md06

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Section 1: Overview and objectives

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OVERVIEW

In the models that we have worked with so far in the module, we have assumed thatindividuals mix randomly. In this session, we illustrate how we can incorporate theassumption that individuals mix non-randomly (heterogeneously) into models. We will alsosee how contact patterns between individuals influence the transmission dynamics andcontrol of infections.

OBJECTIVES

By the end of this session you should:

Be aware of some of the evidence for age-dependent mixing;Be able to define and construct matrices of "Who Acquires Infection from Whom"(WAIFW) to describe non-random mixing patterns between individuals;Know how to use force of infection estimates to calculate WAIFW matrices;Understand how non-random mixing patterns between individuals can affect thetransmission dynamics and control of infections.

This session comprises two parts and will take 2-5 hours to complete.

Part 1 (1-2 hours) describes some of the evidence which shows that people mix non-randomly and the methods for writing down and calculating matrices of Who AcquiresInfection From Whom. Part 2 (1-3 hours) consists of a practical exercise in BerkeleyMadonna, during which you will use a model to explore how different assumptions aboutcontact between individuals influence the impact of vaccination.

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EPM302 Modelling and the Dynamics of Infectious Diseases

MD06 Methods for incorporating non-random (heterogeneous) mixing intomodels

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Section 2: Introduction

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So far in this module, we have assumed that contact between individuals in a population israndom (regardless of characteristics such as age, gender or socio-economic status) andthat all individuals are equally likely to contact any other individual.

However, non-random ("heterogeneous") mixing is especially important in determining theimpact of control strategies, particularly those targeting certain subgroups.




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2.1: Introduction

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For example, shown below are contact patterns between children and adults fortwo different hypothetical populations.

In population A, each child contacts four other children and two adults, and each adultcontacts one child and two adults. In population B, each child contacts one other child andthree adults, and each adult contacts three children and one adult.

Suppose the same proportion of children in both populations have been vaccinated againsta new pandemic strain of influenza, but no adults are vaccinated.

Before proceeding, try to answer the following question: In which population is theincidence likely to be smallest once the new strain is introduced?




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Show

2.2: Introduction

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The previous question is intended to illustrate the fact that predicting the impact of anintervention that is introduced into a non-randomly mixing population is not straightforward.

For example, on the one hand, you might argue that the impact of vaccinating children willbe greater in population A than in population B, since children in population A contactmore individuals than do children in population B (i.e. 6 vs. 4).

However, we need to weigh this up against the fact that children in population A contactfewer adults than do children in population B (i.e. 2 vs. 3). Consequently, reducing thenumber of infectious children through vaccination will have a smaller impact on the numberof new infections occurring per unit time among adults in population A than in populationB.

In fact, using a simple argument which you can see by clicking on the show button below,we can show that we might expect the impact of vaccinating children to be greatest inpopulation B. However, we would need to use a transmission model if we wish to answerdetailed questions such as "How many new infections might be expected to occur per unittime in populations A and B following the introduction of the new strain if, for example,30% of children had been vaccinated?"




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2.3: Introduction

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In order to make reasonable predictions of the impact of interventions against infections inreal populations (in which individuals do not mix randomly) models need to includeassumptions about the amount of contact between individuals in different subgroups of thepopulation.

Before describing the methods for incorporating non-random mixing into models, we firstreview some of the evidence for age-dependent mixing.




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Section 3: Background: Evidence for non-random mixing

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There is much evidence to suggest that contact patterns are age-dependent.

One study in The Netherlands1 , examined the ages of pairs of tuberculosis cases whohad onset during the period 1993-1996 and whose isolates shared identical M tuberculosisDNA fingerprint patterns (Figure 1a).

It is reasonable to assume that these pairs represented a primary and a secondary case,although it is possible that both cases were infected by another person outside of thestudy1 .

Figure 1a shows that the ages of the two cases in the pairs were highly correlated,indicating that individuals are most likely to transmit infection to others of a similar age. Forexample, the average age difference between individuals in a pair was 13.9 years (SD12.2 years), which is statistically significantly smaller than the difference between that ofrandomly paired cases (25.5 years, 95% CI 21.5-29.5 years).

Analogous age patterns have been observed between presumed primary and secondarycases of measles and meningococcal meningitis in the UK from the period 1995-1998(Figure 1b)2 .




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3.1: Background: Evidence for age-dependent mixing

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The largest study of contact patterns (the POLYMOD study3 ) was published in 2008.Individuals from eight European countries completed a diary detailing their physical andnon-physical contacts on a single day between May 2005 and September 2006. A total of7,290 diaries were collected, and the number of diaries collected per country ranged from267 in The Netherlands to 1,328 in Germany3 .

The contact patterns observed were generally consistent across countries: individualswere most likely to contact others of a similar age (see Figure 2 for data from Great Britainand Germany). The study also found that there was much contact between 30-39 yearolds and young children, and between middle-aged adults and older children. Contactbetween these individuals probably reflects contact between parents and their children.

Additional evidence for age-dependent contact is provided in section 7.3.3 of therecommended course text4 . You may prefer to read this section after you havecompleted the session.




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Section 4: Revision of the relationship between , the forceof infection and the number of infectious individuals for arandomly mixing population

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As we saw in MD01 , when we assume that individuals mix randomly, the rate at whichsusceptible individuals are infected at a given time t (the force of infection, (t)) isexpressed in terms of two quantities, namely:

1. The contact parameter, (the rate at which two specific individuals come intoeffective contact per unit time) and

2. The number of infectious individuals at that time, I(t).

i.e. (t) = I(t)

, in turn, is related to the number of contacts that each person makes with others and theproportion of contacts which are sufficient to lead to transmission of infection (whichdepends on the infection).

However, as empirical data on the numbers of contacts that individuals typically make arerare, is usually inferred from epidemiological data.




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4.1: Revision of the relationship between , the force ofinfection and the number of infectious individuals for arandomly mixing population

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For randomly mixing populations (see MD01 ), can be calculated from the basicreproduction number (R0) using the following expression:

=R0

ND

where N is the total population size and D is the duration of infectiousness. As we saw inMD04 , R0 can be calculated using the average force of infection or the average age atinfection.




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Section 5: Methods for incorporating non-random mixing intomodels

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In populations where individuals do not mix randomly, the relationship between the force ofinfection, , and the number of infectious individuals is analogous to that used forrandomly mixing populations, except that the force of infection and must be stratifiedaccording to the subgroups represented in the model.

Example

Suppose we have a population in which the contact patterns of children differ from those ofadults. The rate at which children are infected depends on how closely they interact withother children and with adults. The subscript y will be used to denote children (theyoung) and the subscript o will be used to denote adults (the old).

