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Overview and objectives

page 1 of 78

OVERVIEW

The previous sessions introduced methods for setting up simple models of thetransmission dynamics of immunising infections. This session discusses the insights intothe dynamics of infections provided by these models, and subsequently relates modelpredictions to data.

OBJECTIVES

By the end of this session you should:

Understand what determines whether the number of new infections will increase ordecrease over time;Be aware of methods for calculating R0 for an infection from the growth rate of anepidemic or outbreak;Understand the factors that lead to cycles in the incidence of immunising infections;Be able to calculate the inter-epidemic period for an immunising infection;Know some of the insights into the epidemiology of immunising infections that areprovided by simple models.

This session should take 2-5 hours to complete .

This session comprises two parts. Part 1 (1-2 hours) introduces the insights that modelsprovide into the dynamics of infections; Part 2 (1-3 hours) consists of a practical exerciseusing models set up in Berkeley Madonna, during which you will study the cycles inincidence of immunising infections in detail.

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EPM302 Modelling and the Dynamics of Infectious Diseases

MD03 The natural dynamics of infectiousdiseases

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Section 2: Introduction

page 2 of 78

The following diagram shows the structure of the models that you worked with during thelast two sessions.

The models that you used are among the simplest models that are used to describe thelong-term transmission dynamics of an immunising infection. We assumed that individualsmix randomly and that the population size remains unchanged over time. We have alsonot stratified the population by age, sex or any other subgroup.




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2.1: Introduction

page 3 of 78 2.1

These assumptions are clearly simplifications. However, as shown in Figure 1 (below),despite the simplifications, the models are able to reproduce (at least for a while) cycles inthe number of new infections occurring per unit time that are similar to those seen forimmunising infections (see Figure 2).

The fact that the model's general predictions are reasonably consistent with observed datasuggests that the models, whilst approximations, may be used to help us understand thebehaviour of epidemics.

Figure 1: Predictions of the daily number of new infectious individualswith measles following the introduction of an infectious person into atotally susceptible population, assuming that R0=13, the pre-infectiousperiod = 8 days, the infectious period = 7 days, the total population size= 100,000, the average life expectancy = 70 years, the birth rate = thedeath rate.

Figure 2: Quarterly notification rates of measles and vaccination coveragein England and Wales. Data sources: Office for Population and Censussurveys and the Health Protection Agency.




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2.2: Introduction

page 4 of 78 2.1 2.2

However, the fact that the model predicts that peaks in the epidemics becomeprogressively less pronounced ("damped") over time, but that we do not see this in theactual data, suggests that other factors that are not in the model are needed to sustain theepidemic cycles.

Before discussing the cycles and damping in further detail, we will discuss some of theother insights into the dynamics of infections provided by this model, specifically:

1. What determines whether or not the number of infectious individuals increasesfollowing the introduction of an infectious person into a totally susceptiblepopulation?

2. How fast might we expect the number of infectious individuals to increase followingthe introduction of an infectious person into a totally susceptible population and whatcan we infer from it?

3. Why does the incidence of an immunising infection cycle over time?

4. What other factors lead to cycles in the incidence of immunising infections?

5. What inter-epidemic period might we expect to see for immunising infections?




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Section 3: 1. What determines whether or not the number ofinfectious individuals increases following the introduction ofan infectious person into a totally susceptible population?

page 5 of 78

You may remember from previous sessions that if the basic reproduction number (R0) of a pathogen is greater than 1, the introduction of one infectious person into a susceptiblepopulation should result in an increase in the number of infectious individuals and thepersistence of the infection in the population.

This result can be obtained relatively easily from the differential equations used to describethe transmission dynamics of immunising infections that you have used previously. We willillustrate the derivation of this result by considering a "closed" population, i.e. one in whichthere are no births into or deaths out of the population. The model has the followingstructure:

The parameters in the model are as follows:

is the rate at which two specific individuals come into effective contact per unittime;f is the rate at which those in the pre-infectious category become infectious;r is the rate at which infectious individuals recover from being infectious and becomeimmune;N is the population size;D is the duration of infectiousness.

In this diagram shown above S(t)I(t) is the number of new infections occurring per unittime, fE(t) the number of new infectious individuals occurring per unit time and rI(t) thenumber of individuals who recover per unit time.




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3.1: 1. What determines whether or not the number ofinfectious individuals increases following the introduction ofan infectious person into a totally susceptible population?

page 6 of 78 3.1

When deriving the result, we will use the following equation, which we met in MD01 :

=R0ND

This expression can be rearranged to give the following expression for the basicreproduction number:

R0 = ND

or, equivalently,

R0 =N Equation 1 r

Click the "show" button to see the derivation.

We will show that for the number of infectious individuals to increase following theintroduction of an infectious person into a totally susceptible population, ND must begreater than 1, and that this expression has the literal definition of R0,, i.e. as the averagenumber of secondary infectious individuals resulting from one infectious person followinghis/her introduction into a totally susceptible population.




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page 7 of 78 3.1 3.2

We begin by noting that if the number of infectious individuals increases following theintroduction of an infectious person into a totally susceptible population then the rate ofchange in the number of pre-infectious and infectious individuals must be positive, i.e.

dE > 0 and dl > 0 dt dt

As shown in MD02 , the expressions for the rate of change in the number of pre-infectious and infectious individuals for the model described on page 5 are:

dE = S(t)l(t)- fE(t) Equation 2 dt

dl = fE(t)-rl(t) Equation 3 dt

where f is the rate at which pre-infectious individuals become infectious and r is the rate atwhich the infectious individuals recover to become immune.




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page 8 of 78 3.1 3.2 3.3

For the rate of change in the number of pre-infectious individuals (Equation 2 ) to bepositive, the number of new infections that occur in the population per unit time, S(t)I(t),has to be greater than the number of pre-infectious individuals who become infectious perunit time, fE(t), i.e.

S(t)I(t) > fE(t)

In a similar way, for the rate of change in the number of infectious individuals (Equation 3 ) to be positive, the number of pre-infectious individuals who become infectious per unit

time, fE(t), has to exceed the number of infectious individuals who recover per unit time,rI(t), i.e.

fE(t) > rI(t)




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page 9 of 78 3.1 3.2 3.3 3.4

Combining the logic in the last two expressions , the number of individuals who arenewly infected per unit time must also be larger than the number of individuals whorecover per unit time, i.e.

S(t)I(t) > rI(t)

After dividing both sides of this expression by the number of infectious individuals in thepopulations, I(t), we see that for the rate of change in the number of infected individuals tobe positive and therefore for the number of infected individuals to increase, the followingcondition must hold:

S(t) > r Equation 4




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page 10 of 78 3.1 3.2 3.3 3.4 3.5

Notice that when the infection is introduced into the population, the number of individualswho are susceptible, S(t), is the size of the total population, N. Substituting N for S(t) intothe last expression , we see that for the number of infectious individuals to increasefollowing the introduction of an infectious person into a totally susceptible population, thefollowing must hold:

N > r

If we divide both sides of this expression by the recovery rate r, we see that the followingcondition must hold:

N > 1 Equation 5 r

Substituting for D = 1/r into this expression, we obtain the result that ND >1 for thenumber of infectious individuals to increase after the introduction of one infectious personinto a totally susceptible population. You should recognize that the left side of thisexpression is the same as that of the basic reproduction number, as presented on page 6

.




