7 7.4. Multi-state deterioration models 157 7.4.3 Solving the Multi-State Deterioration Model Figure 6.1 (page 126) defines the Markov chain structure to model deterioration in a sewer main, and Section 7.4.2 introduced the hazard rate functions. The corresponding system of di!erential equations is presented in Eq. 6.2 (page 127) and is solved numerically using the LSODA algorithm from the FORTRANodepack library implemented in SciPy (Virtanen, Gommers, Oliphant, et al., 2020), which employs the Adams/BDF method with automatic sti!ness detection. 7.4.4 Parametric Multi-State Deterioration Models We extract a subset from our case study dataset to construct a cohort with concrete sewer mains carrying mixed and waste content (cohort CMW), representing 37.1% of the sewer network. The model parameters for this cohort are detailed in Appendix D in Tables D.1 and D.2. p1 (t) p 4 (t) p5 (t) pF (t) p2 (t) p3 (t) Pipe Age, t (years) 0.0 0.2 0.4 0.6 0.8 1.0 Pipe Age, t (years) 0.0 0.2 0.4 0.6 0.8 Pipe Age, t (years) 0.0 0.2 0.4 0.6 0.8 0 25 50 75 100 Pipe Age, t (years) 0.00 0.05 0.10 0.15 0.20 0 25 50 75 100 Pipe Age, t (years) 0.0 0.1 0.2 0.3 0.4 0 25 50 75 100 Pipe Age, t (years) 0.0 0.1 0.2 0.3 0.4 Exponential Gompertz Weibull Turnbull Figure 7.2: Probability of being in state k ↗ S at pipe age. The hazard functions are parametrised using the Exponential, Gompertz, and Weibull probability density functions. The Turnbull non-parametric estimator (see Section 6.2.3) indicates the ground truth. The grey circles indicate the frequency based on the inspection dataset. Figure 7.2 illustrates the MSDMs predictions, detailing the stochastic dynamics of sewer main deterioration for pipes in cohort CMW. As Figure 6.1 describes, this deterioration is segmented into five sequentially ordered severity levels (k =1 to k =5), plus a functional failure state (k =F). Di!erences in the y-axis scales are
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