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Section 1: Applying modelling techniques to
analyse(seroprevalence) data
page 1 of 78
OVERVIEW
This session discusses how we can use data on the prevalence of
previous infection (i.e.serological data) to estimate important
epidemiological statistics, some of which are usedin models. These
include the average force of infection, the basic reproduction
number,the average age at infection, and the number of new
infections that might occur in differentage groups per unit
time.
OBJECTIVES
By the end of this session, you should:
Be able to analyse age-specific seroprevalence data for
immunising and non-immunising infections to estimate the average
force of infection;Be able to calculate important epidemiological
statistics (i.e. the average age atinfection, the proportion
susceptible, the basic reproduction number) using estimatesof the
average force of infection;Know how to use graphical and model-free
methods to estimate the force ofinfection;Be familiar with the
basic approaches for fitting catalytic models to
seroprevalencedata.
This session comprises two parts and will take 2-5 hours to
complete .
Part 1 (1-2 hours) describes how we can use seroprevalence data
to estimate the force ofinfection and other useful epidemiological
statistics. Part 2 (1-3 hours) consists of anexercise in Excel,
during which you will estimate the force of infection and these
statisticsfrom age-specific seroprevalence data on rubella from the
UK and China.
NOTE
This module will use the Solver tool in Excel. Check to see if
it is installed already byopening Excel and clicking on the Data
tab at the top of the window. If it is installed, youshould see the
following button on the top right of the window. If it's not
there, click here for instructions on how to install it.
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Section 2: Introduction
page 2 of 78
So far on this module, we have developed a simple model of the
transmission dynamics ofan immunising infection with the following
structure:
This model assumes that individuals mix randomly and uses the
following parameters:
the pre-infectious period;the infectious period;the birth
rate;the death rate;the rate at which two specific individuals come
into effective contact per unit time ().
The first four parameters in this list are usually known for
most infections and populations.However, the parameter is poorly
understood and difficult to measure directly. We haveusually
calculated it from the basic reproduction number, R0, using the
following equation
:
=R0
ND
where N is the total population size and D is the duration of
infectiousness. During theexercises we have generally provided the
value for R0 for you to use in models. By now,you may have been
wondering how we can obtain the value for R0.
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2.1: Introduction
page 3 of 78 2.1
We can often estimate R0 for a given infection using data on the
age-specific proportion ofindividuals who have previously
experienced infection, such as those shown in Figure 1.
Figure 1: The observed proportion of individuals who were
positive for mumps and rubella antibodiesby age in England and
Wales. The sera were collected among unvaccinated individuals
during the1980s. Data were extracted from Farrington (1990)1 .
These show that the proportion of individuals who had antibodies
to mumps or rubellaincreased with age. In the absence of widespread
vaccination, individuals with antibodieshad probably previously
been infected. In addition, the proportion of individuals who
hadantibodies to mumps increased more rapidly with age than that
for rubella, suggesting thatmumps was more infectious than was
rubella.
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2.2: Introduction
page 4 of 78 2.1 2.2
Using data such as those described on the previous page , we can
estimate severalimportant epidemiological statistics, in addition
to the basic reproduction number, namely:
The average force of infection;The average age at infection;The
herd immunity threshold;The number of new infections that might be
seen per unit time in different agegroups; andThe proportion of the
population that is susceptible.
Before discussing the methods for analysing these data, we first
discuss how these dataare related to predictions from the dynamic
model that we worked with in previoussessions.
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Section 3: The relationship between age-specific cross-sectional
seroprevalence data and long-term predictionsfrom a dynamic
model
page 5 of 78
Figure 2 shows predictions of the number of infectious
individuals over time obtained fromthe following measles model that
we worked with during the last two sessions.
Suppose we imagine following individuals who are born in year 80
in the model. Thisgroup is known as a birth cohort. Before
continuing, think about how you would expectthe proportion of
individuals in this cohort who have ever been infected (and
therefore, theproportion seropositive) to change over time as the
cohort ages. You might like to drawthis on a separate piece of
paper - you will need to sketch a graph with Age on the x-axisand
Proportion ever infected on the y-axis. You could also add a line
for Proportionsusceptible.
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Figure 2: Prediction of the number of individuals infectious
with measles, obtained using the model fromthe last two sessions.
If you wish to see the model, start up Berkeley Madonna and click
the followinglinks to open the equation or flowchart editor
versions of these files.
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3.1: The relationship between age-specific
cross-sectionalseroprevalence data and long-term predictions from
adynamic model
page 6 of 78 3.1
Figure 3 shows what the model discussed on the previous page
would predict if we wereto modify it to track individuals who are
born in a given year. In particular, it predicts thatthe proportion
of the cohort born in year 80 that has ever been infected increases
as thecohort ages. If you are interested, click here if you would
like to find out how we canchange the model to make these
predictions.
Figure 3: Predictions of the proportion of a cohort born in year
80 in the model discussed on page 5 that has ever been infected and
the proportion susceptible by age.
The models predictions shown in Figure 3 are consistent with
what we would expect.Typically, in a population, the proportion of
children who are susceptible to an immunisinginfection is greater
than that for adults, as children have had fewer years of exposure
tothe infection than adults.
Note that if the infection is endemic , these patterns should be
similar to those seen incross-sectional data, such as those in
Figure 1 .
In these predictions, the value for R0 is assumed to be 10.
Before continuing, think abouthow this plot will change if R0 is
reduced or increased to 7 and 13, respectively.
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3.2: The relationship between age-specific
cross-sectionalseroprevalence data and long-term predictions from
adynamic model
page 7 of 78 3.1 3.2
Figure 4 shows predictions of the proportion of the cohort in
the model that would haveever been infected as it ages, assuming
different values for R0. As might be expected, forindividuals of a
given age, e.g. 10 years, this proportion increases as the value
for R0, orthe infectiousness of the agent, increases. This suggests
that we can estimate R0 bystudying age-specific serological data
using appropriate methods.
Seroprevalence data areusually analysed using so-called
catalytic models toestimate the average forceof infection. This
averageforce of infection is thenused to calculate otherimportant
epidemiologicalstatistics such as theaverage age at infection,the
proportion of thepopulation that issusceptible, R0, the
herdimmunity threshold and thenumber of new infectionsthat would be
seen perunit time in different agegroups.
Figure 4: Model predictions of the age-specific proportion of
thecohort discussed on pages 5-6 that would have ever beeninfected
assuming different values for R0.
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Section 4: What do we mean by the average force ofinfection?
page 8 of 78
As discussed in previous sessions , the force of infection at
time t, (t), is the rate atwhich susceptible individuals are
infected. It is calculated using the following equation :
(t) = I(t)
where is the rate at which two specific individuals come into
effective contact per unittime, and I(t) is the number of
infectious individuals at time t. For an endemic infection,
theforce of infection changes over time, but on average remains
unchanged, as can be seen ifyou click on the "show" button
below.
We shall now consider how we can estimate the average force of
infection using cross-sectional data, such as those discussed on
page 3 .
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Section 5: Estimating the average force of infection
usingcatalytic models
page 9 of 78
We can estimate the average force of infection from data on the
prevalence of previousinfection by using so-called catalytic
modelling techniques.
Catalytic models are very similar to the SIS, SIR, SIRS models
that we introduced inMD01 , except that they do not explicitly
model transmission between individuals, i.e. theforce of infection
is not expressed in terms of a transmission parameter and the
numberof infectious individuals I(t).
Instead, individuals are assumed to become infected at a fixed
rate, which is either age-dependent or time-dependent.
If you are interested, click here for further information on the
historical origin of the term"catalytic model". Otherwise, you may
prefer to read this material once you havecompleted the
session.
