# Chapter 4 from Bjornstad (2018): Force of infection and age-dependent incidence

Let us assume we test some $n_a$ individuals of each age $a$ and find from serology that $i_a$ individuals have been previously infected. Inferring $\phi$ from this data is a standard(ish) binomial regression problem: $p(a)=1-exp(- \phi a)$ is the expected fraction infected (or seropositive) by age $a$. Thus $\log(-\log(1-p(a)))=\log(\phi)+\log(a)$, so we can estimate a constant log-FoI as the intercept from a glm with binomial error, a complimentary log-log link and log-age as a regression ‘offset. We can illustrate the approach using the pre-vaccination Measles antibody data. The data contain seroprevalence-by-age-bracket of some 300 people from around New Haven, Connecticut from blood drawn in the summer of 1957:

library(epimdr)
data(black)
black

<th scope=col>age</th><th scope=col>mid</th><th scope=col>n</th><th scope=col>pos</th><th scope=col>neg</th><th scope=col>f</th>
<span style=white-space:pre-wrap><1 </span> 0.75 10 8 2 0.8000000
1-4 2.50 21 4 17 0.1904762
5-9 7.00 41 31 10 0.7560976
10-14 12.00 52 50 2 0.9615385
15-19 17.00 30 28 2 0.9333333
20-29 25.00 38 37 1 0.9736842
30-39 35.00 51 49 2 0.9607843
40-49 45.00 35 31 4 0.8857143
<span style=white-space:pre-wrap>>50 </span>60.00 30 26 4 0.8666667

The age-profile of seroprevalence takes the characteristic shape of many pre-vaccination childhood diseases: High seroprevalence of the very young (< 1 yr) due to the presence of maternal antibodies that wanes with age, followed by rapid build-up of immunity to almost 100% seroprevalence by age 20 (fig [fig:black]). There is perhaps some evidence of loss of immunity in the elderly. We use the binomial regression scheme to estimate the log-FoI based on the data for people in the 1-40 yr groups, and compare predicted and observed seroprevalence by age ([fig:black]):

b2 = black[-c(1,8,9),]  #subsetting age brackets
#Estimate log-FoI
fit = glm(cbind(pos,neg) ~ offset(log(mid)),
family = binomial(link = "cloglog"), data = b2)
#Plot predicted and observed
phi = exp(coef(fit))
curve(1-exp(-phi*x), from = 0, to = 60,
ylab = "Seroprevalence", xlab = "Age")
points(black$mid, black$f, pch = "*", col = "red")
points(x = b2$mid, y = b2$f, pch = 8)
exp(fit$coef)  (Intercept): 0.165332852827087 The estimated FoI is 0.16/year, giving a predicted mean age of infection of 6 years. ## More flexible$\phi$-functions The assumption of a constant, age-invariant FoI is usually too simplistic because of age- or time-varying patterns of mixing. We can use data on prevalence of the bacterium Bordetella bronchiseptica in a rabbit breeding facility to illustrate. B. bronchiseptica is a non-immunizing, largely avirulent (though it can cause snuffles), persistent infection of rabbits. Two-hundred and fourteen rabbits of known age were swabbed nasally and tested for the bacterium. data(rabbit) head(rabbit)  <th scope=col>a</th><th scope=col>n</th><th scope=col>inf</th> 1.059 3 2.0 8 7 2.5 4 4 3.0 2 1 3.5 5 1 4.0 2 0 We first calculate the average FoI from the binomial regression scheme introduced above. In the breeding facility the older breeding animals are kept separate from the younger animals, so we restrict ourselves to rabbits$<1$years old. We superimpose our fit on the plot of prevalence by age. In figure [fig:rabbit] the size of the circles is proportional to the sample size: rabbit$notinf = rabbit$n - rabbit$inf
#Binomial regression
fit = glm(cbind(inf, notinf) ~ offset(log(a)),
data = rabbit, subset = a<12)
#Plot data
symbols(rabbit$inf/rabbit$n ~ rabbit$a, circles = rabbit$n,
inches = 0.5, xlab = "Age", ylab = "Prevalence")
#Predicted curves for <1 and all
phi = exp(coef(fit))
curve(1-exp(-phi*x), from = 0, to = 12, add = TRUE)
curve(1-exp(-phi*x), from = 0, to = 30, add = TRUE, lty = 2)
1/phi

(Intercept): 5.91827328566492

The predicted median age of infection is just under 6 months. The constant-FoI model seem to do well for up to about 15 months of age, but the model over-predicts the prevalence in older individuals. To allow for the scenario that the FoI varies with age, we need to implement our own framework (as opposed to using glm) using the maximum likelihood ideas introduced in section [sec:c2cb]. A simple model for age-specific FoI assumes a piecewise constant model , where individuals are classified into discrete age classes. For a piecewise constant model the integrand in equation [eq:p] integrates to $\phi_a (a-c_a)+ \sum_{k<a} \phi_k d_k$, where $\phi_a$ is the FoI of individuals in the $a$’th age bracket, and $c_a$ and $d_a$ are the lower cut-off age and duration of that bracket, respectively. We define a function for the integrand which takes the argument a for age, up is a vector of the upper cut-offs for each age bracket, and foi is the vector of age-specific FoI’s:

integrandpc = function(a, up, foi){
#Find which interval a belongs to
wh = findInterval(a, sort(c(0,up)))
#Calculate duration of each interval
dur = diff(sort(c(0,up)))
#Evaluate integrand
inte = ifelse(wh == 1, foi[1]*a,
sum(foi[1:(wh-1)]*dur[1:(wh-1)])+
foi[wh]*(a-up[wh-1]))
return(inte)
}


The negative log-likelihood function for the piecewise constant model takes arguments corresponding to log-FoI (par), age (age), number of positives (num), number tested in each age group (denom) and age-class cut-offs (up). Estimating the FoI on a log-scale (foi=exp(par)) ensures that all rates will be positive.

llik.pc = function(par, age, num, denom, up) {
ll = 0
for (i in 1:length(age)) {
p = 1 - exp(-integrandpc(a=age[i], up = up,
foi = exp(par)))
ll = ll + dbinom(num[i], denom[i], p, log = T)
}
return(-ll)
}


We use 1, 4, 8, 12, 18, 24 and 30 months as cut-off points for the age-categories and assign arbitrary initial values of 0.1 for each piece of the FoI-function:

x = c(1, 4, 8, 12, 18, 24, 30)
para = rep(0.1, length(x))


For his analysis we use the optim-function (we could also use mle2) to find maximum likelihood estimates:

est = optim(par=log(para),fn=llik.pc, age=rabbit$a, num=rabbit$inf,  denom=rabbit$n, up=x, method="Nelder-Mead", control=list(trace=2))   Nelder-Mead direct search function minimizer function value for initial parameters = 70.627091 Scaled convergence tolerance is 1.05243e-06 Stepsize computed as 0.230259 BUILD 8 73.376577 69.992539 EXTENSION 10 72.524831 68.209065 LO-REDUCTION 12 72.039597 68.209065 EXTENSION 14 71.252242 66.723042 LO-REDUCTION 16 71.023193 66.723042 LO-REDUCTION 18 70.627091 66.723042 EXTENSION 20 70.500600 65.488893 EXTENSION 22 69.992539 63.525916 LO-REDUCTION 24 68.616649 63.525916 LO-REDUCTION 26 68.209065 63.525916 REFLECTION 28 67.443654 63.293935 LO-REDUCTION 30 66.736993 63.293935 LO-REDUCTION 32 66.723042 63.293935 LO-REDUCTION 34 65.488893 63.293935 EXTENSION 36 64.279298 61.584960 LO-REDUCTION 38 64.093291 61.584960 LO-REDUCTION 40 63.954153 61.584960 LO-REDUCTION 42 63.547868 61.584960 EXTENSION 44 63.536244 60.916671 LO-REDUCTION 46 63.525916 60.916671 LO-REDUCTION 48 63.293935 60.916671 LO-REDUCTION 50 62.929887 60.916671 EXTENSION 52 62.382778 59.067972 LO-REDUCTION 54 62.095780 59.067972 LO-REDUCTION 56 61.584960 59.067972 LO-REDUCTION 58 61.565316 59.067972 EXTENSION 60 61.309788 58.044326 LO-REDUCTION 62 61.102808 58.044326 EXTENSION 64 60.916671 56.819122 LO-REDUCTION 66 60.080635 56.819122 LO-REDUCTION 68 59.557204 56.819122 LO-REDUCTION 70 59.368186 56.819122 EXTENSION 72 59.067972 56.209063 LO-REDUCTION 74 58.925916 56.209063 EXTENSION 76 58.044326 54.496513 LO-REDUCTION 78 57.359761 54.496513 LO-REDUCTION 80 57.208951 54.496513 LO-REDUCTION 82 56.991029 54.496513 LO-REDUCTION 84 56.819122 54.496513 LO-REDUCTION 86 56.290804 54.496513 LO-REDUCTION 88 56.209063 54.496513 LO-REDUCTION 90 55.678514 54.496513 LO-REDUCTION 92 55.598439 54.496513 EXTENSION 94 55.435797 54.048577 LO-REDUCTION 96 55.241465 54.048577 LO-REDUCTION 98 55.124332 54.048577 EXTENSION 100 54.942259 53.891665 EXTENSION 102 54.804349 53.326058 LO-REDUCTION 104 54.763115 53.326058 LO-REDUCTION 106 54.525127 53.326058 EXTENSION 108 54.496513 53.211699 LO-REDUCTION 110 54.159923 53.211699 REFLECTION 112 54.048577 53.003487 LO-REDUCTION 114 53.891665 53.003487 LO-REDUCTION 116 53.611130 53.003487 REFLECTION 118 53.352716 52.951346 REFLECTION 120 53.326058 52.932247 LO-REDUCTION 122 53.251431 52.889198 HI-REDUCTION 124 53.211699 52.889198 LO-REDUCTION 126 53.092364 52.889198 LO-REDUCTION 128 53.012348 52.889198 LO-REDUCTION 130 53.003487 52.885510 REFLECTION 132 52.987525 52.855935 LO-REDUCTION 134 52.959845 52.855935 EXTENSION 136 52.951346 52.812438 LO-REDUCTION 138 52.950106 52.812438 LO-REDUCTION 140 52.932247 52.812438 HI-REDUCTION 142 52.889198 52.812438 EXTENSION 144 52.885510 52.754761 LO-REDUCTION 146 52.884167 52.754761 LO-REDUCTION 148 52.856984 52.754761 LO-REDUCTION 150 52.855935 52.754761 HI-REDUCTION 152 52.837965 52.754761 EXTENSION 154 52.835756 52.738193 EXTENSION 156 52.812438 52.710775 LO-REDUCTION 158 52.801602 52.710775 LO-REDUCTION 160 52.801177 52.710775 LO-REDUCTION 162 52.797764 52.710775 REFLECTION 164 52.762494 52.709772 LO-REDUCTION 166 52.754761 52.709772 HI-REDUCTION 168 52.746775 52.709772 EXTENSION 170 52.740426 52.689098 LO-REDUCTION 172 52.738193 52.689098 LO-REDUCTION 174 52.736975 52.689098 REFLECTION 176 52.729361 52.689040 LO-REDUCTION 178 52.710775 52.689040 REFLECTION 180 52.710361 52.683867 LO-REDUCTION 182 52.709772 52.683867 REFLECTION 184 52.694633 52.674964 LO-REDUCTION 186 52.692379 52.674964 EXTENSION 188 52.689258 52.672696 LO-REDUCTION 190 52.689098 52.672696 LO-REDUCTION 192 52.689040 52.672696 LO-REDUCTION 194 52.685191 52.672696 LO-REDUCTION 196 52.683867 52.672696 LO-REDUCTION 198 52.680489 52.672696 LO-REDUCTION 200 52.677307 52.672696 EXTENSION 202 52.677003 52.667837 LO-REDUCTION 204 52.674964 52.667837 LO-REDUCTION 206 52.674548 52.667837 LO-REDUCTION 208 52.674236 52.667837 LO-REDUCTION 210 52.673884 52.667837 LO-REDUCTION 212 52.673157 52.667837 LO-REDUCTION 214 52.672696 52.667837 LO-REDUCTION 216 52.672434 52.667837 EXTENSION 218 52.671837 52.666201 EXTENSION 220 52.670747 52.662825 LO-REDUCTION 222 52.670300 52.662825 LO-REDUCTION 224 52.670144 52.662825 LO-REDUCTION 226 52.669299 52.662825 LO-REDUCTION 228 52.667979 52.662825 LO-REDUCTION 230 52.667837 52.662825 EXTENSION 232 52.666201 52.660660 EXTENSION 234 52.665349 52.658790 LO-REDUCTION 236 52.664502 52.658790 LO-REDUCTION 238 52.664037 52.658790 EXTENSION 240 52.663979 52.657737 EXTENSION 242 52.663215 52.656283 EXTENSION 244 52.662825 52.650946 LO-REDUCTION 246 52.660660 52.650946 LO-REDUCTION 248 52.659968 52.650946 LO-REDUCTION 250 52.659930 52.650946 EXTENSION 252 52.658790 52.646991 LO-REDUCTION 254 52.657737 52.646991 LO-REDUCTION 256 52.657293 52.646991 EXTENSION 258 52.656283 52.638375 LO-REDUCTION 260 52.653954 52.638375 LO-REDUCTION 262 52.653797 52.638375 EXTENSION 264 52.651014 52.630969 LO-REDUCTION 266 52.650946 52.630969 LO-REDUCTION 268 52.649617 52.630969 EXTENSION 270 52.646991 52.613139 LO-REDUCTION 272 52.645055 52.613139 LO-REDUCTION 274 52.641308 52.613139 EXTENSION 276 52.638375 52.609879 LO-REDUCTION 278 52.634698 52.609879 LO-REDUCTION 280 52.633245 52.609879 REFLECTION 282 52.630969 52.608838 HI-REDUCTION 284 52.619231 52.608838 REFLECTION 286 52.618887 52.605377 LO-REDUCTION 288 52.618189 52.605377 HI-REDUCTION 290 52.614203 52.605377 EXTENSION 292 52.613139 52.596329 HI-REDUCTION 294 52.612892 52.596329 LO-REDUCTION 296 52.610034 52.596329 HI-REDUCTION 298 52.609879 52.596329 LO-REDUCTION 300 52.608838 52.596329 LO-REDUCTION 302 52.608141 52.596329 LO-REDUCTION 304 52.607302 52.596329 EXTENSION 306 52.605377 52.591816 EXTENSION 308 52.605191 52.589462 EXTENSION 310 52.601049 52.580808 LO-REDUCTION 312 52.598607 52.580808 LO-REDUCTION 314 52.597895 52.580808 LO-REDUCTION 316 52.597300 52.580808 REFLECTION 318 52.596329 52.579661 REFLECTION 320 52.591816 52.576709 LO-REDUCTION 322 52.589462 52.576709 HI-REDUCTION 324 52.585667 52.576709 LO-REDUCTION 326 52.583747 52.576709 REFLECTION 328 52.582291 52.575985 LO-REDUCTION 330 52.582231 52.575985 LO-REDUCTION 332 52.580808 52.575985 LO-REDUCTION 334 52.579661 52.575985 LO-REDUCTION 336 52.577802 52.575985 LO-REDUCTION 338 52.577507 52.575985 LO-REDUCTION 340 52.577058 52.575985 HI-REDUCTION 342 52.576923 52.575985 LO-REDUCTION 344 52.576731 52.575985 LO-REDUCTION 346 52.576709 52.575941 LO-REDUCTION 348 52.576271 52.575941 REFLECTION 350 52.576214 52.575927 LO-REDUCTION 352 52.576206 52.575897 LO-REDUCTION 354 52.576202 52.575897 LO-REDUCTION 356 52.576144 52.575897 LO-REDUCTION 358 52.576101 52.575860 HI-REDUCTION 360 52.575985 52.575860 LO-REDUCTION 362 52.575952 52.575860 LO-REDUCTION 364 52.575941 52.575860 LO-REDUCTION 366 52.575927 52.575842 HI-REDUCTION 368 52.575925 52.575842 LO-REDUCTION 370 52.575910 52.575842 REFLECTION 372 52.575897 52.575802 HI-REDUCTION 374 52.575879 52.575802 LO-REDUCTION 376 52.575878 52.575802 LO-REDUCTION 378 52.575861 52.575802 HI-REDUCTION 380 52.575860 52.575802 LO-REDUCTION 382 52.575852 52.575802 LO-REDUCTION 384 52.575842 52.575802 LO-REDUCTION 386 52.575841 52.575802 REFLECTION 388 52.575832 52.575801 HI-REDUCTION 390 52.575830 52.575801 EXTENSION 392 52.575817 52.575776 LO-REDUCTION 394 52.575815 52.575776 EXTENSION 396 52.575810 52.575742 LO-REDUCTION 398 52.575807 52.575742 LO-REDUCTION 400 52.575802 52.575742 LO-REDUCTION 402 52.575802 52.575742 LO-REDUCTION 404 52.575801 52.575742 LO-REDUCTION 406 52.575784 52.575742 EXTENSION 408 52.575776 52.575739 REFLECTION 410 52.575776 52.575736 EXTENSION 412 52.575772 52.575686 LO-REDUCTION 414 52.575761 52.575686 LO-REDUCTION 416 52.575753 52.575686 LO-REDUCTION 418 52.575743 52.575686 EXTENSION 420 52.575742 52.575654 LO-REDUCTION 422 52.575739 52.575654 EXTENSION 424 52.575736 52.575630 EXTENSION 426 52.575706 52.575605 EXTENSION 428 52.575694 52.575553 LO-REDUCTION 430 52.575693 52.575553 EXTENSION 432 52.575686 52.575473 LO-REDUCTION 434 52.575668 52.575473 LO-REDUCTION 436 52.575654 52.575473 EXTENSION 438 52.575630 52.575356 LO-REDUCTION 440 52.575605 52.575356 LO-REDUCTION 442 52.575579 52.575356 LO-REDUCTION 444 52.575553 52.575356 EXTENSION 446 52.575494 52.575256 LO-REDUCTION 448 52.575477 52.575256 LO-REDUCTION 450 52.575473 52.575256 EXTENSION 452 52.575437 52.575224 LO-REDUCTION 454 52.575419 52.575224 LO-REDUCTION 456 52.575361 52.575224 LO-REDUCTION 458 52.575356 52.575224 LO-REDUCTION 460 52.575304 52.575224 LO-REDUCTION 462 52.575280 52.575222 LO-REDUCTION 464 52.575258 52.575222 REFLECTION 466 52.575256 52.575216 HI-REDUCTION 468 52.575249 52.575216 LO-REDUCTION 470 52.575231 52.575216 REFLECTION 472 52.575231 52.575216 LO-REDUCTION 474 52.575229 52.575216 LO-REDUCTION 476 52.575224 52.575216 EXTENSION 478 52.575224 52.575210 EXTENSION 480 52.575223 52.575202 LO-REDUCTION 482 52.575222 52.575202 LO-REDUCTION 484 52.575217 52.575202 EXTENSION 486 52.575216 52.575196 LO-REDUCTION 488 52.575216 52.575196 LO-REDUCTION 490 52.575216 52.575196 LO-REDUCTION 492 52.575211 52.575196 LO-REDUCTION 494 52.575210 52.575196 LO-REDUCTION 496 52.575209 52.575196 LO-REDUCTION 498 52.575206 52.575196 REFLECTION 500 52.575203 52.575195 Exiting from Nelder Mead minimizer 502 function evaluations used  The maximum likelihood estimates for the log-FoI is given in est$par. The associated age-specific FoIs are:

round(exp(est$par), 4)  <ol class=list-inline> • 0.0626 • 0.3712 • 0.0573 • 0 • 0 • 0 • 0.0027 • </ol> We can predict the age-prevalence curve and plot it as a step function. #Make space for left and right axes par(mar = c(5,5,2,5)) #Add beginning and ends to x and y for step plot xvals=c(0,x) yvals=exp(c(est$par, est$par[7])) plot(xvals, yvals, type="s", xlab="age", ylab="FoI") #Superimpose predicted curve par(new=T) p = rep(0, 28) for (i in 1:28) { p[i] = 1 - exp(-integrandpc(a=i, up = x, foi = exp(est$par)))
}
plot(p~c(1:28), ylim=c(0,1), type="l", col="red",
axes=FALSE, xlab=NA, ylab=NA)

axis(side = 4)
mtext(side = 4, line = 4, "Prevalence")
legend("right", legend=c("FoI", "Prevalence"),
lty=c(1,1), col=c("black", "red"))


The FoI peaks perinatally and then falls to zero after the 8 month age class. This is likely due to the older breeder females being housed separately and only having contact with their kittens. used this (in combination with some other analyses; see section [sec:c13bb]) to conclude that most infections happen at a young age from infected mothers to their offspring and then among litter mates.

## A log-spline model

An alternative non-parametric approach to the piecewise constant model is to use smoothing splines. A is a smooth curve that can take an arbitrary shape except that it is constrained to be continuous and with continuous 1st and 2nd derivatives . The popularity of splines in non-parametric regression stems from its computational tractability; A spline can be fit by multiple regression on a set of ‘basis function’-decompositions of a covariate. The gam and mgcv packages offers automated ways to fit a variety of spline-variants to binomial data (and any other error distribution within the exponential family). Unfortunately, as with the case of the piecewise constant model, fitting the log-spline model is a bit more involved because of the integration step in equation [eq:p]. The splines package has functions to create various spline-bases that can be used with lm; predict.lm can predict values for the spline given regression coefficients.

