1 Introduction

The complexity and interdependence of (socio-) technical systems increases as designers and planners use new technologies to respond to a demand for more cost effective and efficient infrastructures (Hickford et al. 2018). Increased complexity comes with new dangers such as cascade failures (Buldyrev et al. 2010; Dueñas-Osorio and Vemuru 2009; Kröger and Zio 2011; Little 2002; Lu et al. 2018). However, traditional safety and security assessments cannot keep pace with the rapid evolution of socio-technical systems: While classical risk analysis assesses potential threats, their probability and expected damage effects (Ale 2002; Kaplan and Garrick 1981; Morgan and Henrion 1998), modern resilience assessments additionally consider the recovery of a system (Cimellaro 2016; Häring et al. Häring et al. 2021; Lindell 2013).

Based on (Edwards 2009) and (Thoma et al. 2016), resilience can be addressed in five phases of the resilience cycle, which spans from the preparation in view of an adverse event to the point in time when a given system returns to a steady state: “the ability to repel, prepare for, take into account, absorb, recover from and adapt ever more successfully to actual or potential adverse events. Those events are either catastrophes or processes of change with catastrophic outcome, which can have human, technical or natural causes” (Thoma et al. 2016). Relying on the recommendations of (U.S. Department of Homeland Security), several publications (Cimellaro et al. 2016; Connelly et al. 2017; Francis and Bekera 2014; Häring et al. 2016a, b; Hollnagel 2016; Hosseini et al. 2016; Woods 2015) debate key aspects of resilience engineering, highlighting different fundamentals and aspects in their approaches. A summary of those approaches is given in (Patriarca et al. 2017, 2018), and none of the mentioned models in general contradict the model assumptions presented the in following derivations. Due to the fact that the model presented in the following relies on the existence of a performance over time curve as defined by (Bruneau et al. 2003), we follow this approach in the characterization of resilience engineering, establishing the following key resilience properties:

  • Robustness: The ability to maintain critical operations and functions in the face of crisis, i.e., absorbing and adapting.

  • Resourcefulness: The ability to prepare for, respond to and manage a crisis or disruption as it unfolds, i.e., involving all four of the key resilience properties (Fischer et al. 2018).

  • Rapid recovery: The ability to return to and/or reconstitute normal operations as quickly and efficiently as possible after a disruption, i.e., recovering.

  • Redundancy: The ability that single elements of the investigated system can be substituted, i.e. the capability to satisfy functional requirements in the event of the disruption or loss of functionality.

The question arises, how to assign a quantitative measure to such resilience dimensions. A plethora of approaches exist to determine the value of a given system’s resilience, which are, however, often sector or even system specific (Fischer et al. 2018; Galbusera et al. 2018, 2020; Kaegi et al. 2009; Kollikkathara et al. 2010; Nan and Sansavini 2017; Rose 2004; Zhang and Wang 2023), as summarized in, e.g., the review (Ouyang 2014) or in (Mentges et al. 2023). Hence, there is a need for models and comprehensive frameworks that are broadly applicable to a range of sectors, to support the engineering of resilient systems (Hickford et al. 2018; Koliou et al. 2018; Wu et al. 2023). An established and by now ubiquitous approach to facilitate measuring the resilience of socio-technical systems is to represent these systems’ performance by a single degree of freedom which indicates the system’s performance over time, in the course of the resilience cycle (Bruneau et al. 2003; Kröger and Zio 2011; Poulin and Kane 2021). To trace this so-called resilience curve, models must be devised that capture and reproduce the properties of the respective system’s performance over time, and thus allow to identify or assign model parameters to the relevant resilience dimensions. In engineering disciplines, e.g., mass-spring-damper models (Biggs 1964) successfully condense complex system dynamics and interrelations in a single degree of freedom, to describe, e.g., complicated structural robustness properties in view of extreme loading events (Cormie et al. 2020; Fischer and Häring 2009; Krauthammer 2008; Stolz et al. 2016). Additionally, the use of a single-degree-of-freedom (SDOF) model aligns with the principle of algorithmic complexity, which aims to describe a complex system using minimal information (Richter and Rost 2015). In contrast, the Law of Requisite Variety emphasizes the need for systems to possess sufficient variety or diversity to effectively interact with and regulate their environment. Although these concepts approach complexity from different perspectives, they are not necessarily in conflict. Algorithmic complexity provides a means of quantifying the complexity of individual elements within a system, which can contribute to understanding the overall complexity of the system as described by the Law of Requisite Variety. SDOF models are characterized by a dynamic force equation and describe whether the investigated system is able to return to the unloaded initial situation, i.e. the equilibrium state, which is in alignment with the understanding of resilience of (Holling 1973) or Walker et al. (2004). In the past, the aforementioned SDOF-models have subsequently also been applied to describe the performance of a broad range of socio-technical systems during and after adverse events. (Cimellaro et al. 2010) and (Tao and He 2020) describe the return of a system to its original performance from an initial displacement that corresponds to the disturbance of the system, in terms of an overdamped harmonic oscillator. In a similar oscillator model (Wang 2017), system performance decreases with a given initial velocity before returning to the original values. Several oscillators, the displacement of which is summarized to a performance measure, are driven away from full performance by external forces in (Mahmoud and Chulahwat 2018). In (Reed et al. 2016), parameters of a SDOF model are correlated with multiple weather hazard variables. There also exist other approaches to condense system performance into a single degree of freedom by means of information theoretic measures (Wu et al. 2023). The aforementioned approaches largely have in common that they rely on a description of a system that returns to its original performance, with the exception of external forces that persistently keep the system from the initial performance value (Mahmoud and Chulahwat 2018). In addition, some of these models require the maximum performance loss as an input parameter for the calculation of the recovery trajectory. To overcome this limitation, we propose a model that, besides providing a description of the return to the original performance, is in addition capable to capture permanent performance loss or gain after the recovery phase. While external forces are part of this description, capturing such a performance loss or gain does not rely on these forces, but solely results from the intrinsic model properties. Furthermore, the proposed model allows to quantitatively capture which influence specific resilience dimensions exert on the trajectory of the system’s performance. In this way, the model introduces additional elements to the SDOF description. To demonstrate the broad applicability of the model, we perform a fit to the performance curve for several existing systems on the socio-technical spectrum, subject to historic load events for which the response of the performance was recorded. The model allows to predict how the relevant parameters, which can be associated with the abovementioned resilience dimensions, must be changed so as to steer the model, e.g., from permanent performance loss to full recovery or performance gain. Thus, optimizing those parameters renders a system resilient in an efficient manner.

