of classical deterministic models in capturing random environmental fluctuations and sudden outbreak events is pre sented. The model extends a deterministic seven-compartment framework by introducing both Brownian motion and discrete (Poisson jump) stochastic perturbations in the infected, treated, and recovered compartments. Ana lytical results establish the positivity and well-posedness of solutions, derive explicit expressions for disease-free and endemic equilibria, conditions for reducible nonlinear jump-endemic model. The basic reproduction numbers for both deterministic (R0) and stochastic (Rjp) systems were established. Sensitivity analysis using normalized indices and Partial Rank Correlation Coefficient (PRCC) methods identifies the symptomatic transmission rate (β1), vacci nation rate (ν), and recovery/treatment rates (γ1,δ) as the most influential parameters affecting R0. PRCC results show β1 and α have the strongest positive impact on R0, while γ1 and ν have strong negative effects, highlighting the importance of vaccination and treatment in disease control. Numerical simulations demonstrate that increasing ν or γ1 can reduce R0 below unity, leading to disease eradication, while higher β1 or positive jump intensities can sustain outbreaks even under control measures. The stochastic reproduction number Rjp is shown to decrease with stronger diffusion but increase with higher jump intensity, emphasizing the need for adaptive, non-constant inter ventions. The jump-diffusion framework thus provides a robust tool for epidemic decision support under uncertainty.
References:-
References
Abboubakar, H., & Racke, R. (2021). Mathematical modeling, forecasting, and optimal control of typhoid fever trans-mission dynamics. Chaos, Solitons & Fractals, 149, 111074, 1-20. https://doi.org/10.1016/j. chaos.2021.111074
Adetunde, L. A. (2008). Mathematical methods for the dynamics of typhoid fever in Kassena-Nankana district of Upper East region of Ghana. Journal of Modern Mathematics and Statistics, 2(2), 45-49.
Adeosun, Mabel E., & Ogundunmade, T. P. (2026). Exchange Rate Forecasting in Post-Subsidy Nigeria: A Hybrid Approach Using Asymmetric Jump Diffusion and Deep Learning. Scientific African, e03324.
Adeosun, M. E., & Ugbebor, O. O. (2021). An empirical assessment of symmetric and asymmetric jump-diffusion models for the Nigerian stock market indices. Scientific African, 12, e00733.
Affognon, S. B., Tonnang, H. E., Ngare, P., Kiplangat, B. K., Abelman, S., & Herren, J. K. (2024). Optimizing microbe-infected mosquito release: a stochastic model for malaria prevention. Frontiers in Applied Mathematics and Statistics, 10, 1465153.
Agbata, B. C., Asante-Mensa, F., Abah, E., Kwabi, P. A., Amoah-Mensah, J., Shior, M. M., ... & Obeng-Denteh,
W. (2025). Semi-analytical approach to a fractional-order model for the dynamics and control of Typhoid fever. Journal of Basics and Applied Sciences Research, 3(3), 215-226.
Ali, B., Jubair, S., Ishaq, W., Aljarboa, S., Khalifa, H. A. E. W., Flah, A., & Mohamed, M. (2026). A Data-Driven Deterministic Model for Typhoid Transmission with Stability Analysis and Neural Based Forecasting. Materials Today Communications, 115794.
Ajibola, O., Mshelia, M. B., Gulumbe, B. H., & Eze, A. A. (2018). Typhoid fever diagnosis in endemic countries: a clog in the wheel of progress?. Medicina, 54(2), 23.
Akingbade, J. A., & Bamigbola, O. M. (2025). Transmission Dynamics and Control of a Deterministic Bird-Human Avian Influenza Model.
Akingbade, J. A., & Ayegbusi, F. D. (2025). A Deterministic Mathematical Model and Analysis of Transmission Dynamics of Covid-19 from Reservoir-to-Human. Trends in Computational and Applied Mathematics, 26, e01796.
Akingbade, J. A., & Bamigbola, O. M. (2024). Dynamical Analysis of a Deterministic Bird-Human Avian Influenza Model. Ilorin Journal of Science, 11(3 (Special)), 157-166.
Alharbi, M. H., Alalhareth, F. K., & Ibrahim, M. A. (2023). Analyzing the Dynamics of a Periodic Ty-phoid Fever Transmission Model with Imperfect Vaccination. Mathematics, 11(15), 3298, 1-26. https:// doi.org/10.3390/math11153298
Anazawa, K., (2025). Evaluating a novel reproduction number estimation method: a comparative analysis. Scientific Reports, 15(1), 5423.
