Early warning indicators of epidemics on a coupled behaviour-disease model with vaccine hesitance and incomplete data

Figure 1.  Diagrams showing the epidemiological and opinion dynamics of the model

Figure 2.  Implementation of the model dynamics used for each stochastic realisation. Red blocks represent loops run for each node in the network per time step. Yellow blocks are executed only once per realisation. Blue diamonds represent binary decisions, and grey boxes represent simple instructions

Figure 3.  Diagrams of the trends of overall model behaviour and intertransition distance, and the locations of the social and infection transitions with respect to the sampling proportion $ \beta $. (a) Unsteady trend of the intertransition distance $ K_p-K_s $ (purple), with the inset graph showing the estimated locations of $ K_s^1 $ (red) and $ K_p^1 $ (blue). (b) Trends of model variables of the social and infection dynamics for $ \sigma = 0 $ (left) and $ \sigma = 0.25 $ (right). Hesitance persists in the simulation across all strengths of the social norm $ \sigma $. (c) Estimated locations of the social and infection transitions $ K_s ^\beta $ and $ K_p^\beta $ drift as the proportion of agents randomly sampled $ \beta $ changes

Figure 4.  Changes in the trends of the $ \kappa $-series of the mutual information statistic $ \langle\mathcal{M}\rangle $, the probabilities of having an infected neighbour $ \big\langle {\Gamma_{{\ast}}}\big\rangle $ and the numbers of opinion changes $ \big\langle {\!\Theta_{{\ast}}\!}\big\rangle $ predict transitions in the infection ($ K_p^1 $, dashed vertical line) and social ($ K_s^1 $, dotted vertical black line) dynamics for both social norms $ \sigma = 0 $ (left panels) and $ \sigma = 0.25 $ (right panels). (a) Trend of the mutual information statistic $ \langle\mathcal{M}\rangle $. (b) Probability of hesitant ($ \big\langle {\Gamma_{{H}}}\big\rangle $), pro-vaccine ($ \big\langle {\Gamma_{{N}}}\big\rangle $) and anti-vaccine ($ \big\langle {\Gamma_{{V}}}\big\rangle $) agents of having an infected neighbour. (c) Number of opinion changes committed by hesitant ($ \big\langle {\!\Theta_{{H}}\!}\big\rangle $), anti-vaccine ($ \big\langle {\!\Theta_{{N}}\!}\big\rangle $ and pro-vaccine ($ \big\langle {\!\Theta_{{V}}\!}\big\rangle $) agents, and the total number of changes $ \big\langle {\!\Theta_{{\Sigma}}\!}\big\rangle $

Figure 5.  Lead distances of non-connectivity-based EWS explored in the study, obtained by applying the SNHT. Bar charts on the right of the panels show the proportion of $ \sigma $ values for which each test gave the largest lead distance of all EWS tested. (a) Lead distances of the mutual information statistic $ \langle\mathcal{M}\rangle $ under the Lanzante, Pettitt, Buishand range and standard normal homogeneity tests. (b) Probabilities of hesitant ($ \big\langle {\Gamma_{{H}}}\big\rangle $), pro- ($ \big\langle {\Gamma_{{V}}}\big\rangle $) and anti-vaccine ($ \big\langle {\Gamma_{{N}}}\big\rangle $) agents having an infected neighbour. (c) Opinion changes committed by pro-vaccine ($ V_s $), anti-vaccine ($ N $) and hesitant ($ H $) agents

Figure 6.  Sudden changes in trend of the sizes ($ \!|Z_{{\ast}}| $) and numbers ($ \#Z_{{\ast}} $) of opinion communities with respect to the social norm $ \sigma $ strongly signal upcoming transitions in the model dynamics. Social norm $ \sigma = 0 $ for panels on the left and $ \sigma = 0.25 $ for panels on the right. (a) Sizes of pro-vaccine communities. (b) Sizes of anti-vaccine communities. (c) Sizes of communities of hesitant agents. (d)

Figure 7.  Trends of the lead distances of connectivity-based EWS under the SNHT with respect to the social norm $ \sigma $. Bar charts on the right of the panels show the proportion of $ \sigma $ values for which each test gave the largest lead distance of all EWS tested. Quite a few indicators perform poorly, with near-zero lead distances. (a) Lead distances of the mean sizes of opinion communities. (b) Lead distances of the mean sizes of opinion echo chambers. (c) Lead distances yielded by the numbers of communities and chambers on the network. (d) Trends in the lead distances of the various join count statistics. Bar charts on the right give the percentage of social norm $ \sigma $ values for which the EWS gave the largest lead distance of all EWS