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page 11 of 78 3.1 3.2 3.3 3.4 3.5 3.6

We can use a heuristic argument to see that the quantity (and therefore ND) has the

literal definition of the basic reproduction number :

On average, each infectious person effectively contacts N other individuals per unit time.When the population is entirely susceptible, each of these contacts will generate a newinfection. Multiplying N by the infectious period D (or 1/r), we obtain the total number ofindividuals effectively contacted by the infectious person during their infectious period.

Using this literal definition for R0 in Equation 5 , we conclude that for the introduction ofan infectious person into the population to lead to an increase in the number of infectiousindividuals, the basic reproduction number or ND must be greater than 1.




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page 12 of 78 3.1 3.2 3.3 3.4 3.5 3.6 3.7

Using the result on the previous page, we can see that the number of infected individualsincreases if ND >1. Similarly, the force of infection also increases if ND >1, since therate at which susceptible individuals are infected is directly proportional to the number ofinfectious individuals, i.e.

(t) = I(t)

This is equivalent to Equation 1 in MD01.

We might expect that when the number of infectious individuals increases (or decreases),the infection incidence rate, (t)S(t)=I(t)S(t), will also increase (or decrease). However,whilst this relationship approximately holds, it is not quite exact, since an increase ininfected individuals corresponds to a fall in susceptible individuals, and both I(t) and S(t)appear in the expression for incidence. If we consider outbreaks that only result inrelatively small fluctuations in numbers of susceptible individuals, we can say that,approximately, incidence increases when ND >1.




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Answer

3.8: The epidemic threshold: the proportion susceptible (s)required for the number of infectious individuals to increase(s>1/R0)

page 13 of 78 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8

We can use the logic described on the last few pages to derive the result that theproportion of a population that needs to be susceptible for the number of infectedindividuals to increase at a given time must be greater than 1/R0.

Q1.1 Derive the above result using Equation 4 .

Hint: Divide both sides of Equation 4 by the total population size (N) and and use theresult that R0 = .

Kermack and McKendrick first discussed these results in papers published in 19272 ,although, at the time, they weren't expressed in terms of the basic reproduction number,which was first defined by Macdonald during the early 1950s3-4 .

For the purposes of this module, you do not need to read these references. They arementioned here because of their historical importance.




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3.9 1. What determines whether or not the number ofinfectious individuals increases following the introduction ofan infectious person into a totally susceptible population?

page 14 of 78 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9

Note that the expression ND for the basic reproduction number does not account fordeaths that may occur during the pre-infectious or infectious periods. Anderson and May(1992, Ch 1-4)5 provide the details for expressions that account for deaths. For mostcommon immunising infections, the pre-infectious and infectious periods are generally afew days, whereas the life expectancy (in industrialised populations) is about 70 years. Assuch, since the mortality rate is much smaller than the rate at which individuals becomeinfectious and recover, the effects of these adjustments on the estimate for R0 arerelatively small.

The logic described in this section can be extended to obtain the equations for R0 toaccount for mortality or for non-immunising infections (see e.g. the recommended coursetext, Panel 8.26 ).




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Section 4: 2. How fast might we expect the number ofinfectious individuals to increase following the introductionof an infectious person into a totally susceptible populationand what can we infer from this?

page 15 of 78

It can be shown that, following the introduction of an infectious person into a totallysusceptible population, the number of infectious individuals will increase at a rate (),given by the following expression:

R0 - 1 Equation 6 D

where D is the average duration of infectiousness.

Click the "show" button below for an intuitive explanation of this expression.

is often referred to as the growth rate of an epidemic. The mathematical derivation ofthis result is fairly straightforward although you are not required to know it for this studymodule. If you are interested, you can read Appendix A.2.5 of the recommended coursetext6 , Anderson and May (1991)5 , and Lipsitch, et al (2003)7 . The expression hasalso been extended by Wearing, et al (2005)8 .




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4.1: 2. How fast might we expect the number of infectiousindividuals to increase following the introduction of aninfectious person into a totally susceptible population andwhat can we infer from this?

page 16 of 78 4.1

Equation 6 can be rearranged to give the following expression for R0 in terms of :

R0 D+1 Equation 7

Therefore, given empirical estimates of the growth rate of an epidemic or outbreak (), itshould be possible to infer the R0 of a pathogen. Methods for estimating the growth rateare provided in section 4.2.3.1 of the recommended course text6 . These estimates of R0can then be used to infer future trends in infection incidence.

Equation 7 (and its variants) has been used to derive estimates of the basic reproductionnumber for HIV during the early stages of the HIV epidemic in Kenya and Uganda9 ,which ranged between 4 and 11. As shown on the next page, this theory has been appliedfor Severe Acute Respiratory Syndrome (SARS ), which was caused by a newlyemergent infectious agent which was first identified in 2003.




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page 17 of 78 4.1 4.2

A variant of Equation 7 was used in studies estimating the basic reproduction number ofSevere Acute Respiratory Syndrome (SARS ), which was caused by a newly emergentinfectious agent that was first identified in 2003. The analyses used estimates of theaverage serial interval (calculated as 8.4 days using data from Singapore) and the growthrate in the cumulative numbers of cases7 (see Figure 3). In these analyses, R0 for SARSwas estimated to be in the range 2.0 - 3.6.

If you are interested in seeing the derivation of this result, please read section 4.2.3.2.1 ofthe recommended course text, which provides a simplified description6 . This low valuefor R0 in comparison with that for other infections such as measles or mumps (see MD04

), together with the fact that the peak infectiousness of cases occurs after the onset ofsymptoms, suggested that SARS might be controllable.

Figure 3. a) Number and b) cumulative number of probable or reported cases of SARS in Hong Kong in 2003.Data source: http://www.who.int/csr/sars/country/en/index.html .




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page 18 of 78 4.1 4.2 4.3

The limitations of Equation 7 are that it is only reliable if:

1. The rate of increase in the number of infectious individuals () is calculated duringthe early stages of an outbreak, and

2. The pathogen has only recently been (re)introduced into the population.

Note that if we apply Equation 7 using the growth rate of an epidemic for a pathogen thathas been reintroduced into a population, then we obtain the net reproduction number,defined as the average number of secondary infectious individuals resulting from aninfectious person in a given population (that may not be entirely susceptible).

Further discussion of the application of Equation 7 and its variants can be found in section4.2.3 of the recommended course text6 .




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Section 5: 3. Why does the incidence of an immunisinginfection cycle over time?

page 19 of 78

As we saw earlier (Figure 2 ), the incidence of immunising infections, such as measles,typically cycles over time.

