Interplay between cost and effectiveness in influenza vaccine uptake: a vaccination game approach

Abstract

Pre-emptive vaccination is regarded as one of the most protective measures to control influenza outbreak. There are mainly two types of influenza viruses-influenza A and B with several subtypes-that are commonly found to circulate among humans. The traditional trivalent (TIV) flu vaccine targets two strains of influenza A and one strain of influenza B. The quadrivalent (QIV) vaccine targets one extra B virus strain that ensures better protection against influenza; however, the use of QIV vaccine can be costly, hence impose an extra financial burden to society. This scenario might create a dilemma in choosing vaccine types at the individual level. This article endeavours to explain such a dilemma through the framework of a vaccination game, where individuals can opt for one of the three options: choose either of QIV or TIV vaccine or none. Our approach presumes a mean-field framework of a vaccination game in an infinite and well-mixed population, entangling the disease spreading process of influenza with the coevolution of two types of vaccination decision-making processes taking place before an epidemic season. We conduct a series of numerical simulations as an attempt to illustrate different scenarios. The framework has been validated by the so-called multi-agent simulation (MAS) approach.

Keywords: epidemic model; influenza vaccine; vaccination game.

Figure 1.

The layout of the whole dynamical set-up. The vaccine efficacy of TIV vaccine against influenza B virus is assumed e_T, which is lower than that of QIV vaccine. However, both vaccines are assumed to have same efficacy (e_Q) against influenza A virus. Once the disease spreading process ends, we estimate several fractions such as vaccinated and healthy, vaccinated but infected, infected with influenza A(B), etc., and evaluate their payoffs. These fractions then update their strategies for the next season. This process is repeated until we reach a steady state. Arrows depict the sequence of the evolutionary process. (Online version in colour.)

Figure 2.

Time series of the fractions of infection by two influenza viruses (A and B) for different initial conditions and transmission rates without considering game approach. The vaccination coverage for each vaccine is assumed approximately 33% with vaccine effectiveness e_Q = 0.6 and e_T = 0.4. The recovery rate for each flu virus is chosen as 0.2. Under the current set-up, when β_A = β_B (a,b), B virus is found to dominate because e_T ≤ e_Q, whereas the flu virus with higher transmission rate (c,d) dominates the other one. The transmission rate seems more influential in virus dominance than the degree of initial dominance of each virus infection. (Online version in colour.)

Figure 3.

Six possible combinations of QIV and TIV vaccination coverages at equilibrium attained from the evolutionary equations (2.8) and (2.9). The evolutionary dynamics of the vaccination coverage depends upon the choice of different parameters. Clearly, there is no bi-stability at equilibrium. There are six possible mixtures of dynamics: (a) x = 1, y = 0, when C_Q = C_T = 0.4; e_Q = 0.4, e_T = 0.3. (b) x = 0, y = 0, when C_Q = 0.6, C_T = 0.4; e_Q = 0.4, e_T = 0.3. (c) x = x*, y = y*, (coexistence or internal equilibria) when C_Q = 0.6, C_T = 0.4; e_Q = 0.8, e_T = 0.4. (d) x = x* (internal equilibrium), y = 0, when C_Q = C_T = 0.4; e_Q = 0.7, e_T = 0.4. (e) x = 0, y = 1, when C_Q = 0.6, C_T = 0.4; e_Q = 0.6, e_T = 0.4. (f) x = 0, y = y*, when C_Q = 0.6, C_T = 0.4; e_Q = 0.8, e_T = 0.6. All cases presume, β_A = β_B = 0.5; γ_A = γ_B = 0.2. The initial value of y is kept as, 0.33, while we vary the initial values of x and vice versa. (Online version in colour.)

Figure 4.

C_Q versus C_T (C_T ≤ C_Q) 2D-heatmaps with different effectiveness levels presuming mean-field (with β_A = β_B = 0.5, I_A(0) = 0.00001, I_B(0) = 0.00002) and multi-agent simulation (MAS) approach (in well-mixed population). Chronologically each column represents the final epidemic size for influenza A (FES_A), the final epidemic size for influenza B (FES_B), fraction of QIV vaccinees, fraction of TIV vaccinees and the average social payoff (ASP). Clearly, the infection due to influenza B is dominating for the case, β_A = β_B. The fraction of TIV vaccinees is seen to prevail for certain regime as long as C_T is below some threshold levels, however, QIV vaccinees are predominant whenever C_Q and C_T are comparable. Results from MAS approach show an overall similar tendency to that of mean-field approach. The MAS approach presumes 10 000 agents with I_A(0) = 2 agents, I_B(0) = 4 agents, β_A = β_B = 5.19957 × 10⁻⁵, γ_A = γ_B = 0.2, and takes ensemble average of 100 realizations. (Online version in colour.)

