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Schrick-Noah_CG-Ep-Modeling.tex
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@ -3,9 +3,10 @@
\usepackage{graphicx} % Images
\graphicspath{ {./images/} }
\usepackage{float} % Table captions on top
\floatstyle{plaintop}
\restylefloat{table}
\setcounter{secnumdepth}{3}
\usepackage{caption}
\captionsetup[table]{skip=10pt}
\usepackage{ifpdf} % Detect PDF or DVI mode
\usepackage{babel} % Bibliography
@ -76,6 +77,7 @@ A single-strain approach was originally used for analyzing the compliance and vi
Various works have been presented for a multi-strain epidemiology model. Specifically to computer and information systems, the authors of \cite{9593147} present a S$I_1$$I_2$SD model (Susceptible, Infected-with-Strain-1, Infected-with-Strain-2, Susceptible, Deceased) for modeling malware over Internet of Things Wireless Sensor Networks. This model is able to capture two different strains of malware that each have a dependency on the other. However, for compliance graph analysis, there can, and likely will, be a greater number of violations present. As a result, limiting an epidemiology model to two strains is not fully realistic. The authors of \cite{Lou2021} constructed a multi-strain PDE-SIS (Partial Differential Equation Susceptible - Infected - Susceptible) model that resembles the following and is capable of modeling multiple strains:\\
\begin{math}
\footnotesize
\begin{aligned}
{\left\{ \begin{array}{ll}
\partial _tS=d_\mathrm{S}\Delta S +\sum _{i=1}^k\gamma _i(x)I_i -\frac{S\sum _{i=1}^k\beta _i(x)I_i}{S+\sum _{j=1}^k I_j} &{} x\in \Omega , \ t>0,\\
@ -109,34 +111,34 @@ For initial investigation and testing purposes, the selected epidemiology model
\begin{table}[]
\scriptsize
\centering
\caption[Compartment Descriptors for Compliance Graphs]{Compartment Descriptors for Compliance Graphs. Each compartment is contextualized to compliance graphs, with their meanings and brief explanations of how nodes may fit in compartments.}
\begin{tabular}{|c|c|c|}
\hline
\textbf{Compartment} & \textbf{Description} & \textbf{Contextualization to Compliance Graphs} \\ \hline
\textbf{Compartment} & \textbf{Description} & \textbf{\begin{tabular}[c]{@{}c@{}}Contextualization \\ to Compliance Graphs\end{tabular}} \\ \hline
S & Susceptible & All other nodes. \\ \hline
E & Exposed & \begin{tabular}[c]{@{}c@{}}Nodes flagged as at risk of compliance violation \\ by an IDS, timeframe trigger, or other.\end{tabular} \\ \hline
E & Exposed & \begin{tabular}[c]{@{}c@{}}Nodes flagged as at risk of compliance \\ violation by an IDS, timeframe trigger,\\ or other.\end{tabular} \\ \hline
I & Infected & Nodes that are in violation. \\ \hline
R & Recovered & \begin{tabular}[c]{@{}c@{}}Nodes that were infected and were capable\\ of automatic correction e.g., certificate renewal, \\ scheduled maintenance, or other.\end{tabular} \\ \hline
D & Deceased & \begin{tabular}[c]{@{}c@{}}Nodes removed from the compliance graph;\\ Systems that were removed from the network through \\ quarantine, DMZ, legacy removal, or other \\due to excess risk.\end{tabular} \\ \hline
R & Recovered & \begin{tabular}[c]{@{}c@{}}Nodes that were infected and were\\ capable of automatic correction e.g., \\ certificate renewal, scheduled maintenance,\\ or other.\end{tabular} \\ \hline
