Introduction to Infectious Disease Modelling (EC06)

Module: EPM301 Epidemiology of Communicable Diseases

Course: PG Diploma/ MSc Epidemiology

This document contains a copy of the study material located within the computer assisted learning (CAL) session. The first three columns designate which page, card and screen position the text refers to. If you have any questions regarding this document or your course, please contact DLsupport via DLsupport@lshtm.ac.uk. Important note: this document does not replace the CAL material found on your module CDROM. When studying this session, please ensure you work through the CDROM material first. This document can then be used for revision purposes to refer back to specific sessions. These study materials have been prepared by the London School of Hygiene & Tropical Medicine as part of the PG Diploma/MSc Epidemiology distance learning course. This material is not licensed either for resale or further copying.

London School of Hygiene & Tropical Medicine September 2013 v1.0

Section 1: EC06 Introduction to Infectious Disease Modelling Aims: To introduce the areas of application of models and the methods for setting up models. Objectives: By the end of this session, you should:

know some of the kinds of questions which might be addressed using modelling.

be aware of the different types of models. understand the principles of setting up models and some of the key

input parameters of models. This session should take 35 hours to complete. Section 2: Introduction Mathematical models are increasingly being used to interpret and predict the dynamics and control of infectious diseases.

Applications include predicting the impact of vaccination strategies against common infections such as measles and rubella, and determining optimal control strategies against influenza, HIV and vectorborne diseases. This session illustrates some of the areas of application of modelling and will introduce you to some of the methods for setting up models. Section 3: What are models? Definitions

According to the Oxford English dictionary, a model is any representation of a designed or actual object

Models typically aim to recreate the transmission dynamics of an infection using the smallest possible number of parameters and assumptions

In infectious disease epidemiology, two kinds of models have typically been employed in the past:

Physical or mechanical models Mathematical models

Physical models

The classic example of the physical or mechanical model is the ReedFrost teaching model, developed at Johns Hopkins during the 1930s to teach medical

students about the numbers of cases which are likely to be seen over time, following the introduction of infectious cases into a population. Interaction: Button: Reference (pop up box appears): Fine (1977) A commentary on the mechanical analogue to the Reed-Frost epidemic model. Am J Epidemiol, 1977, 106(2): 87-100 It is named after two famous infectious disease epidemiologists: Reed and Wade Hampton Frost who developed the model. 3.1: What are models? Physical models (cont)

The model consists of a roulette wheel or a trough and some coloured balls (see right), representing susceptible individuals, cases and immune individuals (blue, red and green balls respectively).

The balls are thrown into the trough or roulette wheel; the number of new cases which occur in the next time interval equals the number of blue balls which are in contact with a red ball. These blue balls are replaced with red balls (as they are now "cases"); red balls are replaced with green balls (as they are now "immune").

The whole process of throwing balls is repeated until there are no more red balls in the population.

Nowadays, this whole process of simulating an epidemic can be carried out using mathematical equations, implemented on a computer, within a matter of seconds.

3.2: What are models?

Mathematical models (cont) Mathematical models, as their name suggests, use mathematical equations to describe the transmission dynamics of an infection in a population. The following is an example of one of the simplest mathematical models of the transmission dynamics of measles: St + 1 = St Ct + 1 (+ Bt ) Ct + 1 = St Ct r This particular model is based on reasoning by Hamer in 1906, and was designed to interpret the biennial cycles in measles incidence which are seen in most populations. Interaction: Hyperlink: seen in most populations Output: [pop up]: Weekly measles notifications in England and Wales, 195079 (Fine and Clarkson, 1982)

Interaction: button: References Output [pop up]:

Hamer WH (1906) Epidemic disease in England: the evidence of variability and persistence of type, Br Med J March 17:733739 Fine PEM and Clarkson JA (1982) Measles in England and Wales I: an analysis of factors underlying seasonal patterns. Int J Epidemiol 11(1):514

In these equations: Ct is the number of cases at time t; St is the number of susceptible individuals at time t; Bt is the number of births into the population at time t; r is defined as the proportion of total possible contacts between cases and susceptibles which lead to new infections (nowadays denoted by ).

