Figure 4

Spreading competition between pathogens and therapeutics. (a) As it spreads, the pathogen endogenously reproduces identically at each node through SIR. (b) In contrast, the therapeutic flows in different rates into each node. We analyze the resulting spreading patterns on a weighted random network of \(N={10}^{4}\) nodes (\(k=10\), weights extracted from \({\mathscr{N}}(1.0,{0.2}^{2})\)). (c) The probability density P(j) for a randomly selected node to have infection level j, as obtained at \(t={T}_{{\rm{Peak}}}\). The bounded nature of P(j) indicates that the global demand is homogeneous. (d) At the same time the supply of the therapeutic is extremely heterogeneous with \(P(q)\sim {q}^{-2}\), depicting a coexistence of deprived and oversupplied nodes. (e,f) These distinct spreading patterns are clearly visible in a featured small network of N = 100 nodes: The pathogens (red) impact all nodes roughly simultaneously and homogeneously, while the drug supply (blue) is highly heterogeneous, reaching few nodes early, and leaving most nodes to lag behind. This disparity between homogeneous demand and heterogeneous supply – an intrinsic characteristic of the competing spreading processes – severely limits the effectiveness of centralized mitigation. (g) We measured the rates ξ_sn in (6) and obtained the probability density P(ξ) for \({\xi }_{sn}\in (\xi ,\xi +{\rm{d}}\xi )\) (grey). As predicted we find that \(P(\xi )\sim {\xi }^{-2}\) (black), exposing the topological roots of the observed supply heterogeneity. Dividing the nodes into saved (\({\bf{S}}=1\)) and unsaved (\({\bf{S}}=0\)) we also measured \(P({\bf{S}}\cap \xi )\) under varying capacity levels C (yellow to orange, \(C=0.001,0.005,0.02,0.04,0.1\) days⁻¹). As expected, the saved nodes tend towards the large ξ tail of the distribution, showing that, indeed, the broad distribution of ξ_sn determines the mitigation efficiency. (h) The critical capacity \({C}_{\eta }\) required to save (\({\bf{S}}=1\)) an \(\eta \) fraction of all nodes vs. the number of nodes N. As predicted in Eq. (8) we observe \({C}_{\eta }\sim {N}^{\varphi }\), a scaling relationship that renders mitigation unattainable for large networks (\(N\to \infty \)). For a random network we predict \(\varphi =1\) (solid black line). (i,j) The mean spreading time 〈T〉 of the pathogen (red) and the therapeutic (blue) vs. system size N. The therapeutic spreads in polynomial time (log-log plot, left), while the pathogen in logarithmic time (log-linear plot, right). (k) P(ξ) vs. ξ as obtained from the empirical air-traffic network. We observe similar patterns to those of the random network analyzed above. Indeed, also here the ξ-heterogeneity (\(\nu =1.4\), black solid line) drives the local efficiency levels as observed from \(P({\bf{S}}\cap \xi )\) (yellow to orange, C = 0.01, 0.1, 0.3,1.0, 3.0, 10 days⁻¹). (l) In decentralized mitigation, absent a distribution network, we find that the saved nodes are evenly spread, independently of ξ, an egalitarian increase in saved nodes, in which the rate heterogeneity plays no role (light to dark green, \(C=0.025,\,0.03,\,0.035,\,0.04,\,0.05,\,0.1\) days⁻¹).