Combined effects of prevention and quarantine on a breakout in SIR model

Abstract

Recent breakouts of several epidemics, such as flu pandemics, are serious threats to human health. The measures of protection against these epidemics are urgent issues in epidemiological studies. Prevention and quarantine are two major approaches against disease spreads. We here investigate the combined effects of these two measures of protection using the SIR model. We use site percolation for prevention and bond percolation for quarantine applying on a lattice model. We find a strong synergistic effect of prevention and quarantine under local interactions. A slight increase in protection measures is extremely effective in the initial disease spreads. Combination of the two measures is more effective than a single protection measure. Our results suggest that the protection policy against epidemics should account for both prevention and quarantine measures simultaneously.

Figure 1. The snapshots and dynamics of SIR model with site and/or bond percolations.

The snapshots are (a) site only. (b) bond only. (c) bond and site. Sites are the prevented (white) the susceptible (blue) and the infected (red). The bonds are shown as a thick grey bar. The temporal dynamics of the infected (d) and the recovered (e) are shown for local (blue) and global (red) interactions. Parameter conditions are: β = 1.0, γ = 0.20, and the initial density of I = 0.01.

Figure 2. The final density of the recovered sites (R) for the combined model of both prevention (site percolation) and quarantine (bond percolation) with a constant infection rate (β/γ = 5.0).

(a) local interaction (simulation) and (b) global interaction (mean-field theory analyses) against the density of prevention P_S, where P_B = 0 (blue), 0.2 (red), 0.4 (green), 0.6 (purple) and 0.8 (yellow). (c) local interaction (simulation) and (d) global interaction (mean-field theory analyses) against the density of quarantine P_B, where P_S = 0 (blue), 0.2 (red), 0.4 (green), 0.6 (purple) and 0.8 (yellow). Phase diagrams along with the densities of both prevention (P_S) and quarantine (P_B) for (e) local interaction by simulation and (f) global interaction by mean-field theory (densities of the recovered (R) are shown in colors: blue = 0–0.2; red = 0.2–0.4; green = 0.4–0.6; purple = 0.6–0.8; pail blue = 0.8–1.0). (g) comparisons with the steady state density of SIS models for a half (50%) reduction in infection level for both global and local interactions (Black solid line: SIS for local; black dashed line: SIS for global; red solid line: SIR for local; and red dashed line: SIR for global). The combined effect is the largest for the local SIR model; a half (50%) reduction is achieved merely by P_S = P_B = 0.1 or less.

Figure 3. Dynamics of the recovered (R) when the density of prevention P_S increases in the SIR model.

(a) The temporal profiles of prevention introduction. (b) The temporal dynamics of the recovered density (R). The final proportion of prevention (site blocked) is set at P_S = 0.5 and the rate τ is defined as the time to reach at P_S = 0.5. Four different delayed times τ of protection are used for comparisons: (1) no delay in prevention corresponding to Fig. 1a (τ = 0; red dashed line), (2) a fast rate (τ = 100; green line), (2) a moderate rate (τ = 500; purple line), and (3) a fast rate (τ = 1000; blue line).