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

The paper addresses how to explicitly model the effects of social distancing and isolation on epidemic spread, motivated by COVID-19 where many infections are undetected and nonpharmaceutical interventions dominate. Traditional compartmental SIR models can only represent interventions implicitly via adjusted transmission rates and ignore spatial heterogeneity; individual-based models are detailed but computationally expensive and offer limited analytical insight. The authors propose bridging this gap using a coarse-grained field theory, dynamical density functional theory (DDFT), to capture interactions (e.g., distancing and self-isolation) and spatial dynamics efficiently and with a clearer link to microscopic behavior. The goal is to predict how social interactions and isolation alter infection dynamics, identify conditions under which spread is suppressed, and provide mechanistic insight (e.g., transient phase separation) relevant to pandemic control.

• Compartmental models (e.g., SIR) effectively adjust transmission rates to reflect interventions but cannot model distancing and isolation explicitly or spatial heterogeneity.
• Reaction-diffusion extensions of SIR incorporate spatial spread assuming local transmission, with some phenomenological cross-diffusion to mimic avoidance; however, they lack interaction effects like crowding and general distancing among healthy individuals.
• Individual-based models capture contact networks and detailed movement but are computationally costly and less amenable to analysis.
• DDFT from soft matter physics generalizes diffusion to include interactions via a free-energy functional and has shown strong agreement with microscopic simulations across biological and active matter contexts. Gaussian core models (GCM) and mean-field approximations are well-established for soft, penetrable interactions, making them suitable analogs for social distancing and self-isolation.
• Prior work has analyzed front propagation, spatial pattern formation, and global spread using reaction-diffusion SIR and network-based effective distances; the present study extends these by incorporating explicit inter-person repulsion within a DDFT framework.

Model formulation:

• Population is represented by three interacting density fields: susceptible S(r,t), infected I(r,t), and recovered R(r,t), undergoing diffusion and reactions.
• Core DDFT-based SIR equations (with optional death term m, set to 0 in main results): ∂t S = ΓS ∇·(S ∇ δF/δS) − c S I ∂t I = ΓI ∇·(I ∇ δF/δI) + c S I − ω I − m I ∂t R = ΓR ∇·(R ∇ δF/δR) + ω I
• Free energy F = Fid + Fexc + Fext. Fid is the ideal gas term controlling basic diffusion (DΦ = ΓΦ β−1), Fext encodes external restrictions (e.g., travel bans) as potentials, and Fexc captures interactions.
• Interactions model social measures: • Social distancing: repulsion among healthy individuals. • Self-isolation: stronger repulsion between infected and others. Implemented via Gaussian pair potentials with mean-field excess free energy, yielding convolution kernels: Ksd(r) = exp(−σsd r²), K(r) = exp(−σ r²), with strengths Csd, Csi ≤ 0 (repulsive) and ranges σsd, σ.
• Specific SIR-DDFT form (mean-field, Gaussian kernels) leads to diffusion terms plus interaction-driven fluxes via spatial convolutions, and reaction terms cSI and ωI.

Analysis:

• Linear stability analysis about homogeneous disease-free states (I=0) derives dispersion relations and outbreak criterion: instability when c Shom > ω, consistent with SIR and R0 > 1. Two additional eigenvalues capture interaction-driven instabilities.
• Front propagation speed obtained via marginal stability hypothesis: v = 2√(D (c Shom − ω)).
• Effective transmission rate at large scales emerges from spatial overlap: ceff(t) = ∫ dr εS(r,t) εI(r,t) with ε fields normalized by totals, showing interventions can reduce overlap and thus ceff independently of intrinsic c.

Numerical simulations:

• 2D domains of size L=10. Periodic boundary conditions for outbreak-after-event scenarios; circular domain with Dirichlet boundaries for an “airport” source scenario (outflux at boundaries to prevent overpopulation).
• Discretization: explicit finite differences with spatial steps Δx = 0.05 (phase diagrams), 0.0125 (time evolutions and spatial patterns, incl. airport), 0.02 (temperature dependence); adaptive time stepping.
• Parameters (unless varied): c=1, ω=0.1; Γ=1; DS=DR=0.01; σ=100; interactions Csd, Csi scanned (repulsive). m=0 in main results. For mobility reduction of infected, variation of βi (rescaled inverse temperature) to alter Di while keeping Γi fixed.
• Initial conditions: • Super-spreading scenario: Gaussian S(x,y,0) with amplitude S̄/5 and variance L²/50 centered at (L/2,L/2); I(x,y,0)=0.001 S(x,y,0); R=0. Mean overall density ≈ 0.35. • Airport scenario: homogeneous S=π/5, I=0, R=0 with a local Gaussian source in I of amplitude Isource ∈ (0.05, 0.2), variance L²/1000.
• Outputs: phase diagrams in (Csi, Csd) for peak infected fraction Imax,n and final susceptible fraction S∞,n; time series S(t), I(t), R(t); spatial fields I(x,y,t); effects of βi/βSR on outcomes.

