Mathematical Modeling and Stability Analysis of Bacteria Macrophage Interactions: Insights into Phagocytosis Dynamics and Infection Control

Abstract

The interaction between bacteria and macrophages forms a critical frontline defense in the innate immune response against bacterial infections. This study develops a deterministic nonlinear system of ordinary differential equations to investigate the dynamic interplay between extracellular bacterial populations and macrophage activity. The model incorporates key biological processes including logistic bacterial growth, mass-action phagocytosis, constant and bacteria-stimulated macrophage recruitment, natural macrophage death, and macrophage depletion due to bacterial interaction and inflammatory stress. Steady-state analysis reveals both trivial (bacteria-free) and non-trivial equilibria, while linearization and Jacobian matrix techniques are employed to evaluate local stability. The characteristic equation is derived to determine the conditions under which the infection is cleared or persists. Numerical simulations further illustrate the sensitivity of infection outcomes to key parameters such as phagocytosis rate, recruitment efficiency, and bacterial proliferation. This work bridges a research gap by providing a generalized yet biologically realistic framework that captures the arms race between bacterial proliferation and macrophage-mediated clearance, offering valuable insights for understanding infection persistence and informing potential therapeutic strategies targeting macrophage function. The findings contribute to the growing field of mathematical immunology by highlighting effective factors that govern successful pathogen elimination versus chronic infection.