Since the epidemic model is monotone, every agent will eventually be infected as long as the probability to meet infected agents does not drop to zero. Moreover, we have not discussed decision making by individual agents yet. The activity levels are just predefined in the simulation.

#### Remark

The model we introduced is microscopic, in the sense that interactions among individual agents are considered. The simulated open-loop trajectories are indeed similar to those from a macroscopic model. Since only susceptible and infected populations are considered in the proposed microscopic model, we then compare it with the macroscopic Susceptible-Infected (SI) model. Define the state $s\in[0,1]$ as the fraction of infected population. The growth of the infected population is proportional to the susceptible population and the infected population. Suppose the infection coefficient is $\beta$, the system dynamics in the SI model follow: $$\dot s = \beta s (1-s).$$

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

We simulate the system trajectory under different infection coefficients as shown in \[fig:SI\]. The trajectories also have S shapes, similar to the ones in the microscopic model. However, since this macroscopic SI model is deterministic, there is no “uncertainty" range as shown in the microscopic model. The infection coefficient $\beta$ depends on the 