Fig 1.
The time-evolution of S, I and R for epidemics with no control.
A: and B:
with γ = 1 in both. Horizontal and vertical dashed black lines indicate the peak prevalence Imax and average time of infection
respectively, while green dashed horizontal lines show the attack rate R(∞) found by numerically solving
.
Fig 2.
Illustration of the impact of one-shot intervention in a population with .
The intervention has c = 0.8 for a duration of D = 2 time units. This intervention is introduced at different times as determined by a range of Threshold values. The impact of the threshold (I + R > Tr) for implementing the intervention is shown for A the attack rate R(∞); B S(t); C peak prevalence Imax; D I(t); E average time of infection ; and F plots of I(t) + R(t). In (B,D,F), the no-control case is plotted as a dashed line. The vertical lines in (A,C,E) correspond to the threshold for cumulative infections I + R which yields the intervention leading to the corresponding color in (B, D, F).
Fig 3.
Contour plots for R(∞) (top), Imax (middle) and the mean time of infection (bottom) as a function of parameters for the well-mixed population.
We explore different threshold values of I + R for the intervention to start, from a minimum of 0.05 to a max of 0.9. In the first column duration varies from D = 0.1 to D = 6, holding β = 2.5 and c = 0.8. In the second column, intervention duration is D = 4 and c ranges from 0.2 to 0.9. Finally, in the third column, c = 0.8 and D = 4, and the values of vary from 1 to 4. In all cases γ = 1. In the first row, the black curve denotes the threshold for which
when the intervention completes. In the three regions defined by the two lines in the panels of the second row, the peak prevalence is observed after the intervention has ended (from left to yellow curve), during intervention (area between the curves), or before intervention (from red curve to the end of the figure). Where the two curves align, the prevalence decays as soon as the intervention is implemented and then recovers to the pre-intervention peak.
Fig 4.
Example of an epidemic spreading across 9 subcommunities with different contact rates (see the Appendix A.2 for the precise mixing matrix B).
The epidemic starts from subcommunity 2 and it is run for T = 35 units of time. γ = 1 for all subcommunities. With no control the attack rate or final epidemic size is 0.744.
Fig 5.
Contour plots showing the average attack rate (final epidemic size) over 100 simulated populations for each set of parameter values.
In the first row c is fixed and the duration of control varies on the vertical axis, while in the second row duration is fixed and c varies. Each column corresponds to one of the three strategies: A,D intervention in each subcommunity based on that subcommunity reaching a threshold, B,E global intervention when the first subcommunity breaches the threshold, and C,F global intervention at global threshold for a population consisting of 9 subcommunities. In each plot, the x-axis shows the values that the threshold for intervention can take (from a minimum of 0.05 to a maximum of 0.8). In the first row c = 0.8 is constant, while the duration of control varies from a minimum of T = 1 to a maximum of T = 10. On the second row instead, the duration of control is kept fixed at T = 2, and the values of c varies from c = 0.1 to c = 0.9. The recovery rate is γ = 1 for all subcommunities. In all cases, if the threshold is set too large the intervention is never implemented. The two synchronized interventions can be approximately mapped to one another by noting the largest Ii + Ri at the time the global I + R reaches a given threshold. The subcommunity threshold gives more resolution at small values while the global threshold gives more resolution at large values.
Fig 6.
Illustration of best control strategy (i.e. smallest attack rate) (controlling subcommunities individually but using the same threshold for each) when efficacy and duration of control are fixed at c = 0.8 and D = 2, respectively.
It turns out that the optimal threshold is close to (0.4). This combination represents the point (0.4, 2) in Fig 5A, or equivalently the point (0.4, 0.8) in D. With this strategy, we find that R(∞) goes from R(∞) = 0.75 to R(∞) = 0.63. If we increase control duration from 2 to 10 we would achieve a further reduction to R(∞) = 0.44. The vertical black lines show the onset of control.
Fig 7.
Contour plots of the peak prevalence , averaged across 100 simulated populations each with 9 subcommunities.
Control strategies and setup are the same as in Fig 5.
Fig 8.
Contour plots of the peak prevalence, Ipeak, that is the maximum value achieved by during the time-course of the epidemic for the population used in Fig 4, with the intervention occuring when the global infection count reaches a threshold (as in C and F in earlier figures).
Fig 9.
Contour plots of the global mean infection time, defined as , averaged over 100 simulations.
In terms of control strategies and parameter values have the same setup as in Figs 7 and 5 are used.
Fig 10.
We plot S(t) versus R(t) for A, 2 B, and 4 C.
For given S(t) and R(t), the proportion infected is I(t) = 1 − S(t) − R(t), which equals the vertical or horizontal distance from the point (R(t), S(t)) to the line S + R = 1. The curves and arrows show how a solution to System (1) evolves in time. At points (which occurs only for
) curves move farther from the diagonal, representing an increase in I. Note that the velocity a curve is traversed varies depending on location, and goes to zero close to S + R = 1. Red dots in B–C indicate the point (
, I = 0, R = 1 − S).
Fig 11.
(S,R) phase portrait (arrows indicate growing time) based on an SIR model in a single population with β = 2, γ = 1 (giving ) and initial condition I(0) = 0.01.
The plot shows a trajectory with no control (continuous red line) as well as three other trajectories where β = 0.5 for a time period of length D = 2 but with the intervention setting in only once I + R goes past 0.1 (partially dotted line), 0.3 (partially dashed line) and 0.5 (continuous broken line), respectively. Control for the three different scenarios sets in at the points denoted by A, B and C and control ends at A’, B’ and C’, respectively.
Fig 12.
Contourplots of R∞ for a particular realization of the mixing matrix.
In C we see that there can be multiple peaks in the optimal time [this is also present in B but it is too small to see]. This is because the effectiveness of the interventions depend on the timing of epidemics in the different subpopulations and these are asynchronous.