Competitive epidemic networks with multiple survival-of-the-fittest outcomes

TL;DR

It is proved that, for any arbitrary number of nodes, such networks exist and given almost any network layer of one virus, there exists a network layer for the other virus such that the resulting two-layer network satisfies the aforementioned condition.

Abstract

We use a deterministic model to study two competing viruses spreading over a two-layer network in the Susceptible--Infected--Susceptible (SIS) framework, and address a central problem of identifying the winning virus in a "survival-of-the-fittest" battle. Existing sufficient conditions ensure that the same virus always wins regardless of initial states. For networks with an arbitrary but finite number of nodes, there exists a necessary and sufficient condition that guarantees local exponential stability of the two equilibria corresponding to each virus winning the battle, meaning that either of the viruses can win, depending on the initial states. However, establishing existence and finding examples of networks with more than three nodes that satisfy such a condition has remained unaddressed. In this paper, we prove that, for any arbitrary number of nodes, such networks exist. We do this by proving that given almost any network layer of one virus, there exists a network layer for the other virus such that the resulting two-layer network satisfies the aforementioned condition. To operationalize our findings, a four-step procedure is developed to reliably and consistently design one of the network layers, when given the other layer. Conclusions from numerical case studies, including a real-world mobility network that captures the commuting patterns for people between $107$ provinces in Italy, extend on the theoretical result and its consequences.

Figures (3)

• Figure 1: Schematic of the compartment transitions and two-layer infection network. (a) Each individual exists in one of three health states: Susceptible ($S$), Infected with virus 1 ($I$, orange), or Infected with virus $2$, ($I$, purple). Arrows represent possible transitions between compartments. (b) The two-layer network through which the viruses can spread between populations (nodes). Note that the edge sets of the two layers do not need to match, so that virus $1$ can spread between two nodes but virus $2$ cannot, and vice versa.
• Figure 2: The dynamics of the two-node case study of Eq. (\ref{['eq:bivirus_dynamics']}). In (a), the trajectories $(x_1(t), y_1(t))$ are shown for two different initial states (blue and red); virus 1 and virus 2 win the survival-of-the-fittest battle in the blue and red trajectories, respectively. In (b) and (c), the time evolution of $(x(t), y(t))$ is shown for the blue and red initial states in (a), respectively. In (d), we show the trajectories $(x_1(t), y_1(t))$ for virus $1$ and virus $2$ of the same speed (green, $\gamma = 1$) and virus 1 that is $1.2$ times faster relative to virus 2 (purple, $\gamma = 1.2$), for different initial states. The winning virus for different initial states is recorded when (e) virus 1 and virus 2 are the same speed and (f) when virus 1 is faster than virus 2, with $\gamma = 1.2$. Note the line where the boundaries of the two regions meet forms part of the stable manifold of the unstable coexistence equilibrium.
• Figure 3: The dynamics of the $n = 107$ example for a mobility network of Italian provinces (note the logarithmic scale of time, $t$, along the horizontal axis). In (a) and (b), the time evolution of $(x(t), y(t))$ shows two different initial states yielding two different survival-of-the-fittest outcomes.

Theorems & Definitions (7)

• Proposition 1
• remark 1
• Lemma 1
• proof
• Theorem 1
• Lemma 2
• Lemma 3