D & Deceased & \begin{tabular}[c]{@{}c@{}}Nodes removed from the compliance graph;\\ Systems that were removed from the \\ network through quarantine, DMZ, \\ legacy removal, or other due to \\ excess risk.\end{tabular} \\ \hline
\end{tabular}
\caption[Compartment Descriptors for Compliance Graphs]{Compartment Descriptors for Compliance Graphs. Each compartment is contextualized to compliance graphs, with their meanings and brief explanations of how nodes may fit in compartments.}
\label{table:model-context}
\end{table}
\begin{table}[]
\scriptsize
\centering
\caption[Parameter Descriptors for Compliance Graphs]{Parameter Descriptors for Compliance Graphs. Each parameter is contextualized to compliance graphs, with their meanings and brief explanations of how node relationships affect the parameters.}
\begin{tabular}{|c|c|c|}
\hline
\textbf{Parameter} & \textbf{Description} & \textbf{Contextualization to Compliance Graphs} \\ \hline
\textbf{Parameter} & \textbf{Description} & \textbf{\begin{tabular}[c]{@{}c@{}}Contextualization \\ to Compliance Graphs\end{tabular}} \\ \hline
$\beta$ & Infection Rate & \begin{tabular}[c]{@{}c@{}}Probability of nodes falling out\\ of compliance.\end{tabular} \\ \hline
$\delta$ & Incubation Period & \begin{tabular}[c]{@{}c@{}}Once a node is at risk of falling out\\ of compliance, how long it takes to actually\\ violate a mandate or regulation.\end{tabular} \\ \hline
$\delta$ & Incubation Period & \begin{tabular}[c]{@{}c@{}}Once a node is at risk of falling out\\ of compliance, how long it takes to \\ actually violate a mandate or \\ regulation.\end{tabular} \\ \hline
$\gamma_{R}$ & Recovery Rate & \begin{tabular}[c]{@{}c@{}}Probability of a system to correct\\ its violation status.\end{tabular} \\ \hline
$\gamma_{D}$ & Death Rate & \begin{tabular}[c]{@{}c@{}}Probability of removal for a system \\ in violation.\end{tabular} \\ \hline
$\mu$ & Fatality Ratio & \begin{tabular}[c]{@{}c@{}}Natural rate at which any node may \\ be removed from the network.\end{tabular} \\ \hline
$\epsilon$ & Infected Import Rate & \begin{tabular}[c]{@{}c@{}}Systems that are already in violation;\\ systems that do not fall out of compliance,\\ but are already violating a mandate.\end{tabular} \\ \hline
$\epsilon$ & Infected Import Rate & \begin{tabular}[c]{@{}c@{}}Systems that are already in violation;\\ systems that do not fall out of \\compliance, but are already violating \\ a mandate.\end{tabular} \\ \hline
$\omega$ & Waning Immunity Rate & \begin{tabular}[c]{@{}c@{}}After a system recovers, how long\\ for it to be available for violation.\end{tabular} \\ \hline
\end{tabular}
\caption[Parameter Descriptors for Compliance Graphs]{Parameter Descriptors for Compliance Graphs. Each parameter is contextualized to compliance graphs, with their meanings and brief explanations of how node relationships affect the parameters.}
\label{table:model-params}
\end{table}
@ -164,8 +166,9 @@ For initial investigation and testing purposes, the selected epidemiology model
Converting the single-strain model to function as a multi-strain model requires minimal additional modeling efforts. Rather than using a single SEIRDS model for all violations, each violation is instead represented by a SEIRDS model. In an environment where there are \textit{n} violations, there are \textit{n} SEIRDS models. This is represented in Table \ref{table:multi-strain-context}, which adds an indexing term to allow for multiple violations to be represented in the SEIRDS model.