During the first part of the 20th century there were two main theories for the occurrence ofcycles in the numbers of measles cases:

1. The cycles reflected cycles in infectivity of the measles virus, or2. The cycles resulted from changes in the prevalence of susceptible individuals, as a

result of a constant influx of susceptibles born into the population, and susceptibleindividuals becoming immune after becoming infected10 .

Experimental studies in mice populations carried out in the 1930s found no evidence forchanges in the infectivity11 , and the second argument has since become accepted.

Before discussing how changes in the prevalence of susceptible individuals lead toepidemic cycles for immunising infections, we will first review the relationship between thenet reproduction number (Rn), the trend in incidence and the proportion of individuals in thepopulation who are susceptible.




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5.1: 3. Why does the incidence of an immunising infectioncycle over time?

page 20 of 78 5.1

The net reproduction number (Rn) is defined as the average number of secondaryinfectious individuals resulting from each infectious person in a given population (in whichsome individuals may be immune). Rn is related to R0 as follows:

Rn = R0 s Equation 8

where s is the proportion of the population that is susceptible. For example, if eachinfectious person generates 4 secondary infectious individuals in a totally susceptiblepopulation, then in a population in which only 25% of individuals are susceptible, theinfectious person will generate only 1 (= 4 0.25) secondary infectious individuals.

When the number of infected individuals is increasing, Rn > 1; when the number of infectedindividuals is decreasing, Rn < 1; and when the number of infected individuals is stable, Rn= 1. As discussed on page 12, we can approximately say that when the incidence isincreasing, Rn > 1; when the incidence is decreasing, Rn < 1; and when the incidence isstable, Rn = 1.

Given Equation 8 and the relationship between the trend in incidence and the netreproduction number, we can now infer what proportion of the population is likely to besusceptible when the incidence is increasing, decreasing, or at a peak.

For example, when the incidence is increasing, Rn > 1 and therefore, using the fact that Rn= R0 s, we can say that:

R0 s > 1

Rearranging this expression, we see that whilst the incidence is increasing:

s >1/R0 Equation 9




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Answer

Answer


page 21 of 78 5.1 5.2

Q1.2 Given Equation 8 and the relationship between Rn and the trend in incidence,what can we say about the proportion of the population which is susceptible when theincidence is

a) Decreasing?

b) Stable?




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page 22 of 78 5.1 5.2 5.3

Use of the word incidence (optional reading)

Before continuing, we first reflect on our use of the word incidence

In the above discussion, we have related the "incidence" to the net reproduction numberand the proportion susceptible. As you may recall from your previous epidemiologicaltraining, the incidence is defined as the number of new events (such as new infections orinfectious individuals or other outcome of interest) in the population at risk (usuallysusceptibles) per unit time.

However, most of the figures presented in the rest of this session will plot the number ofnew infectious individuals in the population or per 100,000 population rather than theincidence. We have done this for two main reasons:

1. People are generally more used to working with statistics such as the number of newinfectious individuals per person or per 100,000 population than they are with theincidence, since these statistics are similar to those presented for many infections insurveillance reports or in epidemiological papers.

2. Our goal is to explore the relationship between the incidence, the net reproductionnumber and the proportion of the population that is susceptible. For acuteinfections, this relationship is identical to that between the last two statistics and thenumber of new infectious individuals in the population or per 100,000 population. For example, if the incidence of infectious individuals increases, then the number ofnew infectious individuals per 100,000 population per unit time will also increase.




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page 23 of 78 5.1 5.2 5.3 5.4

Use of the word incidence (optional reading) ctd.

Note that for acute infections, trends in the number of new infectious individuals per unittime will also generally be similar to those in the number of new infections per unit time.For example, any increases in the number of new infectious individuals will start only a fewdays (approximately equal to the average pre-infectious period) after the number of newinfections per unit time starts to increase.

Therefore for acute infections, the relationship between the number of new infections per100,000 population, the net reproduction number and the proportion of the population thatis susceptible will be identical to that between the latter statistics and the number of newinfectious individuals per 100,000 population.




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Answer

Answer

Answer

5.5: Exercise: the relationship between the proportion of thepopulation that is susceptible and trends in the number ofnew infectious individuals per unit time

page 24 of 78 5.1 5.2 5.3 5.4 5.5

Figure 4 shows model predictions of an epidemic curve following the introduction of aninfectious person with measles into a completely susceptible population, assuming that thebasic reproduction number is 13.

Q1.3 What proportion of the population might be expected to be susceptible whilst thenumber of new infectious individuals per day is

a) Increasing?

b) Decreasing?

c) At a peak?




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5.6: Exercise: the relationship between the proportion of thepopulation that is susceptible and trends in the number ofnew infectious individuals per unit time

page 25 of 78 5.1 5.2 5.3 5.4 5.5 5.6

Figure 4. Predictions of the number of new infectious individuals per day (red line, right hand axis) and thenumber of susceptible and immune individuals following the introduction of an infectious person with measles intoa totally susceptible population, assuming that R0 = 13, the pre-infectious period = 8 days, the infectious period =7 days and the total population size = 100,000.




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5.7: Summary of the relationship between the proportionsusceptible, Rn and trends in incidence

page 26 of 78 5.1 5.2 5.3 5.4 5.5 5.6 5.7

The previous exercise highlights the fact that the proportion of the population which issusceptible has to be above a certain threshold value (1/R0) for the incidence (or thenumber of new infections or infectious individuals) to increase, and it has to be below thesame threshold for the incidence (or the number of new infections or infectious individuals)to decrease. When the proportion of individuals who are susceptible equals this thresholdvalue, the incidence is stable.

Table 1 and Figure 5 summarise the relationship between trends in incidence and the sizeof the net reproduction number and the proportion of the population that is susceptible.

Incidence (ornumber of

newinfections or

infectiousindividuals)

Rn Proportion susceptible

Increasing >1 >1/R0

Decreasing

Figure 5. Relationship between Rn, the proportion susceptible and number of new infectious individuals per day.

See the caption to Figure 4 for details of the assumptions in the model.

We will now apply the above results to explain why the incidence of immunising infectionscycles over time.

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page 27 of 78 5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8

During the course of an epidemic following the introduction of an infectious person into atotally susceptible population, we see that the following events occur in succession:

1. The number of new infectious individuals increases as there are sufficient numbersof susceptible individuals for each infectious person to lead to >1 secondaryinfectious people.

2. The proportion of individuals that are susceptible in the population decreases.

3. Once this proportion is sufficiently low (

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page 28 of 78 5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8 5.9

The entry of new susceptibles as a result of new births into the population, however,means that the following occurs:

Stage 1. The proportion of susceptible individuals(s) will eventually start to increase once asufficient number of births have been added (see Figure 6).

Figure 6. Explanation for the epidemic cycles in measles - the relationship between the proportion susceptibleand the number of new infectious individuals. Stages 4 and 5 are identical to the stages discussed on pages 24-26 and are greyed out. We will discuss these stages again once we have discussed stage 3.




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page 29 of 78 5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8 5.9 5.10

Stage 2. At some point, the proportion of susceptible individuals(s) is sufficiently large (i.e.>1/R0) for each infectious person to generate >1 infectious people. Once this occurs thenumber of new infectious individuals starts to increase, as shown in Figure 7.

