Competitive epidemic spreading over arbitrary multilayer networks

This study extends the Susceptible-Infected-Susceptible (SIS) epidemic model for single-virus propagation over an arbitrary graph to an Susceptible-Infected by virus 1-Susceptible-Infected by virus 2-Susceptible (${\mathrm{SI}}_1{\mathrm{SI}}_2\mathrm{S}$) epidemic model of two exclusive, competitive viruses over a two-layer network with generic structure, where network layers represent the distinct transmission routes of the viruses. We find analytical expressions determining extinction, coexistence, and absolute dominance of the viruses after we introduce the concepts of survival threshold and absolute-dominance threshold. The main outcome of our analysis is the discovery and proof of a region for long-term coexistence of competitive viruses in nontrivial multilayer networks. We show coexistence is impossible if network layers are identical yet possible if network layers are distinct. Not only do we rigorously prove a region of coexistence, but we can quantitate it via interrelation of central nodes across the network layers. Little to no overlapping of the layers' central nodes is the key determinant of coexistence. For example, we show both analytically and numerically that positive correlation of network layers makes it difficult for a virus to survive, while in a network with negatively correlated layers, survival is easier, but total removal of the other virus is more difficult.

Figure 1

Schematic of two-layer contact topology $\mathcal{G}(V,E_A,E_B)$, where a group of nodes shares two distinct interactions. In our ${\mathrm{SI}}_1{\mathrm{SI}}_2\mathrm{S}$ model, virus 1 transmits exclusively via $E_A$ links (red) while virus 2 transmits only through $E_B$ links (black). Dotted vertical lines reiterate that individual nodes are the same in both layers of $\mathcal{G}$.

Figure 2

Schematic of a contact network with the node-level stochastic transition diagram for node $i$, according to the ${\mathrm{SI}}_1{\mathrm{SI}}_2\mathrm{S}$ epidemic spreading model. Parameters ${\beta }_1$ and ${\delta }_1$ denote virus 1 infection rate and curing rate, respectively, and $Y_i\left(t\right)$ is the number of node $i$ neighbors in layer $G_A$ infected by virus 1 at time $t$. Similarly, ${\beta }_2$ and ${\delta }_2$ denote virus 2 infection rate and curing rate, respectively, and $Z_i\left(t\right)$ is the number of node $i$ neighbors in layer $G_B$ infected by virus 2 at time $t$.

Figure 3

Phase transition of competitive spreading model ${\mathrm{SI}}_1{\mathrm{SI}}_2\mathrm{S}$ for a single-layer network, i.e., $B=A$. Holding the effective infection rate of virus 2 constant at ${\tau }_2=6\frac{1}{{\lambda }_1\left(B\right)}=6\frac{1}{{\lambda }_1\left(A\right)}$ and varying ${\tau }_1$, (a) the steady-state infection fraction of virus 1, ${\overline{p}}_1^{ss}=\frac{1}{N}\sum p_{1,i}^{ss}$, and (b) the steady-state infection fraction of virus 2, ${\overline{p}}_2^{ss}=\frac{1}{N}\sum p_{2,i}^{ss}$, exhibit abrupt phase transition at ${\tau }_1=6\frac{1}{{\lambda }_1\left(A\right)}={\tau }_2$. Specifically, (a) ${\overline{p}}_1^{ss}$ is zero for ${\tau }_1<{\tau }_2$ and is positive for ${\tau }_1>{\tau }_2$, denoting survival threshold of virus 1, and (b) ${\overline{p}}_2^{ss}$ is positive for ${\tau }_1<{\tau }_2$ and becomes zero for ${\tau }_1>{\tau }_2$, indicating absolute removal of virus 2 and thus the virus 1 absolute-dominance threshold.

Figure 4

Illustration of survival and absolute-dominance thresholds for virus 1 on a multilayer contact network. Holding the effective infection rate of virus 2 constant at ${\tau }_2=6\frac{1}{{\lambda }_1\left(B\right)}$, (a) the steady-state infection fraction of virus 1, ${\overline{p}}_1^{ss}=\frac{1}{N}\sum p_{1,i}^{ss}$, and (b) the steady-state infection fraction of virus 2, ${\overline{p}}_2^{ss}=\frac{1}{N}\sum p_{2,i}^{ss}$, exhibit phase transition at survival threshold ${\tau }_{c1}$ and absolute-dominance threshold ${\tau }_1^{†}$, respectively. Specifically, (a) ${\overline{p}}_1^{ss}$ is zero for ${\tau }_1<{\tau }_{c1}$ and becomes positive for ${\tau }_1>{\tau }_{c1}$, denoting survival threshold of virus 1, and (b) ${\overline{p}}_2^{ss}$ is positive for ${\tau }_1<{\tau }_1^{†}$ and becomes zero for ${\tau }_1>{\tau }_1^{†}$, indicating absolute removal of virus 2 and thus the virus 1 absolute-dominance threshold. Additionally, it is interesting to observe that ${\overline{p}}_2^{ss}$ is constant when ${\tau }_1<{\tau }_{c1}$ while it reduces gradually as ${\tau }_1$ becomes larger than ${\tau }_{c1}$.