At time t, the overall force of infection experienced by children can be expressed asthe sum of two terms:

the force of infection attributable to contact between children (yy(t)), andthe force of infection attributable to contact between children and adults (yo(t)), i.e.

y(t)=yy(t)+yo(t) Equation 1

Similarly, the overall force of infection experienced by adults at time t is given by thesum of:

the force of infection attributable to contact between adults (oo(t)) , andthe force of infection attributable to contact between adults and children (oy(t)), i.e.

o(t)=oo(t)+oy(t) Equation 2




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5.1: Methods for incorporating non-random mixing intomodels

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As we shall show on the next few pages, each of the terms yy(t), yo(t), oo(t), oy(t) canbe expressed in terms of the product of two factors, namely the rate at which effectivecontacts are made between individuals in different subgroups (for example, between onespecific child and another specific child, yy) and the number of infectious individuals in thesubgroup (e.g. children, Iy(t)).




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For example, the force of infection among children that is attributable to contact with otherchildren is given by the following equation:

yy(t) = yy Iy (t) Equation 3

where yy is the rate at which a specific susceptible child and a specific infectious childcome into effective contact per unit time, and Iy(t) is the number of infectious children attime t.

The definition and interpretation of yy is analogous to that for when we assume thatindividuals mix randomly. Click here if you would like to remind yourself of this.

Similarly, the force of infection among children that is attributable to contact with adults isgiven by the following expression:

yo(t) = yo Io (t) Equation 4

where yo is the rate at which a specific susceptible child and a specific infectious adultcome into effective contact per unit time, and Io(t) is the number of infectious adults at timet. The definition and interpretation of yy is analogous to that for when we assume thatindividuals mix randomly. Click here if you would like to remind yourself of this.

These expressions are derived on the following page.




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Proof that yy(t) = yy Iy (t)

As we saw in previous sessions, if we assume that individuals mix randomly, the numberof new infections per unit time is given by the expression:

S(t)I(t)

Extending this logic, the number of new infections among children that is attributable tocontact with other children is given by the expression:

yy Sy(t)Iy(t)

We can also express the number of new infections among children that is attributable tocontact with other children using the following expression:

yy(t)Sy(t)

Equating the expression yy(t)Sy(t) to yy Sy(t)Iy(t) results in the following equation:

yy(t)Sy(t) = yy Sy(t)Iy(t)

Cancelling Sy(t) from both sides of this equation leads to the result that we are looking for:

yy(t) = yy Iy(t)




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Note on subscripts


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By applying a similar argument, we can show that the force of infection among old peoplethat is attributable to contact with children and adults is given by the equations:

oy(t) = oyIy(t) Equation 5

oo(t) = ooIo(t) Equation 6

where oy is the rate at which a specific susceptible adult and a specific infectious childcome into effective contact per unit time;oo is the rate at which a specific susceptible adult and a specific infectious adult comeinto effective contact per unit time; Iy(t) is the number of infectious children at time t; Io(t) is the number of infectious adults at time t.




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We can now use our expressions for yy(t) and yo(t) to obtain an expression for the overallforce of infection among children in terms of yy and yo, as we shall show below.

Substituting yyIy (t) for yy(t) and yoIo (t) for yo(t) into the following equation (Equation 1 ).

y(t)=yy(t)+yo(t)

gives the following:

y(t)=yyIy(t)+yoIo(t) Equation 7

Likewise, substituting oyIy (t) for oy(t) and ooIo (t) for oo(t) into the following equation(Equation 2 )

o(t)=oo(t)+oy(t)

gives the following:

o(t)=ooIo(t)+oyIy(t) Equation 8

Equations 7 and 8 for the force of infection among children and adults respectively, areoften written using the following matrix notation:

Equation 9

The term in brackets, , in this equation is known as the matrix of "Who Acquires

Infection From Whom" or, using its abbreviation, as the WAIFW matrix. This matrix caninclude further subgroups, depending on the number of subgroups in the model.

We shall now revise the notation that is used for matrices. If you are already familiar withmatrices you may like to skip to page 18 .




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Section 6: Revision of matrices

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Matrices lead to a convenient format for summarising sets of equations that must besolved simultaneously.

Example

can be represented as

You should notice several features about how equations are written using matrices:

The terms in front of x and y in the first equation (i.e. 5 and 7) go into the first rowof the matrix. The terms in front of x and y in the second equation (i.e. 3 and 4) go into thesecond row of the matrix.

The variables x and y appear immediately after the matrix using the notation .

The constant terms on the right hand side of the equations (6 and 3 for the first andsecond equations respectively) appear on the right hand side of the matrix equation,

using the notation .




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Answer

Answer

6.1: Revision of matrices

page 17 of 65

In a similar way, when we have three equations with three unknown variables we can usematrix notation to represent the equations, as illustrated below.

4x + 8y + 3z = 182x + y + 5z = 12 x + 3y + 3z = 4

is equivalent to4 8 3 x

=18

2 1 5 y 121 3 8 z 4

Q1.1

a) Rewrite the following equations using matrix notation:

3x + 2y = 6 5x + 5y = 3

b) Write the equations corresponding to the following matrix equation:

8 1 x=

6

2 5 y 2




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Answer

Answer

Answer

Answer

Answer

Section 7: Exercise: Calculating the force of infection fromthe WAIFW matrix and the numbers of infectious individuals

page 18 of 65

We will now practise calculating the force of infection using the parameters in a WAIFWmatrix and estimates of the numbers of infectious individuals.

Suppose we have a population with 20 infectious children and 50 infectious adults, andsuppose the WAIFW matrix describing the rate at which children and adults come intoeffective contact per year with each other is given by the following matrix (in which childrenand adults are denoted using the symbols y and o respectively):

y o y 5.2 10-4 2.8 10-4

o 2.8 10-4 3.6 10-4

Q1.2 Write down the matrix equation relating the force of infection at a given time t amongchildren and adults to the WAIFW matrix and number of infectious children and adults.

Click here if you need to remind yourself of the notation

Q1.3 Calculate the force of infection at time t among

a) children which is attributable to contact with the following (click here to remindyourself of the equations if necessary):

i. other children

ii. adults

iii. both children and adults

b) adults which is attributable to contact with:

i. children




Answer

Answer

ii. other adults

iii. both children and adults

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Section 8: Calculating the parameters in WAIFW matrices

page 19 of 65

We will now consider how we might obtain values for yy, yo, oy and oo to use in ourequation for the force of infection.

As we saw previously, when we assume that individuals mix randomly, can becalculated from R0 which, in turn, is calculated using the average force of infection, asestimated from serological data.

We can also use serological data to estimate yy, yo, oy and oo, except that we use thedata to estimate the average force of infection in children and adults and the averagenumbers of infectious children and adults.

As we shall show on the next few pages, these values for the average force of infection inchildren and adults and the average numbers of infectious children and adults are thensubstituted into the matrix equation for the force of infection, and the equations arerearranged until we obtain values for our parameters.




