Stability Analysis of Malaria Transmission Model, Deterministic and Stochastic using Lyapunov Functions and Numerical Methods (Euler and Euler-Maruyama)

Malaria remains one of the most life-threatening infectious diseases worldwide, causing hundreds of thousands of deaths annually, particularly among children and pregnant women in tropical and subtropical regions. To better understand its complex transmission dynamics, this study develops and analyzes both deterministic and stochastic mathematical models of malaria based on systems of differential equations. The main objective of this research is to study the stability of both the deterministic and stochastic malaria models analytically and numerically using the Lyapunov function, the Euler method, and the Euler–Maruyama method. For the deterministic model, the disease-free and endemic equilibrium points are derived, and their local and global stability are investigated. Numerical simulations are conducted using the mentioned numerical methods to demonstrate the dynamic behavior of the system. The stochastic extension of the model incorporates random perturbations to represent environmental and demographic fluctuations that influence disease spread. The analytical and numerical results reveal that stochastic effects significantly influence malaria dynamics, potentially reducing disease persistence and stabilizing the system under certain conditions. These findings provide deeper insights into malaria transmission mechanisms and contribute to the development of more effective and evidence-based control strategies.