**Project Heads**

*Tim Conrad, Kai Nagel, Christof Schütte
*

**Project Members**

Kristina Maier

**Project Duration**

01.01.2024 – 31.12.2025

**Located at**

ZIB

This project aims to develop a hybrid model that accurately simulates infection spread and the impact of counter-measures. The innovative model couples Agent-Based Models (ABM) with Ordinary Differential Equation (ODE) and Partial Differential

Equation (PDE) models, offering high computational efficiency and precision in modelling large regions.

Equation (PDE) models, offering high computational efficiency and precision in modelling large regions.

In the previous project EF4-13, we coupled PDE and ODE models to simulate infection dynamics in Lombardy, Italy. We conducted additional experiments on a rectangular domain using synthetic populations to test the robustness of the model (see Figures A, B, C), and we selected a more detailed representation of Berlin’s population (see Figures D, E). We decided to maintain a coarse representation of the initial population, similar to our previous experiments in Lombardy, rather than modeling each household individually. This approach proved to be effective. Building on this experience, our next objective is to explore methods for coupling ABMs with PDE models. Ultimately, our goal is to integrate all models to simulate infection dynamics on a larger scale.

Here, we choose Brandenburg as the simulation area for the ABM and Berlin for the PDE model. This decision is based on the fact that 1) Brandenburg has fewer residents than Berlin and 2) Berlin’s population is more evenly distributed. We are provided with trajectories of individuals from Germany who have been in Brandenburg and Berlin at least once – on a weekday, Saturday, and Sunday. From these trajectories, we can derive a landscape to be used in both the ABM and the PDE model. Using grid search, we can determine the diffusion coefficient that best reflects the movement of the agents. Building on this mobility ABM, we incorporate various health statuses to simulate infection dynamics. Next, we deduce a stochastic PDE model with zero-mean noise from the established ABM [1] and further simplify this model to a PDE model. The so-called full-PDE model, lacking stochasticity, is easier to fit, which is why we use it to optimize the missing parameters that control infection dynamics.

Agents are free to walk around the entire domain – including Berlin – as enlarging the simulation domain of the ABM does not directly affect runtime. This approach offers the advantage of integrating agents into the PDE dynamics without losing their health status or tracking their time and location upon leaving the PDE domain. However, the downside of this approach is increased computation time, as we must determine the agents’ locations within the grid of the PDE domain at every time step. Additionally, the infected density of the PDE domain also affects the agents’ health status changes within a given radius.

If an agent enters the PDE domain, we can project its continuous location on the mesh and directly include the number of infected agents into the PDE model.

Integrating over the density of infected individuals results in a real-valued term rather than a whole number. Nonetheless, this term can be incorporated into the adoption rate function of a susceptible agent.

Of course, we still have agent to agent interaction inside the PDE domain.

Similarly, we also take into consideration agents outside the PDE domain which are close to the border of Berlin. There, we only consider half of the ball.

Since we consider a deterministic system and its population density is constant over time, the population does not leave its simulation domain and has no further influence on the agents.

[1] Luzie Helfmann, Nataša Djurdjevac Conrad, Ana Djurdjevac, Stefanie Winkelmann, and Christof Schütte. From interacting agents to density-based modeling with stochastic pdes. Communications in Applied Mathematics and Computational Science, 16(1):1–32, January 2021.

**Project Webpages**

**Selected Publications
**

- Sebastian A. Müller, Michael Balmer, William Charlton, Ricardo Ewert, Andreas Neumann, Christian Rakow, Tilmann Schlenther, and Kai Nagel. Predicting the effects of covid-19 related interventions in urban settings by combining activity-based modelling, agent-based simulation, and mobile phone data. PLoS One, 16(10), 2021. DOI: 10.1371/journal.pone.0259037.
- Hanna Wulkow, T. O. F. Conrad, Natasa Djurdjevac Conrad, Sebastian Alexander Mueller, Kai Nagel, and Ch. Schütte. Prediction of covid-19 spreading and optimal coordination of counter-measures: From microscopic to macroscopic models to pareto fronts. PLoS ONE, 16(4), 2021. DOI: 10.1371/journal.pone.0249676.

**Selected Pictures
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**Selected Pictures
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**Selected Pictures
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**Selected Pictures
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**Selected Pictures
**

A: Number of infectious individuals of full-PDE and hybrid model in rectangular domain for times t ∈ [0,59]. The infectious count decreases with a higher percentage of the ODE domain. This is attributed to the inclusion of the Allee term, which spatially modifies the infection rate in the PDE domain while remaining constant in the ODE domain.

B: Accuracy (mean absolute error) of the full-PDE model and hybrid model in rectangular domain. We can observe an approximately linear growth.

C: Extreme cases of the hybrid model in rectangular domain: the number of infectious individuals is initially equal to the total population number (left) and zero (right). The outcomes for the ODE region appear visually unaffected by the location of the population in the PDE domain, whereas the outcomes for the PDE region vary significantly.

D: Infectious density of PDE model for times t ∈ {1,8,18,60}. Here, the spread can be observed particularly well at the level of home locations.

E: Number of infectious people of full-PDE model (left) and hybrid model (right) in Berlin (simulated and ABM data). The primary reason for the varying number of infectious cases is likely the non-uniform distribution of the population within the PDE domain.

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