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Note

8.1: Calculating the parameters in WAIFW matrices

page 20 of 65

For example, in MD04 we used serological data to estimate the annual force of rubellainfection to be 13% and 4% for children and adults respectively in England and Walesduring the 1980s, i.e. = 0.13 per year and = 0.04 per year.

Suppose we consider a region in England and Wales, such as Cornwall, which comprisesabout 500,000 individuals.

Using methods discussed later in this session , we can estimate that there were 172and 26 infectious children and adults respectively with rubella in Cornwall at any giventime during the 1980s, i.e. Iy = 172 and Io = 26.




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page 21 of 65

Substituting for = 0.13 per year, = 0.04 per year and Iy = 172 and Io = 26 into ourmatrix equation for the force of infection:

leads to the following equation:

As discussed on page 16 , this equation can also be written using the following twoequations:

172 yy + 26 yo = 0.13172 oy + 26 oo = 0.04

However, there are four unknown parameters in two equations. This presents a problemas only two unknown parameters can be calculated from two equations.

This problem can be overcome by constraining the structure of the matrix so there are twoequations with two unknowns.

There are different methods of constraining matrices.




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page 22 of 65

One constraint that is frequently applied is that contact is symmetric.

For example, we can assume that the rate at which a child contacts and transmits infectionto an adult (which equals oy) equals the rate at which an adult contacts and transmitsinfection to a child (which equals yo), i.e. yo = oy. This constraint has reduced thenumber of unknown parameters to three in two equations. Our WAIFW matrix cantherefore be written as follows, where yo and oy have been substituted by the symbol 1.

One additional constraint is required to reduce the number of unknown parameters in ourmatrix to two.




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Answer

Section 9: Matrix structures: symmetric matrices

page 23 of 65

An additional potential constraint is that the rate at which an adult contacts and infects achild equals the rate at which an adult contacts and infects an adult (i.e. yo = oo = 1).Our matrix would then have the following structure:

An alternative constraint is that the rate that a child contacts and infects an adult equalsthe rate a child contacts and infects another child (i.e. oy = yy), which leads to a matrixwith the following structure:

Q1.4 What is assumed about contact between children and adults (denoted by the letters yand o respectively) in the following matrix?




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Answer

9.1: Matrix structures: symmetric matrices

page 24 of 65

The approach described on the last few pages can be extended to deal with more than twosubgroups. A recent publication by Kanaan and Farrington5 , describes the kinds ofWAIFW structures that have typically been used in modelling studies.

An example of a matrix structure from this publication is shown below.

The parameters 1, 2, 3, 4, 5 represent distinct values for the parameters. Contact is

assumed to vary between the age groups 0- 3, 3- 8, 8- 13, 13- 20, 20+ years.

Q1.5 What assumptions does this matrix make about effective contact betweenindividuals?

The increased availability of data, such as those collected through the POLYMOD study , greatly helps in deciding which matrix structure is appropriate in a population. However,

there are several complications which need to be considered when using the POLYMODdata to calculate the parameters for a WAIFW matrix.

First, the POLYMOD data provide information on the number of contacts that individuals indifferent age groups made during the survey, and this number probably differs from thenumber of effective contacts that they made. In theory, the latter could be calculated bymultiplying the number of contacts made by each person by the proportion of the contactsthat are effective. This proportion is likely to differ between infections and may well differbetween different age groups. It may also differ between populations, depending on theinterpretation of the word contact by participants of the contact survey. Ongoing studies




are likely to provide insight into this question in future. Panel 7.6 of the recommendedcourse text4 , summarises attempts to calculate this proportion for mumps by Wallinga etal6 .

Second, the generalisability of the findings from a given survey is unclear. For example,the POLYMOD study found that the amount of contact between individuals in different agegroups were similar in each country studied, with the number of contacts betweenindividuals in the same age group being greater than the corresponding number betweenindividuals in different age groups. However, the absolute numbers of individualscontacted by each person differed between countries . Also, the study focussed onEuropean countries and it is possible that different findings would have been obtained inlow income settings.

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A B C

A B C

A B C

Section 10: Matrix structures: asymmetric matrices

page 25 of 65

The following are examples of asymmetric matrices:

A. B. C.

Such matrices are considered to be unrealistic for many infections as they assume that therate at which a child contacts and infects an adult differs from the rate at which an adultcontacts and infects a child.

On the other hand, asymmetric matrices may be used to describe the transmission of othertypes of infections, such as sexually transmitted infections between males and females,infections transmitted via the faecal-oral route between children and adults, or vector-borne infections between humans and vectors.

Q1.6 Which matrix might we use to describe transmission of:

a) Polio between children and adults?

b) Gonorrhea between heterosexual males andfemales?

c) Malaria between mosquitoes and humans?




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Section 11: Matrix structures

page 26 of 65

Q1.7 Drag the following matrices to the appropriate box.




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A B

A B

11.1: Matrix structures

page 27 of 65

As highlighted on page 25 , matrices can be used to describe mixing patterns betweenmany different types of groups (for example between males and females, humans andvectors, etc.). Matrices can also be used to describe mixing between individuals living indifferent areas, for example, between individuals from urban and rural areas.

Q1.8 Which of the following matrices might we use to describe mixing between individualsfrom:

1. An urban and a rural area (denoted by u and r respectively). Click on the appropriatebutton next to the matrix to select your matrix.

2. Two urban areas (denoted by u1 and u2). Click on the appropriate button next to thematrix to select your matrix.




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Section 12: Estimating parameters for endemic infections,given estimates of the force of infection and the number ofinfectious individuals

page 28 of 65

We now return to our example on page 21 , where we wished to calculate the rate atwhich adults and children come into effective contact for rubella in a region in Englandwhich comprises 500,000 individuals and is similar in size to Cornwall. In that example, weneeded to solve the following matrix equation:

given values for the average force of infection among children and adults of 13% and 4%per year respectively, and estimates of 172 and 26 infectious children and adultsrespectively.




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Note about rounding

12.1: Estimating parameters for endemic infections, givenestimates of the force of infection and the number ofinfectious individuals

page 29 of 65

For now, suppose the WAIFW matrix has the following structure, which assumes that therate at which a child contacts and infects an adult equals the rate at which an adultcontacts and infects either another adult or a child (i.e. oy = oo= yo). In this matrix, yyhas been substituted by 1 and yo, oy, and oo have been substituted by 2.

