Self-initiated behavioral change and disease resurgence on activity-driven networks

Nicolò Gozzi, Martina Scudeler, Daniela Paolotti, Andrea Baronchelli, and Nicola Perra

Phys. Rev. E 104, 014307 – Published 12 July 2021

Abstract

We consider a population that experienced a first wave of infections, interrupted by strong, top-down, governmental restrictions and did not develop a significant immunity to prevent a second wave (i.e., resurgence). As restrictions are lifted, individuals adapt their social behavior to minimize the risk of infection. We explore two scenarios. In the first, individuals reduce their overall social activity towards the rest of the population. In the second scenario, they maintain normal social activity within a small community of peers (i.e., social bubble) while reducing social interactions with the rest of the population. In both cases, we investigate possible correlations between social activity and behavior change, reflecting, for example, the social dimension of certain occupations. We model these scenarios considering a susceptible-infected-recovered epidemic model unfolding on activity-driven networks. Extensive analytical and numerical results show that (i) a minority of very active individuals not changing behavior may nullify the efforts of the large majority of the population and (ii) imperfect social bubbles of normal social activity may be less effective than an overall reduction of social interactions.

Received 16 November 2020
Revised 17 April 2021
Accepted 23 June 2021

DOI:https://doi.org/10.1103/PhysRevE.104.014307

Physics Subject Headings (PhySH)

Epidemic Epidemic growth Evolving networks

Networks

Authors & Affiliations

Nicolò Gozzi ¹, Martina Scudeler ², Daniela Paolotti³, Andrea Baronchelli^4,5, and Nicola Perra^1,*

¹Networks and Urban Systems Centre, University of Greenwich, London SE10 9LS, United Kingdom
²University of Turin, 10124 Turin, Italy
³ISI Foundation, 10126 Turin, Italy
⁴City, University of London, London EC1V 0HB, United Kingdom
⁵The Alan Turing Institute, London NW1 2DB, United Kingdom

^*n.perra@greenwich.ac.uk

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Vol. 104, Iss. 1 — July 2021

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Figure 1
Analytical value of the basic reproductive number $R_{0}$ in different scenarios: (a) $R_{0}$ as a function of $γ$ and $ψ$ as obtained in Eq. (6) in the case of imperfect adoption independent of the nodes' activity, with $p = 0.75$ and $w = 0.8$ ; (b) $R_{0}$ as a function of $p$ and $w$ as obtained in Eq. (6) in the case of imperfect adoption independent of the nodes' activity, with $γ = 0.1$ and $ψ = 0.1$ ; and (c) $R_{0}$ as a function of $p$ and $w$ as obtained in Eq. (20) in the case of adoption dependent of the nodes' activity, with $γ = 0.1$ and $ψ = 0.1$ . The inset highlights a small region of the phase space. In all figures we set $ε = 10^{- 3}$ , $m = 2$ , $α = 2.1$ , and $μ = 10^{- 2}$ and we indicate with a solid red line the threshold $R_{0} = 1$ . The parameters are set so that when $γ = 1$ and $ψ = 1$ (i.e., without behavioral changes) (a) $R_{0} = 2$ and (b) and (c) $R_{0} = 1.2$ .
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Figure 2
(a) Final epidemic size ( $r_{\infty} = R_{\infty} / N$ ) with $95 %$ confidence intervals and (b) normalized relative variance $σ_{r_{\infty}} / σ_{r_{\infty}}^{\max}$ for different values of $λ$ in the case of imperfect random adoption. Vertical dashed lines indicate the analytical threshold derived from Eq. (6) for the different values of $p$ and $w$ considered. (c) and (d) We repeat the analysis in the case of imperfect adoption dependent on the nodes' activity for different values of $p$ and $w$ . In all figures, we represent median and $95 %$ confidence intervals obtained from $10^{2}$ stochastic simulations for each point with the parameters $μ = 10^{- 2}$ , $N = 10^{6}$ , $m = 2$ , $ε = 10^{- 3}$ , $α = 2.1$ , $i_{0} = I_{0} / N = 0.01$ (initial fraction of infected seeds), $γ = 0.8$ , and $ψ = 0.1$ .
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Figure 3
(a) Epidemic size as a function of $λ$ in the presence of communities and behavioral change modeled with activity reduction. We consider both the case in which adoption is assigned randomly (red) and the case in which adoption is assigned in increasing order of activity (blue). Vertical lines indicate the analytical threshold in the absence of communities computed from Eqs. (6) and (20). In the inset we represent the normalized relative variance $σ_{r_{\infty}} / σ_{r_{\infty}}^{\max}$ for different values of $λ$ in the case of imperfect random and activity-based adoption in the presence of communities. We set $γ = 0.8$ , $ψ = 0.1$ , $p = w = 0.6$ , $η = 0.6$ , and $s = 10$ . (b) Epidemic size as a function of $λ$ in the presence of communities and behavioral change modeled with modularity increase. Also in this case we represent in the inset the normalized relative variance $σ_{r_{\infty}} / σ_{r_{\infty}}^{\max}$ for different values of $λ$ . We set $η = 0.6$ , $η_{γ} = 0.68$ , $η_{ψ} = 0.96$ , and $p = w = 0.6$ . (c) Ratio between the final epidemic size obtained in the MI and the AR case as a function of $η_{γ}$ . We set $λ = 0.6$ , $η = 0.6$ , $η_{ψ} = 0.99$ , and $p = w = 0.6$ . In all figures, we represent median and $95 %$ confidence intervals obtained from $10^{2}$ stochastic simulations for each point with the $μ = 10^{- 2}$ , $ε = 10^{- 3}$ , $α = 2.1$ , $N = 10^{5}$ , and $i_{0} = I_{0} / N = 0.01$ (initial fraction of infected seeds).
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Figure 4
Epidemic size as a function of $η$ in the presence of communities and behavioral change modeled with activity reduction (blue) and modularity increase (red) for (a) the case in which adoption is assigned randomly and (b) the case in which adoption is assigned in increasing order of activity. In both figures, we represent median and $95 %$ confidence intervals obtained from $10^{2}$ stochastic simulations for each point with the parameters $μ = 10^{- 2}$ , $ε = 10^{- 3}$ , $α = 2.1$ , $N = 10^{5}$ , $λ = 0.6$ , $p = w = 0.6$ , $s = 10$ , and $i_{0} = I_{0} / N = 0.01$ (initial fraction of infected seeds). Furthermore, in both figures we assume that the susceptible individuals reduce their behaviors by $20 %$ and infected individuals by $90 %$ .
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