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Section 6: Applying simple catalytic models to infection
data
page 10 of 78
In the simplest case, the proportion of individuals who have
ever been infected at differentages can be described using the
following model, known as a "simple catalytic model",which follows
individuals from birth until they become infected:
This assumes that individuals are infected at a constant annual
rate, , which isindependent of age and calendar year, that all
individuals are susceptible at birth, and thatthose susceptible and
those ever infected have similar mortality rates. The rates of
changein the proportion susceptible, s(a), and those (ever)
infected, z(a), with respect to age aregiven by the following
equations:
ds(a) =s(a) Equation 1da
dz(a) = s(a) Equation 2 da
You may recall that we discussed this model in MD02 . As
discussed in that session,Equation 1 can be solved to give the
following equation for the proportion susceptible(s(a)) at age
a:
s(a)=e-a Equation 3
Furthermore, because the proportion susceptible and the
proportion (ever) infected mustadd up to 1, we can obtain the
following equation for the proportion (ever) infected by agea:
z(a)=1 - e-a Equation 4
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6.1: Applying simple catalytic models to infection data
page 11 of 78 6.1
We will now show that this model leads to predictions of the
age-specific proportions ofindividuals that have ever been infected
that are consistent with the data discussedpreviously.
EXERCISE
1. Click here to open up the Excel file "catalytic model.xlsm".
The layout should besimilar to that below. Specifically, you should
see the following:
a) Cell F4 contains the average value for the annual force of
infection. b) The yellow cells represent the age midpoints and the
pink cells hold the
expected proportion (ever) infected, as calculated using the
Excelequivalent of the equation z(a)=1 e-a (Equation 4 ).
c) Figure 1 shows how predictions of the proportion (ever)
infected changeswith increasing age.
Q1.1
a) Click on cell F4 in the spreadsheet and change its value to
be 0.10 (peryear). How does this affect the proportion of
individuals who have everbeen infected?
b) What happens if you change the average annual force of
infection to be 0.2per year?
[The answer is on the next page]
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6.2: Applying simple catalytic models to infection data
page 12 of 78 6.1 6.2
As you vary the values for the force of infection, you should
notice that the model predictsthat the proportion of individuals
who have ever been infected by different ages is similarto the
patterns shown in Figure 6. Specifically, if the force of infection
is high (similar tothat for rubella in high transmission settings3
) the vast majority of individuals arepredicted to have been
infected by age 10 years.
Figure 6. Predictions of the age-specific proportions of
individuals (ever) infected obtained using thesimple catalytic
model, for different values for the force of infection.
These values are similar to those predicted by the transmission
model discussedpreviously (see Figure 4 ). If it is 1% per year
(similar to that for M tuberculosis in partsof Africa during the
1980s (Fine, et al (1999)4 ) less than 50% of adults are predicted
tohave been ever infected.
Q1.2 Based on the plot in Figure 6 and the data shown in Figure
1 , what was likely tohave been the average force of infection for
mumps and rubella during the 1980s inEngland and Wales?
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Section 7: Formal methods for estimating the force
ofinfection
page 13 of 78
As shown in the previous exercise, we can get a reasonably good
idea of what theaverage annual force of infection may have been
simply by comparing predictions from thecatalytic model against
observed data by eye.
However, we can improve our estimates by formally "fitting" our
model to the data,whereby the value of the force of infection is
varied until the smallest distance between themodel prediction and
the observed data (reflected by a goodness of fit statistic)
isobtained, as shown in Figure 7.
Figure 7. Illustration of the fitting process. Typically inthis
process, the value of a parameter is varied untilthe distance
between the model prediction and theobserved data (reflected by the
vertical lines) is assmall as possible.
There are many methods for fitting models to data. The most
widely known method is thatof "Least Squares", whereby we minimise
the statistic , where Oi
represents the value of the ith (observed) data point and Ei
represents the value predicted
by the model for the ith data point (i.e. the expected value).
Another statistic that might be
minimised when fitting models to data is the X2 statistic: .
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A method which is widely applied is that of "Maximum
Likelihood", which is equivalent tominimising the so-called
"log-likelihood" deviance. We will be using this method later in
thesession and we will discuss it further then.
On the next page, we will illustrate how you can fit catalytic
models in Excel.
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Answer
7.1: Formal methods for estimating the force of infection
page 14 of 78 7.1
EXERCISE CTD.
1. Return to the Excel file catalytic model.xlsm that you were
working with. Click here to open it if you have closed it by now.
Select columns A and F together, click
with the right mouse button and select the "Unhide" option"
2. Change the value of the force of infection in cell F4 to be
0.18 per year.
You should now see data on the age-specific proportions of
individuals who hadantibodies to mumps in the UK (Figure 1 )
plotted on the graph. Note that the yellowcells in columns B-D hold
data on the age-specific numbers of individuals who werepositive or
negative for mumps antibodies, and the observed proportion
positive.
Q1.3 From what you observe in the figure, do you think the true
force of infection is higheror lower than that currently
assumed?
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7.2: Formal methods for estimating the force of infection
page 15 of 78 7.1 7.2
We will now fit the model to the data.
3. Unhide column H by selecting columns G and I, clicking with
the right mouse buttonand selecting the "Unhide" option. You should
now see some grey cells in column H.
These hold equations which measure the distance of the model's
prediction of theproportion ever infected at each age from the
observed data, using the so-called "log-likelihood deviance". Do
not worry about this statistic for now; we will revisit it later in
thesession.
4. Unhide rows 6 and 7 by selecting rows 5 and 8 together,
clicking with the rightmouse button and selecting the "Unhide"
option. Your spreadsheet should resemblethe following:
You should now see some grey cells in rows 6-7.
Cell F6 holds the overall goodness of fit statistic, or the
log-likelihood deviance of themodel prediction from the data,
calculated as the sum of all the values in cells H25-H68(currently
364.6).
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7.3: Formal methods for estimating the force of infection
page 16 of 78 7.1 7.2 7.3
You should also see the "Click to fit the model" button which
has been set up to allowyou to fit the model to the data
automatically. Please do not click on it yet!
5. With the force of infection still set at 0.18 per year, write
down the current value forthe deviance (the "Goodness of fit" in
cell F6), and click on the "Click to fit the model"button.
If you obtain an error message saying that Excel cannot run the
macro, click on the optionsbutton above the formula bar and select
the Enable this content option, before clicking onOK.
You should find that the model starts looking for the best value
for the force of infection(we will discuss the mechanics of how
this is done later).
Once the best-fitting value has been found, you should get a
message saying that "Solverfound a solution. All constraints and
optimality conditions are satisfied".
6. Select the option to "Keep Solver Solution".
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7.4: Formal methods for estimating the force of infection
page 17 of 78 7.1 7.2 7.3 7.4
Your spreadsheet should now resemble the following. Note the
values for the force ofinfection and the Goodness of fit
statistic.
You should notice that the value for the deviance (the "Goodness
of fit" in cell F6) hasdecreased to 336.5, i.e. the fit of the
model to the data has improved.
You should find that the best-fitting force of infection is
19.76% per year. In fact, usingmethods discussed later in this
session, the 95% confidence interval for this estimate canbe
calculated to be 19.1-21.5%.
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Section 8: Reflections on catalytic models
page 18 of 78
The distinction between a catalytic model and a transmission
modelAt this point, it is useful to think about what makes
catalytic models different from thetransmission models that we
studied in previous sessions. In transmission models, theforce of
infection is expressed in terms of the number of infectious
individuals in the model,which changes over time, i.e. (t) = I(t).
Catalytic models aim to describe the data withoutexplicitly
modelling the mechanism (i.e. the contact between susceptible and
infectiousindividuals) through which transmission occurs. In other
words the force of infection istaken to be some value which is
independent of the size of other compartments in themodel.