Substituting this matrix into our matrix equation gives the following equation:

This equation can be rewritten as follows:

0.13 = 1721+ 262 Equation 10

0.04 = 1722 + 262 Equation 11

Equation 11 can be simplified to give the following equation:

0.04 = 1982 Equation 12

Dividing both sides of this equation by 198 gives the following value for 2:

2 = 2.02 x 10-4 per year Equation 13

If we divide this value for 2 by 365 we obtain the following value for 2 in units of per day:

2 = 2.02 x 10-4 / 365 = 5.53 x 10-7 per day




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page 30 of 65

As we saw on page 29, 1 and 2 are related according to the following equation:

0.13 = 1721+ 262 Equation 10

We can substitute the calculated value for 2 (2.02 x 10-4 per year) into Equation 10 toobtain an expression in terms of 1:

0.13 = 1721+ 26 x (2.02 x 10-4) Equation 14

After simplifying this equation, we obtain the following:

0.13 = 1721 + 5.252 x 10-3 Equation 15

Rearranging this equation gives:

0.1247 = 1721 Equation 16

After dividing both sides of this equation by 172, we obtain the following:

1 = 7.25 x 10-4 per year Equation 16

Converting this value into a daily rate by dividing by 365 provides the following value for 1:

1 = 7.25 x 10-4 / 365 = 1.99 x 10-6 per day.




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page 31 of 65

The resulting WAIFW matrix, with parameters in units of per day, is:

Figure 3 provides a graphical illustration of this matrix.

Figure 3. Summary of the WAIFW matrix obtained using the forces of rubella infection estimated for children andadults in a region comprising 500,000 individuals (i.e. similar in size to Cornwall) in England and Wales.




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Answer


page 32 of 65

Q1.19 Considering the same population of 500,000 individuals and assuming the averageforce of infection among children and adults to be 13% and 4% per year respectively, with172 infectious children and 26 infectious adults, calculate the parameters describingcontact between children and adults using the following matrix structure:




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Section 13: Methods for calculating the number of infectiousindividuals

page 33 of 65

We will now consider how we can calculate the number of infectious individuals in thepopulation. As we have seen in the last few pages, we can then use the values tocalculate the parameters for a given WAIFW matrix.

In general, we can calculate the average number of infectious individuals in a populationusing the following approximation:

Number of infectious individuals Number of new infections perunit time Duration of infectiousness

assuming further that all individuals are infectious shortly after infection.

As we saw previously , if we assume that individuals mix randomly, the average numberof new infections per unit time in a population can be calculated from the total number ofsusceptible individuals, S, and the average force of infection, , using the expression:

Average number of new infections in the overall population per unit time = S

Similarly, the average number of new infections among children and adults per unit time ina population can be calculated from the average force of infection among children ( ) andadults ( ), and the average number of susceptible children (Sy) and adults (So), using theexpressions:

Average number of new infections among children per unit time = Average number of new infections among adults per unit time =




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13.1: Methods for calculating the number of infectiousindividuals

page 34 of 65

Using the relationship between the number of infectious individuals and the number of newinfections occurring per unit time described before and assuming that the duration ofinfectiousness is D, we obtain the following equation for the average number of infectiouschildren and adults in the population:

Average number of infectious individuals (assuming random mixing) SDAverage number of infectious children Average number of infectious adults

As we saw in MD04 the average number of susceptible individuals, S, in a populationcan be calculated from the average age at infection, which is related to the average forceof infection (see page 22 of MD04 ).

Similarly, the average number of susceptible individuals in a given age group can becalculated from the average age-specific force of infection.

On the next few pages, we will illustrate how we can calculate both the number ofsusceptible and infectious individuals assuming that individuals mix either randomly ornon-randomly. Here we will derive the numbers of young and old infectious individuals withrubella in the region of England (Cornwall) that we presented on page 20 .

If you prefer, you may skip these calculations now and proceed to the second (practical )part of this session. You will probably find it helpful to return to these calculations eitherwhilst you are doing the practical or after you have completed the session.

The next few pages illustrate the calculations for the assumption that individuals mixrandomly.




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Section 14: Methods for calculating the number ofsusceptible and infectious individuals, assuming thatindividuals mix randomly

page 35 of 65

As we saw in MD04, if we assume that individuals mix randomly, the average proportion ofthe population that is susceptible in the absence of an intervention, s, can be calculatedfrom the basic reproduction number using the following equation:

s=1/R0

As we saw in MD04 R0 can be calculated from the average force of infection.




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14.1: Methods for calculating the number of susceptible andinfectious individuals, assuming that individuals mixrandomly

page 36 of 65

EXAMPLE

In MD04 , we estimated that the average force of rubella infection in England and Waleswas about 12% per year during the 1980s and the basic reproduction number, assuming alife expectancy of 75 years, was 9.

Assuming that individuals mix randomly, and applying the equation s=1/R0 implies that theaverage proportion of the population that was susceptible to rubella infection was 1/9 0.111.

We will now use this infomation to calculate the number of susceptible and infectiousindividuals in the region in England (Cornwall) that we considered on page 20 .

The population in that area comprised about 500,000 individuals. Multiplying the proportionsusceptible by 500,000 leads to an estimate of 500,000 0.111 55,500 susceptibleindividuals.




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14.2: Methods for calculating the number of susceptible andinfectious individuals, assuming that individuals mixrandomly

page 37 of 65

EXAMPLE CTD.

We can now calculate the number of infectious individuals in Cornwall by multiplying thenumber of susceptible individuals obtained on the previous page by the force of infectionand the duration of infectiousness, using the equation presented on page 34 .

However, when doing this calculation, we need to ensure that the units that we use for theforce of infection are consistent with the units that we use for the duration ofinfectiousness. For example, if we use the value for the annual force of infection (about12% per year ), the value for the duration of infectiousness must also be in annual units.

The average duration of infectiousness for rubella is 11 days. Dividing this value by 365gives a value for the duration of infectiousness in time units of years of 11/365 = 0.030years.

We can now calculate the number of infectious individuals with rubella in Cornwall duringthe 1980s by substituting 0.12 per year for , 55,500 for S, and 11/365 for D in theequation SD that we discussed on page 34 :

0.12 x 55,500 x (11/365) 201 infectious individuals

On the next few pages, we will illustrate how we can calculate the number of susceptibleand infectious individuals if we assume that individuals mix non-randomly.




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Section 15: Methods for calculating the number ofsusceptible and infectious individuals, assuming thatindividuals mix non-randomly

page 38 of 65

In MD04 , we presented the equations that might be used to calculate the proportion ofindividuals that are susceptible to an immunising infection at different ages, assuming thatthe force of infection is age-dependent.

For example, if the average force of infection differs between those aged less than 15years, and those aged at least 15 years (denoted by and respectively), the proportionsusceptible would be given by the following equations:

We can obtain the number of individuals that are susceptible at each age a (S(a)) bymultiplying s(a) by the number of individuals of that age (N(a)), i.e.

S(a) = N(a)s(a)




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15.1: Methods for calculating the number of susceptible andinfectious individuals, assuming that individuals mix non-randomly

page 39 of 65

Figure 4 shows how the number of susceptible individuals of age a (S(a)) might change byage for a population with a rectangular age distribution of size N and a life expectancyof L (=75 years), assuming that the force of infection changes at age 15 years.