Figure 7. Explanation for the epidemic cycles in measles - the relationship between the proportion susceptibleand the number of new infectious individuals




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page 30 of 78 5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8 5.9 5.10 5.11

Stage 3. At some point, the number of susceptibles who are being removed by infectiousindividuals exceeds the number being added to the population through new births. Oncethis occurs, the increase in the proportion of susceptible individuals slows and thenreverses (Figure 8).

Figure 8. Explanation for the epidemic cycles in measles - the relationship between the proportion susceptibleand the number of new infectious individuals




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page 31 of 78 5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8 5.9 5.10 5.11 5.12

Stage 4. Once the proportion of the population that is susceptible has decreased to reach1/R0, the number of new infectious individuals peaks (so we have now returned to Stage 4in Figures 6-8).

Stage 5. The continuing reduction in the susceptible population means that eventually, theproportion of the population that is susceptible becomes sufficiently low (

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Section 6: Exercise

page 32 of 78

Q1.4 Figure 10 shows how the number of new infectious individuals per unit time changesas the proportion of the population that is susceptible changes. Copy this figure onto apiece of paper. Add plots of the following as they change with the number of new infectiousindividuals per unit time and the proportion susceptible:

a) The net reproduction number

b) The proportion of the population that is immune

Figure 10. The relationship between the cycles in incidence or the number of new infectious individuals per unittime, the proportion susceptible and the net reproduction number.

Please go to the next page to check your answer.




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6.1: Exercise (Answer)

page 33 of 78 6.1

You should have plotted something similar to the plot in Figure 11:

Figure 11. The relationship between the cycles in the numbers of new infectiousindividuals, the proportion susceptible and the net reproduction number

a) Note that the net reproduction number follows the same pattern as the proportion of thepopulation that is susceptible, i.e. the two statistics peak, decrease or reach a troughsimultaneously. This follows from the fact that the net reproduction number is directlyproportional to the proportion susceptible, through the relationship:

Rn = R0s

Also, at a peak or trough in the number of new infectious individuals, Rn is equal to 1;whilst the number of new infectious individuals is increasing Rn is greater than 1 andwhilst the number of new infectious individuals is decreasing Rn is less than 1.

b) The plot of the proportion of the population that is immune is just the mirror image of theplot of the proportion susceptible. This follows from the fact that the proportion immuneis approximately equal to 1-proportion susceptible. This approximation follows from the(reasonable) assumption that, for an immunising infection, the number of infectious orpre-infectious individuals is typically small in a population, in comparison with the




number susceptible or immune.

Therefore, at a peak or trough in the number of new infectious individuals, theproportion immune equals the herd immunity threshold (1-1/R0); whilst the number ofnew infectious individuals is increasing, the proportion immune is less than the herdimmunity threshold (i.e. it is 1-1/R0).

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Section 7: Hamer and the "mass action" principle

page 34 of 78

Hamer used a similar argument to that presented on the previous pages to explain thecycles in measles deaths seen during the early part of the 20th century. His argument,employing the "mass action" principle, was based implicitly on the following differenceequations:

Ct+1 = kCtSt Equation 10

St+1= B + St - Ct+1 Equation 11

where:

Ct and St are the number of cases and susceptibles, respectively, at time t;k is the proportion of the total possible contacts that actually lead to new cases1 ,B is the number of births in each time step.

The time step used in this original model was the serial interval for the infection.

The difference between this model and the other models that we have used so far is thatthe generations of cases in this model do not overlap. In other words, all cases have onsetand infect each other at time steps of 1 serial interval.




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Answer

7.1: Hamer and the "mass action" principle

page 35 of 78 7.1

Hamer argued that when the number of susceptible individuals was at a peak or a trough(i.e. reached a maximum or a minimum), the number of new measles cases occurring inthe population equalled the number of births into the population.

Q1.5 Look at the equations for Hamer's model and think about how you might showthis result.

Hint: Use the result that when the number of susceptible individuals is at a peak or atrough at time t, the number of susceptible individuals at time t+1 equals that at time t (i.e.St+1=St).




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7.2: Hamer and the "mass action" principle

page 36 of 78 7.1 7.2

Figure 12 shows the number of susceptibles and cases predicted by Hamer's model overtime.

In contrast with the models used so far in this module, this model predicts that the cycles inthe numbers of cases should never damp out .

Further theoretical elaborations by Soper (1929)12 found that when the simple massaction model was adapted to assume that the infectious period was not fixed and followeda variable distribution, the cycles damped out. Therefore, the simple mass action modeldid not include sufficient assumptions to fully explain the cycles in incidence for immunisinginfections.

Figure 12. Number of cases and susceptibles over time predicted using Hamer's mass action model, assuming

that C0 = 50, S0 = 1200, B = 50 and k = 0.001 per serial interval. See the Excel file massact.xls for details ofthis model.




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Section 8: 4. What additional factors lead to cycles in theincidence of immunising infections?

page 37 of 78

As we saw in Figure 1 , the model that we worked with during the last few sessionspredicted that the cycles in the number of new infectious individuals damp out, which isinconsistent with the data observed in many populations. For example, as shown in Figure2 , measles epidemics occurred regularly every two years in England and Wales beforethe introduction of vaccination. This discrepancy between model predictions and theobserved data has led to suggestions that other factors must have a role in sustainingthese cycles. These factors are discussed in detail in Anderson and May (1991)1 ; wediscuss the main factors on the next few pages.

You will also find it helpful to read section 4.3.2 of the recommended course text6 .




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8.1: 4. What additional factors lead to cycles in the incidenceof immunising infections?

page 38 of 78 8.1

Seasonality in transmission

Seasonal transmission is the most obvious factor that is likely to lead to cycles in theincidence of immunising infections. There have been many modelling and observationalstudies investigating the effect of seasonality on incidence. Analyses by Fine and Clarkson(1982)1 have found evidence for seasonal transmission of measles in England andWales, occurring as a result of intense mixing between children during term-time, and lessintense mixing during the school holidays (Figure 13).

These analyses used a model-based approach, calculating the transmission parameter k,in Hamer's simple mass action model (i.e. the proportion of the total possible contacts thatactually lead to new cases), using the ratio between the number of cases observed in theUK each week over the time period 1950-77, and the estimates of the number ofsusceptible individuals.

As shown in Figure 13B, the transmission parameter was lowest during the schoolholidays.




Figure 13. Analysis of average biennial measles patterns, based on data from the UK (1950-55). Shaded blocks indicate school summerand Christmas holiday periods1 .

a) Average number of cases notified per week;b) Calculated weekly transmission parameters; c) Estimated numbers of susceptible individuals.

Reproduced from Fine PEM and Clarkson JA (1982) Measles in England and Wales I - an analysis of factors underlying seasonalpatterns. Int J Epidemiol 11(1):5-141 ,. By permission of Oxford University Press.