2 Rheological model analog for the assessment of resilience

The model that traces the system performance with a single degree of freedom comprises several components which are depicted in Fig. 1. Regular load events are absorbed by model components that are active during regular operation, described by an overdamped harmonic oscillator in Sect. 2.1. If the load exceeds a certain threshold, recovery forces are activated, which are described in Sect. 2.2. Regular operation components can break before or after recovering forces are activated. If the threshold performance loss at which recovering forces set in is not surpassed and regular operation components remain intact, the system returns to its original performance after the disruption. In contrast, the system performance change ranges from permanent enhancement to permanent performance loss if the recovery threshold is surpassed which activates recovering forces. A final threshold, defining the maximally available resources for recovery, describes the total depletion of restorative forces.

Fig.1
figure 1

Rheological model for the assessment of system resilience. The performance loss \(\varphi\) is indicated by the location of the inertia \(I\), which can assume positive (+) and negative (−) values. During regular operation, the inertial motion is determined by the spring constant \({C}_{N}\) and the damper \(D\), as outlined in Sect. 1, cf. Figure 2. When \(\varphi\) exceeds the threshold \({\varphi }_{N}\), the spring for regular operation breaks, see Sect. 2.2 and Fig. 3a. As further discussed in Sect. 2.2, if \(\varphi\) exceeds \({\varphi }_{R}\), recovery resources are activated (depicted above by contact of the spring with constant \({C}_{R}\) to the inertia \(I\)) which feature a special evolution of the corresponding force, c.f. Figure 4, and the damper becomes unidirectional, i.e. it does no longer act against performance gain, cf. Figure 3b

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2.1 Regular operation

In the proposed model, the variable \(\varphi\)(t) as function of time \(t\) denotes the respective system’s departure from full performance, \(Q\left({t}_{\text{initial}}\right)={Q}_{\text{initial}}\), where the performance is given by \(Q(t)={Q}_{i\text{nitial}}-\varphi (t)\). For the sake of simplicity, the performance can be assumed to be normalized between \(0\) and \(1\) by rescaling \(\varphi (t)\), such that \({Q}_{\text{initial}}=1\). Thus, the performance loss is initialized to \(\varphi \left({t}_{\text{initial}}\right)=0\) and bounded by \(1\) from above, which corresponds to a total loss of performance.Footnote 1 Here, \({t}_{\text{initial}}\) marks the onset of perturbing forces that act on the system as a result of either regular stresses or adverse events and crises.

During regular operation, the system behaves like a harmonic oscillator subject to a time-dependent external force, i.e., the system obeys the ordinary differential equation.

$$I\ddot{\varphi }(t) + D\dot{\varphi }(t) + C_{{\text{N}}} \varphi (t) = F(t)$$
(1)

Here, \(\dot{\varphi }\) constitutes the change of performance loss, \(\ddot{\varphi }\) the acceleration of performance loss, \(I\) denotes the system’s inertia, \(D\) is a damping constant, \({C}_{N}\) is the regular operation spring rate, and \(F\left(t\right)\) describes the time-dependent external force. The shape of the latter force is not constrained by the model and depends on the specific characteristics of the adverse event or crisis that disturbs the system. Moreover, we demand that the system recovers completely during regular operation and returns to its initial state without oscillations. We therefore assume that the system operates in the overdamped parameter range, that is, we impose.

$$D^{2} - 4C_{{\text{N}}} I > 0$$
(2)

Below the performance loss \({\varphi }_{N}\), the spring coefficient remains constant. In this regime, the regular operation force \({F}_{N}\) traces the diagram displayed in Fig. 2a, and below the recovery threshold \({\varphi }_{R}\), discussed in Sect. 2.2, the dynamics of \(\varphi\) corresponds to the dynamics of the displacement of an overdamped harmonic oscillator. In the present model, the force \({F}_{N}\) thus constitutes a piecewise linear function (which can be modified to a non-linear shape in a generalization of the model). In summary, the coefficients \(D\) and \({C}_{N}\) determine the case in which the system remains within its design limits and the crisis is overcome solely by system intrinsic resources.

Fig. 2
figure 2

Regular operation. a For performance losses \(\varphi\) that do not exceed \({\varphi }_{N}\), the regular operation spring coefficient \({C}_{N}\) causes a linear restorative force \({F}_{N}\). b Force \({F}_{D}\) exerted by the damper during regular operation of the system. The dashed line along the spring and damper functions represents the system dynamics during a regular load event, and visualizes the mobilized portion of the corresponding force

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2.2 Loads exceeding the dimensioning

If \(\varphi\) in Eq. (\(1\)) exceeds \({\varphi }_{N}\) at time \({t}_{N}\), the regular operation spring breaks, and \({C}_{N}\) is set to zero, see Fig. 3a. In that case, if there are no further recovering forces, the system collapses and, in case of the absence of external forces, the performance decreases to a value dictated by the inertia \(I\), damping \(D\), and the residual velocity \(\dot{\varphi }({t}_{N})\).

Fig. 3
figure 3

a If the performance loss exceeds \({\varphi }_{N}\), the regular operation spring breaks and no longer exerts a restorative force. b When recovering forces are activated as soon as \(\varphi\) exceeds \({\varphi }_{R}\), the damper only acts against performance loss but not against performance gain. The dashed line along the spring and damper functions represents the system dynamics during an event in which the regular operation spring breaks, and visualizes the mobilized portion of the corresponding force

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If recovering forces \({F}_{R}\) are present, the system either recovers some of its original performance, or permanently improves or breaks down completely, if the resources for recovery are depleted by the crisis.s

If the performance loss \(\varphi\) exceeds the threshold \({\varphi }_{R}\), recovering forces are activated. In that case, the damper acts only against performance loss, not against performance gain, see Fig. 3b, and an additional term \({F}_{R}\) is added to Eq. (1). The mobilization of this force is displayed in Fig. 4. The recovering force thus also constitutes a piecewise linear function with an abrupt onset (which can be generalized to non-linear functions). If the maximal performance loss \({\varphi }_{M}\) occurs between \({\varphi }_{R}\) and \({\varphi }_{RE}\), the part of the recovery energy between \({\varphi }_{M}\) and \({\varphi }_{RE}\) is left to recover the system in the interval between \({\varphi }_{M}\) and \({\varphi }_{End}={\varphi }_{M}-({\varphi }_{RE}-{\varphi }_{M})\), see Fig. 4b. Depending on system parameters, the final value \({\varphi }_{End}\) of the performance loss either corresponds to permanent performance loss if \({\varphi }_{End}>0\) or permanent improvement if \({\varphi }_{End}<0\). The dynamics is stopped once \({\varphi }_{End}\) is reached, or when the performance loss  \(\dot{\varphi }\) turns from negative to positive.

