world, given a fixed activity level, agents are more likely to meet with close families, friends, colleagues than strangers on the street. To incorporate this non-uniformity into the model, we need to redefine the chance for agent $i$ and agent $j$ to meet each other to be $\beta_{i,j}\min\{u_i, u_j\}$, where $\beta_{i,j}\in[0,1]$ is a coefficient that encodes the proximity between agent $i$ and agent $j$ and will affect the chance for them to meet with each other. For simplicity, we assume that the interaction patterns are uniform in this paper.

3.  Meeting with infected agents will result in immediate infection. To relax this assumption, we may introduce an infection probability to describe how likely it is for a healthy agent to be infected if it meets with an infected agent.

System Dynamic Model
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On day $k$, denote agent $i$’s state and control as $x_{i,k}\in\mathcal{X}$ and $u_{i,k}\in\mathcal{U}$. By definition, the agent state space is $\mathcal{X} = \{0,1\}$ and the agent control space is $\mathcal{U}=[0,1]$. The system state space is denoted $\mathcal{X}^M:= \mathcal{X}\times\cdots \times \mathcal{X}$. The system control space is denoted $\mathcal{U}^M:=\mathcal{U}\times\cdots \times \mathcal{U}$. Define $m_k = \sum_{i} x_{i,k}$ as the number of infected agents at time $k$. The set of infected 