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page 39 of 78 8.1 8.2

In contrast, seasonal transmission seems to be less important in sustaining the epidemiccycles for pertussis than for measles, as the epidemics for pertussis occur at differenttimes each year (see Figure 14). This has led to suggestions that other factors (forexample, relating to climate) may be important in sustaining the cycles.

Figure 14. Seasonal patterns in measles and pertussis notifications (left and right-hand figures respectively),based on weekly case reports in England and Wales, 1941-1991. Based on Figure 11 in Anderson et al (1984)13




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page 40 of 78 8.1 8.2 8.3

Age-dependent contact patterns

Work by Schenzle (1984)14 has shown that if the populations in models are set up so thattransmission is confined to individuals within specific annual birth cohorts (correspondingto classes within school years), then the models predict regular cycles in incidence. Otherassumptions about age-dependent mixing did not lead to predictions of regular cycles inincidence.




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page 41 of 78 8.1 8.2 8.3 8.4

Stochastic effects

The models discussed so far have been deterministic and do not take account of the factthat individuals come in integer (not fractional!) quantities. The work of Bartlett (1956)15and Anderson and May (1986)16 have shown that models that are amended to deal withdiscrete (integer) numbers of individuals and that allow for stochastic variation intransmission predict regular cycles in incidence (so long as the population size issufficiently large), as shown in Figure 15. See chapter 6 of the recommended course text6

for further discussion of stochastic models.

Figure 15. Comparison between the observed notification rates of measles in the UK and Greenland and thosepredicted using a stochastic model (Anderson and May, 1986)16 .




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Section 9: 5. What inter-epidemic period might we expect tosee for immunising infections?

page 42 of 78

The frequency at which epidemics of immunising infections occur differs betweeninfections. For example, epidemics of measles used to occur every two years in Englandand Wales before the introduction of vaccination, whereas those for smallpox used tooccur roughly every 5 years (see Figure 4.16 in the recommended text6 , and Figure 2 ).

It can be shown (see Anderson and May (1991)5 for a detailed proof) that the inter-epidemic period (T) (defined as the time-interval between successive peaks of anepidemic) for immunising infections, predicted by the simple models used in the last twosessions, is given by the following expression:

T 2A(D+D') Equation 12

where A is the average age at infection, D' and D are the average pre-infectious andinfectious periods, respectively.




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9.1: 5. What inter-epidemic period might we expect to see forimmunising infections?

page 43 of 78 9.1

We can adapt equation 12 to express T in terms of R0. For example, in the absence ofvaccination or control programs, the average age at infection A, the average lifeexpectancy L and the basic reproduction number are related through the followingexpression for some types of populations:

R0 1 + L/A

We will discuss the derivation of this expression in MD04.

This equation can be re-arranged to give the expression A L/(R0- 1). If we substitute thisexpression for A into Equation 12, the equation for the inter-epidemic period in theabsence of control becomes:

T 2L(D+D')

Equation 13R0 - 1




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page 44 of 78 9.1 9.2

As shown in Table 2, predictions of the inter-epidemic period obtained using Equation 13 are generally consistent with the observed inter-epidemic period.

Table 2. Estimates of the observed and predicted inter-epidemic periods for different infections in variouslocations (extracted from Anderson and May (1991)5 ).




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page 45 of 78 9.1 9.2 9.3

Equation 12 for the inter-epidemic period suggests that the introduction of vaccinationinto a population may lead to increases in the inter-epidemic period. For example,vaccination reduces the prevalence of infectious individuals in the population, which willreduce the force of infection and thus lead to an increase in average age at infection.According to Equation 12, the inter-epidemic period will therefore increase.

Equations 12 and 13 have been useful in highlighting certain characteristics of thetransmission of pathogens. Considering measles for example, the inter-epidemic periodafter the introduction of vaccination has been shorter than that predicted5 . Thisdiscrepancy between predictions and the observed data has been attributed to the factthat the model makes the simplifying assumption that individuals mix randomly. As weshall see in MD06 , there is much evidence to show that contact between individuals isage-dependent.




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Section 10: Extending the logic to other pathogens...

page 46 of 78

This session has mainly focused on simple immunising infections (such as measles,mumps and rubella), for which the pre-infectious and infectious periods are measured indays, and are therefore short relative to the lifetime of individuals. The pattern seen forthese infections does not necessarily hold for non-immunising infections. Similarly, thelogic is not easily extendible to diseases such as tuberculosis, for which the intervalbetween infection and disease may be long (e.g. decades), as conditions (such as thenumber of individuals contacted by each infectious person) have probably changed overtime.




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Section 11: Further reading

page 47 of 78

We suggest that you now read Chapter 4 of the recommended course text6 , where theissues covered in this session are explained in further depth.

Some of the issues presented in this session are discussed in further detail in the followingarticles:

Anderson RM and May RM (1982) Directly transmitted infectious diseases:control by vaccination. Science, 215:1053-1060 . Re-used with permission fromAAAS.

Fine PEM and Clarkson JA (1982) Measles in England and Wales I - ananalysis of factors underlying seasonal patterns. Int J Epidemiol 11(1):5-14 .By permission of Oxford University Press.

Hamer WH (1906) Epidemic Disease in England: the evidence of variability andpersistency of type. Lancet, 11:733-9 .




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Section 12: Break...

page 48 of 78

The rest of this session (the practical component) is likely to take 1 - 3 hours . You

may wish to take a short break before starting it.




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Section 13: Practical: Analysing the dynamics of infectiousdiseases

page 49 of 78

OVERVIEW

We will now start parts 2 and 3 of this session, which consist of a practical using modelsset up in Berkeley Madonna.

Part 2 of this session revises the relationship between the basic and net reproductionnumbers (R0 and Rn), the herd immunity threshold and the trends in the number of newinfectious individuals. Part 3 (which is optional) explores how the size of the basicreproduction number affects the inter-epidemic period.

OBJECTIVES

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

The basic and net reproduction numbers and the herd immunity threshold;The peaks in the number of new infectious individuals for an immunising infectionand the prevalence of susceptible and immune individuals in the population;The basic reproduction number and the inter-epidemic period.




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Section 14: Part 2 (practical): The relationship between thebasic and net reproduction numbers and trends in thenumbers of new infectious individuals

page 50 of 78

The relationship between the net reproduction number and trends in thenumber of new infectious individuals

We will first focus on how the net reproduction number, Rn changes during the epidemiccycles for a simple immunising infection. As discussed on page 20 ,

Rn= R0 * proportion susceptible

1. Start up Berkeley Madonna and open the file 'measles2 equations.mmd' , or'measles2 flowchart.mmd ' depending on whether you prefer to work with theequation or flowchart editor versions of the models. Unless otherwise stated, theinstructions for this practical are identical for both files.

The structure of the model is as follows:

You should recognise that this model is identical to the model you created in the lastsession, except that it has 2 new variables (prop_sus and prop_imm). You can seethese variables by viewing the equations window (for those using the equationeditor) or by clicking on the globals button in the flowchart window (for those usingthe flowchart editor). prop_sus and prop_imm are defined as the proportion of thepopulation that is susceptible and immune respectively and we are using theapproximation that prop_imm 1- prop_sus. Please see page 33 for the basis ofthis assumption.