Code and data:

• Source data and code are available at Zenodo (DOIs provided).

• Phase suppression with interactions: Phase diagrams in (Csi, Csd) show a clear boundary delineating a regime (large repulsion and higher Csi/Csd ratios) where epidemic spread is significantly suppressed, with low peak infection Imax,n and high final susceptible S∞,n.
• Flattening the curve: Repulsive interactions (social distancing + self-isolation) substantially reduce I(t) peak height and extend pandemic duration. • No interactions (Csi=Csd=0): Imax ≈ 0.49; pandemic ends at t ≈ 58; final recovered fraction ≈ 0.95. • Strong interactions (Csi = 3 Csd = −30): Imax ≈ 0.18; final recovered fraction decreases to ≈ 0.85; pandemic lasts much longer (t ≈ 10^6 reported). • Intermediate interactions (Csi = 2 Csd = −20): Imax ≈ 0.4; end at t ≈ 64; final recovered fraction increases to ≈ 0.97 (i.e., S∞,n slightly lower), illustrating that reduced peak can coincide with slightly larger total infections due to time-varying ceff.
• Mechanisms reducing ceff: Interactions lower infections via (i) demixing that isolates infected from susceptible, reducing spatial overlap εS·εI; (ii) spreading healthy individuals over a larger area, lowering densities and contacts even without full demixing.
• Transient phase separation: For strong interactions (large Csi and ratio Csi/Csd), infection spreads in concentric rings that fragment into isolated spots. This transient demixing reduces S–I overlap, increasing S∞,n, decreasing Imax,n, and lengthening duration. Physically corresponds to infected individuals self-isolating at home.
• Temperature (mobility) effects: Lowering infected individuals’ temperature (higher βi; reduced Di) markedly reduces Imax,n and increases S∞,n; strong interactions case (Csi = 3 Csd = −30) shows monotonic improvement as βi/βSR increases.
• Front dynamics: Initial radial wavefront propagation observed, consistent with reaction-diffusion SIR; analytical front speed v = 2√(D (c Shom − ω)).
• Airport (local source) scenario: With continuous influx, transient rings occur; the system reaches a steady state. For weak influx, stationary steady state; for stronger influx, a periodic oscillatory pattern with period τ ≈ 16 emerges.

The DDFT-augmented SIR model explicitly incorporates social distancing and self-isolation through repulsive interactions, enabling independent control of disease properties (c, ω) and social measures (Csd, Csi, βi). Analytical results recover classical outbreak and front-speed criteria, linking to R0 and highlighting that ceff depends on the spatial overlap of susceptible and infected distributions. Numerically, repulsion suppresses spread, flattens the infection curve, and can induce transient phase separation (rings to spots) that mechanistically explains reduced infections via decreased contact overlap. The model distinguishes between reducing intrinsic transmissibility (e.g., masks lowering c) and reducing contact structure (lower ceff via distancing/isolation), offering insight into optimizing interventions. The airport/source scenario shows how continuous importation shapes steady states and can lead to oscillatory patterns, relevant for sustained introductions or chemical analogs.

The study introduces a DDFT-based extension of the SIR framework that explicitly models social distancing and self-isolation as repulsive interactions among diffusing individuals. It unifies spatial dynamics, interactions, and reactions in a computationally efficient, analytically tractable model. Key contributions include: (i) explicit derivation of how interactions modify effective transmission via spatial overlap; (ii) recovery of classical stability and front-speed results within an interaction-aware setting; (iii) identification of a parameter regime where distancing/isolation suppresses outbreaks; (iv) discovery of transient phase separation (rings and spots) that reduces infections; and (v) demonstration that reducing infected mobility strongly inhibits spread. Future work could integrate more elaborate compartment structures (ages, seasonality), nonlocal transmission kernels, multi-city/country scales with mobility networks, alternative interaction potentials, and improved free-energy approximations.

• DDFT approximations: assumes density is the only slow variable (adiabatic approximation) and uses mean-field excess free energy with Gaussian pair potentials; stochastic fluctuations (noise) are neglected.
• Local diffusion assumption: primary formulation assumes local, Brownian-like motion; long-distance travel is only approximated via source/sink terms in a scenario, not full mobility networks or Lévy flights (suggested as future extensions).
• Parameterization and dimensionality: main simulations are 2D with specific domain size and parameter choices; generalizability to other geometries/scales requires further study.
• Simplified reactions: primary results set death m=0; transmissions are local (no explicit nonlocal transmission kernels); recovery and transmission rates are fixed across space/time except via spatial overlap.
• Interaction forms: social distancing/self-isolation modeled as Gaussian repulsions; real-world behaviors may involve more complex, context-dependent interactions.
• Mobility modeling: infected mobility reduction implemented via temperature (βi) while keeping mobility Γi fixed; alternative choices (e.g., changing Γi) may have different implications.