\begin{table}[]
\scriptsize
\small
\centering
\caption[Multi-Strain Variable Descriptors for Compliance Graphs]{Multi-Strain Variable Descriptors for Compliance Graphs. Each variable from the single-strain model can be represented with an indicator of \textit{n} to denote which SEIRDS violation model the data falls into.}
\begin{tabular}{|c|c|c|}
\hline
\textbf{Model Variable} & \textbf{Description} \\ \hline
@ -182,7 +185,6 @@ Converting the single-strain model to function as a multi-strain model requires
$\epsilon_{n}$ & Infected Import Rate for Violation \textit{n} \\ \hline
$\omega_{n}$ & Waning Immunity Rate for Violation \textit{n} \\ \hline
\end{tabular}
\caption[Multi-Strain Variable Descriptors for Compliance Graphs]{Multi-Strain Variable Descriptors for Compliance Graphs. Each variable from the single-strain model can be represented with an indicator of \textit{n} to denote which SEIRDS violation model the data falls into.}
\label{table:multi-strain-context}
\end{table}
@ -204,10 +206,13 @@ This work made use of both Python and R to support the epidemiology modeling. Th
\end{gathered}
\end{equation}
For the example networks presented in Chapter \ref{ch:example-networks}, certain model parameters are statically defined. For instance, in Equation \ref{eq:param-defs} both \(\delta\) and \(\omega\) are set to 1. As Table \ref{table:multi-strain-context} specifies, \(\delta\) is the incubation period, which defines the incubation period, or how many compliance graph steps (edges) are required before a node can be classified as infected. In a compliance graph format, and specifically for the example networks presented in Chapter \ref{ch:example-networks}, there will always be exactly 1-edge delays between the labeling of an infection and the classification of an infection. Since the edges in a compliance graph are used as the transitionary elements that lead to a change in a system, for node $B$ to be classified as infected, there must be at least one edge between node $A$ and node $B$ for the infected classification to occur. This is also the case for the waning immunity rate, \(\omega\). Since there must be at least one edge to transition a node from a recovered state to a susceptible state, the waning immunity rate is also set to a static value of 1.
This work used the compliance graphs generated from the work presented by the authors of \cite{data}. These example graphs are publicly available and serve as foundational test cases for experimental work. Since these example graphs are generated across multiple industries and have unique properties, it allows for a more thorough investigation and analysis of this work. For these example networks, certain model parameters are statically defined. For instance, in Equation \ref{eq:param-defs} both \(\delta\) and \(\omega\) are set to 1. As Table \ref{table:multi-strain-context} specifies, \(\delta\) is the incubation period, which defines the incubation period, or how many compliance graph steps (edges) are required before a node can be classified as infected. In a compliance graph format, and specifically for the example networks presented in the work of \cite{data}, there will always be exactly 1-edge delays between the labeling of an infection and the classification of an infection. Since the edges in a compliance graph are used as the transitionary elements that lead to a change in a system, for node $B$ to be classified as infected, there must be at least one edge between node $A$ and node $B$ for the infected classification to occur. This is also the case for the waning immunity rate, \(\omega\). Since there must be at least one edge to transition a node from a recovered state to a susceptible state, the waning immunity rate is also set to a static value of 1.
\IncMargin{1em}
\begin{algorithm}[htbp] \label{alg:ep-compart}
\begin{algorithm}[htbp]
\caption{SEIRDS Model compartmentalization and Parameter Derivation}
\label{alg:ep-compart}
\SetKwData{Left}{left}
\SetKwData{This}{this}
\SetKwData{Up}{up}
@ -243,19 +248,20 @@ For the example networks presented in Chapter \ref{ch:example-networks}, certain
{do set $violation$[parameters] = Equation \ref{eq:param-defs}}
}
\caption{SEIRDS Model compartmentalization and Parameter Derivation}
\end{algorithm}
In order to better capture the recurring maintenance events, the recovery parameters can be adjusted based on a periodic function. This function can be adjusted per component and/or per mitigation strategy, and is intended to be interchanged with ease. For this work, the recovery parameters were altered to fit a cosine function. Cosine was chosen in order for the recover parameter minimum to be placed at the final known time step (t=0). A sine function with appropriate shifts could also be used, however a cosine function was used for simplicity. A few adjustments are required. To set the period of the cosine function, the default period (2$\pi$) is divided by the recurring maintenance event's time step of occurrence. Since the recovery rate must be a value between 0 and 1, a piecewise function is defined for the magnitude, where $y=max(0,cosx) \forall {x>0}$. The amplitude of the function was defined by the derived parameter from Algorithm \ref{alg:ep-compart}. This is shown in Equation \ref{eq:recovery-param}.
