# Formal Matters on the Topic of Risk Mitigation: A Mathematical Perspective
## 4 Discussion  
The assessment of risk involves determining the probability of a hazard occurring and estimating the consequences through the quantification of exposure and vulnerability, while mitigation refers to any action aiming at reducing the risk, and includes prevention, preparedness, and response. Prevention is focused on a conscientious land-use planning in order to reduce exposure. Preparedness includes all strategies to better understand the hazardous phenomenon in order to limit its impact, like the development of monitoring and early warning systems. Response consists of the design of actions to contain the threat and for the possible evacuation, reducing vulnerability. The formalization of risk assessment introduced here does not invalidate the more informal approaches normally adopted for natural hazards, but extends them in such a way that the informal approach can be formally recognized as a numerical approximation stemming from epistemic limits [22] or the need to compromise between accuracy and computational complexity. For example, the digital elevation and surface models used as input to numerical fluid dynamics computational models typically employed in hazard and risk assessment for geophysical flows (such as floods [23], landslides [24], pyroclastic density currents [25], or lava flows [26]) are piece-wise constant approximations of the reality, whose higher or lower horizontal resolution and vertical accuracy can influence the models' output used in hazard assessment [27]. Hazard maps for these phenomena are typically assembled from a large number of such simulations: the choice of the combinations of initial conditions, source location, geometry, etc. provide a discretized approximation of all the possible scenarios that may impact the region of interest [28] whose combination is a numerical approximation of the integral form (8). We expect that this kind of insight may be useful in the selection of the representative scenarios for the given problem space, with an eye on well-established numerical integration schemes that may provide higher accuracy or lower computational loads, such as the Clenshaw-Curtis [29] or Gauss-Kronrod [30] quadrature formulas rather than the simpler rectangle formula typically adopted when choosing scenarios on a regular distributed grid. The main benefit of the formalization proposed here, however, is in the more refined functional dependency proposed between the components of risk and the underlying decision variables on which policy makers should act to improve prevention, preparedness, and response. The classification of these variables (environmental properties, elements at risk without direct influence on hazard, elements at risk with an influence on hazard, and resilience of the elements at risk) can provide insights on the extent to which each of them impacts the final risk assessment, and thus guide the decision-making process in risk management and mitigation. For example, the category described by the family \(\mathcal{B}\) of at-risk elements with an impact on hazard (such as buildings and roads in the case of lava flows hazard) presents the unique property of influencing risk assessment through all three of its components (hazard, exposure, and vulnerability), giving it a potentially higher priority over variables in other categories. This is particularly important in land-use and zoning plans (prevention), for which policies should focus not only on prioritizing construction in low-hazard areas, but also on favoring designs with higher resilience (preparedness), as well as ensuring that the associated infrastructure does not amplify hazard itself in the areas of interest (notoriously, for example, roads can become a preferential course for geophysical flows, directing them towards densely populated areas that would otherwise be less threatened by these hazardous phenomena). Such policies must of course take into account a cost-benefit analysis.