## 2 Establishment of a BDIIM for Node Wake-Up Control
### Modeling of the Biomimetic Infectious-Immune Process
#### 2.2.4 Six Sub-Processes of the Proposed BDIIM Algorithm
In the \(t\)th time interval, all the pathogen amounts obtained through CI are defined as  
\[Zc_{k}(t)=\sum_{i\in N_{k}}\varphi_{i\to k}(t)\cdot Z_{i}(t). \tag{12}\]  
The event that the \(k\)th individual is infected by CI is defined as  
\[\varphi_{k}(t)=\left\{ \begin{array}{ll}1&if\quad\quad\quad\quad Zc_{k}(t)> V_{\max}\\ 0&otherwise\end{array} \right.. \tag{13}\]  
To formulate the immune cell production tendency, the immunity intensity factor (IMMF) is first defined according to Rule 11:  
\[IMMF_{k}(t)= \begin{cases}IMMF_{k}(t)+m_{1}\Delta V_{k}(t)&if\ \ (\phi_{k}(t)=1)\cap(\Delta V_{k}(t)\geq 0)\\ m_{2}IMMF_{k}(t)&if\ \ (\phi_{k}(t)=1)\cap(\Delta V_{k}(t)<0)\\ m_{3}IMMF_{k}(t)&if\ \ (\phi_{k}(t)=0)\cap((RDP_{k}(t)\cup RCp_{i\to k}(t))=1)\\ m_{4}&otherwise\end{cases}  \tag{14}\]  
where \(I\in N_{k}\) and \(m_{1}\)-\(m_{4}\) are immune coefficients, with \(m_{1}<m_{2}<m_{3}\). According to Rule 7, the immune cell amount is given as follows:  
\[M_{k}(t)= \begin{cases}M_{\min}&if\ \ M_{k}(t)\leq M_{\min}\\ IMMF_{k}(t)\cdot M_{k}(t)&if\ \ M_{\min}<M_{k}(t)<min(M_{\max},V_{k}(t))\\ M_{\max}&if\ \ M_{k}(t)\geq min(M_{\max},V_{k}(t))\end{cases} . \tag{15}\]  
Whether an immune cell is produced or not is highly related to the individual's newly accumulated pathogen amount. When the individual agents are in the early phase of the mode of \(\phi k(t)=1\), the number of immune cells is small and the pathogen amount increases rapidly. In the last phase of \(\phi k(t)=1\), if the immune cell number is larger than the pathogen amount, then the corresponding individual will be recovered from infectious disease. According to Rule 9, immune memory can be formulated as  
\[IMMT_{k}(t)= \begin{cases}m_{5}&if\ \ (\phi_{k}(t-1)\cap(\phi_{k}(t)=0)\\ IMMT_{k}(t-1)-1&if\ \ IMMT_{k}(t-1)\
eq 0\\ 0&\ otherwise\end{cases}  \tag{16}\]  
where \(m_{5}\geq 0\) denotes the time length of the immune memory. Pathogen Accumulation under Immunity State  
The pathogen amount changes due to three sources: the newly accumulated pathogen amount due to DI and CI, \(\Delta V_{k}(t)\), the residue of the previously accumulated pathogen amount, \(V_{k}(t-1)\), and the decrease in the pathogen amount due to immunity repair, \(M_{k}(t)\). The immune enable signal is given in Equation (17), which refers to the immune mechanism occurring in the \(k\)th individual if they are infected through CI or DI in the \(t\)th time interval:  
\[\theta_{k}(t)= \begin{cases}1&if\ \ (\phi_{k}(t)=1\cap(\underset{i\in N_{k}}{ \cup}\varrho_{i\to k}(t)=1)\\ 0&\ otherwise\end{cases} . \tag{17}\]  
Hence, the accumulated pathogen amount based on Rule 1 is  
\[V_{k}(t)=\min\{V_{\max},\max\{V_{k}(t-1)+\Delta V_{k}(t)-\theta_{k}(t)\cdot M _{k}(t),V_{\min}\}\}. \tag{18}\]  
With the Boolean variable \(RC_{i\to k}(t)\), the newly accumulated pathogen amount is  
\[\Delta V_{k}(t)=Zd_{k}(t)+Zc_{k}(t). \tag{19}\]  
Then, the newly accumulated pathogen amount \(\Delta V_{k}(t)\) and the accumulated pathogen amount \(V_{k}(t)\) measure the probability of superspreader existence and, as shown in Equations (2) and (3), determine the occurrence probability of CI and the probability of immune-deficiency to DI in the next time interval.