Reference model

Compartments

\(d\) — day index.
\(g\) — vaccination lag period (given).
\(w_d\) — vaccine effectiveness on day \(d\) (given).
\(v_d\) — vaccine coverage on day \(d\).
\(r_d\) — flu incidence on day \(d\).
\(b_d\) — number of vaccinations of susceptible on day \(d\).

Calculations

\(d=0\) — initial conditions.
\(d=1\) — first timepoint (day) for which we have data.
\(N\) — starting population size (given).
\(V_d\) — vaccinations on day \(d\) (given).
\(s_d^*\) — infections on day \(d\) in absence of vaccination (given).
\(P_d^*\) — susceptible population on day \(d\) in absence of vaccination.
\(s_d^a\) — averted infections on day \(d\).
\(s_d\) — infections on day \(d\).

For \(d=0\):

\(A_0 = N\)
\(b_0 = 0\)
\(C_0 = 0\)
\(D_0 = 0\)
\(E_0 = 0\)
\(F_0 = 0\)
\(P_0^* = N\)
\(s_0 = 0\)
\(s_0^a = 0\)

For \(d>0\) (order presented corresponds to the order of calculations):

1. \(r_d = \frac{s_d^*}{P_{d-1}^*}\)
   
2. \(s_d=r_dA_{d-1} + r_dC_{d-1} + r_db_{d-1...d-g}\)
   
3. \(P_d^* = P_{d-1}^* - s_d^*\)
   
4. \(s_d^a = s_d^* - s_d\)
   
5. \(v_d = \frac{V_d}{A_{d-1}+E_{d-1}}\)
   
6. \(b_d=v_dA_{d-1}\)
   
7. \(A_d = A_{d-1} - r_dA_{d-1}-b_d\)
   
8. \(b_{d-1} ... b_{d-g}=b_{d-1} ... b_{d-g} - r_db_{d-1} ... b_{d-g}\)
   
9. \(C_d = C_{d-1} - r_dC_d + b_{d-g} - w_db_{d-g}\)
   
10. \(D_d = D_{d-1} + w_db_{d-g}\)
    
11. \(E_d=E_{d-1} + r_dA_{d-1} - v_dE_{d-1}\)
    
12. \(F_d=F_{d-1} + r_dC_{d-1} + r_db_{d-1} ... b_{d-g} + v_dE_{d-1}\)
    

Line \(b_{d-1} ... b_{d-g}\) means that the same calculation is done on each \(b_{d-i}\) from \(i=1\) to \(i=g\). The calculation being (in line 8) \(b_{d-i}=b_{d-i}-r_db_{d-i}\).