agents is denoted: $$\mathcal{I}_k := \{i: x_{i,k}= 1\}.$$

The state transition probability for the multi-agent system is a mapping $$\mathbb{T}: \mathcal{X}^M \times \mathcal{U}^M \times \mathcal{X}^M \mapsto [0,1].$$

According to the assumptions, an infected agent will always remain infected. Hence the state transition probability for an infected agent $i$ does not depend on other agents’ states or any control. However, the state transition probability for a healthy agent $i$ depends on others. The chance for a healthy agent $i$ to not meet an infected agent $j\in\mathcal{I}_k$ is $1-\min\{u_i,u_j\}$. A healthy agent can stay healthy if and only if it does not meet any infected agent, the probability of which is $\Pi_{j\in\mathcal{I}_k} (1-\min\{u_i,u_j\})$. Then the probability for a healthy agent to be infected is $1-\Pi_{j\in\mathcal{I}_k} (1-\min\{u_i,u_j\})$. From the expression $\Pi_{j\in\mathcal{I}_k} (1-\min\{u_i,u_j\})$, we can infer that: the chance for a healthy agent $i$ to stay health is higher if

-   the agent $i$ limits its own activity by choosing a smaller $u_i$;

-   the number of infected agents is smaller;

-   the infected agents in $\mathcal{I}_k$ limit their activities.

The state transition probability for an agent $i$ is summarized in \[table:dynamics\].

\[table:dynamics\]

#### Example

Consider a four-agent system shown in \[fig:example\]. Only agent 