This model is simplistic in that it describes how the number of infectious individuals changes from one generation to the next, taking time steps of 1 serial interval (time interval between successive cases in a chain of transmission). Interaction: button: Note Output [pop up]: This is a simple way of thinking conceptually about epidemics. For example, in a real epidemic, generations of cases overlap each other, and so the model predictions do not exactly correspond to what we would observe in a real outbreak. We will illustrate how this particular model can be refined to allow generations of cases to overlap later in this session. End Note button interaction. Interaction: button: References Output [pop up]:

HopeSimpson RE (1948) The period of communicability of certain epidemic diseases. Lancet (Nov 13), 755760. Fine PE (2003). The Interval between successive cases of an infectious disease. Am J Epidemiol 158, 10391047.

3.3: What are models? As shown in the graph button opposite, this particular model, which incorporates new susceptible individuals being born into the population, and susceptibles being removed from the population, as a result of their becoming cases and then immune, is able to recreate these cycles in incidence.

Number of cases and susceptibles over time predicted using Hamer's mass action argument

Interaction: button: Graph Output:

Section 4: Applications of modelling Models have been applied to many problems in infectious disease epidemiology. Their main (though not exclusive) use has been in predicting the impact of control programs and the future numbers of cases.

However, models may also provide useful insight into the epidemiology of an infection, since they (usually!) aim to recreate the transmission dynamics of an infection using the smallest number of parameters. Any discrepancy between the models output and real data may then help to elucidate other factors which may be important in the epidemiology of that infection.

We first describe the use of models for predicting the impact of control programs. Section 5: Applications of modelling predicting the critical levels of treatment or vaccination coverage The most obvious area of application of modelling is in predicting the impact of control programs and the future number of cases.

Much of the work centres around the theme of thresholds and the idea that, in order to control transmission, it is not necessary to treat all the cases or vaccinate every susceptible individual: it is sufficient to reduce the numbers of cases or susceptibles to a sufficiently low threshold level and transmission will eventually cease.

This idea was first conceived by Ronald Ross in 1908, who reasoned that to control malaria transmission, the density of mosquitoes in the population needed to be reduced to a sufficiently low level. Interaction: button: References (1): Output: Ross (1911) The Prevention of Malaria, 2nd ed., Murray, London. End interaction. This theme was subsequently developed by Kermack and McKendrick in 1927, and then by Macdonald in 1957, who defined the basic reproduction rate as the average number of secondary cases resulting from each infectious case when introduced into a totally susceptible population. Interaction: button: References (2): Output: Kermack WO and McKendrick AG (1927). Contributions to the mathematical theory of epidemics. I Proceedings of the Royal Society of Medicine, 115A: 700721. MacDonald G (1957). The Control of Epidemiology and Control of Malaria. Oxford University Press, London. 5.1: Applications of modelling predicting the critical levels of treatment or vaccination coverage Macdonald referred to this basic reproduction rate Interaction: hyperlink: basic reproduction rate Output: Basic reproduction rate Techinically, R0 is a number or ratio rather than a rate, since it has no time units. Nowadays, epidemiologists refer to it as the basic reproduction number. End interaction. as Z0, which subsequently became known as R0 and is now more commonly known as the basic reproduction number or ratio.

His observation that this quantity had to exceed 1 for malaria to persist in a population led to optimism that malaria could be eradicated.

Although this optimism proved to be shortlived, the reproduction number concepts were developed and applied extensively during the 1980s to many immunizing infections, such as measles, mumps, rubella, to calculate the herd immunity threshold Interaction: hyperlink: herd immunity threshold Output: Herd immunity threshold The proportion of the population which needs to be immune in order to control transmission, i.e., for the net reproduction number to equal 1. See EC03 if you would like to revise this concept. End interaction. and hence the proportion of the population which needed to be vaccinated in order to control transmission.

(See Anderson and May (1991) for a review.) Interaction: Button: Reference (pop up box appears): Anderson and May (1991) Infectious Disease of Humans. Dynamics and Control. Oxford University Press. Oxford. 5.2: Applications of modelling predicting the critical levels of treatment or vaccination coverage As you saw in EC03, the herd immunity threshold is calculated using the equation:

Herd immunity threshold = 11/R0. The panel opposite shows the relationship between R0 and the herd immunity threshold, and the values for the herd immunity threshold for several infections.

