## 4 The Proposed Approach
### System Design
#### 4.1.2 Foraging Task Model  
For a swarm of \(R\) robots searching for \(N_{g}\) resources that have deadlines \(T_{\xi}\), the task can be formulated as follows:  
**Definition 1**. : _The robots are divided into two types: searching robots (\(R_{s}\)) and foraging robots (\(R_{f}\)). The searching robots are responsible for searching for resources and moving them to the depots. The foraging robots are responsible for delivering resources from the depots to the central collection zone._  
**Definition 2**. : _The velocity of \(R_{f}\) is constant across all experiments, and \(R_{s}\) can always carry one resource at a time._  
**Definition 3**. : _The collection zones are divided into two types: a central collection zone \(N_{c}\) and distributed zones \(N_{d}\) (depots). These zones are distributed uniformly and are known to the robots. The collected resources are moved to the closest zone; then, they are moved to the central collection zone. For each search region, there is a specified area \(A_{r}\) with a collection zone in the center. The number of search regions is \(N_{r}\), so the total arena area is \(A=N_{r}A_{r}\)._  
**Definition 4**. : _A resource is associated with a list of events (\(L_{t}\)) representing its states, where \(L_{t}\in\{S,\,E,\,I,\,R,\,D\}\). Each event is constrained by an online deadline (\(T_{\xi}\)) calculated using the DD_Model to predict future states and their deadlines._ \begin{table}
\begin{tabular}{c c} \hline \hline
**State** & **Deadline** \\ \hline Exposed & \(T_{0}<T<T_{I}\) \\ Infected & \(T_{I}<T<T_{I}\) \\ Recovered & \(T_{I}<T<T_{r}\) \\ Susceptible & \(T_{r}<T<T_{s}\) \\ Dead & \(T>T_{d}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Deadlines in the DD_model. **Definition 5**. : \(T_{\zeta}\) _is the time representing the deadline of each resource, where \(\zeta\in\) [l, t, r, s, d]. Each deadline is updated continuously using the DD_Model, as explained in Equations (1)-(5)._  
**Definition 6**. : _The site density (\(D_{t}\)) is the density of an area \(A_{r}\) based on the number of resources (\(N_{g}\)) in \(A_{r}\) weighted by their current states, where \(N_{g}\) is distributed uniformly in \(A_{r}\). \(D_{t}\) can be measured using (7), where \(\infty_{state(i)}\) is the state weight of a resource (i) in proportion to the states in the DD_Model: {\(\infty_{S}=1\), \(\infty_{E}=2\), \(\infty_{I}=3\), \(\infty_{R}=4\), \(\infty_{D}=5\)}._  
\[D_{t}=\Sigma_{n=1}^{N_{g}}\prod_{i=N_{g}}^{n}\infty_{state(i)} \tag{7}\]  
**Definition 7**. : _The spot intensity (\(H_{t}\)), which is also called the hotspot level, is the predicted maximum energy of a spot in an area \(A_{r}\) based on the average time of resource expiration in that spot predicted by the DD_Model. \(H_{t}\) is computed using (8), where f(\(n\)) is the expected expiration time \(T_{d}\) of the expected expired events in \(A_{r}\) and C(\(d\))\({}_{N_{g}}\) is the number of resources \(N_{g}\) that are predicted to die in \(A_{r}\)._  
\[H_{t}=\frac{\Sigma_{n=1}^{N_{g}}f(n)}{C(d)_{N_{g}}}\text{; where }f(n)=\left\{ \begin{array}{c}T_{d};\text{ if }D\in L_{t}(n)\\ 0\text{ ; else }\end{array} \right. \tag{8}\]  
**Definition 8**. : _The collected resources are replenished after collection or expiration to maintain a consistent density. When a resource is collected, another resource is placed in a location drawn from a uniform random distribution. Moreover, when a resource expires, another susceptible resource is placed in another location with a new constraint based on the DD_Model._  
**Definition 9**. : _The foraging rate (\(F\)) is the number of resources collected in the central collection zone \(N_{c}\) per unit time. The foraging rate in a region is \(F_{d}\), which represents the number of resources transported to the regional collection zone \(N_{d}\)._  
**Definition 10**.