Asamoah, A., Otoo, H., & Nyarko, P. K. (2024). Transmission Dynamics of Typhoid Fever Outbreak: A Mathematical Modelling and Optimal Control Approach. AMERICAN JOURNAL OF BIOSCIENCE AND BIOINFORMATICS E-palli, 3(1), 17-35.
Baisa, L. A., & Kotola, B. S. (2024). Dynamics and control of typhoid fever in Sheno town, Ethiopia: A comprehensive nonlinear model for transmission analysis and effective intervention strategies. Plos one, 19(8), e0306544.
Bao, J., Yuan, C.: Stochastic population dynamics driven by Levy noise. J. Math. Anal. Appl. 391(2), 363-375 (2012)
Boakye Okyere, P., Twumasi-Ankrah, S., Newton, S., Nkansah Darko, S., Owusu Ansah, M., Darko, E., ... & Owusu-Dabo, E. (2025). Risk factors for typhoid fever: systematic review. JMIR Public Health and Surveillance, 11, e67544.
Dixit, R., Mishra, S., Chandra, V., & Banday, S. (2025). of Communicable Diseases. COVID-19 Impact on Economy.
Environment and Healthcare, 43.
Edward, S. (2024). A fractional order model for the transmission dynamics of shigellosis. Heliyon, 10(10).
Elsaid, M., Nasef, M. A., & Huy, N. T. (2021). R0 of COVID-19 and its impact on vaccination coverage: compared with previous outbreaks. Human Vaccines & Immunotherapeutics, 17(11), 3850-3854.
Falodun, M. O., Olorunfemi, O., & Irinoye, O. O. (2025). Infectious diseases: Addressing global challenges and preven-tion strategies for national health improvement. Community Acquired Infection, 12.
Gao, Q., Liu, Z., Xiang, J., Zhang, Y., Tong, M. X., Wang, S.,& Bi, P. (2021). Impact of temperature and rainfall on typhoid/paratyphoid fever in Taizhou, China: effect estimation and vulnerable group identification. The American journal of tropical medicine and hygiene, 106(2), 532.
Ibrahim, H. K. (2025). Emerging Viral Threats in the Post-Pandemic Era: Surveillance, Climate Dynamics, and Global Health Preparedness. Libyan Journal of Health, Science, and Development (LJHSD), 08-15.
Idowu, O. K., Erinle-Ibrahim, L. M., Agbomola, J. O., & Olawale-Shosanya, S. O. (2024). Effect of Environmental Precaution on the Transmission of Typhoid Fever: A Mathematical Modelling Ap-proach. EDUCATUM Journal of Science, Mathematics and Technology, 12(1), 89-106. Retrieved from https://ejournal.upsi.edu.my/index.php/EJSMT/article/view/9769
Jalija, E., Amos, J., Atokolo, W., Abah, E., Celestine, A. B., Acheneje, G. O., ... & Bolaji, B. (2026). Numerical solution of fractional order typhoid fever model via the generalized fractional Adams-Bashforth-Moulton approach. Network Modeling Analysis in Health Informatics and Bioinformatics, 15(1), 68.
Tsafack, T. J., Kwa Kum, C., Tass´e, A. J. O., & Tsanou, B. (2025). Mathematical modelling of the dynamics of typhoid fever and two modes of treatment in a Health District in Cameroon.Mathematical Biosciences and Engineering, vol. 22, no. 2, pp. 477-510. DOI: 10.3934/mbe.2025018.
Jan, R., Boulaaras, S., Alnegga, M., & Abdullah, F. A. (2024). Fractional?calculus analysis of the dynamics of typhoid fever with the effect of vaccination and carriers. International Journal of Numerical Modelling: Electronic Networks, Devices and Fields, 37(2), e3184.
Kailan Suhuyini, A., & Seidu, B. (2023). A mathematical model on the transmission dynamics of typhoid fever with treatment and booster vaccination. Frontiers in Applied Mathematics and Statistics, 9, Article 1151270. https://doi.org/10.3389/fams.2023.1151270
Khadija, O. U. B. O. U. S. K. O. U. R., KHASSAL, S., KHAJJI, B., & BALATIF, O. (2026). Mathematical modeling
and optimal control strategies of the typhoid transmission with cost-effectiveness analysis. Iranian Journal of Numerical Analysis & Optimization, 16(1).