Note the small value for the herd immunity threshold for smallpox, which, some have argued, greatly facilitated the eradication of smallpox during the 1970s.

The R0 for malaria has been estimated to be very high (approaching 200)hence the high value for its herd immunity threshold.

5.3: Applications of modelling predicting the critical levels of treatment or vaccination coverage The basic reproduction number (and hence, the herd immunity threshold) is straightforward to calculate from serological data if it can be assumed that individuals mix randomly.

Interaction: hyperlink: randomly

Output:

Everyone is as likely to contact, e.g. people of different ages, social groups, etc., as they are to contact people of their own age, social group, etc. This is also referred to as homogenous mixing.

End interaction.

See the tabs to revise these methods. Calculating the R0 Interaction: tabs: Tab 1: Revision I As you saw in EC03, if a population has a rectangular Interaction: hyperlink: rectangular Output:

End interaction. age distribution (also known as a Type I distribution), such as that seen in many Western populations, R0 can be calculated using the expression R0 = L/A

Here, L is the life expectancy and A is the average age at infection. Tab 2: Revision II The expression for the R0 for a population with a negative exponential

Interaction: hyperlink: negative exponential Output:

End interaction. age distribution (also known as a Type II distribution), such as that seen in some developing countries is analogous to that for a population with a rectangular age distribution: R0 = 1 + L/A

Here, L is the average life expectancy and A is the average age at infection. See Anderson and May (1991) for the derivation of this expression. Interaction: Button: "References": Anderson and May (1991) Infectious Disease of Humans. Dynamics and Control. Oxford University Press. Oxford 5.4: Applications of modelling predicting the critical levels of treatment or vaccination coverage Estimates for the R0 based on assumptions of random mixing are helpful for providing a rough estimate of the level of coverage required to control transmission.

However, few populations mix randomly and estimates of the R0 based on assumptions of random mixing may either under or overestimate the true proportion of the population which would need to be vaccinated in order to control transmission. 5.5: Applications of modelling predicting the critical levels of treatment or vaccination coverage Example:

The diagram opposite shows the contact patterns in two populations (A and B).

In population A, each child contacts 1 other child and 1 adult and each adult contacts 5 adults and 1 child.

In population B, individuals mix randomly: each child contacts the same number of adults and children as does an adult.

In both populations, each case leads to an average of 4 cases.

Question:

If vaccination is available only for children, is the level of coverage required to control transmission in population A higher, equal to or lower than that in population B? Interaction: hotspots: Hotspot 1: Higher Output [correct response]: Correct Control is easiest to achieve if it is targeted at the individuals who are the most active transmitters. In population A, young individuals are responsible for a smaller proportion of all the transmission events (2/8) than in population B (4/8). As a

result, it will be harder to control transmission in population A through vaccinating children than in population B. A higher proportion of children will therefore need to be vaccinated in population A than in B in order to control transmission. Hotspot 2: Lower Output [incorrect response]: No, thats not right. Control is easiest to achieve if it is targeted at the individuals who are the most active transmitters. In population A, young individuals are responsible for a smaller proportion of all the transmission events (2/8) than in population B (4/8) and therefore a higher proportion of individuals need to be vaccinated in population A than in population B to control transmission. Hotspot 3: Equal Output [incorrect response]: No, thats not right. Control is easiest to achieve if its targeted at the individuals who are the most active transmitters. In population A, young individuals are responsible for a smaller proportion of all the transmission events (2/8) than in population B (4/8) and therefore a higher proportion of individuals need to be vaccinated in population A than in population B to control transmission. 5.6: Applications of modelling predicting the critical levels of treatment or vaccination coverage Estimates of the level of vaccination coverage which would be required to control transmission need to take into account both how many people each child or adult contacts and who they contact.

It can be shown (using modelling techniques which are beyond the scope of this introductory session!) that approximately 81% of population A would need to be vaccinated to control transmission, as compared with 75% of population B. It is not necessary to use a model to see that control is easiest to achieve if its targetted at the individuals who are the most active transmitters.

