Hybrid metapopulation agent-based epidemiological models for efficient insight on the individual scale: A contribution to green computing

Abstract

Emerging infectious diseases and climate change are two of the major challenges in 21st century. Although over the past decades, highly-resolved mathematical models have contributed in understanding dynamics of infectious diseases and are of great aid when it comes to finding suitable intervention measures, they may need substantial computational effort and produce significant CO₂ emissions. Two popular modeling approaches for mitigating infectious disease dynamics are agent-based and population-based models. Agent-based models (ABMs) offer a microscopic view and are thus able to capture heterogeneous human contact behavior and mobility patterns. However, insights on individual-level dynamics come with high computational effort that scales with the number of agents. On the other hand, population-based models (PBMs) using e.g. ordinary differential equations (ODEs) are computationally efficient even for large populations due to their complexity being independent of the population size. Yet, population-based models are restricted in their granularity as they assume a (to some extent) homogeneous and well-mixed population. To manage the trade-off between computational complexity and level of detail, we propose spatial- and temporal-hybrid models that use ABMs only in an area or time frame of interest. To account for relevant influences to disease dynamics, e.g., from outside, due to commuting activities, we use population-based models, only adding moderate computational costs. Our hybridization approach demonstrates significant reduction in computational effort by up to 98% - without losing the required depth in information in the focus frame. The hybrid models used in our numerical simulations are based on two recently proposed models, however, any suitable combination of ABM and PBM could be used, too. Concluding, hybrid epidemiological models can provide insights on the individual scale where necessary, using aggregated models where possible, thereby making a contribution to green computing.

Keywords: Agent-based modeling; Computational efficiency; Energy reduction; Hybrid modeling; Infectious disease dynamics; Metapopulation model.

Fig. 1

Flow chart of the infection state adoption model. The infection states are Susceptible (S), Exposed (E), Carrier (C), Infected (I), Recovered (R) and Dead (D). The dotted lines shows the influences for the second-order adoption.

Fig. 2

Design sketch of a spatial-hybrid model (a) and a temporal-hybrid model (b). Large gray disks represents (sub)regions, with possible population exchanges indicated by arrows. The population is represented by small disks for agents and bar charts for ODE compartments. Red indicates infected agents or population shares, gray indicates noninfected agents or population shares. The exchange in (a) only moves small parts of a local population to another region, the exchange in (b) moves the entire population to the other model.

Fig. 3

Quadwell potentialEq. (11)used for the diffusion process in the ABM. The four metaregions are separated by the axes x = 0 and y = 0. The figure shows an (a) isometric view and (b) a contour plot of the potential.

Fig. 4

Position distribution for a simulation of800agents for50days in the quadwell scenario for noise termσ = 0.55. Agents are initialized with positions having 0.3 distance from the axes x = 0, ±2 and y = 0, ±2.

Fig. 5

Spatial hybridization for the quadwell potential: Focus region, sum of all regions and Region 2. Number of infectious agents (compartments C and I) for (a) the focus region Ω₁ (b) the sum of all regions and (c) Region 2. The figures show the mean outcomes in solid lines with a partially transparent face between the p25 and p75 percentiles from 500 runs. Additionally, MAPE, MAE, and MSE between the ABM mean and the mean of PDMM and spatial-hybrid model are displayed.

Fig. 6

Log-scaled runtime (in seconds) for ABM, PDMM and spatial-hybrid model. Shown is the mean runtime of 56 runs (one compute node) for (a) the quadwell example with the setup according to Table D.8, Table D.9 and (b) the Munich example with the setup according to Table D.8, Table D.10.

Fig. 7

Runtime distribution for ABM, PDMM and spatial-hybrid model for a varying proportion of initially infected and varying values ofρ⁽²⁾. Shown are the runtimes of 112 runs for (a) five different values for the proportion of initially infected with the corresponding proportion distributed equally to compartments E, C and I and (b) seven different values for the transmission rate in Ω₂. In (a) ρ⁽²⁾ = 0.3 for all runs and in (b) 1% of the population (0.2% Exposed, 0.3% Carrier and 0.5% Infected) is initially infected in all runs.

Fig. 8

Potential defined for the ABM in and around Munich. Black areas have value zero and white parts value one. We obtain the potential F by discretizing the map, resulting in a matrix P ∈ [0,1]^p×p and then interpolating between matrix entries.

Fig. 9

Daily commuting from and to Munich City for different surrounding counties. 5% and 95% percentiles of daily transitions in simulations shown as colored area together with median as solid line. The static register data from Federal Agency of Work, scaled to number of simulated agents, is shown as dotted lines.

Fig. 10

Spatial hybridization for the Munich potential: Focus region and sum of all regions. Number of infectious agents (compartments C and I) for (a) Munich City (focus region) and (b) the sum of all regions. The figures show the mean outcomes in solid lines with a partially transparent face between the p25 and p75 percentiles from 500 runs. If possible, MAPE, MAE, and MSE between the ABM mean and the mean of PDMM and spatial-hybrid model are displayed. Since the number of infected agents in the focus region is zero at the beginning, the MAPE cannot be calculated for the left figure.

Fig. 11

Single well potential Eq. (16) used for the diffusion process in the ABM of the temporal-hybrid model.

Fig. 12

Temporal hybridization for the single well example. Shown is the sum of compartments E, C and I for (a) all runs, (b) extinction runs with virus extinction, and (c) survival runs without virus extinction for ABM, PDMM and temporal-hybrid models with s = 2 and s = 5. The figures show the mean outcomes in solid lines with a partially transparent face between the p25 and p75 percentiles from 10,000 runs with 10,000 agents.