Fig. 4
figure 4

a If the performance loss \(\varphi\) remains below the threshold \({\varphi }_{R}\), recovery forces remain deactivated. b Upon passing the threshold \({\varphi }_{R}\), recovery forces set in. First, performance loss is attenuated until \(\varphi\) reaches the maximal performance loss \({\varphi }_{M}\). If \({\varphi }_{M}\) remains below the threshold \({\varphi }_{RE}\), the remaining recovery resources \({\varphi }_{RE}-{\varphi }_{M}\) provide a force to recover or improve the system in the interval from \({\varphi }_{M}\) to \({\varphi }_{End}={\varphi }_{M}-({\varphi }_{RE}-{\varphi }_{M})\). c If the maximal performance loss \({\varphi }_{M}\) exceeds \({\varphi }_{RE}\), the crisis completely depletes the resources for recovery, and the system collapses. The dashed line along the spring function represents the system dynamics, and visualizes the mobilized portion of the corresponding force

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If the maximal performance loss \({\varphi }_{M}\) exceeds \({\varphi }_{RE}\), cf. Figure 4c, the system breaks down entirely and suffers permanent performance loss, or the performance is lost completely,\(Q=0.\)

2.3 System dynamics and intrinsic scales

The model and its parameters are characterized by three basic units:

  1. (i)

    The performance loss \({\varphi }_{0}=1\), which corresponds to a full loss of performance. If, e.g., at a specific airport the departure of \({n}_{0}\) airplanes (cf. Sect. 3.2) corresponds to full performance, then \(\varphi\) scales such that \({\varphi }_{0}\) corresponds to the loss of all \({n}_{0}\) departures

  2. (ii)

    A characteristic time scale \({t}_{0}\)

  3. (iii)

    The system specific inertia \({I}_{0}\) which determines how the system responds to external forces

All relevant quantities can be expressed in terms of these basic units, where, e.g., the model parameters \({C}_{N}\) and \({C}_{R}\) are measured in units of \({I}_{0}/{t}_{0}^{2}\), and \(D\) is measured in units of \({I}_{0}/{t}_{0}\).

The parameters of the model can be associated with resilience phases and dimensions, cf. Sect. 1. The inertia governs the response of the system to all acting forces. As such, the inertia is relevant for all phases within the resilience cycle as introduced in Sect. 1. Other system parameters have more specific roles in this regard. The normal operation spring constant determines the system’s robustness following (Francis and Bekera 2014) and can mainly be associated with the protect and absorb phases. The recovery spring constant, in addition, determines the system’s response, recovery and adaptation. When the recovering force is activated, the damping constant can be assigned to the absorption and response phases, and can be associated with what is sometimes called graceful degradation (Klutke and Yang 2002). The threshold at which recovering forces set in quantifies the rapidity with which the systems responds to the disruption and its recovery is initiated. If the system undergoes several crises, the associated change of the relevant parameters quantifies the adaptation where the system is adjusted to be better prepared to future disruptions. In this way, fitting the model to the time evolution of the performance constitutes a novel approach to attribute a quantitative measure to these resilience phases. The parameter values that result from the fit allow a comparison among one another of the resilience of systems subject to an extreme event, or to probe the development of resilience properties of a single system subject to a series of consecutive crises.

Figure 5 shows the four most important cases for the system dynamics discussed in Sect. 2.2 and in this section, which are covered completely by the model described above. The blue curve shows complete system recovery during regular operation. The teal curve corresponds to permanent system improvement. Displayed in dark yellow, the system shows permanent performance loss after some recovery. Finally, the red curve corresponds to the complete breakdown of the system. Note that these four cases, which have previously been qualitatively sketched (see, e.g., Fig. 1 in (Klimek et al. 2019)), here result dynamically from the equations of motion.

Fig. 5
figure 5

Four qualitatively different system dynamics of the model introduced in Sect. 2, cf. Ref. (Klimek et al. 2019). Robust behavior (blue curve) corresponds to the system’s return to its original performance. The teal curve shows a system that improves as a result of the crisis. The damaged system (yellow curve) suffers permanent performance loss. In the collapsed system (red curve), lost performance is not recovered

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3 Application and interpretation

To demonstrate the wide applicability of the model introduced in Sect. 2, a least-square fit of this model is performed to three systems that vary widely on the socio-technical spectrum:

  1. 1.

    The continental Europe power grid synchronous area separation of 8th January 2021 resides on the technical side of this spectrum and is discussed in Sect. 3.1.

  2. 2.

    The number of flight departures from German airports in the course of the Covid-19 pandemic, discussed in Sect. 3.2, has pronounced sociological and technical aspects.

  3. 3.

    The revenue of the German gastronomy sector in the course of the Covid-19 pandemic, discussed in Sect. 3.3, thematically resides on the sociological side of the socio-technical spectrum.

The fit determines the value of the parameters of the model, as well as an error band. The respective errors of the fit parameters, in cases in which these values are not predetermined or constrained, are obtained by increasing or decreasing these parameters until the model leaves the error band.Footnote 2 The relation between these fit parameter values and real-world quantities is subsequently discussed in Sect. 3.4.

3.1 The continental Europe power grid synchronous area separation

(© ENTSO-E 2021) provides a report that covers the separation of the European power grid synchronous area of the 8th of January 2021: the tripping of network transmission elements caused a network split into an area (northwest area) with a deficit of power (and lowered net frequency) and an area (southeast area) with a surplus of power (with increased net frequency), see Fig. 6. Figure 7 shows the time development of the net frequency at the station in Bassecourt located in the northwest area and at the Temelli station located in the southeast area shortly before and after the split. Before the split, the frequencies measured at both stations nearly coincide. After the split, the frequency measured in Bassecourt remains stable at a lower value than before the split, and the frequency measured in Temelli remains stable at a higher value.

