Comparative Analysis of Runge–Kutta and Backward Euler Methods for Modeling Monkeypox Transmission Dynamics

Background: Monkeypox (Mpox) is a re-emerging zoonotic viral disease that poses an increasing threat to global public health. Mathematical modeling is a key tool for understanding transmission dynamics of diseases like Mpox and supporting effective control strategies. Reliable numerical methods are essential for solving the nonlinear differential equations arising from such models. Materials: We developed a deterministic compartmental model to describe Mpox transmission between human and small mammal populations. Human compartments include susceptible, exposed, infected, isolated, and recovered individuals, with corresponding classes for small mammals. We solved the system of nonlinear ordinary differential equations using the fourth-order Runge–Kutta (RK4) method and the Backward Euler method. We validated the model using real outbreak data from the U.S. Clade II Mpox cases reported by the Centers for Disease Control and Prevention (CDC, 2025). Results: Simulation results demonstrate that RK4 provides higher accuracy and faster convergence in non-stiff scenarios, making it suitable for short-term epidemic predictions. The Backward Euler method exhibits superior numerical stability for stiff systems, allowing reliable long-term simulations with larger time steps. Error and computational analyses confirm RK4’s efficiency, while Backward Euler ensures robustness in unstable dynamic regions. Data fitting verifies that RK4 produces closer short-term approximations, whereas Backward Euler yields smoother long-term trends. Conclusion: Both numerical methods are effective for modeling Mpox transmission. RK4 is recommended for accurate short-term analysis, while Backward Euler is preferable for stiff epidemic dynamics requiring high stability. These results highlight the importance of appropriate numerical method selection in computational epidemiology.