Figure 4: A. Illustration of the relationship between the number of susceptible individuals and age for a populationof size N with a rectangular age distribution (as shown in figure B), assuming that the force of infection differsbetween those aged under and over 15 years. If the life expectancy is L (=75 years), the proportion of the totalpopulation that is in each single year age band equals 1/L (=1/75) and the number of individuals in each singleyear age band equals N/L (=N/75). The shaded portion of these figures reflects individuals who are aged 0-15years (i.e. children).




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page 40 of 65

The number of individuals in a given age range that are susceptible can be obtained bysumming up the values for S(a) for all values of a in the age ranges of interest.

For example, to calculate the number of 0-15 year olds that are susceptible, we would sumup the values for S(a) for values for a of between 0 and 15 years. This is equivalent tosumming the area under the plot of S(a) between the ages 0 and 15 years, shown by theshaded area in Figure 4A .

You may recall from your previous mathematical training that we can calculate the areaunder a curve by integrating the expression which leads to that curve. If you are interested,further details about how we can integrate expressions are provided on page 17 of themaths refresher and the Basic maths section of the recommended course text4 . Byintegrating the expression for the number of susceptible individuals of age a shown onpage 38 , we can show that the numbers of susceptible children and adults for theexample in Figure 4 are given by the following equations:

Note that we do not expect you to be able derive these equations(although you may try to do this, if you wish!). For the purposes of thisstudy module, it is sufficient for you to be aware of the methods that areused.




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page 41 of 65

EXAMPLE

We will now return to the example that we discussed on page 20 , and illustrate how wecan calculate the number of children and adults who were susceptible or infectious withrubella during the 1980s at a given time in a region in England, which is similar in size toCornwall.

Substituting for the force of infection estimated for children and adults ( = 0.13 per yearand = 0.04 per year), the total population size (N = 500,000) and the life expectancy (L= 75 years) into our equations for Sy and So leads to the following:

Sy =500,000 (1 - e -0.1315) 43,986 individuals 0.13 75

So =500,000 e -0.1315 (1 - e -0.04(75-15)) 21,561 individuals 0.04 75

We can now calculate the number of infectious children and adults in the region bysubstituting for = 0.13 per year and = 0.04 per year and the duration of infectiousnessfor rubella (D = 11 days = 11/365 years) into the expressions for the average number ofinfectious children and adults as follows:

You should notice that these values are identical to the values presented on page 20 .




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Section 16: Break...

page 42 of 65

We have now completed part 1 of this session, in which we covered the theory forincluding assumptions about non-random mixing between individuals into models.

The rest of this session (part 2) consists of a practical exercise using Berkeley Madonna. It is likely to take 1-3 hours to complete.

You may like to take a short break before starting it.




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Section 17: Part 2 (practical): The effect of non-randommixing on rubella transmission and control

page 43 of 65

OVERVIEW

We will now start part 2 of this session in which we incorporate non-random mixing into amodel of the transmission dynamics of rubella, which is set up in Berkeley Madonna, andexplore the effect of non-random mixing on the impact of vaccinating children.

OBJECTIVES

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

Be able to define a WAIFW matrix;Be able to use estimates of the force of infection and the number of infectiousindividuals to calculate the WAIFW matrix for different assumptions about mixingbetween individuals;Understand the effect of non-random mixing patterns on the impact of a vaccinationstrategy.




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Section 18: Part 2 (practical): Introduction to the model

page 44 of 65

In MD04 , we presented estimates for the force of rubella infection for the UK for thoseaged

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Answer

18.1: Part 2 (practical): Introduction to the model

page 45 of 65

To describe the transmission dynamics of rubella in the UK, which accounts for the age-dependency in the force of infection that we discussed on the previous page, we will use amodel with the structure shown below.

In this model, the population is stratified into three age groups: the young, the middle-agedand the old. Individuals move between age categories at a constant rate i.e. youngindividuals enter the middle-aged category at a constant rate, middle-aged individualsbecome old at a constant rate, etc.

Q2.1 How would you calculate the rate at which young and middle-aged individuals age,assuming that they spend an average of 15 years in the young and middle-aged groups?




Optional reading about ageing

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page 46 of 65

1. Start up Berkeley Madonna and click here to open the file rubwaifw flowchart.mmd or rubwaifw equations.mmd , depending on whether you prefer to work with the

flowchart or equation editor versions of the models. Unless otherwise stated, theinstructions for this practical are identical for both files.

We will begin by reviewing the key features of the model.

Demography

a) The demography of the population is determined by the parameters in the sectioncalled "Demographic parameters and variables" which can be found in the globalswindow (flowchart version) or equations window (equation editor version). This sectionalso includes variables which calculate the total numbers of young, middle-aged andold individuals (specified by tot_young, tot_mid and tot_old respectively).

b) Young and middle-aged individuals spend an average of 15 years in their respectiveage groups. Old individuals spend an average of 30 years in their age compartmentprior to death. In the model, only individuals in the old age group can die. Therefore,the overall average life expectancy in the model is assumed to be 60 years.

c) The total population size remains constant over time with 60,000 individuals (15,000young, 15,000 middle-aged and 30,000 old individuals). You can see this by runningthe model and looking at page 1 of the figures window, which plots the total numbers ofyoung (tot_young), middle-aged (tot_mid) and old (tot_old) individuals over time. Clickhere to see the figure that you should see at this stage. Note that the red line for themiddle-aged category overlaps with the black line for the young category. You can seethe line for the young category by deselecting the plot for the middle-aged category.




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page 47 of 65

Infection and transmission

a) Parameters relating to rubella and transmission are defined in the section calledInfection-related parameters (in the globals window or equation editor window).

b) The pre-infectious and infectious periods are set to those for rubella (10 and 11 daysrespectively).

c) Separate variables for the force of infection for each age group have been set up(force_of_infn_y, force_of_infn_m and force_of_infn_o). At present, these have beenassigned fixed values corresponding to the values estimated for the force of infectionfor rubella in the UK. We will change the settings for the force of infection later(please do not do this yet!).

d) parameters defining mixing between individuals in the three different age groups havebeen set up (b_yy, b_ym, b_yo, etc). These values are currently set to zero; we willamend them later in the session.




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page 48 of 65

Vaccination

a) Parameters relating to vaccination are defined in the section called Vaccination-relatedparameters (in the globals window or equation editor window).

b) Vaccination of newborns is introduced 100 years after the start of the simulations at alevel specified by the parameters birth_cov and prop_vacc. The time at whichvaccination is introduced is determined by the value of yr_start_vacc. At present, thevaccination coverage is zero.

