## 3 Mathematical Description of the Medical Rescue Operations after Mass Casually Incident
### Quantities Characterizing Medical Rescue Operations of DES during MCI  
A random variable \(W_{1,g}\) is assumed to be the probability of death of the \(l\)-th injured person, which was qualified to the \(g\)-th medical treatment priority group. \(W_{1,g}\) takes values from the interval \([0;1]\), where value one means the death of the injured person, zero--otherwise.  
The travel time of the \(r\)-th mode of transportation between the current place \(y\) and the place of the event \(z_{1}\) is described by the following formula:  
\[t^{track}(y,z_{1},r)=\frac{d_{z_{1}}(r,y)}{V_{r}}\, \tag{33}\]  
where \(d_{z_{1}}(r,y)\) is the distance which the \(r\)-th mode of MRT has to travel from place \(y\) to the MCI scene \(z_{1}\) according to Equation (9). Here \(y\) means the actual location of the \(s\)-th HED, i.e., \(h_{z,1}\) according to Equation (2) or the location of the \(b\)-th MRTB, i.e., \(a_{b,1}\) according to \((6),V_{r}\) is the average speed of the \(r\)-th mode of MRT.  
The time measured from the moment the incident occurred to the moment of arrival of the \(r\)-the mode of MRT from the \(b\)-th MRTB to the \(s\)-th HED is described by the following equation:  
\[\tau(b,s,r)=t^{z}+t^{track}(a_{b,1},z_{1},r)+t^{travel}(h_{s,1},z_{1},r), \tag{34}\]  
where \(t^{z}\) is the time elapsed from the moment of the incident to the moment of notifying the DES about the incident according to Equation (9) and \(t^{travel}\) is the travel time of the MRT, according to Equation (33).  
The expected value of the death in the MCI depends on the severity of the traumatic state and, in most cases, may be characterized by the logistic function [16]. Let \(E(W_{1,g}/\tau)\) be the expected value of random variable \(W_{1,g}\) after time \(\tau\) has elapsed:  
\[E(W_{1,g}/\tau)=\frac{1}{1+e^{-k_{g}(\tau-v_{g})}}, \tag{35}\]  
where \(k_{g}v_{g}\) are the parameters of the logistic function depending on the \(g\)-th medical treatment priority group, where the \(k_{g}\) value is responsible for the exponential growth of the function vertically, and expression \(\left(\tau-v_{g}\right)\) specifies the horizontal translation of the function plot.  
The logistic function of the form of the Equation (35) takes values from zero (for which the expected value of the death is minimal) to one (for which the expected value is maximal and means a certain death).  
Injuries with a state of severity over three (\(n\geq 3\)) are a direct threat to life.  
Survival of people injured in the MCI additionally depends on the capabilities of HEDs, which will take care of these people.  
Let \(\mathbf{S}_{l}^{tranna}\) be a set of HEDs, which can effectively treat the \(l\)-th injured person because of his/her worst injuries (\(n\geq 3\)):  
\[\mathbf{S}_{l}^{tranna}=\left\{s:\mathbf{U}_{l,n}^{inc}\subset h_{s,3}\text{ for }n\geq 3\text{ and }s\in S\right\},\ l\in\mathbf{L}. \tag{36}\]  
In regard to the above, the expected value of the \(l\)-th injured person's death depending on the moment of arrival of the \(r\)-th mode of MRT from the \(b\)-th MRTB to the \(s\)-th HED is yielded as:  
\[f^{death}(l,b,s,r)=\left\{\begin{array}{ll}E\big{(}W_{1,g}/\tau(b,s,r)\big{)},&l\in\mathbf{L}_{g},g\in\mathbf{G}&\text{if }s\in\ \mathbf{S}_{l}^{tranna}\\ 1,&\text{otherwise},\end{array}\right., \tag{37}\]  
where \(f^{death}(l,b,s,r)=1\) means that the \(l\)-th injured person will not survive if he/she will be transported by the \(r\)-th mode of MRT from the \(b\)-th MRTB to the \(s\)-th HED.  
Description of other expected values (e.g., the expected value of disabilities or the expected value of the cost of emergency medical care), which determine the effectiveness of the DES during the rescue undertaking of the MCI, is neglected because they are similar to the expected value of death.