Fig. 6
figure 6

[Adapted from Ref. (© ENTSO-E 2021)]: On January 8th 2021, the tripping of network transmission elements caused a network split of the European power grid synchronous area into an area located in the northwest (orange) with a power deficit and an area in the southeast (green) with a surplus of power

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Fig. 7
figure 7

[Data from Ref. (© ENTSO-E 2021)]: Network frequencies measured in the northwest area by the Bassecourt station (green curve) and in the southeast area by the Temelli station (orange curve). Before the split, the network frequencies at both stations nearly conincide at a value of \(\sim 50.025Hz\). Following the split of the network, the Bassecourt station frequency remains stable below the frequency before the split, and the Temelli station frequency remains stable above the frequency before the split

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Network operators aim to keep the network frequency at \(50Hz\) (Neidhöfer 2011), and a deviation of \(\sim 0.2Hz\) is considered within the limits of normal operation. If deviations exceed \(\sim 1Hz\), load (at lowered net frequencies) or generators (at increased net frequencies) must be disconnected from the network (Meyer 2007). In the present case, before the split the network frequency is stable at \(\approx 50.025Hz\). In view of the model parameters and scales discussed in Sect. 2, we thus take the absolute value of the deviation from this net frequency as the performance indicator of the power grid (note that, thus, in the present example, improvement as discussed in Sect. 2.3 is not relevant or applicable), and identify.

  1. (i)

    \(\varphi = 0\) with full performance (\(50.025Hz\))

  2. (ii)

    \(\varphi_{N}\) with \(50.025Hz\pm 0.2Hz\) the threshold for normal operation

  3. (iii)

    \(\varphi_{0} = 1\) with complete breakdown (\(50.025Hz\pm 1Hz\)).

The unit of time \({t}_{0}\) was chosen to correspond to one minute. Figures 8a and 9a show the results of the fits for the model introduced in Sect. 2 to the ensuing performance over time curves for the Bassecourt and the Temelli stations, respectively. Due to the abruptness of the onset of the perturbation in the data, the shape of the external force \(F(t)\) that acts in this example is chosen to correspond to a Dirac delta peak. To compare the data of the two regions, the magnitude of the force that acts upon both regions is set equal. This allows to express the coefficients of the model of Sect. 2 for both regions in terms of the chosen unit of time \({t}_{0}\), the breakdown performance loss \({\varphi }_{0}\), and the inertia \({I}_{0}\) of the Bassecourt region. Results for the fit parameters, expressed in terms of these units, are displayed in Figs. 8b and 9b.

Fig. 8
figure 8

a The model of Sect. 2 is fitted to the performance over time curve which corresponds to the frequency over time measurement at the Bassecourt station, see orange curve in Fig. 7. b The parameters of the model fit (blue dots: fit parameters, red area: parameter uncertainties) in (a) are expressed in terms of the basic units \({t}_{0}=1min\), \({\varphi }_{0}=1,\) which corresponds to a complete performance loss of the system at \(50.025Hz\pm 1Hz\), and \({I}_{0}\), which denotes the inertia of the northwest area

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Fig. 9
figure 9

a Model fit of the performance over time curve that corresponds to the frequency over time measurement at the Temelli station, displayed in Fig. 7 in green. b Parameters of this model fit (blue dots: fit parameters, red area: parameter uncertainties), expressed in terms of \({t}_{0}=1min\), \({\varphi }_{0}\), which corresponds to complete performance loss, and the inertia \({I}_{0}\) of the northwest area. The inertia of the southeast area is smaller than the inertia of the northwest area, which qualitatively mirrors the power served in these areas, cf. Sect. 3.4

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The inertia of the southeast area is smaller than the inertia of the northwest area, which qualitatively mirrors the amount of power served in the respective areas, cf. the discussion in Sect. 3.4. The recovery spring constant \({C}_{R}\) and the damping \(D\) follow the same pattern. Both regions emerge in the damaged state (the original state of the system is restored on a much longer timescale when the two regions are reconnected (© ENTSO-E 2021)) at a comparable performance loss. The onset \({\varphi }_{R}\) and magnitude \({\varphi }_{RE}\) of recovery resources is larger for the southeast region, which is reflected in the larger amplitude of the performance loss during the crisis. For the southeast region, the fit does not capture a subsequent oscillation of the performance. The regular operation spring constant \({C}_{N}\) is not shown in Figs. 8b and 9b since the dynamics is largely independent of this constant for a wide range of values and uncertainties are thus comparatively large, cf. Appendix 1. This point is picked up upon in connection with the discussion of Fig. 16 in Sect. 4.

In summary, the deviation from the sought net frequency of about \(50Hz\) in the network regions that result from the network split corresponds to the damaged state in the introduced model outlined in Sect. 2. The values of inertia that result from the fit for these regions also qualitatively mirror the sizes of the respective grid portions.

3.2 The number of flight departures from German airports in the course of the Covid-19 pandemic

Travel restrictions in the course of the Covid-19 pandemic caused a massive drop in the number of flights all over the world (Bureau of Transportation Statistics). In this section, the resilience of German airports in view of the pandemic is investigated. To this end, we fit the model of Sect. 2 to the number of flight departures over time for several German airports in the respective timeframe. In descending order, according to the monthly number of departing flights \({n}_{\text{initial}}\) at \({t}_{\text{initial}}\), where \({t}_{\text{initial}}\) is chosen to coincide with the onset of the German worldwide travel warning (Amt 2020), we consider the airports of Frankfurt, Munich, Düsseldorf, Hamburg, and Leipzig. Before the onset of the pandemic, the data displayed in Fig. 10 shows a clear periodic annual variation in the number of departing flights, cf. Appendix 2. To correct for this effect, a least-square fit of the data is performed starting January 2018 and lasting until \({t}_{\text{initial}}\), January 2020, with a yearly sinusoidal oscillation to which we add a constant \({n}_{K}\), \({n}_{A}\text{sin}(2\pi t/T + p) +{n}_{K}\). This constant together with the amplitude \({n}_{A}\) and phase \(p\) is determined by the aforementioned fit, \(T=1 \text{year}\). The numbers of flight departures \({n}_{A}\) and \({n}_{K}\) translate into performance losses \({\varphi }_{A}\) and \({\varphi }_{K}\), and \({n}_{K}\) will in the following be referred to as the average number of pre-pandemic flight departures which serves as the measure of full performance. The resulting sine function is then subtracted from the data (for all \(t\)). The original as well as the corrected dataFootnote 3 are displayed in Fig. 10, where also the onset \({t}_{\text{initial}}\) and duration of the German worldwide travel warning imposed due to the Covid-19 pandemic (Amt 2020) is indicated, which lasted from March 17th 2020 to June 15th 2020.

