A Statistical Reliability Framework for Markov Epidemic Networks
- 1Department of Statistics and Operations Research, College of Science, Qassim University, Buraydah 51452, Saudi Arabia
Abstract
Disease progression can be statistically represented as a network of patient states connected by transition probabilities, providing a natural basis for quantifying progression toward favorable and unfavorable outcomes. This study develops a statistical reliability framework for disease progression networks based on discrete-time Markov transitions, integrating maximum-likelihood estimation, first-passage reliability, bootstrap uncertainty quantification, and probability-simplex-constrained sensitivity analysis. The framework converts patient-level longitudinal trajectories into an estimated transition structure and subsequently quantifies the probability of reaching a favorable target state before an absorbing unfavorable outcome. A row-preserving perturbation strategy is used to assess the influence of individual transitions while maintaining the probability constraints, and a dimensionless elasticity measure is introduced to rank transitions according to their contribution to network reliability. Monte Carlo experiments demonstrate stable statistical performance of the proposed estimator: for the specified baseline network, the true reliability is approximately 0.7483, while the RMSE decreases from 0.0465 for n=100 to 0.0098 for n=2000, indicating substantial improvement in estimation accuracy as the sample size increases. The sensitivity analysis further identifies the most influential transitions, with progression from localized disease to treatment and treatment to remission showing the strongest positive effects on reliability, whereas transitions toward death exhibit negative effects. The framework is also applied to a real longitudinal breast cancer dataset comprising 169 patient records. The empirical analysis identifies 16 recurrence events and four deaths and characterizes the observed treatment, surgery, recurrence, and survival-time patterns. Importantly, because the available clinical variables do not unambiguously identify all states of the theoretical seven-state model, the real-data analysis is restricted to directly observable patient events rather than imposing unsupported clinical state assignments. Overall, the results demonstrate that the proposed framework provides a statistically interpretable pipeline linking patient-level transition data, network reliability, uncertainty quantification, and transition importance, with potential applications to reliability-based analysis of multistate disease progression and other longitudinal health processes.
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BibTeX
@article{sohaly2026statistical,
title = {{A Statistical Reliability Framework for Markov Epidemic Networks}},
author = {M. A. Sohaly and M. F. Al-Harbi and M. M. Al-Buhairi and K. A. Alwshmi and N. A. Alqussyer and T. M. Alnoushan},
journal = {Mathematics},
year = {2026},
month = oct,
doi = {10.3390/math14213897},
url = {https://www.mdpi.com/2227-7390/14/21/3897},
note = {CC BY 4.0},
}
RIS
TY - JOUR
TI - A Statistical Reliability Framework for Markov Epidemic Networks
AU - M. A. Sohaly
AU - M. F. Al-Harbi
AU - M. M. Al-Buhairi
AU - K. A. Alwshmi
AU - N. A. Alqussyer
AU - T. M. Alnoushan
PY - 2026
DA - 2026/10/26
JO - Mathematics
DO - 10.3390/math14213897
UR - https://www.mdpi.com/2227-7390/14/21/3897
AB - Disease progression can be statistically represented as a network of patient states connected by transition probabilities, providing a natural basis for quantifying progression toward favorable and unfavorable outcomes. This study develops a statistical reliability framework for disease progression networks based on discrete-time Markov transitions, integrating maximum-likelihood estimation, first-passage reliability, bootstrap uncertainty quantification, and probability-simplex-constrained sensitivity analysis. The framework converts patient-level longitudinal trajectories into an estimated transition structure and subsequently quantifies the probability of reaching a favorable target state before an absorbing unfavorable outcome. A row-preserving perturbation strategy is used to assess the influence of individual transitions while maintaining the probability constraints, and a dimensionless elasticity measure is introduced to rank transitions according to their contribution to network reliability. Monte Carlo experiments demonstrate stable statistical performance of the proposed estimator: for the specified baseline network, the true reliability is approximately 0.7483, while the RMSE decreases from 0.0465 for n=100 to 0.0098 for n=2000, indicating substantial improvement in estimation accuracy as the sample size increases. The sensitivity analysis further identifies the most influential transitions, with progression from localized disease to treatment and treatment to remission showing the strongest positive effects on reliability, whereas transitions toward death exhibit negative effects. The framework is also applied to a real longitudinal breast cancer dataset comprising 169 patient records. The empirical analysis identifies 16 recurrence events and four deaths and characterizes the observed treatment, surgery, recurrence, and survival-time patterns. Importantly, because the available clinical variables do not unambiguously identify all states of the theoretical seven-state model, the real-data analysis is restricted to directly observable patient events rather than imposing unsupported clinical state assignments. Overall, the results demonstrate that the proposed framework provides a statistically interpretable pipeline linking patient-level transition data, network reliability, uncertainty quantification, and transition importance, with potential applications to reliability-based analysis of multistate disease progression and other longitudinal health processes.
SN - 2227-7390
ER -