SIR-Based Modeling and Stability Analysis of Defect Spread in Production Systems

This work proposes a nonlinear dynamical framework to model, analyze, and control the propagation of defects in production systems using a compartmental SIR (Susceptible-Infected-Recovered) structure. Machines or production units are partitioned into three dynamic states: susceptible (operational but vulnerable), defective (actively propagating quality problems), and repaired (inspected or quarantined). A system of ordinary differential equations describes the transitions between these states based on a defect transmission rate and a repair/recovery rate. We derive a defect reproduction number, R0 which plays a threshold role analogous to epidemiological models: if R0 > 1, defect levels can exhibit an outbreak before decaying; if R0 < 1., defects are immediately driven to extinction. Fundamental properties of the model, such as positivity, boundedness, and global existence and uniqueness of solutions, are rigorously established. We identify the defect-free invariant manifold and prove both global asymptotic stability and an exponential (second-order) stability result via Lyapunov methods. A parameter estimation procedure is proposed using industrial indicators, combining scrap/rework behavior in automotive manufacturing and typical mean time to repair (MTTR) values reported for traditional manufacturing plants. A numerical case study with a 100-machine production line illustrates defect outbreaks, peak defective load, and the effect of mitigation strategies. The proposed framework provides a mathematically rigorous and practically interpretable tool that supports defect containment, predictive maintenance, and digital-twin based monitoring in smart manufacturing environments.