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

Abstract

For preventing the spread of epidemics such as the coronavirus disease COVID-19, social distancing and the isolation of infected persons are crucial. However, existing reaction-diffusion equations for epidemic spreading are incapable of describing these effects. In this work, we present an extended model for disease spread based on combining a susceptible-infected-recovered model with a dynamical density functional theory where social distancing and isolation of infected persons are explicitly taken into account. We show that the model exhibits interesting transient phase separation associated with a reduction of the number of infections, and allows for new insights into the control of pandemics.

Fig. 1. Phase diagrams and time evolutions for the SIR-DDFT model.

a The shown phase diagrams reveal the dependence of the maximal fraction of infected persons ${Ī}_{\max ,\mathrm{n}}$ and the final fraction of susceptible persons ${\overline{S}}_{\infty ,\mathrm{n}}$ on the strength of self-isolation C_si and social distancing C_sd. A phase boundary is clearly visible. Blue points correspond to the time evolutions presented in the following subfigures. b Time evolutions of the fractions of susceptible (${\overline{S}}_{\mathrm{n}}$), infected (${Ī}_{\mathrm{n}}$), and recovered (${\overline{R}}_{\mathrm{n}}$) persons are shown for no interactions (C_si = C_sd  = 0), moderate interactions (C_si = 2C_sd = −20), and strong interactions (C_si = 3C_sd = −30). It can be seen that a reduction of social contacts flattens the curve ${Ī}_{\mathrm{n}}\left(t\right)$. c The density of infected persons I(x, y, t) is shown for the same three interaction strengths at different times t. For the strongest interactions, phase separation (first into rings, then into single spots) is observed. The color bar applies to Figs. 1a and 1c.

Fig. 2. Dependence of infection numbers on the amount of motion of infected persons.

The maximal fraction of infected persons ${Ī}_{\max ,\mathrm{n}}$ and the final fraction of susceptible persons ${\overline{S}}_{\infty ,\mathrm{n}}$ are shown as functions of β_I/β_S,R, where β_I and β_S,R are the rescaled inverse temperatures corresponding to the amount of motion of the infected and healthy persons, respectively. The plot corresponds to strong interactions with C_si = 3C_sd = −30. A decrease of the temperature, i.e., a reduction of the amount of motion of the infected persons, inhibits the outbreak.

Fig. 3. Time evolution of the density of infected persons I(x, y, t) for a local source of infected persons (airport).

The time evolution is shown for strong interactions with C_si = 3C_sd = −30 and different influx strengths I_source. Initially, one observes radial spreading (transient regime). Later, a steady state is reached, which is stationary for a small influx. A larger influx leads to a periodic regime (I(x, y, t) = I(x, y, t + nτ) with $n\in \mathbb{N}$ and period duration τ ≈ 16), involving the formation of a complex oscillating structure.