The approach taken here is a bit cheeky in that it ‘hi-jacks’ a spline-regression object created using the bs-spline basis functions in combination with lm and use optim to update/override the regression coefficients in the lm-object until a maximum likelihood solution is found. First we set the number of degrees-of-freedom for the spline. The dl-object will end up as the hi-jacked object for the age-specific FoI .

require(splines)
# Degrees-of-freedom
df = 7
# Construct dummy lm-object
dl = lm(inf ~ bs(a, df), data = rabbit)


We write a tmpfn-function to predict the spline on a log-transformed scale to ensure that the force-of-infection (FoI) is strictly positive:

tmpfn = function(x, dl) {
x = predict(dl, newdata = data.frame(a = x))
exp(x)
}


The tmpfn2-function calculates the negative log-likelihood of the FoI as we did in the foipc-function above. In contrast to the piecewise constant model, the integrated splines do not have a closed form solution so we use Rs inbuilt numerical integrator, integrate:

tmpfn2 = function(par, data, df){
#Dummy lm-object
dl = lm(inf ~ bs(a,df), data = data)
#Overwrite spline coefficients with new values
dl$coefficients = par #Calculate log-likelihood ll = 0 for(i in 1:length(data$a)){
p = 1 - exp(-integrate(tmpfn, 0, i, dl = dl)$value) ll = ll + dbinom(data$inf[i], data$n[i], p ,log = T) } return(-ll) }  We use arbitrary initial values and minimize the negative log-likelihood using optim. (Warnings are turned off to avoid excessive output) options(warn=-1) para = rep(-1, df + 1) dspline = optim(par = para, fn = tmpfn2, data = rabbit, df = df, method = "Nelder-Mead", control = list(trace = 2, maxit = 2000)) options(warn=0)   Nelder-Mead direct search function minimizer function value for initial parameters = 143.107967 Scaled convergence tolerance is 2.13247e-06 Stepsize computed as 0.100000 BUILD 9 208.890158 136.838368 LO-REDUCTION 11 160.486050 136.838368 EXTENSION 13 154.497562 113.605482 LO-REDUCTION 15 145.168488 113.605482 LO-REDUCTION 17 144.918156 113.605482 LO-REDUCTION 19 144.569912 113.605482 LO-REDUCTION 21 143.395341 113.605482 LO-REDUCTION 23 143.107967 113.605482 LO-REDUCTION 25 140.960762 113.605482 EXTENSION 27 136.838368 98.745777 LO-REDUCTION 29 131.266359 98.745777 LO-REDUCTION 31 129.073267 98.745777 LO-REDUCTION 33 126.571882 98.745777 EXTENSION 35 123.390222 85.412874 LO-REDUCTION 37 121.494057 85.412874 EXTENSION 39 118.281390 74.140531 LO-REDUCTION 41 113.605482 74.140531 LO-REDUCTION 43 105.059128 74.140531 LO-REDUCTION 45 103.755458 74.140531 EXTENSION 47 100.176186 63.644364 LO-REDUCTION 49 98.745777 63.644364 REFLECTION 51 90.193830 63.407645 REFLECTION 53 85.412874 62.612496 REFLECTION 55 78.453322 62.479692 REFLECTION 57 75.706534 62.171198 LO-REDUCTION 59 74.340958 62.171198 LO-REDUCTION 61 74.140531 62.121451 LO-REDUCTION 63 65.870336 62.121451 LO-REDUCTION 65 63.644364 62.121451 LO-REDUCTION 67 63.407645 62.121451 LO-REDUCTION 69 62.666244 62.121451 LO-REDUCTION 71 62.612496 62.121451 HI-REDUCTION 73 62.493571 62.121451 HI-REDUCTION 75 62.481437 62.121451 LO-REDUCTION 77 62.479692 62.121451 EXTENSION 79 62.347523 61.923681 LO-REDUCTION 81 62.337621 61.923681 LO-REDUCTION 83 62.293940 61.923681 LO-REDUCTION 85 62.273725 61.923681 EXTENSION 87 62.267333 61.869867 LO-REDUCTION 89 62.221955 61.869867 EXTENSION 91 62.171198 61.733317 EXTENSION 93 62.121451 61.327030 LO-REDUCTION 95 62.051158 61.327030 LO-REDUCTION 97 62.047967 61.327030 LO-REDUCTION 99 62.014560 61.327030 EXTENSION 101 61.945396 60.883681 LO-REDUCTION 103 61.923681 60.883681 LO-REDUCTION 105 61.869867 60.883681 EXTENSION 107 61.733317 60.558256 LO-REDUCTION 109 61.484953 60.558256 LO-REDUCTION 111 61.426583 60.558256 LO-REDUCTION 113 61.422825 60.558256 LO-REDUCTION 115 61.406159 60.558256 EXTENSION 117 61.327030 60.224650 LO-REDUCTION 119 61.171690 60.224650 LO-REDUCTION 121 60.883681 60.224650 REFLECTION 123 60.871040 60.215465 LO-REDUCTION 125 60.730754 60.215465 LO-REDUCTION 127 60.693176 60.215465 LO-REDUCTION 129 60.594718 60.215465 LO-REDUCTION 131 60.558256 60.215465 LO-REDUCTION 133 60.368485 60.211146 REFLECTION 135 60.336319 60.175697 LO-REDUCTION 137 60.274739 60.175697 LO-REDUCTION 139 60.273785 60.173571 LO-REDUCTION 141 60.232724 60.173571 EXTENSION 143 60.224650 60.089510 LO-REDUCTION 145 60.217783 60.089510 LO-REDUCTION 147 60.215465 60.089510 LO-REDUCTION 149 60.211146 60.089510 LO-REDUCTION 151 60.204384 60.089510 LO-REDUCTION 153 60.185507 60.089510 EXTENSION 155 60.175795 60.038540 LO-REDUCTION 157 60.175697 60.038540 LO-REDUCTION 159 60.173571 60.038540 EXTENSION 161 60.157371 59.966298 LO-REDUCTION 163 60.125594 59.966298 LO-REDUCTION 165 60.116567 59.966298 LO-REDUCTION 167 60.099834 59.966298 LO-REDUCTION 169 60.096702 59.966298 LO-REDUCTION 171 60.089510 59.966298 LO-REDUCTION 173 60.083027 59.966298 REFLECTION 175 60.042864 59.966293 EXTENSION 177 60.038540 59.910884 LO-REDUCTION 179 60.031123 59.910884 LO-REDUCTION 181 60.026525 59.910884 REFLECTION 183 60.016209 59.905031 LO-REDUCTION 185 59.981918 59.905031 EXTENSION 187 59.976385 59.873996 LO-REDUCTION 189 59.966298 59.873996 LO-REDUCTION 191 59.966293 59.873996 LO-REDUCTION 193 59.946586 59.873996 EXTENSION 195 59.937851 59.851807 LO-REDUCTION 197 59.917763 59.851807 LO-REDUCTION 199 59.917157 59.851807 EXTENSION 201 59.910884 59.829838 LO-REDUCTION 203 59.908333 59.829838 LO-REDUCTION 205 59.905031 59.829838 LO-REDUCTION 207 59.877899 59.829838 LO-REDUCTION 209 59.873996 59.829838 REFLECTION 211 59.872123 59.826785 HI-REDUCTION 213 59.853902 59.826785 HI-REDUCTION 215 59.852644 59.826785 LO-REDUCTION 217 59.851807 59.826785 EXTENSION 219 59.848868 59.807038 LO-REDUCTION 221 59.846838 59.807038 LO-REDUCTION 223 59.846601 59.807038 LO-REDUCTION 225 59.842601 59.807038 LO-REDUCTION 227 59.840188 59.807038 EXTENSION 229 59.836438 59.789206 EXTENSION 231 59.829838 59.775566 LO-REDUCTION 233 59.826785 59.775566 EXTENSION 235 59.820769 59.744107 LO-REDUCTION 237 59.819168 59.744107 LO-REDUCTION 239 59.813848 59.744107 LO-REDUCTION 241 59.808522 59.744107 EXTENSION 243 59.807038 59.713660 LO-REDUCTION 245 59.793337 59.713660 LO-REDUCTION 247 59.789206 59.713660 EXTENSION 249 59.775566 59.677198 LO-REDUCTION 251 59.774099 59.677198 EXTENSION 253 59.762318 59.636254 LO-REDUCTION 255 59.759325 59.636254 LO-REDUCTION 257 59.745618 59.636254 EXTENSION 259 59.744107 59.611910 EXTENSION 261 59.722300 59.573051 LO-REDUCTION 263 59.713660 59.573051 EXTENSION 265 59.695126 59.531587 EXTENSION 267 59.677198 59.497138 LO-REDUCTION 269 59.662559 59.497138 LO-REDUCTION 271 59.648718 59.497138 REFLECTION 273 59.636254 59.492002 LO-REDUCTION 275 59.611910 59.492002 REFLECTION 277 59.576619 59.474978 LO-REDUCTION 279 59.573051 59.474978 LO-REDUCTION 281 59.531587 59.474978 LO-REDUCTION 283 59.517237 59.474978 LO-REDUCTION 285 59.506390 59.474978 LO-REDUCTION 287 59.502819 59.474978 LO-REDUCTION 289 59.498525 59.474698 EXTENSION 291 59.497138 59.458901 LO-REDUCTION 293 59.492002 59.458901 HI-REDUCTION 295 59.485318 59.458901 LO-REDUCTION 297 59.484464 59.458901 EXTENSION 299 59.483056 59.438019 LO-REDUCTION 301 59.482313 59.438019 HI-REDUCTION 303 59.475503 59.438019 HI-REDUCTION 305 59.474978 59.438019 LO-REDUCTION 307 59.474698 59.438019 LO-REDUCTION 309 59.467571 59.438019 EXTENSION 311 59.464814 59.433976 REFLECTION 313 59.463691 59.432107 REFLECTION 315 59.461679 59.428852 LO-REDUCTION 317 59.458901 59.428852 EXTENSION 319 59.456410 59.412002 LO-REDUCTION 321 59.442917 59.412002 LO-REDUCTION 323 59.439308 59.412002 LO-REDUCTION 325 59.438019 59.412002 EXTENSION 327 59.433976 59.393514 LO-REDUCTION 329 59.432107 59.393514 EXTENSION 331 59.431142 59.379118 LO-REDUCTION 333 59.428852 59.379118 LO-REDUCTION 335 59.417955 59.379118 LO-REDUCTION 337 59.414438 59.379118 LO-REDUCTION 339 59.414073 59.379118 LO-REDUCTION 341 59.412002 59.379118 EXTENSION 343 59.408231 59.363015 LO-REDUCTION 345 59.398108 59.363015 EXTENSION 347 59.393514 59.344297 LO-REDUCTION 349 59.392588 59.344297 LO-REDUCTION 351 59.391621 59.344297 LO-REDUCTION 353 59.387371 59.344297 EXTENSION 355 59.383540 59.316214 LO-REDUCTION 357 59.379708 59.316214 LO-REDUCTION 359 59.379118 59.316214 EXTENSION 361 59.365807 59.285413 LO-REDUCTION 363 59.363015 59.285413 LO-REDUCTION 365 59.350240 59.285413 EXTENSION 367 59.345186 59.263699 LO-REDUCTION 369 59.344297 59.263699 LO-REDUCTION 371 59.326443 59.263699 LO-REDUCTION 373 59.323015 59.263699 EXTENSION 375 59.316214 59.204538 LO-REDUCTION 377 59.306696 59.204538 LO-REDUCTION 379 59.288397 59.204538 HI-REDUCTION 381 59.285413 59.204538 LO-REDUCTION 383 59.277933 59.204538 EXTENSION 385 59.267642 59.167399 LO-REDUCTION 387 59.266261 59.167399 LO-REDUCTION 389 59.263699 59.167399 EXTENSION 391 59.259571 59.140120 LO-REDUCTION 393 59.255134 59.140120 EXTENSION 395 59.216421 59.102754 EXTENSION 397 59.206032 59.058506 LO-REDUCTION 399 59.204538 59.058506 EXTENSION 401 59.185126 58.993605 LO-REDUCTION 403 59.178405 58.993605 LO-REDUCTION 405 59.167399 58.993605 LO-REDUCTION 407 59.142041 58.993605 LO-REDUCTION 409 59.140120 58.993605 EXTENSION 411 59.102754 58.954918 LO-REDUCTION 413 59.100601 58.954918 EXTENSION 415 59.080444 58.915734 EXTENSION 417 59.058506 58.791010 LO-REDUCTION 419 59.039087 58.791010 LO-REDUCTION 421 59.030646 58.791010 LO-REDUCTION 423 59.028152 58.791010 LO-REDUCTION 425 58.993605 58.791010 LO-REDUCTION 427 58.966737 58.791010 LO-REDUCTION 429 58.954918 58.791010 REFLECTION 431 58.915734 58.770748 EXTENSION 433 58.905273 58.638114 LO-REDUCTION 435 58.887693 58.638114 LO-REDUCTION 437 58.855010 58.638114 LO-REDUCTION 439 58.829838 58.638114 LO-REDUCTION 441 58.818448 58.638114 EXTENSION 443 58.817206 58.508691 EXTENSION 445 58.791010 58.349242 LO-REDUCTION 447 58.770748 58.349242 LO-REDUCTION 449 58.728280 58.349242 LO-REDUCTION 451 58.667731 58.349242 LO-REDUCTION 453 58.658305 58.349242 LO-REDUCTION 455 58.650596 58.349242 EXTENSION 457 58.638114 58.081425 LO-REDUCTION 459 58.586752 58.081425 LO-REDUCTION 461 58.508691 58.081425 EXTENSION 463 58.405573 57.879435 LO-REDUCTION 465 58.383924 57.879435 LO-REDUCTION 467 58.378741 57.879435 EXTENSION 469 58.359318 57.680746 LO-REDUCTION 471 58.349242 57.680746 REFLECTION 473 58.176254 57.631407 LO-REDUCTION 475 58.130983 57.631407 LO-REDUCTION 477 58.081425 57.631407 EXTENSION 479 57.997343 57.485909 LO-REDUCTION 481 57.923014 57.485909 LO-REDUCTION 483 57.879435 57.485909 LO-REDUCTION 485 57.815812 57.485909 LO-REDUCTION 487 57.784974 57.485909 LO-REDUCTION 489 57.680746 57.485909 HI-REDUCTION 491 57.664354 57.485909 LO-REDUCTION 493 57.663447 57.485909 LO-REDUCTION 495 57.631407 57.485909 LO-REDUCTION 497 57.623283 57.485909 LO-REDUCTION 499 57.608771 57.485909 LO-REDUCTION 501 57.604495 57.485909 LO-REDUCTION 503 57.565838 57.485909 LO-REDUCTION 505 57.562502 57.485909 EXTENSION 507 57.557571 57.448131 LO-REDUCTION 509 57.550478 57.448131 LO-REDUCTION 511 57.539655 57.448131 LO-REDUCTION 513 57.537945 57.448131 EXTENSION 515 57.534937 57.415980 EXTENSION 517 57.508752 57.344479 LO-REDUCTION 519 57.486134 57.344479 LO-REDUCTION 521 57.485909 57.344479 LO-REDUCTION 523 57.480293 57.344479 EXTENSION 525 57.468841 57.281398 LO-REDUCTION 527 57.461756 57.281398 LO-REDUCTION 529 57.448131 57.281398 LO-REDUCTION 531 57.415980 57.281398 LO-REDUCTION 533 57.407478 57.281398 EXTENSION 535 57.400192 57.278900 REFLECTION 537 57.393838 57.262797 EXTENSION 539 57.344479 57.116846 LO-REDUCTION 541 57.341200 57.116846 LO-REDUCTION 543 57.313052 57.116846 LO-REDUCTION 545 57.301857 57.116846 LO-REDUCTION 547 57.289368 57.116846 LO-REDUCTION 549 57.281398 57.116846 LO-REDUCTION 551 57.278900 57.116846 LO-REDUCTION 553 57.262797 57.116846 EXTENSION 555 57.231237 57.047477 LO-REDUCTION 557 57.227540 57.047477 LO-REDUCTION 559 57.211860 57.047477 EXTENSION 561 57.201065 56.983878 LO-REDUCTION 563 57.183216 56.983878 LO-REDUCTION 565 57.180599 56.983878 REFLECTION 567 57.119954 56.967707 EXTENSION 569 57.116846 56.930068 EXTENSION 571 57.087183 56.908420 REFLECTION 573 57.068935 56.904428 REFLECTION 575 57.047477 56.885085 LO-REDUCTION 577 57.007651 56.885085 LO-REDUCTION 579 56.988545 56.885085 LO-REDUCTION 581 56.983878 56.885085 HI-REDUCTION 583 56.967707 56.885085 LO-REDUCTION 585 56.930068 56.885085 EXTENSION 587 56.922211 56.865029 HI-REDUCTION 589 56.908420 56.865029 LO-REDUCTION 591 56.904428 56.865029 LO-REDUCTION 593 56.903365 56.865029 LO-REDUCTION 595 56.899519 56.865029 LO-REDUCTION 597 56.896169 56.865029 LO-REDUCTION 599 56.893840 56.865029 REFLECTION 601 56.886385 56.859734 REFLECTION 603 56.885085 56.857207 LO-REDUCTION 605 56.881598 56.857207 LO-REDUCTION 607 56.872493 56.857207 LO-REDUCTION 609 56.870715 56.857207 LO-REDUCTION 611 56.866955 56.857207 REFLECTION 613 56.866011 56.856704 HI-REDUCTION 615 56.865029 56.856704 LO-REDUCTION 617 56.862298 56.856704 EXTENSION 619 56.861313 56.853993 LO-REDUCTION 621 56.859964 56.853993 LO-REDUCTION 623 56.859734 56.853993 EXTENSION 625 56.859108 56.846771 LO-REDUCTION 627 56.857972 56.846771 LO-REDUCTION 629 56.857239 56.846771 LO-REDUCTION 631 56.857207 56.846771 LO-REDUCTION 633 56.856704 56.846771 LO-REDUCTION 635 56.855970 56.846771 LO-REDUCTION 637 56.855256 56.846771 EXTENSION 639 56.853993 56.842603 LO-REDUCTION 641 56.853154 56.842603 LO-REDUCTION 643 56.852762 56.842603 EXTENSION 645 56.851667 56.841160 EXTENSION 647 56.851577 56.834939 LO-REDUCTION 649 56.848503 56.834939 LO-REDUCTION 651 56.847492 56.834939 EXTENSION 653 56.846771 56.826029 LO-REDUCTION 655 56.845477 56.826029 LO-REDUCTION 657 56.843339 56.826029 LO-REDUCTION 659 56.842603 56.826029 EXTENSION 661 56.841160 56.821426 EXTENSION 663 56.835918 56.817763 EXTENSION 665 56.835734 56.814940 LO-REDUCTION 667 56.834939 56.814940 LO-REDUCTION 669 56.831928 56.814940 REFLECTION 671 56.828767 56.811790 LO-REDUCTION 673 56.827513 56.811790 LO-REDUCTION 675 56.826029 56.811790 LO-REDUCTION 677 56.821426 56.811790 LO-REDUCTION 679 56.817763 56.811790 LO-REDUCTION 681 56.816791 56.811790 LO-REDUCTION 683 56.816522 56.811790 LO-REDUCTION 685 56.814940 56.811790 LO-REDUCTION 687 56.814729 56.811790 EXTENSION 689 56.814653 56.810089 LO-REDUCTION 691 56.813617 56.810089 LO-REDUCTION 693 56.813531 56.810089 LO-REDUCTION 695 56.813522 56.810089 EXTENSION 697 56.813118 56.809215 LO-REDUCTION 699 56.812627 56.809215 LO-REDUCTION 701 56.812586 56.809215 EXTENSION 703 56.811790 56.808212 LO-REDUCTION 705 56.811570 56.808212 EXTENSION 707 56.811553 56.805091 LO-REDUCTION 709 56.810575 56.805091 LO-REDUCTION 711 56.810230 56.805091 LO-REDUCTION 713 56.810089 56.805091 LO-REDUCTION 715 56.809640 56.805091 LO-REDUCTION 717 56.809215 56.805091 LO-REDUCTION 719 56.809080 56.805091 LO-REDUCTION 721 56.808212 56.805091 EXTENSION 723 56.807708 56.804624 EXTENSION 725 56.807555 56.802539 LO-REDUCTION 727 56.806892 56.802539 EXTENSION 729 56.806877 56.800421 LO-REDUCTION 731 56.806420 56.800421 EXTENSION 733 56.805670 56.796606 LO-REDUCTION 735 56.805463 56.796606 LO-REDUCTION 737 56.805091 56.796606 LO-REDUCTION 739 56.804624 56.796606 LO-REDUCTION 741 56.802856 56.796606 LO-REDUCTION 743 56.802539 56.796606 EXTENSION 745 56.801523 56.793056 LO-REDUCTION 747 56.800511 56.793056 EXTENSION 749 56.800421 56.790183 LO-REDUCTION 751 56.799818 56.790183 LO-REDUCTION 753 56.798413 56.790183 LO-REDUCTION 755 56.798257 56.790183 EXTENSION 757 56.796786 56.786921 EXTENSION 759 56.796606 56.782818 LO-REDUCTION 761 56.794597 56.782818 LO-REDUCTION 763 56.793285 56.782818 LO-REDUCTION 765 56.793056 56.782818 EXTENSION 767 56.792275 56.780840 LO-REDUCTION 769 56.791397 56.780840 LO-REDUCTION 771 56.790183 56.780840 LO-REDUCTION 773 56.788719 56.780840 EXTENSION 775 56.786921 56.779171 LO-REDUCTION 777 56.785385 56.779171 EXTENSION 779 56.785297 56.777228 LO-REDUCTION 781 56.783405 56.777228 REFLECTION 783 56.782818 56.776172 LO-REDUCTION 785 56.782734 56.776172 LO-REDUCTION 787 56.781500 56.776172 HI-REDUCTION 789 56.780840 56.776172 LO-REDUCTION 791 56.779573 56.776172 LO-REDUCTION 793 56.779171 56.776172 LO-REDUCTION 795 56.778373 56.776172 EXTENSION 797 56.778178 56.772581 LO-REDUCTION 799 56.777733 56.772581 LO-REDUCTION 801 56.777228 56.772581 LO-REDUCTION 803 56.776626 56.772581 LO-REDUCTION 805 56.776468 56.772581 LO-REDUCTION 807 56.776352 56.772581 LO-REDUCTION 809 56.776190 56.772581 EXTENSION 811 56.776172 56.771280 EXTENSION 813 56.775506 56.769115 LO-REDUCTION 815 56.774904 56.769115 LO-REDUCTION 817 56.774147 56.769115 LO-REDUCTION 819 56.774009 56.769115 EXTENSION 821 56.773962 56.765807 LO-REDUCTION 823 56.773111 56.765807 EXTENSION 825 56.772581 56.761620 LO-REDUCTION 827 56.771280 56.761620 LO-REDUCTION 829 56.770188 56.761620 EXTENSION 831 56.769376 56.758036 LO-REDUCTION 833 56.769197 56.758036 LO-REDUCTION 835 56.769115 56.758036 LO-REDUCTION 837 56.767302 56.758036 REFLECTION 839 56.765807 56.757328 EXTENSION 841 56.764143 56.754977 EXTENSION 843 56.762835 56.749433 LO-REDUCTION 845 56.761620 56.749433 LO-REDUCTION 847 56.760808 56.749433 LO-REDUCTION 849 56.758900 56.749433 LO-REDUCTION 851 56.758325 56.749433 LO-REDUCTION 853 56.758036 56.749433 LO-REDUCTION 855 56.757328 56.749433 LO-REDUCTION 857 56.754977 56.749433 EXTENSION 859 56.753448 56.746252 LO-REDUCTION 861 56.752619 56.746252 LO-REDUCTION 863 56.750962 56.746252 EXTENSION 865 56.750556 56.746022 EXTENSION 867 56.750435 56.741703 LO-REDUCTION 869 56.750367 56.741703 LO-REDUCTION 871 56.749584 56.741703 EXTENSION 873 56.749433 56.736134 LO-REDUCTION 875 56.746406 56.736134 LO-REDUCTION 877 56.746371 56.736134 LO-REDUCTION 879 56.746252 56.736134 LO-REDUCTION 881 56.746022 56.736134 LO-REDUCTION 883 56.743384 56.736134 EXTENSION 885 56.743204 56.730637 LO-REDUCTION 887 56.741703 56.730637 LO-REDUCTION 889 56.740722 56.730637 LO-REDUCTION 891 56.740060 56.730637 LO-REDUCTION 893 56.739949 56.730637 EXTENSION 895 56.739498 56.723634 LO-REDUCTION 897 56.737462 56.723634 LO-REDUCTION 899 56.736134 56.723634 EXTENSION 901 56.735096 56.719998 EXTENSION 903 56.733738 56.714760 LO-REDUCTION 905 56.731995 56.714760 EXTENSION 907 56.731080 56.702156 LO-REDUCTION 909 56.730637 56.702156 LO-REDUCTION 911 56.726154 56.702156 LO-REDUCTION 913 56.725860 56.702156 EXTENSION 915 56.723634 56.687292 LO-REDUCTION 917 56.719998 56.687292 LO-REDUCTION 919 56.714974 56.687292 EXTENSION 921 56.714760 56.679752 EXTENSION 923 56.708971 56.664625 LO-REDUCTION 925 56.706147 56.664625 LO-REDUCTION 927 56.704021 56.664625 EXTENSION 929 56.702156 56.646642 LO-REDUCTION 931 56.691505 56.646642 LO-REDUCTION 933 56.689243 56.646642 EXTENSION 935 56.687292 56.640029 EXTENSION 937 56.679752 56.632868 EXTENSION 939 56.670447 56.622539 LO-REDUCTION 941 56.665197 56.622539 LO-REDUCTION 943 56.664625 56.622539 LO-REDUCTION 945 56.659363 56.622539 REFLECTION 947 56.652768 56.614194 LO-REDUCTION 949 56.646642 56.614194 HI-REDUCTION 951 56.640029 56.614194 EXTENSION 953 56.633924 56.601877 LO-REDUCTION 955 56.632868 56.601877 LO-REDUCTION 957 56.630477 56.601877 EXTENSION 959 56.627765 56.584607 LO-REDUCTION 961 56.624295 56.584607 LO-REDUCTION 963 56.622539 56.584607 LO-REDUCTION 965 56.621726 56.584607 EXTENSION 967 56.614194 56.559337 LO-REDUCTION 969 56.612284 56.559337 LO-REDUCTION 971 56.606098 56.559337 LO-REDUCTION 973 56.603550 56.559337 LO-REDUCTION 975 56.601877 56.559337 EXTENSION 977 56.595432 56.546002 LO-REDUCTION 979 56.594102 56.546002 LO-REDUCTION 981 56.587257 56.546002 EXTENSION 983 56.584607 56.542417 EXTENSION 985 56.576886 56.525335 EXTENSION 987 56.576738 