Figure 8.  Values of the global clustering coefficient $ \big\langle {C_{{\ast}}}\big\rangle $, triad census $ \big\langle {\!\Delta_{{\ast}}\!}\big\rangle $ and opinion network diameter $ \big\langle {\Omega_{{\ast}}}\big\rangle $ with respect to the perceived vaccine risk $ \kappa $. Social norm $ \sigma = 0 $ for the panels on the left and $ \sigma = 0.25 $ for the panels on the right. (a) Clustering coefficient of the various opinion networks. (b) Number of opinion triads on the network. (c) Diameters of the various opinion networks

Figure 9.  Trends in the lead distances given by other connectivity-based EWS investigated using the SNHT. (a) Lead distances given by the clustering coefficients of the various opinion networks. (b) Lead distances given by the numbers of opinion triads. (c) Lead distances of the diameters of the various opinion networks

Figure 10.  Side-by-side bar charts comparing the relative performance of each EWS under the SNHT. Green bars give the proportion of $ \sigma $ values for which the EWS gave the maximum lead distance of all EWS and the red bars give the proportion for which the lead distances were minima, failures or undefined

Figure 11.  Heat map plots showing the ensemble means of model variables with respect to the social norm $ \sigma $ and the perceived vaccine risk $ \kappa $. All agents were observed ($ \beta = 1 $). (a) $ \big\langle H\big\rangle $ (hesitant agents). (b) $ \big\langle N\big\rangle $ (anti-vaccine agents). (c) $ \big\langle {V_s}\big\rangle $ (pro-vaccine agents). (d) $ \big\langle {V_p}\big\rangle $ (vaccinated agents). (e) $ \big\langle {I}\big\rangle $ (infected agents). (f) $ \big\langle {S}\big\rangle $ (susceptible agents)

Figure 12.  The dynamics of the model shows bistability about the vaccine risk $ \kappa = 0 $ for all parameter values tested. (a) Final number of recovered agents. (b) Final number of vaccinated agents. (c) Final number of susceptible agents

Figure 13.  Locations of the physical and social transitions ($ K_p^\beta $ and $ K_s^\beta $), and trends of the intertransition distance $ K_p^\beta-K_s^\beta $ obtained by sampling different proportions $ \beta $ of agents in the network. (a) Locations of the physical ($ K_p^{0.4} $, inset, red) and social ($ K_s^{0.4} $, inset, blue) transitions found by sampling proportion $ \beta = 0.4 $ of the network. Intertransition distance $ K_p^{0.4}-K_s^{0.4} $ (purple) is shown in the foreground. (b) Locations of the physical ($ K_p^{0.6} $, inset, red) and social ($ K_s^{0.6} $, inset, blue) transitions found by sampling proportion $ \beta = 0.6 $ of the network. Intertransition distance $ K_p^{0.6}-K_s^{0.6} $ (purple) is shown in the foreground. (c) Locations of the physical ($ K_p^{0.8} $, inset, red) and social ($ K_s^{0.8} $, inset, blue) transitions found by sampling proportion $ \beta = 0.8 $ of the network. Intertransition distance $ K_p^{0.8}-K_s^{0.8} $ (purple) is shown in the foreground. (d) Locations of the physical ($ K_p^{1} $, inset, red) and social ($ K_s^{1} $, inset, blue) transitions found by sampling proportion $ \beta = 1 $ of the network. Intertransition distance $ K_p^{1}-K_s^{1} $ (purple) is shown in the foreground

Figure 14.  Time series demonstrating the sensitivity of system dynamics to small changes in perceived vaccine risk $ \kappa $ in the absence of a social norm ($ \sigma = 0 $). Time in week (time steps) is given by $ \tau $. (a) $ [I]_{\tau\le6} $, with $ \kappa = 0.00625 $. (b) $ [I]_{\tau\le6} $, with $ \kappa = 0 $. (c) $ [I]_{\tau\le6} $, with $ \kappa = -0.0125 $. (d) $ [V_s] $, with $ \kappa = 0.05 $. (e) $ [V_s] $, with $ \kappa = 0 $. (f) $ [V_s] $, with $ \kappa = -0.025 $. (g) $ [V_p] $, with $ \kappa = 0.05 $. (h) $ [V_p] $, with $ \kappa = 0 $. (i) $ [V_p] $, with $ \kappa = -0.025 $