Simplifications of the simple catalytic modelThe simple
catalytic model makes several simplifying assumptions. For example,
itassumes that all individuals are susceptible at birth and that
the force of infection is neitherage- nor time-dependent. The model
also makes assumptions about what the infectionmarker represents:
i.e. past infection, current infection, recent infection, etc., and
this willinfluence the interpretation of the force of infection and
the age-specific marker prevalenceprofiles. We will discuss methods
for modifying these assumptions later in this session.
We will now discuss how estimates of the average force of
infection can be applied toestimate several useful epidemiological
measures.
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Section 9: Applying estimates of the average force
ofinfection
page 19 of 78
Estimates of the average force of infection are typically
applied to estimate the followingepidemiological measures:
The average age at infection;The proportion susceptible;R0 and
the herd immunity threshold;The number of new infections in
different age groups per unit time.
On the next few pages, we shall illustrate how each of these
statistics can be estimatedfrom the average force of infection.
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Section 10: The average age at infection
page 20 of 78
The average age at infection, A, is a helpful summary measure
that can be used tocompare the "infectiousness" of different
infections, or of the same infection acrossdifferent populations
(see Anderson and May (1992)5 ). Low average ages at infection
aresuggestive of increased transmission. For example, the average
age of measles infectionin the USA (1955-8) was 5-6 years, as
compared with 2-3 years in Ghana (1960-8),suggesting that
transmission was more intense in Ghana than in the USA (Anderson
andMay (1992)5 ).
Changes in the average age at infection also provide insight
into whether an intervention(e.g. treatment of infectious persons)
has had any impact. The average age at infection isalso very
important in guiding vaccination policy. For example, if the
average age atinfection is 4 years, vaccinating children aged over
4 years will have little impact ontransmission.
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10.1: The average age at infection
page 21 of 78 10.1
The simplest method for calculating the average age at infection
is to calculate the medianage at infection, defined as the age by
which 50% of individuals have been infected. Thiscan be read off a
scatter plot of the data. For example, as shown in the figure below
whichis a replica of Figure 1 , 50% of individuals had antibodies
to mumps by about age 5years in England and Wales during the 1980s,
suggesting that the average age at mumpsinfection was about 5
years. The median age at infection for rubella was slightly
higherthan that for mumps, i.e. around 8 years.
Note that this method for calculating the median age at
infection implicitly assumes thateveryone is infected in their
lifetime, which is unrealistic for settings for which this does
notoccur.
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10.2: The average age at infection
page 22 of 78 10.1 10.2
We can also obtain an expression for the average age at
infection (A) by using thefollowing relationship between the
average time to an event and the average rate at whichit occurs
(see MD01 ).
average time to event = 1/(average rate at which it occurs)
Applying this relationship, we obtain the following
equation:
A1/
The relationship between the average age at infection and the
force of infection isapproximate, since the average age at
infection depends on how many individuals diebefore being infected
and hence on the mortality rate. It also assumes that the force
ofinfection is independent of age, and that individuals mix
randomly.
However, for most practical purposes, this equation provides a
reasonably good estimateof the average age at infection. Equations
for improving the estimates are discussed inAnderson and May(1992)5
. In general, these can be derived from the generalmathematical
formula for the average:
or
where (a) is the force of infection among individuals of age a,
and therefore (a)S(a) isthe number of new infections among
individuals of age a. Section 5.2.3.1 of therecommended course
text6 discusses the equations for the average age at infection
forpopulations with specific age distributions in further
detail.
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10.3: The average age at infection
page 23 of 78 10.1 10.2 10.3
Q1.3 As calculated on page 17 , the average annual force of
infection for mumps inEngland and Wales during the 1980s was about
19.8% per year.
What was the approximate average age at infection?
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Section 11: The average proportion susceptible: populationswith
a rectangular age distribution
page 24 of 78
We shall now consider how we can estimate the second statistic
that we mentioned onpage 19 , namely the proportion of the
population that is susceptible.
The proportion of the population that is susceptible depends
both on the rate at whichindividuals become infected and the age
distribution of the population.
If we assume that individuals mix randomly, then in populations
with "rectangular" or so-called Type I age distributions (similar
to those in industrialised countries today - seebelow), in which it
can be assumed that everybody lives only until age L, the
proportionsusceptible is related to the average age at infection A
through the following expression:
s A/L
We can obtain this formula using the argument that in a
population with a rectangular agedistribution, on average everyone
is susceptible until an average age A and everyoneabove that age is
immune. As illustrated in Figure 8a, the proportion susceptible is
thengiven by the area of the shaded box (i.e. A) divided by the sum
of the shaded and whiteareas (i.e. L).
Figure 8: a) Illustration of the relationship between the
proportion of the population that is susceptibleand the life
expectancy in a population with a rectangular age distribution. The
shaded area reflects theproportion of the population that is
susceptible. b) Population in England and Wales, 1991. Datasource:
Office for National Statistics. Adapted from Fine PEM (1993)4
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11.1: The average proportion susceptible: populations with atype
II (exponential) age distribution
page 25 of 78 11.1
When the age distribution follows a so-called "type II" pattern
(i.e. it is exponential, similarto that in "developing" countries -
see figure 9 below), the proportion susceptible is givenby the
expression:
s 1 1+L/A
or equivalently,
s AA+L
The expression assumes that individuals mix randomly and
therefore that the force ofinfection is the same for all age
groups. It also assumes that if individuals live longenough, they
will be infected. Unfortunately, there is no intuitive explanation
for thisequation. However, you can see its derivation in section
5.2.3.2 and Appendix A.3.1 of therecommended text6 .
Note that if a population has an exponential age distribution,
then the mortality rate isassumed to be constant (i.e. it is
identical for all individuals). You do not need to worryabout the
derivation of this result. For further reading, please refer to
section 3.5 of therecommended text6 for further details.
Figure 9: a) Illustration of how the proportion of the
population that is susceptible changes with age in apopulation with
an exponential age distribution. The shaded area reflects the
proportion of the
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population that is susceptible. b) Population in Malawi, 1998.
Data source: National Statistics Office,Malawi. Adapted from Fine
PEM (1993)4
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Answer
11.2: The average proportion susceptible: Exercise
page 26 of 78 11.1 11.2
As calculated on page 23 , the average age at mumps infection in
England and Walesduring the 1980s was about 5 years.
Q1.4
a) Assuming a life expectancy (L) of 70 years, that the age
distribution is rectangular andthat individuals mix randomly, what
was the average proportion of the population that
wassusceptible?
b) Calculate what proportion of the population would have been
susceptible if thepopulation had an exponential age distribution
and if the average life expectancy had been70 years.
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Section 12: The basic reproduction number, R0
page 27 of 78
We shall now consider how we can estimate the third statistic
that we mentioned on page19 , namely the basic reproduction number
(R0).
If we assume that individuals mix randomly, so that the force of
infection is identical for allage groups, we can calculate the
basic reproduction number R0 of an endemic infection asthe
reciprocal of the proportion susceptible:
R0 = 1/s Equation 6
We can obtain this equation by using the relationship that we
discussed in MD03 between the net reproduction number, R0 and the
proportion of the population that issusceptible. As we saw in that
session, the net reproduction number, Rn, is equal to R0s,and for
an endemic infection, Rn= 1. After rearranging the expression R0s =
1, we obtainthe result R0 = 1/s.
We can apply equation 6 to obtain equations for R0 for
populations with either rectangularor exponential age
distributions.