In fact, some control strategies e.g. shutting schools and thereby limiting contact between schoolchildren (who are considered to be the most active transmitters of influenza infection) have been introduced in some countries during influenza epidemics, without any prior modelling being carried out. 5.7: Applications of modelling predicting the critical levels of treatment or vaccination coverage On the other hand, models are necessary to calculate the actual level of treatment or vaccination coverage required to control transmission for a given assumption about contact between individuals.

Given that the true degree of contact between different population groups (e.g. the young and the old) is poorly understood, in some countries e.g. the UK, models are used to explore the effect of several different assumptions about contact between individuals on the impact of vaccination against measles, pneumococcal, meningococcal and other infections. Interaction: tabs Tab 1: Example In the UK, uptake of MMR vaccination has been variable (and sometimes below the herd immunity threshold for measles), as a result of public concerns over its safety.

Changes in the proportion of individuals who are susceptible to measles infection, as a result of MMR vaccination and ongoing transmission are used to calculate the overall net reproduction number for different assumptions about contact between individuals. Interaction: Button: "References": Gay NJ, Hesketh LM, Morgan-Capner P, Miller E (1995) Interpretation of serological surveillance data for measles using mathematical models: implications for vaccine strategy. Epidemiol Infect. 1995; 115(1): 139-56. Tab 2: Example (cont) The size of the net reproduction number then provides an indication of the chance of an outbreak of measles; models incorporating different assumptions about contact can provide insight into the numbers of cases which might occur in different age groups. Section 6: Applications of modelling predicting the impact of control strategies Besides answering questions about the critical level of vaccination or treatment coverage, models can also be used to answer other policyrelated questions, such as:

For how long do you need to vaccinate in order to control transmission?

Is mass vaccination at periodic intervals more effective at reducing transmission than vaccinating a fixed proportion of individuals each year?

If no cases have been observed e.g. for 1 year, what is the probability that control has been achieved? These questions have been explored in relation to rubella, measles, and more recently polio.

Interaction: Button: References Output: Anderson RM and Grenfell BT (1986). Quantitative investigation of different vaccination policies for the control of congenital rubella syndrome (CRS) in the UK. J Hyg. Camb, 96, 30533. Anderson RM and May RM (1985). Agerelated changes in the rate of disease transmission: implications for the design of vaccination programmes. J Hyg. Camb, 94, 365436. Eichner M, Hadeler KP, Ditz K. Stochastic models for the eradication of poliomyelitis: minimum population size for polio virus persistence. Model for Infectious Diseases: Their Structure and Relation to Data. Eds. V Isham, G Medley. Cambridge University Press, 1996. Trotter CL, Gay NJ, Edmunds WJ (2005). Dynamic models of meningococcal carriage, disease, and the impact of serogroup C conjugate vaccination. Am J Epidemiol. 2005, July 1; 162(1):89100.

End interaction.

To illustrate this use of modelling, we shall focus on one specific example, namely the relationship between rubella, the incidence of Congenital Rubella Syndrome and rubella vaccination coverage.

6.1: Applications of modelling predicting the impact of control strategies Background

Rubella is a mild immunizing infection. However, if a woman is infected with rubella when pregnant, the child may be born with Congenital Rubella Syndrome (CRS).

In countries in which the incidence of rubella is high, most women will have been infected with rubella when young and will be immune by childbearing age. In such countries, few women are newly infected with rubella during childbearing age and the burden of CRS is typically low. Exercise: The following chart shows the agespecific proportion of individuals seropositive to rubella antibodies in the UK and China during the 1980s.

