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Section 1: Overview and objectives

page 1 of 97

OVERVIEW

In MD06, we saw how assumptions about the amount of contact between individualsinfluenced model predictions of the effect of vaccination. This suggests that contactpatterns determine the herd immunity threshold and therefore R0. In this session weillustrate how we can calculate R0 to take account of different contact patterns.

OBJECTIVES

By the end of this session you should:

Know the importance of accounting for non-random mixing between individuals whencalculating the basic reproduction number and the critical vaccination coveragerequired for controlling transmission.Be able to calculate the basic reproduction number and herd immunity threshold,assuming either that individuals mix randomly or non-randomly.Know how reproduction number estimates are currently used to estimate thepotential for a measles epidemic to occur in England.

This session is made up of 2 parts and is likely to take 2-5 hours to complete.

The first part (1-2 hours) describes the theory for calculating R0 and describes howcalculations of the reproduction number are presently used in England to assess thepotential for a measles epidemic to occur.

The second part (1-2 hours) consists of a practical exercise in Excel, providing you withfurther practice in calculating the basic and net reproduction numbers.

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EPM302 Modelling and the Dynamics of Infectious Diseases

MD07 Calculating R0 for non-randomlymixing populations

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Section 2: Introduction

page 2 of 97

In the previous session , we explored the impact of different levels of rubella vaccinationcoverage among newborns in a population in which individuals were stratified into theyoung, middle-aged and old (denoted by the symbols y, m and o) as shown in the modeldiagram below.




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2.1: Introduction

page 3 of 97

We assumed that the force of infection differed between the young, middle-aged, and old,and was given by the following equations:

Force_of_infn_y = b_yy*Infous_y + b_ym*Infous_m + b_yo*Infous_o

Force_of_infn_m = b_my*Infous_y + b_mm*Infous_m + b_mo*Infous_o

Force_of_infn_o = b_oy*Infous_y + b_om*Infous_m + b_oo*Infous_o

Here, b_yy, b_ym etc represent the rate at which individuals in a given age group comeinto effective contact with those in other age groups per unit time.

Individuals in different age categories were assumed to contact each other according toWho Acquires Infection From Whom (WAIFW) matrices A and B (as shown below) wherethe parameters (in units of per day) were calculated using the force of infection estimatedfrom data on rubella seroprevalence for England and Wales:

WAIFW A y m o

y 1.81 10-5 0 0m 0 2.92 10-5 0o 0 0 3.35 10-5

WAIFW B y m o

y 1.66 10-5 4.16 10-6 4.16 10-6

m 4.16 10-6 4.16 10-6 4.16 10-6

o 4.16 10-6 4.16 10-6 4.16 10-6




Figure 1. Comparison between the WAIFW matrices A and B, which were calculated using the force of infectionestimated from rubella seroprevalence data from England and Wales in MD06 .

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2.2: Introduction

page 4 of 97

For both assumptions about contact, predictions of the age-specific proportion ofindividuals who were susceptible and the daily number of new infections in the absence ofvaccination were identical. However, predictions of these statistics differed oncevaccination of newborns was introduced, as shown in Figure 2.

For example, if individuals were assumed to contact each other according to matrixWAIFW A, then vaccination of newborns with a coverage of 86% was insufficient to controltransmission. In contrast, if individuals were assumed to contact each other according tothe WAIFW B matrix, transmission stopped shortly after the introduction of 86%vaccination coverage among newborns.

The differences between the predictions obtained using matrices WAIFW A and WAIFW Breflect differences in R0 and therefore the herd immunity threshold that is associated withthese two matrices. In fact, as we shall see later, R0 for WAIFW A and WAIFW B is about10.9 and 3.64 respectively, which correspond to values of the herd immunity threshold of91% and 73% respectively.

In this session, we will show you how we can estimate R0 for these and other assumptionsabout contact between individuals.

Figure 2. Comparisonbetween predictions of thedaily number of new rubellainfections per 100,000population, obtainedassuming that individualscontacted each otheraccording to WAIFWmatrices A and B, using theBerkeley Madonna modelsfrom MD06 (see pages 54 and 58 respectively inMD06). Vaccination ofnewborns, with an effectivecoverage of 86% isintroduced 100 years afterthe start.




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Section 3: R0, Rn and the herd immunity threshold

page 5 of 97

Before illustrating how we can calculate R0 to take non-random mixing between individualsinto account, we first review the methods that are used to calculate R0 and how it is relatedto Rn and the herd immunity threshold when we assume that individuals mix randomly.

You may find that you can skip through the next few pages quickly as you will have comeacross most of the content previously.




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3.1: The basic reproduction number (R0)

page 6 of 97

By now, you are probably familiar with the definition of the basic reproduction number (R0)as the average number of secondary infectious individuals generated by a single typicalinfectious person following his/her introduction into a totally susceptible population.

As we saw in MD03 , if we assume that individuals mix randomly, R0 can be calculatedusing the following equation:

R0 = ND Equation 1

where

is the rate at which two specific individuals come into effective contact per unittime;N is the population size;D is the duration of infectiousness.

In a deterministic model, for the incidence to increase following the introduction of aninfectious person into a totally susceptible population, R0 must be bigger than one.




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3.2: The net reproduction number (Rn)

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If we assume that individuals mix randomly, R0 and the net reproduction number, Rn, arerelated as follows:

Rn = R0s Equation 2

where s is the proportion of individuals who are susceptible in the population.

This expression provides a useful method for calculating the basic reproduction number.For example, when the infection is at equilibrium, each infectious person will be leading toone secondary infectious person. Thus the net reproduction number will then be equal to1.

i.e.

Rn = 1

If we substitute for Rn = 1 into Equation 2 and then rearrange it slightly, we obtain thefollowing equation:

R0 = 1/s* Equation 3

where s* is the proportion of the population that is susceptible to infection at equilibrium.

This expression leads to several other expressions for the basic reproduction number forrandomly-mixing populations, as we will show on the next few pages.




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a) Rectangular age distribution b) England and Wales: population by age, 1991

Figure 3. a) Illustration of the relationship between the proportion of the population that is susceptible and the lifeexpectancy in a population with a rectangular age distribution. The shaded area reflects the proportion of the population thatis susceptible. Adapted from Fine PEM (1993)1 b) Population in England and Wales, 1991. Data source: Office forNational Statistics.

3.3: Other expressions for R0, assuming that individuals mixrandomly

page 8 of 97

As we saw in MD04 , the following expressions for R0 can be obtained for populationswhose age distributions are approximately rectangular or exponential, if we assume thatindividuals mix randomly.

Rectangular age distributions: R0 L/AExponential age distributions: R0 1+L/A

where A is the average age at infection and L is the average life expectancy. Figure 3 andFigure 4 summarise the relationship between A, L, and the proportion of the populationthat is susceptible for populations with rectangular and exponential age distributions.


Figure 3 Figure 4



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3.4: Other expressions for R0 assuming that individuals mixrandomly: N/(B(A-M))

page 9 of 97

When the average age at infection A is small, another equation can be used regardless ofthe age distribution in the population:

R0 N Equation 4B(A-M)

where

N is the population size;B is the number of surviving infants;M is the duration of maternal immunity;andA is the average age at infection (and issmall).

