Semiparametric response surface model (GAM)

Response surface models were built to determine the optimum radius while accounting for variation in the other uncertain explanatory variables (i.e., resource, effectiveness, CIP and EDR). A response surface model is an approximation model that best represents, or mimic the relationship between the decision variable and the objective function of the response system. As an objective function, a net present value (NPV), i.e., reduction in the sum of the direct costs and the macroeconomic costs (total costs) of an epidemic, was used as this variable took into account all epidemiological and economic consequences of an FMD epidemic.

Generalised additive models (GAMs) were used to fit models that represented the response of the system while controlling for variability. A GAM is an extension of a generalised linear model (GLM), with difference being that a GAM uses some smooth functions of covariates. The structure of GAMs in general is as follows (Wood, 2006):

݃{ॱ(ݕ_௜)} =ߚ_଴+ߚܺ_௜כ_{+෍ ܮ}

௜௝݂௝൫ݔ௜௝൯ ௝ୀଵ

(1)

where g is a link function, yi is a response variable of ith observation, Ƣ0 is an intercept, Xi*

is a row of the model matrix for the parametric model components, Ƣ is the corresponding parameter vector, fj is a smooth function of the covariate xij, and Lij is a

linear function. Smooth functions can be estimated using a method within the spline family. For example, if cubic regression splines were used, piecewise cubic polynomials are fitted within each segment of the covariate and then polynomial lines are connected smoothly at each knot. The optimal smoothness, i.e., effective degrees of freedom (the number of knots), was estimated from the data within a specified maximum degree of freedom, k. All the GAM parameters were estimated using package ‘mgcv’ (Wood, 2015) in R version 3.2.2 (R Development Core Team, 2015).

For each policy and for each uncertain variable, a multivariable GAM was fitted using the following predetermined formula:

ݕ= ߚ_଴ +݂(ݎܽ݀݅ݑݏ,ݔ) (2) where y is the NPV, x is an explanatory variable (resource, effectiveness, CIP and EDR), and f is the smooth function for a product of radius and x. Based on the fitted GAMs, the vaccination radii that minimised the predicted NPVs were determined for the quantiles of the sampled values for resource and effectiveness, or simulated values for CIP and EDR.

7.4. Results

7.4.1. Descriptive statistics of the simulated epidemics

Of 18,000 simulated epidemics randomly seeded in Auckland Region, 26% (n = 4,766) were contained within 21 days after detection of the index case by SO. Throughout the following analyses, only the results of the simulated epidemics in which at least one new case was detected on day 21 or later (n = 13,234) were used. The summary statistics of the simulated FMD epidemics and their economic outcomes, for SO, VTD, VTL and VTL* are shown in Table 7-1. Emergency vaccination reduced the simulated total number of

IPs from 108 to 63, and time till eradication from 77 to 46 days. The estimated median direct costs of simulated epidemics ranged from USD 79 million to 221 million, and the order from lowest to highest was SO, VTL/VTL* and VTD. The estimated median macroeconomic costs ranged from USD 7.1 billion to 11.3 billion, and the order from lowest to highest was VTL*, SO/VTD and VTL.

Figure 7-1 shows the cumulative density functions of the two epidemiological outcomes and two economic outcomes. All the distributions of the four outcome variables were right-skewed, indicating occurrence of extreme outcomes with a small probability. The long-tails for the total number of IPs (>520 IPs, <1%) and time to eradication (>230 days, <1%) for SO were not observed with emergency vaccination (Figure 7-1A and B). For the distributions of the direct costs, VTD had a relatively longer tail (>USD 1,850 million, <1%), whereas for the macroeconomic costs, a longer tail (>USD 16 billion, <1%) was observed for both SO and VTD.

Table 7-1 The median and the 5th and 95th percentiles (in parenthesis) of simulated foot- and-mouth disease (FMD) epidemics lasting for >21 days in the Auckland Region, controlled by stamping-out only (SO), vaccinate-to-die (VTD), vaccinate-to-live (VTL), and VTL with a hypothetical 3-month waiting period for recognition of FMD-free status (VTL*) (13,459 iterations).

SO VTD VTL VTL*

Silent spread phase 1 _{(days) 14}

(7 and 25) ȩ ȩ ȩ EDR 2 _1.4 (0.3 and 8.0) ȩ ȩ ȩ CIP 3 ₁₇ (2 and 70) ȩ ȩ ȩ

Total number of vaccinated premises 0 (0 and 0) 2,733 (273 and 7,676) ȩ ȩ

Total number of vaccinated animals (×103₎ 0 (0 and 0) 159 (13 and 818) ȩ ȩ

Total number of IPs 108 (12 and 397)

63 (11 and 222)

ȩ ȩ

Time till eradication 4 _{(days) 77}

(28 and 176)

46 (28 and 75)

ȩ ȩ

Time to recover the OIE’s FMD-free status 167 (118 and 266) 167 (132 and 248) 226 (209 and 257) 136 (119 and 167) Direct cost (USD million) 79

(33 and 216) 221 (58 and 958) 177 (57 and 660) 177 (57 and 660) Macroeconomic cost (USD

billion) 8.5 (6.2 and 13.2) 8.5 (6.9 and 12.3) 11.3 (10.5 and 12.7) 7.1 (6.3 and 8.5) Total cost (USD billion) 8.6

(6.2 and 13.3) 8.7 (6.9 and 13.2) 11.5 (10.6 and 13.2) 7.3 (6.4 and 9.0) 1 The interval between infection in the primary case and detection of the index case

2 Estimated dissemination rate on the 7-day basis on day 21 3 Cumulative number of detected infected premises (IPs) on day 14

4 The number of days taken from detection of the index case until depopulation of the last case

Figure 7-1 Cumulative distribution functions of the [A] total number of infected premises (IPs), [B] time till eradication, [C] direct costs and [D] macroeconomic costs for simulated foot-and-mouth disease (FMD) epidemics lasting for >21 days in the Auckland Region, controlled by stamping-out (SO), vaccinate-to-die (VTD), vaccinate-to-live (VTL), and vaccinate-to-live with a 3-month waiting period (VTL*) (n = 13,459).