Optimizing Hybrid Spreading in Metapopulations

TL;DR

This study develops a theoretical framework for studying hybrid epidemics and examines the optimum balance between spreading mechanisms in terms of achieving the maximum outbreak size and shows the existence of critically hybrid Epidemics where neither spreading mechanism alone can cause a noticeable spread but a combination of the two spreading mechanisms would produce an enormous outbreak.

Abstract

Epidemic spreading phenomena are ubiquitous in nature and society. Examples include the spreading of diseases, information, and computer viruses. Epidemics can spread by local spreading, where infected nodes can only infect a limited set of direct target nodes and global spreading, where an infected node can infect every other node. In reality, many epidemics spread using a hybrid mixture of both types of spreading. In this study we develop a theoretical framework for studying hybrid epidemics, and examine the optimum balance between spreading mechanisms in terms of achieving the maximum outbreak size. We show the existence of critically hybrid epidemics where neither spreading mechanism alone can cause a noticeable spread but a combination of the two spreading mechanisms would produce an enormous outbreak. Our results provide new strategies for maximising beneficial epidemics and estimating the worst outcome of damaging hybrid epidemics.

Figures (5)

• Figure 1: Hybrid epidemic spreading in a metapopulation. At each time step, an infected node has a fixed total spreading effort which must be allocated between local spreading and global spreading. The proportion of spreading effort spent in local spreading is $\alpha$ and that in global spreading is $1-\alpha$. Local spreading occurs between infected and susceptible nodes that are connected in individual subpopulations; global spreading happens between an infected node and any susceptible node in the metapopulation.
• Figure 2: Theoretical predictions and simulation results for hybrid epidemics in a single-population. The final outbreak size $r_{\infty}$ is shown as a function of the hybrid tradeoff $\alpha$. Three network topologies are considered: (1) a fully connected network (i.e. fully mixed); (2) a random network with an average degree of 5; (3) a scale-free network with a power-law degree distribution $p_k \sim 2m^2k^{-3}$ which is generated by the configuration model Newman_Book_2010 with the minimum degree $m=3$. The population has 1000 nodes. The global infection rate $\beta_2=10^{-4}$ and recovery rate $\gamma=1$ are the same for epidemics on these three types of networks. The local infection rate $\beta_1$ is $6\times10^{-3}$ for epidemics on the fully connected network; and it is $0.8$ for epidemics on the random and scale-free networks. Initially 5 random nodes are infected. Simulation results are shown as points and theoretical predictions of equation (\ref{['eq-rinf0']}) are dashed curves. The simulation results are averaged over 1000 runs with bars showing the standard deviation. The epidemic threshold values of $\alpha$ are predicted by equation (\ref{['eq:threshold']}) and marked as vertical lines.
• Figure 3: Simulation results of hybrid epidemics in a metapopulation where (a) each subpopulation is a random network and (b) each subpopulation is a scale-free network. Three quantities are shown as a function of the hybrid tradeoff $\alpha$, including the final outbreak size as the fraction of recovered nodes $r_{\infty}$ (squares); the final outbreak size as the fraction of recovered subpopulations $R_{\infty}$ (circles); and the population reproduction number, $R_p$ (triangles, right y-axis). The metapopulation contains 500 subpopulations each with 100 nodes. In (a) each subpopulation is a random network with an average degree of 5; and In (b) each subpopulation is a scale-free network with a power-law degree distribution $p_k \sim 2m^2k^{-3}$ which is generated by the configuration model Newman_Book_2010 with the minimum degree $m=3$.. The local infection rate $\beta_1=0.8$, the global infection rate $\beta_2=10^{-6}$ and the recovery rate $\gamma=1$. Initially 3 random nodes in a subpopulation are infected. Simulation results are shown as points and each result is averaged over 1000 runs.
• Figure 4: The population reproduction number $R_p$ as a function of the hybrid tradeoff $\alpha$. Theoretical predictions from equation (\ref{['eq:R_N0m']}) are shown as a dashed curve. Simulation results are shown as points (average over 1,000 runs) and bars (one standard deviation). The metapopulation and epidemic parameters are the same as Figure \ref{['fig-ob-RN0-a']}a.
• Figure 5: The estimated optimal hybrid tradeoff $\alpha^*$ and the maximal population reproduction number $R_p^*$ for hybrid epidemics in a metapopulation. (a) $\alpha^*$ as a function of local infection rate $\beta_1$ and recovery rate $\gamma$; (b) $R_p^*$ as a function of $\beta_1$ and $\gamma$; (c) $\alpha^*$ as a function of $\beta_1/\gamma$, which is fitted by a dash line of $ln(\alpha^*)=-0.84-0.57\cdot ln(\beta_1/\gamma)$. (d) Population reproduction number $R_p$ as a function of $\alpha$ and $\beta_1$ with $\gamma=0.1$, where the points are the corresponding optimal $\alpha^*$ for given $\beta_1$. We fix $\beta_2=10^{-6}$ and the metapopulation is as in Figure \ref{['fig-ob-RN0-a']}a.