# Measuring the Seismic Resilience of Housing Systems in Italy
## 2 Resilience Measurement  
According to the resilience definition provided by some authors [10, 11, 41] and widely adopted in resilience assessment studies (e.g., [13, 26, 34, 42]), the resilience index \(R\) of a community can be defined as the area underneath the functionality function of a given system normalized by the control time (\(T_{LC}\)) for the period of interest. Analytically, \(R\) is commonly defined as follows:  
\[R=\int\limits_{t_{0E}}^{t_{0E}+T_{LC}}Q(t)/T_{LC}dt \tag{1}\]  
where \(t_{0E}\) is the time instant when the disastrous earthquake happens, \(T_{LC}\) is the analyzed control time and \(Q(t)\) is the functionality function of the considered system (urban subsystem, city, city's system, region). The latter can be evaluated considering all the subsystems of the PEOPLES resilience framework, or only some of them. As evidenced in some studies [26, 28], this \(R\) index is geometrically equivalent to an average height. In fact, a height is obtained by normalizing the area underneath the functionality function \(Q(t)\) with a control time that represents the base of this area (see Figure 1). Consequently, the resilience of a system can be considered as an average functionality on the assigned control time over which the area underneath the functionality function is assessed and normalized. Generally, the functionality function \(Q(t)\) of a system is a piecewise function in which its constituent functions are continuous on the corresponding time intervals (subdomains) and there is no discontinuity at each endpoint of the subdomains within that interval. As shown in Figure 1, after a disastrous earthquake, the theoretical functionality of a resilient system can be described through four main intervals: (i) at \(t\)=0, a drop in functionality linked to the seismic risk of the exposed assets in which a residual functionality (robustness) is obtained through the usable structures and infrastructures; (ii) a pseudo-horizontal phase (downtime) comprising the emergency management, the seismic damage evaluation and the planning and implantation of preparatory activities for starting the reconstruction process; (iii) a third phase of recovery with a progressive increase in functionality thanks to the gradual allocation of funds and the consequent restoration works; and (iv) a fourth phase in which pre-earthquake functionality is roughly restored. The different urban subsystems of a city can be considered as dominant or non-dominant according to their impact on urban resilience. In particular, during the downtime and recovery phase, the global functionality of a city depends on the functionality of its dominant urban subsystems, which work in series (i.e., when one dominant subsystem fails, the urban system fails). As a result, the urban functionality is most affected by the more vulnerable dominant subsystem. Investigating the resilience of Italian cities against earthquakes, [PERSON] et al. [24] highlight that the vulnerability of residential buildings is predominant in determining the resilience of cities, especially for longer control times. Thus, several interesting studies were performed to analyze the functionality of housing systems during the reconstruction process that took place after the 2009 L'Aquila Earthquake [25; 34; 43] or the 2012 Northern Italy Earthquake [44]. For most of these studies, the functionality function of a housing system in the recovery phase is similar in shape and slope to the temporal curve of the cumulative financial grant allocated to cope with the disaster.