# Vulnerability analysis#

Vulnerability curves provide the expected level of loss for a given level of ground shaking. The most frequent method to estimate vulnerability is through the convolution between fragility functions and a damage-to-loss model.

## Damage to loss models#

Damage-to-loss models (also know as consequence models) establish the connection between the level of damage and the expected loss (see examples in Figure 1). Figure. 1 Example of damage-to-loss models. From left to right: Di Pasquale and Goretti (2001) (Italy); Kappos et al. (2006) (Greece); Bal et al. (2008) (Turkey) and HAZUS (FEMA 2014) (California).

Uncertainty around the mean loss can be modelled through a beta distribution (Martins et al 2016).

## Computing vulnerability curves#

Converting fragility into vulnerability functions is then numerically performed from the equation below using a damage-to-loss model that correlates each damage state with the respective probability distribution of loss ratio.

$E[LR|IM]=\sum_{i=1}^{nDS}\sum_{j=1}^{m}\left(P[DS=ds_i|IM]*LR_{i,j}*P[LR_{i,j}\right)$

## Standard deviation on expected loss#

When not explicitly modelled, the uncertainty in the mean loss ratio can be estimated from the equation by Silva (2019) (see below)

$\sigma_{LR}|GMR=\sqrt{E[LR_{GMR}|IM]*\left(-0.7-2*E[LR_{GMR}|IM]+\sqrt{6.8*E[LR_{GMR}|IM]+0.5}\right)}$