\begin{equation}
\begin{gathered}
\gamma_{r} = \gamma_{r, derived} * \max(0,cosine(\frac{2*\pi*t}{RecurringMaintenanceTimeStep}))
\small
\begin{split}
\gamma_{r} &= \gamma_{r, derived} * \\
&\max(0,cosine(\frac{2*\pi*t}{RecurringMaintenanceTimeStep}))
\label{eq:recovery-param}
\end{gathered}
\end{split}
\end{equation}
For accurately capturing the pre-defined events presented in Chapter \ref{ch:example-networks}, the infection rate parameters are also designed to be dynamic. Due to the degradation of parts, health, or quality of the asset over time, the infection parameter is expected to steadily increase. The pre-defined events can be used as a baseline for increasing the infection rate to support this functionality. For instance, for the automobile maintenance network, the drive belts have a pre-defined event at time step 3 where a replacement or inspection is required. A function can be defined that increases the infection rate of the drive belt violation as the component reaches its replacement or inspection lifespan. When data is available for the lifespan of a given component, an infection rate parameter increase could model the behavior of $\lim_{t \to expectedPartLifeSpan} f(t) = 1 $. For components with pre-defined events, the ``expectedPartLifeSpan" can be obtained by identifying the event time of the part failure, repair, or replacement. For components without a pre-defined event, identifying the ``expectedPartLifeSpan" would require additional data for each component represented in the compliance graph. As previously stated, this work intends to be self-contained and independent of likelihood estimations. Though tire replacement data may be readily available, identifying the likelihood of forcibly decrypting a database (or the point of its occurrence) requires transitional estimation that is outside of this work. Instead, this work made a simplification of $ \lim_{t \to \infty} f(t) = 1 $ when the component lifespan data was unavailable.
For accurately capturing the pre-defined events presented in the compliance graphs of \cite{data}, the infection rate parameters are also designed to be dynamic. Due to the degradation of parts, health, or quality of the asset over time, the infection parameter is expected to steadily increase. The pre-defined events can be used as a baseline for increasing the infection rate to support this functionality. For instance, for the automobile maintenance network, the drive belts have a pre-defined event at time step 3 where a replacement or inspection is required. A function can be defined that increases the infection rate of the drive belt violation as the component reaches its replacement or inspection lifespan. When data is available for the lifespan of a given component, an infection rate parameter increase could model the behavior of $\lim_{t \to expectedPartLifeSpan} f(t) = 1 $. For components with pre-defined events, the ``expectedPartLifeSpan" can be obtained by identifying the event time of the part failure, repair, or replacement. For components without a pre-defined event, identifying the ``expectedPartLifeSpan" would require additional data for each component represented in the compliance graph. As previously stated, this work intends to be self-contained and independent of likelihood estimations. Though tire replacement data may be readily available, identifying the likelihood of forcibly decrypting a database (or the point of its occurrence) requires transitional estimation that is outside of this work. Instead, this work made a simplification of $ \lim_{t \to \infty} f(t) = 1 $ when the component lifespan data was unavailable.