56.502465 LO-REDUCTION 989 56.564714 56.502465 LO-REDUCTION 991 56.559337 56.502465 LO-REDUCTION 993 56.554832 56.502465 LO-REDUCTION 995 56.551507 56.502465 LO-REDUCTION 997 56.546002 56.502465 LO-REDUCTION 999 56.542417 56.502465 REFLECTION 1001 56.525335 56.498582 LO-REDUCTION 1003 56.518489 56.498582 EXTENSION 1005 56.518236 56.494694 LO-REDUCTION 1007 56.516248 56.494694 EXTENSION 1009 56.514561 56.488616 LO-REDUCTION 1011 56.509203 56.488616 REFLECTION 1013 56.508478 56.487274 LO-REDUCTION 1015 56.504871 56.487274 LO-REDUCTION 1017 56.502465 56.487274 LO-REDUCTION 1019 56.498582 56.487274 LO-REDUCTION 1021 56.496919 56.487274 LO-REDUCTION 1023 56.494694 56.487274 REFLECTION 1025 56.493138 56.486362 REFLECTION 1027 56.492382 56.486222 LO-REDUCTION 1029 56.489185 56.486222 LO-REDUCTION 1031 56.489144 56.486083 REFLECTION 1033 56.488616 56.485800 REFLECTION 1035 56.487890 56.485621 HI-REDUCTION 1037 56.487817 56.485621 LO-REDUCTION 1039 56.487356 56.485517 REFLECTION 1041 56.487274 56.485397 REFLECTION 1043 56.486362 56.484474 HI-REDUCTION 1045 56.486222 56.484474 HI-REDUCTION 1047 56.486199 56.484474 LO-REDUCTION 1049 56.486083 56.484474 LO-REDUCTION 1051 56.485800 56.484474 LO-REDUCTION 1053 56.485621 56.484474 REFLECTION 1055 56.485517 56.484403 EXTENSION 1057 56.485397 56.483500 HI-REDUCTION 1059 56.485255 56.483500 LO-REDUCTION 1061 56.485230 56.483500 LO-REDUCTION 1063 56.485086 56.483500 LO-REDUCTION 1065 56.485013 56.483500 REFLECTION 1067 56.484903 56.483465 REFLECTION 1069 56.484611 56.483158 LO-REDUCTION 1071 56.484474 56.483158 REFLECTION 1073 56.484403 56.483157 EXTENSION 1075 56.484023 56.481826 LO-REDUCTION 1077 56.483811 56.481826 LO-REDUCTION 1079 56.483523 56.481826 LO-REDUCTION 1081 56.483500 56.481826 LO-REDUCTION 1083 56.483465 56.481826 LO-REDUCTION 1085 56.483369 56.481826 LO-REDUCTION 1087 56.483158 56.481826 LO-REDUCTION 1089 56.483157 56.481826 LO-REDUCTION 1091 56.483041 56.481826 EXTENSION 1093 56.482815 56.481455 LO-REDUCTION 1095 56.482489 56.481455 EXTENSION 1097 56.482428 56.481281 EXTENSION 1099 56.482331 56.480959 EXTENSION 1101 56.482190 56.480780 LO-REDUCTION 1103 56.482011 56.480780 HI-REDUCTION 1105 56.481899 56.480780 LO-REDUCTION 1107 56.481826 56.480780 LO-REDUCTION 1109 56.481647 56.480780 LO-REDUCTION 1111 56.481455 56.480780 REFLECTION 1113 56.481284 56.480704 HI-REDUCTION 1115 56.481281 56.480704 LO-REDUCTION 1117 56.481277 56.480704 LO-REDUCTION 1119 56.481004 56.480704 EXTENSION 1121 56.480959 56.480640 EXTENSION 1123 56.480959 56.480577 REFLECTION 1125 56.480915 56.480538 LO-REDUCTION 1127 56.480909 56.480538 HI-REDUCTION 1129 56.480868 56.480538 REFLECTION 1131 56.480780 56.480501 LO-REDUCTION 1133 56.480733 56.480501 REFLECTION 1135 56.480704 56.480468 LO-REDUCTION 1137 56.480671 56.480468 HI-REDUCTION 1139 56.480666 56.480468 REFLECTION 1141 56.480640 56.480443 LO-REDUCTION 1143 56.480577 56.480443 LO-REDUCTION 1145 56.480554 56.480443 LO-REDUCTION 1147 56.480538 56.480443 LO-REDUCTION 1149 56.480535 56.480443 LO-REDUCTION 1151 56.480513 56.480443 EXTENSION 1153 56.480501 56.480416 REFLECTION 1155 56.480470 56.480407 LO-REDUCTION 1157 56.480468 56.480407 LO-REDUCTION 1159 56.480466 56.480407 REFLECTION 1161 56.480464 56.480402 LO-REDUCTION 1163 56.480449 56.480402 LO-REDUCTION 1165 56.480444 56.480402 LO-REDUCTION 1167 56.480443 56.480402 LO-REDUCTION 1169 56.480423 56.480402 EXTENSION 1171 56.480421 56.480370 LO-REDUCTION 1173 56.480416 56.480370 LO-REDUCTION 1175 56.480411 56.480370 LO-REDUCTION 1177 56.480409 56.480370 LO-REDUCTION 1179 56.480407 56.480370 LO-REDUCTION 1181 56.480406 56.480370 LO-REDUCTION 1183 56.480403 56.480370 EXTENSION 1185 56.480402 56.480349 LO-REDUCTION 1187 56.480394 56.480349 LO-REDUCTION 1189 56.480389 56.480349 EXTENSION 1191 56.480387 56.480326 LO-REDUCTION 1193 56.480384 56.480326 EXTENSION 1195 56.480377 56.480295 LO-REDUCTION 1197 56.480371 56.480295 LO-REDUCTION 1199 56.480370 56.480295 LO-REDUCTION 1201 56.480359 56.480295 EXTENSION 1203 56.480353 56.480288 EXTENSION 1205 56.480349 56.480254 EXTENSION 1207 56.480342 56.480207 LO-REDUCTION 1209 56.480326 56.480207 LO-REDUCTION 1211 56.480310 56.480207 EXTENSION 1213 56.480309 56.480167 LO-REDUCTION 1215 56.480299 56.480167 EXTENSION 1217 56.480295 56.480115 LO-REDUCTION 1219 56.480288 56.480115 LO-REDUCTION 1221 56.480254 56.480115 EXTENSION 1223 56.480224 56.480042 LO-REDUCTION 1225 56.480214 56.480042 LO-REDUCTION 1227 56.480207 56.480042 LO-REDUCTION 1229 56.480193 56.480042 EXTENSION 1231 56.480174 56.479955 LO-REDUCTION 1233 56.480167 56.479955 LO-REDUCTION 1235 56.480131 56.479955 LO-REDUCTION 1237 56.480115 56.479955 EXTENSION 1239 56.480083 56.479939 EXTENSION 1241 56.480080 56.479907 LO-REDUCTION 1243 56.480075 56.479907 REFLECTION 1245 56.480042 56.479879 LO-REDUCTION 1247 56.480020 56.479879 EXTENSION 1249 56.479991 56.479815 LO-REDUCTION 1251 56.479982 56.479815 LO-REDUCTION 1253 56.479955 56.479815 LO-REDUCTION 1255 56.479939 56.479815 LO-REDUCTION 1257 56.479909 56.479815 LO-REDUCTION 1259 56.479907 56.479815 LO-REDUCTION 1261 56.479897 56.479815 LO-REDUCTION 1263 56.479879 56.479815 REFLECTION 1265 56.479860 56.479812 EXTENSION 1267 56.479858 56.479766 LO-REDUCTION 1269 56.479849 56.479766 LO-REDUCTION 1271 56.479833 56.479766 LO-REDUCTION 1273 56.479831 56.479766 LO-REDUCTION 1275 56.479826 56.479766 EXTENSION 1277 56.479816 56.479731 LO-REDUCTION 1279 56.479815 56.479731 LO-REDUCTION 1281 56.479812 56.479731 LO-REDUCTION 1283 56.479803 56.479731 REFLECTION 1285 56.479794 56.479730 EXTENSION 1287 56.479775 56.479718 LO-REDUCTION 1289 56.479774 56.479718 EXTENSION 1291 56.479767 56.479693 LO-REDUCTION 1293 56.479766 56.479693 REFLECTION 1295 56.479750 56.479686 LO-REDUCTION 1297 56.479732 56.479686 LO-REDUCTION 1299 56.479731 56.479686 LO-REDUCTION 1301 56.479731 56.479686 REFLECTION 1303 56.479730 56.479684 LO-REDUCTION 1305 56.479718 56.479684 LO-REDUCTION 1307 56.479707 56.479684 REFLECTION 1309 56.479705 56.479677 REFLECTION 1311 56.479700 56.479672 REFLECTION 1313 56.479697 56.479672 EXTENSION 1315 56.479693 56.479646 LO-REDUCTION 1317 56.479687 56.479646 LO-REDUCTION 1319 56.479686 56.479646 LO-REDUCTION 1321 56.479686 56.479646 LO-REDUCTION 1323 56.479684 56.479646 LO-REDUCTION 1325 56.479677 56.479646 EXTENSION 1327 56.479672 56.479640 LO-REDUCTION 1329 56.479672 56.479640 EXTENSION 1331 56.479671 56.479620 LO-REDUCTION 1333 56.479664 56.479620 LO-REDUCTION 1335 56.479658 56.479620 LO-REDUCTION 1337 56.479654 56.479620 LO-REDUCTION 1339 56.479654 56.479620 EXTENSION 1341 56.479651 56.479607 EXTENSION 1343 56.479646 56.479575 LO-REDUCTION 1345 56.479640 56.479575 LO-REDUCTION 1347 56.479627 56.479575 LO-REDUCTION 1349 56.479626 56.479575 LO-REDUCTION 1351 56.479625 56.479575 LO-REDUCTION 1353 56.479623 56.479575 EXTENSION 1355 56.479620 56.479550 LO-REDUCTION 1357 56.479607 56.479550 EXTENSION 1359 56.479594 56.479528 LO-REDUCTION 1361 56.479584 56.479528 EXTENSION 1363 56.479581 56.479514 LO-REDUCTION 1365 56.479580 56.479514 LO-REDUCTION 1367 56.479580 56.479514 LO-REDUCTION 1369 56.479575 56.479514 LO-REDUCTION 1371 56.479559 56.479514 REFLECTION 1373 56.479550 56.479509 EXTENSION 1375 56.479541 56.479457 LO-REDUCTION 1377 56.479534 56.479457 LO-REDUCTION 1379 56.479528 56.479457 LO-REDUCTION 1381 56.479519 56.479457 LO-REDUCTION 1383 56.479518 56.479457 EXTENSION 1385 56.479516 56.479441 EXTENSION 1387 56.479514 56.479407 LO-REDUCTION 1389 56.479509 56.479407 LO-REDUCTION 1391 56.479489 56.479407 EXTENSION 1393 56.479483 56.479372 LO-REDUCTION 1395 56.479478 56.479372 LO-REDUCTION 1397 56.479467 56.479372 EXTENSION 1399 56.479457 56.479342 LO-REDUCTION 1401 56.479454 56.479342 EXTENSION 1403 56.479441 56.479295 LO-REDUCTION 1405 56.479417 56.479295 LO-REDUCTION 1407 56.479412 56.479295 EXTENSION 1409 56.479407 56.479262 EXTENSION 1411 56.479382 56.479218 LO-REDUCTION 1413 56.479372 56.479218 LO-REDUCTION 1415 56.479349 56.479218 LO-REDUCTION 1417 56.479342 56.479218 LO-REDUCTION 1419 56.479329 56.479218 REFLECTION 1421 56.479306 56.479215 LO-REDUCTION 1423 56.479295 56.479215 LO-REDUCTION 1425 56.479272 56.479215 EXTENSION 1427 56.479264 56.479186 LO-REDUCTION 1429 56.479262 56.479186 LO-REDUCTION 1431 56.479241 56.479186 EXTENSION 1433 56.479232 56.479148 LO-REDUCTION 1435 56.479225 56.479148 LO-REDUCTION 1437 56.479222 56.479148 EXTENSION 1439 56.479218 56.479137 HI-REDUCTION 1441 56.479215 56.479137 LO-REDUCTION 1443 56.479198 56.479137 EXTENSION 1445 56.479192 56.479118 LO-REDUCTION 1447 56.479186 56.479118 LO-REDUCTION 1449 56.479184 56.479118 EXTENSION 1451 56.479179 56.479064 LO-REDUCTION 1453 56.479155 56.479064 LO-REDUCTION 1455 56.479151 56.479064 LO-REDUCTION 1457 56.479148 56.479064 LO-REDUCTION 1459 56.479141 56.479064 LO-REDUCTION 1461 56.479137 56.479064 LO-REDUCTION 1463 56.479123 56.479064 LO-REDUCTION 1465 56.479118 56.479064 LO-REDUCTION 1467 56.479116 56.479064 EXTENSION 1469 56.479105 56.479034 LO-REDUCTION 1471 56.479100 56.479034 LO-REDUCTION 1473 56.479090 56.479034 EXTENSION 1475 56.479086 56.479000 LO-REDUCTION 1477 56.479079 56.479000 EXTENSION 1479 56.479073 56.478960 LO-REDUCTION 1481 56.479071 56.478960 LO-REDUCTION 1483 56.479064 56.478960 LO-REDUCTION 1485 56.479042 56.478960 EXTENSION 1487 56.479038 56.478912 LO-REDUCTION 1489 56.479034 56.478912 EXTENSION 1491 56.479007 56.478877 LO-REDUCTION 1493 56.479000 56.478877 EXTENSION 1495 56.478986 56.478830 EXTENSION 1497 56.478978 56.478761 LO-REDUCTION 1499 56.478969 56.478761 LO-REDUCTION 1501 56.478960 56.478761 EXTENSION 1503 56.478919 56.478658 LO-REDUCTION 1505 56.478912 56.478658 LO-REDUCTION 1507 56.478886 56.478658 LO-REDUCTION 1509 56.478877 56.478658 EXTENSION 1511 56.478830 56.478568 LO-REDUCTION 1513 56.478802 56.478568 LO-REDUCTION 1515 56.478791 56.478568 EXTENSION 1517 56.478761 56.478403 LO-REDUCTION 1519 56.478712 56.478403 LO-REDUCTION 1521 56.478697 56.478403 LO-REDUCTION 1523 56.478659 56.478403 LO-REDUCTION 1525 56.478658 56.478403 LO-REDUCTION 1527 56.478609 56.478403 LO-REDUCTION 1529 56.478584 56.478403 REFLECTION 1531 56.478568 56.478384 LO-REDUCTION 1533 56.478551 56.478384 EXTENSION 1535 56.478480 56.478243 LO-REDUCTION 1537 56.478462 56.478243 LO-REDUCTION 1539 56.478415 56.478243 LO-REDUCTION 1541 56.478411 56.478243 LO-REDUCTION 1543 56.478410 56.478243 EXTENSION 1545 56.478403 56.478175 LO-REDUCTION 1547 56.478395 56.478175 LO-REDUCTION 1549 56.478384 56.478175 EXTENSION 1551 56.478341 56.478099 LO-REDUCTION 1553 56.478316 56.478099 EXTENSION 1555 56.478282 56.478025 EXTENSION 1557 56.478271 56.477898 LO-REDUCTION 1559 56.478251 56.477898 LO-REDUCTION 1561 56.478243 56.477898 LO-REDUCTION 1563 56.478217 56.477898 LO-REDUCTION 1565 56.478175 56.477898 LO-REDUCTION 1567 56.478105 56.477898 LO-REDUCTION 1569 56.478099 56.477898 REFLECTION 1571 56.478037 56.477892 LO-REDUCTION 1573 56.478025 56.477892 EXTENSION 1575 56.478021 56.477805 EXTENSION 1577 56.477943 56.477717 LO-REDUCTION 1579 56.477940 56.477717 LO-REDUCTION 1581 56.477921 56.477717 EXTENSION 1583 56.477906 56.477628 LO-REDUCTION 1585 56.477900 56.477628 EXTENSION 1587 56.477898 56.477528 LO-REDUCTION 1589 56.477892 56.477528 LO-REDUCTION 1591 56.477809 56.477528 LO-REDUCTION 1593 56.477805 56.477528 LO-REDUCTION 1595 56.477745 56.477528 LO-REDUCTION 1597 56.477718 56.477528 LO-REDUCTION 1599 56.477717 56.477528 REFLECTION 1601 56.477683 56.477515 EXTENSION 1603 56.477672 56.477383 LO-REDUCTION 1605 56.477628 56.477383 LO-REDUCTION 1607 56.477587 56.477383 EXTENSION 1609 56.477542 56.477208 LO-REDUCTION 1611 56.477536 56.477208 LO-REDUCTION 1613 56.477534 56.477208 LO-REDUCTION 1615 56.477528 56.477208 LO-REDUCTION 1617 56.477515 56.477208 EXTENSION 1619 56.477404 56.476938 LO-REDUCTION 1621 56.477394 56.476938 LO-REDUCTION 1623 56.477388 56.476938 LO-REDUCTION 1625 56.477383 56.476938 LO-REDUCTION 1627 56.477331 56.476938 LO-REDUCTION 1629 56.477317 56.476938 EXTENSION 1631 56.477208 56.476659 LO-REDUCTION 1633 56.477208 56.476659 LO-REDUCTION 1635 56.477172 56.476659 EXTENSION 1637 56.477107 56.476439 LO-REDUCTION 1639 56.477073 56.476439 LO-REDUCTION 1641 56.476968 56.476439 LO-REDUCTION 1643 56.476948 56.476439 EXTENSION 1645 56.476938 56.476289 EXTENSION 1647 56.476789 56.476146 LO-REDUCTION 1649 56.476786 56.476146 REFLECTION 1651 56.476659 56.476101 EXTENSION 1653 56.476503 56.475811 LO-REDUCTION 1655 56.476498 56.475811 LO-REDUCTION 1657 56.476479 56.475811 LO-REDUCTION 1659 56.476439 56.475811 LO-REDUCTION 1661 56.476289 56.475811 LO-REDUCTION 1663 56.476160 56.475811 LO-REDUCTION 1665 56.476146 56.475811 LO-REDUCTION 1667 56.476101 56.475811 EXTENSION 1669 56.476073 56.475729 EXTENSION 1671 56.475928 56.475256 LO-REDUCTION 1673 56.475919 56.475256 LO-REDUCTION 1675 56.475888 56.475256 LO-REDUCTION 1677 56.475857 56.475256 LO-REDUCTION 1679 56.475847 56.475256 LO-REDUCTION 1681 56.475817 56.475256 EXTENSION 1683 56.475811 56.475035 LO-REDUCTION 1685 56.475729 56.475035 LO-REDUCTION 1687 56.475580 56.475035 EXTENSION 1689 56.475540 56.474664 LO-REDUCTION 1691 56.475494 56.474664 LO-REDUCTION 1693 56.475478 56.474664 EXTENSION 1695 56.475338 56.474192 LO-REDUCTION 1697 56.475256 56.474192 LO-REDUCTION 1699 56.475125 56.474192 LO-REDUCTION 1701 56.475097 56.474192 EXTENSION 1703 56.475035 56.473687 EXTENSION 1705 56.474749 56.472972 LO-REDUCTION 1707 56.474709 56.472972 LO-REDUCTION 1709 56.474664 56.472972 LO-REDUCTION 1711 56.474358 56.472972 LO-REDUCTION 1713 56.474353 56.472972 EXTENSION 1715 56.474236 56.472507 EXTENSION 1717 56.474192 56.471585 LO-REDUCTION 1719 56.473687 56.471585 LO-REDUCTION 1721 56.473512 56.471585 EXTENSION 1723 56.473260 56.470923 LO-REDUCTION 1725 56.473164 56.470923 LO-REDUCTION 1727 56.472997 56.470923 LO-REDUCTION 1729 56.472972 56.470923 LO-REDUCTION 1731 56.472507 56.470923 EXTENSION 1733 56.472207 56.470435 EXTENSION 1735 56.471776 56.470159 LO-REDUCTION 1737 56.471585 56.470159 LO-REDUCTION 1739 56.471498 56.470159 LO-REDUCTION 1741 56.471340 56.470159 LO-REDUCTION 1743 56.471290 56.470159 REFLECTION 1745 56.471237 56.470019 REFLECTION 1747 56.470923 56.469930 LO-REDUCTION 1749 56.470617 56.469930 LO-REDUCTION 1751 56.470539 56.469930 LO-REDUCTION 1753 56.470435 56.469930 HI-REDUCTION 1755 56.470391 56.469930 LO-REDUCTION 1757 56.470343 56.469930 LO-REDUCTION 1759 56.470159 56.469930 HI-REDUCTION 1761 56.470143 56.469930 LO-REDUCTION 1763 56.470080 56.469930 HI-REDUCTION 1765 56.470079 56.469930 EXTENSION 1767 56.470058 56.469828 LO-REDUCTION 1769 56.470020 56.469828 LO-REDUCTION 1771 56.470019 56.469828 LO-REDUCTION 1773 56.470015 56.469828 LO-REDUCTION 1775 56.470007 56.469828 EXTENSION 1777 56.470004 56.469739 LO-REDUCTION 1779 56.469932 56.469739 LO-REDUCTION 1781 56.469930 56.469739 LO-REDUCTION 1783 56.469922 56.469739 EXTENSION 1785 56.469914 56.469666 EXTENSION 1787 56.469876 56.469590 LO-REDUCTION 1789 56.469861 56.469590 LO-REDUCTION 1791 56.469828 56.469590 EXTENSION 1793 56.469791 56.469540 EXTENSION 1795 56.469775 56.469481 LO-REDUCTION 1797 56.469747 56.469481 LO-REDUCTION 1799 56.469739 56.469481 LO-REDUCTION 1801 56.469666 56.469481 EXTENSION 1803 56.469646 56.469410 REFLECTION 1805 56.469614 56.469409 LO-REDUCTION 1807 56.469590 56.469409 LO-REDUCTION 1809 56.469572 56.469409 EXTENSION 1811 56.469540 56.469353 LO-REDUCTION 1813 56.469530 56.469353 LO-REDUCTION 1815 56.469490 56.469353 EXTENSION 1817 56.469489 56.469257 LO-REDUCTION 1819 56.469481 56.469257 LO-REDUCTION 1821 56.469448 56.469257 LO-REDUCTION 1823 56.469438 56.469257 LO-REDUCTION 1825 56.469410 56.469257 LO-REDUCTION 1827 56.469409 56.469257 LO-REDUCTION 1829 56.469391 56.469257 EXTENSION 1831 56.469360 56.469148 LO-REDUCTION 1833 56.469353 56.469148 LO-REDUCTION 1835 56.469337 56.469148 EXTENSION 1837 56.469306 56.469067 LO-REDUCTION 1839 56.469295 56.469067 LO-REDUCTION 1841 56.469295 56.469067 LO-REDUCTION 1843 56.469262 56.469067 LO-REDUCTION 1845 56.469257 56.469067 EXTENSION 1847 56.469217 56.469028 REFLECTION 1849 56.469193 56.469024 REFLECTION 1851 56.469177 56.468998 EXTENSION 1853 56.469148 56.468926 LO-REDUCTION 1855 56.469108 56.468926 LO-REDUCTION 1857 56.469095 56.468926 REFLECTION 1859 56.469069 56.468924 LO-REDUCTION 1861 56.469067 56.468924 LO-REDUCTION 1863 56.469028 56.468924 LO-REDUCTION 1865 56.469024 56.468924 LO-REDUCTION 1867 56.468998 56.468924 LO-REDUCTION 1869 56.468984 56.468924 EXTENSION 1871 56.468944 56.468836 LO-REDUCTION 1873 56.468937 56.468836 LO-REDUCTION 1875 56.468936 56.468836 LO-REDUCTION 1877 56.468935 56.468836 LO-REDUCTION 1879 56.468933 56.468836 LO-REDUCTION 1881 56.468927 56.468836 LO-REDUCTION 1883 56.468926 56.468836 LO-REDUCTION 1885 56.468924 56.468836 LO-REDUCTION 1887 56.468903 56.468836 EXTENSION 1889 56.468896 56.468796 EXTENSION 1891 56.468892 56.468747 LO-REDUCTION 1893 56.468885 56.468747 LO-REDUCTION 1895 56.468875 56.468747 LO-REDUCTION 1897 56.468847 56.468747 LO-REDUCTION 1899 56.468838 56.468747 EXTENSION 1901 56.468838 56.468729 EXTENSION 1903 56.468836 56.468640 LO-REDUCTION 1905 56.468796 56.468640 LO-REDUCTION 1907 56.468785 56.468640 EXTENSION 1909 56.468768 56.468574 LO-REDUCTION 1911 56.468755 56.468574 LO-REDUCTION 1913 56.468753 56.468574 LO-REDUCTION 1915 56.468747 56.468574 LO-REDUCTION 1917 56.468729 56.468574 EXTENSION 1919 56.468697 56.468534 REFLECTION 1921 56.468666 56.468533 EXTENSION 1923 56.468656 56.468456 LO-REDUCTION 1925 56.468640 56.468456 LO-REDUCTION 1927 56.468614 56.468456 EXTENSION 1929 56.468584 56.468348 LO-REDUCTION 1931 56.468579 56.468348 LO-REDUCTION 1933 56.468574 56.468348 LO-REDUCTION 1935 56.468534 56.468348 EXTENSION 1937 56.468533 56.468316 LO-REDUCTION 1939 56.468472 56.468316 LO-REDUCTION 1941 56.468462 56.468316 REFLECTION 1943 56.468456 56.468307 REFLECTION 1945 56.468424 56.468273 EXTENSION 1947 56.468414 56.468156 LO-REDUCTION 1949 56.468410 56.468156 EXTENSION 1951 56.468348 56.467968 LO-REDUCTION 1953 56.468346 56.467968 LO-REDUCTION 1955 56.468326 56.467968 LO-REDUCTION 1957 56.468316 56.467968 LO-REDUCTION 1959 56.468307 56.467968 EXTENSION 1961 56.468273 56.467885 LO-REDUCTION 1963 56.468190 56.467885 EXTENSION 1965 56.468158 56.467650 LO-REDUCTION 1967 56.468156 56.467650 LO-REDUCTION 1969 56.468097 56.467650 LO-REDUCTION 1971 56.468094 56.467650 LO-REDUCTION 1973 56.467994 56.467650 LO-REDUCTION 1975 56.467968 56.467650 EXTENSION 1977 56.467950 56.467509 EXTENSION 1979 56.467885 56.467227 EXTENSION 1981 56.467775 56.466935 LO-REDUCTION 1983 56.467706 56.466935 LO-REDUCTION 1985 56.467676 56.466935 LO-REDUCTION 1987 56.467655 56.466935 EXTENSION 1989 56.467654 56.466624 LO-REDUCTION 1991 56.467650 56.466624 LO-REDUCTION 1993 56.467509 56.466624 EXTENSION 1995 56.467227 56.466056 LO-REDUCTION 1997 56.467171 56.466056 EXTENSION 1999 56.467133 56.465595 Exiting from Nelder Mead minimizer 2001 function evaluations used  We can plot the resultant maximum likelihood fits. options(warn=-1) par(mar = c(5, 5, 2, 5)) #Room for two axes #Overwrite dummy-objects coefficients with MLEs dl$coefficients = dspline$par #Age-prevalce plot plot(tmpfn(rabbit$a,dl) ~ rabbit$a, type = "l", ylab = "FoI", xlab = "Age (mos)", las = 1) #Overlay FoI par(new = TRUE) p = rep(0, 28) for (i in 1:28) { p[i] = 1 - exp(-integrate(tmpfn, 0, i, dl = dl)$value)
}
plot(p ~ c(1:28), ylim = c(0,1), type = "l", col = "red",
axes = FALSE, xlab = NA, ylab = NA)
axis(side = 4, las = 1)
mtext(side = 4, line = 4, "Prevalence")
legend("topright", legend = c("FoI", "Prevalence"),
lty = c(1,1), col = c("black", "red"))
options(warn=0)


