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:

A₀ = N
b₀ = 0
C₀ = 0
D₀ = 0
E₀ = 0
F₀ = 0
P₀^* = N
s₀ = 0
s₀^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.