BCEA

In a population made up of \(N\) individuals, consider the number of patients taking up the vaccine when available \(V_1 \sim \mbox{Binomial}(\phi,N)\) where \(\phi\) is the vaccine coverage rate. Obviously, \(V_0=0\) as the vaccine is not available in the status quo. For convenience, we denote the total number of patients in the two groups, vaccinated ($v=1$) and non-vaccinated ($v=0$), by \(n_{tv}\) with \(n_{t1}:=V_t\) and \(n_{t0}:=N-V_t\) respectively.

The relevant clinical are \(j=1\): influenza infection; \(j=2\): GP visit; \(j=3\): minor complications; \(j=4\): major complications; \(j=5\): hospitalisation; \(j=6\): death; and \(j=7\): adverse events of influenza vaccination. For each clinical outcome \(j\), \(\beta_j\) is its baseline rate of occurrence and \(\rho_{v}\) is the proportional reduction due to the vaccine in the chance of infection. Vaccinated patients ($v=1$) will experience a reduction in the chance of infection by a factor \(\rho_1\); conversely, for \(v=0\), individuals are not vaccinated and so the chance of infection is just the attack rate \(\beta_1\). This is equivalent to setting \(\rho_0:=0\).

Under these assumptions, the number of individuals becoming infected in each group is \(I_{tv}\sim\mbox{Binomial}(\pi_v,n_{tv})\), where \(\pi_v := \beta_1(1-\rho_v)\) is the probability of infection. Among the infected subjects, the number visiting a GP for the first time is \(G\!P^{(1)}_{tv} \sim \mbox{Binomial}(\beta_2,I_{tv})\). Using a similar reasoning, among those who have had a GP visit, we can define: the number of individuals with minor complications \(G\!P^{(2)}_{tv} \sim \mbox{Binomial}(\beta_3,G\!P^{(1)}_{tv})\); the number of those with major complications \(P_{tv} \sim \mbox{Binomial}(\beta_4,G\!P^{(1)}_{tv})\); the number of hospitalisations \(H_{tv} \sim \mbox{Binomial}(\beta_5,G\!P^{(1)}_{tv})\); and the deaths \(D_{tv} \sim \mbox{Binomial}(\beta_6,G\!P^{(1)}_{tv})\). The number of individuals experiencing adverse events due to vaccination is computed as \(A\!E_{tv} \sim\mbox{Binomial}(\beta_7,n_{tv})\) — obviously, this will be identically 0 in the status quo ($t=0$) and among those individuals who choose not to take the vaccine up in the vaccination scenario ($t=1,v=0$).

The model also includes other parameters, such as the chance of receiving a prescription after the first GP visit ($\gamma_1$) or following minor complications ($\gamma_2$) for a number of antiviral drugs ($\delta$); of taking OTC medications ($\xi$); and of remaining off-work ($\eta$) for a number of days ($\lambda$). Combining these with the relevant populations at risk, we can then derive the expected number of individuals experiencing each of these events.

As for the costs, we consider the relevant as \(h=1\): GP visits; \(h=2\): hospital episodes; \(h=3\): vaccination; \(h=4\): time to receive vaccination; \(h=5\): days off work; \(h=6\): antiviral drugs; \(h=7\): OTC medications; \(h=8\): travel to receive vaccination. For each, we define \(\psi_h\) to represent the associated unit cost for which we assume informative lognormal distributions, a convenient choice to model positive, continuous variables such as costs.

Finally, we include in the model suitable parameters to represent the loss in quality of life generated by the occurrence of the clinical outcomes. Let \(\omega_j\) represent the QALYs lost when an individual experiences the \(j-\)th outcome. We assume that GP visits do not generate loss in QALYs and therefore set \(\omega_2=\omega_3:=0\); the remaining \(\omega_j\)’s are modelled using informative lognormal distributions.

The assumptions encoded by this model are that we consider a population parameter \(\boldsymbol\theta = (\boldsymbol\theta^0,\boldsymbol\theta^1)\), with the two components being defined as \(\boldsymbol\theta^0=(\beta_j,\gamma_1,\gamma_2,\delta,\xi,\eta,\lambda,\psi_h,\omega_j)\) and \(\boldsymbol\theta^1=(\phi,\beta_j,\rho_v,\gamma_1,\gamma_2,\delta,\xi,\eta,\lambda,\psi_h,\omega_j)\). We assume that the components of \(\boldsymbol\theta\) have the distributions specified in Table XXX. The distributions of Table XXX are derived by using suitable “hyper-parameters” that have been set to encode knowledge \(\mathcal{D}\) available from previous studies and expert opinion — cfr. Baio and Dawid (2011) for a more detailed discussion of this point.

In order to perform the economic analysis, we need to define suitable summary measures of cost and effectiveness. The total cost associated with each clinical resource can be computed by multiplying the unit cost \(\psi_h\) by the number of patients consuming it. For instance, the overall cost of GP visit is \(\left(G\!P^{(1)}_{tv}+G\!P^{(2)}_{tv}\right)\times \psi_1\). If, for convenience of terminology, we indicate with \(N_{tvh}\) the total number of individuals consuming the \(h-\)th resource under intervention \(t\) and in group \(v\), we can then extend this reasoning and compute the under intervention \(t\) as

Similarly, the total QALYs lost due to the occurrence of the relevant outcomes can be obtained by multiplying the number of individuals experiencing them by the weights \(\omega_j\). For example, the total number of QALYs lost to influenza infection can be computed as \(I_{tv}\times \omega_1\). If we let \(M_{tvj}\) indicate the number of subjects with the \(j-\)th outcome in intervention \(t\) and group \(v\), we can define the for intervention \(t\) as

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