Question: Assuming that neither country had introduced rubella or MMR vaccination during the 1980s, in which country would you have expected to see a higher incidence (per live birth) of Congenital Rubella Syndrome ? Interaction: Buttons: Button 1: UK Output: UK Correct. The proportion of women of childbearing age who have antibodies (i.e. immunity) to rubella is higher in China than in the UK. You might therefore expect the number of new infections per capita among women in China to be lower than that in the UK, and for the incidence of CRS per 1000 live births to be correspondingly lower in China. Button 2: China Output: China Thats not quite right. The proportion of women of childbearing age who have antibodies (i.e. immunity) to rubella is higher in China than in the UK. You might therefore expect the number of new infections per capita among women in China to be lower than that in the UK, and for the incidence of CRS per 1000 live births to be correspondingly lower in China. Button 3: References Output: Farrington CP (1990) Modelling forces of infection for measles, mumps and rubella. Statistics in Medicine, 9, 95367.

Wannian (1985). Rubella in the Peoples Republic of China. Rev Inf Dis; 7 (S72). 6.2: Applications of modelling predicting the impact of control strategies The introduction of any vaccination or treatment programme into a population reduces the prevalence of infectious cases.

With the reduced opportunity for individuals to become infected, the risk of infection goes down over time, and as a result, the average age at infection can increase.

This is important for infections such as rubella. If vaccination (such as the MMR vaccine) is provided at below the herd immunity threshold and only among infants, then the proportion of individuals who reach childbearing age still susceptible to infection increases over time. This, in turn, may lead to an increase in the burden of Congenital Rubella Syndrome.

Question: Considering the rubella seroprevalence data which you saw on the previous panel, might you be more cautious about introducing infant MMR vaccination in China or in the UK? Interaction: Buttons: Button 1: UK Output: UK (Probably!) incorrect. In the UK, the burden of CRS was likely to have been greater in the absence of infant MMR vaccination than in China, where the burden was very low or non existent. In China, the introduction of infant MMR vaccination at below the herd immunity threshold might therefore result in the emergence of a new problem of CRS and, potentially, a greater relative increase in the burden of CRS than in the UK. Button 2: China Output: China (Probably!) correct. In the UK, the burden of CRS was likely to have been greater in the absence of infant MMR vaccination than in China, where the burden was very low or non existent.

In China, the introduction of infant MMR vaccination at below the herd immunity threshold might therefore result in the emergence of a new problem of CRS and, potentially, a greater relative increase in the burden of CRS than in the UK. 6.3: Applications of modelling predicting the impact of control strategies

In practice, this question is very difficult to answer without using modelling techniques e.g. setting up a model which describes rubella transmission in the population and simulating the introduction of different levels of coverage. For example, the incidence of CRS is proportional to:

(proportion of women of childbearing age who are susceptible)

risk of infection

The introduction of infant vaccination will lead to an increase in the proportion of women of childbearing age who are susceptible.

On the other hand, it will lead to a reduction in the risk of infection.

The net effect on the incidence of CRS will depend on whether the increase in the proportion of women of childbearing age who are susceptible outweighs the reduction in the risk of infection 6.4: Applications of modelling predicting the impact of control strategies Extensive modelling work on this question was carried out by many researchers (e.g. Anderson, Grenfell, Knox) during the 1980s, as shown in the diagrams opposite.

According to Figure a, vaccination among infants is associated with decreases in the CRS incidence, if the average age at infection before introduction of vaccination is 12 years or more.

As shown in Figure b, for settings in which the average age at infection before the introduction of vaccination is very low e.g. 3 years (such as that in The Gambia), the introduction of vaccination at any level of coverage below 85% results in an increased burden of CRS.

Predictions of the ratio between the numbers of cases of CRS after the introduction of vaccination infants at different levels of coverage (x-axis) and that before the introduction of vaccination (Anderson and May (1983)

Interaction: button: Show

Output:

Typed out by M. A. Predictions of the ratio between the numbers of cases of CRS after the introduction of vaccination among infants at different levels of coverage (X-axis) and that before the introduction of vaccination (Anderson and May (1983) Proportion vaccinated End interaction. The different lines are predictions for settings in which the average age at infection before the introduction of vaccination ranged between 3 and 18 years. Interaction: button: References Output:

Anderson RM and May RM (1983). Vaccination against rubella and measles: quantitative investigations of different policies. J Hyg Camb. 90, 259325. 6.5: Applications of modelling predicting the impact of control strategies These predictions were proved correct during the 1990s in Greece, where MMR vaccination had been available in the private sector since the mid 1970s.