This work intended for $f(t)$ to be fully replaceable. Better cost models can replace the $f(t)$ functions to better fit to the expected changes in infection parameters. Rather than identifying unique $f(t)$ functions for each component, this work implemented two functions. When the expected component lifespan was known, Equation \ref{eq:known_ft} was used as the $f(t)$ function. When the expected component lifespan was not known, Equation \ref{eq:unknown_ft} was used as the $f(t)$ function. Equation \ref{eq:known_ft} was intended to be simplistic. Due to the exponential function, the limit approaches 1 as the component reaches its lifespan. There is a shift in the function for axis alignment, but no other modifications are implemented. This function does have a quick rise toward 1, so future adjustments to its gain may be necessary. For Equation \ref{eq:unknown_ft}, more attention was given to its rise. Since the component has an unknown life span, a gradual increase toward 1 at $t=\infty$ may be unrealistic. Likewise, a rapid gain toward 1 in the first few time steps may also be unrealistic. This function was chosen and tuned to allow for a rise toward a infection rate parameter value of 1 near 30 time steps. Since time steps are variable based on the step size selection in the network generation process, this function can be tuned to better fit specific components or networks. Figure \ref{fig:unknown-infec-rate} displays a plot of this function. After a mitigation or correction scheme is implemented for a component, the infection rate is reset to its original, derived value.
@ -285,7 +291,9 @@ This work intended for $f(t)$ to be fully replaceable. Better cost models can re
After the Python model preparation was called with the Reticulate library \cite{reticulate} (which acts as an interface between R and Python), R code was used to evaluate the problem space using a differential equation solver. The implemented solver was ODE45, provided by the pracma library \cite{pracma}. Equation \ref{eq:seirds} was defined as a function in R, with inputs of model parameters and model compartments. This function was called through ODE45 over a specified range of time, and would produce an output of epidemic curves. Unless otherwise specified, the epidemic curves would be generated starting with time = ``0" (the last known step of the compliance graph), and would evaluate data for the next 5 years. Since this work is approaching the compliance graph through a multi-strain lens, the differential equation solver was applied over each violation. As a result, epidemic curves were produced for each violation that provided insight on the violation trends over time. This process is displayed in Algorithm \ref{alg:ep-process}.
\IncMargin{1em}
\begin{algorithm}[htbp] \label{alg:ep-process}
\begin{algorithm}[htbp]
\caption{Producing Epidemic Curves for all Possible Violations in a Compliance Graph}
\label{alg:ep-process}
\SetKwData{Left}{left}
\SetKwData{This}{this}
\SetKwData{Up}{up}
@ -318,7 +326,6 @@ After the Python model preparation was called with the Reticulate library \cite{
{do plot vioOut;}
}
\caption{Producing Epidemic Curves for all Possible Violations in a Compliance Graph}
\end{algorithm}
\subsection{Results and Analysis} \label{sec:ep-res}
@ -342,7 +349,7 @@ Some Figures also displayed results with no changes in compartment trends. Figur
\centering
\vspace{.2truein} \centerline{}
\caption[Epidemic Curve for the Automobile Maintenance Network's Drive Belt Violation]{Epidemic Curve for the Automobile Maintenance Network's Drive Belt Violation.
This epidemic curve highlights the effectiveness of the routine maintenance for the automobile maintenance network, evident through the regular decreases in the Infected compartments of the SEIRDS epidemiology model. These maintenance events take place every 1 year and 6 months, and successfully correct the vehicle to a state of compliance with respect to the drive belt violations. However, there is a steady increase of violation (the population of I) due to the increased infection rate parameter caused by the pre-defined events presented in Chapter \ref{ch:example-networks}. This Figure also displays the effectiveness at preventing the spread of a drive belt violation. If the Infected population rose to 1.0, then every node in a compliance graph or subgraph would be expected to posses that violation. Since the Infected population remains below 10\%, the mitigation strategy can be considered effective at correcting the drive belt violation before it spreads through the compliance graph or subgraph.}