For example, as shown on the previous page , for a population
with a rectangular agedistribution,
s A/L
Substituting this expression for s into Equation 6 leads to the
expression:
R0 1/ (A/L) L/A Equation 7
If we then use the following approximation A 1/ for the average
age at infection (seepage 22 ) in this expression, we obtain the
following expression for R0: R0 L.
In contrast, if the population has an approximately exponential
age distribution, then usingthe equation for the average proportion
of the population that is susceptible that wediscussed on page 25 ,
, we obtain the following equation for R0.
R0 1 + L/A Equation 8
It can be shown (see references below) that if the age
distribution is approximately
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exponential, then R0 is related to the average force of
infection and the life expectancythrough the following
equation:
R0 = 1 + L
The mathematical details for why this equation has an equals (=)
sign rather than anapproximately equals () sign can be found in
sections 5.2.3.2 and 5.2.3.3 in therecommended course text6 , and
the Appendices mentioned in those sections.
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Answer
12.1: The basic reproduction number, R0
page 28 of 78 12.1
Q1.5 The average proportion of the population in England and
Wales that was susceptibleto mumps during the 1980s was 0.07 (see
Q1.4 ). Use an appropriate expression toestimate R0.
Click here if you wish to remind yourself of the equations for
R0.
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12.2: The basic reproduction number, R0
page 29 of 78 12.1 12.2
The expressions R0 L/A and R0 1+ L/A have been used to estimate
the basicreproduction numbers for many different infections. Some
of these are summarised inTable 1.
Infection Location Timeperiod
R0
Measles Cirencester,England
1947-50 13-14
England and Wales 1950-68 16-18
Kansas, USA 1918-21 5-6
Ghana 1960-8 14-15
Eastern Nigeria 1960-8 16-17
Pertussis England and Wales 1944-78 16-18
Ontario, Canada 1912-13 10-11
Chickenpox Maryland, USA 1913-17 7-8
Baltimore, USA 1943 10-11
Diphtheria New York, USA 1918-19 4-5
Maryland, USA 1908-17 4-5
Scarlet fever Maryland, USA 1908-17 7-8
New York, USA 1918-17 5-6
Pennsylvania, USA 1910-16 6-7
Mumps Baltimore, USA 1943 7-8
England and Wales 1960-80 11-14
Rubella England and Wales 1960-70 6-7
Poland 1970-7 11-12
Gambia 1976 15-16
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Answer
Poliomyelitis USA 1955 5-6
Netherlands 1960 6-7
Table 1. Summary of estimated values for the basic reproduction
number for different infections.Reproduced from Anderson and May
(1992)5 .
You will notice that R0 for measles was much higher in the UK
(1950-1968) than in Kansasin 1918.
Q1.6 Why do you think this was the case?
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Section 13: The (age-specific) number of new infections perunit
time
page 30 of 78
We shall now briefly consider how we can estimate the fourth
statistic that we mentionedon page 19 , namely the (age-specific)
number of new infections per unit time.
The force of infection can be used to predict the age-specific
proportion of individuals whoare susceptible to infection, and
subsequently, the number of new infections which mightoccur in
different age groups per unit time. This is especially relevant if
infection at aspecific age is associated with adverse outcomes.
For example, rubella infection during pregnancy can result in
the child being born withCongenital Rubella Syndrome; infection
with the polio virus during adulthood is associatedwith an
increased risk of paralytic polio; the risk of developing measles
encephalitisdepends on the age at measles infection. See Anderson
and May (1992)5 for furtherdetails.
In general, as seen in previous sessions (MD01 ) the number of
new infections per unittime in a population is given by the
expression:
(t)S(t)
where S(t) is the number of susceptible individuals at time t
and (t) is the force of infectionat time t. Adapting this
expression, we obtain the following expression for the number ofnew
infections per unit time among individuals of age a:
(a)S(a)
We will apply these expressions extensively in MD05. To see an
example of thiscalculation in the meantime, you can try Exercise
5.1 of the recommended course text6 ,which provides practice in
calculating the number of new infections per unit time. Solutionsto
the exercises are provided on the books
website:www.anintroductiontoinfectiousdiseasemodelling.com .
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13.1: Recap: Why do we need the average force of infection?
page 31 of 78 13.1
At this point, it is useful to summarise what we have covered so
far.
We have shown that the average force of infection of a given
infection can be estimated byfitting a simple catalytic model to
age-specific serological data.
In turn, the average force of infection can be used to estimate
several important statistics,namely the average age at infection,
the average proportion of the population that issusceptible, R0,
and the age-specific number of new infections occurring per unit
time. Examples of these calculations are provided below:
To estimate the average age at infection:
Applying the formula A 1/ using the force of infection, ,
estimated for mumps, weobtain the following estimate for the
average age at infection:
A 1/0.198 5 years
To estimate the average proportion susceptible:
Assuming that life expectancy (L) = 70 years, that the
population has a rectangular agedistribution and substituting for A
5 years into the equation s A/L , we obtain thefollowing for the
average proportion of the population that is susceptible:
s A/L 5/70 0.07
To estimate R0:
Substituting for s 0.07 into the equation R0= 1/s , we obtain
the following for R0:
R0= 1/s, which implies that R0 1/0.07 14
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Refining estimates of the force of infection the next steps
page 32 of 78
We have now seen how estimates of the average force of infection
can be applied. Weshall next consider how we can refine estimates
to the force of infection to take account ofthe following:
a) The presence of maternal antibodies during the first few
months of life.
b) Differences in the force of infection between different age
groups.
c) Changes in the force of infection over time.
d) Non-permanent immunity.
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Section 14: Refining estimates of the force of
infection:Maternally-derived immunity
page 33 of 78
The assumption that individuals are susceptible from birth may
result in a poor fit of thesimple catalytic model to the
seroprevalence data for youngest age groups, as it does notaccount
for any maternally-derived immunity that they may have.
It also means that the force of infection estimated by fitting a
simple catalytic model to thedata may underestimate the true force
of infection, since the model overestimates the totalperson years
of time that individuals are actually at risk of infection.
For example, considering 10 year olds, a simple catalytic model
would assume thatindividuals could have been infected during the
previous 10 years, whereas in reality theywould only have been
infected during the previous 9.5 years, if maternal immunity lasts
for6 months.
As we shall show later, in order to match the seroprevalence
data, the best-fitting force ofinfection estimated using a simple
catalytic model does not need to be as high as thatrequired if it
is assumed that individuals are susceptible a few months after
birth.
The models also assume that all individuals are born with
maternal immunity. This is areasonable assumption to make for
endemic infections, for which most women are likely tobe
immune.
The duration of maternally-derived immunity relative to the
force of infection hasimplications for designing vaccination
strategies. For example, vaccinating at too young anage, when
infants are still protected by maternal antibodies, means that many
doses ofvaccine are wasted. Vaccinating individuals when they are
older in a population with a highforce of infection may have little
effect on reducing morbidity, as many children will havealready
been infected.
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14.1: Refining estimates of the force of infection:
Maternally-derived immunity
page 34 of 78 14.1
In order to incorporate maternally-derived immunity, the simple
catalytic model needs to beadapted to track individuals from birth
as they lose their maternal protection to becomesusceptible and
then infected, as follows:
In these models, maternally-derived immunity is assumed to be
either:
lost at a constant rate from birth (as specified by the average
duration of protectionthat is provided by maternal antibodies),
orpresent during the first few months of life, after which
individuals become fullysusceptible to infection.
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14.2: Refining estimates of the force of infection:
Maternally-derived immunity
page 35 of 78 14.1 14.2
If we assume that individuals lose maternal immunity at a
constant rate , it can be shownthat the expression for the
proportion susceptible at age a is given by the expression:
(e-ae-a)
-
Do not worry about the details of this equation. You can refer
to the solution to Exercise5.6a in the recommended text for further
details6 .
