md04

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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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EPM302 Modelling and the Dynamics of Infectious Diseases

MD04 Applying modelling techniques toanalyse seroprevalence data

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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.




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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Answer

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: .




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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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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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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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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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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Answer


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




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

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




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




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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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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Show

Answer


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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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




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.




Figure 12 shows what happens if we do this for another perfect 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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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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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




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?


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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Answer

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.


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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Answer

Answer

Answer

Answer

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




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.




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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Hint Answer

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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Answer


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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Answer


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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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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Check

Answer

Answer


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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Answer


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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Answer


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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Answer


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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Answer


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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Answer

Check


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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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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Answer

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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Answer


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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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

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