Both the piecewise and spline model show strong evidence of age-specificity in the FoI with a peak in transmission somewhere between 1 and 5 months of age, suggesting that circulation is mainly among the young and among litter-mates . We revisit on this case study in section [sec:c13bb].

### Rubella

Rubella is a relatively mild, vaccine-preventable infection except that infection during pregnancy leads to stillbirths or . The main public health objective is therefore to minimize the FoI in women of childbearing age. The issue was made clear because of a surprising surge in CRS cases in Greece in the mid-90s following a low-intensity vaccination campaign .

Age-intensity data is less ideal than seroprevalence data for catalytic analysis, however it is more common and therefore worth considering. studied age-intensity curves for rubella across the provinces of Peru between 1997 and 2009. There were $24,116$ reported cases during the period. The data are -monthly to age 1 and yearly thereafter (fig. [fig:peru]). With age-incidence data on immunizing infections, we can use the catalytic framework to estimate the relative age-specific FoI using the cumulative incidence by age (in place of age-seroprevalence or age-prevalence). For the analysis we use the total number of cases as our denominator because the actual number of susceptibles in each age group is not monitored. Hence, the estimate is a relative FoI because of the unknown baseline. Using the total cases as a denominator, further leads to sever biases of the FoI at old age classes (because exactly all of the assumed susceptibles in the final age class will be presumed to be infected at the time), so it should only be applied to the younger portion of the data. Its application also assumes a uniform age-distribution, so a correction for the age-pyramid may be necessary for a more refined analysis .

data(peru)
#Calculate cumulative incidence
peru$cumulative=cumsum(peru$incidence)
#Define denominator
peru$n=sum(peru$incidence)
par(mar = c(5,5,2,5)) #Make room for two axes and plot
#Plot incidence with cumulative overlaid
plot(peru$incidence~peru$age, type="b", xlab="Age",
ylab="Incidence")
par(new=T)
plot(peru$cumulative~peru$age, type="l", col="red",
axes=FALSE, xlab=NA, ylab=NA)
axis(side = 4)
mtext(side = 4, line = 4, "Cumulative")
legend("right", legend=c("Incidence", "Cumulative"),
lty=c(1,1), col=c("black", "red"))

<th scope=col>age</th><th scope=col>incidence</th><th scope=col>cumulative</th><th scope=col>n</th><th scope=row>2</th><th scope=row>3</th><th scope=row>4</th><th scope=row>5</th><th scope=row>6</th><th scope=row>7</th>
0.010958901 56 24116
0.013698631 57 24116
0.016438361 58 24116
0.019178082 60 24116
0.035616441 61 24116
0.038356162 63 24116

We first apply the piecewise model assuming a separate FoI for each year up to age 20 and 10 year classes there after. Convergence of the piecewise model with this many segments is very slow so the actual figure (Fig. [fig:peru2]) was produce by doing repeat calls to optim using different optimization methods (Nelder-Mead, BFGS and SANN), feeding the estimates from each call as starting values for the next. However, the basic analysis is:

#Upper age cut-offs
up = c(1:20, 30, 40, 50, 60, 70,100)
para = rep(.1, length(up)) #Inital values
#Minimize log-likelihood
est2 = optim(par = log(para),fn = llik.pc, age = peru$age, num = peru$cumulative, denom = peru$n, up = up, method = "Nelder-Mead", control = list(trace = 2, maxit = 2000)) #Step plot x = c(0, up) y = exp(c(est2$par, est2$par[26])) plot(x, y, ylab = "Relative FoI", xlab = "Age", type = "s", ylim = c(0, 0.25), xlim = c(0, 80))   Nelder-Mead direct search function minimizer function value for initial parameters = 30792.290816 Scaled convergence tolerance is 0.000458841 Stepsize computed as 0.230259 BUILD 27 32644.012507 27891.708805 EXTENSION 29 30792.290816 26148.012817 LO-REDUCTION 31 30785.524198 26148.012817 LO-REDUCTION 33 30762.120773 26148.012817 LO-REDUCTION 35 30693.446532 26148.012817 LO-REDUCTION 37 30445.317869 26148.012817 LO-REDUCTION 39 30244.073674 26148.012817 LO-REDUCTION 41 30183.815776 26148.012817 LO-REDUCTION 43 30114.622720 26148.012817 LO-REDUCTION 45 30035.359103 26148.012817 LO-REDUCTION 47 29946.057015 26148.012817 LO-REDUCTION 49 29846.604916 26148.012817 LO-REDUCTION 51 29757.054415 26148.012817 LO-REDUCTION 53 29735.263764 26148.012817 LO-REDUCTION 55 29663.961312 26148.012817 LO-REDUCTION 57 29611.449145 26148.012817 LO-REDUCTION 59 29479.559477 26148.012817 LO-REDUCTION 61 29433.000113 26148.012817 LO-REDUCTION 63 29342.705126 26148.012817 EXTENSION 65 29202.140241 24056.568565 LO-REDUCTION 67 29133.054983 24056.568565 LO-REDUCTION 69 29065.276771 24056.568565 LO-REDUCTION 71 28944.240445 24056.568565 LO-REDUCTION 73 28930.966484 24056.568565 LO-REDUCTION 75 28854.762184 24056.568565 LO-REDUCTION 77 28838.073244 24056.568565 LO-REDUCTION 79 28311.139331 24056.568565 LO-REDUCTION 81 28132.378257 24056.568565 LO-REDUCTION 83 27955.387260 24056.568565 LO-REDUCTION 85 27891.708805 24056.568565 LO-REDUCTION 87 27839.602535 24056.568565 LO-REDUCTION 89 27813.522829 24056.568565 LO-REDUCTION 91 27750.129399 24056.568565 EXTENSION 93 27612.837620 22051.973101 LO-REDUCTION 95 27477.620198 22051.973101 LO-REDUCTION 97 27346.367239 22051.973101 LO-REDUCTION 99 27220.202702 22051.973101 LO-REDUCTION 101 27100.483895 22051.973101 LO-REDUCTION 103 27038.620414 22051.973101 LO-REDUCTION 105 26795.426891 22051.973101 LO-REDUCTION 107 26782.687516 22051.973101 LO-REDUCTION 109 26453.869549 22051.973101 LO-REDUCTION 111 26332.616836 22051.973101 LO-REDUCTION 113 26175.961611 22051.973101 EXTENSION 115 26148.012817 20009.219162 LO-REDUCTION 117 25727.584601 20009.219162 LO-REDUCTION 119 25562.591445 20009.219162 LO-REDUCTION 121 25339.929462 20009.219162 LO-REDUCTION 123 25184.946973 20009.219162 LO-REDUCTION 125 24953.147857 20009.219162 LO-REDUCTION 127 24880.617444 20009.219162 LO-REDUCTION 129 24806.230567 20009.219162 LO-REDUCTION 131 24735.238881 20009.219162 LO-REDUCTION 133 24735.079918 20009.219162 LO-REDUCTION 135 24682.573428 20009.219162 LO-REDUCTION 137 24481.421316 20009.219162 EXTENSION 139 24399.043464 17088.367412 LO-REDUCTION 141 24167.420309 17088.367412 LO-REDUCTION 143 24056.568565 17088.367412 LO-REDUCTION 145 23629.846423 17088.367412 LO-REDUCTION 147 23471.826478 17088.367412 LO-REDUCTION 149 23307.698290 17088.367412 LO-REDUCTION 151 23135.649432 17088.367412 LO-REDUCTION 153 22953.395647 17088.367412 LO-REDUCTION 155 22871.092426 17088.367412 LO-REDUCTION 157 22729.259484 17088.367412 LO-REDUCTION 159 22368.721844 17088.367412 LO-REDUCTION 161 22357.980371 17088.367412 LO-REDUCTION 163 22216.371496 17088.367412 LO-REDUCTION 165 22161.783754 17088.367412 LO-REDUCTION 167 22051.973101 17088.367412 LO-REDUCTION 169 21575.846655 17088.367412 LO-REDUCTION 171 21432.660605 17088.367412 EXTENSION 173 21296.925663 14745.923237 LO-REDUCTION 175 21174.648476 14745.923237 LO-REDUCTION 177 21115.446898 14745.923237 LO-REDUCTION 179 21045.421036 14745.923237 LO-REDUCTION 181 20971.493618 14745.923237 LO-REDUCTION 183 20441.856023 14745.923237 LO-REDUCTION 185 20401.454676 14745.923237 LO-REDUCTION 187 20289.657273 14745.923237 LO-REDUCTION 189 20009.219162 14745.923237 LO-REDUCTION 191 19584.986722 14745.923237 LO-REDUCTION 193 19471.639961 14745.923237 LO-REDUCTION 195 19254.052729 14745.923237 EXTENSION 197 19230.293589 11968.209117 LO-REDUCTION 199 19226.076504 11968.209117 LO-REDUCTION 201 19038.899258 11968.209117 LO-REDUCTION 203 18856.872993 11968.209117 LO-REDUCTION 205 18685.026754 11968.209117 LO-REDUCTION 207 18530.293531 11968.209117 LO-REDUCTION 209 18314.039037 11968.209117 LO-REDUCTION 211 18079.036629 11968.209117 LO-REDUCTION 213 18001.936507 11968.209117 LO-REDUCTION 215 17686.420266 11968.209117 LO-REDUCTION 217 17374.059931 11968.209117 LO-REDUCTION 219 17236.924681 11968.209117 LO-REDUCTION 221 17225.219641 11968.209117 LO-REDUCTION 223 17088.367412 11968.209117 LO-REDUCTION 225 16713.848489 11968.209117 LO-REDUCTION 227 16705.399333 11968.209117 EXTENSION 229 16280.571364 9392.576398 LO-REDUCTION 231 16017.050847 9392.576398 LO-REDUCTION 233 16014.793251 9392.576398 LO-REDUCTION 235 15913.049117 9392.576398 LO-REDUCTION 237 15740.314206 9392.576398 LO-REDUCTION 239 15626.256773 9392.576398 LO-REDUCTION 241 15502.208214 9392.576398 LO-REDUCTION 243 15317.884950 9392.576398 LO-REDUCTION 245 14986.733812 9392.576398 LO-REDUCTION 247 14880.373612 9392.576398 LO-REDUCTION 249 14745.923237 9392.576398 LO-REDUCTION 251 14288.611666 9392.576398 LO-REDUCTION 253 14099.573362 9392.576398 LO-REDUCTION 255 13904.424283 9392.576398 LO-REDUCTION 257 13699.934485 9392.576398 LO-REDUCTION 259 13594.167502 9392.576398 EXTENSION 261 13504.912960 8264.085411 LO-REDUCTION 263 13482.636423 8264.085411 LO-REDUCTION 265 13061.580991 8264.085411 LO-REDUCTION 267 12988.333790 8264.085411 LO-REDUCTION 269 12763.378443 8264.085411 LO-REDUCTION 271 12446.414238 8264.085411 LO-REDUCTION 273 12327.217596 8264.085411 EXTENSION 275 12176.021980 6696.563310 LO-REDUCTION 277 12079.538043 6696.563310 LO-REDUCTION 279 11968.209117 6696.563310 LO-REDUCTION 281 11407.510874 6696.563310 LO-REDUCTION 283 11387.299084 6696.563310 LO-REDUCTION 285 11358.187266 6696.563310 LO-REDUCTION 287 11219.356451 6696.563310 LO-REDUCTION 289 11110.891311 6696.563310 LO-REDUCTION 291 10973.455382 6696.563310 LO-REDUCTION 293 10690.352475 6696.563310 LO-REDUCTION 295 10299.276005 6696.563310 LO-REDUCTION 297 10261.932587 6696.563310 LO-REDUCTION 299 10067.942245 6696.563310 LO-REDUCTION 301 10041.581195 6696.563310 LO-REDUCTION 303 9927.200297 6696.563310 LO-REDUCTION 305 9799.508604 6696.563310 LO-REDUCTION 307 9690.325702 6696.563310 LO-REDUCTION 309 9612.754853 6696.563310 LO-REDUCTION 311 9392.576398 6696.563310 LO-REDUCTION 313 9000.259823 6696.563310 LO-REDUCTION 315 8855.847012 6696.563310 LO-REDUCTION 317 8711.900350 6696.563310 LO-REDUCTION 319 8673.527221 6696.563310 LO-REDUCTION 321 8665.992147 6696.563310 LO-REDUCTION 323 8557.092335 6696.563310 LO-REDUCTION 325 8320.272070 6696.563310 LO-REDUCTION 327 8309.155816 6696.563310 LO-REDUCTION 329 8264.085411 6696.563310 LO-REDUCTION 331 8253.167147 6696.563310 EXTENSION 333 8150.111699 6100.064178 LO-REDUCTION 335 8142.881103 6100.064178 LO-REDUCTION 337 8133.631177 6100.064178 LO-REDUCTION 339 8124.549832 6100.064178 LO-REDUCTION 341 8113.873828 6100.064178 LO-REDUCTION 343 8083.382148 6100.064178 LO-REDUCTION 345 8044.363324 6100.064178 LO-REDUCTION 347 7976.245842 6100.064178 LO-REDUCTION 349 7851.956985 6100.064178 EXTENSION 351 7695.111457 5358.611322 LO-REDUCTION 353 7592.086454 5358.611322 LO-REDUCTION 355 7533.614080 5358.611322 LO-REDUCTION 357 7528.110572 5358.611322 HI-REDUCTION 359 7484.437260 5358.611322 LO-REDUCTION 361 7464.280206 5358.611322 LO-REDUCTION 363 7432.527233 5358.611322 LO-REDUCTION 365 7379.383631 5358.611322 LO-REDUCTION 367 7319.050777 5358.611322 LO-REDUCTION 369 7287.565549 5358.611322 LO-REDUCTION 371 7254.284078 5358.611322 LO-REDUCTION 373 7245.985799 5358.611322 LO-REDUCTION 375 7183.356347 5358.611322 EXTENSION 377 7125.160760 4451.331861 LO-REDUCTION 379 7100.659475 4451.331861 LO-REDUCTION 381 6946.184063 4451.331861 LO-REDUCTION 383 6923.856910 4451.331861 LO-REDUCTION 385 6898.220501 4451.331861 LO-REDUCTION 387 6826.116413 4451.331861 LO-REDUCTION 389 6787.061468 4451.331861 LO-REDUCTION 391 6734.691449 4451.331861 LO-REDUCTION 393 6720.634064 4451.331861 HI-REDUCTION 395 6696.563310 4451.331861 LO-REDUCTION 397 6526.273860 4451.331861 LO-REDUCTION 399 6383.811473 4451.331861 LO-REDUCTION 401 6364.934414 4451.331861 LO-REDUCTION 403 6100.064178 4451.331861 LO-REDUCTION 405 6008.494804 4451.331861 LO-REDUCTION 407 5996.256806 4451.331861 LO-REDUCTION 409 5944.654491 4451.331861 EXTENSION 411 5909.330389 3896.546242 LO-REDUCTION 413 5889.719039 3896.546242 LO-REDUCTION 415 5822.164958 3896.546242 LO-REDUCTION 417 5723.399149 3896.546242 LO-REDUCTION 419 5716.476148 3896.546242 LO-REDUCTION 421 5675.947343 3896.546242 LO-REDUCTION 423 5596.079664 3896.546242 LO-REDUCTION 425 5582.147537 3896.546242 EXTENSION 427 5548.322961 3430.292439 LO-REDUCTION 429 5509.276424 3430.292439 LO-REDUCTION 431 5381.772635 3430.292439 LO-REDUCTION 433 5358.611322 3430.292439 LO-REDUCTION 435 5141.179455 3430.292439 LO-REDUCTION 437 5126.251662 3430.292439 LO-REDUCTION 439 4971.422111 3430.292439 LO-REDUCTION 441 4967.907627 3430.292439 LO-REDUCTION 443 4847.075287 3430.292439 LO-REDUCTION 445 4788.387546 3430.292439 LO-REDUCTION 447 4766.505111 3430.292439 EXTENSION 449 4711.370125 2808.838057 LO-REDUCTION 451 4707.960120 2808.838057 LO-REDUCTION 453 4671.031179 2808.838057 LO-REDUCTION 455 4570.388802 2808.838057 LO-REDUCTION 457 4555.449446 2808.838057 LO-REDUCTION 459 4516.422898 2808.838057 LO-REDUCTION 461 4461.901225 2808.838057 LO-REDUCTION 463 4451.331861 2808.838057 LO-REDUCTION 465 4336.184117 2808.838057 LO-REDUCTION 467 4237.581311 2808.838057 LO-REDUCTION 469 4147.577229 2808.838057 LO-REDUCTION 471 4089.763221 2808.838057 LO-REDUCTION 473 4018.297573 2808.838057 LO-REDUCTION 475 3992.646734 2808.838057 LO-REDUCTION 477 3915.951787 2808.838057 LO-REDUCTION 479 3896.546242 2808.838057 EXTENSION 481 3864.750999 2697.767876 EXTENSION 483 3804.531375 2349.101570 LO-REDUCTION 485 3765.390939 2349.101570 LO-REDUCTION 487 3734.052945 2349.101570 LO-REDUCTION 489 3653.404850 2349.101570 LO-REDUCTION 491 3597.606758 2349.101570 LO-REDUCTION 493 3569.635449 2349.101570 LO-REDUCTION 495 3536.341413 2349.101570 LO-REDUCTION 497 3513.473224 2349.101570 LO-REDUCTION 499 3479.523400 2349.101570 LO-REDUCTION 501 3431.356341 2349.101570 LO-REDUCTION 503 3430.292439 2349.101570 LO-REDUCTION 505 3407.300808 2349.101570 LO-REDUCTION 507 3175.942472 2349.101570 LO-REDUCTION 509 3153.165375 2349.101570 LO-REDUCTION 511 3099.327372 2349.101570 LO-REDUCTION 513 3097.347833 2349.101570 LO-REDUCTION 515 3026.989011 2349.101570 EXTENSION 517 2988.474441 2250.382557 LO-REDUCTION 519 2987.212331 2250.382557 LO-REDUCTION 521 2986.251268 2250.382557 LO-REDUCTION 523 2943.979840 2250.382557 LO-REDUCTION 525 2933.494482 2250.382557 LO-REDUCTION 527 2926.602162 2250.382557 LO-REDUCTION 529 2882.895202 2250.382557 REFLECTION 531 2808.838057 2218.400248 LO-REDUCTION 533 2747.798594 2218.400248 EXTENSION 535 2697.767876 2064.635073 LO-REDUCTION 537 2690.500495 2064.635073 LO-REDUCTION 539 2684.092885 2064.635073 LO-REDUCTION 541 2656.518484 2064.635073 LO-REDUCTION 543 2653.948659 2064.635073 LO-REDUCTION 545 2619.028918 2064.635073 LO-REDUCTION 547 2618.990511 2064.635073 LO-REDUCTION 549 2579.815171 2064.635073 LO-REDUCTION 551 2571.893278 2064.635073 LO-REDUCTION 553 2565.052163 2064.635073 LO-REDUCTION 555 2563.055191 2064.635073 LO-REDUCTION 557 2545.106551 2064.635073 LO-REDUCTION 559 2542.075589 2064.635073 LO-REDUCTION 561 2523.038458 2064.635073 LO-REDUCTION 563 2509.013524 2064.635073 LO-REDUCTION 565 2495.025087 2064.635073 EXTENSION 567 2457.647497 1995.393988 LO-REDUCTION 569 2444.011777 1995.393988 LO-REDUCTION 571 2435.852345 1995.393988 LO-REDUCTION 573 2416.663558 1995.393988 EXTENSION 575 2415.957806 1973.490974 LO-REDUCTION 577 2350.761153 1973.490974 LO-REDUCTION 579 2349.101570 1973.490974 LO-REDUCTION 581 2338.809655 1973.490974 EXTENSION 583 2328.194642 1941.813813 EXTENSION 585 2313.001054 1863.389793 LO-REDUCTION 587 2307.558538 1863.389793 LO-REDUCTION 589 2307.371663 1863.389793 EXTENSION 591 2301.173896 1712.847904 LO-REDUCTION 593 2250.382557 1712.847904 LO-REDUCTION 595 2218.400248 1712.847904 LO-REDUCTION 597 2215.145661 1712.847904 LO-REDUCTION 599 2206.090801 1712.847904 HI-REDUCTION 601 2197.066677 1712.847904 LO-REDUCTION 603 2179.586978 1712.847904 LO-REDUCTION 605 2166.074881 1712.847904 LO-REDUCTION 607 2164.023731 1712.847904 LO-REDUCTION 609 2157.026265 1712.847904 LO-REDUCTION 611 2143.948130 1712.847904 LO-REDUCTION 613 2121.357969 1712.847904 LO-REDUCTION 615 2116.791304 1712.847904 LO-REDUCTION 617 2103.638510 1712.847904 LO-REDUCTION 619 2094.994788 1712.847904 LO-REDUCTION 621 2080.762015 1712.847904 LO-REDUCTION 623 2064.635073 1712.847904 LO-REDUCTION 625 2045.726717 1712.847904 LO-REDUCTION 627 2040.062975 1712.847904 LO-REDUCTION 629 2035.137594 1712.847904 LO-REDUCTION 631 2019.512646 1712.847904 LO-REDUCTION 633 2016.604950 1712.847904 LO-REDUCTION 635 1995.393988 1712.847904 LO-REDUCTION 637 1974.973345 1712.847904 LO-REDUCTION 639 1973.490974 1712.847904 LO-REDUCTION 641 1962.664417 1712.847904 LO-REDUCTION 643 1957.903134 1712.847904 LO-REDUCTION 645 1941.813813 1712.847904 LO-REDUCTION 647 1912.045476 1712.847904 LO-REDUCTION 649 1903.960851 1712.847904 LO-REDUCTION 651 1900.291164 1712.847904 LO-REDUCTION 653 1893.252226 1712.847904 LO-REDUCTION 655 1889.747732 1712.847904 LO-REDUCTION 657 1885.507470 1712.847904 LO-REDUCTION 659 1882.047085 1712.847904 LO-REDUCTION 661 1881.977388 1712.847904 EXTENSION 663 1877.497160 1674.090697 LO-REDUCTION 665 1874.764867 1674.090697 LO-REDUCTION 667 1874.492941 1674.090697 LO-REDUCTION 669 1863.389793 1674.090697 LO-REDUCTION 671 1851.552919 1674.090697 LO-REDUCTION 673 1846.190450 1674.090697 LO-REDUCTION 675 1839.770568 1674.090697 LO-REDUCTION 677 1838.799975 1674.090697 LO-REDUCTION 679 1835.103711 1674.090697 LO-REDUCTION 681 1830.658015 1674.090697 LO-REDUCTION 683 1827.679533 1674.090697 LO-REDUCTION 685 1821.368992 1674.090697 LO-REDUCTION 687 1815.439650 1674.090697 EXTENSION 689 1789.342608 1587.943089 LO-REDUCTION 691 1786.525899 1587.943089 LO-REDUCTION 693 1785.838849 1587.943089 LO-REDUCTION 695 1783.713996 1587.943089 LO-REDUCTION 697 1779.337539 1587.943089 LO-REDUCTION 699 1759.833633 1587.943089 LO-REDUCTION 701 1759.191531 1587.943089 LO-REDUCTION 703 1756.846122 1587.943089 LO-REDUCTION 705 1744.660538 1587.943089 LO-REDUCTION 707 1743.348497 1587.943089 LO-REDUCTION 709 1742.565167 1587.943089 LO-REDUCTION 711 1740.663813 1587.943089 LO-REDUCTION 713 1738.022641 1587.943089 LO-REDUCTION 715 1737.488912 1587.943089 LO-REDUCTION 717 1736.543132 1587.943089 EXTENSION 719 1736.360164 1557.907468 LO-REDUCTION 721 1729.447126 1557.907468 LO-REDUCTION 723 1727.909460 1557.907468 LO-REDUCTION 725 1720.002482 1557.907468 EXTENSION 727 1715.353642 1494.501207 LO-REDUCTION 729 1712.847904 1494.501207 LO-REDUCTION 731 1711.855947 1494.501207 LO-REDUCTION 733 1689.327972 1494.501207 LO-REDUCTION 735 1682.207061 1494.501207 LO-REDUCTION 737 1678.042619 1494.501207 LO-REDUCTION 739 1676.422756 1494.501207 LO-REDUCTION 741 1674.090697 1494.501207 LO-REDUCTION 743 1660.224159 1494.501207 LO-REDUCTION 745 1656.860957 1494.501207 LO-REDUCTION 747 1655.850140 1494.501207 LO-REDUCTION 749 1651.886861 1494.501207 LO-REDUCTION 751 1632.828981 1494.501207 LO-REDUCTION 753 1631.963446 1494.501207 LO-REDUCTION 755 1629.197031 1494.501207 LO-REDUCTION 757 1626.969780 1494.501207 LO-REDUCTION 759 1626.639782 1494.501207 LO-REDUCTION 761 1625.744684 1494.501207 LO-REDUCTION 763 1614.516589 1494.501207 EXTENSION 765 1613.534933 1468.582638 EXTENSION 767 1613.529436 1424.775652 LO-REDUCTION 769 1598.658147 1424.775652 LO-REDUCTION 771 1598.467803 1424.775652 LO-REDUCTION 773 1592.923028 1424.775652 LO-REDUCTION 775 1592.211748 1424.775652 LO-REDUCTION 777 1589.113596 1424.775652 LO-REDUCTION 779 1587.943089 1424.775652 LO-REDUCTION 781 1578.965152 1424.775652 EXTENSION 783 1557.907468 1369.512689 LO-REDUCTION 785 1546.654380 1369.512689 LO-REDUCTION 787 1545.232149 1369.512689 LO-REDUCTION 789 1543.467071 1369.512689 LO-REDUCTION 791 1542.031225 1369.512689 LO-REDUCTION 793 1532.016355 1369.512689 LO-REDUCTION 795 1524.926301 1369.512689 LO-REDUCTION 797 1524.265837 1369.512689 LO-REDUCTION 799 1522.356328 1369.512689 LO-REDUCTION 801 1522.236037 1369.512689 LO-REDUCTION 803 1521.025015 1369.512689 EXTENSION 805 1518.097498 1310.736061 LO-REDUCTION 807 1512.141707 1310.736061 LO-REDUCTION 809 1500.450954 1310.736061 LO-REDUCTION 811 1500.092795 1310.736061 LO-REDUCTION 813 1498.545794 1310.736061 LO-REDUCTION 815 1494.501207 1310.736061 LO-REDUCTION 817 1473.655226 1310.736061 LO-REDUCTION 819 1471.367251 1310.736061 LO-REDUCTION 821 1468.582638 1310.736061 LO-REDUCTION 823 1461.217368 1310.736061 LO-REDUCTION 825 1455.227840 1310.736061 LO-REDUCTION 827 1442.343128 1310.736061 LO-REDUCTION 829 1441.071361 1310.736061 LO-REDUCTION 831 1437.361768 1310.736061 LO-REDUCTION 833 1432.603142 1310.736061 EXTENSION 835 1432.508830 1259.430996 LO-REDUCTION 837 1426.387633 1259.430996 LO-REDUCTION 839 1424.775652 1259.430996 LO-REDUCTION 841 1422.157271 1259.430996 LO-REDUCTION 843 1419.781065 1259.430996 LO-REDUCTION 845 1413.578432 1259.430996 LO-REDUCTION 847 1403.092069 1259.430996 LO-REDUCTION 849 1395.458369 1259.430996 LO-REDUCTION 851 1389.175901 1259.430996 LO-REDUCTION 853 1386.826495 1259.430996 LO-REDUCTION 855 1382.566044 1259.430996 LO-REDUCTION 857 1376.754889 1259.430996 EXTENSION 859 1375.545701 1218.177396 LO-REDUCTION 861 1369.512689 1218.177396 LO-REDUCTION 863 1366.293466 1218.177396 LO-REDUCTION 865 1365.021556 1218.177396 LO-REDUCTION 867 1361.664968 1218.177396 LO-REDUCTION 869 1348.743506 1218.177396 LO-REDUCTION 871 1348.176368 1218.177396 LO-REDUCTION 873 1345.115600 1218.177396 LO-REDUCTION 875 1341.763449 1218.177396 LO-REDUCTION 877 1332.891748 1218.177396 EXTENSION 879 1328.385536 1182.348289 LO-REDUCTION 881 1324.715133 1182.348289 LO-REDUCTION 883 1323.921488 1182.348289 LO-REDUCTION 885 1322.347217 1182.348289 LO-REDUCTION 887 1320.555751 1182.348289 LO-REDUCTION 889 1317.859360 1182.348289 LO-REDUCTION 891 1310.736061 1182.348289 LO-REDUCTION 893 1293.751591 1182.348289 EXTENSION 895 1291.950784 1157.097084 LO-REDUCTION 897 1281.769216 1157.097084 LO-REDUCTION 899 1277.939320 1157.097084 LO-REDUCTION 901 1275.353401 1157.097084 LO-REDUCTION 903 1274.807127 1157.097084 LO-REDUCTION 905 1272.790917 1157.097084 EXTENSION 907 1269.990780 1152.516814 LO-REDUCTION 909 1262.234607 1152.516814 LO-REDUCTION 911 1260.250594 1152.516814 LO-REDUCTION 913 1259.430996 1152.516814 REFLECTION 915 1242.001840 1137.335760 LO-REDUCTION 917 1236.825880 1137.335760 LO-REDUCTION 919 1235.358477 1137.335760 REFLECTION 921 1232.471339 1135.374802 LO-REDUCTION 923 1230.710875 1135.374802 EXTENSION 925 1229.303307 1119.731017 LO-REDUCTION 927 1220.586771 1119.731017 LO-REDUCTION 929 1218.177396 1119.731017 LO-REDUCTION 931 1215.837377 1119.731017 LO-REDUCTION 933 1215.317885 1119.731017 EXTENSION 935 1209.543636 1080.802619 LO-REDUCTION 937 1209.041630 1080.802619 LO-REDUCTION 939 1204.439191 1080.802619 LO-REDUCTION 941 1192.198866 1080.802619 LO-REDUCTION 943 1189.500860 1080.802619 LO-REDUCTION 945 1185.661005 1080.802619 LO-REDUCTION 947 1182.348289 1080.802619 LO-REDUCTION 949 1179.125625 1080.802619 LO-REDUCTION 951 1178.619342 1080.802619 LO-REDUCTION 953 1177.571594 1080.802619 HI-REDUCTION 955 1172.329948 1080.802619 LO-REDUCTION 957 1165.843115 1080.802619 LO-REDUCTION 959 1163.222075 1080.802619 LO-REDUCTION 961 1157.097084 1080.802619 LO-REDUCTION 963 1156.248042 1080.802619 EXTENSION 965 1154.565041 1073.360761 LO-REDUCTION 967 1152.516814 1073.360761 LO-REDUCTION 969 1140.825289 1073.360761 LO-REDUCTION 971 1138.724907 1073.360761 EXTENSION 973 1137.968972 1060.709194 LO-REDUCTION 975 1137.943087 1060.709194 LO-REDUCTION 977 1137.335760 1060.709194 LO-REDUCTION 979 1135.926979 1060.709194 LO-REDUCTION 981 1135.652643 1060.709194 LO-REDUCTION 983 1135.374802 1060.709194 LO-REDUCTION 985 1125.912797 1060.709194 LO-REDUCTION 987 1125.209428 1060.709194 LO-REDUCTION 989 1123.981456 1060.709194 REFLECTION 991 1120.522926 1059.765035 LO-REDUCTION 993 1119.731017 1059.765035 LO-REDUCTION 995 1115.525742 1059.765035 LO-REDUCTION 997 1115.177781 1059.765035 REFLECTION 999 1112.959665 1059.445316 REFLECTION 1001 1112.925938 1056.130845 REFLECTION 1003 1108.288355 1054.959477 LO-REDUCTION 1005 