As shown in the figure opposite, throughout this time, the proportion of pregnant women who were susceptible increased from about 12% during the early 1970s to reach about 35% by 19901.

This gradual increase in the proportion susceptible led to an increase in the incidence of CRS during the 1990s, with an outbreak of 25 cases occurring in 1993. A further outbreak occurred in 1999. In contrast, only 8 cases were reported during the period 19801990.

Interaction: button: references Output: Panagiotopoulos T, Antoniadou I, ValassiAdam E. Increase in congenital rubella occurrence after immunisation in Greece: retrospective survey and systematic review. BMJ, 1999 Dec 4; 319 (7223):14627. 6.6: Applications of modelling predicting the impact of control strategies The issues discussed in relation to rubella are also relevant for other infections:

Polio increases in the incidence of paralytic polio were seen during the 1950s in many Western populations.

(see e.g. Anderson and May (1991)).

These were attributed to improvements in hygiene and hence a reduction in the risk of infection and an increasing proportion of individuals being infected with polio virus for the first time when adults. (Note that infection during adulthood carries a greater risk of paralytic polio than does infection as a child). Interaction: button: references Output: Anderson and May (1991). Infectious Diesases of Humans: Dynamics and Control. Oxford University Press, Oxford.

Mumps outbreaks of mumps have been seen among adolescents and young adults in the UK and other Western countries since the year 2000.

This has been attributed to the introduction of MMR vaccination among young children, leading to a reduced risk of infection in the overall population, with individuals born before the introduction of vaccination reaching adult life still susceptible to infection. Interaction: button: references Output: Savage E, Ramsay M, White J, Beard S, Lawson H, Hunjan R, Brown D. Mumps outbreaks across England and Wales in 2004: Observational study. BMJ, 2005;330(7500):111920. Section 7: Applications of modelling elucidating the epidemiology of infection

Another use of modelling, which often receives little attention, is in providing useful insight into the epidemiology of an infection.

For example, models (usually!) aim to recreate the transmission dynamics of an infection using the smallest number of parameters and assumptions.

Any discrepancy between the models output and real data may therefore help to elucidate other factors which may be important in the natural history of that infection. Interaction: tabs: Tab 1: This diagram Interaction: hyperlink: diagram Output:

shows predictions of the Hamer model described earlier, adapted to take daily time steps, rather than time steps of 1 serial interval.

Note that this model, which is more realistic than the Hamer model, predicts that the cycles should ultimately damp out. The fact that this damping out is inconsistent with the observed data, suggests that extrinsic factors, which had not been incorporated in the model, help to sustain the cycles. Tab 2: The diagram Interaction: hyperlink: diagram Output:

shows predictions of the numbers of cases of CJD in the UK by Ghani et al during the late 1990s. Interaction: button: References Output: Ghani AC, Ferguson NM, Donnelly CA, Hagenaars TJ, Anderson RM. Epidemiological determinants of the pattern and magnitude of the vCJD epidemic in Great Britain. Proc Biol. Sci. 1998 Dec 22;265(1413):244352 End interaction. Each of the scenarios shown were based on different assumptions, but led to predictions of the numbers of cases seen until then, which were consistent with the observed data.

The work highlighted the fact that too little was known about the epidemiology of CJD in order to make reliable predictions of the likely scale of the epidemic. Section 8: Classification of models More than 75% of published models have been socalled deterministic models, as these are easiest to set up.

We will first describe what we mean by deterministic models and return to the other kinds of models (stochastic models) later in this session. Section 9: Setting up deterministic models Deterministic models describe what happens on average in a population.

The parameters in the model (e.g. the rate at which individuals recover, the number of individuals contacted by each person per unit time etc) are fixed. Hence the outcome (e.g. the number of cases seen at any given time) is predetermined.

The majority of deterministic models are compartmental models Compartmental models stratify individuals into broad epidemiologically meaningful categories, which depend on the disease, and describe the transitions between these categories. We will focus mainly on compartmental models.

