Epidemic prevalence information on social networks can mediate emergent collective outcomes in voluntary vaccine schemes

Abstract

The effectiveness of a mass vaccination program can engender its own undoing if individuals choose to not get vaccinated believing that they are already protected by herd immunity. This would appear to be the optimal decision for an individual, based on a strategic appraisal of her costs and benefits, even though she would be vulnerable during subsequent outbreaks if the majority of the population argues in this manner. We investigate how voluntary vaccination can nevertheless emerge in a social network of rational agents, who make informed decisions whether to be vaccinated, integrated with a model of epidemic dynamics. The information available to each agent includes the prevalence of the disease in their local network neighborhood and/or globally in the population, as well as the fraction of their neighbors that are protected against the disease. Crucially, the payoffs governing the decision of agents vary with disease prevalence, resulting in the vaccine uptake behavior changing in response to contagion spreading. The collective behavior of the agents responding to local prevalence can lead to a significant reduction in the final epidemic size, particularly for less contagious diseases having low basic reproduction number [Formula: see text]. Near the epidemic threshold ([Formula: see text]) the use of local prevalence information can result in divergent responses in the final vaccine coverage. Our results suggest that heterogeneity in the risk perception resulting from the spatio-temporal evolution of an epidemic differentially affects agents' payoffs, which is a critical determinant of the success of voluntary vaccination schemes.

Fig 1. Schematic representation of the coupling between the spread of an epidemic and strategic vaccine uptake behavior by individuals.

Agents are classified according to their state with respect the disease as Susceptible (S), Infected (I), Recovered (R) and Vaccinated (V). The two layers represent the states of the nodes at two time instants. The broken lines represent the change in the state of agents and grey solid lines represent the flow of information about the state (infected and removed) of agents in the network. The curved arrow between the two layers represents the update (time evolution) of the system. The broken curve encloses the game-theoretic process that determines whether an agent decides to vaccinate or not, based on the probability of an agent choosing to get vaccinated P(S → V). The table inside the broken curve is a payoff matrix used by an agent to make decisions. Here the “opponent” is a hypothetical agent having identical information, choices of actions and associated payoffs. The payoff received by the focal player is represented by a function of the form U_xy(f_i, f_p), where x and y are the actions of the focal player and opponent respectively. The fractions of infected and protected (immune) agents are represented by f_i and f_p, respectively. By varying these two parameters the nature of the game can change between different classes, as shown in the inset to the lower left.

Fig 2. Simulation results for the co-evolution of epidemic spreading and vaccine uptake behavior in the largest connected component of a social network in a village of southern India.

A snapshot of the network for village 55 (see Ref. [57], data obtained from Ref. [55]) with N = 1180 and 〈k〉 = 9.78, showing the final states of nodes following a simulated epidemic with β = 0.025 and τ_I = 10 for (a) α = 0 and (b) α = 1. The colors of the nodes are representative of the final state: blue, susceptible; yellow, vaccinated; red, recovered (i.e., infected during the epidemic). (c) A sample time series showing the evolution of S, I, R and V for a simulated epidemic with β = 0.025 and τ_I = 10 for α = 0 (left) and α = 1 (right). The inset of (c) provides a closer view of the sudden emergence of vaccination when the prevalence becomes sufficiently high. A comparison of the final fraction of agents (d) infected inf_∞ and (e) vaccinated vac_∞ during a simulated epidemic with different values of ${\mathcal{R}}_0$, for α = 0 and α = 1. Each of the points represents the median of 1000 simulation runs and the patches indicate the interquartile range (IQR).

Fig 3. Simulation results for the co-evolution of epidemic spreading and vaccine uptake behavior in Erdős-Rényi networks.

(a) A sample time series showing the evolution of S, I, R and V for a simulated epidemic with β = 0.02 and τ_I = 10 in random networks with N = 1024 and 〈k〉 = 10 is displayed for α = 0 (left) and α = 1 (right). We display a comparison of the final fraction of agents that are (b) infected inf_∞ and (c) vaccinated vac_∞ during a simulated epidemic with different values of ${\mathcal{R}}_0$, for α = 0 and α = 1. Each point represents the median of 1000 simulation runs and the patches indicate the corresponding IQR. (d) Dependence of crossover area $\mathcal{A}$ on average node degree 〈k〉 behaves similarly in empirical social networks and model random networks. The solid line and patch shows the median and IQR of the 1000 simulated epidemics on Erdős-Rényi networks respectively. The circle and error bars represent the median and IQR of the 1000 simulated epidemic on social network of villages in southern India that have a largest connected component greater than 1000.

Fig 4. Use of local or global information by agents can qualitatively alter the collective vaccination outcome to epidemics.

(a) Assessing the dependence of vac_∞ in a network having average node degree 〈k〉 = 10 on population size N. The results are shown for α = 0 (top) and α = 1 (bottom). (b) Bimodality coefficient $\mathcal{B}\mathcal{C}$ for the probability distribution of V_∞ calculated over 2000 trials for Erdős-Rényi networks with N = 16384 and 〈k〉 = 10, and shown over the range of values of ${\mathcal{R}}_0$ and α. (c) Probability distribution of V_∞ as a function of ${\mathcal{R}}_0$, calculated over 2000 trials, and shown for different values of α.