# Cyber Physical Systems Dependability Using CPS-IOT Monitoring
## 3 The CPS Dependability Model
### Epidemic Modelling of the CPS-IoT
Similarly, Table 1 reveals that the average performance in terms of dependability is achieved in Case 2, when the IoT nodes stay longer in the susceptible and attacked states before being removed. Case 1 and Case 3 depict low dependability situations with the worst scenario being Case 3, where the nodes spend less time in both the susceptible and attacked states. Case 2 applies to \begin{table}
\begin{tabular}{c c c c} \hline \hline  & **S-to-A** & **A-to-R** & \\
**Safety** & **Availability** & **Reliability** & **Case** \\ \hline Low dependability & & & \\ Low availability, low reliability & 1 & 0 & 1 \\ Average dependability & & & \\ High availability, low reliability & 0 & 0 & 2 \\ Worse dependability & & & \\ Low availability, high reliability & 1 & 1 & 3 \\ High dependability & & & \\ High availability, high reliability & 0 & 1 & 4 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Safety perspective to CPS dependability. non-sensitive applications with soft real-time constraints that should be deployed with low migration rates from susceptible to attacked status \(a_{i}\) and from attacked to removed state \(b_{i}\). Such applications will require less surveillance in terms of anomaly detection and slower response/actuation by implementing cost aware data collection algorithms such as Routing Protocol for Low-Power and Lossy Networks (RPL) [6]. The analysis of the dependability/safety cases when triggered by transition rates adjustments is presented in Section 5. The comparison between the CTPs in terms of epidemic status levels and their relevance in CPS dependability will also be evaluated in Section 5. In modelling these transitions epidemically, Figure 3b presents a finite state machine of the epidemic model upon which the dependability of the CPS relies upon. It depicts a network that has been partitioned into a number of diffusion sets, with nodes in the same diffusion set assumed to be infectiously similar to each others. The figure reveals the states of the diffusion sets and for each state its associated transitions as well as the actions that trigger such state transitions. Susceptible nodes in the diffusion set \(I_{t}\) may be attacked at the rate \(a_{i}\), while the attacked nodes from \(I_{t}\) are removed at rate \(c_{i}\). Susceptible nodes in the diffusion set \(I_{t}\) may experience high interference, enough to move them directly to the removed status without transiting through the attacked status. On the other hand, removed nodes my cause some of the susceptible or attacked nodes to leave the network because of the destruction of connection links. We consider \(b_{i}\) to be the rate at which susceptible nodes in \(I_{t}\) are removed. Susceptible nodes from diffusion set \(I_{t}\) migrate to diffusion set \(I_{j}\) with rate \(\lambda_{ij}\), and attacked (or infected) nodes in \(I_{t}\) migrate to \(I_{j}\) at a rate of \(\rho_{ij}\). In contrast to the classic SIR epidemic model, our model allows susceptible nodes to move directly into the removed state. This reflects the case of nodes that are removed from the system for reasons other than the infection/attack, such as component failure or malfunction. The difference in equations resulting from these assumptions will be presented as constraints of the CPS monitoring model proposed in Section 4. Using the Euler method, this system of differential equations can be solved to produce three graphs representing the evolution of the populations \(S(t)\), \(A(t)\), and \(R(t)\) over time as presented in Figure 3c. With regards to CPS safety, these cases may be translated into an objective optimisation problem, consisting of maximising or minimising the surface below the curves \(S(t)\), \(A(t)\), and \(R(t)\). This can be expressed by the integral of these curves over time and defined as:  
\[f(t)=\max\int_{0}^{T}S_{t}(t)\pm\int_{0}^{T}A_{t}(t)\mp\int_{0}^{T}R_{t}(t) \tag{1}\]  
where \(T\) is the study period of the epidemic model and \(\pm\) express a maximisation or minimisation of the surface below the epidemic curve in relation to Table 1.