Intervention threshold for epidemic control in susceptible-infected-recovered metapopulation models

Metapopulation epidemic models describe epidemic dynamics in networks of spatially distant patches connected via pathways for migration of individuals. In the present study, we deal with a susceptible-infected-recovered (SIR) metapopulation model where the epidemic process in each patch is represented by an SIR model and the mobility of individuals is assumed to be a homogeneous diffusion. We consider two types of patches including high-risk and low-risk ones under the assumption that a local patch is changed from a high-risk one to a low-risk one by an intervention. We theoretically analyze the intervention threshold which indicates the critical fraction of low-risk patches for preventing a global epidemic outbreak. We show that an intervention targeted to high-degree patches is more effective for epidemic control than a random intervention. The theoretical results are validated by Monte Carlo simulations for synthetic and realistic scale-free patch networks. The theoretical results also reveal that the intervention threshold depends on the human mobility network and the mobility rate. Our approach is useful for exploring better local interventions aimed at containment of epidemics.

Figure 1

Schematic illustration describing the SIR metapopulation model with high-risk patches (red thick circles) and low-risk ones (blue thin circles).

Figure 2

The theoretically derived global reproduction number $R_c^{\mathrm{rn}}$ [Eq. (35)] as a function of the intervention rate $u$ and the mobility rate $p$ for a scale-free network with a degree distribution $p\left(k\right)\sim k^{-\gamma }$ with $\gamma =2.1$, under the random intervention. The level of the global reproduction number $R_c^{\mathrm{rn}}$ is also indicated by gray-scale color. The value of $R_c^{\mathrm{rn}}$ increases monotonically with decreasing $u$ and increasing $p$. The yellow filled circles represent the critical intervention threshold $u_c^{\mathrm{rn}}$ as a function of $p$, which is theoretically derived from Eq. (36). The value of $u_c^{\mathrm{rn}}$ increases with $p$.

Figure 3

The theoretically derived intervention threshold $u_c^{\mathrm{rn}}$ as a function of the degree exponent $\gamma$ and the mobility rate $p$. The level of the intervention threshold $u_c^{\mathrm{rn}}$ is also indicated by color. The intervention threshold increases with decreasing $\gamma$ and increasing $p$.

Figure 4

(a) The piecewise function $q_l\left(k\right)$ defined in Eq. (38). (b) The probability density function $q^{\mathrm{tg}}\left(k\right)$ for $u<{\widehat{q}}_{k_{\max }}$ in Eq. (40).

Figure 5

The theoretically derived global reproduction number $R_c^{\mathrm{tg}}$ as a function of the intervention rate $u$ and the mobility rate $p$ for a scale-free network with a degree distribution $p\left(k\right)\sim k^{-\gamma }$ with $\gamma =2.1$, under the targeted intervention. The level of the global reproduction number $R_c^{\mathrm{tg}}$ is also indicated by gray-scale color. $R_c^{\mathrm{tg}}$ increases with decreasing $u$ and increasing $p$. The yellow filled circles represent the critical intervention threshold $u_c^{\mathrm{tg}}$ as a function of $p$, which is theoretically derived from Eqs. (30) and (40). The value of $u_c^{\mathrm{tg}}$ increases with $p$.

Figure 6

The theoretically derived intervention threshold $u_c^{\mathrm{tg}}$ as a function of the degree exponent $\gamma$ and the mobility rate $p$. The level of the intervention threshold $u_c^{\mathrm{tg}}$ is also indicated by color.

Figure 7

Numerical results for the final epidemic size $R_{\infty }/N$ with different values of the intervention rate $u$. The mobility rate is fixed at $p=0.05$. The crosses and open circles indicate the average values over 50 simulations for the random and targeted interventions, respectively. The error bar indicates the standard deviation. (a) Scale-free patch networks generated with the configuration model [50]. (b) The U.S. airport network [40].

Figure 8

Numerical results for the final epidemic size $R_{\infty }/N$ plotted against the intervention rate $u$ and the mobility rate $p$. The average value over 50 simulations is plotted for each parameter condition. The level of the epidemic size $R_{\infty }/N$ is also indicated by color. The yellow filled circles indicate theoretically obtained critical intervention thresholds $u_c$ for different values of $p$. (a) Random intervention in synthetic scale-free patch networks. (b) The same as (a), but for targeted intervention. (c) Random intervention in the US airport network. (d) The same as (c) but for targeted intervention. The theoretical values of $u_c$ are computed from Eq. (36) for (a) and (c), and from Eqs. (30) and (40) for (b) and (d).