Stochastic Delay Differential Equations: A Comprehensive Approach for Understanding Biosystems with Application to Disease Modelling

Abstract

Mathematical models have been of great importance in various fields, especially for understanding the dynamical behaviour of biosystems. Several models, based on classical ordinary differential equations, delay differential equations, and stochastic processes are commonly employed to gain insights into these systems. However, there is potential to extend such models further by combining the features from the classical approaches. This work investigates stochastic delay differential equations (SDDEs)-based models to understand the behaviour of biosystems. Numerical techniques for solving these models that demonstrate a more robust representation of real-life scenarios are presented. Additionally, quantitative roles of delay and noise to gain a deeper understanding of their influence on the system’s overall behaviour are analysed. Subsequently, numerical simulations that illustrate the model’s robustness are provided and the results suggest that SDDEs provide a more comprehensive representation of many biological systems, effectively accounting for the uncertainties that arise in real-life situations.

Keywords

• stochastic delay differential equation
• biosystems
• disease modelling
• Euler–Maruyama
• Milstein
• numerical techniques and simulations