## 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 this work, both are considered to be unknown. Therefore, a priori knowledge \(P_{d}\) and \(P_{f}\) are used to formulate \(Dlp_{k}(t)\)._  
Figure 3: The relationships among the six sub-processes in the BDIIM. Immunity/Immunue-Deficiency of Direct-Infection and Cross-Infection  
According to Rule 2, a greater amount of carried pathogen indicates a worse health situation, so the probability of immune-deficiency of DI of \(I_{k}\) at the \(t\)th time interval is formulated as follows:  
\[RDp_{k}(t)=f_{d}(\ \ V_{k}(t),\ \ \ \ \Delta V_{k}(t-1)|V_{\min},\ \ \ \ V_{\max}\ \ ). \tag{3}\]  
\(I_{k}\) will be infected through CI by \(I_{i}\) if three conditions are satisfied. First, \(I_{i}\) has infectivity. In other words, according to Rule 5, \(I_{i}\) is infected through DI. Second, according to Rule 10, the distance between \(I_{i}\) and \(I_{k}\) is within the infectious distance of \(I_{i}\). Third, according to Rule 10, \(I_{k}\) is out of immune memory. Taking all three conditions into consideration, the probability of immune-deficiency of CI from \(I_{i}\) to \(I_{k}\) in the \(t\)th time interval is formulated as follows:  
\[\text{RC}_{i\to k}(t)= \begin{cases}1&if\ \ (EC_{i\to k}(t)=1)\cap(D(I_{i},I_{k})\leq R_{c})\cap(IMMT_{k}(t)=0)\\ 0&otherwise\end{cases}  \tag{4}\]  
where \(IMMT_{k}(t)\) denotes the immune memory of the \(k\)th individual at the \(t\)th time interval. Its definition will be detailed later. According to Rule 5, the cross-infectivity is defined as  
\[EC_{i\to k}(t)= \begin{cases}1&\phi_{i}(t)=1\\ 0&otherwise\end{cases}  \tag{5}\]  
**Remark 4**. : _It is worth mentioning that RC \(p_{i\to k}(t)\) represents the immunity of an individual to CI. One important condition of CI occurring is immune deficiency; however, this was ignored in our previous work, which leads to the event that some individuals residing outside of the infectious distance should be considered as immune-deficient._  
Pathogen Amount of Direct-Infection and Cross-Infection  
Taking Equations (1) and (3) into consideration, the probability of \(DI\) occurrence is formulated as  
\[P\{\phi_{k}(t)=1\}=DIp_{k}(t)\cdot RDpk(t) \tag{6}\]  
The transmit pathogen amount of \(DI\) is defined as a constant \(\kappa_{d}\), then in the \(t\)th time interval, the pathogen amount obtained through \(DI\) is defined as  
\[Zd_{k}=\phi_{k}(t)\cdot\kappa_{d} \tag{7}\]  
Taking Equations (2) and (4) into consideration, the \(CI\) occurrence probability from the \(i\)th individual to the \(k\)th individual is formulated as  
\[P\{\varphi_{i\to k}(t)=1\}=CIp_{k}(t)\cdot RCp_{i\to k}(t)\cdot EC_{i\to k}(t) \tag{8}\]  
Before defining the transmit pathogen amount of \(CI\), according to Rule 4, we first construct the cross-infection infectivity level (CIIL) as follows:  
\[CIIL_{k}(t)=\sum_{j\in N_{k},EC_{j\to k}=1}\frac{V_{j}(t)/V_{\max}}{D(I_{j}, I_{k})/R_{c}} \tag{9}\]where \(D(I_{i},I_{k})/R_{c}\) and \(V_{j}(t)/V_{max}\) are the normalized distance and normalized pathogen amount, respectively. According to Rule 6, the cross-infection intensity factor (CIIF) is constructed as follows:  
\[CIIF_{k}(t)= \begin{cases}c_{1}&CIIL_{k}(t-2)\leq CIIL_{k}(t-1)\leq CIIL_{k}(t)\\ c_{3}&CIIL_{k}(t-2)>CIIL_{k}(t-1)>CIIL_{k}(t)\\ c_{2}&else\end{cases} . \tag{10}\]  
According to Rules 5 and 10, each individual with infectivity may infect their neighbors with immune deficiency, and the pathogen amount corresponding to the individual's neighbors increases. The pathogen increment increases monotonically with CIIF and the infective individual's carried pathogen amount. The pathogen increment is described as follows:  
\[Z_{k}(t)=\frac{V_{k}(t)-V_{\min}}{V_{\max}-V_{\min}}CIIF_{k}(t). \tag{11}\]  
Taking Equation (11), the released pathogen amount \(Z_{k}(t)\) is proportionate to the carried pathogen amount. If \(V_{k}(t)=V_{max}\), \(Z_{k}(t)\) will be largest; if \(V_{k}(t)=V_{min}\), \(Z_{k}(t)\) will be zero, so CI will not occur.