Note that the R0 is currently equal to 13.




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14.1: Part 2 (practical): The relationship between the basicand net reproduction numbers and trends in the numbers ofnew infectious individuals

page 51 of 78 14.1

2. If you are using the equation editor, set up Rn in the equations window using theappropriate expression. If you are using the flowchart editor, set up a new variablecalled Rn in the globals window for the net reproduction number using theappropriate expression.

3. Open the parameters window and run the model, which has been set to run for73,000 days.

Click here to check your expression.

Click here to see the figure that you should have obtained.

PLEASE AVOID CLICKING ON THE BUTTON ON THE TOP RIGHTHAND CORNER OF THE FIGURES WINDOW during this session, as thiswill result in the contents of the figures window being deleted.Unfortunately, the figures are then irrecoverable and will need to be setup again, unless you use one of the solution files that are available inthis session.




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page 52 of 78 14.1 14.2

We first focus on the relationship between Rn and the daily number of infectious individualsover time.

4. Set up a new figure (called page 2) by clicking on the "New Page" button on thetoolbar of the figures page. We will now add a plot of Rn to this figure.

The simplest way to add another variable, such as Rn to the graph is to select theappropriate button for that variable at the bottom of the window. Berkeley Madonnaautomatically includes buttons for the first few variables in the model, but has notincluded a button for Rn. We therefore need to add a button for Rn manually.

5. Follow the instructions provided here to set up a button for Rn to the plot.

At present, it is difficult to see how Rn changes with the number of new infectiousindividuals per day. We will therefore change the scales on the x- and y-axes beforetrying to interpret the figure.

6. Change the scale on the y-axis for Rn to range from 0 to 1.5. Click here forinstructions if you don't recall how to do this.




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page 53 of 78 14.1 14.2 14.3

7. We are interested in how the net reproduction number changes during an epidemiccycle (e.g. over a 10 year period). Therefore, in a similar way, change the x-axis togo from time t = 14600 days (i.e. year 40) to time t = 18250 days (i.e. year 50).

The axis settings window should now look as follows:

Once you have entered the appropriate options into the axis settings window, click on OKto continue.

Click here to see the output you should have by this stage.

If your output does not resemble this figure, you can check your settings against those inthe files measles2 - equations_solna.mmd or measles2 - flowchart_solna.mmd .




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Answer

Answer


page 54 of 78 14.1 14.2 14.3 14.4

Q2.1 How does the net reproduction number change over time? What is the value of thenet reproduction number when the number of new infectious individuals per day peaks?What is the value when the number of new infectious individuals per day reaches atrough?

Q2.2 What is the trend in the number of new infectious individuals per day when

a) Rn 1?c) Rn =1?

Are these trends reasonable?




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Section 15: Part 2 (practical): The relationship between theproportion susceptible and trends in the numbers of newinfectious individuals

page 55 of 78

We will now explore how the number of new infectious individuals per day changes withthe proportion of the population that is susceptible.

8. Copy the figure on Page 2 to Page 3 by clicking on the "New Page" button . Addprop_sus to the new plot (on the left hand y-axis) by clicking on the prop_sus buttonat the bottom of the figure window. Remove Rn from the plot by clicking on the Rnbutton at the bottom of the window.

9. Change the scale of the left hand y-axis to range from 0 to 0.15, in the same waythat you did before.

Click here if you need to remind yourself of how you can change the scales on axes inBerkeley Madonna.


If your output does not resemble this figure, you can check your settings against those inthe files measles2 - equations_solnb.mmd or measles2 - flowchart_solnb.mmd .




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Answer

15.1: Part 2 (practical): The relationship between theproportion susceptible and trends in the numbers of newinfectious individuals

page 56 of 78 15.1

Q2.3 What proportion of the population is susceptible to infection when the number of newinfectious individuals per day is at a peak or trough? Is this what you would expect andwhy?




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Answer

Section 16: Part 2 (practical): The relationship between theherd immunity threshold and trends in the number of newinfectious individuals

page 57 of 78

For transmission of an infection to cease, the proportion of the population which is immunemust be kept higher than the herd immunity threshold (H), which is given by the followingexpression:

H = 1 - 1/R0

Q2.4 What is the herd immunity threshold in the population in the model?

1. Copy the figure on Page 3 to a new page (called Page 4). Add prop_imm to the ploton the left hand y-axis and remove prop_sus from the plot. Change the scale of theleft-hand y-axis to range from 0.8 to 1.0 to see the prop_imm.


If your output does not resemble this figure, you can check your settings against those inthe files measles2 - equations_solnc.mmd or measles2 - flowchart_solnc.mmd .




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Answer

Answer

16.1: Part 2 (practical): The relationship between the herdimmunity threshold and trends in the number of newinfectious individuals

page 58 of 78 16.1

Q2.5 What is the value of the proportion immune when the number of new infectiousindividuals per day peaks or reaches a trough? What do you notice about the value ofprop_imm when the number of new infectious individuals per day is declining or when it isincreasing? How does this relate to your estimated value for the herd immunity threshold?

2. Return to Page 1 of the Figures window.

Q2.6 What is the long-term equilibrium value for the proportion of the population which issusceptible or immune? How do these values relate to the herd immunity thresholdcalculated in Q2.4 ?




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Section 17: Part 2 (practical): The relationship between theherd immunity threshold and the impact of vaccination

page 59 of 78

We will now adapt the model to explore how vaccinating a fixed proportion of thepopulation at a level of coverage close to the herd immunity threshold affects transmission.Before continuing, deselect prop_imm and prop_sus from the plot on Page 1 of the Figurewindow.

1. Set up a new parameter (in the Equation window if you're using the equation editoror in the Globals window if you're using the flowchart editor) called prop_vacc,reflecting the vaccination coverage in the population. Set this equal to 0.75.

We will assume that vaccination is introduced some time (e.g. 50 years) afterthe infection has started circulating in the population.

2. Add the following text to your model on the line immediately after the definition forprop_vacc:

eff_cov = if (time>18250) then prop_vacc else 0

This indicates that the parameter eff_cov (which we will take to reflect theproportion of newborns which are effectively vaccinated) takes the value ofprop_vacc 18250 days (i.e. 50 years) after the start of the simulations;otherwise the value is zero.




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Answer

17.1: Part 2 (practical): The relationship between the herdimmunity threshold and the impact of vaccination

page 60 of 78 17.1

The following is the general structure of the model that we are currently working with:

3. On a separate piece of paper, draw arrow(s) to show how you would change themodel diagram to show newborns being vaccinated.




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Answer


page 61 of 78 17.1 17.2

Q2.7 Set up the equation for the following in terms of the total population, birth rate, andeffective coverage:

a) The number of newborns entering the population per day who aresusceptible (denoted by the arrow "Susceptible births").

b) The number of newborns entering the population per day who are immune(denoted by the arrow "Immunised births").

Use the terms provided in the drop down menus to complete the expressions that youmight use in your model for these terms. Terms may be used more than once or not at all.