Karunditu, J. W., Kimathi, G., & Osman, S. (2019). Mathematical modeling of typhoid fever disease incorporating unprotected humans in the spread dynamics. Journal of Advances in Mathematics and Computer Science, 32(3), 1-11.
Kou, S. G. (2002). A jump-diffusion model for option pricing. Management science, 48(8), 1086-1101.
Lahrouz, A., Omari, L., Kiouach, D., & Belmaati, A. (2011). Deterministic and stochastic stability of a mathematical model of smoking. Statistics & Probability Letters, 81(8), 1276-1284.
Lawal, O. F., Yusuf, T. T., & Abidemi, A. (2025). On the efficiency of optimal vaccination, environmental sanitation and treatment controls for typhoid fever: a mathematical study approach. Journal of Mathematical Analysis and Modeling, 6(1), 15-34.
Liu, Q., Jiang, D., Hayat, T., Ahmad, B.: Analysis of a delayed vaccinated SIR epidemic model with temporary immunity and Levy jumps. Nonlinear Anal. Hybrid Syst. 27, 29-43 (2018)
Maaji, Y. M., Akpan, C. E., Emmanuel, A. Y., Collins, A. E., & Salihu, A. I. (2026). A MATHEMATICAL MODELING OF THE DYNAMICS OF TYPHOID FEVER. FULafia Journal of Science and Technology, 10(1), 148-155.
Makinde O. D., Getachew T. T., David M. Modeling & Optimal control of typhoid disease with cost-effective strategies. Mathematical and Computational Methods in Medicine, 2017, Volume 2017 —Article ID 2324518 — 16 pages — https://doi.org/10.1155/2017/2324518.
Marks, F., Im, J., Park, S. E., Pak, G. D., Jeon, H. J., Nana, L. R. W., ... & Rakotozandrindrainy, R. (2024). Incidence of typhoid fever in Burkina Faso, Democratic Republic of the Congo, Ethiopia, Ghana, Madagascar, and Nigeria (the Severe Typhoid in Africa programme): a population-based study. The Lancet Global Health, 12(4), e599-e610.
Merton, R. C. (1976). Option pricing when underlying stock returns are discontinuous. Journal of financial economics, 3(1-2), 125-144.
Moatlhodi K., Gosalamang K. Mathematical analysis of typhoid infection with treatment. J.Math. Sci. Adv. Appl., 2016, 40 , 75-91.
Mogasale, V. V., John, J., Sahai, N., Ray, A., Farooqui, H. H., Mogasale, V., ... & Abbas, K. (2026). Burden of typhoid fever and antimicrobial resistance in India (2023): a modelling study. The Lancet Regional Health-Southeast Asia, 44.
Ntagalinda, E., Mureithi, E., Marijani, T., & Alendal, G. (2025). Mathematical Modeling of Antimicrobial Resistance of Typhoid Fever Incorporating Public Health Education. Tanzania Journal of Science, 51(2), 429-446.
Ochu, C. L., Kamateeka, M., Ananaba, N., Ndukwu, C. I., Akanbi, O. A., Mba, S. C., ... & Husain, F. (2026). Prospective surveillance of acute febrile illness in two tertiary health facilities in Nigeria, March 2021-September 2022.
Ogunlade, T. O., Ogunmiloro, O. M., & Fatoyinbo, G. E. (2021). On the deterministic and stochastic model applications to typhoid fever disease dynamics. In Journal of Physics: Conference Series (Vol. 1734, No. 1, p. 012048). IOP Publishing.
Oksendal, B. (2013). Stochastic differential equations: an introduction with applications. Springer Science & Business Media.
Omame, A., Umana, R. A., Iheonu, N. O., & Chioma, S. (2015). On the existence of a stochastic model of typhoid fever. Mathematical Theory and Modeling, 5(8),2015.
Oyiza Rabiu, H., Amos, J., Egbemhenghe, J., Atokolo, W., Omale, D., & Bolaji, B. (2025). Modeling and Simulation of Typhoid Fever Using a Fractional-Order Approach with the Generalized Adams Bashforth-Moulton Method. GPH-International Journal of Applied Science, 8(10), 119-147. https://doi.org/10.5281/zenodo.17621302
Payagala, S.,& Pozniak, A. (2024). The global burden of HIV. Clinics in dermatology, 42(2), 119-127.
