## 4 The Proposed Approach
### System Design
#### 4.1.1 Dynamic Deadline Model  
In the past few months, due to the global pandemic of the novel coronavirus disease 2019 [16], epidemiological modeling has been evolving quickly to enable researchers to study virus transmission and model it as accurately as possible. The way the virus behaves and dynamically affects different people is very interesting. Thus, different aspects of this epidemic model can be used to generalize the adaptation and change over time to construct a mathematical model for swarm foraging to predict unknown dynamic deadlines. To understand how the proposed mathematical model is adapted from epidemiological modeling, it is important to first identify similarities between the two concepts. In swarm foraging, robots move resources into collection zones before they expire. When the deadlines are dynamic, the robot must know its transmission rate to predict the likelihood of deadline expiration in a certain region. Epidemiological modeling can be utilized to predict such transmission, where predicting possible exposure in a population to help victims before they die is an analogy for predicting deadline expiration in an area to move resources into collection zones before they expire. Thus, infected resources are an analogy for constrained resources. As shown in Table 1, ED_Foraging performs its task in a 2D arena (\(A\)). The size of this arena represents the epidemic size in epidemiological modeling. Multiple robots (\(R\)) are used to collaboratively accomplish the task, which involves moving resources from one place to another. The resources should be moved to a depot collection zone (\(N_{d}\)) temporarily until they are moved to the central collection zone (\(N_{c}\)). The resources (\(N_{g}\)) in ED_Foraging represent the individuals in an epidemic population. Moreover, the deadlines are represented by \(T_{\xi}\), where \(\xi\) represents the resource state. Hence, the infection state is mapped to a certain time constraint that represents the deadlines and can be updated online using the proposed epidemic model. Note that each infection state is taken as a deadline, and the death state symbolizes deadline expiration. The transmission of an infection throughout the population can be used to update the deadlines dynamically and predict the likelihood of epidemic recurrence to assign the robots to the most critical regions. Inspired by the SEIRD model [12], a dynamic deadline model called the DD_Model is proposed in this paper to model the dynamics of the time constraints. The DD_Model is used to predict the state of each resource and update their deadlines online during the task. In particular, there are six states that a resource can be in (transmission, susceptible, exposed, infected, recovered, and dead). The following parameters are used in the DD_Model:  
* \(\alpha\) corresponds to the incubation rate;
* \(\beta\) corresponds to the exposure rate;
* \(\gamma\) corresponds to the recovery rate;
* \(\eta\) corresponds to the mortality rate; and
* \(\varepsilon\) corresponds to the duration of immunity for a resource after moving to a recovery state. Each resource has its own deadline, which is computed with respect to the current time and state. A resource state is governed by one of the deadlines below, where \(T\) is the current time, \(T_{0}\) is the initial time before any state starts, and \(R_{exp}\) is a uniform random exponential function with a value that ranges from zero to the rate parameter. Incubation period \(T_{I}\)--the average time for a resource to enter the infected state after the exposed state. \[T_{I}=T_{0}+R_{exp}(\alpha)\,^{*}\,T \tag{1}\]  
Transmission time \(T_{I}\)--the average time for a resource to contact other resources, during which it can infect its neighbors, before the recovered state. \[T_{t}=T_{I}+R_{exp}(\beta)\,^{*}\,T \tag{2}\]  
Recovery time \(T_{r}\)--the time needed to reach the recovered state after entering the infected state. \[T_{r}=T_{I}+R_{exp}(\gamma)\,^{*}\,T \tag{3}\]