Fig 1.
We model the vaccination dynamics under imperfect vaccine as a two-stage game. At stage 1 (vaccination choice), a proportion V0 of the population decides to vaccinate. Vaccination costs and provides imperfect protection against the infectious disease. At stage 2 (health outcome), we use the Susceptible-Infected-Recovered (SIR) model with preemptive vaccination to simulate the epidemiological process. Every individual faces the risk of infection, which depends on their vaccination status and the number of infectious neighbors,
, they have. The transmission rate of the disease (per day per infectious neighbor) to unvaccinated individuals is β, as compared to
for vaccinated. Here, the parameter
denotes the level of vaccine effectiveness. The cost of infection is CI. Without loss of generality, we use the relative cost of vaccination,
in the remainder of this paper. Those unvaccinated individuals who remain healthy are free-riders off the vaccination efforts of others, as they are indirectly protected to some extent by herd immunity.
Fig 2.
Bistability of equilibrium vaccination levels and the emergence of hysteresis loops in well-mixed populations.
We simulate the social imitation of vaccination dynamics in a finite, well-mixed population and obtain hysteresis loops, composed of the ascending and descending paths, in response to variations to model parameters: ( A) the relative cost of vaccination, c, and ( B) vaccine effectiveness, ε. The population can exhibit bistability within certain ranges of the model parameters c and ε. Stochastic agent-based simulation results align with theoretical analysis, with noticeable discrepancies attributable to finite population effects. Parameters: population size N = 1000, infection seeds I0 = 10. For the first simulation, the initial number of vaccinated individuals is . For subsequent simulations, the results of the preceding simulations are used as the initial conditions, while the model parameters are varied in a prescribed sequence of increasing and then decreasing values. Disease transmission rate
, recovery rate
, intensity of selection K = 1. (A) effectiveness
, (B) c = 0.35. Simulation results are averaged over 100 independent runs.
Fig 3.
Hysteresis and sensitivity of vaccination dynamics in lattice populations.
(A) and (B) depict the hysteresis loops, represented by the descending and ascending paths of the population’s equilibrium vaccination level, in response to variations in the relative cost of vaccination, c, and vaccine effectiveness, ε, respectively. Overall, the critical parameter region where the hysteresis occurs depends on specific model parameter choices. The spatial population is highly sensitive to changes in and abruptly transitions between complete opt-out and universal coverage of vaccination. (C) and (D) show spatial snapshots of population states along the descending and ascending paths of Fig 3A for a fixed c = 0.015, respectively. The color codes of individuals are the same as in Fig 1: Blue represents individuals who were vaccinated and remained healthy during the season, yellow represents individuals who were vaccinated but still became infected, grey represents individuals who were unvaccinated yet remained healthy, and red represents individuals who were unvaccinated and became infected. Parameters: Square lattice
with the von Neumann neighborhood, infection seeds I0 = 30, initial number of vaccinated
, disease transmission rate
, recovery rate
, intensity of selection K = 1. (A) effectiveness
. (C) (D): c = 0.015,
. Simulation results are averaged over 150 independent runs.
Fig 4.
Impact of pre-existing, fixed vaccine beliefs on vaccination dynamics.
The presence of vaccine-averse (skeptical) beliefs, even at low frequencies, can render the population more sensitive to the cost of vaccination and vaccine effectiveness. Shown are the hysteresis loops of vaccination levels with respect to changes in (A) the cost of vaccination and (B) the vaccine effectiveness. For comparison sake, the grey lines are the results in Fig 2 without any vaccine beliefs. (C) and (D) show spatial snapshots of population states in the descending and ascending paths respectively. The color of individuals indicates their specific combinations of vaccine beliefs and uptake behaviors: blue: vaccinated individuals with vaccine-neutral attitude; yellow: unvaccinated individuals with vaccine-neutral attitude; green: vaccinated individuals with vaccine-averse attitude; red: unvaccinated individuals with vaccine-averse attitude. Parameters: square lattice with von Neumann neighborhood, initial number of infection seeds I0 = 30, initial number of vaccinated
, fixed number of vaccine skeptical individuals 50, disease transmission rate
, recovery rate
,
, K = 1. (A) effectiveness
, (C) (D): c = 0.01,
. Simulation results are averaged over 150 independent runs.
Fig 5.
Coevolution of vaccine beliefs and uptake behavior.
Plotted are the hysteresis loops of equilibrium vaccination levels as a function of ( A) the relative cost of vaccination, c, and ( B) vaccine effectiveness, ε. For comparison, we include the simulation results in the absence of any vaccine beliefs (Fig 3) as well as those with fixed vaccine beliefs (Fig 4). Compared to the case without vaccine beliefs, the concurrent spreading of beliefs–where a vaccine-neutral attitude competes with a vaccine-averse attitude alongside the social contagion (imitation) process of vaccine behavior choices–lead to slightly less favorable condition for vaccination. However, this impact is much less severe than in the scenario where a small proportion of individuals hold a vaccine-averse attitude and remain unchanged. Parameters: ( A, B) square lattice with von Neumann neighborhood, initial number of infection seeds I0 = 30, initial number of vaccinated
(50%), initial number of vaccine skeptical individuals 1250 (50%), disease transmission rate
, recovery rate
,
, K = 1. (A) for fixed effectiveness
, and (B) for fixed c = 0.01. Simulation results are averaged over 150 independent runs.
Fig 6.
Microscopic view of the hysteresis loops arising from the coevolution of vaccine beliefs and uptake behavior.
Shown are hysteresis loops, indicated by the corresponding descending and ascending paths of the equilibrium fractions of individuals grouped into four types based on the combinations of their vaccine beliefs and behavior choices, as a function of ( A) the relative vaccination cost c and ( B) the vaccine effectiveness ε. The equilibrium fraction of individuals with a vaccine-averse attitude who also opt for vaccination is almost zero across the parameter space studied. The proportion of vaccine-averse individuals almost exclusively opt out of vaccination, and their fraction can reach a maximum of around 6% in the population. Parameters: square lattice with von Neumann neighborhood, initial number of infection seeds I0 = 30, initial number of vaccinated
(50%), initial number of vaccine skeptical individuals 1250 (50%), disease transmission rate
, recovery rate
,
, K = 1. ( A) for fixed effectiveness
( B) for fixed c = 0.01. Simulation results are averaged over 150 independent runs.