Fig 1.
Illustration of the model describing opinion formation and disease propagation.
Blue circles represent neutral individuals, green circles represent pro-vaccine individuals, and red circles represent anti-vaccine individuals. The first stage involves the generation of the social network and the initialization of agent opinion states as agents with neutral opinions. Then, external exposures to positive and negative information triggers the initial seed sets for both anti-vaccine and pro-vaccine contagion. Opinion diffusion continues until a stopping criterion is reached. In this stage, a vaccination takes place for all non-negative individuals. Subsequently, a randomly chosen non-vaccinated individual is infected, and the spread of the disease continues until no further newly infected agents are generated. Finally, we record the number of recovered agents to measure the epidemic size.
Fig 2.
Illustration of opinion propagation and campaigning methods.
The figure shows the exchange of vaccine-related opinions and external exposures, as well as the positive campaign types. (A) Random dissemination of negative and positive vaccine-related sentiments from external campaigns to the public. (B) Targeted positive campaign. μ−, and μ+ are the general exposure rates for negative and positive sentiments, respectively. ω−, ω+ are the social exposure rates for negative and positive opinions, respectively.
Table 1.
Model parameters and descriptions.
Table 2.
Campaign strategies and descriptions.
Fig 3.
Average epidemic size for the random campaign (StatRandAll) as a function of the social rate . (A) τ = ∞ (B) τ = 400.
The figure shows results for different positive exposure rates μ+ with fixed negative exposure rate μ− = 0.001.
Fig 4.
Average epidemic size for static campaigns as a function of the social rate .
(A) and (B) for the targeted random campaign (StatRandT). (C) and (D) for the centrality-based campaign (StatCentT). (A) and (C) τ = ∞, (B) and (D) τ = 400. The figures show different positive exposure rates μ+ with fixed negative exposure rate μ− = 0.001. Target set size is T = 500. For each scenario we generate 300 different networks, and perform 300 SIR model runs for each network.
Fig 5.
Dependence of the average epidemic size on the campaign updating interval using dynamic campaigns.
(A) and (B) represent the dynamic random campaign DynRandT, and (C) and (D) represent the DynAntiT campaign. (A) and (C) τ = ∞, (B) and (D) τ = 400. The targets set T = 50, social rate is . The figures show different positive exposure rates μ+ with fixed negative exposure rate
.
Fig 6.
Dependence of the average epidemic size on the target number of anti-vaccine neighbors ζ using the DynLocT campaign.
The updating time is , the social rate is ω = 0.006 for both negative and positive
, and the size of the target set is T = 50. The figures show different positive exposure rates μ+ with fixed negative exposure rate
.
Fig 7.
Average epidemic size using the DynAdvLocT dynamic campaign in the long-run setting of τ = ∞.
The figure shows the performance of the dynamic campaigns with T = 50 targets. The epidemic size is shown as a function of the target number of anti-vaccine neighbors ζ and the target number of neutral neighbors Z a neutral has at time t. The updating time is =1, and the social rate is ω = 0.006 for both negative and positive
. The general exposure influence rate for negative is
and the positive rate is shown in the figures captions.
Fig 8.
Targeting scheme for DynAdvLocT campaign.
The top panels illustrate the neighborhood structure of the target set at a single time step t=350 during the opinion diffusion stage, specifically showing the number of anti-vaccine neighbors and pro-vaccine neighbors. These panels represent the average number of agents with x anti-vaccine neighbors and y neutral neighbors for various settings: (A) ζ = 1 , Z = 8, (B) ζ = 8 , Z = 1, (C) ζ = 6 , Z = 6. (D) and (E) illustrate the evolution of anti-vaccine opinion adopters and pro-vaccine opinion adopters, respectively, while panel (F) represents the corresponding epidemic size. ‘x’ indicates that no agent exists for that neighborhood pattern. The targeting analysis is an average of 15 simulations.
Fig 9.
Epidemic size obtained with varying target sizes for targeted campaigns in the long-run setting where τ = ∞.
The figure illustrates the epidemic size obtained for all campaigns with general exposure rates . The social rate is
for all scenarios. For dynamic campaigns, the updating time interval is
=1, for DynLocT ζ = 1, and DynAdvLocT campaigns the target numbers of negative and neutral neighbours are ζ = 10 , Z = 10. For each scenario we generate 500 different networks, and perform 500 SIR model runs for each network.
Fig 10.
Cross-campaign epidemic size comparison with τ = ∞.
The figure illustrates the epidemic size obtained for all campaigns with different positive rates μ+ compared to negative rate μ−. For all campaigns, . First group:
, second group:
, and third group:
. Social rate is ω = 0.006 for all scenarios. Target set size T = 500 for static campaigns, i.e., StatRandT, and StatCentT, and T = 50 for the other dynamic campaigns. For dynamic campaigns, the updating time is
, in addition we include
for DynLocT and DynAdvLocT campaigns. In addition, for DynLocT ζ =1, for DynAdvLocT ζ = Z =10.
Fig 11.
Cross-campaign epidemic size comparison with τ = ∞.
The figure illustrates the epidemic size obtained for the best campaigns with different positive rates μ+ compared to negative rate μ− and for lower positive social rate. For all campaigns, . First group:
and second group:
. Social rate is
for all scenarios. Target set size T = 500 for static campaigns, i.e., StatRandT, and StatCentT, and T = 50 for the other dynamic campaigns. For dynamic campaigns, the updating time is
for DynRandT and
for DynAdvLocT. For DynAdvLocT ζ = Z =10. The results are the average of 50 simulations.