If we assume that individuals are immune for the first six
months of life and susceptiblethereafter, then we can adapt
Equation 3 on page 10 to obtain the following equation forthe
proportion of individuals who are susceptible at a given age a (so
long as a>0.5 years):
s(a) = e-(a-0.5)
Since the proportion of people of age a who have ever been
infected, z(a), is given by 1 -proportion of individuals of age a
who are susceptible, we obtain the following equation forz(a):
z(a) = 1 s(a) =1 e-(a-0.5)
We can interpret the quantity a-0.5 in these equations as the
number of years duringwhich persons of age a had been
susceptible.
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Answer
14.3: Refining estimates of the force of infection:
Maternally-derived immunity
page 36 of 78 14.1 14.2 14.3
We will now return to our Excel file to explore how
incorporating maternal immunity intoour catalytic model affects the
value for the force of infection that we estimated from themumps
data that we analysed earlier. You may recall that when we assumed
thatindividuals did not have any maternal immunity, the best
fitting value for the force ofinfection was around 19.8% per year
.
1. Return to the spreadsheet catalytic model.xlsm that you were
working withpreviously. If you have already closed it, click here
to reopen it. Click on the sheet"maternal immunity".
The layout of this sheet is identical to that of the simple
catalytic sheet, except that theexpression for the proportion ever
infected in column F holds the Excel equivalent of theequation that
we discussed on the previous page :
z(a) = 1-e-(a-0.5)
You should notice that the curve for the proportion ever
infected is shifted slightly to theright in the figure, with the
value of this proportion at age 0.5 years being zero.
2. Click the Click to fit the model button to fit the model.
Click the Show button below if you want to check the graph
against the one that youshould have by this stage.
Q1.6 How does the estimated value for the force of infection
compare against the best-fitting value obtained using the simple
catalytic model?
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14.4: Refining estimates of the force of infection:
Maternally-derived immunity
page 37 of 78 14.1 14.2 14.3 14.4
Figure 10 compares predictions of the proportions of individuals
who have ever beeninfected with mumps by different ages, as
obtained using the best-fitting catalytic model forthe two
different assumptions about how maternal immunity is lost, against
data on theobserved proportion who had antibodies to mumps during
the 1980s.
Figure 10. Comparison between best-fitting predictions of the
proportion ever infected with rubellaagainst observed data for the
UK, obtained using different assumptions about protection from
maternalantibodies
The two assumptions lead to similar predictions of the
age-specific proportion everinfected, but slightly different
estimates of the force of infection (although the
confidenceintervals overlap) as illustrated in Table 2.
Type of model Force of infection (%per year) (95% CI)
Simple catalytic 19.8 (19.1-20.5)
Constant decline in maternalimmunity
20.3 (19.6-21.0)
100% susceptible after age 6months
21.0 (20.3-21.8)
Table 2. Comparison between the best-fitting values for the
force of infection, obtained by fittingcatalytic models using
different assumptions about the protection derived from maternal
antibodies
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Also notice that, as discussed on Page 33 , the estimated force
of infection that wasobtained assuming that individuals have
maternal immunity for the first 6 months of life ishigher than that
obtained using a simple catalytic model.
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Section 15: Age-dependency in the force of infection
page 38 of 78
We now consider the second refinement in the list outlined on
page 32 that can beincorporated when analysing seroprevalence data,
namely an age-dependent force ofinfection.
The catalytic model that we have considered so far has assumed
that the force of infectionis identical for all age groups. This
assumption may be inappropriate.
For example, considering the mumps data that we analysed earlier
(see Figure 10 ), themodel underestimates the proportion
seropositive among 5-14 year olds, suggesting thatthe true force of
infection may have been higher in this age group than that
estimated bythe model.
For older individuals, the model overestimates the observed
proportion seropositive,probably because the best-fitting value for
the force of infection may be higher than thetrue force of
infection experienced by older persons.
Differences between the actual and the assumed sensitivity of
the antibody test could havealso contributed to differences between
the best-fitting model predictions and the observeddata, especially
for older persons who may have low antibody titres. This issue is
notcovered in this session; methods for dealing with this
assumption are mentioned in section5.2.4 of the recommended course
text6 .
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15.1: Assessing age-dependency in the force of infectionfrom a
dataset: graphical technique
page 39 of 78 15.1
Before describing how we can change the catalytic model to
account for an age-dependent force of infection, we first describe
how we can identify whether the force ofinfection is age-dependent
from our age-stratified serological data.
We can determine whether the force of infection is age-dependent
using a graphicalmethod7 . This involves plotting (-ln{observed
proportion seronegative}) for each datapoint against the
corresponding age midpoint.
Figure 11 shows what happens if we do this for a perfect dataset
(shown in Figure 11A),which is generated assuming that the force of
infection had been constant over time at10% per year and identical
for all ages and that the proportion susceptible equals
theproportion seronegative. If we take the -ln(proportion
seronegative) of each datapoint andplot it against the age group
corresponding to each data point, we obtain a straight linewith a
gradient of 10%, i.e. the gradient is equal to the assumed force of
infection.
Figure 11: A. Predictions of the proportion of individuals that
are seronegative at different ages, assuming thatthe force of
infection has been constant over time at 10% per year and identical
for all ages and that theproportion susceptible equals the
proportion seronegative. These predictions can be generated using a
simplecatalytic model, namely by evaluating the expression e-0.1a
using values of a between 0 and 30. B. Predictions of-ln(proportion
seronegative) using values for the proportion seronegative from
figure A.
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dataset (shown in Figure12A), which is generated assuming that the
force of infection had been 10% per year forthose aged under 15
years and 5% per year for those age over 15 years. As shown
inFigure 12B, the plot of -ln(proportion seronegative) is made up
of two straight lines, whichjoin at the age at which the force of
infection changes. The gradient of the first line is 10%,and that
of the second line if 5%. In other words, the gradients of the
lines are equal to theassumed force of infection for
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15.2: Assessing age-dependency in the force of infectionfrom a
dataset: graphical technique
page 40 of 78 15.1 15.2
The plots in Figures 11 and 12 that we saw on the previous page
suggest that we canidentify whether or not the force of infection
in a population is age-dependent from cross-sectional
age-stratified serological data simply by plotting ln(observed
proportionseronegative) and looking to see whether the plot falls
on a straight line.
We can summarise the steps that we need to follow to do this as
follows:
1. Estimate the proportion susceptible in age group a, s(a),
using the expressionSa/Na, where Sa is the number of persons in age
group a who were seronegative,and Na is the number of persons in
age group a who were tested. If Sa = 0, replaceSa with 0.5.
2. Calculate the values -ln(Sa/Na).
3. Plot these values for -ln(Sa/Na) against the corresponding
age midpoints of agegroups a.
If the force of infection is identical for all age groups, your
plot should approximate to astraight line through the origin. The
slope of the straight line is the force of infection, .
If the plot does not fall on a straight line, then the force of
infection cannot be assumed tobe the same for all age groups. Note
that data for individuals in the first few months of lifeshould be
excluded from these plots if most of these individuals are
seropositive due to thepresence of maternal antibodies.
The next page provides some optional reading for why this
graphical test works. You mayprefer to skip this for now and go to
page 42 , where we will follow the above steps usingthe mumps data
that we studied earlier in the session.