For immunizing infections, for example, it would be useful to stratify people into those who are Susceptible, Infected (but not yet infectious), Infectious and Immune, as follows: Note that the InfEcted class is also sometimes referred to as the _xposed_class; this however, can be confusing since everyone can be considered to be exposed. 9.1: Setting up deterministic models Compartmental models describe the transmission dynamics in terms of the total number of individuals in these categories.

Question: Which of the diagrams opposite might describe the model of the transmission dynamics of an infection which does not confer complete immunity?

Model A is defined as a SIS model; Model B is referred to as a SIR model; Model C is defined as a SEIR model and Model D is defined as a SIRS model.

9.2: Setting up deterministic models Deterministic compartmental models can be set up using either difference equations or differential equations.

We will first describe the approach based on difference equations.

Difference equations describe changes in the numbers of individuals in given categories using discrete time steps e.g. of 1 day, 2 days etc. Suppose that we would like to predict the average number of measles cases which we might see in a closed population (i.e. in which there are no births, deaths or immigrants or emigrants) following the introduction of a measles case into that population.

As we are setting up a compartmental model, we first need to stratify individuals into epidemiologically meaningful categories (compartments and decide on the transitions between these categories. 9.3: Setting up deterministic models Question: Based on the assumption that measles is an immunizing infection, which of the diagrams opposite reflects a possible structure for our model?

9.4: Setting up deterministic models Once the model structure has been decided, the next stage in the model development involves writing down the equations for the number of susceptible, infected, infectious and immune individuals at a given time t + 1 (e.g. tomorrow) in terms of the number present at a previous time t (e.g. today).

Returning to our measles model, and considering the susceptible individuals, the equations need to specify the fact that:

Number susceptible at time t + 1 =

Number susceptible at time t Number of individuals who are newly infEcted between t and t + 1

The number of individuals who are newly infEcted between t and t + 1, in turn, is just the product of the risk that a susceptible individual is infEcted between time t and t + 1 (traditionally denoted by the symbol) and the number of susceptible individuals at time t. i.e. tSt

Interaction: hyperlink: tSt

Output:

For example, if 10% (= t) of susceptible individuals are newly infected between today and tomorrow, and we have 500 (St) susceptible individuals present today, then the number of individuals who are newly infected between today and tomorrow is 0.1 * 500 = 50.

9.5: Setting up deterministic models The risk of infection (also referred to as the "force of infection", for historical reasons) depends on the number of infectious individuals in the population and on the extent to which they contact other individuals.

If its assumed that individuals mix randomly, then the number of individuals newly infected in each time is given by the following expression: StIt Here, is defined formally as the probability of an effective contact between two specific individuals in each time step, and

It is the prevalence of infectious individuals in the population at time t. We will return to the definition and derivation of later in this session.

This particular equation is known as the massaction equation, first used by Hamer in 1906,

Interaction: hyperlink: Hamer in 1906

Output:

See also page 3 of 23, page of 3 of this session. [Note: this is exactly how this text appears meaning uncertain!]

End interaction.

which considered individuals coming into contact with each other in the same (random) way that gas molecules come into contact with each other.

As we shall see later in this session, though simplistic, this formulation leads to reasonable predictions of time trends in the overall incidence of an immunizing infection. 9.6: Setting up deterministic models

Equating the two expressions for the number of individuals who are newly infEcted between time t and t + 1, i.e. StIt = tSt

we see that, after cancelling St on both sides of this equation: It = t

We can now formally write down the equations for the number of susceptible individuals at time t + 1 as follows: St + 1 = St tSt

or as St + 1 = St StIt

9.7: Setting up deterministic models

Returning to our model diagram and now considering the individuals in the infEcted category, we see that the equations need to specify the fact that: Number of individuals in the infEcted category at time t + 1

= Number of individuals in the infEcted category at time t

+ Number of individuals who are newly infEcted between time t and t + 1 Number of individuals who develop infectious disease between time t and t + 1.

The number of individuals who develop infectious disease between time infected and t + 1, in turn, is just the product of the proportion of infected individuals who develop infectious disease between time t and t + 1 (we will denote this proportion by f for now) and the number of infected individuals at time t. We will return to the definition and derivation of f later in this session.