Figure 5. Illustration of the relationship between theproportion of the population that is susceptible, the durationof maternally-derived immunity and the average age atinfection for a population with an unspecified demography.

Click here to see the derivation.

.




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Show

3.5: Other expressions for R0 assuming that individuals mixrandomly: 1/(1-i*)

page 10 of 97

Another equation can be applied when the infectiondoes not confer immunity, i.e. when the appropriatemodel would be of the SIS type, as seen in MD01 and illustrated in Figure 6.

In this instance R0 is given by the equation

R0=1/(1-i*) Equation 6

Figure 6. The general structure of a Susceptible Infectious Susceptible (SIS) model

where i* is the equilibrium prevalence of infection (or equivalently, of infectious individuals) in the population.

Click the show button below to see the derivation of this equation.

IMPORTANT NOTE: These relatively simple equations assume that individuals in thepopulation mix randomly. If we cannot assume that individuals mix randomly, we need touse an alternative approach, which we will discuss later.




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Section 4: Herd immunity threshold

page 11 of 97

Whilst R0 is useful for indicating the transmissibility of an infection, it can also be used todetermine the proportion of the overall population at which an intervention, such asvaccination, needs to be targeted in order to control transmission.

As you will have learned in your previous training, the herd immunity threshold (H) is givenby the equation:

H=1-s* = 1-1/R0

For example, the basic reproduction number for measles in several settings has beenestimated to be about 13 (click here to see examples of estimated values for R0), whichsuggests that the herd immunity threshold for measles is about:

100 x(1-

1 ) 92%13

This suggests that, in these settings, over 92% of the population would need to beeffectively vaccinated in order to control transmission.




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Section 5: Summary of the steps in calculating R0 - randomlymixing population

page 12 of 97

As we saw in MD04 , the following steps are required to calculate R0 if we assume thatindividuals in a population mix randomly and that the infection is at equilibrium:

1. Measure the prevalence of previous infection in the population using aseroprevalence survey.

2. Assuming that the population is at equilibrium, calculate s*, the proportion of thepopulation that is susceptible.

3. Calculate R0 using the expression R0= 1/s* (Equation 3 ).

Note that it is also possible to estimate R0 using the growth rate of an epidemic oroutbreak (see MD03 ) or using data on the secondary attack rate (see Fine, et al 2 ).




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5.1: Summary of the steps in calculating R0 - assuming non-random mixing

page 13 of 97

However, the situation is more complicated than that described on the previous pages ifwe assume that individuals do not mix randomly. We then need to follow the steps belowin order to calculate R0:

1. Measure the prevalence of previous infection in the population using a serosurvey.2. Estimate the forces of infection in different subgroups (e.g. age strata).3. Choose the structure of the matrix of Who Acquires Infection From Whom (WAIFW).4. Calculate the parameters for the WAIFW matrix.5. Formulate the Next Generation Matrix.6. Calculate R0 from the Next Generation Matrix.

Steps 1 - 4 have been covered in sessions MD04 and MD06. In this session we will dealwith steps 5 and 6.




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5.2: Non-random mixing and the Next Generation Matrix

page 14 of 97

When we say that a population mixes nokn on-randomly, we imply that the population canbe divided into two or more subgroups and individuals in one subgroup have a differentamount of contact (e.g. mix more or less intensively) with individuals from their ownsubgroup than with individuals from another subgroup.

The number of secondary infectious individuals generated by one infectious personintroduced into a totally susceptible population then depends on the subgroup to whichthat person belongs.




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page 15 of 97

For example, in MD06 we considered the population in the diagram below, in whichindividuals were stratified into the young and the old (or children and adults). In thispopulation, each infectious child generated a different number of secondary infectiousindividuals from the number generated by an infectious adult.

Also, the number of secondary infectious children generated by each child differed from thenumber of infectious adults that they generated.

As we shall show on the next few pages, we can summarise the number of infectiouschildren and adults generated by each child and adult using the Next Generation Matrix. We can then use the matrix to calculate the basic reproduction number.

Population A Population B

A represents a child and a represents an adult.

Figure 7: Diagram showing the contact patterns of two hypothetical populations considered in MD06 . Inpopulation A, an infectious child could generate six secondary infectious individuals, four of which would bechildren and two of which would be adults. An infectious adult in the same population, on the other hand, wouldgenerate three infectious individuals, two of whom would be adults and one of whom would be a child.




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page 16 of 97

The Next Generation Matrix is defined as the matrix that summarises the number ofsecondary infectious individuals in a given category resulting from infectious individuals ineach of the categories. For a population in which individuals are stratified into either the"young" or the "old", it would be given by the following matrix:

Here,

Ryy is the number of young secondary infectious individuals generated by eachinfectious young person;Ryo is the number of young secondary infectious individuals generated by eachinfectious old person;Roy is the number of old secondary infectious individuals generated by eachinfectious young person;Roo is the number of old secondary infectious individuals generated by eachinfectious old person.




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page 17 of 97

Notice the order of the subscripts used in the notation for the numbers (or "elements") inthe Next Generation Matrix on the last page. In each expression (e.g. Roy) the firstcomponent of the subscript reflects the category of individuals among whom the secondaryinfectious individuals occur (i.e. old individuals when considering Roy), and the secondcomponent of the subscript reflects the category of the infectious person who istransmitting the infection (i.e. young individuals when considering Roy).

As shown on the next page, the order of the subscripts used for Roy, Ryo, etc. is identical tothat used for the parameters.




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Section 6: Summary of the notation used for the subscripts

page 18 of 97

The following summarises the notation used for the parameters:

Therefore yo is the rate at which a specific young (susceptible) individual comesinto effective contact with a specific old (infectious) individual per unit time.

The following summarises the notation used for the Ryy, Ryo etc. elements of the NextGeneration Matrix:

Therefore Ryo is the number of secondary infectious individuals among youngindividuals resulting from one old (infectious) individual in a totally susceptiblepopulation.




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Population A1.Nextgenerationmatrix

child adult

child

adult

a) c) e) g)

b) d) f) h)

Section 7: Next Generation Matrix example - exercisepage 19 of 97

We will now practice using the Next Generation Matrix notation. Below we show the two hypothetical mixing patterns that wediscussed on page 15 in which individuals are either children or adults.

Population A Population B

Q1.1 The tabs below show examples of Next Generation Matrices. Click on the appropriate Next Generation Matrix for

1) Population A2) Population B

from the options in the appropriate tab (tab 1 for population A, tab 2 for population B). Not every option will be used, and nooption will be used more than once.

Hint: The template for the Next Generation Matrix is shown at the top of the tab. The titles above the columns of the matrix referto the source of infection and the titles next to the rows of the matrix refer to the recipient of the infection.


1 2


0

Section 7: Next Generation Matrix example - exercise file:///I:/DEP/SHARED/2013-14Final_ RESTRUCTURE/EPM302/EP...

1 of 1 04/04/2014 16:06

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Section 8: Writing the Next Generation Matrix

page 20 of 97

Each of the numbers in the Next Generation Matrix can be expressed in terms of thefollowing:

a) The parameters of the WAIFW matrix describing contact between the young andold;b) The numbers of individuals in each age group; andc) The duration of infectiousness.