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15.3: Example: Assessing age-dependency in the force ofinfection
from a dataset: graphical technique (optionalreading)
page 41 of 78 15.1 15.2 15.3
The rationale for this graphical test is as follows:
If the force of infection is not age-dependent, then, as
discussed on page 10 , theproportion of individuals of a given age
a who are susceptible, s(a), is given by theexpression:
s(a) = e-a
Taking the natural logarithms of both sides (see the maths
refresher to revise logs ifnecessary), we obtain the result
ln{s(a)} = ln(e-a) Equation 9
According to the laws of logarithms (see maths refresher for
further details),
ln(e-a) = -a
Using this result in Equation 9 and multiplying both sides of
the equation by -1, we obtainthe following equation:
-ln{s(a)} = a
This equation is analogous to that of a straight line "y=mx+c"
if we replace y by -ln{s(a)},m by , x by a and c by 0, i.e. the log
of the proportion susceptible at age a is linearlyrelated to the
force of infection.
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15.4: Assessing age-dependency in the force of infectionfrom a
dataset: graphical technique
page 42 of 78 15.1 15.2 15.3 15.4
We will now return to the mumps data discussed previously to see
whether the force ofinfection in the UK was age dependent.
1. Return to the Excel file catalytic model.xlxm that you were
working with. Click here to open it if you have closed it by now.
Select the simple catalytic worksheet.
2. Select columns I and K together, click with the right mouse
button and select theUnhide option.
3. You should now see the contents of column J. This holds the
Excel equivalent of theequation ln(Sa/Na) for each age group
included in the dataset.
Select columns Q and Y together, click with the right mouse
button and select the"Unhide" option. You should see a figure
labelled "Figure 2: Graphical check for aconstant force of
infection", which is similar to that shown below. This plots
thevalues for ln(Sa/Na) against the midpoint for the corresponding
age group a.
Figure 13: Plot of ln(Sa/Na) against the age midpoint for the
mumps dataset
Q1.7 From your observation of the plot, do you think the force
of infection is identical for all
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Answer
age groups?
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15.5: Exercise: graphical method for estimating a constantforce
of infection
page 43 of 78 15.1 15.2 15.3 15.4 15.5
Q1.8 Complete the following table for measles using the data
provided. Using pen andpaper, plot -ln(Sa/Na) against the age
midpoint. What do you conclude about how theforce of infection
changes with age in this population?
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Answer
Age(yrs)
Na Sa Sa/Na -ln(Sa/Na)
1-5 52 32
6-10 63 25
11-20 46 4
21-50 58 1
51-80 42 0
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Section 16: Age dependency in the force of infection
page 44 of 78
Once the way in which the force of infection depends on age has
been determined, wethen need to write down expressions for the
age-specific proportion susceptible, and fitthem to the serological
data to estimate the actual force of infection. The
expressiondepends on assumptions of how the force of infection
changes with age. For example, thefollowing is the expression that
we would use if we assume that the force of infectiondiffers
between those aged
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16.1: Age dependency in the force of infection
page 45 of 78 16.1
As shown in figure 14, the assumption that the force of
infection is age dependent leads toan improved fit to the mumps
data described above, with a higher force of infectionestimated for
those aged
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16.2: Age dependency in the force of infection
page 46 of 78 16.1 16.2
Similar patterns of a higher force of infection among children,
as compared with thatamong adults, have been found for several
infections (Farrington, 1990)1 . The agedifferences are probably
due to differences in susceptibility or in exposure (e.g.
youngindividuals may be more likely to contact other young
individuals, than older individuals).As we shall see in MD06 ,
contact patterns between children and adults are likely todiffer
substantially and the differences greatly affect the impact of
interventions targeted atchildren.
The age-specific pattern will also depend on the study
population e.g. whether it isurban/rural, developed/developing etc.
Data on measles serology from New Haven, 1957(Grenfell and Anderson
(1985)8 ) also suggested that there are differences in
age-specificpatterns between large and small families.
Methods for dealing with different assumptions about the force
of infection (e.g. that itchanges continuously with age) are
discussed in Farrington (1990)1 .
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Section 17: Time dependency in the force of infection
page 47 of 78
On the next few pages, we briefly consider the third and fourth
refinements in the listoutlined on page 32 that can be incorporated
when analysing seroprevalence data,namely a time-dependent force of
infection, and non-permanent immunity followinginfection.
For infections for which the incidence cycles over time, the fit
of a simple catalytic modelmay be particularly poor for the younger
age groups if the data have been collectedimmediately following an
epidemic.
In these situations, estimates of the average force of infection
are best obtained by fittingthe catalytic model to the data of
older individuals, who would have been present duringboth epidemic
and non-epidemic periods, when the high and low forces of
infectionoccurred. These issues are discussed further in Whitaker
and Farrington (2004)9 .
For some infections, apparent age differences in the force of
infection may be attributed tosecular changes in the force of
infection. Insight into whether this is the case or not can
beobtained by analysing data collected over several years (see
Sutherland (1983)10 forDutch tuberculin data analyses, Nokes et al
(1993)11 , Ljungstrm et al (1995)12 fortoxoplasmosis data).
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Section 18: Variants of the simple catalytic model
page 48 of 78
There are other structures of catalytic models that may be more
appropriate than a simplecatalytic model for some infections, such
as those which confer non-permanent immunity.
The reversible or SIS (susceptible - infected/infectious -
susceptible) model assumes thatindividuals become
infected/infectious at a rate , and once infected, return to
beingsusceptible at a rate rs.
This leads to theage-specific patternin the prevalence
ofinfection that isshown on the right,i.e. the proportion
ofindividuals that isinfected increaseswith age andreaches a
plateauwith a value of/(+rs). Note thatthis result isobtained
assumingthat both and rs donot vary over time.
The reversible model has been applied to cross-sectional
tuberculin data (Mnch (1959)2, Fine et al (1999)4 ), diptheria,
malaria (Kitua et al (1996)13 ) and filariasis (Vanamail etal
(1989)14 ).
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Hint Answer
18.1: Variants of the simple catalytic model:
Optionalexercise
page 49 of 78 18.1
EXERCISE
Prove the result that the level of the plateau for the
reversible model, assuming that andrs do not vary over time, is
given by the equation /(+rs).
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18.2: Variants of the simple catalytic model
page 50 of 78 18.1 18.2
The two-stage or SIR (susceptible - infected/infectious -
recovered) model
and the compound model
are two other variations of the simple catalytic model, and have
been applied to yaws andhistoplasmosis data (Mnch, 1959)2 . These
models have the following patterns in theage-specific proportion
infected:
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Section 19: Optional reading: Model-free analyses
page 51 of 78
On the next few pages, we describe ways in which serological
data can be analysedwithout using a model. These pages are optional
reading and if you wish, you may skip topage 55 to see the
summary.
Several useful parameters can be calculated without using a
model. If we can assume thatthe proportion of individuals who test
negative for the infection is equal to the proportionsusceptible s,
then s can be estimated by using the following equation:
where
a is the proportion of the population in the age group a
(obtained from life tables),Sa is the number of susceptibles among
those tested in age group a, andNa is the number of individuals of
age a who were tested.
This expression is equivalent to the weighted average of the
age-specific proportion ofindividuals who are susceptible.
If we need to account for maternally-derived immunity in
infancy, it is convenient toassume that the proportion susceptible
is zero until some age M, after which the maternalantibodies wear
off (e.g. M = 0.5 years).
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19.1: Optional reading: Model-free analyses
page 52 of 78 19.1
If we can assume that the population has arectangular age
distribution, such as thatshown in Figure 8b for England and
Waleswith average life expectancy L, then a isgiven by:
a = wa/L
where wa is the width of age group a.
