## 3 CCT-Based Traffic Flow Model
### CCT-Based Traffic Flow Model under Sudden Fire Disaster  
Two critical issues need to be considered in constructing the regional traffic flow model based on cusp catastrophe theory: (1) the selection of traffic flow parameters and (2) the corresponding relationship between traffic flow parameters and variables in the cusp model of catastrophe theory. (1) Traffic flow parameters selection. When a sudden fire disaster happens, the range of affected traffic flow includes not only the location where fire disaster happened, but also the adjacent road segments and intersections. It is a regional affection to the traffic network. Therefore, the parameters of regional traffic flow are suitable to describe the status and degree of affection and changing for a disaster-affected traffic network. The cusp catastrophe model has three variables, state variables \(x\) and control variables \(y\) and \(z\). Considering the possibility of traffic data collection and analysis under sudden fire disasters, the regional average volume, occupancy and speed of traffic flow are selected to describe the characteristics of traffic flow in the sudden fire disaster-affected area. (2) Relationship between traffic flow parameters and variables in the cusp model of catastrophe theory. To matchup the relationship between traffic flow parameters and variables in the cusp model of catastrophe theory, the most important thing is to determine the traffic flow parameter corresponding to the state variables. In reality, the most intuitive and visible indicator of traffic state is speed, so the regional average speed is chosen as the state variable. Furthermore, from the perspective of traffic flow theory, the catastrophic change of speed may be caused by a slow change of traffic flow occupancy or traffic flow volume. Therefore, it is feasible to select the regional average speed \(\
u\) as the state variable \(x\). Then, the remaining parameters are regional average flow \(q\) and regional average occupancy \(o\), corresponding to the control variables \(y\) and \(z\) in the cusp model of catastrophe theory. To form a CCT-based model with traffic flow parameters, we assume that there are three conversion relations between traffic flow parameters and cusp model variables. We make the hypothesis that:  
\[x=a\cdot v \tag{3}\]  
\[y=b\cdot q \tag{4}\]  
\[z=c\cdot o \tag{5}\]  
where: \(a\), \(b\), \(c\) is the conversion coefficients of regional average speed \(\
u\), volume \(q\) and occupancy \(o\) of traffic flow, respectively. Putting formula (3)-(5) into formula (1), formula (1) can be rewritten as:  
\[E(v,q,o)=a^{4}\cdot v^{4}+b\cdot q\cdot a^{2}\cdot v^{2}+c\cdot o\cdot a\cdot v \tag{6}\]  
Based on cusp catastrophe theory, if the state of traffic flow \(v\) satisfies \(\frac{dE}{dv}=0\), the traffic flow is in equilibrium. The derivative of formula (6) is zero. \[4a^{4}\cdot v^{3}+2b\cdot a^{2}\cdot q\cdot v+c\cdot a\cdot o=0 \tag{7}\]  
where \(\beta=4a^{4}/(c\cdot a)=4a^{3}/c\), \(\gamma=2b\cdot a^{2}/(c\cdot a)=2 ba/c\). Formula (7) can be updated and simplified as:  
\[\beta\cdot v^{3}+\gamma\cdot q\cdot v+o=0 \tag{8}\]  
The parameters \(a,b,c,\beta,\gamma\) can be estimated by using maximum likelihood procedures. According to the catastrophe dynamics method, if the state of traffic flow \(v\) satisfies \(\frac{d^{2}E}{dv^{2}}=0\), the traffic flow is also in equilibrium, but in edge of discontinues changing. The second derivative of formula (6) is also zero. \[3\beta\cdot v^{2}+\gamma\cdot q=0 \tag{9}\]  
Then, catastrophe border can be identified in the singularity set of the equilibrium surface. The solution formula of the equilibrium surface for singularity set is:  
\[\left\{ \begin{array}{c}\beta\cdot v^{3}+\gamma\cdot q\cdot v+o=0\\ 3\beta\cdot v^{2}+\gamma\cdot q=0\end{array} \right. \tag{10}\]  
The singularity set of the equilibrium surface is two creases in the folded part of the equilibrium surface.