Figure 5

Steady-state infection fraction curve of virus 1 in the ${\mathrm{SI}}_1{\mathrm{SI}}_2\mathrm{S}$ competing spreading model (red). While increasing ${\tau }_1$, the steady-state infection fraction of virus 1 in the ${\mathrm{SI}}_1{\mathrm{SI}}_2\mathrm{S}$ model becomes nonzero at the survival threshold ${\tau }_{1c}$, while it coincides with that of the SIS model (black curve) at the absolute-dominance threshold ${\tau }_1^{†}$. In this simulation, the steady-state infection fraction of virus 1 (${\overline{p}}_1^{ss}$) is zero for ${\tau }_1\le {\tau }_{1c}\simeq 3\frac{1}{{\lambda }_1\left(A\right)}$, an extinction region for virus 1. Interestingly, for ${\tau }_1>{\tau }_1^{†}\simeq 6.6\frac{1}{{\lambda }_1\left(A\right)}$, ${\overline{p}}_1^{ss}$ for the competitive scenario (red curve) is identical to the case of single-virus propagation (black curve), suggesting extinction of virus 2, hence marking this region the absolute-dominance range for virus 1. For ${\tau }_1\in ({\tau }_{1c},{\tau }_1^{†})$, virus 1 and virus 2 both persist in the population, marking this range a coexistence region.

Figure 6

Illustration of survival and absolute-dominance threshold curves in the ${\mathrm{SI}}_1{\mathrm{SI}}_2\mathrm{S}$ model. (a) Virus 1 survives if its effective infection rates is larger than the survival threshold, i.e., ${\tau }_1>{\tau }_{c1}={\Phi }_1\left({\tau }_2\right)$. A similar argument holds for the survival threshold curve of virus 2, as depicted in (b). The absolute-dominance threshold curves can be obtained from the survival curves shown in (a) and (b). Specifically, the region virus 1 is the absolute winner is where virus 1 survives and virus 2 does not survive, as shown in (c). Likewise, the region virus 2 is the absolute winner is where virus 1 does not survive while virus 2 survives (d).

Figure 7

The ${\mathrm{SI}}_1{\mathrm{SI}}_2\mathrm{S}$ model with two-layer contact topology exhibits four possibilities: extinction region N, where both viruses die out; virus 1 absolute-dominance region I, where virus 1 survives and virus 2 dies out; virus 2 absolute-dominance region II, where virus 2 survives and virus 1 dies out; and, finally, coexistence region III, where both viruses survive and persist in the population.

Figure 8

The survival regions diagram in ${\mathrm{SI}}_1{\mathrm{SI}}_2\mathrm{S}$ model for values of $({\tau }_1,{\tau }_2)$ close to $[\frac{1}{{\lambda }_1\left(A\right)},\frac{1}{{\lambda }_1\left(B\right)}]$ (left) and for very large values of $({\tau }_1,{\tau }_2)$ (right). Regions N, I, II, and III are as defined in Fig. 7. The red arrow shows the survival region of virus 1 (regions I and III) and the green arrow shows the survival region of virus 2 (regions II and III). For the aggressive viruses scenario, axes have inverse values of $({\tau }_1,{\tau }_2)$ so that the origin represents infinitely large values. Equations (11) and (13) analytically find the separating lines between the survival regions in explicit expressions.

Figure 9

Two-layer network generation for numerical simulations. The contact network $G_A$ through which virus 1 propagates is a random geometric graph where pairs of nodes with a distance less than $r_c$ are connected to each other. For visualization convenience, the number of nodes is $N=100$, which is different from the actual $N=1000$ used for numerical simulation results. For the contact graph of virus 2 ($G_B$), we first generated a scale-free network according to the B-A model, then associated the nodes of this graph with the nodes of $G_A$ to achieve a certain degree correlation coefficient with $G_A$. Specifically, we obtained three different permutations such that the resultant graphs are negatively, neutrally, or positively correlated with $G_A$. These three graphs are equivalent if isolated and yet distinct in their interrelation with $G_A$. The high degree nodes in the positively correlated $G_B$ (bottom right) also have high degree in $G_A$ (top left), while the high degree nodes in the negatively correlated $G_B$ (top right) have low degree size in $G_A$. The uncorrelated $G_B$ (bottom left) shows no clear association.

Figure 10

Comparison of steady-state infection fraction curves of virus 1 in the${\mathrm{SI}}_1{\mathrm{SI}}_2\mathrm{S}$ competitive spreading model. Survival threshold ${\tau }_{1c}$ is larger for positively correlated $G_B$, indicating that it is more difficult to survive positively correlated $G_B$, while ${\tau }_1^{†}$ is larger for negatively correlated $G_B$, indicating that it is more difficult to completely suppress the other virus in negatively correlated $G_B$.

Figure 11

Steady-state fraction of infection for virus 1 (left) and virus 2 (right) as a function of ${\tau }_1$and${\tau }_2$. The white lines are theoretical threshold curves accurately separating the survival regions.

Figure 12

Standardized threshold diagram for the case where $G_B$ is negatively correlated with $G_A$ (left) and the case where $G_B$ is positively correlated with $G_A$ (right). Dashed lines are the predictions from analytical approximation formulas explicitly expressed in (21). The standardized threshold diagram shows three survival regions: virus 1 absolute-dominance region I, where only virus 1 survives and virus 2 dies out; virus 2 absolute-dominance region II, where only virus 2 survives and virus 1 dies out; and finally, coexistence region III, where both viruses survive and persist in the population.