## 2 Loss Model (LM)  
In this section, a loss model is proposed to assess the reduction in the road infrastructure's functionality due to earthquakes at the time of occurrence. In this regard, it is fundamental to divide the direct and indirect losses. Traditionally, direct costs depend on the vulnerable elements that may reach several failure/damage states after the earthquakes. Indirect losses are associated with system failures, as shown in [22]. Recently, [PERSON]. [23] divided these losses into two sources: economic costs, and those related to casualties. Moreover, [PERSON]. [24] proposed a methodology based on resilience to assess the vulnerability of bridges, including direct and indirect losses. In particular, the assessment of indirect losses may be a difficult issue, as shown in [7], for several reasons. Mainly, these losses are extremely variable, since they depend on the system components and network conditions. In this regard, [PERSON]. [25] summarized various methods used to assess the indirect costs of natural hazards. Other approaches (i.e., [26]) can be used to estimate indirect losses considering the impact of natural disasters on public finances in terms of the government's capacity to cope with large amounts of expenditure due to natural disasters, and their subsequent ability to deliver basic services in the aftermath. In addition, indirect losses can be estimated with idealized models that emphasize the role or one or more particular relation(s) or mechanism(s) in economic systems [27; 28].  
The loss model adopted in the present paper proposes to assess the direct and indirect losses in a transportation network. In this regard, this paper considers the direct costs that derive from the structural costs of the bridge by adopting the performance-based earthquake engineering (PBEE) methodology in order to apply a probabilistic-based approach.  
Indirect losses are divided into the two typologies proposed in [5] (i.e., prolongation time and connectivity losses). Indirect losses are computed from the assessment of repair time, in order to reduce the uncertainties due to traffic flow estimations (before and after the earthquakes), as described in [29]. As shown in [25], risk assessments require probabilistic-based methodologies to consider all of the possible uncertainties in the definition of the damage and failure scenarios of each component of the infrastructure. In particular, indirect losses are calculated by referring to the PBEE framework, which is based on the application of the total probability theorem to disaggregate the problem into several intermediate models by calculating the repair quantities with the probabilistic-based local linearization of repair cost and time (LLRCT) methodology [30; 31], which is based on the closed-form \"Fourway method\", Ref. [32] and Monte Carlo simulations.  
Furthermore, road infrastructures are similar with regard to dimensions (e.g., size, area coverage), complexity, and interconnectedness, but may significantly differ in specific details. Most such differences depend on (inter)dependencies that should not be neglected in vulnerability assessments [33]. Interdependencies are fundamental in the evaluation of connectivity, and Ref. [34] provides an overview of how to identify, understand, and analyze them. To provide a detailed description and modeling of interdependent road infrastructures, many relevant data are required, and are often inaccessible--for example, due to confidentiality and privacy issues and a reluctance to share data [35]. However, the assessment of indirect costs needs to consider the mutual effects of different infrastructures that can supply one another when the functionality of one of them is reduced or fails.