Susceptible births = Choose... Choose... Choose...

Immunised births = Choose... Choose... Choose...




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Answer


page 62 of 78 17.1 17.2 17.3

Q2.8 Identify the correct differential equations from the list below that incorporate i)immunised newborns and ii) susceptible newborns entering the population.

a) d/dt(Susceptible) = beta*Susceptible*Infectious - total_popn*b_rate*(1-eff_cov) - Susceptible*m_rate

b) d/dt(Susceptible) = beta*Susceptible*Infectious - total_popn*b_rate*(1-eff_cov) + Susceptible*m_rate

c) d/dt(Susceptible) = -beta*Susceptible*Infectious + total_popn*b_rate*(1-eff_cov) - Susceptible*m_rate

d) d/dt(Susceptible) = -beta*Susceptible*Infectious + total_popn*b_rate*(1-eff_cov) + Susceptible*m_rate

e) d/dt(Immune) = Infectious*rec_rate - Immune*m_rate +total_popn*b_rate*eff_cov

f) d/dt(Immune) = Infectious*rec_rate + Immune*m_rate +total_popn*b_rate*eff_cov

g) d/dt(Immune) = - Infectious*rec_rate + Immune*m_rate +total_popn*b_rate*eff_cov

h) d/dt(Immune) = - Infectious*rec_rate - Immune*m_rate +total_popn*b_rate*eff_cov




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Show


page 63 of 78 17.1 17.2 17.3 17.4

4. Change the model (either in the equations or flowchart editor, depending on whichapproach you're working with) so that:

a) A proportion eff_cov of newborn individuals are effectively vaccinated (i.e.enter the immune compartment) after they are born.

b) The remaining proportion (1 - eff_cov) of newborns enter the susceptiblecompartment.

Click the show button below to check your model if you are using the flowchart editor.

Click here to check your model if you are using the equation editor.




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page 64 of 78 17.1 17.2 17.3 17.4 17.5

5. Run the model. Your output on Page 1 of the Figures window should resemble thefollowing at this stage - you should notice that the epidemic cycles change aftervaccination of 75% of newborns is introduced on day 18250.

If your model is failing to run or your figure looks different from this figure, click measles -equations_solnd.mmd or measles - flowchart_solnd.mmd to access the file thatyou should have developed by this stage.




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Answer


page 65 of 78 17.1 17.2 17.3 17.4 17.5 17.6

6. Run the model for values of prop_vacc that are either above or below the herdimmunity threshold, either by setting up and using the sliders or by changing thevalue in the parameter window. If you are using the sliders, set the maximum toequal 1.0 and the increment for prop_vacc to equal 0.01.

Click here to remind yourself how you can set up and use sliders.

Q2.9 Look at Page 1 of the Figures window. What happens to the number of newinfectious individuals per day if the proportion of the population which is effectivelyvaccinated is above the herd immunity threshold? What happens to the number of newinfectious individuals per day if the value is below the herd immunity threshold?

If you have time, try Part 3 of this session, in which we explore how R0 and other factors(e.g. the vaccination coverage and the birth rate in the population) affect the epidemiccycles. Please save your Berkeley Madonna files before continuing.

Figure 16. Model predictions of the effect of different levels of effective vaccination coverage among newborns,introduced in year 50, on the long-term number of new infectious individuals per day.




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Answer

Section 18: Part 3 (practical): The effect of the basicreproduction number and other factors on the inter-epidemicperiod (optional)

page 66 of 78

1. Reset the proportion of the population which is vaccinated to be zero. Re-run themodel and look at Page 1 of the figures window.

Q3.1 What is the inter-epidemic period 50-100 years following the introduction of oneinfectious person into this population? Note that we are focusing on the time period 50 -100 years after the introduction of one infectious person since at this time, the cyclesshould have stabilized and epidemics should be occurring at approximately regularintervals.

Hint: How many cycles in the number of new infectious individuals per day occur in each10 year period?

You may wish to change the x-axis to use annual, rather than daily time units.

Click here to remind yourself of how you can change what is plotted on the x-axis.




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Answer

18.1: Part 3 (practical): The effect of the basic reproductionnumber and other factors on the inter-epidemic period(optional)

page 67 of 78 18.1

Anderson and May (1992)5 , provide the following formula for estimating the inter-epidemic period for immunising infections:

T 2L(D+D')

Equation 14R0 - 1

where D' is the average duration of the pre-infectious period, D is the average duration ofinfectiousness, L is the life expectancy and is the constant (3.14...).

Q3.2 Are your results from Q3.1 consistent with this formula? Note that the lifeexpectancy currently equals 70 years.




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Answer


page 68 of 78 18.1 18.2

2. Run the model for values of R0 of 5 and 18. R0 of measles was estimated to be 5-6in Kansas (US 1918-9), 13-14 in Cirencester (UK 1947-50) and 18 in England andWales (1950-68).

Q3.3 How does the inter-epidemic period resulting from an R0 of 18 compare against thatresulting from an R0 of 5? Why might this occur?

Note that, as mentioned in MD02 , you can see the effect of changing a given parametervalue on model predictions by clicking on the overlay button and then running themodel.




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Answer


page 69 of 78 18.1 18.2 18.3

Q3.4 How should the introduction of vaccination affect the inter-epidemic period? Checkyour answer by changing the value of prop_vacc and re-running the model.




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Answer


page 70 of 78 18.1 18.2 18.3 18.4

Q3.5 How should the birth rate in the population influence the inter-epidemic period? Testyour hypothesis by changing the birth rate assuming that the population size remainsconstant over time.




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Answer

Answer


page 71 of 78 18.1 18.2 18.3 18.4 18.5

Q3.6 Would you expect the inter-epidemic period for measles to be longer or shorter thanthat for:

a) Chickenpox?b) Mumps?c) Rubella?

Hint: R0 for measles is typically higher than that for rubella and mumps; the average pre-infectious and infectious periods for these infections are shorter than those for measles.

3. Change the parameters in the model to be those for influenza (pre-infectious andinfectious period = 2 days, R0 = 2). Reset the birth rate to be that in the originalmodel (i.e. corresponding to a life expectancy of 70 years) and, making sure that noone is vaccinated in the population, run the model.

Q3.7 Why might you be cautious about using the predictions of the inter-epidemic periodfor influenza from this model?




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Section 19: Further exercises

page 72 of 78

To consolidate your understanding, we suggest that you try the exercises associated withthe model files for chapter 4 in the recommended course text6 .




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Section 20: Summary

page 73 of 78

1. For immunising infections, for the numbers of infectious individuals to increase oncean infectious person enters a totally susceptible population, the basic reproductionnumber, R0, which is given by the expression ND, must be bigger than 1.

Here N is the total population size, D is the duration of infectiousness and is therate at which two specific individuals come into effective contact per unit time.

2. During the early stages of an epidemic of a new infection, R0 can be estimated fromthe epidemic growth rate () using the expression: R0 1+D, and variants of thisexpression.

3. These equations have been used to estimate R0 for HIV during the 1980s, for SARSwhen it first emerged and for pandemic influenza.

