A mathematical model of tuberculosis diabetes co-infection with optimal control
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Date
2025-05
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Kenyatta University
Abstract
Tuberculosis and diabetes co-infection is a complex health issue that requires effective management. Understanding the dynamics and interactions between these two diseases is crucial for developing optimal control strategies. Diabetes’ immune dysregulation can compromise the host’s ability to control TB infection, leading to higher TB and DM-related complications and Kenya being a third world country, faces this challenge. Kenya ranks 13th in TB burden; diabetes triples TB risk. To understandthedynamicsofthis, aninecompartmentalmodelfortuberculosis-diabetes co-infection is formulated. The Non-standard finite difference Scheme (NSFD) of themodelisformulatedfromthefirst-orderordinarydifferentialequations(ODE)to avoidfullimplicitschemesthatarecomputationallyexpensive. Theoverlysmallstep sizes in NSFD give the userautonomy in controlling the accuracy of the results, making it suitable for disease control applications. This paper expands existing model to incorporate the co-infection of diabetes and tuberculosis to understand disease complications better. Numerical simulations with different step sizes of the NSFD for the TB-Diabetes model are carried out to find the optimal step size, h. A comparison of the best resultant numerical simulation based on optimal h in NSDF indicates NSFD gives better results when compared with the corresponding first-order ode. The phase-plane analysis revealed that the NSFD formulated for tuberculosis and diabetes co-infection is generally asymptotically stable. Stability analysis of the ODE model based on RC indicated that RC ≤ 1 suggests potential epidemic eradication, whileRC ≥ 1indicatesariskofepidemicspread. Thesensitivityanalysishighlighted the significant impact of TB transmission coefficient β and diabetes acquisition rate α1 on RC, emphasizing the need for optimal control measures targeting these factors. A decrease in TB transmission coefficient led to a reduction inRC from 1.0863 to 0.1845, suggesting the potential effectiveness of control strategies. Future studies should consider formulating the proposed model with varied control parameters such as medication to compare the results with those from first-order ode. The study also recommends exploring models considering different diabetes types in future research. Thestudy’sfindingsbenefitvariousstakeholders,fromtheMinistryofHealth and health institutions to planners and other researchers.
Description
Thesis submitted in fulfillment of the award of the degree of Doctor of Philosophy (Applied Mathematics) in the school of pure and applied sciences of Kenyatta University, May 2025
Supervisors:
Prof. Winfred Nduku Mutuku
Dr. Nancy Matendechere Imbusi
Dr. Paul Wanjau