We derive the expressions on the next few pages.




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8.1: Writing the Next Generation Matrix

page 21 of 97

As discussed previously , if we assume that individuals mix randomly, R0 is calculatedusing Equation 1:

R0 = ND Equation 1

where is the rate at which two specific individuals come into effective contact per unittime, N is the total population size, and D is the duration of infectiousness.

However, as we saw in MD06 , if we assume that individuals do not mix randomly, needs to be stratified according to the subgroups in the population. For example, the rateat which an infectious child comes into effective contact with an adult may differ from therate at which an infectious child comes into effective contact with another child.




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page 22 of 97

Extending the logic of Equation 1 , and considering a population in which individuals arestratified into the young or old, we can write down the number of young infectiousindividuals resulting from the introduction of one infectious person into a totally susceptiblepopulation as follows:

Ryy = yy Ny D Equation 7

where yy is the rate at which two specific young individuals come into effective contact perunit time, Ny is the number of young individuals in the population and D is the duration ofinfectiousness.

Similarly, the number of infectious old individuals resulting from the introduction of oneinfectious young person into a totally susceptible population is given by:

Roy = oy No D Equation 8

where oy is the rate at which a specific young infectious person comes into effectivecontact with a specific old susceptible individual, and No is the total number of oldindividuals in the population.




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page 23 of 97

The expressions for the number of young secondary infectious individuals resulting fromeach old infectious person and the number of old secondary infectious individuals resultingfrom each old infectious person (Ryo and Roo respectively) are analogous:

Ryo = yo Ny D Equation 9

and

Roo = oo No D Equation 10




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page 24 of 97

As we saw on page 16 , the Next Generation Matrix is defined as the matrix thatsummarises the number of secondary infectious individuals in a given category resultingfrom infectious individuals in each of the categories. We can now write down our NextGeneration Matrix for our population comprising young and old individuals as follows:

The following diagram summarises the notation used for the subscripts:

Notice that the first subscript of Ryo (i.e. "y") reflects the category of the recipient of theinfection and matches both the first subscript of the parameter and that of the size of thepopulation subgroup on the right-hand side of the equation.

The second subscript (i.e."o") reflects the category of the source of infection and matchesthe second subscript of the parameter on the right-hand side of this equation.




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Answer

8.5: Writing the Next Generation Matrix: Exercise

page 25 of 97

During the last session we calculated the following WAIFW matrix describing effectivecontact between the young and old in a region in England with 500,000 individuals, usingdata on rubella, where the parameters are in units of per day:

In this population, , and the

duration of infectiousness was 11 days.

Optional reading - explanation of the equations for Ny and No

Q1.2 Write down the Next Generation Matrix corresponding to the above WAIFW matrix.




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page 26 of 97

In general terms, if we have a population in which individuals are stratified into multiplegroups, the number of secondary infectious individuals in group i produced by an infectiousperson in group j in a totally susceptible population is denoted as Rij. Extending the logicon page 24 , it can be calculated using the following equation:

Rij= ij Ni D

where ij is the rate at which an infectious individual from group j comes into effectivecontact with a specific susceptible individual from group i per unit time, and Ni is the totalnumber of individuals of group i.

The Next Generation Matrix simply contains all possible values for Rij as its entries. In thismatrix Rij is the value in the i

th row and the jth column.




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Answer

Answer

Section 9: Exercise: Writing down the Next Generation Matrixfor a population comprising 3 groups

page 27 of 97

Q1.3 Suppose we have a population in which individuals are stratified into the young,middle-aged and old (denoted by the letters y, m, o) and that there are Ny, Nm, Noindividuals in these groups.

a) Write down the equivalent of the matrix provided on page 16 for this population.

b) Write down the expressions for Ryy, Rym and Ryo, in terms of Ny, yy, ym, yo andD.




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Answer

Answer

9.1: Exercise: Writing down the Next Generation Matrix for apopulation comprising 3 groups

page 28 of 97

Q1.3 ctd

c) Write down the expressions for Rmy, Rmm, and Rmo in terms of Nm, my, mm, moand D.

d) Write down the expressions for Roy, Rom, and Roo in terms of No, oy, om, oo andD.

Click here to see the Next Generation Matrix.




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Section 10: Calculating R0

page 29 of 97

The number of secondary infectious individuals resulting from the introduction of a "typical"infectious person into a totally susceptible population will be some average of each of thenumbers of the Next Generation Matrix.

As demonstrated by an example in section 7.5.2 of the recommended course text5 ,taking the average of the number of infectious individuals resulting from each young andold person does not lead to the basic reproduction number. Instead, alternative methodsneed to be used.

The theory to calculate the basic reproduction number was developed by Heesterbeek etal during the 1990s (see Diekmann et al (1990)3 ), and we will explore its applications inthe next pages. Unfortunately the mathematical proof is beyond the scope of this course.

Before presenting the method for calculating the basic reproduction number, we first showhow R0 can be calculated from some simple Next Generation Matrices.




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Answer

Answer

10.1: Calculating R0 from the Next Generation Matrix -exercise

page 30 of 97

EXAMPLE 1

Consider the following example of a Next Generation Matrix for a given population in whichindividuals are either young or old, denoted by the letters y and o respectively:

Q1.4a

i. How many infectious young individuals does each infectious young person lead to?ii. How many infectious old individuals does each infectious young person lead to?

Q1.4b

i. How many infectious young individuals does each infectious old person lead to?ii. How many infectious old individuals does each old infectious person lead to?

Q1.4c Given your answers to parts a and b, which of the values listed below is the R0?

Hint: think about the total number of secondary infectious individuals generated by either ayoung or old person.

R0 = 1 R0 = 2 R0 = 3 R0 = 6




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Answer

10.2: Calculating R0 from the Next Generation Matrix -Example

page 31 of 97

EXAMPLE 2

Consider the following Next Generation Matrix:

Q1.5a

i. How many secondary infectious individuals in total does each young infectiousperson lead to?

ii. How many secondary infectious individuals in total does each old infectious personlead to?

Q1.5b Which of the following is the basic reproduction number for this Next GenerationMatrix?

R0 = 1 R0 = 2 R0 = 3 R0 = 4




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10.3: Calculating R0 for more complicated Next GenerationMatrices

page 32 of 97

Calculating R0 for the previous examples was relatively straightforward because eachinfectious young and old person led to the same total number of infectious individuals.However, in practice, this is rarely the case.

Consider the following example:

The diagram on the right provides a visualrepresentation of this Next Generation Matrix. Thishighlights the following:

One infectious child will lead to 1secondary infectious child and 1 secondaryinfectious adult.

One infectious adult will lead to 1secondary infectious child and 4 secondaryinfectious adults.

We will think about how we can calculate R0 on the next few pages.




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10.4: Calculating R0 for more complicated Next GenerationMatrices

page 33 of 97

To calculate R0, we need to define a "typical" infectious person, who is some suitableaverage of the subgroups.

