Transition from growth to decay of an epidemic due to lockdown

Abstract

We study the transition of an epidemic from growth phase to decay of the active infections in a population when lockdown health measures are introduced to reduce the probability of disease transmission. Although in the case of uniform lockdown, a simple compartmental model would indicate instantaneous transition to decay of the epidemic, this is not the case when partially isolated active clusters remain with the potential to create a series of small outbreaks. We model this using the Gillespie stochastic simulation algorithm based on a connected set of stochastic susceptible-infected-removed/recovered networks representing the locked-down majority population (in which the reproduction number is less than 1) weakly coupled to a large set of small clusters in which the infection may propagate. We find that the presence of such active clusters can lead to slower than expected decay of the epidemic and significantly delayed onset of the decay phase. We study the relative contributions of these changes, caused by the active clusters within the population, to the additional total infected population. We also demonstrate that limiting the size of the inevitable active clusters can be efficient in reducing their impact on the overall size of the epidemic outbreak. The deceleration of the decay phase becomes apparent when the active clusters form at least 5% of the population.

Figure 1

SIR model. (A) SIR networks in the locked-down (green) and not-socially-distancing (red) subpopulations. The latter is composed of N clusters, each has its own SIR dynamics. $S,I,$ and R are the proportions of susceptible, infected, and recovered individuals, respectively. Dashed curves represent infections transmitted by infected individuals to the susceptible individuals. (B) The total number of infected individuals versus time calculated using the Gillespie stochastic simulation algorithm. Solid curves represent five example runs of the algorithm (in different colors) for $N=100$ clusters of size $M=10$. Black dashed line represents $I_{\text{tot}}\left(t\right)$, average of 10 runs of the algorithm. Other simulation parameters are summarized in Table 1. To see this figure in color, go online.

Figure 2

Total number of infected individuals ($I_{\text{tot}}$(t)) versus time for different numbers (N) and sizes (M) of active clusters. Circles represent $I_{\text{tot}}\left(t\right)$ calculated using the Gillespie algorithm after averaging over 10 repeated simulations. The solid red line is obtained by fitting $I_{0_2}{e}^{-\alpha t}$ to the simulation data points highlighted by orange using the Mathematica routine NonlinearModelFit. To perform the fitting with minimal effects of the stochasticity, in most cases, the fitting range is set to start at a time after lockdown time $t_{\text{LD}}$ where $100<I_{\text{tot}\left(t\right)}<3000$ (see Table S1 for the fit parameters $I_{0_2}$ and α). Dashed green line represents $I_{\text{LD}}{e}^{\gamma \left(R_{0_2}-1\right)\left(t-t_{\text{LD}}\right)}$ that passes through $I_{\text{LD}}=I_{\text{tot}}\left(t_{\text{LD}}\right)$, where $t_{\text{LD}}$ is the time of lockdown. Blue line represents $I_{0_1}{e}^{\gamma \left(R_{0_1}-1\right)t}$. Black dashed line represents $t_{\text{LD}}$. Other simulation parameters are in Table 1. To see this figure in color, go online.

Figure 3

Analysis of the decay phase of an epidemic. (A) A sketch illustrating three properties of the decay phase (for example, with $N=2000$ and $M=70$). The decay rate α corresponds to the slope of the fit $I_{0_2}{e}^{-\alpha t}$ (red solid line). The delay τ in the epidemic decay, because of the presence of the active clusters, is the time-shift relative to an exponential decay with the same slope (α) that passes through the lockdown point $\left(t_{\text{LD}},I_{\text{LD}}\right)$ (brown dotted line); see Eq. 5. Finally, the number of extra infected individuals $I_{\text{extra}}$, calculated using Eq. 6, is represented by the area of the yellow shaded region. Green dashed line: $I_{\text{LD}}\text{exp}\left(\gamma \left(R_{0_2}-1\right)\left(t-t_{\text{LD}}\right)\right)$. (B–D) Decay rate α, delay τ, and relative increase of total infections μ versus the active population ratio not locked-down, $1-L$. Dashed line in (B) represents${\alpha }_0=\gamma \left(1-R_{0_2}\right)$. Other simulation parameters are in Table 1. To see this figure in color, go online.

Figure 4

Analysis of the extra number of infected individuals, $I_{\text{extra}}$ calculated using Eq. 6. (A and B) $I_{\text{extra}}$ vs N and M in linear (A) and log (B) scales. (C and D) Dashed lines represent fits $r{N}^a{M}^b$ to $I_{\text{extra}}\left(N;M\right)$ simulation data points, where $a=1.08\pm 0.02$, $b=1.86\pm 0.02$, and $r=0.01\pm 0.001$. Parameter values correspond to mean $\pm$ SE. Dotted lines represent the reference value of total infected from the start of lockdown in a population without active clusters (N = 0), i.e., the area under the dashed green curve in Fig. 3A. To see this figure in color, go online.