2.2. Modelling methodology

Our epidemiological game-theoretic analysis is based on a population-level epidemiological model for influenza transmission and an individual-level calculation of payoff associated with infection and/or vaccination. Our epidemiological game theoretic model also incorporated the public perception of the key epidemiological parameters of an influenza epidemic and vaccine efficacy. To determine the likely impact of public perceptions about influenza and its vaccine, we compared the results based on normative epidemiological parameter values with those based on perceived parameter values estimated as the median value from our survey study () [

25].

The population was divided into susceptible (

*S*), vaccinated and uninfected (

*V*), vaccinated and infectious (

*I*_{V}), unvaccinated and infectious (

*I*_{U}) and naturally immune (

*R*) compartments. We assumed no residual immunity, because seasonal influenza rapidly evolves new antigenic variants, and immunity tends to wane before the next influenza season [

26]. Susceptible individuals become infected with probability

*β*{

*I*_{U}(

*t*) +

*I*_{V}(

*t*)}

*/N*(

*t*) per unit of time, where

*N*(

*t*) is the size of the population and

*β* the transmission parameter. A fraction,

*ϕ*, of the population was assumed to be vaccinated before the influenza epidemic started.

Vaccination is assumed to reduce the probability of infection by

*σ*, where

*σ* is vaccine efficacy. It was shown that individuals tend to underestimate the efficacy of influenza vaccines; the perceived value of vaccine efficacy was 70 per cent compared with an actual value of 78 per cent (). It was also assumed that vaccination reduced the disease severity in the event of infection, which is consistent with epidemiological data [

3]. Specifically, when vaccinated individuals become infected, the probabilities of various medical outcomes are assumed to be lower than those of unvaccinated ones who are infected (). Thus, the severity of infection was incorporated into the calculation of the payoffs. For instance, the probability of hospitalization following infections with influenza is 1.2 per cent for unvaccinated individuals, but it is 0.61 per cent for those who are vaccinated and infected [

3,

27,

28]. Similarly, vaccination reduces the probability of outpatient visits (conditional on infection) from 42 to 17 per cent [

3,

27–

30].

| **Table 4.**Cost parameters regarding vaccination and infection, and normative probability of infection outcomes, if infected with epidemic influenza [3,20]. |

Upon infection, individuals enter an infectious period, the perceived duration of which is 10 days, compared with an actual value of 4.5 days () [

24]. The perceived infection probability was 34 per cent (), compared with an actual infection probability of 30 per cent [

21–

23]. We assumed that the perceived and normative values of the case fatality proportion for seasonal influenza among the unvaccinated (

*α*_{U}*/*(

*γ* +

*α*_{U})) are equal and estimated at 0.065 per cent [

3,

20]. The case fatality proportion was not collected in our survey, because unlike a pandemic influenza strain, the case fatality associated with seasonal influenza across all age/risk groups is negligible. Nevertheless, the cost associated with disease-related death, i.e. lost productivity, was large enough to be incorporated into the calculation of the utility of a vaccine. Similarly, both the perceived and normative values of the case fatality proportion among the vaccinated (

*α*_{V}*/*(

*γ* +

*α*_{V})) are estimated at 0.041 per cent [

3,

20]. Based on this assumption and normative and perceived values of recovery rate for non-fatal cases (

*γ*), the normative and perceived value of disease-related death rate,

*α*, is estimated at 1.44 × 10

^{−4} d

^{−1} and 6.50 × 10

^{−5} d

^{−1}, respectively.

Given these assumptions, the epidemiological model can be expressed by the following deterministic system of ordinary differential equations:

where

*S*(0) = (1 −

*ϕ*)

*N*(0),

*V*(0) =

*ϕN*(0),

*I*_{U}(0) = 0

^{+},

*I*_{V}(0) = 0

^{+},

*R*(0) = 0.

From our model, the cumulative incidence in the absence of vaccination (

*ϕ* = 0) is calculated as

which is effectively equivalent to an individual infection probability. Based on the cumulative incidence in the absence of vaccination (), the transmission rate (

*β*) was estimated at 0.12 per day using perceived parameters and 0.26 per day using normative parameters.

