**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.

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 have access to 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. Using a mobility ABM with this landscape, we incorporate various health statuses to simulate infection dynamics. Most parameters are taken from the literature [2]. For the infection model, we adopt the probability for an agent to become infected from [2], which considers various activities within buildings, making transmission dependent not only on agent coordinates but also on the type of activity, such as home, work, or leisure. Additionally, the model estimates factors like room size and air exchange rate for these different activity types.

Next, we derive a stochastic PDE model with zero-mean noise [1] from the established ABM and further simplify this model to a PDE model. The so-called full-PDE model, which lacks stochasticity, is easier to fit, making it suitable for optimizing the missing parameters that control infection dynamics. For the reduced PDE system, we consider the mean values of factors like room size and air exchange rate, as we do not differentiate between different activity types.

Agents are free to walk around the entire domain, as enlarging the simulation domain of the ABM does not directly affect runtime. These agents follow their real day-to-day trajectories across Germany (see Agents outside of Brandenburg in Figure F). Their locations are continuous and not bound to a grid. However, we use a rectangular grid to determine in which pixels the agents are located by simply mapping their location to the grid indices. This allows us to determine whether they have entered Berlin and, if so, to identify the closest grid point of the triangulated PDE domain. These computations add to the computational time, which are not present in a full-ABM model.

If an agent enters the PDE domain, we can project its continuous location onto the mesh and directly adjust the density of the corresponding compartment at one grid point, adding a new person to the PDE system. This can be interpreted as the “birth” of a whole person in the PDE system (fractional births are generally possible). 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.

Similar to births, a person leaving the PDE domain can be interpreted as the “death” of a whole person. The same adjustments as before are performed, while keeping in mind that the density should not become negative. Additionally, the health status of the departing person must be determined.

One possibility is to consistently track when, where, and how many agents are entering the ABM domain and estimate their realistic health state, trying to find the closest grid point in the PDE domain where this density equivalent of a person may be subtracted from, or even subtract this density over the entire domain. This might not always be possible, as the density might become negative. In this case, one might opt for the next most likely health state, and so on.

The other possibility is to compute for every grid point or boundary point, the number of people that might migrate, and then randomly pick a health state, depending on how these are distributed at that time step. This approach would almost always result in a susceptible person leaving, as the compartments diffuse and advect in the same way, following the dynamics of a deterministic PDE system.

Then, depending on the chosen approach, either the exact trajectory corresponding to the person is selected, or a random trajectory is chosen, which may result in the agent re-entering the PDE domain.

[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.

[2] Sebastian Müller, Michael Balmer, William Charlton, Ricardo Ewert, Andreas Neumann, Christian Rakow, Tilmann Schlenther, and Kai Nagel. A realistic agent-based simulation model for covid-19 based on a traffic simulation and mobile phone data, 11 2020.

**Project Webpages**

**Selected Publications
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- 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.
- K. Maier, M. Weiser, T. Conrad. (2024) “Hybrid PDE-ODE Models for Efficient Simulation of Infection Spread in Epidemiology.” DOI: 10.48550/arXiv.2405.12938. (Under Review)
- S. Müller, S. Paltra, J. Rehmann, K. Nagel, T. O. F. Conrad. (2023) “Explicit Modelling of Antibody Levels for Infectious Disease Simulations in the Context of SARS-CoV-2.” iScience, 26 (9). DOI: 10.1016/j.isci.2023.107554.
- K. Sherratt, A. Srivastava, T. O. F. Conrad, C. Schuette, K. Nagel, Grah et al. (2024) “Characterising information gains and losses when collecting multiple epidemic model outputs.” Epidemics, 47. DOI: 10.1016/j.epidem.2024.100765.
- S. Paltra, T. O. F. Conrad. (2024) “Clinical Effectiveness of Ritonavir-Boosted Nirmatrelvir—A Literature Review.” Adv. Respir. Med. 92(1). DOI: 10.3390/arm92010009.
- P. Dönges, J. Wagner, S. Contreras, E. N. Iftekhar, S. Bauer, S. B. Mohr, J. Dehning, A. C. Valdez, M. Kretzschmar, M. Mäs, K. Nagel, V. Priesemann. (2022) “Interplay between risk perception, behavior, and COVID-19 spread.”
*Front. Phys.*10:842180. DOI: 10.3389/fphy.2022.842180. - K. Nagel, C. Rakow, S. A. Müller. (2021) “Realistic agent-based simulation of infection dynamics and percolation.” Physica A: Statistical Mechanics and its Applications, Volume 584, 126322, ISSN 0378-4371. DOI: 10.1016/j.physa.2021.126322.

**Selected Pictures
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**Selected Pictures
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**Selected Pictures
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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.

F: Distribution of agents across Germany, zoomed into Brandenburg, and the triangular grid over Berlin.

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