Peter E Kloeden & Eckhard Platen. (1999). Numerical solution of stochastic differential equations. Springer Verlag, Qasim, Z., Aqeel, A., Ali, N., Abbas, Z., Mehdi, M. M., Gajian, W., ... & Ali, S. (2026). Food borne Salmonellosis:
Global Trends, Antibiotic Resistance, and Public Health Implications with a Focus on Baluchistan, Pakistan. Global Research Journal of Natural Science and Technology.
Ramdhania, K. F. (2026). Development of Deterministic-Stochastic Mathematical Models for Predicting Zoonotic Disease Transmission Dynamics in Tropical Regions. Science Get Journal, 3(1), 78-89.
Sabir, Z., Akkilic, A. N., Bulut, H., Umar, M., Salahshour, S., & Saba, I. (2025). A stochastic neural network pro-cedure for the nonlinear typhoid fever disease system. Network Modeling Analysis in Health Informatics and Bioinformatics, 14(1), 102.
Sabir, Z., Abdelkawy, M. A., Mehmood, M. A., & Bayram, M. (2025). Modeling typhoid dynamics using recurrent neural networks with Bayesian regularization. Computational Biology and Chemistry, 108790.
Saul, A., Smith, T., & Maire, N. (2013). Stochastic simulation of endemic Salmonella enterica serovar Typhi: the importance of long lasting immunity and the carrier state. PloS one, 8(9), e74097.
Shi, K., & You, T. (2025). Global trends in typhoid and paratyphoid, and invasive non-typhoidal salmonella, and the burden of antimicrobial resistance: a trend analysis study from 1990 to 2021. Frontiers in medicine, 12, 1588507.
Sinan, M., Shah, K., Kumam, P., Mahariq, I., Ansari, K. J., Ahmad, Z., & Shah, Z. (2022). Fractional order mathe-matical modeling of typhoid fever disease. Results in Physics, 32, 105044.
Tesfay, A., Saeed, T., Zeb, A., Tesfay, D., Khalaf, A., & Brannan, J. (2021). Dynamics of a stochastic COVID-19 epidemic model with jump-diffusion. Advances in Difference Equations, 2021(1), 228.
Thomas, N. E., Malini, B. T., Dileepan, P., & John, J. (2026). Exploring barriers to and motivations for vaccine uptake in a typhoid vaccine trial in Vellore, South India: a qualitative study. BMC Public Health.
Tijani, K. A., Madubueze, C. E., & Gweryina, R. I. (2023). Typhoid fever dynamical model with cost-effective opti-malcontrol. Journal of the Nigerian Society of Physical Sciences, 1579-1579.
Tilahun, G. T., Woldegerima, W. A., & Wondifraw, A. (2020). Stochastic and deterministic mathematical model of cholera disease dynamics with direct transmission. Advances in Difference Equations, 2020(1), 670.
Tsafack, T. J., Kum, C. K., Tasse, A. J. O., & Tsanou, B. (2025). Mathematical modelling of the dynamics of typhoid fever and two modes of treatment in a Health District in Cameroon. Mathematical Biosciences and Engineering, 22(2), 477-510.
Veggalam, S., & Kandi, V. (2026). Elimination of tuberculosis by 2025, an Indian perspective. Discover Public Health, 23(1), 118.
Watson C. H., Edmunds W. J. A review of typhoid fever transmission dynamics model and economic evaluation of vaccination. Vaccine, Elsevier, 2015, Volume 33, Supplement 3, pp C42- C54.
WHO (2007) Background paper on Vaccination against typhoid fever using new generation Vaccines. Presented at the SAGE meeting 11
Williams, M. M., Wankir, S. S., Yilgwan, P. L., Williams, J. M., Wetben, L. Y., Yiljika, J. D., ... & Iliya, G. B. (2026). Environmental Persistence of Typhoid and Barriers to Treatment-Seeking: An Eco-Health Study in Pankshin Local Government Area of Plateau state, Nigeria. EJSMT, 2(1), 169-183.
Wu, J., Dhingra, R., Gambhir, M., & Remais, J. V. (2013). Sensitivity analysis of infectious disease models: methods, advances and their application. Journal of The Royal Society Interface, 10(86), 20121018.
Yadav, S., & Tripathi, M. K. (2025). Fundamentals of infectious diseases and global health threats. In Drug Discovery and One Health Approach in Combating Infectious Diseases (pp. 21-43). Elsevier.
Zhang, X., Jiang, D., Hayat, T., Ahmad, B.: Dynamics of a stochastic SIS model with double epidemic diseases driven by Levy jumps. Phys. A, Stat. Mech. Appl. 471, 767-777 (2017)