For the matrices considered in the last few pages, this means that the "typical" infectiousperson will be partly young and partly old. In mathematical terms, if a fraction x of thistypical infectious person is young, then by definition, a fraction (1-x) must be old. We canrepresent this infectious individual using the following vector notation:

See page 6 of the maths refresher to review the definition of vectors and theirrelationship to matrices.You may prefer to do this at a later stage, after you have had thechance to read through the next few pages.

You will see how we can find the values for x and R0 on the next page.




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Section 11: Defining the typical infectious person and findingR0

page 34 of 97

According to mathematical theory (the proof of which is beyond the scope of this course,see Diekmann 19903 ), if we simulate the introduction of an infectious person into a totallysusceptible population in which there is an unlimited supply of susceptible individuals andin which individuals contact each other according to some Next Generation Matrix, thentwo things happen:

1. The number of secondary infectious individuals resulting from each infectious personin each generation converges to R0; and

2. The distribution of the infectious individuals in each generation converges to somedistribution, which reflects that of the "typical" infectious person.

Therefore if we divide the number of infectious individuals in one generation by that in theprevious generation, we will get R0. Likewise, if we calculate the proportion of theindividuals in each generation who belong to each of the subgroups, we can obtain theproportion of the typical infectious person which belongs to these subgroups and obtain x.

In this module, we shall refer to this method for calculating R0 as the simulationapproach. The application of this theory is fairly straightforward, as we shall show on thenext few pages.




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11.1: Defining the typical infectious person and finding R0

page 35 of 97

On the next few pages, we will show that the simulation process described on the previouspage is equivalent to repeatedly multiplying some vector representing an initial infectiousperson introduced into a totally susceptible population by the Next Generation Matrix.

We will illustrate how we can do this in Excel for the Next Generation Matrix that weintroduced on page 32 :




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11.2: Exercise - Finding R0 and x

page 36 of 97

1. Open the Excel file NGM_demo.xlsx .

You should see something resembling the the image below:

a) Pink and blue cells containing the Next Generation Matrix (cells B4:C5). Thesecells have been assigned the names R_yy, R_yo, R_oy and R_oo.b) Cells F4 and F5 contain the number of infectious young and old individualsrespectively at the start.c) Cell F7 contains the total number of infectious individuals at the start.




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Answer a

Answer b

Answer c


page 37 of 97

Q1.6 Using pen and paper, calculate the following:

a) The number of secondary infectious young and old individuals that you wouldexpect to see in the first generation.b) The total number of infectious individuals that you would expect to see in the firstgeneration.c) The ratio between the number of infectious individuals in the first generation andthat at the start.

Click here if you would like to revise how matrices can be multiplied to vectors.

2. If you wish, use pen and paper to calculate the numbers mentioned in Q1.6 for thesecond generation. Alternatively, return to the spreadsheet, select columns F andAA together, click with the right mouse button and select the unhide option.Similarly, select rows 8 and 10 together, click with the right mouse button and selectthe unhide option.

You should now see how the number of infectious individuals changes with eachgeneration. You should also see how the ratio between the number of infectiousindividuals in a given generation and that in the preceding generation converges to somevalue (4.3). This value equals R0.




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Answer


page 38 of 97

We will now explore how the proportion of infectious individuals who are young and oldchanges in each generation.

3. Select rows 10 and 13 together, click with the right mouse button and select the"Unhide" option.

4. You should now see cells showing the proportions of the infectious individuals ineach generation who are young (pink) and old (blue). You should notice that theseproportions converge to the values 0.232408 (young) and 0.767592 (old). Thesevalues reflect the proportions of the typical infectious person that is young and old.

5. Change the number of infectious individuals introduced into the population at thestart (in cells F4 and F5) to be the following, and look at the value for R0 in eachcase.

a) 1 young person and 1 old person.b) 50 young people and 20 old people.c) Any value for each category that you choose.

Q1.7 How does changing the number of infectious individuals introduced at the startchange your estimate of R0 (see cell Z9)?




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Answer


page 39 of 97

We will now see whether this method for calculating R0 works for another matrix.

6. Change the Next Generation Matrix in cells B4:C5 in the spreadsheet to be asfollows and identify the value for R0.




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page 40 of 97

As shown in the previous exercise, if we simulate the introduction of an infectious personinto a totally susceptible population in which different subgroups contact each otheraccording to some Next Generation Matrix, then the ratio between the number of infectiousindividuals in successive generations, and the proportions of individuals in each generationthat belong to different subgroups converge to some values. As shown by Heesterbeek etal 3-4 , these numbers represent the values for R0 and the proportion of the typicalinfectious person that belongs to each of the subgroups respectively.

For example, Figure 7 plotsthe estimates for the fractionof the infectious person ineach generation that isyoung (=x) or old (=1-x) ineach generation, which weobtain through thesimulation processdescribed above if we usethe following Next

Generation Matrix: .

The graph shows that afterseveral (in this case 4)generations the distributionof young and old infectiousindividuals in eachgeneration converges to avalue that gives thedistribution of the "typicalinfectious person".

For this Next GenerationMatrix, the typical infectiousperson is 23% young and77% old.

Figure 7. Proportion of individuals in each generation that are young orold, obtained by simulating the introduction of infectious individuals into atotally susceptible population, with unlimited numbers of susceptibleindividuals, in which individuals contact each other according to matrix

.




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page 41 of 97

The same applies when we plot the ratio between the numbers of infectious individuals ina given generation and that in the previous generation.

We can see this in Figure 8, which shows that after several generations have occurred,the ratio between the numbers of infectious individuals in successive generationsconverges to some value, which equals R0. In this case R0 4.3.

Figure 8. Ratio between the number of infectious individuals in successive generations, obtained by simulatingthe introduction of infectious individuals into a totally susceptible population, with unlimited numbers of

susceptible individuals, in which individuals contact each other according to matrix .




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Section 12: Other methods for calculating R0

page 42 of 97

There are alternative ways to calculate R0. We discuss two of them here.

Simultaneous equations approach

It can be shown that the simulation approach that we introduced on page 34 forcalculating R0 and the proportion of the typical infectious person that is young is equivalentto finding the values for R0 and x which satisfy the following two equations simultaneously:

Ryyx + Ryo (1-x) = R0x Equation 16

Roy x + Roo(1-x) = R0 (1-x) Equation 17

For the purposes of this study module, you do not need to be able to derive theseequations. However, if you would like to find out how we can obtain them, a simplifiedderivation of these equations, based on the work of Diekmann et al 3-4 is provided in theAppendix A.5.1 of the recommended course text5 .

Equation 16 and Equation 17 can also be written in matrix form as follows:

Ryy Ryo x= Ro

xEquation 18Roy Roo

1 -x 1 - x

You can look at page 6 of the maths refresher or page 16 of MD06 if you would liketo revise how simultaneous equations can be written using matrix notation.

On the next few pages, we will use the equations to find the R0 and x for the followingNext Generation Matrix that we saw on page 32 :




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Explanation

12.1: Other methods for calculating R0: Illustration of thesimultaneous equation approach

page 43 of 97

We begin by substituting the matrix into Equation 18 to obtain the

following:

1 1

x = Ro

xEquation 191 4 1 - x 1 -x

After applying the rules for multiplying a matrix to a vector (see page 6 of the mathsrefresher , we see that the left-hand side of this equation simplifies to the following:

Using this simplification, we see that Equation 19 can be written equivalently as

or, using simultaneous equations, as:

1 = R0x Equation 20

4 - 3x = R0 (1-x) Equation 21

After some rearranging (click on the button below for the explanation), these equations canbe solved to give the following two sets of possible values for x and R0, respectively.