We assumed that an individual's expected payoff consists of the costs associated with influenza vaccination and infection. The individuals' decisions to be vaccinated are assumed to rely upon how individuals weigh the payoff of vaccination and the cost associated with infection, and such an approach has been adopted by game theory or related behavioural modelling in the past [

3,

6–

8,

31–

33]. Alternatively, the payoff to individuals can be expressed in terms of quality-adjusted life year (QALY). However, switching utility scales to QALY would not allow us to incorporate the cost of vaccine or the cost of vaccine administration. Thus, we used monetary costs to estimate an individual's payoff, and included the cost of vaccines and administration into the costs of vaccination (

*C*_{3}; ). We calculated the costs of infection by summing the products of the costs and probabilities of each possible medical outcome (). It was assumed that the economic burden of influenza mortality depends on both the mortality rates and the economic costs associated with mortality. Here, the cost of mortality is measured as the economic opportunity cost associated with the loss of a statistical life [

3]. Assigning the personal disutility to mortality is nearly impossible, because individuals would give infinitely low values, when asked to assign the disutility to their mortality. As a result, in economic analyses of vaccination, including health decision modelling, it is standard to assign monetary values to all outcomes, including health or metaphysical outcomes [

6–

8,

34].

Vaccination is assumed to reduce the severity of infection, and thus reduce the probability of suffering from complications associated with influenza. As a result, the cost associated with infection for a vaccinated individual (*C*_{1}) is lower than that for an unvaccinated one (*C*_{2}).

In the presence of influenza vaccination,

and

describe the cumulative number of infections among vaccinated and unvaccinated individuals, respectively. Thus, the probability of infection among vaccinated individuals is

whereas the probability of infection among unvaccinated individuals is

Here,

*ϕN*(0) is the number of total vaccinated individuals, and (1 −

*ϕ*)

*N*(0) is the number of total unvaccinated individuals. Thus, the expected net personal payoffs of vaccine acceptance (

*Q*_{A}) and refusal (

*Q*_{R}) are

and

respectively. Here,

*C*_{1} is the cost of infection among those who are vaccinated,

*C*_{2} is the cost of infection among the unvaccinated, and

*C*_{3} is the cost of vaccination (). The costs associated with vaccination and infection were indicated as negative terms in the payoff calculation, because they decrease the payoffs.

The probability of infection for both vaccinated and unvaccinated individuals decreases as vaccine coverage increases. Thus,

*Q*_{A}/

*ϕ* ≥ 0 and

*Q*_{R}/

*ϕ* ≥ 0, indicating the direct and indirect protection by influenza vaccination. Here, the increased marginal payoffs for vaccinated and unvaccinated population are defined as

*Q*_{A}/

*ϕ* and

*Q*_{R}/

*ϕ*, respectively. It has previously been defined that, at the selfish equilibrium, no individual can improve their expected payoff by changing their vaccination probability [

4]. Thus, the expected net payoff of vaccine acceptance (

*Q*_{A}) is equal to the net payoff of vaccine refusal (

*Q*_{R}) at the selfish equilibrium. However, if we incorporate both selfish and altruistic motivation for influenza vaccination into the payoff calculation, the payoff of vaccine acceptance is increased by the externalities of vaccination. Here, the externalities of vaccination are defined as the marginal payoff of additional vaccination to the population. Specifically, the externality of vaccination,

*ϕ* ·

*Q*_{A}/

*ϕ* + (1−

*ϕ*) ·

*Q*_{R}/

*ϕ*, is weighted according to vaccine coverage in the community. This is based on the fact that when an individual is vaccinated, benefits accrue not only to that individual directly, but also to the society as a whole through herd immunity.

Given these assumptions, we used our payoff calculation to examine how the expected vaccine coverage level changes as the degree of altruism (denoted by

*θ*, 0 ≤

*θ* ≤ 1) varies. When individuals act according to pure self-interest, the resulting population vaccination coverage is expected to converge to the selfish equilibrium. On the other hand, the community optimum is achieved if vaccination decisions are solely driven by altruistic motivation. Finally, when vaccination decisions are driven by both self-interest and altruism, the resulting vaccination probability,

*ϕ*, satisfies the following equation:

where

*θ* is the degree of altruism (0 ≤

*θ* ≤ 1) and

*ϕ* is calculated through single parameter optimization. We searched for a probability of vaccination (

*ϕ*) that minimizes the difference between

*Q*_{A} + α(

*ϕQ*_{A}/

*ϕ* + (1−

*ϕ*)

*Q*_{R}/

*ϕ*) and

*Q*_{R} at different values of

*θ*. By varying the degree of altruism (

*θ*), we can examine the impact of altruism in the resulting vaccine coverage level. Here, the selfish equilibrium and community optimum are obtained by setting

*θ* = 0 and

*θ* = 1, respectively.

We also evaluated the sensitivity of our modelling results to survey data sampling by re-estimating epidemiological parameters in 10 000 bootstrap samples. Using each bootstrap sample, we parametrized our model and predicted the resulting final epidemic size. This size, in turn was used to estimate the individual probability of infection based on their vaccination status, and thus their expected cost associated with vaccine acceptance and refusal was estimated.