Earthquakes can cause interconnected disruptions to the built environment, population health, emergency response systems, and community recovery, with consequences that may persist long after the initial seismic event. These post-earthquake dynamics are inherently time-dependent and may exhibit memory and hereditary effects that are not adequately represented by conventional integer-order differential equations. A physics-informed fractional framework was therefore developed to characterize the coupled evolution of post-earthquake disaster impacts and recovery processes. Five state variables were introduced to represent physical damage, population health burden, shelter demand, emergency response capacity, and community recovery. Their interactions were described through a coupled system of fractional-order differential equations. A physics-informed neural network was subsequently formulated by embedding the governing fractional-order equations and initial-condition constraints directly into the learning objective. Reference numerical trajectories were generated using the fractional Adams–Bashforth–Moulton method to assess the internal numerical consistency of the scenario-based simulations. Synthetic datasets informed by publicly available earthquake-related indicators were used to examine the computational behavior of the proposed framework under representative post-earthquake scenarios. The resulting formulation provides an integrated computational representation of the temporal dependencies among disaster impacts, public health burden, emergency response capacity, shelter demand, and community recovery while explicitly accounting for memory effects through fractional-order dynamics. The framework establishes a methodological basis for investigating post-earthquake system evolution and for examining how persistent disaster effects may influence recovery trajectories. Following validation against empirical observations from real earthquake events, the proposed approach could support scenario analysis, disaster preparedness, public health planning, emergency resource allocation, and quantitative assessment of post-earthquake recovery.
Manufacturing resilience is strongly influenced by the ability of production systems to maintain operational continuity under stochastic equipment failures and constrained maintenance capacity. Building on an entropy-based stochastic framework, a modeling approach was developed to quantify operational uncertainty and characterize the resilience of a manufacturing system comprising $M$ identical and independently operating machines supported by a single maintenance crew. Machine failures and repairs were assumed to occur randomly at specified failure and repair rates, respectively, while no more than one failed machine could be repaired at any given time. The resulting machine repair system was formulated as a birth-death process of finite size $M+1$, in which each state was defined by the number of operational machines. Steady-state probabilities were derived as functions of the repair rate-to-failure rate ratio and were subsequently used to quantify information entropy, the expected numbers of operational and failed machines, maintenance crew utilization, and system operational availability. Information entropy was employed as a quantitative measure of uncertainty in the distribution of system states and, consequently, as an indicator of changes in operational resilience. An information stiffness parameter was further introduced. Then entropy hysteresis was identified, and its hysteresis width was interpreted as a measure of information asymmetry associated with transitions between operating regimes. The entropy hysteresis width increased with the number of available machines, $M$. By integrating stochastic machine-repair dynamics with information-theoretic measures, the proposed framework provides a quantitative basis for evaluating the relationships among equipment reliability, maintenance capacity, operational uncertainty, and manufacturing resilience. The resulting measures can support maintenance capacity planning, downtime mitigation, operational cost control, and production-delivery management, thereby facilitating more resilient and sustainable manufacturing operations.
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