Numerical Analysis of Typhoid Fever Spread Using Runge-Kutta and Nonstandard Finite Difference Methods
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Abstract
Background: Typhoid fever, caused by Salmonella typhi, spreads through food or water contaminated with manure, posing significant individual and public health risks. This study analyzes typhoid fever dynamics using a mathematical model with susceptible, unprotected, infected, and recovered populations.
Methods: The next-generation matrix was employed to compute the threshold quantity, evaluating the existence and stability of equilibrium points. Two numerical schemes were developed: a conditionally stable fourth-order Runge–Kutta (RK-4) scheme and an unconditionally stable nonstandard finite difference (NSFD) scheme. The NSFD scheme was designed to ensure dynamic reliability by preserving the continuous model's key properties. Numerical simulations were conducted using MATLAB R2015b.
Results: The RK-4 scheme maintained reliability only for smaller step sizes and did not preserve all essential properties of the original model. In contrast, the NSFD scheme accurately captured the dynamics of the model, maintaining positivity, boundedness, and monotonicity of solutions. Stability analyses revealed that the NSFD scheme converges locally and globally, irrespective of step size, for both disease-free and endemic equilibrium points.
Conclusion: The NSFD scheme preserves all critical dynamic properties of the continuous model and demonstrates its effectiveness in predicting the spread of typhoid fever. This study highlights the NSFD scheme as a robust numerical tool for modeling infectious disease dynamics, offering accurate and reliable results in alignment with the theoretical model.
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