Summary variables

a) Summary variables have been set up in the "Useful summary variables" section of theglobals or equations window.

b) The summary variables include the proportion of young, middle-aged and oldindividuals who are susceptible (specified by (prop_sus_y, prop_sus_m, prop_sus_o)and the daily number of new infections among young, middle-aged and old individualsper 100,000 population (specified by new_infn_y_p100000, new_infn_m_p100000 andnew_infn_o_p100000).




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page 49 of 65

2. Run the model and look at pages 2 and 3 of the figures window to check that you obtainthe values for the average proportion of individuals in different age groups that aresusceptible and the daily number of new infections per 100,000 that are consistent with thevalues in the table below. Recall that you can view the actual values by clicking on theTable button in the window.

You should notice that the values for the age-specific proportion of individuals that aresusceptible and the daily number of new infections per 100,000 are constant over time.This is to be expected since the force of infection, which is used to calculate thesestatistics, is currently set to be constant over time.

Agecategory

Average %susceptible

Average dailynumber of new

infections/100,000

Averagedaily

force ofinfection

Young 33.41 12.163.64 10-4

Middle-aged

20.56 2.351.14 10-4

Old 9.13 1.041.14 10-4

Table 2: Long-term values for the age-specific percentage of individuals that are susceptible and the dailynumber of new infections per 100,000. See Appendix for details of the equations underlying these values.

If we wanted to explore the effect of vaccination, we would be unable to do this adequatelyusing the existing model, as it does not describe contact between children and adults, andassumes that the force of infection remains unchanged over time.

To incorporate contact between individuals, we first need to calculate the values for the parameters (b_yy, b_yo, etc) and then change our expression for the force of infection forthe young, middle-aged and old. We will do this on the next few pages.




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Answer

Optional reading

Section 19: Part 2 (practical): Calculating the averagenumber of infectious individuals in the model

page 50 of 65

In order to calculate the parameters for this model, we first need to calculate the averagenumber of infectious children, middle-aged and old individuals.

As we saw before , the average number of infectious children at a given time is given bythe following expression:

There are 15,000 young, 15,000 middle-aged and 30,000 old individuals in the population.

Q2.2 Using pen and paper, use the values for the percentage of individuals in differentage groups who are susceptible in Table 2 to calculate the average number of young,middle-aged and old individuals who are susceptible.

Q2.3 Using pen and paper, use your answer to the previous question, the fact that theaverage duration of infectiousness is 11 days and the age-specific values for the force ofinfection in Table 2 to calculate the number of infectious young, middle-aged and oldindividuals in the population.

Check: number of infectious children

Check: number of infectious middle-aged individuals

Check: number of infectious old individuals




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Check

Check

Check

Section 20: Part 2 (practical): Changing the expression forthe force of infection

page 51 of 65

As we saw earlier , when the population is stratified into just the young and the old, theexpression for the force of infection is given by the following:

y(t)=yyIy(t)+yoIo(t) Equation 7

o(t)=oyIy(t)+ooIo(t) Equation 8

where

Iy (t) is the number of infectious children at time t;Io(t) is the number of infectious adults at time t;yy is the rate at which a susceptible child and an infectious child come into effectivecontact;yo is the rate at which a susceptible child and an infectious adult come into effectivecontact;oy is the rate at which a susceptible adult and an infectious child come into effectivecontact; andoo is the rate at which a susceptible adult and an infectious adult come into effectivecontact.

3. Return to your Berkeley Madonna model and do the following (in the globals window ifyou are using the flowchart model and in the equations window if you are using theequation editor model):

a) Change the expression for the force of infection among young individualsto be in terms of b_yy, b_ym, b_yo, Infous_y, Infous_m, Infous_o.

b) Change the expression for the force of infection among middle-agedindividuals to be in terms of b_mm, b_my, b_mo, Infous_y, Infous_m,Infous_o.

c) Change the expression for the force of infection among old individuals tobe in terms of b_oo, b_oy, b_om, Infous_y, Infous_m, Infous_o.

If you wish to check your equations for the force of infection further, click rubwaifw flowchart_solna.mmd or rubwaifw equations_solna.mmd , which hold the modelsthat you should have by now. Note that we are not yet ready to run the model,as the values for b_yy, b_yo etc are still zero.




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Answer

Section 21: Part 2 (practical): Calculating parameters fordifferent matrix structures

page 52 of 65

We will now calculate the parameters for our model. The relationship between the age-specific forces of infection, contact parameters and numbers of infectious cases can bedescribed with the following equation:

Suppose we assume that contact between individuals in our population is described usingthe following matrix:

Matrix A

Q2.4 Using pen and paper, use the values for the numbers of infectious young, middle-aged and old individuals that you calculated on page 50 , and the daily force of infection(see page 49 ), to calculate the values for this matrix.




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Hint

Show

21.1: Part 2 (practical): Calculating parameters for differentmatrix structures

page 53 of 65

5. Incorporate the values for that you have just calculated into the model and save thismodel as WAIFWA.mmd. Click on the button below for a hint about how to incorporate thevalues (you will get the chance to check your model shortly).

6. Run the model and look at the figures showing the proportion of individuals that aresusceptible in each age group and the daily number of new infections per 100,000.

Click the button below if you would like to check the output that you should be getting atthis stage.

If your model has failed to run click WAIFWA flowchart_solna.mmd or WAIFWA equations_solna.mmd to see the model that you should have by now.

Click here to see Table 2, which shows the percentage of the population that issusceptible and the daily number of new infections per 100,000 that our original modelpredicted. You should find that the models predictions of these statistics are very similar tothose found in this table. This is to be expected, given that the parameters in the modelwere calculated using the values for the force of infection that are in Table 2. If the parameters are correctly incorporated into the model, the model should generatepredictions of the force of infection and other statistics that are similar (depending onrounding) to the values used to calculate the parameters.

At this stage, you may also want to plot the value for the force of infection to check that thevalue that the model is predicting is consistent with the value that you used to calculate the parameters. Click here if you need to remind yourself of how you can do this.




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Note

21.2: Part 2 (practical): The impact of vaccinating newbornsif individuals contact each other according to WAIFWA

page 54 of 65

As we saw in MD04 , if we used the average force of rubella infection for England andWales for the 1980s and assumed a life expectancy of 60 years (i.e. identical to that in themodel population), we estimated that R0 was about 7. This value for R0 leads to a to aherd immunity threshold of 86%. Therefore, in a population in which individuals mixrandomly, we would need to vaccinate over 86% of the population in order to controltransmission.

7. Now run your model assuming 86% of newborns in the population are vaccinated.

At this stage your output should resemble the images shown below. If it does not, clickWAIFWA flowchart_solnb.mmd or WAIFWA equations_solnb.mmd to seethe model that you should have by now.

Proportion susceptible Daily no. of new infections/100,000

Q2.5 What happens to the age-specific proportions of individuals who are susceptible and




Answer

the daily number of new infections per 100,000?