Figure 15.  Grid comparing the performance of each EWS at each value of the social norm $ \sigma $ under the SNHT. Green tiles represent maximum lead distances (of all EWS), grey tiles represent intermediate lead distances, red tiles represent minimum positive lead distances (of all EWS), white tiles represent missing/undefined values, black tiles represent failures (negative lead distances) and yellow tiles indicate that all EWS gave the same lead distance

Figure 16.  Variance in the trends of the lead times of some EWS caused by sampling a proportion $ \beta $ of the social network. (a-c) $ {\text{SNHT}}\{\langle {C_{{\ast}}^{{\beta}}} \rangle\} $, (d-f) $ {\text{SNHT}}\{\langle {\!\Delta_{{\ast}}^{{\beta}}} \rangle\} $, (g-i) $ {\text{SNHT}}\{\langle {\Omega_{{\ast}}^{{\beta}}} \rangle\} $. In all cases, $ \beta\in\{0.4, 0.6, 0.8\} $ respectively

Figure 17.  Changes in the sizes ($ \left|{J_{{\ast}}}\right| $) and numbers ($ \big\langle {\!\#{J_{{\ast}}\!}}\big\rangle $) of echo chambers in the network warn of the transitions $ K_s^1 $ and $ K_p^1 $. Social norm $ \sigma = 0 $ for the panels on the left and $ \sigma = 0.25 $ for panels on the right. (a) Sizes of pro-vaccine echo chambers. (b) Sizes of anti-vaccine echo chambers. (c) Sizes of echo chambers of hesitant agents. (d) Numbers of different types of echo chambers on the network

Figure 18.  Trends in the join count statistics of the network also give clear warnings of approaching social and infection transitions. Social norm $ \sigma = 0 $ for panels on the left and $ \sigma = 0.25 $ for panels on the right. (a) The proportions of all join counts on the network. (b) The proportion of $ \big\langle {N, V_s}\big\rangle $ dissimilar joins. (c) The proportions of joins involving hesitant agents

Figure 19.  Changes in the trends of the lead times of some EWS caused by sampling a proportion $ \beta $ of the social network. (a-d) $ {\text{SNHT}}\{\langle { \mathcal M^{{\beta}}} \rangle\} $. (e-h) $ {\text{SNHT}}\!\left\{\langle {\!\Gamma_{{\ast}}^{{\beta}}} \rangle\right\} $. (i-l) $ {\text{SNHT}}\{\langle {\!\Theta_{{\ast}}^{{0.6}}} \rangle\} $. In each case, $ \beta\in\{0.4, 0.6, 0.8, 1\} $ respectively

Figure 20.  Changes in the trends of the lead times of some EWS caused by sampling a proportion $ \beta $ of the social network. (a-c) $ {\text{SNHT}}\{\langle {|Z_{{\ast}}^{{\beta}}|} \rangle\} $, (d-f) $ {\text{SNHT}}\{\langle {|J_{{\ast}}^{{\beta}}|} \rangle\} $, (g-i) $ {\text{SNHT}}\{\langle {\#Z_{{\ast}}^{{\beta}}} \rangle\} $ and $ {\text{SNHT}}\{\langle {\#J_{{\ast}}^{{\beta}}} \rangle\} $, (j-l) $ {\text{SNHT}}\{\langle {{\ast, \ast}} \rangle^{{\beta}}\} $. In all cases, $ \beta\in\{0.4, 0.6, 0.8\} $ respectively

Figure 21.  Grid showing the performance of EWS for each value of the social norm $ \sigma $ when $ 40\% $ of the network is sampled

Figure 22.  Bar chart showing the summary performance of EWS with respect to the value of the social norm $ \sigma $ when $ 40\% $ of the network is sampled

Figure 23.  Grid showing the performance of EWS for each value of the social norm $ \sigma $ when $ 60\% $ of the network is sampled

Figure 24.  Bar chart showing the summary performance of EWS for each value of the social norm $ \sigma $ when $ 60\% $ of the network is sampled

Figure 25.  Grid showing the performance of EWS for each value of the social norm $ \sigma $ when $ 80\% $ of the network is sampled

Figure 26.  Bar chart showing the summary performance of EWS when $ 80\% $ of the network is sampled