Fig. 10
figure 10

(Data from Destatis : database number 46421–0012): Number of departing flights (green, dot-dashed) at the German airports a Frankfurt, b Munich, c Düsseldorf, d Hamburg, and e Leipzig between the years 2019 and 2023. The worldwide travel warning issued by Germany lasting from March 17th 2020 to June 15th 2020 (yellow area) in response to the Covid-19 pandemic caused a clear downturn in the number of flights. After the travel warning, the number of departing flights slowly recovers, and only in the case of the airport of Leipzig this number reaches pre-pandemic levels in the timeframe under consideration. The blue curve shows the data after a correction of annual variations (cf. Sect. 3.2). Note that this correction briefly causes one slightly negative datapoint for the number of flights at the airport of Düsseldorf. https://www-genesis.destatis.de/genesis/online

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To facilitate the comparison between the fits to the data of the above airports, we demand that \({F}_{i}/{n}_{Ki} ={F}_{j}/{n}_{Kj}\), where \({F}_{i}\) denotes the force that affects airport \(i\). That is, we assume that airports are affected by a force proportional to the respective average number of pre-pandemic flight departures. For all airports, this force is taken to be uniform over the time of the worldwide travel warning. The results of the fit of the model are displayed in Fig. 11 for the above-mentioned airports, alongside the respective parameter values of the fits. Parameters are expressed in terms of complete performance loss \({\varphi }_{0}=1\), the unit of time \({t}_{0}=1 \text{month}\), and in terms of the inertia \({I}_{0}\) of the largest airport, Frankfurt.

Fig. 11
figure 11

Left: Fit of the model introduced in Sect. 2 to the normalized flight departure data corrected for annual variations, depicted in Fig. 10. Right: Fit parameter values (blue dots: fit parameters, red area: parameter uncertainties) expressed in terms of complete performance loss \({\varphi }_{0}=1\), the unit of time \({t}_{0}=1\) month, and the inertia \({I}_{0}\) of the largest airport, Frankfurt. The values for quantities that are measured in terms of the inertia of the airports, i.e. \(I\) itself, \({C}_{R}\), and \(D\), follow the same sequence as the pre-pandemic average number of flights, with the exception of the airport of Leipzig which, e.g., features a disproportionately larger inertia, cf. Figure 10

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The magnitude of the inertia \(I\) of the airports, of the spring constant \({C}_{R}\), as well as of the damping constants \(D\) largely follow the same sequence as the magnitude of the average number of pre-pandemic flight departures. The airport of Leipzig constitutes an exception, which features disproportionately large \(I\), \(D,\) and \({C}_{R}\). A possible explanation for the special role of the airport of Leipzig is that the airport features a significantly larger ratio of freight to passenger flights as compared to all of the other airports, which will be investigated in detail in Sect. 3.4. Values for the regular operation spring constant \({C}_{N}\) and threshold \({\varphi }_{N}\) are not shown, since the rapid downturn in the number of flight departures at the beginning of the pandemic causes this spring to break almost immediately, that is, on scale shorter than the temporal resolution of the data. The dynamics is thus largely independent of the values of these parameters except for very large values of \({C}_{N}\). This point will be taken up again in Sect. 4 in connection with Fig. 16. The onset of recovering forces is not displayed, since here, the dynamics depends mostly upon whether this onset is surpassed, cf. Fig. 17.

All airports discussed in this section emerge from the crisis in the damaged state, with the exception of the airport of Leipzig which emerges improved. The former airports feature similar values of \({\varphi }_{R}\) and \({\varphi }_{RE}\) among one another, cf. Appendix 1.

The airports discussed in this section thus provide examples for the final states damaged and improved of the model introduced in Sect. 2. Section 3.4 contains a discussion of the relation between these and further airports’ inertia and real-world parameters. In Sect. 4 it will be discussed at the example of the airport of Leipzig how parameter variations drive the model into different final states.

3.3 The revenue of the German gastronomy sector in the course of the Covid-19 pandemic

In the course of the Covid-19 pandemic, the revenue of the German gastronomy sector shows two clear downturns that coincide with a first lockdown, as well as a second lockdown together with the subsequent ‘emergency break’ (German: ‘Notbremse’) law (Welle 2021; Bundestag 2021), respectively, as displayed in Fig. 12a. In view of the model introduced in Sect. 2, we assume that the same force \(F\) acts during the first lockdown, in the time span from March 16th 2020 until May 5th 2020, and during the second lockdown lasting from November 2nd 2020 until April 24th 2021 together with the directly ensuing ‘emergency break’ phase from April 24th 2021 until June 30th 2021, as shown in Fig. 12b. It is stressed that this choice for the shape of the external force is arbitrary, and that more sophisticated approaches could further improve the results of the model fits qualitatively. Note that, in contrast to the systems discussed so far in Sects. 3 and 3.2, the system discussed in this section is disturbed twice in the timespan comprised by the recorded data.

Fig. 12
figure 12

(Data from Destatis (https://www-genesis.destatis.de/genesis/online): database number 45213–0005): a Revenue of the German gastronomy sector in a timeframe that includes the Covid-19 pandemic. Two clear downturns are observed that commence with the first (yellow) and second (red) lockdowns coming into effect March 16th 2020 and November 2nd 2020, respectively. The time span depicted in grey corresponds to the ‘emergency break’ law phase from April 24th 2021 until June 30th 2021. b External force that enters the model introduced in Sect. 2 during the lockdown and emergency break phases. The force is taken to be uniform during these phases. c Fits of the model introduced in Sect. 2 to the normalized revenue data. Two fits are performed, one fit that considers the data in the period from the beginning of the first lockdown to the beginning of the second lockdown, and one fit in the time span from the beginning of the second lockdown to the end of the available data. https://www-genesis.destatis.de/genesis/online

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The data displayed in Fig. 12a allows for a comparison of the properties of the system initialized at the beginning of the first and at the beginning of the second lockdown. To this end, two fits of the model introduced in Sect. 2 are performed: For the first fit, the period lasting from March 16th 2020 until November 2nd 2020, and for the second fit, the period from November 2nd 2020 until the end of the available data is considered. The results of these fits are displayed in Fig. 12c. The corresponding fit parameters are shown in Fig. 13a for the former period, and in Fig. 13b for the latter period.

Fig. 13
figure 13

a Parameters for the fit to the data from March 16th 2020 until November 2nd 2020 which encompasses the first lockdown phase. b Parameters for the fit to the data (blue dots: fit parameters, red area: parameter uncertainties) starting November 2nd 2020 until the end of the available data which takes into account the second lockdown as well as the emergency break phases

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The comparison between the two phases shows that, in the second phase, the inertia \(I\) and recovery spring constant \({C}_{R}\) are increased in comparison to the first lockdown phase. The damping \(D\), on the other, is decreased in the second period. Interestingly, the threshold for recovery \({\varphi }_{R}\) and the available resources for recovery \({\varphi }_{RE}\) remain largely unchanged between the lockdown phases. \({C}_{N}\) and \({\varphi }_{N}\) are not shown, cf. Sects. 3.1 and 3.2 and Appendix 1

In summary, a comparison of the first and second lockdown phases shows a significant change in the model parameters between these two phases, which is an indication for how the system adapts between successive crises.