R0 4.3 and x 0.23 orR0 0.69 and x 1.23

Since it is not possible for the fraction of the typical infectious person that is young to bebigger than 1, we are led to accept the value for R0 and x of 4.3 and 0.23, respectively.

These values are consistent with those estimated using the simulation approach seenpreviously (see pages 40 and 41 ).




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12.2: Optional - other methods for calculating R0

page 44 of 97

As illustrated by the example on page 43, there may be more than one value for "R0"which satisfies Equation 18 . According to mathematical theory (which is beyond thescope of this course), R0 is then taken to be the largest value which satisfies that equation.

A hand-waving explanation for why R0 is taken as the largest value which satisfiesEquation 18 is that, if we substitute it into the equation for the herd immunity threshold (1-1/R0), it leads to the higher value for the proportion of the population which needs to beimmune to control transmission. If coverage of the intervention were to be introduced atthis level in the overall population, it is likely to be sufficiently high to control transmissioneven in the highest risk group.

The mathematical name for the largest value satisfying Equation 18 is the "dominanteigenvalue of the Next Generation Matrix".

If you are interested in reading more about this, see Diekmann and Heesterbeck3 , andthe recommended course text5 , pages 212-215 for further details.

We will now consider another approach for calculating R0. We will refer to this method asthe Matrix determinant approach




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page 45 of 97

Matrix determinant approach

If we have a population comprising 2 subgroups (denoted using the subscripts y and orespectively), then it can be shown that finding the value for R0 which satisfies Equation18 is equivalent to finding the value of which satisfies the following equation:

(Ryy - )(Roo - ) - RyoRoy = 0 Equation 24

As R0 is given by the largest value which satisfies Equation 18, R0 must also be the largestvalue of which satisfies Equation 24.

For the purposes of this study module, you are not expected to be able to derive Equation24. However, if you are interested, further details of how Equation 24 can be derived fromEquation 18 and why this approach can be referred to as the "matrix determinantapproach" are provided in the recommended course text5 on pages 212-215 andAppendix A.5.1.2.

We will illustrate how we can apply Equation 24 on the next page.




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page 46 of 97

We return to the matrix that we considered on page 32 , namely .

In this matrix Ryy = 1, Ryo = 1, Roy = 1 and Roo = 4.

Substituting for these values into Equation 24 , we obtain the following equation:

(1 - )(4 - ) - 1*1 = 0

This equation can be rearranged to give the following:

2 - 5 + 3 = 0

This equation is analogous to Equation 22 which is discussed in the derivation of the valuefor R0 using the simultaneous equations approach (click on the "Explanation button" onpage 43 ). The remainder of the derivation presented on that page, together with the factthat R0 is the largest value of which satisfies Equation 24 leads to the result that R0 4.3.




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Section 13: Calculating the net reproduction number

page 47 of 97

The methods that we have described so far in this session to calculate Ro can be appliedto calculate the net reproduction number (Rn) for a non-randomly mixing population, inwhich some individuals may already be immune, as a result of vaccination or previousinfection. In this instance, the Next Generation Matrix is written down using the number ofsusceptible (rather than all) individuals in each group. For example, considering apopulation in which individuals are stratified into the young and old, Ryy, Ryo, Roy, and Roowould be given by the following equations (see also the diagram below):

Ryy = yySyD

Ryo = yoSyD

Roy = oySoD

Roo = ooSoD

where Sy and So are thenumber of susceptible youngand old individuals.

Rn is then calculated usingthe resulting NextGeneration Matrix in thesame way we calculated R0above. We will illustrate thisin further detail in thepractical part of this session.




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13.1: Extending the method to deal with more than twosubgroups

page 48 of 97

The methods that we have described to calculate R0 can also be extended relatively easilyto deal with populations consisting of more than two subgroups.

For example, if the population comprised 3 subgroups, a fraction x, w and 1-x-w of thetypical infectious person would belong to the first, second and third subgroups. For a NextGeneration Matrix such as the following, in which the population is stratified into the young,middle-aged and the old (denoted by the letters y, m and o respectively):

we could calculate R0 by simulating the introduction of an infectious person into a totallysusceptible population in which there is an unlimited supply of susceptible individuals andin which individuals contact each other according to this Next Generation Matrix.




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13.2: Extending the method to deal with more than twosubgroups

page 49 of 97

As in the case when the population is stratified into two age groups , R0 then equals theratio between the number of infectious Individuals in a given generation and that in theprevious generation after the ratio had converged. x, w and (1-x-w) can then be calculatedas the proportion of each generation which are in the first, second and third subgroupsrespectively.

Adapting the simultaneous equations approach described for two age groups to dealwith three age groups, we see that R0 can also be calculated by solving the followingmatrix equation:

or, for the matrix described on the previous page, as follows:

This equation could be written equivalently using the following simultaneous equations:

x+5w+4(1-x-w) = R0x

2x+3w+6(1-x-w) = R0w

3x+7w+8(1-x-w) = R0(1-x-w)




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Section 14: Practical application of the Next GenerationMatrix to calculate Rn for measles in England and Wales

page 50 of 97

The methods described above have been and are still used to estimate the netreproduction number for various diseases. In particular, during the mid-1990s thesetechniques were used by the then Public Health Laboratory Service to assess the potentialfor a measles epidemic to occur in England and the need for further measles vaccination.The work was carried out by Gay and colleagues, and has since been published6-7 .

We discuss this application on the next few pages.




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14.1: Practical applications: calculating Rn for measles inEngland and Wales - background

page 51 of 97

Measles vaccine was introduced in England in 1968, but its uptake was variable (seeFigure 9, dotted line). The introduction of vaccination resulted in a decline in the notificationrates for measles (see Figure 9, bars), and by the early 1990s they had reached a verylow level.

During the first half of 1994, slight increases were seen in the notification rates (see Figure9, red circle); a large outbreak had occurred in Scotland during the period 1993-4 andthere were concerns that a large epidemic, with more than 100,000 cases, was imminent.

Work was carried out by Gay et al to estimate the net (or effective) reproduction number(Rn) in England and Wales and whether it was likely that an epidemic would occur

6 .

Figure 9. Measles notifications and deaths in England and Wales. Extracted from Gay et al6 , .




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14.2: Practical applications: calculating Rn for measles inEngland and Wales the model

page 52 of 97

Figure 10 shows examples of the WAIFW matrices calculated using values for the force ofmeasles infection estimated from data from England and Wales from before theintroduction of measles vaccination6 . The force of infection among those aged over 10years was not reliably known, due to the high prevalence of immunity in this agegroup. This meant that the parameters for a given structure for the WAIFW matrix couldnot be reliably estimated. For a given WAIFW structure, Gay et al therefore used severaldifferent assumptions about the amount of contact between individuals in this age groupand those in other age groups.