1105.317933 1054.959477 LO-REDUCTION 1007 1103.404050 1054.959477 LO-REDUCTION 1009 1100.061193 1054.959477 LO-REDUCTION 1011 1096.712054 1054.959477 REFLECTION 1013 1092.735483 1048.242717 LO-REDUCTION 1015 1091.192415 1048.242717 LO-REDUCTION 1017 1087.452302 1048.242717 REFLECTION 1019 1086.669206 1047.979953 LO-REDUCTION 1021 1085.755325 1047.979953 LO-REDUCTION 1023 1083.066188 1047.979953 LO-REDUCTION 1025 1081.914674 1047.979953 LO-REDUCTION 1027 1081.719538 1047.979953 LO-REDUCTION 1029 1080.802619 1047.979953 LO-REDUCTION 1031 1080.762416 1047.979953 REFLECTION 1033 1078.708879 1045.960150 LO-REDUCTION 1035 1073.360761 1045.960150 LO-REDUCTION 1037 1071.740174 1045.960150 REFLECTION 1039 1069.256959 1041.232977 LO-REDUCTION 1041 1065.791686 1041.232977 LO-REDUCTION 1043 1063.368117 1041.232977 LO-REDUCTION 1045 1062.069204 1041.232977 LO-REDUCTION 1047 1061.618545 1041.232977 LO-REDUCTION 1049 1061.338802 1041.232977 LO-REDUCTION 1051 1060.960725 1041.232977 LO-REDUCTION 1053 1060.709194 1041.232977 LO-REDUCTION 1055 1060.095448 1041.232977 LO-REDUCTION 1057 1059.765035 1041.232977 LO-REDUCTION 1059 1059.445316 1041.232977 LO-REDUCTION 1061 1057.205332 1041.232977 LO-REDUCTION 1063 1056.130845 1041.232977 EXTENSION 1065 1055.905771 1038.266051 LO-REDUCTION 1067 1055.342847 1038.266051 LO-REDUCTION 1069 1055.204324 1038.266051 LO-REDUCTION 1071 1054.959477 1038.266051 REFLECTION 1073 1053.866612 1036.890302 LO-REDUCTION 1075 1051.830593 1036.890302 LO-REDUCTION 1077 1050.999248 1036.890302 LO-REDUCTION 1079 1050.419123 1036.890302 LO-REDUCTION 1081 1050.201780 1036.890302 EXTENSION 1083 1049.140523 1032.157608 LO-REDUCTION 1085 1048.685037 1032.157608 LO-REDUCTION 1087 1048.409717 1032.157608 LO-REDUCTION 1089 1048.388421 1032.157608 LO-REDUCTION 1091 1048.242717 1032.157608 LO-REDUCTION 1093 1047.979953 1032.157608 LO-REDUCTION 1095 1047.389348 1032.157608 LO-REDUCTION 1097 1046.364306 1032.157608 EXTENSION 1099 1045.960150 1030.944801 LO-REDUCTION 1101 1044.801401 1030.944801 LO-REDUCTION 1103 1044.645382 1030.944801 EXTENSION 1105 1044.484821 1025.407113 LO-REDUCTION 1107 1044.306020 1025.407113 LO-REDUCTION 1109 1044.229784 1025.407113 LO-REDUCTION 1111 1043.613857 1025.407113 LO-REDUCTION 1113 1043.483421 1025.407113 EXTENSION 1115 1043.086306 1018.436161 LO-REDUCTION 1117 1041.996408 1018.436161 LO-REDUCTION 1119 1041.232977 1018.436161 LO-REDUCTION 1121 1041.205194 1018.436161 LO-REDUCTION 1123 1039.964177 1018.436161 LO-REDUCTION 1125 1039.703791 1018.436161 LO-REDUCTION 1127 1038.266051 1018.436161 LO-REDUCTION 1129 1037.877434 1018.436161 LO-REDUCTION 1131 1037.405255 1018.436161 LO-REDUCTION 1133 1037.204247 1018.436161 LO-REDUCTION 1135 1037.104053 1018.436161 LO-REDUCTION 1137 1036.921254 1018.436161 LO-REDUCTION 1139 1036.890302 1018.436161 EXTENSION 1141 1036.127425 1011.058907 LO-REDUCTION 1143 1036.005452 1011.058907 LO-REDUCTION 1145 1035.750968 1011.058907 LO-REDUCTION 1147 1035.429755 1011.058907 LO-REDUCTION 1149 1032.192667 1011.058907 LO-REDUCTION 1151 1032.157608 1011.058907 LO-REDUCTION 1153 1032.066169 1011.058907 LO-REDUCTION 1155 1032.065014 1011.058907 LO-REDUCTION 1157 1031.650689 1011.058907 LO-REDUCTION 1159 1030.944801 1011.058907 LO-REDUCTION 1161 1030.785826 1011.058907 LO-REDUCTION 1163 1029.424975 1011.058907 LO-REDUCTION 1165 1028.527475 1011.058907 LO-REDUCTION 1167 1028.416889 1011.058907 LO-REDUCTION 1169 1027.588012 1011.058907 EXTENSION 1171 1026.705048 1001.639022 LO-REDUCTION 1173 1025.550370 1001.639022 LO-REDUCTION 1175 1025.407113 1001.639022 LO-REDUCTION 1177 1024.680715 1001.639022 LO-REDUCTION 1179 1024.234158 1001.639022 LO-REDUCTION 1181 1023.621928 1001.639022 LO-REDUCTION 1183 1023.008359 1001.639022 LO-REDUCTION 1185 1021.615379 1001.639022 LO-REDUCTION 1187 1021.415688 1001.639022 LO-REDUCTION 1189 1020.991470 1001.639022 LO-REDUCTION 1191 1020.297846 1001.639022 LO-REDUCTION 1193 1019.976965 1001.639022 LO-REDUCTION 1195 1019.933688 1001.639022 EXTENSION 1197 1019.541476 998.154531 LO-REDUCTION 1199 1018.436161 998.154531 LO-REDUCTION 1201 1017.426794 998.154531 LO-REDUCTION 1203 1017.032267 998.154531 EXTENSION 1205 1015.941848 993.192601 LO-REDUCTION 1207 1015.912396 993.192601 LO-REDUCTION 1209 1013.483891 993.192601 LO-REDUCTION 1211 1012.904583 993.192601 LO-REDUCTION 1213 1012.282351 993.192601 EXTENSION 1215 1011.765080 988.754006 LO-REDUCTION 1217 1011.720574 988.754006 LO-REDUCTION 1219 1011.229341 988.754006 LO-REDUCTION 1221 1011.083814 988.754006 LO-REDUCTION 1223 1011.058907 988.754006 LO-REDUCTION 1225 1009.158261 988.754006 LO-REDUCTION 1227 1008.907004 988.754006 LO-REDUCTION 1229 1007.206134 988.754006 LO-REDUCTION 1231 1006.094824 988.754006 LO-REDUCTION 1233 1006.033051 988.754006 EXTENSION 1235 1005.136889 984.344970 LO-REDUCTION 1237 1004.884196 984.344970 LO-REDUCTION 1239 1004.725267 984.344970 LO-REDUCTION 1241 1004.126221 984.344970 LO-REDUCTION 1243 1003.311043 984.344970 LO-REDUCTION 1245 1002.273812 984.344970 LO-REDUCTION 1247 1001.639022 984.344970 LO-REDUCTION 1249 1001.453074 984.344970 LO-REDUCTION 1251 1001.156469 984.344970 LO-REDUCTION 1253 999.659784 984.344970 LO-REDUCTION 1255 998.154531 984.344970 LO-REDUCTION 1257 996.958437 984.344970 LO-REDUCTION 1259 995.177613 984.344970 LO-REDUCTION 1261 995.143887 984.344970 EXTENSION 1263 994.647444 982.500139 LO-REDUCTION 1265 994.230251 982.500139 LO-REDUCTION 1267 994.228715 982.500139 LO-REDUCTION 1269 993.569382 982.500139 LO-REDUCTION 1271 993.556633 982.500139 HI-REDUCTION 1273 993.397873 982.500139 LO-REDUCTION 1275 993.192601 982.500139 LO-REDUCTION 1277 991.411857 982.500139 HI-REDUCTION 1279 991.282900 982.500139 LO-REDUCTION 1281 990.531080 982.500139 LO-REDUCTION 1283 990.514486 982.500139 REFLECTION 1285 989.924531 980.817387 LO-REDUCTION 1287 989.133664 980.817387 LO-REDUCTION 1289 988.754006 980.817387 LO-REDUCTION 1291 988.436401 980.817387 LO-REDUCTION 1293 988.072240 980.817387 EXTENSION 1295 987.655166 979.480907 LO-REDUCTION 1297 987.550072 979.480907 EXTENSION 1299 987.436766 977.631769 LO-REDUCTION 1301 987.192478 977.631769 LO-REDUCTION 1303 986.060143 977.631769 LO-REDUCTION 1305 985.986343 977.631769 LO-REDUCTION 1307 985.337155 977.631769 LO-REDUCTION 1309 985.267465 977.631769 LO-REDUCTION 1311 985.020827 977.631769 LO-REDUCTION 1313 984.898615 977.631769 LO-REDUCTION 1315 984.783000 977.631769 LO-REDUCTION 1317 984.777110 977.631769 LO-REDUCTION 1319 984.562801 977.631769 LO-REDUCTION 1321 984.505476 977.631769 LO-REDUCTION 1323 984.361409 977.631769 LO-REDUCTION 1325 984.344970 977.631769 EXTENSION 1327 983.370089 975.908389 LO-REDUCTION 1329 983.166722 975.908389 LO-REDUCTION 1331 983.019158 975.908389 LO-REDUCTION 1333 983.016623 975.908389 LO-REDUCTION 1335 982.863973 975.908389 LO-REDUCTION 1337 982.500139 975.908389 LO-REDUCTION 1339 982.185064 975.908389 LO-REDUCTION 1341 982.063662 975.908389 LO-REDUCTION 1343 981.765901 975.908389 LO-REDUCTION 1345 981.573178 975.908389 LO-REDUCTION 1347 981.197759 975.908389 LO-REDUCTION 1349 981.127973 975.908389 LO-REDUCTION 1351 980.926694 975.908389 HI-REDUCTION 1353 980.898005 975.908389 EXTENSION 1355 980.817387 975.739572 LO-REDUCTION 1357 980.435882 975.739572 REFLECTION 1359 980.256482 975.675894 EXTENSION 1361 980.180627 974.371914 LO-REDUCTION 1363 980.128288 974.371914 LO-REDUCTION 1365 979.930987 974.371914 LO-REDUCTION 1367 979.866758 974.371914 LO-REDUCTION 1369 979.520698 974.371914 EXTENSION 1371 979.480907 973.480715 LO-REDUCTION 1373 979.204554 973.480715 LO-REDUCTION 1375 978.645388 973.480715 LO-REDUCTION 1377 978.457902 973.480715 LO-REDUCTION 1379 978.375425 973.480715 LO-REDUCTION 1381 978.124121 973.480715 LO-REDUCTION 1383 977.954338 973.480715 EXTENSION 1385 977.747913 972.939411 LO-REDUCTION 1387 977.732806 972.939411 LO-REDUCTION 1389 977.631769 972.939411 LO-REDUCTION 1391 977.030339 972.939411 LO-REDUCTION 1393 976.984479 972.939411 LO-REDUCTION 1395 976.840747 972.939411 LO-REDUCTION 1397 976.776362 972.939411 LO-REDUCTION 1399 976.594938 972.939411 EXTENSION 1401 976.491778 971.732308 LO-REDUCTION 1403 976.240381 971.732308 LO-REDUCTION 1405 976.233119 971.732308 LO-REDUCTION 1407 975.908389 971.732308 LO-REDUCTION 1409 975.877938 971.732308 LO-REDUCTION 1411 975.739572 971.732308 LO-REDUCTION 1413 975.675894 971.732308 HI-REDUCTION 1415 975.673287 971.732308 LO-REDUCTION 1417 975.067530 971.732308 LO-REDUCTION 1419 974.887318 971.732308 LO-REDUCTION 1421 974.886868 971.732308 LO-REDUCTION 1423 974.864416 971.732308 LO-REDUCTION 1425 974.860887 971.732308 LO-REDUCTION 1427 974.711227 971.732308 LO-REDUCTION 1429 974.613795 971.732308 LO-REDUCTION 1431 974.545149 971.732308 LO-REDUCTION 1433 974.371914 971.732308 LO-REDUCTION 1435 974.110906 971.732308 LO-REDUCTION 1437 974.089209 971.732308 LO-REDUCTION 1439 974.047696 971.732308 LO-REDUCTION 1441 973.997904 971.732308 EXTENSION 1443 973.972559 970.740844 LO-REDUCTION 1445 973.758452 970.740844 LO-REDUCTION 1447 973.638967 970.740844 LO-REDUCTION 1449 973.577850 970.740844 LO-REDUCTION 1451 973.540714 970.740844 LO-REDUCTION 1453 973.540226 970.740844 LO-REDUCTION 1455 973.480715 970.740844 LO-REDUCTION 1457 973.346271 970.740844 LO-REDUCTION 1459 973.339201 970.740844 LO-REDUCTION 1461 973.204161 970.740844 LO-REDUCTION 1463 973.129233 970.740844 LO-REDUCTION 1465 972.939526 970.740844 LO-REDUCTION 1467 972.939411 970.740844 LO-REDUCTION 1469 972.928939 970.740844 EXTENSION 1471 972.593014 969.779295 LO-REDUCTION 1473 972.563274 969.779295 LO-REDUCTION 1475 972.548453 969.779295 LO-REDUCTION 1477 972.422767 969.779295 LO-REDUCTION 1479 972.408620 969.779295 LO-REDUCTION 1481 972.360753 969.779295 LO-REDUCTION 1483 972.352697 969.779295 LO-REDUCTION 1485 972.333037 969.779295 LO-REDUCTION 1487 972.309560 969.779295 LO-REDUCTION 1489 971.971389 969.779295 LO-REDUCTION 1491 971.925732 969.779295 LO-REDUCTION 1493 971.826756 969.779295 LO-REDUCTION 1495 971.804425 969.779295 EXTENSION 1497 971.732308 968.805382 LO-REDUCTION 1499 971.702424 968.805382 LO-REDUCTION 1501 971.567321 968.805382 LO-REDUCTION 1503 971.519720 968.805382 LO-REDUCTION 1505 971.358040 968.805382 LO-REDUCTION 1507 971.347642 968.805382 LO-REDUCTION 1509 971.345117 968.805382 LO-REDUCTION 1511 971.336388 968.805382 LO-REDUCTION 1513 971.324130 968.805382 LO-REDUCTION 1515 971.181968 968.805382 LO-REDUCTION 1517 971.138899 968.805382 LO-REDUCTION 1519 970.978798 968.805382 LO-REDUCTION 1521 970.950297 968.805382 LO-REDUCTION 1523 970.903238 968.805382 LO-REDUCTION 1525 970.740844 968.805382 LO-REDUCTION 1527 970.722685 968.805382 EXTENSION 1529 970.685010 967.546521 LO-REDUCTION 1531 970.674717 967.546521 LO-REDUCTION 1533 970.508681 967.546521 LO-REDUCTION 1535 970.503436 967.546521 LO-REDUCTION 1537 970.363931 967.546521 LO-REDUCTION 1539 970.250704 967.546521 LO-REDUCTION 1541 970.229768 967.546521 LO-REDUCTION 1543 970.155114 967.546521 LO-REDUCTION 1545 970.148019 967.546521 LO-REDUCTION 1547 970.118608 967.546521 LO-REDUCTION 1549 969.856452 967.546521 LO-REDUCTION 1551 969.853277 967.546521 LO-REDUCTION 1553 969.779295 967.546521 LO-REDUCTION 1555 969.666683 967.546521 LO-REDUCTION 1557 969.588503 967.546521 EXTENSION 1559 969.446638 967.116089 LO-REDUCTION 1561 969.416161 967.116089 LO-REDUCTION 1563 969.362679 967.116089 EXTENSION 1565 969.327957 966.139302 LO-REDUCTION 1567 969.205369 966.139302 LO-REDUCTION 1569 969.184177 966.139302 LO-REDUCTION 1571 969.097741 966.139302 LO-REDUCTION 1573 969.035292 966.139302 LO-REDUCTION 1575 969.029977 966.139302 LO-REDUCTION 1577 968.906601 966.139302 LO-REDUCTION 1579 968.805382 966.139302 LO-REDUCTION 1581 968.744975 966.139302 LO-REDUCTION 1583 968.351407 966.139302 LO-REDUCTION 1585 968.307359 966.139302 LO-REDUCTION 1587 968.299310 966.139302 EXTENSION 1589 968.296814 965.315846 LO-REDUCTION 1591 968.279765 965.315846 LO-REDUCTION 1593 968.276206 965.315846 LO-REDUCTION 1595 968.259557 965.315846 LO-REDUCTION 1597 968.247763 965.315846 LO-REDUCTION 1599 968.136217 965.315846 EXTENSION 1601 967.898753 964.312616 LO-REDUCTION 1603 967.807599 964.312616 LO-REDUCTION 1605 967.742021 964.312616 LO-REDUCTION 1607 967.575042 964.312616 LO-REDUCTION 1609 967.546521 964.312616 LO-REDUCTION 1611 967.455164 964.312616 LO-REDUCTION 1613 967.434143 964.312616 LO-REDUCTION 1615 967.371413 964.312616 LO-REDUCTION 1617 967.296415 964.312616 LO-REDUCTION 1619 967.186108 964.312616 LO-REDUCTION 1621 967.116089 964.312616 LO-REDUCTION 1623 966.988566 964.312616 LO-REDUCTION 1625 966.988270 964.312616 EXTENSION 1627 966.846266 963.320516 LO-REDUCTION 1629 966.818230 963.320516 LO-REDUCTION 1631 966.606761 963.320516 LO-REDUCTION 1633 966.458064 963.320516 LO-REDUCTION 1635 966.406238 963.320516 LO-REDUCTION 1637 966.367406 963.320516 LO-REDUCTION 1639 966.178744 963.320516 LO-REDUCTION 1641 966.139302 963.320516 LO-REDUCTION 1643 966.075780 963.320516 LO-REDUCTION 1645 965.949834 963.320516 EXTENSION 1647 965.718146 962.446629 LO-REDUCTION 1649 965.592596 962.446629 LO-REDUCTION 1651 965.583944 962.446629 LO-REDUCTION 1653 965.412168 962.446629 LO-REDUCTION 1655 965.315846 962.446629 LO-REDUCTION 1657 965.169697 962.446629 LO-REDUCTION 1659 965.101252 962.446629 LO-REDUCTION 1661 965.093009 962.446629 LO-REDUCTION 1663 965.081974 962.446629 EXTENSION 1665 964.903824 961.020361 LO-REDUCTION 1667 964.806549 961.020361 LO-REDUCTION 1669 964.768263 961.020361 LO-REDUCTION 1671 964.758887 961.020361 LO-REDUCTION 1673 964.672174 961.020361 LO-REDUCTION 1675 964.628404 961.020361 LO-REDUCTION 1677 964.312616 961.020361 LO-REDUCTION 1679 964.179201 961.020361 LO-REDUCTION 1681 963.956710 961.020361 LO-REDUCTION 1683 963.834654 961.020361 LO-REDUCTION 1685 963.824164 961.020361 LO-REDUCTION 1687 963.712822 961.020361 LO-REDUCTION 1689 963.661470 961.020361 LO-REDUCTION 1691 963.649170 961.020361 EXTENSION 1693 963.446557 959.862590 LO-REDUCTION 1695 963.368096 959.862590 LO-REDUCTION 1697 963.333458 959.862590 LO-REDUCTION 1699 963.320516 959.862590 LO-REDUCTION 1701 963.064127 959.862590 LO-REDUCTION 1703 962.977972 959.862590 LO-REDUCTION 1705 962.774649 959.862590 LO-REDUCTION 1707 962.754773 959.862590 LO-REDUCTION 1709 962.713908 959.862590 LO-REDUCTION 1711 962.655114 959.862590 LO-REDUCTION 1713 962.630365 959.862590 LO-REDUCTION 1715 962.484250 959.862590 LO-REDUCTION 1717 962.446629 959.862590 LO-REDUCTION 1719 962.242497 959.862590 LO-REDUCTION 1721 962.194012 959.862590 LO-REDUCTION 1723 962.087631 959.862590 LO-REDUCTION 1725 961.990438 959.862590 LO-REDUCTION 1727 961.962658 959.862590 EXTENSION 1729 961.812204 959.646082 LO-REDUCTION 1731 961.675173 959.646082 LO-REDUCTION 1733 961.607891 959.646082 LO-REDUCTION 1735 961.509939 959.646082 LO-REDUCTION 1737 961.488282 959.646082 LO-REDUCTION 1739 961.458421 959.646082 LO-REDUCTION 1741 961.360490 959.646082 LO-REDUCTION 1743 961.358082 959.646082 EXTENSION 1745 961.217481 959.335567 LO-REDUCTION 1747 961.076529 959.335567 LO-REDUCTION 1749 961.020361 959.335567 LO-REDUCTION 1751 960.920065 959.335567 REFLECTION 1753 960.895823 959.317870 LO-REDUCTION 1755 960.867950 959.317870 LO-REDUCTION 1757 960.852437 959.317870 LO-REDUCTION 1759 960.798913 959.317870 LO-REDUCTION 1761 960.578487 959.317870 REFLECTION 1763 960.526298 959.290446 LO-REDUCTION 1765 960.493740 959.290446 LO-REDUCTION 1767 960.456658 959.290446 EXTENSION 1769 960.411175 958.890988 LO-REDUCTION 1771 960.343791 958.890988 LO-REDUCTION 1773 960.327956 958.890988 LO-REDUCTION 1775 960.321628 958.890988 LO-REDUCTION 1777 960.307817 958.890988 LO-REDUCTION 1779 960.179583 958.890988 LO-REDUCTION 1781 960.151391 958.890988 LO-REDUCTION 1783 959.995690 958.890988 LO-REDUCTION 1785 959.961710 958.890988 LO-REDUCTION 1787 959.920878 958.890988 EXTENSION 1789 959.906388 958.639683 LO-REDUCTION 1791 959.867618 958.639683 EXTENSION 1793 959.862590 957.944839 LO-REDUCTION 1795 959.827095 957.944839 LO-REDUCTION 1797 959.655641 957.944839 LO-REDUCTION 1799 959.646082 957.944839 LO-REDUCTION 1801 959.612178 957.944839 LO-REDUCTION 1803 959.578756 957.944839 LO-REDUCTION 1805 959.550677 957.944839 LO-REDUCTION 1807 959.549431 957.944839 LO-REDUCTION 1809 959.429639 957.944839 LO-REDUCTION 1811 959.390213 957.944839 LO-REDUCTION 1813 959.339492 957.944839 LO-REDUCTION 1815 959.335567 957.944839 HI-REDUCTION 1817 959.317870 957.944839 LO-REDUCTION 1819 959.290446 957.944839 LO-REDUCTION 1821 959.283603 957.944839 LO-REDUCTION 1823 959.256079 957.944839 LO-REDUCTION 1825 959.248688 957.944839 LO-REDUCTION 1827 959.221326 957.944839 LO-REDUCTION 1829 959.177374 957.944839 EXTENSION 1831 959.130663 957.652139 LO-REDUCTION 1833 959.086635 957.652139 LO-REDUCTION 1835 959.065986 957.652139 LO-REDUCTION 1837 959.022480 957.652139 LO-REDUCTION 1839 959.009205 957.652139 EXTENSION 1841 958.988732 957.343080 LO-REDUCTION 1843 958.890988 957.343080 LO-REDUCTION 1845 958.767024 957.343080 LO-REDUCTION 1847 958.639683 957.343080 LO-REDUCTION 1849 958.616652 957.343080 LO-REDUCTION 1851 958.616132 957.343080 LO-REDUCTION 1853 958.594233 957.343080 LO-REDUCTION 1855 958.547726 957.343080 EXTENSION 1857 958.529265 956.901292 LO-REDUCTION 1859 958.501962 956.901292 LO-REDUCTION 1861 958.465786 956.901292 LO-REDUCTION 1863 958.460923 956.901292 LO-REDUCTION 1865 958.432344 956.901292 LO-REDUCTION 1867 958.431655 956.901292 LO-REDUCTION 1869 958.344017 956.901292 LO-REDUCTION 1871 958.337842 956.901292 LO-REDUCTION 1873 958.311910 956.901292 LO-REDUCTION 1875 958.272253 956.901292 EXTENSION 1877 958.116495 956.234285 LO-REDUCTION 1879 958.001364 956.234285 LO-REDUCTION 1881 957.978814 956.234285 LO-REDUCTION 1883 957.944839 956.234285 LO-REDUCTION 1885 957.924576 956.234285 LO-REDUCTION 1887 957.818801 956.234285 LO-REDUCTION 1889 957.753165 956.234285 LO-REDUCTION 1891 957.719212 956.234285 LO-REDUCTION 1893 957.713782 956.234285 LO-REDUCTION 1895 957.706441 956.234285 LO-REDUCTION 1897 957.664611 956.234285 LO-REDUCTION 1899 957.652139 956.234285 LO-REDUCTION 1901 957.591327 956.234285 LO-REDUCTION 1903 957.576134 956.234285 EXTENSION 1905 957.469752 956.014464 LO-REDUCTION 1907 957.434526 956.014464 LO-REDUCTION 1909 957.418386 956.014464 LO-REDUCTION 1911 957.371179 956.014464 EXTENSION 1913 957.343080 955.726396 LO-REDUCTION 1915 957.246457 955.726396 LO-REDUCTION 1917 957.200104 955.726396 LO-REDUCTION 1919 957.048383 955.726396 LO-REDUCTION 1921 957.033759 955.726396 LO-REDUCTION 1923 957.005088 955.726396 LO-REDUCTION 1925 956.934436 955.726396 LO-REDUCTION 1927 956.901292 955.726396 LO-REDUCTION 1929 956.828480 955.726396 LO-REDUCTION 1931 956.826266 955.726396 LO-REDUCTION 1933 956.796424 955.726396 EXTENSION 1935 956.764747 955.532850 LO-REDUCTION 1937 956.714004 955.532850 LO-REDUCTION 1939 956.695218 955.532850 LO-REDUCTION 1941 956.694019 955.532850 EXTENSION 1943 956.686285 955.389268 LO-REDUCTION 1945 956.632085 955.389268 LO-REDUCTION 1947 956.604181 955.389268 LO-REDUCTION 1949 956.479658 955.389268 LO-REDUCTION 1951 956.392614 955.389268 LO-REDUCTION 1953 956.358611 955.389268 LO-REDUCTION 1955 956.334556 955.389268 LO-REDUCTION 1957 956.325083 955.389268 EXTENSION 1959 956.291804 955.085118 LO-REDUCTION 1961 956.288597 955.085118 LO-REDUCTION 1963 956.234285 955.085118 LO-REDUCTION 1965 956.155974 955.085118 LO-REDUCTION 1967 956.134995 955.085118 EXTENSION 1969 956.072531 954.885738 LO-REDUCTION 1971 956.041954 954.885738 LO-REDUCTION 1973 956.014464 954.885738 LO-REDUCTION 1975 956.001108 954.885738 LO-REDUCTION 1977 955.995755 954.885738 LO-REDUCTION 1979 955.993674 954.885738 LO-REDUCTION 1981 955.900719 954.885738 LO-REDUCTION 1983 955.885319 954.885738 LO-REDUCTION 1985 955.745945 954.885738 EXTENSION 1987 955.726396 954.267731 LO-REDUCTION 1989 955.707032 954.267731 LO-REDUCTION 1991 955.660783 954.267731 LO-REDUCTION 1993 955.631740 954.267731 LO-REDUCTION 1995 955.572693 954.267731 LO-REDUCTION 1997 955.551922 954.267731 LO-REDUCTION 1999 955.532850 954.267731 Exiting from Nelder Mead minimizer 2001 function evaluations used  We see a clear peak in FoI in the 8-10 age-group. The pattern makes sense given the biology of rubella and the assortative mixing commonly seen in the human host with most contacts being among same-aged individuals (see section [sec:waifw]). Peru has a life-expectancy of around 75 years, and the$R_0$of rubella is typically quoted in the 4-10 range, so according to$\bar{a} \simeq L /(R_0 -1)$the peak in circulation is predicted to be in an interval around 10 yrs of age. We can do a more refined scenario-analyses regarding consequences of vaccination using the spline model. We focus on the 0-45 year age-range as this spans the pre- to post- child-bearing age: data3 = peru[peru$age < 45, ]
df = 5
para = rep(0.1, df + 1)