4. For an immunising infection, the proportion of the population susceptible for thenumber of infectious individuals to increase (or equivalently, for an epidemic tooccur) must be bigger than 1/R0. This result can be obtained by using the facts that:

For the number of infectious individuals to increase, the net reproductionnumber, Rn (defined as the average number of secondary infectiousindividuals resulting from an infectious person in a given population) must bebigger than 1.

Rn is related to R0 and the proportion of the population that is susceptible (s)through the equation Rn = R0 s.

If Rn>1, then R0 s must be bigger than 1 for the number of infectiousindividuals to increase. Rearranging the expression R0 s >1 leads to the resultthat s>1/R0 for the number of infectious individuals to increase.




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20.1: Summary

page 74 of 78 20.1

5. In general, for the incidence or the number of new infectious individuals to decreasefor an immunising infection, the proportion of the population that is susceptible has tobe below 1/R0.

6. The following table summarises the relationship between trends in incidence or thenumber of new infections per unit time, Rn, the proportion susceptible and theproportion immune:

Incidence(or number

of newinfections

orinfectious

individuals)

Rn Proportionsusceptible

Proportionimmune

Increasing >1 >1/R0

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20.2: Summary

page 75 of 78 20.1 20.2

9. For immunising infections, the inter-epidemic period, T, is given by the followingapproximation:

T 2A(D+D')

where A is the average age at infection, D' and D are the durations of the pre-infectious and infectious periods respectively, L is the average life expectancy.

10. The following approximation can be used to calculate the inter-epidemic period in theabsence of vaccination:

T 2L(D+D')R0 - 1

11. This approximation is obtained by using the fact that, in the absence of vaccinationR0, A and L are related through R01+L/A (see EP301 and discussed further inMD04 ), or after rearranging, AL/(R0 -1), where L is defined as the average lifeexpectancy. After substituting the latter expression for A into the approximationdescribed in point 9 above, we obtain the equation:

T 2L(D+D')R0 - 1

12. These approximations have usually led to estimates of the inter-epidemic periodwhich are reasonably consistent with those observed for many infections.




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20.3: Summary

page 76 of 78 20.1 20.2 20.3

13. In general, the inter-epidemic period for immunising infections increases as R0decreases. This follows from the fact that, if R0 is low, the proportion of thepopulation that needs to be susceptible for an epidemic to occur (1/R0) is greaterthan that required when R0 is high. This means that it takes longer for the size of thesusceptible population to grow, through the accumulation of susceptible births, to therequired threshold for the epidemic to occur when R0 is low than when R0 is high.

14. Likewise, the introduction of vaccination (e.g. of very young children) leads to anincrease in the inter-epidemic period. This follows from the fact that it takes longerfor the size of the susceptible population to grow, through the accumulation ofsusceptible births, to the threshold required for an epidemic to occur than when noindividuals have been vaccinated.

15. The results described in this session largely relate to acute immunising infectionsand assume that individuals mix randomly. The results will not necessarily hold fornon-immunising infections and for infections which have long incubation periods forwhich conditions (such as the amount of contact between individuals) has probablychanged over time, at least in many Western populations.




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1. Double click in the middle of the graph. This opens a new window called "ChooseVariables".

2. Double click on the variable that you would like to plot from the variables list on theleft hand side of this window, so that the variable appears in the list under the "YAxes" section on the right hand side. This section lists the variables for whichbuttons will be set up at the bottom of the figures window.

3. Click on "OK" to continue and re-run the model. Your selected variable should now

Section 21: Working with graphs in Berkeley Madonna -summary of key features

page 77 of 78

Overview of the toolbar available in the figures window


Setting up buttons for a new variable Changing the scale on the x- or y-axis

Changing what is plotted on the x--axis How to use Sliders



be included in the plot (appearing on the left hand side y-axis).

Note: Unless you re-run the model, the variable will not appear in the plot!

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References

page 78 of 78

1. Fine, P.E. and J.A. Clarkson, Measles in England and Wales--I: An analysis offactors underlying seasonal patterns. Int J Epidemiol, 1982. 11(1): p. 5-14.

2. Kermack, W.O. and A.G. McKendrick, Contributions to the mathematical theory ofepidemics--I. 1927. Bull Math Biol, 1991. 53(1-2): p. 33-55.

3. Macdonald G. The analysis of equilibrium in malaria. Trop Dis Bull 1952; 49:81329.

4. Macdonald G. The epidemiology and control of malaria. London: Oxford UniversityPress, 1956.

5. Anderson, R.M. and R.M. May, Infectious diseases of humans: dynamics andcontrol. 1991, Oxford: Oxford University Press.

6. Vynnycky, E. and R.G. White, An Introduction to Infectious Disease Modelling. 2010,Oxford: Oxford University Press.

7. Lipsitch, M., et al., Transmission dynamics and control of severe acute respiratorysyndrome. Science, 2003. 300(5627): p. 1966-70.

8. Wearing, H.J., P. Rohani, and M.J. Keeling, Appropriate models for the managementof infectious diseases. PLoS Med, 2005. 2(7): p. e174.

9. Anderson, R.M., May, R.M. and A.R. McLean, Possible demographic consequencesof AIDS in developing countries. Nature, 1988. 332(6161): p. 228-34.

10. Hamer, W.H., Epidemic disease in England - the evidence of variability and ofpersistency of type. Lancet, 1906(i): p. 733-9.

11. Fine, P.E., John Brownlee and the Measurement of Infectiousness: An HistoricalStudy in Epidemic Theory. Journal of the Royal Statistical Society, 1979. 142(3): p.16.

12. Soper, M.A., The interpretation of periodicity in disease prevalence. J.R. Stat. Soc.Ser. A, 1929. 92: p. 34-61.

13. Anderson RM, Grenfell BT, May RM. Oscillatory fluctuations in the incidence ofinfectious disease and the impact of vaccination: time series analysis. J Hyg (Lond).1984 Dec;93(3):587-608.

14. Schenzle, D., An age-structured model of pre- and post-vaccination measlestransmission. IMA J Math Appl Med Biol, 1984. 1(2): p. 169-91.

15. Bartlett, M., On theoretical models for competitive and predatory biological systems.Biometrika, 1957. 44: p. 27-42.

16. Anderson, R.M. and R.M. May, The invasion, persistence and spread of infectiousdiseases within animal and plant communities. Philos Trans R Soc Lond B Biol Sci,1986. 314(1167): p. 533-70.




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MwMi9tZDAzL3BhZ2VfNjQuaHRtAA==: form0: zoom_query: input1:

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MwMi9tZDAzL3BhZ2VfNjEuaHRtAA==: Q1Combo1: Choose...Q1Combo2: Choose...Q1Combo3: Choose...Q1Combo1_(1): Choose...Q1Combo2_(1): Choose...Q1Combo3_(1): Choose...

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MwMi9tZDAzL3BhZ2VfMjcuaHRtAA=

md03

Documents

simplest models

number of new infections

forimmunising infections

models general predictions

infectious period

practical exerciseusing

infectious person

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