Figure 10. Examples of WAIFW matrices describing contact between different age groups in England and Walesobtained by Gay et al, using estimates of the force of infection for measles, calculated using data collectedbefore the introduction of vaccination6 . reflects the factor by which the rate at which 10-14 year olds comeinto contact with each other differs from that between individuals aged 5-9 years. Extracted from Gay et al6 .




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14.3: Practical applications: calculating Rn for measles inEngland and Wales the model

page 53 of 97

Gay et al then combined these estimates with estimates of the proportion of individuals indifferent age groups who were susceptible to measles infection (see Figure 11) in 1994.

Figure 11. Estimates of the percentage of different birth cohorts alive in England and Wales who weresusceptible to measles in 19946 , calculated using haemagglutination inhibition (HI) data for those born before1984 and different assumptions about the proportion susceptible for those born after 1985/6 (labelled ScenarioA" and "Scenario B). See Gay et al6 for further details of these assumptions.




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14.4: Practical applications: estimates of Rn for measles inEngland and Wales

page 54 of 97

The figure below shows the estimates for the net reproduction number (Rn) obtained byGay et al using different assumptions about contact between 10-14 year olds. Theseresults highlighted that the net reproduction number in 1994 in England and Wales wasvery close to 1 (the horizontal dashed line) and that there was potential for an epidemic tooccur.

Further calculations carried out by Gay et al suggested that if an outbreak occurred, itcould involve more than 100,000 cases. These conclusions were supported by otherstudies using dynamic models 6 .

As a result of these analyses, a measles-rubella vaccination campaign was carried out inNovember 1994 in England and Wales, targeting 95% of the 7 million 5-16 yr olds. Nomeasles epidemic was recorded during the subsequent few years.

This was perhaps the first time that modelling was used to guide vaccination policy in theUK. Since then, the potential for a measles epidemic to occur in England and Walescontinues to be evaluated in the same way7 .

If you have time, try Exercise 7.4 and the exercises associated with model 7.5 of the onlineexercises associated with the course text5 , where you can try doing these calculationsyourself.

Figure 12. Estimates for Rn during the 1990s in England and Wales obtained using different assumptions about




contact between individuals aged 10-14 years. See page 52 and Gay et al6 for further details.

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Section 15: Break

page 55 of 97

We have now completed part 1 of this session, in which we covered the theory for how wecan calculate R0 to account for non-random mixing between individuals.

The rest of this session consists of a practical exercise, during which you will be able toapply the theory, using Excel and Berkeley Madonna. This exercise is likely to take 1-3hours . You may like to take a break before continuing.




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Section 16: Practical: calculating the basic reproductionnumber for a non-randomly mixing population

page 56 of 97

OVERVIEW

We will now start parts 2 and 3 of this session during which you will apply the theorydiscussed in the first part to calculate the Next Generation Matrix and R0 for a given set ofassumptions about contact between individuals.

OBJECTIVES

By the end of the practical part of this session you should:

Be able to write down the "Next Generation Matrix" for given assumptions aboutcontact between individuals.Understand the relationship between the basic reproduction number and the NextGeneration Matrix.Be able to calculate R0 by using the simulation approach, i.e. simulating transmissionfollowing the introduction of one infectious person into a totally susceptiblepopulation mixing according to given contact patterns.Be able to calculate the net reproduction number from the Next Generation Matrix.Be able to calculate R0 using the simultaneous equations approach in Excel.

Part 2 of this session provides further practice in writing down the Next Generation Matrixand using it to calculate R0. Part 3 of this session illustrates how R0 can be calculatedusing the simultaneous equations and matrix determinant approaches.




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16.1: Part 2 (Practical): Introduction

page 57 of 97

In this exercise, we will revisit the population that we worked with in the practicalcomponent of MD06 , in which individuals were stratified into the young, middle-agedand old, and illustrate how we can calculate R0 associated with matrices WAIFW A andWAIFW B.

Click here to remind yourselves of these matrices.




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16.2: Part 2 (Practical): Calculating the Next GenerationMatrix

page 58 of 97

We first focus on how we can calculate the Next Generation Matrix associated withWAIFW B.

Calculating the Next Generation Matrix for populations mixing accordingto WAIFW B

1. Open up the Excel file R0waifb.xls . The layout of the spreadsheet shouldresemble what you see in the image below:

The blue cells (rows 2-20) contain the parameters required to calculate the basicreproduction number, namely:

a) The average duration of infectiousness (ave_infous).




b) The total number of individuals in the young, middle-aged and old categories(N_y, N_m and N_o respectively).c) The proportion immune (prop_imm, currently set to be 0).d) The number of young, middle-aged and old susceptible individuals (S_y, S_mand S_o respectively).e) The daily rate at which specific infectious and susceptible individuals in differentage categories come into effective contact, namely

b_yy, b_ym, b_yo, b_my, b_mm, b_mo, b_oy, b_om, b_oo,

located in cells F17:H19.

NOTE: You can see the name of a given cell by clicking on that cell and looking in thebox in the top left hand corner of your sheet, just below the menu bar.

The orange cells in cells F25:H27 will contain the entries for the Next Generation Matrix,and have been assigned the names R_yy, R_ym, R_yo, R_my, R_mm, R_mo, R_oy,R_om and R_oo.

We will set up expressions in these cells later. Please do not do so just yet!

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Answer


page 59 of 97

Q2.1 Looking at the orange cells in the spreadsheet (cells F25:H27), how many secondaryinfectious individuals among young individuals will occur as a result of the introduction of:

i) 1 infectious young personii) 1 infectious middle-aged person andiii) 1 infectious old person?

You may have noticed that the equations in cells F25, G25 and H25 have been set upusing the number of susceptible young, middle-aged and old individuals, rather than interms of the total number of individuals in each category. We could have set up theseequations to be in terms of the total numbers of individuals in each category, i.e. N_y, N_mand N_o. However, later in this practical, we will vary the proportion of the population thatis immune in the population and use these cells to calculate the net reproduction number.For this reason, the numbers in these cells have been expressed in terms of the number ofsusceptible individuals in each category.




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Hint Answer


page 60 of 97

2. Using pen and paper, write down the appropriate expressions for the number ofsecondary infectious individuals which would occur among middle-aged susceptibleindividuals as a result of the introduction of:

i) 1 young infectious person,ii) 1 middle-aged infectious person, andiii) 1 old infectious person.

3. Now set up the appropriate expressions in cells F26, G26 and H26 of thespreadsheet.




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Answer


page 61 of 97

4. Select the appropriate terms (from the drop-down menus below) to complete theequations for the number of secondary infectious individuals which would occuramong old susceptible individuals as a result of the introduction of:

i) 1 young infectious person = * *

ii) 1 middle-aged infectious person=

* *

iii) 1 old infectious person = * *

5. Now set up the appropriate equations in cells F27, G27 and H27 of the spreadsheet.




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Answer


page 62 of 97

Q2.2 How many secondary infectious individuals does each young, middle-aged and oldinfectious person generate in a totally susceptible population? Select the correct option foreach age group from the drop down menus.

a) Each young infectious person generates infectious individuals in atotally susceptible population.

b) Each middle-aged infectious person generates infectiousindividuals in a totally susceptible population.

c) Each old infectious person generates infectious individuals in atotally susceptible population.

