Extended Weibull-Cox model survival function

Question

I'm reading an article where the authors use an extended Cox-Weibull hazard model with time-varying covariates to model survival time of an individual. This individual is allowed to start at week $m$, and covariates are available at weekly granularity.

Assuming the hazard is represented as $\lambda\left(t ; Z(t)\right)=\lambda_0(t) e^{\beta^\top Z(t)}$, where the baseline is Weibull, they provide an expression for the probability an individual survives at least $m^\prime-m$ weeks, given they started at week $m$, through $$\tag{1} S(m^\prime-m|m)=\exp(-\lambda B(m,m^\prime))$$ where $$ B(m,m^\prime)=\sum_{i=m+1}^{m^\prime}[(i-m)^c-(i-m-1)^c]\exp(\beta^\top Z(i)). $$ I'm having trouble justifying expression (1), however, and feel it should follow from the relation between survival and hazard functions, namely $$ S(m^\prime -m|m)=\exp\left(-\int_0^{m^\prime -m} \exp (\beta^\top Z(u)) \lambda_0(u)~\mathrm d u\right). $$ Using the fact the covariates $Z(t)$ are a step function, as they change only at the start of each week, I feel as if I can calculate what I'm after through something like $$ \exp\left(-\left[\exp (\beta^\top Z(0)) \int_0^{1} \lambda_0(u) ~\mathrm d u+\ldots+\exp(\beta^\top Z(m^\prime-m-1)) \int_{m^\prime-m-1}^{m^\prime-m} \lambda_0(u) ~\mathrm d u\right]\right), $$ but it doesn't seem like this will correspond to the expression in the paper.

Answer 1

The authors may be expressing that if an individual starts at week $m$ then they have survived up until week $m$, that is, they have the survival function $S(t | m) = P(T \ge t | T \ge m)$. This conditional survival function can be simplified to look like: $S(t | m) = \frac{S(t)} {S(m)}$. Their expression looks like this ratio.