$1$ is infected. And the agents choose the following activity levels: $u_1 = 0.1, u_2 = 0.2, u_3 = 0.3, u_4 = 0.4$. Then the chance $p_{i,j}$ for agents $i$ and $j$ to meet with each other is $p_{1,2} = p_{1,3} = p_{1,4} = 0.1$, $p_{2,3} = p_{2,4} = 0.2$, and $p_{3,4} = 0.3$. Note that $p_{i,j} = p_{j,i}$. The chance for agents $2$, $3$, and $4$ to stay healthy is $0.9$, although they have different activity levels.

![image](example_4_agent.pdf){width="3.5cm"}

Case Study
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Before we start to derive the optimal strategies for individual agents and analyze the closed-loop multi-agent system, we first characterize the (open-loop) multi-agent system dynamics by Monte Carlo simulation according to the state transition probability in \[table:dynamics\].

Suppose we have $M=1000$ agents. At the beginning, only agent $1$ is infected. We consider two levels of activities: normal activity level $u$ and reduced activity level $u^*$. The two activity levels are assigned to different agents following different strategies as described below. In particular, we consider “no intervention" case where all agents continue to follow the normal activity level, “immediate isolation" case where the activity levels of infected agents immediately drop to the reduced level, “delayed isolation" case where the 