Network structure-based interventions on spatial spread of epidemics in metapopulation networks

Mathematical modeling of epidemics is fundamental to understand the mechanism of the disease outbreak and provides helpful indications for effectiveness of interventions for policy makers. The metapopulation network model has been used in the analysis of epidemic dynamics by taking individual migration between patches into account. However, so far, most of the previous studies unrealistically assume that transmission rates within patches are the same, neglecting the nonuniformity of intervention measures in hindering epidemics. Here, based on the assumption that interventions deployed in a patch depend on its population size or economic level, which have shown a positive correlation with the patch's degree in networks, we propose a metapopulation network model to explore a network structure-based intervention strategy, aiming at understanding the interplay between intervention strategy and other factors including mobility patterns, initial population, as well as the network structure. Our results demonstrate that interventions to patches with different intensity are able to suppress the epidemic spreading in terms of both the epidemic threshold and the final epidemic size. Specifically, the intervention strategy targeting the patches with high degree is able to efficiently suppress epidemics. In addition, a detrimental effect is also observed depending on the interplay between the intervention measures and the initial population distribution. Our study opens a path for understanding epidemic dynamics and provides helpful insights into the implementation of countermeasures for the control of epidemics in reality.

Figure 1

Transmission rates $\beta \left(t\right)$ in cities of China fitted with real data during the period from 10 January 2020 to 30 January 2020 for COVID-19 transmission. The horizontal axis represents the number of days starting from 10 January 2020. The transmission rates evolve with time to stable values due to the involvement of intervention measures.

Figure 2

(a) Correlation between a city's GDP value and its degree $k$ in the high-speed railway network of China. The correlation coefficient is $r=0.659$. (b) Correlation between a city's population and its degree $k$ in the high-speed railway network of China. The correlation coefficient is $r=0.442$.

Figure 3

The intervention factor ${\theta }_i(k_i,\alpha )$ of a patch with degree $k_i$ for choices of (a) $\alpha <0$ and (b) $\alpha >0$. (c) The average transmission rate $〈{\beta }_i〉$ versus parameter $\alpha$ for different intervention strategies. The red, orange, and blue curves are the results for $\beta =0.0008$, 0.0004, and 0.0002, respectively.

Figure 4

Schematic framework of a metapopulation model with a heterogeneous distribution of transmission rate $\beta$ induced by intervention to patches with different intensity. Inside each patch, the SEIR model is used to describe the reaction process. The flux between patches $i$ and $j$ is proportional to $w_{ij}$. Individuals in patch $i$ move to a neighboring patch $j$ with probability $p$ according to the transition matrix element $W_{ij}=\frac{w_{ij}}{{\sum }_k{w}_{ik}}$ and return to their residence $i$ at each time step.

Figure 5

The final fraction of recovered individuals for different interventions controlled by parameter $\alpha$ starting from different initial population distributions. (a) Heterogeneous initial population distribution (HED). (b) Homogeneous initial population distribution (HOD). The solid curves correspond to the numerical solutions of the Markovian evolution equations [Eq. (3)], whereas the points represent the results of MC simulations. Each point is the average over more than 100 MC simulations.

Figure 6

The time evolution of infectious ratio for different interventions controlled by $\alpha$ starting from different initial population distributions. (a) Heterogeneous distribution of initial population (HED). (b) Homogeneous distribution of initial population (HOD). The solid curves are the results obtained by MC simulations. Each curve is the average over 100 independent experiments.

Figure 7

The final fraction of recovered individuals $R$ as a function of $\beta$ and $\alpha$. (a) Heterogeneous distribution of initial population (HED). (b) homogeneous distribution of initial population (HOD).

Figure 8

The distribution of recovered individuals in patch $k$ versus $\beta$ for different choices of $\alpha$, starting from different initial population distributions. Top: Heterogeneous distribution of initial population (HED). Bottom: Homogeneous distribution of initial population (HOD). (a, d) $\alpha =-1$, (b, e) $\alpha =0$, and (c, f) $\alpha =1$.

Figure 9

The final fraction of recovered individuals versus $\beta$ for different mobility rate $p$ when intervention strategies are introduced. (a–c) Heterogeneous distribution of initial population (HED). (d–f) Homogeneous distribution of initial population (HOD). (a, d) $\alpha =-1$, (b, e) $\alpha =0$, and (c, f) $\alpha =1$.

Figure 10

The final fraction of recovered individuals $R$ obtained as a function of $p$ and $\beta$ for different intervention strategies. The color code represents the density of recovered individuals $R$ obtained by Monte Carlo simulations, and the white solid curves correspond to the theoretical analysis of the epidemic threshold with Eq. (10). Top: Heterogeneous distribution of initial population (HED). Bottom: Homogeneous distribution of initial population (HOD). (a, d) $\alpha =-1$, (b, e) $\alpha =0$, and (c, f) $\alpha =1$.