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21.3: Part 2 (practical): Calculating parameters for differentmatrix structures

page 55 of 65

Alternatively, suppose contact between individuals in our population is described using adifferent matrix:

Matrix B

If you wish, calculate the values for this matrix using the same method that you used toobtain Matrix A, using the numbers of infectious children, middle-aged and old individualsand age-specific forces of infection. If not, click here for the values.




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Answer

21.4: Part 2 (practical): Comparing predictions obtained usingmatrices A and B

page 56 of 65

The following figure compares the values for matrices A and B.

Figure 5: Comparison between the values for WAIFWA and WAIFWB.

Q2.6 Which matrix structure is more realistic (WAIFWA or WAIFWB)?

Before continuing, think about how the impact of vaccinating 86% of individuals in apopulation mixing according to matrix B would differ from that predicted for a populationmixing according to matrix A.




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Answer


page 57 of 65

8. Incorporate the following values for Matrix B into the model and save the model asWAIFWB. Click here if you would like to remind yourself about the methods forincorporating parameters.

1 = 1.66 x 10-5 per day

2 = 4.16 x 10-6 per day

3 = 4.16 x 10-6 per day

9. Set the value for prop_vacc to be zero and run the model. Click WAIFWB flowchart_solna.mmd or WAIFWB equations_solna.mmd if you wish to checkthat you have incorporated your values correctly.

Q2.7 How do predictions of the age-specific proportions of individuals who are susceptibleand the daily number of new infections per 100,000 compare against those obtained usingthe original model and WAIFWA?




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Answer


page 58 of 65

10. Run your WAIFWB model assuming that 86% of the newborns are vaccinated. At thisstage your output should resemble the following. If it does not, click WAIFWB flowchart_solnb.mmd or WAIFWB equations_solnb.mmd to see the model thatyou should have by now.

Proportion susceptible Daily no. of new infections/100,000

Q2.8 How does vaccination affect predictions of the population susceptible and the dailynumber of new infections per 100,000?




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Section 22: Conclusions from the practical exercise

page 59 of 65

We will now reflect on what this practical exercise tells us about how contact patterns affectthe impact of control strategies. Figure 6 summarises predictions of the effect ofvaccinating 86% of newborns each year from year 100 in the model on the age-specificproportion of individuals who are susceptible and the daily number of new infections per100,000, assuming that individuals contact each other according to matrix A or B.

Proportion susceptible Daily number of newinfections/100,000

Year

Figure 6. Summary of model predictions of the age-specific proportion of individuals who are susceptible and thedaily number of new infections per 100,000, obtained using models WAIFWA and WAIFWB, assuming that 86%of newborns are vaccinated from year 100. For improved clarity, the lines for the youngest age groups have beenomitted from the plot of the daily number of new infections per 100,000.

Both matrices A and B were calculated using identical values for the force of infection and,in the absence of vaccination, lead to identical values for the daily number of newinfections per 100,000. However, the impact of vaccination is greater for populations




mixing according to matrix B than matrix A, highlighting that contact strongly determinesthe impact of control.

One problem with interpreting these findings is that it is impossible to be sure whichWAIFW matrix best describes the mixing patterns in your population. Several WAIFWmatrices may be able to reproduce the same average force of infection in the absence ofan intervention, but will lead to different predictions of the impact of control.

Consequently, modellers typically use many different assumptions about contact patternswhen using models to predict the impact of an intervention. This situation may change infuture years, as an increasing number of studies (such as the POLYMOD study) try tocollect data on the actual mixing patterns that occur in populations.

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Answer

22.1: Conclusions from the practical exercise

page 60 of 65

We shall continue thinking about the effect of non-random mixing in MD07 , when weshall calculate R0 for matrices A and B. We will then identify the actual level of coveragethat would be required to control transmission if individuals were to contact each otheraccording to these matrices.

Final Optional Exercise

As you shall see in the next session, the values for R0 for populations with mixing patternsdescribed by WAIFWA and WAIFWB are about 10.9 and 3.6, respectively. Use thesevalues of R0 to calculate the critical levels of vaccination coverage for these populations.Check that vaccination at these levels in the model population results in patterns in theage-specific daily number of new infections per 100,000 that you would expect.

Hint: Remember that the equation for the herd immunity threshold is 1- 1/R0.




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Section 23: Further reading and exercises

page 61 of 65

In this session, we have illustrated how we can use estimates of the force of infection toestimate contact parameters and have explored how assumptions about contact betweenindividuals influence predictions of the impact of an intervention.

For further reading and to consolidate your understanding, we recommend that you readthe Chapter 7, sections 7.1-7.4 of the recommended course text4 .

You should also try the following exercises:

1. Exercises accompanying models 7.1 and 7.2 of the recommended course text4

(see www.anintroductiontoinfectiousdiseasemodelling.com ).

2. The paper and pen exercises 7.1, 7.2, 7.3 at the end of chapter 7 of therecommended course text4 . Solutions are available from the books website.




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23.1: Summary

page 62 of 65

We will now summarise the key messages that you should have obtained from thissession.

1. Several studies suggest that contact patterns between individuals are strongly age-dependent, and the nature of this age-dependency probably varies betweenpopulations.

2. Contact patterns greatly influence the impact of interventions against infections, andtherefore models need to take account of this if they are to be used to predict theeffect of control.

3. Assumptions of non-random mixing between individuals are incorporated intomodels using matrices of "Who Acquires Infection from Whom".




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23.2: Summary

page 63 of 65

4. The steps for calculating a matrix of "Who Acquires Infection From Whom" todescribe transmission between individuals in different age groups, for examplebetween the young and old (denoted using the subscripts y and o respectively)for endemic infections are as follows:

a) Calculate the average forces of infection in specific age groups (the young andold) prior to the introduction of any intervention ( and ).

b) Use the estimates of the force of infection to calculate the numbers of young andold susceptible individuals, as follows:

Sy = proportion of young individuals who are susceptible thenumber of young individuals in the population. So = proportion of old individuals who are susceptible the numberof old individuals in the population.

Sy can be calculated by summing up the number of susceptible individuals of agea (S(a)) for all values of a between the lowest and maximum limits of the agerange of children. This is equivalent to calculating (or integrating) the area underthe curve of S(a) between the lower and upper limits of the age range of children.The approach for calculating So is analogous. See page 40 for referencesrelating to methods for integration.

c) Use the age group-specific values for the force of infection ( and respectively), the number of susceptible individuals and the duration ofinfectiousness to calculate the average number of infectious individuals in eachage group (Iy and Io for the young and old respectively) as follows:

d) Choose an appropriate WAIFW matrix to describe the mixing pattern for the agegroups.

e) Calculate the parameters for the matrix using the estimates of the forces ofinfection and the numbers of infectious individuals.