This epidemic curve highlights the effectiveness of the routine maintenance for the automobile maintenance network, evident through the regular decreases in the Infected compartments of the SEIRDS epidemiology model. These maintenance events take place every 1 year and 6 months, and successfully correct the vehicle to a state of compliance with respect to the drive belt violations. However, there is a steady increase of violation (the population of I) due to the increased infection rate parameter caused by the pre-defined events presented in the compliance graphs of \cite{data}. This Figure also displays the effectiveness at preventing the spread of a drive belt violation. If the Infected population rose to 1.0, then every node in a compliance graph or subgraph would be expected to posses that violation. Since the Infected population remains below 10\%, the mitigation strategy can be considered effective at correcting the drive belt violation before it spreads through the compliance graph or subgraph.}
\label{fig:car-ep-drivebelt}
\end{figure}
@ -393,9 +400,10 @@ In order to validate the epidemiology modeling approach, the following character
Epidemiology modeling successfully provides insight on the predicted trends and behaviors of compliance violations over time. These trends are useful for identifying the utility of any correction schemes, for future mitigation planning, or for determining which periods of time may require additional attention to prevent noncompliance. However, for presenting quantitative metrics to advisory boards, upper level staff, or management, alternative figures may yield better results, especially when requesting increases in funding. One common method for determining a quantitative metric with respect to expected or predicted challenges is risk assessment. Risk assessment allows for further insight regarding potential risks, damages, or losses, and considers any mitigations, likelihood of occurrences, and can provide information on fault-tolerance in the instance of a risk event \cite{RA}. This Section discusses the risk assessment strategies that are implemented for compliance graph analysis in Section \ref{sec:ra-imp}, with results presented in Section \ref{sec:ra-res}.
\subsection{Implementation Details} \label{sec:ra-imp}
A primary motivator for this work is to provide analysis on compliance graphs that is independent of (but compatible with) any external probability or state transition estimations. As Chapters \ref{ch:Intro}, \ref{ch:rel-works}, and \ref{ch:example-networks} discuss, though works exist for determining transitions of nodes and edges through Markov Decision Processes, empirical analysis, or investigations of the National Vulnerability Database (NVD) and Common Vulnerabilities and Exposures (CVE) list, this work aimed to stay as self-contained and self-sufficient as possible. In order to achieve this, this work presents a modification to Annualized Loss Expectancy (ALE) computations. Equation \ref{eq:ALE} displays the traditional approach for computing ALE.
A primary motivator for this work is to provide analysis on compliance graphs that is independent of (but compatible with) any external probability or state transition estimations. Though works exist for determining transitions of nodes and edges through Markov Decision Processes, empirical analysis, or investigations of the National Vulnerability Database (NVD) and Common Vulnerabilities and Exposures (CVE) list, this work aimed to stay as self-contained and self-sufficient as possible. In order to achieve this, this work presents a modification to Annualized Loss Expectancy (ALE) computations. Equation \ref{eq:ALE} displays the traditional approach for computing ALE.
\begin{equation}
\tiny
\begin{gathered}
ALE = Annual Rate of Occurrence * Single Loss Expectancy, \quad where: \\
Single Loss Expectancy = Asset Value * Exposure Factor
@ -423,7 +431,7 @@ Using the modified ALE equation, a process can be defined for obtaining necessar
\end{figure}
\subsection{Results and Analysis} \label{sec:ra-res}
For each example network presented in Chapter \ref{ch:example-networks}, ALE was computed for all possible violations. To display the results, bar graphs were used to highlight the magnitudes of expected losses incurred from an event of noncompliance. This display allows for a simplistic view of violations that are expected to yield unfavorable penalties, as well as those that may yield little to no penalties. The bar graphs were plotted in terms of both monetary and time penalties. Figures \ref{fig:car-ale-mon} and \ref{fig:car-ale-t} present the results for the Automobile Maintenance Network (Section \ref{sec:Automotive}) for its monetary and time expected losses, respectively. Figures \ref{fig:hip-ale-mon} and \ref{fig:hip-ale-t} present the results for the HIPAA Network (Section \ref{sec:Healthcare}) for its monetary and time expected losses, respectively. Figures \ref{fig:osha-ale-mon} and \ref{fig:osha-ale-t} present the results for the OSHA 1910H Network (Section \ref{sec:OSHA}) for its monetary and time expected losses, respectively.