For example, if the life expectancy is 75years, then the width
of the age bands 0-4,5-9 and 10-19 years is 5, 5 and 10
yearsrespectively, which gives the following:
0-4 = 5/755-9 = 5/7510-19 = 10/75
Figure 8b) Population in England and Wales, 1991.Reproduced for
convenience from page 24 . Datasource: Office for National
Statistics
If it is reasonable to assume that the force of infection is
constant with age, then once wehave estimated the proportion
susceptible using the model-free method discussed on page51 , we
can also estimate the average force of infection, the average age
at infection, R0and the herd immunity threshold, as illustrated in
the next exercise. Estimates obtainedusing these model-free methods
are useful for checking that the estimates obtained usingformal
methods are realistic. Other quick model-free methods are discussed
in section5.2.2.1 of the recommended course text6 .
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19.2: Optional reading: Model-free analyses: Exercise
page 53 of 78 19.1 19.2
EXERCISE
The following are data on the numbers of individuals who are
negative for measlesantibodies in a given population. Assuming that
the age distribution of the population isrectangular and that the
life expectancy is 80 years, complete the following table.
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Age (yrs) Na Sa Sa/Na a a Sa/Na
1-5 52 32
6-10 63 25
11-20 46 4
21-50 58 1
51-80 42 0
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19.3: Optional extras: Model-free analyses: Exercise ctd
page 54 of 78 19.1 19.2 19.3
Age(yrs)
Na Sa Sa/Na a axSa/Na
1-5 52 32 0.615 5/80 0.0385
6-10 63 25 0.397 5/80 0.0248
11-20 46 4 0.087 10/80 0.0109
21-50 58 1 0.017 30/80 0.0065
51-80 42 0 0 30/80 0
Using your answer to the last question (reproduced in the table
above), estimate thefollowing:
1. The proportion of individuals who aresusceptible
2. The average age at infection
3. The force of infection (assuming thatit is not
age-dependent)
4. The basic reproduction number
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Section 20: Summary
page 55 of 78
We have now completed the first part of this session. We will
first summarise the keymessages that you should have obtained by
now, before proceeding to the second(practical) part of this
session, in which you will practise working with a catalytic model
andserological data.
1. We study age-specific serological data to obtain estimates
for the average force ofinfection , the average age at infection,
the proportion susceptible and R0.
2. Serological data are typically analysed using so-called
catalytic models. The simplecatalytic model has the following
structure:
3. If the average force of infection () is assumed to be
identical for all age groups, theproportion of individuals of age a
that are susceptible or (ever) infected (denoted bys(a) and z(a)
respectively) for an endemic immunising infection are given by
thefollowing equations:
s(a) = e-a z(a) = 1-e-a
4. The average force of infection can be estimated formally by
fitting predictions of theage-specific proportion ever infected to
data on the age-specific proportion ofindividuals who are
seropositive.
5. If the force of infection is assumed to be the same for all
age groups, the averageforce of infection is related to the average
age at infection (A) through the followingexpression:
1/A
6. The expression for the proportion susceptible depends on
whether the agedistribution of the population is assumed to be
rectangular or exponential. Assumingrandom mixing, the expressions
are as follows:
s A/L Rectangular age distribution
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s A/(A+L) Exponential age distribution
7. Assuming that individuals mix randomly, the basic
reproduction number can becalculated from the proportion
susceptible (s) using the equation R0=1/s.
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20.1: Key messages from this session
page 56 of 78 20.1
8. R0 can also be calculated using the following expressions,
depending on whether theage distribution of the population is
assumed to be rectangular or exponential:
R0L/A Rectangular age distribution
R0 1 + L/A Exponential age distribution
9. The simple catalytic model can be adapted to take account of
several complications,such as the presence of maternal immunity, an
age or time-dependency in the forceof infection, and non-permanent
immunity following infection.
10. Estimates of the force of infection that are obtained
assuming that individuals havematernal immunity are higher than
those obtained assuming that individuals aresusceptible from
birth.
11. We can assess whether the force of infection is
age-dependent by plotting -ln{sa}against the age-midpoint, where sa
= observed proportion seronegative in age groupa. If all the points
fall on a straight line, then the force of infection is probably
thesame for all age groups; otherwise, it is probably
age-dependent.
12. Approximate values for the proportion susceptible can also
be obtained directly fromthe data using the following equation:
13. Such estimates can be used to obtain approximate values for
the force of infection,the average age at infection and R0. These
estimates are useful for checking thatthe values obtained by
modelling methods are reasonable.
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Section 21: Break...
page 57 of 78
The remainder of this session (the practical component) will
consolidate some of the ideascovered in the first part of the
session and will provide you with practice in fitting models
todata, and calculating the force of infection, the average age at
infection and otherstatistics. It is likely to take 1 - 3
hours.
You may wish to take a short break before starting it.
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Section 22: Practical: Estimating forces of infection by
fittingcatalytic models to seroprevalence data
page 58 of 78
OVERVIEW
We will now begin the practical component of this session, in
which we will set up a simplecatalytic model that we discussed in
the first part of this session, in Excel. The model willthen be
fitted to age-stratified serological data for rubella from two
settings (China and theUK) to estimate the average force of
infection. We will then use the force of infection toestimate other
useful epidemiological statistics, such as the average age at
infection andR0.
OBJECTIVES
By the end of this practical, you should:
Understand the basic methods for fitting catalytic models to
seroprevalence data.Be able to apply the methods to estimate the
force of infection and the basicreproduction number of an
infection.
If you did not install Solver at the beginning of the session,
click here for the installationinstructions.
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Section 23: Practical: Mechanics of calculating forces
ofinfection from seroprevalence data
page 59 of 78
1. Start by opening the Excel spreadsheet "rubcrude.xlsm" . The
file comprises 2sheets, one with data on the age-specific
proportion of individuals who hadantibodies to rubella in the UK,
and the other with similar data from China. Thelayouts of these
sheets is similar. You should see something resembling
thefollowing:
a) Blue cells containing a value for the force of infection.
This has been assigned thename "foi_uk" or "foi_c" depending on
whether the sheet has data for the UK or China.At present, the
value for the force of infection in the UK in the spreadsheet is
set to be0.12 per year (or, equivalently, 12% per year).
b) Yellow cells containing data on the age-specific numbers of
individuals positive forrubella antibodies in the UK (sheet UK) and
in China (sheet China) in different timeperiods.
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c) Pink cells in which you will soon set up equations for the
age-specific proportion ofindividuals who have ever been
infected.
NB. You are not expected to type in anything just yet...
d) A graph plotting the observed proportion seropositive in the
UK or China populations.
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23.1: Practical: Mechanics of calculating forces of
infectionfrom seroprevalence data
page 60 of 78 23.1
2. We will begin by analysing the data from the UK. If you have
not yet done so, selectthe sheet "UK".
Q2.1 Ignoring the contribution of maternal antibodies for now
and assuming that the forceof infection is identical for all ages,
type in an appropriate Excel formula for the proportionof 0.5 year
olds who have ever been infected in cell F25.
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23.2: Practical: Mechanics of calculating forces of
infectionfrom seroprevalence data
page 61 of 78 23.1 23.2
3. Copy the formula that you have set up for 0.5 year olds down
for all the age groupsin the data set.
A pink line showing the proportion ever infected (predicted for
the current value for theforce of infection) should have now
appeared in the graph, as shown below.
If this has not yet happened, click here to check the
expressions that you should haveset up by now.
Q2.2 Do you think the true value for the force of infection in
the UK was greater or smallerthan that currently assumed?
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23.3: Practical: Mechanics of calculating forces of
infectionfrom seroprevalence data
page 62 of 78 23.1 23.2 23.3
We will now find the best fitting value (as identified by this
model) for the force of infectionfor the UK. We will do this using
a similar approach to the approach that we used in thefirst part of
this session, namely by finding the lowest value for an appropriate
goodness offit statistic. If you have changed the value for the
force of infection since opening thespreadsheet, change it to be
0.12 (per year).