If you find that your answers differ from those provided, click here to open upR0waifwb_solna.xlsx, where you can check your expressions.




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Section 17: Part 2 (Practical): Calculating R0 using thesimulation approach

page 63 of 97

Our Next Generation Matrix is now as follows:

We will now calculate the basic reproduction number corresponding to this NextGeneration Matrix using the simulation approach that we introduced on page 34 .




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Answer

17.1: Part 2 (Practical): Calculating R0 using the simulationapproach

page 64 of 97

6. Select rows 43 and 58 together, click with the right mouse button and choose the"Unhide" option.

You should now see some pink cells containing statistics relating to the number ofinfectious individuals which result over time as a result of the introduction of one infectiousperson into a totally susceptible population. At present, this person has been specified tobe young.

Q2.3 According to cell B48, how many young infectious individuals will occur in the firstgeneration as a result of the introduction of this infectious individual?




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Hint Answer


page 65 of 97

7. Set up an appropriate expression for the number of middle-aged infectiousindividuals (in cell C48), which will occur in the first generation as a result of theintroduction of the initial infectious person.




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Answer


page 66 of 97

8. Similarly, set up appropriate expressions for:

i) the number of old infectious individuals (in cell D48), andii) the total number of infectious individuals (in cell E48)

which will occur in the first generation as a result of the introduction of the initialinfectious person.

If your answer differs from the one provided above, refer to R0waifb_solnb.xlsx whichholds the expressions that you should have set up by now.

NOTE: There is an alternative method for setting up these equations in Excel, which isexplained on the next page.




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Section 18: Part 2 (Practical): Calculating R0 using thesimulation approach

page 67 of 97

Alternative expressions for the number of infectious individuals in agiven age category

As an alternative, you could have also typed the following into the cell for the number ofyoung infectious individuals in the first generation:

=SUMPRODUCT($F$25:$H$25,B47:D47)

This calculates the sum of the cross-product of the cells in the range F25:H25 andB47:D47 i.e.

F25*B47 + G25*C47 + H25* D47

The dollar signs have been inserted for the cell range F25:H25 so that, when this formulais copied down to the next row, it still refers to the same cells F25:H25.

The corresponding formulae for the number of middle-aged and old infectious individualsin the first generation would be:

=SUMPRODUCT($F$26:$H$26,B47:D47) (middle-aged infectiousindividuals)=SUMPRODUCT($F$27:$H$27,B47:D47) (old infectious individuals)




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Answer

Answer

18.1: Practical: Part 2 (Practical): Calculating R0 using thesimulation approach

page 68 of 97

Q2.4 According to cells G48-I48, what proportion of infectious individuals in the firstgeneration are young, middle-aged and old?

Click the button below to see what you should see at this point.

Q2.5 According to cell K48, how many secondary infectious individuals resulted directlyfrom the initial infectious person introduced into the population?




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page 69 of 97

We will now explorewhat happens in thesecond andsubsequentgenerations.

9. Copy all theexpressions forthe firstgenerationdown until the10thgeneration.

10. Now selectcolumns N andY together,click with theright mousebutton andselect theunhide option.

You shouldnow see twofiguresresemblingthose shownon the right.

The top graphshows the agedistribution of theinfectious individualsin each generation.




The bottom graphshows the averagenumber of secondaryinfectious individualsresulting from eachinfectious person.

If the graphs that yousee at this stagediffer from thoseshown here, clickhere to open upR0waifwb_solnc.xlsx,where you can checkyour expressions.

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Answer

Answer


page 70 of 97

Q2.6 What happens to the age distribution of the new infectious individuals in eachgeneration after a few generations have occurred?

Q2.7 What is the average number of secondary infectious individuals resulting from eachinfectious person after a few generations have occurred?




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Answer

Section 19: Part 2 (Practical): Effective vaccination coveragerequired to control transmission

page 71 of 97

11. Calculate the level of effective vaccination coverage that would be required to controltransmission in a population which mixed according to WAIFW B, assuming that thebasic reproduction number was equal to the value obtained in the previous question.




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19.1: Part 2 (Practical): Testing the estimated value for R0using the Berkeley Madonna model

page 72 of 97

We will now return to the model that we worked with in MD06, which described thetransmission of rubella in a population in which individuals were stratified into the young,middle-aged and old. We will test to see if the value for R0 that we estimated for WAIFW Bis correct by exploring what happens when we incorporate vaccination at levels ofcoverage which are similar to those calculated in the previous step.

12. Start up Berkeley Madonna and open the Berkeley Madonna file waifwb_ R0 -flowchart.mmd or waifwb_R0 - equations.mmd . The model in this file isidentical to the one which you used in in MD06, except for the fact all the parameters for WAIFW B have already been set up. Click here if you would like toremind yourself of the key features of this model. Run the model and click on Page 3of the Figures window to see the daily number of new infections per 100,000population for the three age groups.

Vaccination of newborns is currently introduced on day 36500 (or year 100) in the model.




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Answer


page 73 of 97

Q2.8 How do you think predictions of the daily number of new infections per 100,000among young, middle-aged and old individuals will change if you introduce vaccination ofnewborns at the following levels of effective coverage:

i) 72%ii) 72.2%iii) 72.3%iv) 72.4%v) 72.5%

13. Check your hypothesis by running the model for the levels of effective coveragespecified in the previous question. Remember that you can modify the vaccinecoverage by moving the slider to the desired level, or by typing in the value in theparameters window.

You can check that you are getting the correct figures by clicking on the tab which islabelled with the vaccination coverage that you are interested in. If your model has crashedor you are not getting these results, open up the file waifwb_ R0 - flowchart.mmd orwaifwb_R0 - equations.mmd again and ensure that you are entering the vaccinationcoverage as a proportion and not as a percentage.


72% vaccination coverage

72.2% vaccination coverage

72.3% vaccinationcoverage





According to these graphs, transmission appears to stop if the level of effective vaccinationcoverage is 72.3%, which is below the herd immunity threshold that we calculated in step11 . However, this conclusion is incorrect. If we click on the table button , we see thata tiny number of new infections are still occurring even on day 300,000.

As we shall show on the next page, we can see this in detail if we change the figure sothat the number of new infections per 100,000 is plotted on a log scale.

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Show


page 74 of 97

14. Copy the figure on Page 3 to a new figure by clicking on the button and changethe scale on the y-axis to go from a minimum of 10-40 to a maximum of 20, and to belogarithmic. See page 5 of the guide to Berkeley Madonna if you would like toremind yourself of how to change the scale of the y-axis.

Click the button below to see the figure that you should see by this stage for levels ofeffective coverage of 72.3%, 72.4% and 72.5%.

If your output fails to match these figures, check your settings against those in the fileWAIFWB_R0 - flowchart_solna.mmd or WAIFWB_R0 - equations_solna.mmd .




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Answer


page 75 of 97

Q2.9 What can we conclude about the herd immunity threshold from our BerkeleyMadonna model?




