Spatial early warning signals of social and epidemiological tipping points in a coupled behaviour-disease network

Abstract

The resurgence of infectious diseases due to vaccine refusal has highlighted the role of interactions between disease dynamics and the spread of vaccine opinion on social networks. Shifts between disease elimination and outbreak regimes often occur through tipping points. It is known that tipping points can be predicted by early warning signals (EWS) based on characteristic dynamics near the critical transition, but the study of EWS in coupled behaviour-disease networks has received little attention. Here, we test several EWS indicators measuring spatial coherence and autocorrelation for their ability to predict a critical transition corresponding to disease outbreaks and vaccine refusal in a multiplex network model. The model couples paediatric infectious disease spread through a contact network to binary opinion dynamics of vaccine opinion on a social network. Through change point detection, we find that mutual information and join count indicators provided the best EWS. We also show the paediatric infectious disease natural history generates a discrepancy between population-level vaccine opinions and vaccine immunity status, such that transitions in the social network may occur before epidemiological transitions. These results suggest that monitoring social media for EWS of paediatric infectious disease outbreaks using these spatial indicators could be successful.

Figure 1

Representation of the infection (a) and opinion (b) dynamics of the model occurring on the physical and social layers of the 2-layer network, respectively. (a) Effective contacts occur between susceptible $S$ and infected $I$ agents with probability $p$ per time step ($1$ week). Upon deciding to vaccinate (with probability ${\mathbb{P}}_n(N\to V_s)$), a susceptible agent $n$ becomes physically vaccinated ($S\to V_p$). Infection lasts $\ell =2$ weeks after which agents recover ($I\to R$). Upon death (with probability $\mu$ per week), an agent is “rebirthed” with either vaccinated (probability $\alpha \cdot \mu$) or susceptible (probability $\mathrm{(1}-\alpha )\cdot \mu$) status. (b) Per time step, each agent switches between pro- ($V_s$) and anti-vaccine ($N$) opinion with probabilities ${\mathbb{P}}_n(N\to V_s)$ and ${\mathbb{P}}_n(V_s\to N)$ respectively upon interaction with a dissenting neighbour. $\alpha$ gives the probability of being birthed with pro-vaccine opinion $V_s$.

Figure 2

Contour plots of the region $(\sigma ,\kappa )\in [0,2.4]\times [-1,0.2]$ of the parameter plane, capturing the transition dynamics of both the social and physical dynamics averaged over 20 realisations of each set of parameters; $\sigma$ represents the strength of the social norm, and $\kappa$ the vaccine risk. (a) $\langle V_s\rangle$ (proportion of pro-vaccine agents), (b) $\langle V_p\rangle$ (vaccine coverage), and the corresponding trends in (c) $\langle \mathcal{M}\rangle$ (mutual information) and (d) $\langle N,V_s\rangle$ (dissimilar join count).

Figure 3

Time series demonstrating high sensitivity of the social dynamics to small changes (both positive and negative) in vaccine risk $\kappa$ when the strength of the social norm $\sigma =0$. All panels show the results of $100$ realisations of respective parameter combinations. (a) $\left[V_s\right]$, $\kappa =0.03125$. (b) $\left[V_s\right]$, $\kappa =0$. (c) $\left[V_s\right]$, $\kappa =-0.03125$. (d) $\left[V_p\right]$, $\kappa =0.03125$. (e) $\left[V_p\right]$, $\kappa =0$. (f) $\left[V_p\right]$, $\kappa =-0.03125$.

Figure 4

Trends of the EWS’ equilibrium values approaching the transitions of the social and physical dynamics $K_s$ and $K_p$ (marked in order by the first and second vertical black lines) respectively, demonstrating the signals given by each tool with respect to the perceived vaccine cost $\kappa$. The intervals in each panel represent one standard deviation of the mean equilibrium value in each stochastic realisation of the model. Social norm $\sigma =0$ for column (A), and $\sigma =0.25$ for column (B). (a,b) Social dynamics $\langle V_s\rangle$ (green, solid), $\langle N\rangle$ (red, solid) and physical dynamics $\langle R\rangle$ (black, dashed), $\langle V_p\rangle$ (blue, dashed). (c,d) Join counts: $\langle N,N\rangle$ (blue), $\langle N,V_s\rangle$ (red), $\langle V_s,V_s\rangle$ (green). (e,f) Dissimilar join count $\langle N,V_s\rangle$ alone (red). (g,h) Mutual information $\langle \mathcal{M}\rangle$ (red). (i,j) Moran’s I $\langle \mathcal{J}\rangle$ (red). (k,l) Geary’s C $\langle \mathcal{C}\rangle$ (red).

Figure 5

(a) Demonstration of the shrinking intertransition distance $K_p-K_s$ (purple), with the inset graph showing the estimated locations of $K_s$ (blue) and $K_p$ (red). (b) Under the Lanzante change point test, the lead time of each EWS $\left(B_{\mathrm{W}\mathrm{S}}^{\mathrm{L}\mathrm{a}\mathrm{n}}\right)$ varies substantially with the strength of the social norm $\sigma$; trends corresponding to each EWS are represented by the different colours in the legend; the bar chart on the right gives the number of $\sigma$ values for which each individual EWS gave the maximum lead time. The same for (c), which shows the variance of lead times $\left(B_{\mathrm{S}/\mathrm{P}}^{\mathrm{L}\mathrm{a}\mathrm{n}}\right)$ of the model variables with social norm $\sigma$, with the bar chart on the right giving the number of times each model variable gave the maximum lead time of all variables (over all values of sigma).

Figure 6

Graph of the trends of ${\chi }_{\min }^{\mathrm{Lan}}$ (blue) and ${\chi }_{\max }^{\mathrm{Lan}}$ (red) with respect to the value of the social norm $\sigma$, allowing us to do maximin and maximax comparisons of the two classes of warning signals (WS and model variables S/P). The green-shaded region shows where ${\chi }_{⁎}^{\mathrm{Lan}}>0$, and the red-shaded region shows where ${\chi }_{⁎}^{\mathrm{Lan}}<0$. The inset table shows the percentage of $\sigma$ values for which ${\chi }_{⁎}^{\mathrm{Lan}}>{\epsilon }_{⁎}^{\mathrm{Lan}}$ (pos: tracking EWS works better), $\left|{\chi }_{⁎}^{\mathrm{L}\mathrm{a}\mathrm{n}}\right|\le {\epsilon }_{⁎}^{\mathrm{L}\mathrm{a}\mathrm{n}}$ (zero: both approaches work equally well) and ${\chi }_{⁎}^{\mathrm{Lan}}<-{\epsilon }_{⁎}^{\mathrm{Lan}}$ (neg: monitoring simple trends works better). (blue curve, row 1 of inset table) Positive values (green-shaded region) of ${\chi }_{\min }^{\mathrm{Lan}}$ occur at the $\sigma$ (social norm) values where the worst-performing (least lead time) EWS still gives higher lead time than the worst-performing model variable. (red curve, row 2 of inset table) Similar to above, positive values of ${\chi }_{\max }^{\mathrm{Lan}}$ occur (in the red-shaded region) when the EWS perform absolutely better than the model variables.