We use a log-transformation to constrain the FoI to be positive, create the ‘dummy’ lm-object, and define the function to evaluate the negative log-likelihood of the FoI curve given the data:

#Prediction function
tmpfn=function(x,dl){
x=predict(dl, newdata=data.frame(age=x))
exp(x)}
#Dummy lm-object
dl=lm(cumulative~bs(age,df), data=data3)
#Log-likelihood function
tmpfn2=function(par,data, df){
dl=lm(cumulative~bs(age,df), data=data)
dl$coefficients=par ll=0 for(a in 1:length(data$age)){
p=((1-exp(-integrate(tmpfn,0,data$age[a], dl=dl)$value)))
ll=ll+dbinom(data$cumulative[a],data$n[a],p,log=T)
}
return(-ll)
}


Getting a good fit is, again, computationally expensive, but reveals and interesting two-peaked force-of-infection. A dominant peak just under 10 years and a subdominant peak around 35. A plausible scenario is that most people get infected in school but the fraction that escapes this dominant mode of infection are most likely to contract the virus from their children when they reach school age.

#Fit model
options(warn=-1)
dspline.a45.df5=optim(par=log(para),fn=tmpfn2,
control=list(trace=4, maxit=5000))
#Overwrite dummy-objects coefficients with MLEs
dl$coefficients=dspline.a45.df5$par
plot(exp(predict(dl))~data3$age, xlab="Age", ylab="Relative FoI", type="l") options(warn=0)   Nelder-Mead direct search function minimizer function value for initial parameters = 1141035.737231 Scaled convergence tolerance is 0.0170028 Stepsize computed as 0.230259 BUILD 7 1141035.737231 970950.627465 LO-REDUCTION 9 1137877.634033 970950.627465 LO-REDUCTION 11 1121934.640856 970950.627465 EXTENSION 13 1103727.484754 894493.206451 LO-REDUCTION 15 1094400.861158 894493.206451 EXTENSION 17 1079917.487811 761540.193954 LO-REDUCTION 19 1029493.054676 761540.193954 EXTENSION 21 996693.178557 619430.190272 LO-REDUCTION 23 970950.627465 619430.190272 EXTENSION 25 918394.487677 408688.548477 LO-REDUCTION 27 894493.206451 408688.548477 EXTENSION 29 783649.934481 182510.957601 LO-REDUCTION 31 761540.193954 182510.957601 EXTENSION 33 658497.237544 47258.246308 LO-REDUCTION 35 619430.190272 47258.246308 REFLECTION 37 468878.161804 31095.755010 LO-REDUCTION 39 408688.548477 31095.755010 LO-REDUCTION 41 261614.367647 31095.755010 LO-REDUCTION 43 182510.957601 31095.755010 LO-REDUCTION 45 66959.772115 31095.755010 HI-REDUCTION 47 52966.279618 31095.755010 HI-REDUCTION 49 47258.246308 31095.755010 REFLECTION 51 43604.874615 28776.840485 HI-REDUCTION 53 42346.151702 28776.840485 HI-REDUCTION 55 40424.787118 28776.840485 LO-REDUCTION 57 38066.900745 28776.840485 LO-REDUCTION 59 35451.202011 28776.840485 LO-REDUCTION 61 34814.010063 28776.840485 REFLECTION 63 34143.055709 28506.940407 EXTENSION 65 31445.429957 21176.909535 LO-REDUCTION 67 31207.697066 21176.909535 LO-REDUCTION 69 31095.755010 21176.909535 LO-REDUCTION 71 29098.474935 21176.909535 LO-REDUCTION 73 28776.840485 21176.909535 LO-REDUCTION 75 28506.940407 21176.909535 EXTENSION 77 27408.403349 18366.599514 LO-REDUCTION 79 27096.774006 18366.599514 EXTENSION 81 26052.414300 15284.134064 EXTENSION 83 24026.686318 13111.402965 LO-REDUCTION 85 22449.874820 13111.402965 LO-REDUCTION 87 21176.909535 13111.402965 LO-REDUCTION 89 19537.352869 13111.402965 REFLECTION 91 18366.599514 12586.831755 LO-REDUCTION 93 15284.134064 12586.831755 LO-REDUCTION 95 14190.658465 12586.831755 LO-REDUCTION 97 13939.482253 12577.831769 LO-REDUCTION 99 13275.391433 12577.831769 EXTENSION 101 13161.410838 11577.146834 LO-REDUCTION 103 13111.402965 11577.146834 LO-REDUCTION 105 12876.586151 11577.146834 LO-REDUCTION 107 12666.355041 11577.146834 LO-REDUCTION 109 12586.831755 11577.146834 EXTENSION 111 12577.831769 10590.968896 LO-REDUCTION 113 12418.402837 10590.968896 LO-REDUCTION 115 12180.684631 10590.968896 LO-REDUCTION 117 11742.337489 10590.968896 EXTENSION 119 11593.281076 10172.903525 EXTENSION 121 11577.146834 9533.393929 EXTENSION 123 10739.809207 7962.424225 LO-REDUCTION 125 10665.176354 7962.424225 LO-REDUCTION 127 10614.654952 7962.424225 LO-REDUCTION 129 10590.968896 7962.424225 LO-REDUCTION 131 10172.903525 7962.424225 EXTENSION 133 9533.393929 7150.111287 LO-REDUCTION 135 9505.111273 7150.111287 LO-REDUCTION 137 9214.893047 7150.111287 EXTENSION 139 8594.588817 6596.633387 EXTENSION 141 8456.567112 6067.016373 LO-REDUCTION 143 7962.424225 6067.016373 LO-REDUCTION 145 7380.577804 6067.016373 REFLECTION 147 7233.347930 5933.121935 LO-REDUCTION 149 7150.111287 5933.121935 LO-REDUCTION 151 6596.633387 5933.121935 HI-REDUCTION 153 6318.080595 5933.121935 LO-REDUCTION 155 6201.346820 5930.467955 LO-REDUCTION 157 6134.151016 5930.467955 LO-REDUCTION 159 6073.762972 5930.467955 LO-REDUCTION 161 6067.016373 5930.467955 LO-REDUCTION 163 6062.670160 5930.467955 LO-REDUCTION 165 6001.672658 5922.318282 HI-REDUCTION 167 5972.542566 5922.318282 REFLECTION 169 5946.950869 5915.101412 HI-REDUCTION 171 5940.925650 5915.101412 LO-REDUCTION 173 5937.226996 5915.101412 EXTENSION 175 5933.121935 5883.760920 HI-REDUCTION 177 5930.467955 5883.760920 EXTENSION 179 5922.318282 5857.294098 LO-REDUCTION 181 5917.852532 5857.294098 LO-REDUCTION 183 5916.222641 5857.294098 LO-REDUCTION 185 5915.101412 5857.294098 REFLECTION 187 5909.987086 5842.045879 EXTENSION 189 5883.760920 5813.310104 EXTENSION 191 5873.500097 5792.104271 EXTENSION 193 5870.161746 5763.437681 EXTENSION 195 5860.075843 5710.543964 LO-REDUCTION 197 5857.294098 5710.543964 EXTENSION 199 5842.045879 5648.989094 EXTENSION 201 5813.310104 5583.707062 LO-REDUCTION 203 5792.104271 5583.707062 LO-REDUCTION 205 5763.437681 5583.707062 LO-REDUCTION 207 5740.700966 5583.707062 REFLECTION 209 5710.543964 5561.988414 LO-REDUCTION 211 5657.388879 5561.988414 LO-REDUCTION 213 5648.989094 5561.988414 REFLECTION 215 5604.169243 5545.560302 LO-REDUCTION 217 5594.315468 5545.560302 HI-REDUCTION 219 5584.183963 5545.560302 REFLECTION 221 5583.707062 5545.514183 HI-REDUCTION 223 5570.854400 5545.514183 LO-REDUCTION 225 5561.988414 5544.025590 LO-REDUCTION 227 5561.861342 5542.837341 REFLECTION 229 5556.165142 5540.985397 LO-REDUCTION 231 5546.353619 5540.985397 HI-REDUCTION 233 5545.560302 5539.803463 LO-REDUCTION 235 5545.514183 5538.843731 LO-REDUCTION 237 5544.025590 5538.224451 LO-REDUCTION 239 5542.837341 5537.506031 LO-REDUCTION 241 5541.396140 5537.073016 LO-REDUCTION 243 5540.985397 5537.073016 HI-REDUCTION 245 5539.803463 5537.073016 LO-REDUCTION 247 5538.843731 5537.073016 HI-REDUCTION 249 5538.224451 5537.073016 LO-REDUCTION 251 5538.041671 5537.053088 EXTENSION 253 5537.977325 5535.829720 LO-REDUCTION 255 5537.543782 5535.829720 LO-REDUCTION 257 5537.506031 5535.829720 LO-REDUCTION 259 5537.236770 5535.829720 REFLECTION 261 5537.073016 5535.682087 LO-REDUCTION 263 5537.053088 5535.682087 LO-REDUCTION 265 5536.616032 5535.682087 REFLECTION 267 5536.234784 5535.456893 EXTENSION 269 5536.083439 5534.630453 EXTENSION 271 5535.937781 5533.841274 LO-REDUCTION 273 5535.834275 5533.841274 LO-REDUCTION 275 5535.829720 5533.841274 EXTENSION 277 5535.682087 5533.451985 LO-REDUCTION 279 5535.456893 5533.451985 EXTENSION 281 5534.767577 5531.144436 LO-REDUCTION 283 5534.630453 5531.144436 EXTENSION 285 5534.026940 5530.320350 EXTENSION 287 5533.841274 5528.297363 LO-REDUCTION 289 5533.683622 5528.297363 LO-REDUCTION 291 5533.451985 5528.297363 EXTENSION 293 5531.688770 5523.459832 LO-REDUCTION 295 5531.144436 5523.459832 LO-REDUCTION 297 5530.320350 5523.459832 LO-REDUCTION 299 5530.259005 5523.459832 LO-REDUCTION 301 5530.118847 5523.459832 EXTENSION 303 5528.297363 5516.912103 LO-REDUCTION 305 5527.973868 5516.912103 LO-REDUCTION 307 5526.551832 5516.912103 LO-REDUCTION 309 5526.035695 5516.912103 EXTENSION 311 5524.005433 5511.388785 EXTENSION 313 5523.459832 5505.313381 LO-REDUCTION 315 5520.714541 5505.313381 EXTENSION 317 5517.708138 5496.798019 LO-REDUCTION 319 5517.027072 5496.798019 LO-REDUCTION 321 5516.912103 5496.798019 EXTENSION 323 5511.388785 5488.706153 EXTENSION 325 5505.875596 5470.032942 LO-REDUCTION 327 5505.313381 5470.032942 LO-REDUCTION 329 5499.140655 5470.032942 LO-REDUCTION 331 5497.158880 5470.032942 EXTENSION 333 5496.798019 5467.593306 LO-REDUCTION 335 5488.706153 5467.593306 EXTENSION 337 5483.331213 5441.048621 LO-REDUCTION 339 5475.510890 5441.048621 LO-REDUCTION 341 5474.436289 5441.048621 EXTENSION 343 5470.032942 5434.943656 HI-REDUCTION 345 5469.201029 5434.943656 EXTENSION 347 5467.593306 5421.359733 EXTENSION 349 5453.193140 5387.642786 LO-REDUCTION 351 5448.338777 5387.642786 LO-REDUCTION 353 5446.888799 5387.642786 LO-REDUCTION 355 5441.048621 5387.642786 LO-REDUCTION 357 5434.943656 5387.642786 EXTENSION 359 5421.359733 5377.674771 EXTENSION 361 5414.101334 5358.436699 LO-REDUCTION 363 5405.162225 5358.436699 EXTENSION 365 5399.755414 5326.200930 LO-REDUCTION 367 5396.550299 5326.200930 LO-REDUCTION 369 5387.642786 5326.200930 LO-REDUCTION 371 5377.674771 5326.200930 EXTENSION 373 5360.461492 5310.155212 LO-REDUCTION 375 5358.436699 5310.155212 LO-REDUCTION 377 5343.547456 5310.155212 EXTENSION 379 5338.555786 5291.813898 EXTENSION 381 5336.724680 5269.845085 LO-REDUCTION 383 5326.200930 5269.845085 LO-REDUCTION 385 5311.277585 5269.845085 HI-REDUCTION 387 5310.272786 5269.845085 LO-REDUCTION 389 5310.155212 5269.845085 EXTENSION 391 5293.432839 5256.024110 LO-REDUCTION 393 5292.655012 5256.024110 LO-REDUCTION 395 5291.813898 5256.024110 LO-REDUCTION 397 5291.241910 5256.024110 LO-REDUCTION 399 5282.485885 5256.024110 LO-REDUCTION 401 5278.548184 5256.024110 EXTENSION 403 5276.461104 5241.084076 EXTENSION 405 5269.845085 5208.354985 LO-REDUCTION 407 5260.418308 5208.354985 LO-REDUCTION 409 5258.271365 5208.354985 LO-REDUCTION 411 5257.574344 5208.354985 LO-REDUCTION 413 5256.024110 5208.354985 EXTENSION 415 5241.084076 5192.458025 EXTENSION 417 5240.199767 5154.772302 EXTENSION 419 5223.839443 5114.163198 LO-REDUCTION 421 5221.260437 5114.163198 LO-REDUCTION 423 5210.039756 5114.163198 LO-REDUCTION 425 5208.354985 5114.163198 EXTENSION 427 5192.458025 5022.885648 EXTENSION 429 5154.772302 4950.582457 LO-REDUCTION 431 5139.016073 4950.582457 LO-REDUCTION 433 5117.048914 4950.582457 EXTENSION 435 5114.951352 4844.916333 EXTENSION 437 5114.163198 4698.619137 LO-REDUCTION 439 5022.885648 4698.619137 LO-REDUCTION 441 5002.992710 4698.619137 EXTENSION 443 4965.591772 4426.763578 LO-REDUCTION 445 4950.582457 4426.763578 LO-REDUCTION 447 4844.916333 4426.763578 EXTENSION 449 4734.111236 4047.998943 LO-REDUCTION 451 4705.284925 4047.998943 LO-REDUCTION 453 4698.619137 4047.998943 EXTENSION 455 4504.672389 3943.940121 REFLECTION 457 4434.862243 3840.300019 EXTENSION 459 4426.763578 3555.333802 LO-REDUCTION 461 4238.437563 3555.333802 LO-REDUCTION 463 4093.754423 3555.333802 LO-REDUCTION 465 4047.998943 3555.333802 HI-REDUCTION 467 3943.940121 3555.333802 LO-REDUCTION 469 3840.300019 3555.333802 REFLECTION 471 3802.820469 3547.069626 LO-REDUCTION 473 3753.716743 3547.069626 REFLECTION 475 3686.129599 3511.647546 EXTENSION 477 3668.523943 3447.738213 LO-REDUCTION 479 3558.508847 3447.738213 HI-REDUCTION 481 3555.333802 3447.738213 EXTENSION 483 3551.538937 3351.397924 HI-REDUCTION 485 3547.069626 3351.397924 LO-REDUCTION 487 3511.647546 3351.397924 LO-REDUCTION 489 3496.737035 3351.397924 LO-REDUCTION 491 3494.345029 3351.397924 EXTENSION 493 3475.258807 3284.154780 LO-REDUCTION 495 3447.738213 3284.154780 EXTENSION 497 3400.654154 3139.154721 LO-REDUCTION 499 3396.822396 3139.154721 LO-REDUCTION 501 3352.753454 3139.154721 LO-REDUCTION 503 3351.397924 3139.154721 EXTENSION 505 3340.855627 3092.419580 EXTENSION 507 3284.154780 2981.039481 EXTENSION 509 3271.024558 2766.784153 EXTENSION 511 3147.603184 2496.490897 LO-REDUCTION 513 3139.658498 2496.490897 EXTENSION 515 3139.154721 2287.579725 LO-REDUCTION 517 3092.419580 2287.579725 LO-REDUCTION 519 2981.039481 2287.579725 EXTENSION 521 2766.784153 2095.150748 LO-REDUCTION 523 2649.765522 2095.150748 EXTENSION 525 2596.239716 1938.296847 LO-REDUCTION 527 2496.490897 1938.296847 LO-REDUCTION 529 2352.391230 1920.351517 EXTENSION 531 2287.579725 1647.827291 LO-REDUCTION 533 2172.168076 1647.827291 LO-REDUCTION 535 2095.150748 1647.827291 HI-REDUCTION 537 1975.416001 1647.827291 LO-REDUCTION 539 1938.296847 1647.827291 LO-REDUCTION 541 1920.351517 1647.827291 LO-REDUCTION 543 1863.598582 1647.827291 LO-REDUCTION 545 1823.345274 1647.827291 LO-REDUCTION 547 1805.893329 1647.827291 REFLECTION 549 1719.004875 1615.837836 LO-REDUCTION 551 1714.239381 1615.837836 HI-REDUCTION 553 1688.462421 1615.837836 HI-REDUCTION 555 1686.337018 1615.837836 LO-REDUCTION 557 1673.330446 1615.837836 EXTENSION 559 1656.750214 1588.652255 LO-REDUCTION 561 1647.827291 1588.652255 HI-REDUCTION 563 1646.757464 1588.652255 LO-REDUCTION 565 1628.067462 1588.652255 LO-REDUCTION 567 1624.094263 1588.652255 REFLECTION 569 1616.890353 1584.112321 LO-REDUCTION 571 1615.837836 1584.112321 REFLECTION 573 1608.665463 1579.619639 EXTENSION 575 1607.427211 1560.323465 LO-REDUCTION 577 1591.069049 1560.323465 LO-REDUCTION 579 1590.182366 1560.323465 LO-REDUCTION 581 1588.652255 1560.323465 LO-REDUCTION 583 1584.112321 1560.323465 HI-REDUCTION 585 1579.619639 1560.323465 LO-REDUCTION 587 1577.557780 1560.323465 EXTENSION 589 1570.350004 1534.924799 LO-REDUCTION 591 1569.087327 1534.924799 LO-REDUCTION 593 1563.716654 1534.924799 LO-REDUCTION 595 1562.815776 1534.924799 LO-REDUCTION 597 1561.542390 1534.924799 EXTENSION 599 1560.323465 1514.237911 EXTENSION 601 1553.047313 1492.170950 LO-REDUCTION 603 1548.754483 1492.170950 LO-REDUCTION 605 1541.711428 1492.170950 EXTENSION 607 1536.968568 1483.166008 EXTENSION 609 1534.924799 1475.865364 EXTENSION 611 1514.237911 1416.963542 LO-REDUCTION 613 1503.244145 1416.963542 LO-REDUCTION 615 1499.746373 1416.963542 LO-REDUCTION 617 1492.170950 1416.963542 HI-REDUCTION 619 1483.166008 1416.963542 EXTENSION 621 1475.865364 1395.645823 LO-REDUCTION 623 1456.240570 1395.645823 LO-REDUCTION 625 1449.880366 1395.645823 EXTENSION 627 1447.659177 1351.861281 LO-REDUCTION 629 1426.915163 1351.861281 LO-REDUCTION 631 1416.963542 1351.861281 LO-REDUCTION 633 1412.331042 1351.861281 LO-REDUCTION 635 1396.649284 1351.861281 LO-REDUCTION 637 1395.645823 1351.861281 REFLECTION 639 1374.999672 1351.859771 REFLECTION 641 1372.459597 1346.417030 REFLECTION 643 1368.304444 1345.626273 EXTENSION 645 1357.802008 1319.563568 LO-REDUCTION 647 1355.877828 1319.563568 LO-REDUCTION 649 1351.861281 1319.563568 HI-REDUCTION 651 1351.859771 1319.563568 EXTENSION 653 1346.417030 1300.656730 LO-REDUCTION 655 1345.626273 1300.656730 EXTENSION 657 1336.783063 1265.557202 LO-REDUCTION 659 1335.717223 1265.557202 EXTENSION 661 1320.427270 1237.910994 LO-REDUCTION 663 1319.563568 1237.910994 EXTENSION 665 1303.036273 1207.932243 EXTENSION 667 1300.656730 1172.460931 LO-REDUCTION 669 1284.753828 1172.460931 EXTENSION 671 1265.557202 1151.593162 REFLECTION 673 1248.113424 1144.650974 REFLECTION 675 1237.910994 1139.635655 LO-REDUCTION 677 1207.932243 1139.635655 LO-REDUCTION 679 1179.253316 1137.508782 LO-REDUCTION 681 1172.460931 1137.508782 LO-REDUCTION 683 1151.593162 1137.508782 REFLECTION 685 1145.064462 1136.343641 LO-REDUCTION 687 1144.650974 1136.343641 HI-REDUCTION 689 1144.513776 1136.343641 LO-REDUCTION 691 1144.300348 1136.343641 LO-REDUCTION 693 1139.635655 1136.343641 HI-REDUCTION 695 1137.896960 1136.343641 REFLECTION 697 1137.508782 1136.123295 REFLECTION 699 1137.445475 1135.360052 LO-REDUCTION 701 1137.083167 1135.303084 LO-REDUCTION 703 1136.640411 1135.125282 LO-REDUCTION 705 1136.611019 1134.923517 HI-REDUCTION 707 1136.343641 1134.923517 LO-REDUCTION 709 1136.123295 1134.923517 HI-REDUCTION 711 1135.360052 1134.923517 HI-REDUCTION 713 1135.303084 1134.900330 HI-REDUCTION 715 1135.176744 1134.875873 REFLECTION 717 1135.125282 1134.790280 LO-REDUCTION 719 1135.039568 1134.787099 HI-REDUCTION 721 1135.018855 1134.774858 