Erratic flu vaccination emerges from short-sighted behavior in contact networks

Abstract

The effectiveness of seasonal influenza vaccination programs depends on individual-level compliance. Perceptions about risks associated with infection and vaccination can strongly influence vaccination decisions and thus the ultimate course of an epidemic. Here we investigate the interplay between contact patterns, influenza-related behavior, and disease dynamics by incorporating game theory into network models. When individuals make decisions based on past epidemics, we find that individuals with many contacts vaccinate, whereas individuals with few contacts do not. However, the threshold number of contacts above which to vaccinate is highly dependent on the overall network structure of the population and has the potential to oscillate more wildly than has been observed empirically. When we increase the number of prior seasons that individuals recall when making vaccination decisions, behavior and thus disease dynamics become less variable. For some networks, we also find that higher flu transmission rates may, counterintuitively, lead to lower (vaccine-mediated) disease prevalence. Our work demonstrates that rich and complex dynamics can result from the interaction between infectious diseases, human contact patterns, and behavior.

Figure 1. In a heterogeneous population, an individual's decision to vaccinate depends on the number of his or her contacts (degree) and the perceived epidemiological risk in the prior season.

(A) When the costs of vaccination and infection are the same for everybody, an individual should only choose to vaccinate if his or her risk exceeds a calculated threshold depending on the person's degree (here ). (B) The proportion of the population with each given degree are different for a homogeneous network (magenta histogram), an urban network (blue bimodal histogram), and an exponentially-scaled power-law network (steeply descending green histogram). For a log-linear plot of the degree distributions, see the online supplement.

Figure 2. Risk from one season to the next and equilibria in the homogeneous, urban and power-law networks.

(A) The inter-seasonal risk map showing the relationship between risk in one season and risk in the next season, assuming everybody acts to maximize his or her payoff (as in Figure 1A). The line indicates constant level of risk from one season to the next, and an intersection of the response risk curve with this line represents a Nash Equilibrium. The stairstep shape seen in the homogeneous and urban networks is also present in the power-law example but appears smooth here as the steps are very small in comparison to the line width. (B) Each of these intersection points corresponds to an equilibrium level of risk (horizontal lines) and vaccination threshold degrees (vertical lines). Both figures assume that transmissibility is .

Figure 3. The effect of transmissibility on risk and vaccination dynamics.

(A) The effect of transmissibility (T) on inter-seasonal change in risk in the urban network. Equilibria occur at intersections with black () line. (B) As the transmissibility increases, the equilibrium vaccination threshold and risk change non-monotonically. For low T (, magenta) the equilibrium level of risk is less than 0.2 per year, and the vaccination threshold is greater than the maximum degree in the network. Consequently, nobody in the population is expected to vaccinate. For intermediate T (, blue), the equilibrium risk is near 0.5 per year and only a small fraction of the most connected individuals vaccinate. At high T (, orange), a large fraction of individuals vaccinate, leaving an intermediate level of risk. (C) As both risk and transmissibility increase, vaccination behavior increases. (D) Consequently, the equilibrium level of vaccination is an increasing function of the transmission rate.

Figure 4. Cobwebbing diagrams of risk and vaccination rates.

(A) shows the inter-seasonal risk relation and (B) the corresponding fractions of the population expected to vaccinate as a best response to the perceived risk of infection for the urban network when and . Whether vaccination rate (or ) is stable from season to season depends on the slope of the inter-seasonal risk relation at the equilibrium (the slope of the intersections above or likewise in Figures 2A and 3A. When this slope is zero, there is a partially vaccinating degree class at equilibrium and the system is dynamically unstable, and otherwise (infinite slope) there are no partially vaccinating classes and it is dynamically stable. Additionally, when the “average” slope has magnitude less than one, the system is convergently stable. Conversely, if the magnitude of the average slope is greater than one, it is not convergently stable. The dynamics shown are both dynamically and convergently unstable.

Figure 5. The effect of transmissibility on risk.

The impact of transmissibility on risk (steady-states and orbits) for the (A) homogeneous (B) urban and (C) power-law networks. Nash equilibria are unstable for the majority of the interval between and ; some individuals waver between accepting and rejecting vaccination.

Figure 6. The effect of transmissibility on prevalence.

The impact of transmissibility on prevalence (steady-states and orbits) for the (A) homogeneous (B) urban and (C) power-law networks.

Figure 7. The effect of memory on vaccination decisions.

(A) Contributions of past seasons to current perceptions of risk, under different s values. (B) The impact of prior information on oscillations when : longer memory decreases variability.

Figure 8. Longer-term memory reduces oscillations for various transmissibilities.

As increases, individuals integrate more of their prior epidemiological experiences into their decision-making and two-cycles disappear. (A) (B) (C) (D) .