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Answer

Section 20: Part 2 (Practical): Effect of the initial numbers ofinfectious individuals on estimates of R0

page 76 of 97

15. Return to the spreadsheet that you were working with. If you have already closed thespreadsheet, click here to open up the spreadsheet that you should havedeveloped by now. Change the numbers of infectious individuals introduced into thepopulation at the start to take the following values and look at the figures showingthe age distribution of infectious individuals in each generation, and the ratiobetween the numbers of infectious individuals in successive generations:

i) 20, 50, 30 young, middle-aged and old infectious individuals respectively.

ii) 30, 20 and 2 young middle-aged and old infectious individuals respectively.

iii) 0.5, 0.2 and 0.3 young, middle-aged and old infectious individualsrespectively.

Q2.10 How does changing the values for the numbers of infectious individuals introducedinto the population at the start affect the age distribution of infectious individuals and theratio between the number of infectious individuals in successive generations, after a fewgenerations have occurred?




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Expected results

20.1: Part 2 (Practical): Net reproduction number

page 77 of 97

The methods applied in this session can also be used to calculate the net reproductionnumber if, for example, a proportion of the population is immunised.

16. Still working with your spreadsheet, change the value for prop_imm (cell F8) to thevalues listed below to see what happens to the average number of secondaryinfectious individuals resulting from each infectious person after a few generationshave occurred if the following proportions of the population are immune:

i) 25%ii) 50%iii) 72.5%iv) 75%

Before continuing, you may like to save your Excel file.




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Section 21: Part 2 (Practical): Additional exercises

page 78 of 97

We have now completed part 2 of this session.

Before continuing to part 3 of the session, you may like to repeat your calculations usingthe transmission parameters relating to WAIFW A . You should find that the basicreproduction number corresponding to this matrix is about 10.9. When doing this, you willneed to ensure that infectious individuals are introduced into each of the young, middle-aged and old age groups at the start, as otherwise, the answer that you get may bemisleading.




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21.1: Part 3 (Practical): Calculating R0 using the simultaneousequations approach

page 79 of 97

As mentioned on page 49 , if we have a population in which individuals are stratified intothree groups, e.g. the young, middle-aged and old, the basic reproduction number can befound by identifying the values x, w and R0 for which the following matrix equation holds:

Equation A

This equation can be written out in full as follows:

Ryyx+Rymw+Ryoz = R0x Equation B

Rmyx+Rmmw+Rmoz = R0w Equation C

Royx+Romw+Rooz = R0z Equation D

where z = 1 - x - w.

Equation B is equivalent to saying that the total number of young infectious individualsresulting from each infectious person equals R0x.

Equation C is equivalent to saying that the total number of middle-aged infectiousindividuals resulting from each infectious person equals R0w.

Equation D is equivalent to saying that the total number of old infectious individualsresulting from each infectious person equals R0(1-x-w).

The values for x, w and z (=1-x-w) when all three equations B - D hold simultaneouslyprovide the proportions of the typical infectious person which is young, middle-aged andold respectively.

We will now illustrate how values for x, w and R0 which satisfy equations B - D can beobtained using the "Solver" option in Excel.




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Answer


page 80 of 97

1. Return to the spreadsheet that you were working with. If you have already closed thefile, click here to open the file which you should have developed by now. If youhave not yet done so, reset the value for the proportion of the population which isimmune (in cell F8) to zero. Select rows 30 and 43 together, click with the rightmouse button and choose the "Unhide" option.

The layout of this section of the spreadsheet should resemble what you see in the imagebelow. It contains the following:

a) Yellow cells containing the proportions x, w and z which are currently set toequal 0.8, 0.15 and 0.05 respectively (see cells F33, F34 and F35). These cellshave been assigned the names x_, y_ and z_ respectively.b) Lilac cells containing the following:

i) an initial estimate of 4 for R0 (called R0_est), andii) cells which will eventually contain expressions for the number of young,middle-aged and old infectious individuals resulting from each infectiousperson using the left and right-hand side of equations B-D .

Q3.1 With the current value for R0_est, x, w and z, how many young infectious individualswill result from the introduction of an infectious individual?




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page 81 of 97

We now need to set up the left and right-hand side of equations C and D in theappropriate cells in our spreadsheet and find values for x, w, z and R0_est for which theleft-hand sides of Equations C-D equal their right-hand sides.

2. Set up the appropriate expression in cell F40 for the number of middle-agedindividuals resulting from the introduction of an infectious individual into a totallysusceptible population, using the left-hand side of equation C.

Click the button below to check your expressions for cell F40.

3. Set up the appropriate expression in cell F41 for the number of old infectiousindividuals resulting from the introduction of an infectious individual into a totallysusceptible population, using the left-hand side of equation D.

Click the button below to check your expression for cell F41.

If your answers differ from the ones provided, click here to open R0waifwb_solnd.xlsx,where you can check your equations.




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Show

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page 82 of 97

4. Set up the appropriate expression in cell H40 for the number of middle-agedindividuals resulting from the introduction of an infectious individual into a totallysusceptible population, using the right-hand side of equation C.

Click the button below to check your expression for cell H40.

5. Set up the appropriate expression in cell H41 for the number of old infectiousindividuals resulting from the introduction of an infectious individual into a totallysusceptible population, using the right-hand side of equation D.

Click the button below to check your expression for cell H41.

If your answers differ from those provided, click here to open R0waifwb_solne.xlsx,where you can check your equations.




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page 83 of 97

We will now try to find the values for x, w, z and R0_est for which the left-hand sides ofequations B-D equal the right-hand side of these equations using Excels Solver option.

You may recall from MD04 that to identify optimal parameter values using the Solverroutine, you need to provide the following:

a) The location of a single cell which contains an expression whose value you wantto maximise, minimise or to be set to some value (specified under the "Set targetcell" option) andb) The cell(s) which are allowed to change so that this maximum, minimum, etc isattained (under the "By changing cells" option).

We first consider how we can set up a single cell containing an expression which, whenminimised, contains the values for x, w and R0_est that we need. We shall use anexpression obtained by summing the squares of the difference between the values on theleft and right hand side of the equations. The rationale for this expression is described onthe next page.




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page 84 of 97

The expressions that we will be using have been hidden and we will first unhide them.

6. Return to your spreadsheet, select the range of cells J36:L42 and change the colorof these cells to be black.

Reminder: to change the colour of cells which you have selected, click on the Home tab,click on the arrow part of the Fill color button and choose the black box from thepalette of cell colours available. Click on OK to continue.

The layout of the spreadsheet should resemble what you see in the image below. Itcontains the following:

a) Cell K39 contains a formula for the sum squared of the difference between thenumber of young infectious individuals predicted using the left hand-side of equationB and that predicted using the right-hand side of equation B.b) Cells K40 and K41 have analogous formulae relating to equations C and D;c) Cell K42 holds the sum of the cells K39, K40 and K41.




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Answer


page 85 of 97

Q3.2 If R0_est were to equal the basic reproduction number and if x, w, and z were toreflect the proportion of the typical infectious person which is young, middle-aged and oldrespectively, what should be the values of the expressions in the following cells:

a) K39, K40, K41?b) K42 (i.e. the sum of cells K39, K40, and K41)?

OPTIONAL CHECKYou can test your answer by typing in the values for x, w and R0 which you obtai

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