query on hazard rate

[jasepark2]

Hi,
I'm studying LPPLS and while I was studying through various papers to understand this theory, I saw in the paper below that I drew a graph using Hazard rate, and I had some questions.
https://www.researchgate.net/figure/Summary-of-various-indicators-obtained-by-the-LPPLS-analysis-performed-at-time-t-2-May_fig1_282183920
According to the LPPLS model of Johanssen-ledoit-Sornette (JLS 2000), a description of the crash hazard rate (Equation 17) is given, and the formula is as follows. arXiv:cond-mat/9810071v2 19 Oct 1998
h(t) =α*(t_c-t)^{m-1}(1+betacosine(omegaln(t_c-t)-phi’)) where alpha, beta, omega and phi’ are parameters and I can get those parameters from LPPLS model fitting or optimization).

Firstly, thanks to your Github code, I can estimate parameters for LPPLS.
LPPLS(t) =A + B*(t_c-t)^{m}+C(t_c-t)^{m}acosine(\omegaln(t_c-t)-\phi))

fit the model to the data and get back the params

tc, m, w, a, b, c, c1, c2, O, D = lppls_model.fit(MAX_SEARCHES)
Boulder-Investment-Technologies/lppls: Library for fitting the LPPLS model to data. (github.com)

Now I tried to get the hazard rate(t) using the parameters generated from the code described above, but I don't know how to handle the alpha variable. Can you give me some hints for the answer?