EF45 – Multi-Agent Complex Systems

Next, we derived a stochastic PDE model with zero-mean noise [1] from the established ABM and further simplify this model to a PDE model. In contrast to the ABM, implementing the infection dynamics in the PDE model is more difficult, as the ABM’s infection process depends on numerous factors — such as shedding and intake rates, duration of interaction with infected agents, contact intensity, a calibration parameter corresponding to the infection rate in a compartmental model [2], and daily activity change rates — where some of these factors vary over time and across different facility categories. The implementation of activity reductions has the desired effect of reducing infection numbers, resulting, e.g., in a lower infection rate in the PDE system. For this reason, we directly incorporated changes in activity participation into the infection rate. School closures are handled differently, as they typically occur over a clearly defined time interval, leading to a more pronounced impact. Rather than scaling the infection rate or adjusting it to the changed landscape, we divide the simulation time interval into two phases — one without school closures and one with school closures — fitting the infection rates separately for these subintervals. All other transition parameters are assumed to be the same across the ABM and PDE model.

The coupling between the ABM and PDE components is implemented by dynamically exchanging individuals between the two model domains. To ensure well-posedness of the PDE system and avoid artificial fluxes across the boundary, we impose zero Neumann boundary conditions for the PDE contribution of the hybrid model across the entire boundary.

When an agent leaves the ABM domain and enters the PDE domain, the agent is removed from the ABM and their contribution is added to the PDE model. The agent’s health state is read and used to place them into the corresponding compartment of the PDE system. This procedure ensures consistent mass transfer across the model boundary and preserves the total population in the hybrid model. Adjusting the density of each compartment across the entire grid can be interpreted as new initial conditions for the system, which is solved multiple times over the entire simulation period.

To simulate agents transitioning from the PDE domain (Berlin) into the ABM domain (Brandenburg), we precompute the expected number of individuals present in the PDE model for each hour of the day. If the time step corresponds to one hour, this allows us to determine how many individuals should leave the PDE model and be represented as agents in the ABM. For smaller time steps, we interpolate these values accordingly. When an individual enters the ABM domain, we randomly pick a free agent ID, and the corresponding plan from the mobile phone-based event data is activated. This plan defines the agent’s activities and locations for the current day. Occasionally, the agent’s first or final position from that plan might be inside Berlin, i.e. outside of the ABM domain.

Additionally, the health status of the departing person must be determined. Their health status is not sampled stochastically but determined deterministically based on the current distribution of the PDE compartment densities. Instead of drawing random health states, we compute the ratio of individuals to be removed from each compartment, proportional to their current densities. These initial ratios are adjusted so that we end up rounding down to whole individuals. Any remaining individuals are assigned to the susceptible compartment, which dominates the distribution. These ratios are then used to proportionally reduce the corresponding densities at each grid point, ensuring that more individuals are removed where the density is higher. We deliberately avoid fractional individuals, as they could distort the dynamics depending on the implementation, and because individuals in the model represent countable persons, not fractions thereof.