Effects of social distancing and isolation on epidemic spreading modeled via dynamical density functional theory

The COVID-19 pandemic highlighted the critical need for effective strategies to control infectious disease spread. While simple compartmental models like the SIR model offer a basic framework, they lack the ability to explicitly represent crucial non-pharmaceutical interventions such as social distancing and isolation. Individual-based models, while more detailed, are computationally expensive and limit analytical insights. This research addresses this gap by employing a DDFT approach, a coarse-grained field theory commonly used in physics to bridge the gap between microscopic and macroscopic descriptions. DDFT offers a computationally efficient method while maintaining a clear connection to the underlying microscopic dynamics, making it well-suited for modeling complex systems like epidemic spreading. The study aims to develop and analyze a DDFT-based model that explicitly incorporates social distancing and isolation, enabling a more realistic and nuanced understanding of their impact on disease transmission and the potential for pandemic control. The importance of this research lies in providing a powerful tool for predicting the effects of various interventions, which is crucial for informed policy decisions during outbreaks.

Existing mathematical models for disease spreading include simple compartmental models like the SIR model, which can incorporate social restrictions by adjusting transmission rates but cannot model them explicitly. Individual-based models offer more detail but are computationally expensive. The authors highlight the potential of DDFT, a coarse-grained field theory successfully used in soft matter physics, as a more efficient alternative for modeling complex systems. Previous work has applied DFT and dynamical models to social systems, suggesting the suitability of DDFT for epidemiological modeling. The application of DDFT has been successful in various biological contexts such as cancer growth, protein adsorption, ecology, and active matter, indicating its potential value for understanding epidemic spreading.

The researchers developed a SIR-DDFT model by extending the SIR model to incorporate spatial dynamics and social interactions. Individuals are represented as diffusing particles, with social distancing and self-isolation modeled as repulsive interactions between susceptible, infected, and recovered individuals. The dynamics are described by DDFT equations, with reaction terms accounting for disease transmission and recovery. Different mobilities are allowed for each population group (susceptible, infected, recovered). A term is added to account for the death of infected individuals, though this is set to zero in most of the analysis to focus on infection spread. The conserved current is defined, demonstrating that the total number of people is constant. The free energy incorporates the ideal gas term (representing non-interacting individuals) and the excess free energy, representing interactions through social distancing and self-isolation. Gaussian pair potentials are used to model these interactions, offering an effective way to capture the soft repulsive nature of social distancing (between all individuals) and self-isolation (stronger repulsion between infected and others). A mean-field approximation is used to simplify the excess free energy calculation. The resulting model comprises three coupled partial differential equations describing the temporal and spatial evolution of susceptible, infected, and recovered individuals. The linear stability analysis is performed to determine the conditions for a disease outbreak, recovering the standard SIR model's outbreak criterion. The marginal stability hypothesis is used to derive the front propagation speed. The concept of the basic reproduction number (R0) is connected to the model, explaining how it can be modified by the spatial distribution of individuals. Numerical simulations in two dimensions (and supplementary simulations in one dimension) are conducted using an explicit finite-difference scheme with periodic (Figs. 1 and 2) and Dirichlet boundary conditions (Fig. 3), with Gaussian initial conditions representing a localized outbreak (Figs. 1 and 2) or a constant influx (Fig. 3). The parameters of the model (transmission rate, recovery rate, diffusion constants, interaction strengths) are carefully chosen to allow for disease outbreaks in the absence of social distancing and self-isolation.

The linear stability analysis of the SIR-DDFT model recovers the standard outbreak criterion (R0 > 1) but provides additional insights into the effective transmission rate, demonstrating how it is influenced by spatial overlap between susceptible and infected individuals. Numerical simulations reveal that repulsive interactions significantly reduce both the peak number of infections (Imax) and the total number of infections, effectively flattening the infection curve. Social distancing and self-isolation, modeled through repulsive interactions, exhibit a phase transition-like behavior, leading to phase separation where infected individuals cluster together, reducing contact with susceptible individuals. Phase diagrams show clear regions where social restrictions dramatically reduce Imax and increase the final fraction of susceptible individuals (S∞). Time evolution simulations show that, while strong social restrictions flatten the curve, intermediate interaction strengths can lead to a lower Imax but slightly higher overall infection rates. Reducing the mobility of infected persons further inhibits outbreaks. Spatial simulations show radial spreading for the case of initial outbreak at a central point. In the presence of strong interactions, this spreads out into concentric circles then separates into isolated infection spots—a visual representation of individuals self-isolating at home. Simulations with a localized source of infection (airport) demonstrate transient radial spreading followed by a steady state, which is stationary for weak influx but oscillates for stronger influx.

The findings demonstrate that the SIR-DDFT model successfully captures the effects of social distancing and self-isolation on epidemic spread. The results are consistent with observations in real-world pandemics, showing that the model accurately reflects the impact of interventions. The model improves upon simpler SIR models by explicitly incorporating spatial dynamics and interactions, enabling the study of social distancing and disease properties separately. The observed transient phase separation, a novel feature of the model, offers a mechanistic explanation for the effectiveness of self-isolation. The model's success in predicting the shape of infection curves and the impact of interventions suggests its potential as a valuable tool for pandemic preparedness and response. The results also highlight potential applications in other areas involving interacting diffusing particles, such as chemical reaction dynamics.

This paper presents a novel SIR-DDFT model for epidemic spreading, successfully incorporating social distancing and self-isolation. The model yields crucial insights into the interplay between spatial dynamics, social interactions, and disease transmission. The findings demonstrate the efficacy of social restrictions in flattening the infection curve and reducing the overall number of infections. Future work could involve extending the model to incorporate more sophisticated compartmental models, transmission kernels, and larger-scale simulations. Investigating different interaction potentials and free energy approximations could also yield further insights.

The model relies on several simplifying assumptions, such as the use of Gaussian pair potentials for social interactions and the mean-field approximation for the excess free energy. The assumption of homogeneous mixing in spatial simulations could be refined by explicitly modeling urban environments with varied population densities and transportation networks. The model's accuracy depends on the accuracy of the parameters used, which can be challenging to estimate in real-world situations. The impact of individual behaviors and heterogeneity in compliance with social distancing measures is not explicitly considered.