agents’ choices of activity levels. However, there is not an explicit relationship yet. It is better to directly use the microscopic model to analyze the consequences of individual agents’ choices.

Distributed Optimal Control
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This section tries to answer the following question: in the microscopic multi-agent epidemic model, what is the best control strategy for individual agents? To answer that, we need to first specify the knowledge and observation models as well as the cost (reward) functions for individual agents. Then we will derive the optimal choices of agents in a distributed manner. The resulting system dynamics correspond to a Nash Equilibrium of the system.

Knowledge and Observation Model
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A knowledge and observation model for agent $i$ includes two aspects: what does agent $i$ know about itself, and what does agent $i$ know about others? The knowledge about any agent $j$ includes the dynamic function of agent $j$ and the cost function of agent $j$. The observation corresponds to run-time measurements, i.e., the observation of any agent $j$ includes the run-time state $x_{j,k}$ and the run-time control $u_{j,k}$. In the following discussion, regarding the knowledge and observation model, we make the following assumptions:

-   An agent knows its own 