4. Select rows 5 and 10 together, click with the right mouse
button and select the"Unhide" option.
You should see some grey cells containing an expression for the
goodness of fit of themodel to the data. This is technically known
as the "loglikelihood deviance".
Further information about the deviance
Q2.3 What is the current value for the deviance?
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23.4: Practical: Mechanics of calculating forces of
infectionfrom seroprevalence data
page 63 of 78 23.1 23.2 23.3 23.4
We will now use Excels Solver tool to find the force of
infection which results in thesmallest possible deviance between
the observed proportion seropositive and modelpredictions.
To run the Solver tool, we need to enable some macros which have
built into Excel. Ifyou see a security warning below the toolbar,
stating that "Macros have been disabled",click on the "Options"
button next to this warning and then select the "Enable this
content"option, before clicking on "OK". If this option is not
available, you will need to close andreopen the spreadsheet to see
it. If you need to reopen your spreadsheet, click here toaccess the
file that you should have set up by now.
5. Select the "Solver" option (located under the Data tab, to
the farright). If it is not available, click here for the
instructions to set it up.
In order to use the "Solver" tool, we need to specify:
The location of the cell whose value we want to maximise,
minimise or set to beequal to some value. This is defined under the
"Set Target Cell" option.The cell(s) which are allowed to change so
that the maximum/minimum, etc., isattained (under the "By Changing
Cells" option).
In our case, we want to identify the value for the force of
infection (cell D4) which results inthe minimum deviance (cell D6).
We will do this on the next page.
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23.5: Practical: Mechanics of calculating forces of
infectionfrom seroprevalence data
page 64 of 78 23.1 23.2 23.3 23.4 23.5
6. Set up the "Target cells" and the "By Changing Cells" options
to refer to theappropriate cells in the Excel file. Click here if
you wish to remind yourself of whatthese options represent. Select
the "Min" option under the "Equal to" option and clickon the
"Solve" button.
At this point, a "Solver Results" window should appear,
indicating that "Solver has found asolution." Click OK to keep the
Solver solution. If you have not found a solution, check
thesettings that you've specified in Solver (see Step 6).
Q2.4 What is the best-fitting value for the force of
infection?
Q2.5 For which age groups does the model overestimate or
underestimate the proportionof seropositive individuals?
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23.6: Practical: Mechanics of calculating forces of
infectionfrom seroprevalence data
page 65 of 78 23.1 23.2 23.3 23.4 23.5 23.6
Q2.6
a) Assuming that the force of infection is independent of age,
what is theaverage age at infection in the UK? Use the expression
presented on page22 .
b) Assuming that the average life expectancy (L) is 60 years,
what is the R0for this population according to the expression R0
L/A?
c) What is the herd immunity threshold? You should recall that
the Herdimmunity threshold (H) is given by the expression: H = 1 -
1/R0.
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23.7: Practical: Mechanics of calculating forces of
infectionfrom seroprevalence data
page 66 of 78 23.1 23.2 23.3 23.4 23.5 23.6 23.7
7. Now, click on the China worksheet (the tab at the bottom of
the Excel window),return to page 60 and repeat steps 2-6 to
calculate the following:
a) The best-fitting force of infection;b) The average age at
infection;c) R0 (assuming the same life expectancy as in the UK);
and
d) The herd immunity threshold for China.
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23.8: Practical: Mechanics of calculating forces of
infectionfrom seroprevalence data
page 67 of 78 23.1 23.2 23.3 23.4 23.5 23.6 23.7 23.8
Q2.7 How would your answers have changed if you assumed that the
life expectancy was75 years rather than 60 years in China and the
UK?
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23.9: Practical: Mechanics of calculating forces of
infectionfrom seroprevalence data
page 68 of 78 23.1 23.2 23.3 23.4 23.5 23.6 23.7 23.8 23.9
Below is a table comparing the parameters obtained for the UK
and for China.
Population Force ofinfection
(% pa)
Averageage at
infection(yrs)
R0 Herdimmunitythreshold
UK (L=60years)
11.59 8.6 6.95 86.0%
UK (L=75years)
8.72 88.5%
China(L=60years)
20.32 4.9 12.19 92.0%
China(L=75years)
15.31 93.5%
Q2.8 How do the values for the force of infection, the average
age at infection, R0 and theherd immunity threshold in China
compare against those for the UK? Suggest possibleexplanations for
these differences.
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23.10: Practical: Mechanics of calculating forces of
infectionfrom seroprevalence data
page 69 of 78 23.1 23.2 23.3 23.4 23.5 23.6 23.7 23.8 23.9
23.10
Q2.9 How should your estimate for the force of infection change
if we were to assume thatindividuals are immune for the first 6
months of life?
8. Return to the spreadsheet and update your expressions for the
proportion everinfected in each age group for China and the UK to
include maternal antibodies(assume that individuals are immune for
the first 6 months of life and are thensusceptible).
Click here to remind yourself of the expression that you may
need to use at this stage.
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23.11: Practical: Mechanics of calculating forces of
infectionfrom seroprevalence data
page 70 of 78 23.1 23.2 23.3 23.4 23.5 23.6 23.7 23.8 23.9 23.10
23.11
At this stage you should have obtained figures similar to the
ones below for China and theUK, i.e. the proportion ever infected
is zero at age 0.5 years. The curve is shifted to theright as
compared with the previous version. Click here to check your
spreadsheet ifnecessary.
9. Refit the model using Solver for both China and the UK and
see if the new estimatesfor the force of infection are consistent
with your response to the last question. Clickhere to see the
best-fitting values for the force of infection.
Optional technical note: You should notice that, when you change
the expression for the prevalence of previous infection toaccount
for maternal immunity for all age groups, the deviance becomes
negative, which is statisticallyunacceptable. This negative
deviance results from the fact that, for 0.5 years olds, the
contribution tothe loglikelihood from the catalytic model is zero.
To overcome this problem, we can adjust theexpression for the
deviance so that it only uses the data and estimates for
individuals aged over 0.5years. However, you should find that the
best-fitting value for the force of infection is identical to
thevalue obtained when data for all age groups were used when
fitting the model.
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Section 24: Practical: Assessing an age-dependency in theforce
of infection
page 71 of 78
We will now investigate whether our assumption that the force of
infection is independentof age is appropriate for these
settings.
10. If you have not already done so, unhide column I (by
selecting columns F and Jtogether, clicking with the right mouse
button and selecting the "Unhide" option).Enter the appropriate
formula into the lavender cells in cell I25 and copy theexpression
down for all age groups. Complete this process for both the China
andthe UK sheets.
At this stage, Figure 2 on the Excel sheet should resemble one
of the following, dependingon whether you're considering China or
the UK. If it does not, click here to see thecorrect version of the
file.
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24.1: Practical: Assessing an age-dependency in the force
ofinfection
page 72 of 78 24.1
Q2.10 According to these figures , is the assumption that the
force of infection isidentical for all age groups in these
populations justified? At what age does it appear asthough the
force of infection changes in these populations?
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24.2: Practical: Assessing an age-dependency in the force
ofinfection
page 73 of 78 24.1 24.2
If we were to change our model so that it assumed that the force
of infection differsbetween those aged
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24.3: Practical: Assessing an age-dependency in the force
ofinfection
page 74 of 78 24.1 24.2 24.3
Other notable characteristics of these estimates include the
following:
1. The very low (almost non-existent!) force of infection
estimated for older individualsin China is unrealistic. This
estimate is unreliable since most individuals areseropositive by
age 20 years and therefore, both high and low values for the force
ofinfection among adults will lead to similar levels (i.e. 100%)
for the percentage whoare seropositive.
2. Both the age-specific and crude estimates for the force of
infection in China arehigher than for the UK. As mentioned b