Figure 5.

e_Q versus e_T (e_T ≤ e_Q) 2D-heatmaps with similar (upper panels) and different (lower panels) cost levels presuming mean-field framework with parameters choices, β_A = β_B = 0.5, γ_A = γ_B = 0.2, and initial conditions, I_A(0) = 0.00001, I_B(0) = 0.00002. Similar cost encourages people (with a higher degree) to commit QIV vaccine, consequently, showing sensitivity in the direction of e_Q. However, in case of different costs (C_Q > C_T), QIV vaccine is seen to favour over TIV vaccine when e_Q ≥ 0.5 (approximately), on the other hand, TIV is favoured over QIV when e_T ≥ 0.4 (approximately). The average social payoff for similar cost seems to be better than that of the different costs. Moreover, infection due to influenza A is lower than influenza B. (Online version in colour.)

Figure 6.

The heatmaps (a–d) show the fractions of infected individuals with influenza A and B viruses (FES_A, FES_B), fraction of QIV and TIV vaccinees as a function of C_Q and C_T (C_T ≤ C_Q), when the transmission rate for virus A is higher than B virus (β_A > β_B;β_A = 0.6, β_B = 0.4). (e–h) The same objects for the case β_A < β_B; β_A = 0.4, β_B = 0.6. We choose, e_Q = 0.6 and e_T = 0.4 for both cases. We can perceive the infection dominance of A and B viruses according to the conditions β_A > β_B and β_A < β_B. With the current setting, the case, β_A > β_B implies the total predominance of TIV vaccinees (e,f). However, if β_A < β_B, then QIV vaccinees prevail whenever the cost difference is not so high and both costs are below 0.6 (approximately) but the prevalence of TIV vaccinees surges with the increase of C_Q and the cost difference. The region where QIV is dominant shows more impact on disease suppression comparing to that of TIV (f,g,h). (Online version in colour.)

Figure 7.

The effect of transmission rates (β_A, β_B) on disease spreading (a), the vaccine-dependent basic reproduction number-R₀ (b), and the vaccination coverage of both vaccines (c,d). The deep blue region inside the box in (a) represents a DFE for both viruses, where R₀ is below one (b), and accordingly, there is no vaccination coverage of any type inside the box. Outside of that box, influenza A(B) becomes prevalent with the increase of β_A(β_B). The red dotted line depicts the threshold level below(above) which influenza A(B) is dominant. TIV vaccinees predominate when β_A gets larger and QIV vaccinees predominate when β_B increases. The relevant parameters are set as γ_A = γ_B = 0.2; e_Q = 0.6, e_T = 0.4; C_Q = 0.4, C_T = 0.2. (Online version in colour.)

Figure 8.

The schematic of SED derivation. At first, we choose a point ($C_Q^{\ast }$,$C_T^{\ast }$) on the heatmap (C_Q versus C_T) illustrating the average social payoff (ASP) at equilibrium (EQM) (a), in which there is a corresponding point, (x*, y*), i.e. a corresponding pair of vaccinees (QIV, TIV). With fixed ($C_Q^{\ast }$,$C_T^{\ast }$) and y*, we vary x (0 ≤ x ≤ 1) and estimate the maximum average payoff, which we term as social optimum (SO) payoff for x at ($C_Q^{\ast }$,$C_T^{\ast }$) and y*, i.e. ${\text{ASP}}_{(C_Q^{\ast },C_T^{\ast }\text{|}{y}^{\ast })}^{\text{S}{\text{O}}_x}$ (b). After that we derive SED for x at ($C_Q^{\ast },C_T^{\ast }$) by subtracting ASP at equilibrium from the payoff at social optimum. We follow the same procedure to derive SED for y using (a) and (c). (Online version in colour.)

Figure 9.

Representation of social optimum payoff (a,b) and SED (c,d) for x (fraction of QIV vaccinees) and y (fraction of TIV vaccinees) as a function of (C_Q, C_T). SED heatmaps are attained by subtracting the ASP at equilibrium (figure 4a-v) from the social optimum (SO) payoffs for x and y (a,b). The relevant parameters are chosen as: e_Q = 0.6, e_T = 0.4, β_A = β_B = 0.5 and γ_A = γ_B = 0.2. Referring to figure 4a-i–a-iv, these parameter choices depict the prevalence of influenza B virus and lower vaccination coverage for QIV vaccine (especially when the cost difference is high). The whiteout regions in (c,d) depict that there is no gap between SO payoff (for x or y) and ASP at equilibrium. More areas in SED-heatmap for x appears to have darker regime than that of the case for y, which indicates the higher gap between SO payoff for x and ASP at equilibrium. (Online version in colour.)