activity levels of infected agents drop to the reduced level after several days, and “lockdown" case where the activity levels of all agents drop to the reduced level immediately.

![image](monte_carlo.pdf){width="\linewidth"}

For each case, we simulate 200 system trajectories and compute the average, maximum, and minimum $m_k$ (number of infected agents) versus $k$ from all trajectories. A system trajectory in the “no intervention" case is illustrated in \[fig:monte\_carlo\], where $u=1/M$ for all agents. The $m_k$ trajectories under different cases are shown in \[fig:open\_loop\], where the solid curves illustrate the average $m_k$ and the shaded area corresponds to the range from min $m_k$ to max $m_k$. The results are explained below.

-   Case 0: no intervention.

    All agents keep the normal activity level $u$. The scenarios for $u= 1/M$ and $u= 2/M$ are illustrated in \[fig:open\_loop\]. As expected, a higher activity level for all agents will lead to faster infection. The trajectory of $m_k$ has a $S$ shape, whose growth rate is relatively slow when either the infected population is small or the healthy population is small, and is maximized when $50\%$ agents are infected. It will be shown in the following discussion that (empirical) macroscopic models also generate $S$-curves.

-   Case 