Fig 2. Effects of process and observation error on the S and SI methods.

Plotted are the estimates [Row A] Z_k, [Row B] I_k, and [Row C] β_k of true incidence Z(t), prevalence I(t), and the seasonally forced transmission rate β(t) (Eq (27)) obtained by applying the [Left] S and [Right] SI methods without input error to each of four simulated reported incidence time series (indicated by the legend; Δt = 1 week, n = ⌊3 × 365/7⌋ = 156). The first simulation was purely deterministic [dark grey] (ϵ = 0, p_rep = 1), while the remaining three accounted for (i) environmental stochasticity [ES, light grey] (ϵ = 0.5, p_rep = 1), (ii) ES and demographic stochasticity [ES+DS, blue] (ϵ = 0.5, p_rep = 1), or (iii) ES, DS, and observation error [ES+DS+OE, red] (ϵ = 0.5, p_rep = 0.25). Reference values (Table 1) were assigned to all other data-generating parameters, in all four simulations. The left and right panels in Row A are identical, because the S and SI methods compute Z_k identically (compare Eqs (25a) and (26a)). RRMSE in the β_k time series is (0.0239, 0.0375, 0.1126, 0.1432) with the S method and (0.0021, 0.0153, 0.0494, 0.0591) with the SI method (order follows the legend). Note that the underlying β(t) was the same in all simulations; it is not plotted in Row C, but is close to perfectly represented by the dark grey curve in the right panel (RRMSE ≈ 0.2%). Due to process error, the underlying Z(t) and I(t) (also not shown) varied between the deterministic, ES, and ES+DS simulations.