Incentives for self-isolation based on incidence rather than prevalence could help to flatten the curve: a modelling study

TL;DR

This study couples an SIS-type epidemic with imitation-driven self-isolation decisions, comparing prevalence- versus incidence-based information as drivers of behavior. The authors derive a nondimensional ODE system with compartments $S,U,I,Q,F$ and an evolving cooperation level $x$, revealing five equilibria and simple stability criteria that separate regimes of no, partial, and full compliance. They find that equilibria do not depend on whether prevalence or incidence is used at equilibrium, but incidence information can reduce transient peak prevalence, especially under fast epidemic or high-volatility conditions; an optimal isolation duration can further flatten the curve in the partial-compliance regime. Simulations on an empirical two-layer network show fair agreement with the analytic predictions, while highlighting how network structure can modulate dynamics and oscillations. The work offers insight for policy design, suggesting that incentives to reduce isolation cost and favor incidence-based information can help manage peak demand during outbreaks.

Abstract

In recent years, numerous advances have been made in understanding how epidemic dynamics is affected by changes in individual behaviours. We propose an SIS-based compartmental model to tackle the simultaneous and coupled evolution of an outbreak and of the adoption by individuals of the isolation measure. The compliance with self-isolation is described with the help of the imitation dynamics framework. Individuals are incentivised to isolate based on the prevalence and the incidence rate of the outbreak, and are tempted to defy isolation recommendations depending on the duration of isolation and on the cost of putting social interactions on hold. We are able to derive analytical results on the equilibria of the model under the homogeneous mean-field approximation. Simulating the compartmental model on empirical networks, we also do a preliminary check of the impact of a network structure on our analytical predictions. We find that the dynamics collapses to surprisingly simple regimes where either the imitation dynamics no longer plays a role or the equilibrium prevalence depends on only two parameters of the model, namely the cost and the relative time spent in isolation. Whether individuals prioritise disease prevalence or incidence as an indicator of the state of the outbreak appears to play no role on the equilibria of the dynamics. However, it turns out that favouring incidence may help to flatten the curve in the transient phase of the dynamics. We also find a fair agreement between our analytical predictions and simulations run on an empirical multiplex network.

Figures (7)

• Figure 1: Schematic description of the compartmental model for an SIS-like epidemic evolution with an imitation dynamics affecting the probability that individuals cooperate with the isolation measure.
• Figure 2: The 'one way' implementation is more efficient than the 'two ways' implementation. a) Violin plots of the distribution of $50$ combined mean errors for the 'one way' (green) and the 'two ways' (blue) implementations, and for different numbers of runs. The red lines are the median (dashed) and the lower and upper quartiles (dotted). b) Distribution of the execution time for both the implementations. We consider a complete network of size $N = 500$.
• Figure 3: Equilbrium prevalence. Heatmaps of the equilibrium prevalence depending on the cost $c$ of quarantine, the lack of coverage $q$ of the infectious period by quarantine, the rapidity of entrance in isolation $u$ and the degree of caution $\varepsilon$ when breaking quarantine. The limits separating the ENC and EPC regimes, as well as the ones separating the EPC and EFC regimes are shown (black lines). The basic reproduction number is set to $\mathcal{R}_0 = 5$.
• Figure 4: Conditions for an optimal duration of quarantine. a): Different choices of the parameter $c$: $c = 0.15$ (dark green), $c(q)=0.1*\exp\left(0.4*\left(1 + \frac{1}{q}\right)\right)$ (light green), $c = 0.6$ (yellow), $c(q)=\sqrt{1+q}$ (blue), $c = 1$ (orange) and $c(q)=\frac{1}{\sqrt{1+q}}$ (red). b): Equilibrium prevalence as a function of $q$ when considering the values of $c$ shown in panel a. Parameter values: $\mathcal{R}_0 = 5$, $u = 3$, $\varepsilon = 0.5$.
• Figure 5: The incidence as an indicator of the state of the outbreak can help to flatten the curve. We show the value of the peak prevalence (purple) and its temporal average (dark cyan) as functions of the parameter $p$, for various choices of the basic reproduction number $\mathcal{R}_0$ and of the volatility of opinions $\kappa$. In all panels the system is in the 'Partial Compliance' regime. We show both the effect of $p$ in the ODE formulation (continuous line) and in the stochastic simulations run over an empirical two-layers network (dot-dashed line). Parameter values: $u = 5$, $q = 0.25$, $c = 0.3$, $p = 0.4$, $\varepsilon = 0.5$, $x_0 = 0.4$.

Theorems & Definitions (1)

• Lemma 1: Routh-Hurwitz Conditions for Orders 3 and 5 Allen_2007