For each example network presented in \cite{data}, ALE was computed for all possible violations. To display the results, bar graphs were used to highlight the magnitudes of expected losses incurred from an event of noncompliance. This display allows for a simplistic view of violations that are expected to yield unfavorable penalties, as well as those that may yield little to no penalties. The bar graphs were plotted in terms of both monetary and time penalties. Figures \ref{fig:car-ale-mon} and \ref{fig:car-ale-t} present the results for the Automobile Maintenance Network for its monetary and time expected losses, respectively. Figures \ref{fig:hip-ale-mon} and \ref{fig:hip-ale-t} present the results for the HIPAA Network for its monetary and time expected losses, respectively. Figures \ref{fig:osha-ale-mon} and \ref{fig:osha-ale-t} present the results for the OSHA 1910H Network for its monetary and time expected losses, respectively.
\begin{figure}[htp]
\includegraphics[width=\linewidth]{"./images/car-ale-mon.png"}
@ -498,7 +506,7 @@ In order to validate ALE, the following characteristics were examined, and test
\section{Future Works}
\section{Conclusions}
This work presented and implemented a SEIRDS epidemiology model used for analyzing the trends (and expected trends) of a compliance graph. This model is able to compartmentalize a compliance graph into five unique and related sets of populations. This model also functions with a variety of tunable, customizable parameters that assist in the analysis of the spread of violations in an environment. These parameters are intended to be used alongside any user-provided data, such as the maintenance or replacement schedules or the expected lifespans of components. Each example compliance graph presented in Chapter \ref{ch:example-networks} was analyzed using this model. The results were validated as part of the validation process shown in Section \ref{sec:ep-valid}. The results presented in Section \ref{sec:ep-res} display notable visualizations about the analysis process. The methodology of this work is able to uncover valuable information regarding the compliance standing of an environment, such as the effectiveness of any mitigation, correction, or prevention strategies, and the exposure to and spread of violations. Additionally, this work allows for the testing and experimentation of mitigation strategies to provide quantifiable results as to the successes or failures of any alterations, removals, or additions toward maintaining compliance. This approach is shown to be successful in all three unique example compliance graphs, and boasts the ability to be modified or fine-tuned given additional data inputs.
This work presented and implemented a SEIRDS epidemiology model used for analyzing the trends (and expected trends) of a compliance graph. This model is able to compartmentalize a compliance graph into five unique and related sets of populations. This model also functions with a variety of tunable, customizable parameters that assist in the analysis of the spread of violations in an environment. These parameters are intended to be used alongside any user-provided data, such as the maintenance or replacement schedules or the expected lifespans of components. Each example compliance graph presented in \cite{data} was analyzed using this model. The results were validated as part of the validation process shown in Section \ref{sec:ep-valid}. The results presented in Section \ref{sec:ep-res} display notable visualizations about the analysis process. The methodology of this work is able to uncover valuable information regarding the compliance standing of an environment, such as the effectiveness of any mitigation, correction, or prevention strategies, and the exposure to and spread of violations. Additionally, this work allows for the testing and experimentation of mitigation strategies to provide quantifiable results as to the successes or failures of any alterations, removals, or additions toward maintaining compliance. This approach is shown to be successful in all three unique example compliance graphs, and boasts the ability to be modified or fine-tuned given additional data inputs.
In addition, this work presented an approach for risk assessment through a modified Annualized Loss Expectancy (ALE) computation. This technique allows for analysis on a complex environment without the need for estimation regarding rate of occurrence that may otherwise be difficult or impossible to obtain. The results as shown in Section \ref{sec:ra-res} were presented in the form of bar graphs that display the expected losses for each resource across all examined violations. The results were validated as part of the validation process shown in Section \ref{sec:ale-valid}, and demonstrate that this approach is functional even when no noncompliance instances are observed.

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@ -28,8 +28,8 @@
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