LO-REDUCTION 723 1134.923517 1134.764351 HI-REDUCTION 725 1134.900330 1134.761159 LO-REDUCTION 727 1134.875873 1134.709734 LO-REDUCTION 729 1134.790280 1134.709734 LO-REDUCTION 731 1134.787099 1134.707387 LO-REDUCTION 733 1134.774858 1134.694921 HI-REDUCTION 735 1134.764351 1134.694921 HI-REDUCTION 737 1134.761159 1134.694921 REFLECTION 739 1134.729969 1134.669616 HI-REDUCTION 741 1134.709734 1134.669616 HI-REDUCTION 743 1134.707387 1134.669616 LO-REDUCTION 745 1134.699665 1134.666660 HI-REDUCTION 747 1134.698842 1134.666660 LO-REDUCTION 749 1134.694921 1134.663843 LO-REDUCTION 751 1134.687157 1134.663843 REFLECTION 753 1134.680710 1134.660228 EXTENSION 755 1134.676271 1134.619078 LO-REDUCTION 757 1134.670516 1134.619078 LO-REDUCTION 759 1134.669616 1134.619078 HI-REDUCTION 761 1134.666660 1134.619078 LO-REDUCTION 763 1134.663843 1134.619078 LO-REDUCTION 765 1134.660228 1134.619078 EXTENSION 767 1134.647096 1134.581237 LO-REDUCTION 769 1134.642666 1134.581237 LO-REDUCTION 771 1134.635834 1134.581237 EXTENSION 773 1134.632582 1134.561441 EXTENSION 775 1134.625835 1134.540023 LO-REDUCTION 777 1134.619078 1134.540023 EXTENSION 779 1134.595019 1134.473523 LO-REDUCTION 781 1134.594715 1134.473523 LO-REDUCTION 783 1134.581237 1134.473523 EXTENSION 785 1134.561441 1134.435120 LO-REDUCTION 787 1134.546221 1134.435120 EXTENSION 789 1134.540023 1134.420880 LO-REDUCTION 791 1134.505060 1134.420880 LO-REDUCTION 793 1134.494007 1134.420880 EXTENSION 795 1134.477079 1134.359953 LO-REDUCTION 797 1134.473523 1134.359953 LO-REDUCTION 799 1134.435120 1134.359953 LO-REDUCTION 801 1134.427335 1134.359953 LO-REDUCTION 803 1134.424762 1134.359953 LO-REDUCTION 805 1134.420880 1134.359953 LO-REDUCTION 807 1134.410191 1134.359953 EXTENSION 809 1134.400218 1134.333270 LO-REDUCTION 811 1134.398536 1134.333270 LO-REDUCTION 813 1134.383500 1134.333270 LO-REDUCTION 815 1134.377403 1134.333270 EXTENSION 817 1134.371625 1134.279266 LO-REDUCTION 819 1134.359953 1134.279266 LO-REDUCTION 821 1134.340869 1134.279266 LO-REDUCTION 823 1134.339025 1134.279266 LO-REDUCTION 825 1134.337215 1134.279266 EXTENSION 827 1134.333270 1134.256034 LO-REDUCTION 829 1134.300333 1134.256034 EXTENSION 831 1134.293160 1134.228896 EXTENSION 833 1134.292174 1134.206121 EXTENSION 835 1134.282502 1134.163885 LO-REDUCTION 837 1134.279266 1134.163885 LO-REDUCTION 839 1134.256674 1134.163885 LO-REDUCTION 841 1134.256034 1134.163885 EXTENSION 843 1134.228896 1134.108154 LO-REDUCTION 845 1134.206121 1134.108154 LO-REDUCTION 847 1134.201287 1134.108154 EXTENSION 849 1134.190136 1134.029647 LO-REDUCTION 851 1134.177044 1134.029647 LO-REDUCTION 853 1134.163885 1134.029647 LO-REDUCTION 855 1134.146907 1134.029647 EXTENSION 857 1134.124670 1134.006163 LO-REDUCTION 859 1134.108154 1134.006163 EXTENSION 861 1134.079046 1133.885003 LO-REDUCTION 863 1134.052496 1133.885003 LO-REDUCTION 865 1134.047491 1133.885003 EXTENSION 867 1134.029647 1133.808712 LO-REDUCTION 869 1134.007647 1133.808712 LO-REDUCTION 871 1134.006163 1133.808712 EXTENSION 873 1133.932441 1133.634358 LO-REDUCTION 875 1133.915611 1133.634358 LO-REDUCTION 877 1133.911032 1133.634358 EXTENSION 879 1133.885003 1133.553320 EXTENSION 881 1133.843916 1133.471948 EXTENSION 883 1133.808712 1133.270246 EXTENSION 885 1133.723090 1133.006633 LO-REDUCTION 887 1133.659845 1133.006633 LO-REDUCTION 889 1133.634358 1133.006633 LO-REDUCTION 891 1133.553320 1133.006633 EXTENSION 893 1133.471948 1132.880986 EXTENSION 895 1133.318145 1132.822046 EXTENSION 897 1133.270246 1132.362764 LO-REDUCTION 899 1133.233945 1132.362764 LO-REDUCTION 901 1133.175861 1132.362764 EXTENSION 903 1133.006633 1132.273535 HI-REDUCTION 905 1132.880986 1132.273535 EXTENSION 907 1132.822046 1131.947413 EXTENSION 909 1132.796102 1131.355856 LO-REDUCTION 911 1132.606611 1131.355856 LO-REDUCTION 913 1132.396144 1131.355856 LO-REDUCTION 915 1132.362764 1131.355856 LO-REDUCTION 917 1132.273535 1131.355856 EXTENSION 919 1131.947413 1130.584904 LO-REDUCTION 921 1131.709672 1130.584904 LO-REDUCTION 923 1131.616013 1130.584904 LO-REDUCTION 925 1131.612787 1130.584904 EXTENSION 927 1131.545709 1130.153166 LO-REDUCTION 929 1131.355856 1130.153166 LO-REDUCTION 931 1130.961597 1130.153166 EXTENSION 933 1130.797580 1129.521900 LO-REDUCTION 935 1130.739481 1129.521900 LO-REDUCTION 937 1130.584904 1129.521900 LO-REDUCTION 939 1130.311130 1129.521900 LO-REDUCTION 941 1130.298305 1129.521900 LO-REDUCTION 943 1130.153166 1129.521900 EXTENSION 945 1129.957309 1129.137128 LO-REDUCTION 947 1129.808033 1129.137128 LO-REDUCTION 949 1129.776329 1129.137128 LO-REDUCTION 951 1129.656977 1129.137128 EXTENSION 953 1129.573229 1128.881090 LO-REDUCTION 955 1129.521900 1128.881090 LO-REDUCTION 957 1129.183771 1128.881090 EXTENSION 959 1129.156728 1128.791940 LO-REDUCTION 961 1129.152120 1128.791940 LO-REDUCTION 963 1129.137128 1128.786706 REFLECTION 965 1128.947347 1128.775787 EXTENSION 967 1128.894137 1128.397226 LO-REDUCTION 969 1128.881090 1128.397226 LO-REDUCTION 971 1128.823434 1128.397226 EXTENSION 973 1128.791940 1128.276105 LO-REDUCTION 975 1128.786706 1128.276105 LO-REDUCTION 977 1128.775787 1128.276105 EXTENSION 979 1128.601578 1127.760485 LO-REDUCTION 981 1128.572586 1127.760485 LO-REDUCTION 983 1128.538665 1127.760485 EXTENSION 985 1128.404162 1127.384810 LO-REDUCTION 987 1128.397226 1127.384810 LO-REDUCTION 989 1128.276105 1127.384810 EXTENSION 991 1128.188936 1127.226693 EXTENSION 993 1127.939928 1126.300374 LO-REDUCTION 995 1127.760485 1126.300374 LO-REDUCTION 997 1127.550491 1126.300374 EXTENSION 999 1127.442610 1125.572620 LO-REDUCTION 1001 1127.384810 1125.572620 LO-REDUCTION 1003 1127.226693 1125.572620 EXTENSION 1005 1126.506775 1124.475433 LO-REDUCTION 1007 1126.351569 1124.475433 LO-REDUCTION 1009 1126.300374 1124.475433 LO-REDUCTION 1011 1126.114819 1124.475433 LO-REDUCTION 1013 1125.702062 1124.475433 LO-REDUCTION 1015 1125.572620 1124.475433 REFLECTION 1017 1125.005074 1124.471609 EXTENSION 1019 1124.869698 1123.801604 LO-REDUCTION 1021 1124.646753 1123.801604 LO-REDUCTION 1023 1124.528303 1123.801604 LO-REDUCTION 1025 1124.512564 1123.801604 LO-REDUCTION 1027 1124.475433 1123.801604 HI-REDUCTION 1029 1124.471609 1123.801604 LO-REDUCTION 1031 1124.317917 1123.801604 LO-REDUCTION 1033 1124.302587 1123.801604 EXTENSION 1035 1124.203866 1123.498131 LO-REDUCTION 1037 1124.131139 1123.498131 EXTENSION 1039 1124.085823 1123.249412 EXTENSION 1041 1124.046131 1122.892335 LO-REDUCTION 1043 1123.842332 1122.892335 EXTENSION 1045 1123.801604 1122.386518 LO-REDUCTION 1047 1123.775224 1122.386518 EXTENSION 1049 1123.498131 1121.479822 LO-REDUCTION 1051 1123.249412 1121.479822 EXTENSION 1053 1123.042147 1120.964767 EXTENSION 1055 1122.892335 1119.726442 LO-REDUCTION 1057 1122.465119 1119.726442 EXTENSION 1059 1122.386518 1118.526425 LO-REDUCTION 1061 1121.734891 1118.526425 EXTENSION 1063 1121.479822 1117.463519 LO-REDUCTION 1065 1120.964767 1117.463519 EXTENSION 1067 1120.201777 1116.646677 EXTENSION 1069 1119.726442 1115.858090 LO-REDUCTION 1071 1118.680485 1115.858090 REFLECTION 1073 1118.526425 1115.758837 LO-REDUCTION 1075 1117.693870 1115.758837 LO-REDUCTION 1077 1117.463519 1115.758837 LO-REDUCTION 1079 1116.646677 1115.758837 LO-REDUCTION 1081 1116.360563 1115.758837 LO-REDUCTION 1083 1116.280683 1115.758837 REFLECTION 1085 1116.267162 1115.682324 REFLECTION 1087 1115.922658 1115.668265 HI-REDUCTION 1089 1115.905732 1115.668265 EXTENSION 1091 1115.878553 1115.396746 LO-REDUCTION 1093 1115.858090 1115.396746 LO-REDUCTION 1095 1115.758837 1115.396746 LO-REDUCTION 1097 1115.729874 1115.396746 LO-REDUCTION 1099 1115.682324 1115.396746 REFLECTION 1101 1115.668265 1115.296885 REFLECTION 1103 1115.534482 1115.201957 LO-REDUCTION 1105 1115.422885 1115.201957 LO-REDUCTION 1107 1115.408957 1115.201957 LO-REDUCTION 1109 1115.402899 1115.201957 LO-REDUCTION 1111 1115.396746 1115.201957 REFLECTION 1113 1115.296885 1115.140268 HI-REDUCTION 1115 1115.281631 1115.140268 LO-REDUCTION 1117 1115.268173 1115.140268 REFLECTION 1119 1115.257235 1115.139601 EXTENSION 1121 1115.220661 1115.063333 LO-REDUCTION 1123 1115.219322 1115.063333 LO-REDUCTION 1125 1115.201957 1115.063333 REFLECTION 1127 1115.176954 1115.051538 EXTENSION 1129 1115.140268 1114.905163 LO-REDUCTION 1131 1115.139601 1114.905163 LO-REDUCTION 1133 1115.109714 1114.905163 LO-REDUCTION 1135 1115.101368 1114.905163 EXTENSION 1137 1115.063333 1114.878012 REFLECTION 1139 1115.051538 1114.860950 EXTENSION 1141 1115.013025 1114.755490 LO-REDUCTION 1143 1114.958662 1114.755490 LO-REDUCTION 1145 1114.921700 1114.755490 LO-REDUCTION 1147 1114.905163 1114.755490 LO-REDUCTION 1149 1114.878012 1114.755490 EXTENSION 1151 1114.860950 1114.630647 LO-REDUCTION 1153 1114.795734 1114.630647 LO-REDUCTION 1155 1114.792240 1114.630647 LO-REDUCTION 1157 1114.771378 1114.630647 LO-REDUCTION 1159 1114.770379 1114.630647 LO-REDUCTION 1161 1114.755490 1114.630647 REFLECTION 1163 1114.717513 1114.628657 EXTENSION 1165 1114.678534 1114.529310 LO-REDUCTION 1167 1114.657294 1114.529310 LO-REDUCTION 1169 1114.648613 1114.529310 LO-REDUCTION 1171 1114.643401 1114.529310 LO-REDUCTION 1173 1114.630647 1114.529310 REFLECTION 1175 1114.628657 1114.527769 REFLECTION 1177 1114.593662 1114.518980 EXTENSION 1179 1114.562972 1114.459125 LO-REDUCTION 1181 1114.559562 1114.459125 LO-REDUCTION 1183 1114.555667 1114.459125 EXTENSION 1185 1114.529310 1114.451448 LO-REDUCTION 1187 1114.527769 1114.451448 LO-REDUCTION 1189 1114.518980 1114.451448 REFLECTION 1191 1114.500006 1114.434577 EXTENSION 1193 1114.484523 1114.401205 LO-REDUCTION 1195 1114.464943 1114.401205 LO-REDUCTION 1197 1114.459125 1114.401205 HI-REDUCTION 1199 1114.457634 1114.401205 LO-REDUCTION 1201 1114.451448 1114.401205 LO-REDUCTION 1203 1114.434577 1114.401205 EXTENSION 1205 1114.429067 1114.383715 LO-REDUCTION 1207 1114.424552 1114.383715 LO-REDUCTION 1209 1114.414199 1114.383715 LO-REDUCTION 1211 1114.413063 1114.383715 LO-REDUCTION 1213 1114.410441 1114.383715 LO-REDUCTION 1215 1114.406845 1114.383715 EXTENSION 1217 1114.401205 1114.360781 LO-REDUCTION 1219 1114.394889 1114.360781 LO-REDUCTION 1221 1114.393311 1114.360781 LO-REDUCTION 1223 1114.390793 1114.360781 EXTENSION 1225 1114.384982 1114.337347 LO-REDUCTION 1227 1114.383715 1114.337347 LO-REDUCTION 1229 1114.378560 1114.337347 EXTENSION 1231 1114.372778 1114.320535 LO-REDUCTION 1233 1114.362912 1114.320535 LO-REDUCTION 1235 1114.362316 1114.320535 LO-REDUCTION 1237 1114.360781 1114.320535 EXTENSION 1239 1114.348200 1114.295520 LO-REDUCTION 1241 1114.337347 1114.295520 LO-REDUCTION 1243 1114.327657 1114.295520 REFLECTION 1245 1114.322141 1114.295386 LO-REDUCTION 1247 1114.321219 1114.295386 REFLECTION 1249 1114.320535 1114.287898 Exiting from Nelder Mead minimizer 1251 function evaluations used  The fraction of cases that is predicted to occur in the child-bearing age-bracket (say, 15-40 years of age) is the joint probability of not being infected by age 15 and the probability of being infected in the 15-40 age range. $exp(-\int_0^{15} \phi (a) \, da) (1-exp(-\int_{15}^{40} \phi (a) \, da))$ We can predict this fraction from the spline model. (exp(-integrate(tmpfn,0,15,dl=dl)$value))*(1-
exp(-integrate(tmpfn,15,40,dl=dl)$value))  0.0896859073784746 Thus, with the current pattern of circulation just over$9\%$of the cases are predicted to occur in the at-risk age group. Let us ask how this fraction will change with a flat$50\%$reduction in FoI. redn=0.5 (exp(-redn*integrate(tmpfn,0,15,dl= dl)$value))*(1-exp(-redn*integrate(tmpfn,
z=matrix(mossong$contact.rate, ncol=30, nrow=30) image(x=x, y=y, z=z, xlab="Contactor", ylab="Contactee", col=gray((12:32)/32)) contour(x=x, y=y, z=z, add=TRUE)  <th scope=col>contactor</th><th scope=col>contactee</th><th scope=col>contact.rate</th> 1 1 120.37234 2 1 33.45833 3 1 23.13380 4 1 24.33333 5 1 29.00662 6 1 14.50331 The reported contact rates are not symmetrical – which a WAIFW matrix will be – because of age-specific biases in diary entry rates as well as the age-profile of the contactors versus contactees. Before we ‘symmetrize’ the matrix we, look at the reported marginal contact rate for each age group . Most contacts are among same-aged individuals and school-age children have the greatest number of contacts. We do, however, also see off-diagonal ridges resulting from for example parent-offspring or teacher-pupil interactions. plot(apply(z,1,mean)~x, ylab="Total contact rate", xlab="Age")  The symmetrized contact rate matrix is an estimate of the ’WAIFW’-matrix. ## Advanced: RAS model discussed the importance of age-structured mixing when modeling infectious disease dynamics. extended this model to the ‘realistic age-structured (RAS) model’ which in its full elaboration is an age-structured compartmental model with discrete aging of each birth cohort (at the beginning of each school year) and seasonality in transmission. Seasonality is the topic of chapter 4. We can incorporate the POLYMOD contact matrix in a simpler age-structured model. We will make the simplifying assumptions that individuals age exponentially with rates set such that they will on average spend the right amount in each age-bracket. This allows us to formulate the model using chains of ordinary differential equations. The upper-age cut-offs and age-progression rates for the$n=30$age-categories are x and a = c(1/diff(x), 0)  We can in principle use the raw symmetrized WAIFW matrix in our model, but we will use a thin-plate spline smoothed matrix using the Tps-function in the fields-package. The smoothing protocol also allow interpolation to use different age-brackets for the model than used in the contact survey whenever necessary. require("fields") n = length(x) z2 = (z + t(z))/2 z3 = as.vector(z2) xy = data.frame(x = rep(x[1:n], n), y = rep(y[1:n], each = n)) polysmooth = Tps(xy, z3, df = 100) surface(polysmooth, xlab = "", ylab = "", col=gray((12:32)/32))  For our age-structured SIR model we first normalize the WAIFW matrix: W = matrix(polysmooth$fitted.values[,
1]/mean(polysmooth$fitted.values), nrow = n)  The age-specific force-of infection is$\vec{\phi} = \beta \vec{W} \vec{I} / N$. The age-structured SIR model is thus (in log-coordinates)[6]: siragemod = function(t, logx, params){ n=length(params$a)
x = exp(logx)
S = x[1:n]
I = x[(n+1):(2*n)]
R = x[(2*n+1):(3*n)]
with(as.list(params), {
phi = (beta*W%*%I)/N
dS = c(mu,rep(0,n-1)) - (phi+a)*S +
c(0,a[1:n-1]*S[1:n-1])*(1-p) - mu*S
dI = phi*S + c(0,a[1:n-1]*I[1:n-1]) -
(gamma+a)*I - mu*I
dR =  c(0,a[1:n-1]*S[1:n-1])*p +
c(0,a[1:n-1]*R[1:n-1]) + gamma*I -
a*R - mu*R
res = c(dS/S, dI/I, dR/R)
list((res))
})
}


where S, I and R are vectors of length $n$, $\phi$ is the age-specific force of infection predicted by the WAIFW matrix and $p$ is a vector of length $n$ that allows for age-specific vaccination rates (we will assume no vaccination). The $a$-vector sets appropriate aging rates if age-groups varies in duration. We use the following parameters and initial conditions:

p.pre = rep(0,n)
pars.pre = list(N = 1, gamma = 365/14, mu = 0.02, sigma = 0.2,
beta = 100, W = W,p = p.pre, a = a)
ystart = log(c(S = rep(0.099/n,n), I = rep(0.001/n,n),
R = rep(0.9/n,n)))


and integrate to plot the age-specific I-dynamics and equilibrium age-specific prevalence for the polymod matrix.

times = seq(0, 500, by = 14/365)
#Polymod mixing
out=as.data.frame(ode(ystart, times=times,
func = siragemod, parms = pars.pre))
par(mfrow = c(1,2)) #Room for side-by-side plots
#Time series
matplot(times, exp(out[,32:61]), type = "l", xlab = "Time",
ylab = "Prevalence", xlim = c(50, 90), ylim = c(0, 0.0005))
#Final age-prevalence curve
plot(x, t(exp(out[13036, 32:61])*a), ylab = "Prevalence",
xlab = "Age", ylim = c(0, 4E-5))
#Homogenous mixing:
pars.pre\$W = matrix(1, ncol = 30, nrow = 30)
out2=as.data.frame(ode(ystart, times=times,
func=siragemod, parms=pars.pre))
points(x, t(exp(out2[13036,32:61])*a), col=2, pch="*")