3.4 Interpretation of fit parameters

Based on the results for the fit parameters in Sects. 3.1–3.3, the question arises whether the obtained values of the fit parameters can directly be connected to characteristic properties of the systems under consideration which, so far, has not been investigated in the context of SDOF models. In this section, we consider the connection between the model’s inertia to the power served in the grid portions of Sect. 3.1, and the model’s inertia and the relative amount of freight at the airports investigated in Sect. 3.2, as well as for further airports.

First, we want to test the assumption whether the inertia that results from the model fit reflects the system size of the respective grid portion in the case of the grid separation. To this end, we choose as an indicator of the system size the power served in the respective grid portion. While the power served in the northwest region of the continental Europe power grid discussed in Sect. 3.1 amounts to \({P}_{\text{northwest}}=326\) GW, the power in the southeast region amounts to \({P}_{\text{southeast}}=70.5\) GW (© ENTSO-E 2021). This corresponds to a ratio \({P}_{\text{southeast}}/{P}_{\text{northwest}}\) of about \(0.22\). The ratio of the inertias of the model fits to the Temelli and Bassecourt stations \({I}_{\text{Temelli}}/{I}_{\text{Bassecourt}}\) amounts to about \(0.10\). Thus, the model’s inertia is in this case indeed qualitatively related to the size of the underlying grid portion (as determined by the power served). To test whether this finding holds for further examples and to establish how the model’s inertia is quantitatively related to grid size requires further test cases.

For the airports discussed in Sect. 3.2, we found that the magnitude of those parameters that are measured in units of the inertia roughly follow the same sequence as the pre-pandemic average number of flight departures at the respective airports, with the notable exception of the airport of Leipzig which serves as a major air cargo hub in the world (Port Authority NY NJ). As such, the ratio of the weight of the cargo per year to the number of passengers at the airport of LeipzigFootnote 4 significantly exceeds the respective ratio at the other airports that have been considered in Sect. 3.2 (Statistisches Bundesamt), see Fig. 14. Since we naturally expect the travel restrictions of the pandemic to disproportionately affect passenger related air traffic as compared to freight related air traffic, we expect a less pronounced reaction to the pandemic at the airport of Leipzig as compared to at the other airports under consideration, which should be reflected in a larger relative inertia. Indeed, in Sect. 3.2 we found that the inertia of the airport of Leipzig exceeds the inertia of the other airports when normalized by the pre-pandemic number of flight departures.

Fig. 14
figure 14

Inertia, derived from the model introduced in Sect. 2, normalized by the average number of pre-pandemic flight departures vs. tonnes of cargo per year, normalized by number of passengers, for the airports investigated in Sect. 3.2 and all other airports from the same database (Statistisches Bundesamt). The double-logarithmic plot shows a clear increase of the normalized inertia with the proportionate amount of cargo at the respective airports. The yellow bars correspond to the normalized inertias’ uncertainties

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To test whether there exists a relation between the aforementioned proportionate amount of cargo and the normalized inertia for the airports considered in Sect. 3.2, we plot these quantities against one another in Fig. 14. In addition to the airports considered in Sect. 3.2, we include airports with departures of freight flights that appear in the database (Statistisches Bundesamt) that serves as the basis for that part of the study. The double-logarithmic plot in Fig. 14 shows a clear increase of the normalized inertia with the proportionate amount of cargo at the respective airports. A weighted regression to the logarithm (to base 10) of the data (blue dot-dashed) for all of the airports, in which the errors for the inertia (yellow bars in Fig. 14) that result from fitting the model to the data are considered, shows a slope of \(0.29\pm 0.07\) and an intercept of \(- 5.7\pm 1.1\). An unweighted regression (red dashed) here produces a slope of \(0.21\pm 0.03\) and an intercept of \(- 7.4\pm 0.5\). Remarkably, these regression curves, which correspond to a power-law dependence, where the slopes determined above correspond to the respective exponent, thus allow a prediction of the model’s inertia based on quantities that are not intrinsic to the model (pre-pandemic average number of flights, cargo, and number of passengers) for further airports without performing a fit.

The results of this section thus highlight the fact that the model parameters can be estimated from real-world quantities with some uncertainty, with no indications that would suggest otherwise, provided the availability of sufficient data to determine the relevant correlations.

4 Parameter variations

In Sects. 3.1 to 3.3, the model introduced in Sect. 2 has been calibrated via fits of the performance curves of the respective systems on the socio-technical spectrum. In this section, we investigate changes of the model dynamics under changes of model parameters, and how the parameters drive the model from one of the regimes discussed in Sect. 2.3 and displayed in Fig. 5 to another (such as, e.g., from damaged to robust behavior). The following discussion is based on the example of the performance of the airport of Leipzig investigated in Sect. 3.2, which has improved in the course of the Covid-19 pandemic. The respective parameter ranges are thereby chosen so as to cover the full range of possible system states, to cover all interesting realistic cases.

In Fig. 15, the final performance \({Q}_{End}\) of the airport of Leipzig is displayed as a function of the damping constant \(D\) and the threshold performance loss \({\varphi }_{R}\) after which recovering forces are activated. The type of system dynamics (improved, robust, damaged, collapsed) according to the scheme of Sect. 2.3 is indicated in the same color code as is used in Fig. 5. The parameter values of the airport of Leipzig, as determined in Sect. 3.2, are indicated by the black pin. Since the system ends up in an improved state, the pin is located in the teal area of the plot. In this state, the maximal performance loss surpasses the threshold for regular operation \({\varphi }_{N}\) such that the regular operation spring is no longer active. Decreasing the damping \(D\) (moving to the right on the \(D\)-axis in Fig. 15) from this state drives the model from the original improved to a damaged state, since here resources for recovery are used up to compensate for the decreased absorption of the shock by the damping. Upon a further decrease of \(D\), lost performance can no longer be recovered such that the model enters the collapsed state.

