Active risk aversion in SIS epidemics on networks

Abstract

We present and analyze an actively controlled Susceptible-Infected-Susceptible (actSIS) model of interconnected populations to study how risk aversion strategies, such as social distancing, affect network epidemics. A population using a risk aversion strategy reduces its contact rate with other populations when it perceives an increase in infection risk. The network actSIS model relies on two distinct networks. One is a physical contact network that defines which populations come into contact with which other populations and thus how infection spreads. The other is a communication network, such as an online social network, that defines which populations observe the infection level of which other populations and thus how information spreads. We prove that the model, with these two networks and populations using risk aversion strategies, exhibits a transcritical bifurcation in which an endemic equilibrium emerges. For regular graphs, we prove that the endemic infection level is uniform across populations and reduced by the risk aversion strategy, relative to the network SIS endemic level. We show that when communication is sufficiently sparse, this initially stable equilibrium loses stability in a secondary bifurcation. Simulations show that a new stable solution emerges with nonuniform infection levels.

Figures (6)

• Figure 1: Simulation for network actSIS \ref{['eq:pdot']},\ref{['eq:phat']} with $10$ nodes and regular contact graph and communication graph, using rackauckas2017differentialequations. The degree of $\hat{A}$ is constant but $A$ is sparse (left) or dense (right). $\bar{\beta} = 0.3, \delta = 0.5, \mu = 0.5, \nu = 1.7, \tau_s = 1$.
• Figure 2: Plot of \ref{['eq:stab_cond']} over 40 nodes with fixed sparse contact graph: $d = 5$ (left), and fixed dense contact graph: $d=30$ (right). Each curve in each plot corresponds to an $\hat{A}$ with different $\hat{d}$. $\hat{A}$ is generated randomly using hagberg2008exploring. $\delta = 1, \mu = 0.1, \nu=3$.
• Figure 3: Distribution of the critical point $\bar{\beta}_2$ of Theorem \ref{['thm:uniform_EE_stab']} for 40 nodes with (a) sparse contact graph; (b) dense contact graph; and communication graphs with range of $d$. Each box plot contains $\bar{\beta}_2$ from 1000 simulations. For each simulation, $A$ and $\hat{A}$ of appropriate $d$, $\hat{d}$ are generated randomly using hagberg2008exploring. $\delta = 1, \mu = 0.1, \nu=3$.
• Figure 4: Trajectories of network SIS and network actSIS dynamics over $26$-node regular graphs with $d = 5$ and $\hat{d} = 21$. $\delta = 0.5, \mu = 0.2, \nu=8$ and $\tau_s=1$. Created using rackauckas2017differentialequations.
• Figure 5: Simulations using rackauckas2017differentialequations for a network with $10$ nodes. $\bar{\beta} = 0.3, \delta = 0.5, \tau_s = 1, \mu = 0.2$, $\nu = 8$.

Theorems & Definitions (12)

• Theorem 4.1: Well-Definedness
• proof
• Theorem 4.2: Local bifurcation of EE
• proof
• Corollary 4.1
• Lemma 5.1: Risk aversion lowers EE in well-mixed model
• proof
• Lemma 5.2: Uniform Endemic Equilibrium (UEE)
• Theorem 5.3: Risk aversion lowers UEE on regular graphs
• proof