5. The steps described in point 4 can be extended to deal with several different agegroups or to describe transmission between individuals in different settings, e.g.between individuals living in urban and rural areas.

6. Several different WAIFW matrices can be calculated that are consistent with the age-specific (or group-specific) value for the force of infection before the introduction ofan intervention.

7. Since the effect of an intervention depends on the assumed mixing patterns,modelling studies typically use several different WAIFW structures when predictingthe impact of control.

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References

page 64 of 65

1. Borgdorff MW, Nagelkerke NJ, van Soolingen D, Broekmans JF. Transmission oftuberculosis between people of different ages in The Netherlands: an analysis usingDNA fingerprinting. Int J Tuberc Lung Dis, 1999. 3(3): p.202-6.

2. Edmunds WJ, Kafatos G, Wallinga J, Mossong JR. Mixing patterns and the spread ofclose-contact infectious diseases. Emerg Themes Epidem, 2006. 3(10).

3. Mossong J, Hens N, Jit M, Beutels P, Auranen K, Mikolajczyk R, Massari M,Salmaso S, Tomba GS, Wallinga J, Heijne J, Sadkowska-Todys M, Rosinska M,Edmunds WJ. Social contacts and mixing patterns relevant to the spread ofinfectious diseases. PLoS Medicine, 2008. 5(3).

4. Vynnycky E, White RG. An introduction to infectious disease modelling. OxfordUniversity Press, 2010. Oxford

5. Kanaan MN, Farrington CP. Matrix models for childhood infections: a Bayesianapproach with applications to rubella and mumps. Epidemiol Infect, 2005. 133(6):p.1009-1021.

6. Wallinga J, Teunis P, Kretzschmar M. Using data on social contacts to estimate age-specific transmission parameters for respiratory-spread infectious agents. Am JEpidemiol. 2006 Nov 15;164(10):936-44.

7. Anderson, R.M. and R.M. May, Infectious diseases of humans. Dynamics andcontrol. Oxford University Press, 1991, Oxford.




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Appendix: Expressions for the number of young, middle-agedand old susceptible individuals in the population

page 65 of 65

The number of young, middle-age and older individuals who are susceptible to infection(Sy(t), Sm(t) and So(t) respectively) at a given time t in the population described in themodel satisfy the following differential equations, assuming that no individuals arevaccinated.

ay is the rate at which young individuals become middle-aged;am is the rate at which middle-aged individuals become old;B is the number of births into the population per unit time;

is the force of infection among the young, middle-aged and oldrespectively;mo is the mortality rate among old individuals.

The long-term (equilibrium) average number of young individuals who are susceptible toinfection is then obtained by equating these differential equations to zero and rearrangingthe resulting expressions. This leads to the following equations:

Note that these expressions are specific to the population with the age distribution used inthis model. See Anderson and May (1991) [p178]7 and the recommended course text4for expressions for the number of individuals susceptible to infection in a population with




other more realistic mortality patterns.

Local DiskSection 1: Overview and objectivesSection 2: Introduction2.1: Introduction2.2: Introduction2.3: IntroductionSection 3: Background: Evidence for non-random mixing3.1: Background: Evidence for age-dependent mixingSection 4: Revision of the relationship between , the force of infection and the number of infectious individuals for a randomly mixing population4.1: Revision of the relationship between , the force of infection and the number of infectious individuals for a randomly mixing populationSection 5: Methods for incorporating non-random mixing into models5.1: Methods for incorporating non-random mixing into models5.2: Methods for incorporating non-random mixing into models5.3: Methods for incorporating non-random mixing into models5.4: Methods for incorporating non-random mixing into models5.5: Methods for incorporating non-random mixing into modelsSection 6: Revision of matrices6.1: Revision of matricesSection 7: Exercise: Calculating the force of infection from the WAIFW matrix and the numbers of infectious individualsSection 8: Calculating the parameters in WAIFW matrices8.1: Calculating the parameters in WAIFW matrices8.2: Calculating the parameters in WAIFW matrices8.3: Calculating the parameters in WAIFW matricesSection 9: Matrix structures: symmetric matrices9.1: Matrix structures: symmetric matricesSection 10: Matrix structures: asymmetric matricesSection 11: Matrix structures11.1: Matrix structuresSection 12: Estimating parameters for endemic infections, given estimates of the force of infection and the number of infectious individuals12.1: Estimating parameters for endemic infections, given estimates of the force of infection and the number of infectious individuals12.2: Estimating parameters for endemic infections, given estimates of the force of infection and the number of infectious individuals12.3: Estimating parameters for endemic infections, given estimates of the force of infection and the number of infectious individuals12.4: Estimating parameters for endemic infections, given estimates of the force of infection and the number of infectious individualsSection 13: Methods for calculating the number of infectious individuals13.1: Methods for calculating the number of infectious individualsSection 14: Methods for calculating the number of susceptible and infectious individuals, assuming that individuals mix randomly14.1: Methods for calculating the number of susceptible and infectious individuals, assuming that individuals mix randomly14.2: Methods for calculating the number of susceptible and infectious individuals, assuming that individuals mix randomlySection 15: Methods for calculating the number of susceptible and infectious individuals, assuming that individuals mix non-randomly15.1: Methods for calculating the number of susceptible and infectious individuals, assuming that individuals mix non-randomly15.2: Methods for calculating the number of susceptible and infectious individuals, assuming that individuals mix non-randomly15.3: Methods for calculating the number of susceptible and infectious individuals, assuming that individuals mix non-randomlySection 16: Break...Section 17: Part 2 (practical): The effect of non-random mixing on rubella transmission and controlSection 18: Part 2 (practical):Introduction to the model18.1: Part 2 (practical):Introduction to the model18.2: Part 2 (practical):Introduction to the model18.3: Part 2 (practical): Introduction to the model18.4: Part 2 (practical):Introduction to the model18.5: Part 2 (practical): Introduction to the modelSection 19: Part 2 (practical):Calculating the average number of infectious individuals in the modelSection 20: Part 2 (practical):Changing the expression for the force of infectionSection 21: Part 2 (practical): Calculating parameters for different matrix structures21.1: Part 2 (practical): Calculating parameters for different matrix structures21.2: Part 2 (practical): The impact of vaccinating newborns if individuals contact each other according to WAIFWA21.3: Part 2 (practical):Calculating parameters for different matrix structures21.4: Part 2 (practical):Comparing predictions obtained using matrices A and B21.5: Part 2 (practical): Comparing predictions obtained using matrices A and B21.6: Part 2 (practical):Comparing predictions obtained using matrices A and BSection 22: Conclusions from the practical exercise22.1: Conclusions from the practical exerciseSection 23: Further reading and exercises23.1: Summary23.2: In the next session...ReferencesAppendix: Expressions for the number of young and old susceptible individuals in the population

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