Fig. 15
figure 15

The final performance \({Q}_{End}\) of the model introduced in Sect. 2 for the parameter values determined at the airport of Leipzig (black pin) is displayed as a function of the damping \(D\) and the threshold performance loss for the activation of recovering forces \({\varphi }_{R}\). Varying these parameters drives the model through all of its possible final states, improved (teal), robust (blue), damaged (orange), and collapsed (red), cf. Sect. 4. The grey contours at \({Q}_{End}=0\) indicate parameter values for which the final performance changes discontinuously

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Increasing the damping \(D\) starting from the parameter values determined for the airport of Leipzig leads to stronger improvement until a small discontinuous fall of the final performance is observed, to a distinct region of improvement in the parameter landscape. Here, the damping becomes strong enough such that the regular operation threshold \({\varphi }_{N}\) is no longer surpassed. In this case, the regular operation spring remains active and pulls the system towards \({Q}_{End}=1\), acting against performance increase once the model’s original performance is exceeded. Since for the parameter values determined for the airport of Leipzig the regular operation spring \({C}_{N}\) is very weak in comparison to the recovery spring \({C}_{R}\), this pull, and thus the concomitant fall in performance, are comparatively small. A further increase of the damping causes the threshold for the activation of recovering forces to no longer be surpassed and the system to end up in a region of robustness.

Increasing the threshold \({\varphi }_{R}\) starting from the parameter values of the airport of Leipzig, the model discontinuously transitions from an improved state to a collapsed state. Here, the threshold for the activation of recovering forces \({\varphi }_{R}\) is no longer reached during the dynamics of the model, while also the regular operation spring is no longer active since the performance loss has surpassed \({\varphi }_{N}\), i.e. \({\varphi }_{N}< {\varphi }_{M}<{\varphi }_{R}\). Decreasing the damping \(D\) starting from this collapsed state drives the model to the regime where recovering forces are activated, \({\varphi }_{M}>{\varphi }_{R}\), leading to a damaged and, then again, collapsed state. This transition from a collapsed to a damaged state and, once again, collapsed state might appear counterintuitive. The cause is that, while in the former case, recovering forces remain inactive, in the latter case, recovering resources are fully depleted. Finally, increasing the damping \(D\) from the first mentioned collapsed state causes the regular operation spring to no longer fail, \({\varphi }_{M}<{\varphi }_{N}\), and the model dynamics becomes robust.

Figure 16 shows the final performance \({Q}_{End}\) of the airport of Leipzig as a function of the regular operation spring constant \({C}_{N}\) and the threshold \({\varphi }_{N}\) for the failure of that spring. Upon increasing \({C}_{N}\) from the parameter values of the airport of Leipzig, \({Q}_{End}\) at first remains largely independent of this parameter. Only in the range of comparably large values of \({C}_{N}\) (note the logarithmic scale) the regular operation spring increasingly absorbs the shock before the spring breaks down, causing an increase of \({Q}_{End}\). At values of \({\varphi }_{N}\) right below the value observed for the airport of Leipzig, and upon a further increase of \({C}_{N}\), there exists a very narrow parameter range in which the regular operation spring breaks down and recovering forces are not yet activated, causing a collapsed system state. Increasing \({C}_{N}\) following this parameter range, the regular operation spring no longer breaks down and the system remains robust.

Fig. 16
figure 16

The final performance \({Q}_{End}\) of the model introduced in Sect. 2 for the parameter values determined at the airport of Leipzig (black pin) is displayed as a function of the regular operation spring constant \({C}_{N}\) and the threshold \({\varphi }_{N}\) for the breaking of that spring. The failure of this normal operation spring is marked by the discontinuity of the teal surfaces, where the parameter values for Leipzig is located in a region where the normal operation spring breaks during system dynamics

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Increasing the threshold \({\varphi }_{N}\) for the breaking of the normal operation spring also leaves \({Q}_{End}\) invariant until a value is reached for which this spring no longer fails. Similar to the case of increasing \(D\) from its original value at the airport of Leipzig in Fig. 15, this causes a small decrease of the final performance \({Q}_{End}\). Increasing \({C}_{N}\) starting from these values of \({\varphi }_{N}\) causes an increase of the performance, until the pull of the regular operation spring towards \({Q}_{End}=1\) becomes stronger and causes a performance drop. After a further increase of \({C}_{N}\), the threshold for the activation of recovering forces is no longer surpassed and the system becomes robust.

In Fig. 17, the dependence of the final performance \({Q}_{End}\) on the threshold for normal operation \({\varphi }_{N}\) and the onset of recovering forces \({\varphi }_{R}\) is displayed. Increasing \({\varphi }_{N}\) from the original parameter values of the airport of Leipzig shows the same transition to slightly decreased improvement as in Fig. 16. Increasing \({\varphi }_{R}\) from this latter improved region (e.g., at \({\varphi }_{N}\sim 0.4{\varphi }_{0}\)) eventually causes the system to end up in a robust state, since here \({\varphi }_{M}<{\varphi }_{N/R}\). Decreasing \({\varphi }_{N}\) from this robust state or increasing \({\varphi }_{R}\) from the original parameter values of the airport of Leipzig causes the system to eventually end up in the region where the normal operation spring fails and recovering forces have not been activated, \({\varphi }_{N}< {\varphi }_{M}<{\varphi }_{R}\), and thus in a collapsed state. Since \({\varphi }_{N}\) and \({\varphi }_{R}\) only mark threshold values, the final performance \({Q}_{End}\) remains constant as a function of these parameters except at those values at which the respective threshold is surpassed during system dynamics.

Fig. 17
figure 17

The final performance \({Q}_{End}\) remains largely constant as a function of the threshold performance loss \({\varphi }_{N}\) at which the regular operation spring breaks and the threshold \({\varphi }_{R}\) at which recovering forces are activated, except when these values are surpassed during the system dynamics. In that case the system, starting from the parameter values of the airport at Leipzig, ends up either with smaller improvement, or in the robust or collapsed states

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The dependence of \({Q}_{End}\) as a function of \({\varphi }_{RE}\) and \({C}_{R}\) is displayed in Fig. 18. Starting from the parameter values of the airport of Leipzig indicated by the black pin, a decrease of the available resources for recovery, which corresponds to decreasing \({\varphi }_{RE}\), causes the system to transition from the originally improved final state over a damaged to a collapsed final state.Footnote 5 The influence of the spring constant \({C}_{R}\) of the recovery spring is marginal in comparison, and causes a broadening of the improved final state for comparatively large values of this parameter.

Fig. 18
figure 18

Lowering the amount of available resources for recovery by lowering \({\varphi }_{RE}\) causes the transition from the originally improved state at the airport of Leipzig (black pin) over a damaged to a collapsed state. The parameter \({C}_{R}\) only has a significant influence on the final state for comparatively large parameter values

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Figure 19 shows the dependence of the final performance \({Q}_{End}\) on the damping constant \(D\) and the recovery spring constant \({C}_{R}\). In analogy to Fig. 15, starting from the black pin that marks the parameter values determined for the airport of Leipzig, decreasing the damping takes the system from the improved final state through damaged to collapsed final states. Increasing the damping from the initial value takes the system through a slightly less improved state once the damping is such that the regular operation spring does no longer break, cf. the discussion with regard to Fig. 15. Finally, further increasing \(D\) causes a robust state when the damping is strong enough such that recovering forces are no longer activated. For large values, the recovery spring constant has little influence on the final state, and only changes the thresholds for the above-described transitions for parameter values that exceed the initial value by several orders of magnitude. For small values of the recovery spring constant, comparable to the regular operation spring, the performance gain is decreased as long as both springs are active.

Fig. 19
figure 19

Changing the values for the damping \(D\) starting from the parameter values determined for the airport of Leipzig (black pin) takes the system through all possible final states, cf. Figure 15. The recovery spring damping constant has little influence of the final state, and only affects the threshold for the transition between possible final system states for parameter values that exceed the initial value by several orders of magnitude, and lowers improvement when comparable to the magnitude of the regular operation spring

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5 Summary and outlook

In order to describe the performance of socio-technical systems, a novel generalized model has been introduced that features a single degree of freedom. While models with a single degree of freedom presented in the literature so far were largely limited to systems that fully recover from the consequences of a disruption, the model presented here also describes permanent improvement or deterioration in the aftermath of a disruption. The model has been applied to three systems that cover a wide range on the socio-technical spectrum, to demonstrate the model’s versatility:

  • The splitting of the European power grid synchronous area of January 8th 2021, for which the ensuing sub-areas where investigated

  • The number of flight departures in the course of the Covid-19 pandemic, where a comparison of different airports was performed

  • The revenue of the German gastronomy sector in the course of the Covid-19 pandemic, which was compared for different lockdown phases.

Fitting the model to these systems determines the model’s parameter values and allows to develop an understanding of how these parameters must be adjusted so as to change the model’s final state after the crisis. Note that this could be a first step in identifying the most efficient system changes towards increased system resilience. One advantage of the presented model is that the parameters required to describe different socio-technical systems can be stated in a generalized, but consistent way, which in the future allows the cross- sectoral comparison of the behavior of these systems’ resilience.

While the proposed model captures the time evolution of the system performance fairly well in most cases, there are exceptions that call for a refinement of the model. A notable example is the oscillation of the frequency deviation in the southeast area of the European power grid separation that has not been captured in its entirety. Modifications are necessary also to capture, e.g., the ripple pattern observed in trade networks in the aftermath of natural disasters (Middelanis et al. 2021). The question arises whether the versatility of the model can be increased accordingly without the introduction of or by the introduction of only a few additional parameters (such as, e.g., additional friction terms), which will be subject of further investigation.

So far, the model is reliant on the availability of crisis data to calibrate the model’s parameters. Thus, the question arises whether these parameters’ values can also be derived from statistical or analytical considerations, or boundary conditions. Hence, on the basis of two examples, it was investigated whether the fit parameters that ensued by fitting the model to the performance of the socio-technical systems under consideration could be connected to underlying properties of these systems. First, in case of the power grid separation, it has been shown that the larger (in terms of power served) of the two areas ensuing from the separation came with a larger inertia than the respective smaller area. Second, for the airport of Leipzig, the inertia resulting from the model fit (normalized by the average number of pre-pandemic flight departures) significantly exceeded the normalized inertia of all other airports under consideration. This was found to be reflected in the cargo-heavy nature of this airport, in that the relative amount of annual cargo (measured in terms of weight, and divided by the number of passengers in the same timeframe) similarly exceeded the relative amount of cargo at the other airports. An increase of the normalized inertia of the model with the relative amount of cargo was indeed observed when this relation was investigated for all airports in Germany above a certain size, that feature the departure of freight flights: here, the data suggested a power-law dependence of the normalized inertia on the relative amount of cargo, and respective fits that were performed resulted in small exponents for these power laws. Remarkably, these fits allow for a prediction of the model’s inertia for a given airport by means of parameters that are not directly related to the model itself (in this case, the average number of pre-pandemic flights, amount of cargo, and number of passengers). This suggests that such predictions might be possible also for the remaining model parameters, provided the availability of sufficient data to determine the respective correlations. Predictions for the remaining parameters would enable the estimation of the performance curves also of airports for which crisis data is currently unavailable.

The impact of parameter changes on the behavior of the model has been studied by means of the example of the airport of Leipzig. Here, the final performance and the respective system state (improved, robust, damaged, collapsed) after the time evolution of the model has been investigated as a function of pairs of the model parameters. Most notably, this final performance undergoes discontinuous transitions whenever the parameter combinations are such that the system dynamics passes one of the threshold values for the breaking of the regular operation spring or the activation of recovering forces. The investigation furthermore explored the parameter ranges that drive the model for the airport of Leipzig from the improved state that resulted from the performance fit into robust, damaged, or collapsed states. While the final state of the model proved quite sensitive to parameters such as the damping and the availability, activation threshold, and total potential of recovery resources, other parameters such as the spring constants required comparatively large relative change to drive the model into a different regime.

The results of the paper can be viewed as the initial steps toward a toolkit that supports informed design decisions, to steer given socio-technical systems that would emerge from a crisis in a damaged or collapsed state towards robustness or a strengthened state. To this end, to identify the relevant control parameters, the connection between the model’s fit parameters and underlying properties of the system must be established in further works. While it has been demonstrated that such a connection can be anticipated for the system’s inertia in certain cases, it seems apparent that similar relations also exist for the other model parameters such as the damping or spring constants. While the inertia significantly affects the entire time evolution of the system and thus all resilience phases described in the introduction, it might be helpful to consider that, e.g., the damping’s role is more centered on the absorb phase of the time evolution or that the recovery spring constant is most important for the recover and adapt phases of the resilience cycle.

Another application of the SDOF model proposed in this work is as a surrogate for more complex physics-based models. Several simulation runs of the latter models might serve to establish a connection between these models’ physical parameters and the fit parameters of the SDOF model.

As a concluding remark, extensions of the model to further scenarios might likely prove insightful. Non-linear spring models might capture serial redundancies, and thereby add this resilience dimension to the model. An extension to coupled degrees of freedom in a Kuramoto approach can be used to describe the interdependence of performances of coupled systems. In this way, e.g., synchronization effects between subsystems can be captured. Furthermore, the performance measure might be subdivided into categories following the TOSE (technical, organization, social, economic) concept (Bruneau et al. 2003), which would involve a model with several degrees of freedom. In such an approach, the recovering forces can be modelled by springs that act in parallel.