9.8: Setting up deterministic models

We can now formally write down the equations for the number of infected individuals at time t + 1 as follows: Et + 1 = Et + tSt f Et or

Et + 1 = Et + StIt f Et Here we are assuming that the proportion of infected individuals who develop infectious disease in each time step is the same over time we will return to this assumption later. 9.9: Setting up deterministic models

Returning to our model diagram and now considering the individuals in the infectious category, we see that the equations need to specify the fact that:

Number of individuals who are infectious at time t + 1

= Number of individuals who were infectious at time t

+ Number of individuals who develop infectious disease between time t and

t + 1.

Number of individuals who recover from infectious disease between time t and t + 1 The number of individuals who recover from infectious disease between time t and t + 1, in turn, is just the product of the proportion of infectious individuals who recover from infectious disease between time t and t + 1 (we will denote this proportion by r for now) and the number of infectious individuals at time t. i.e. r It

We will return to the definition and derivation of r later in this session. 9.10: Setting up deterministic models Exercise:

Write down the equation for the number of infectious individuals at time t + 1 in terms of It, Et, f and r. Check your answer by clicking on the answer box. Interaction: button: Answer Output: Answer

It + 1 = It + fEt rIt Exercise:

Write down the equation for the number of immune individuals at time t + 1 (Rt + 1) in terms of Rt, It and r, and check your answer by clicking on the answer box. Interaction: Button: Answer: Answer The equations need to specify the fact that: Number of individuals who are immune at time t + 1 = Number of individuals who were immune at time t + Number of individuals who recover from infectious disease between time t and t+1 Using mathematical notation, this equation can be written as: Rt+1 = Rt + rIt

9.11: Setting up deterministic models

The difference equations for this model can be summarized as follows: St + 1 = St St It or equivalently St + 1 = St tSt Et + 1 = Et + St It fEt Et + 1 = Et + tSt fEt It + 1 = It + fEt r It It + 1 = It + fEt r It

Rt + 1 = Rt + r It Rt + 1 = Rt + r It

At present, this model does not incorporate births entering the population or individuals dying.

To incorporate births into the population, we would need to change the equation for the number of susceptible individuals (assuming, for now, that individuals are born susceptible as follows:

St + 1 = St St It + Bt where Bt is the number of individuals born between time t and t + 1. 9.12: Setting up deterministic models

[diagram static]

Exercise: Rewrite the difference equations for the above model, incorporating the assumption that the proportion of individuals who die during each time step (m) is the same for the susceptible, infected, infectious and immune individuals (again, this may not be realistic).

Interaction: button: Answer

Output: Answer St + 1 = St St It + Bt mSt Et + 1 = Et + St It fEt mEt It + 1 = It + fEt r It mIt Rt + 1 = Rt + r It mRt

If we assume that the proportion of individuals who die during each time step (m) is the same for the susceptible, infected, infectious and immune individuals, then the number of susceptible, infected, infectious and immune individuals who die between time t and t + 1 is mSt, mEt, mIt, and mRt, respectively. We therefore need to subtract mSt individuals from the susceptible population at time t to adjust for the susceptible individuals dying. Similarly, we need to subtract mEt individuals from the infected population at time t to adjust for the infected individuals dying, etc.

Exercise

Write down the difference equations corresponding to the following model diagram, assuming that no individuals are born into or die from the population.

Interaction: button: Answer

Output: Answer The equations should be as follows: St+1 = St tSt + r1Ct Ct+1 = Ct + tSt + r1Ct fCt r2Ct Dt+1 = Dt + fCt r3Dt Rt+1 = Rt + r3Dt + r2Ct 9.13: Setting up deterministic models Technical issue 1: Directions of the arrows

You should notice that the direction of the arrows linking the different compartments in the model diagram determines whether individuals are added or taken away from the number of individuals in that compartment.

For example, there is an arrow going out of the susceptible compartment and entering the infected compartment, reflecting the fact that susceptible individuals are being newly infected. Hence, there is a minus sign in front of the term for the number of individuals who are newly infected ( StIt) in the equation for the number of susceptible individuals at time t + 1, and there is a plus sign in front of this term in the equation for the number of infected individuals at time t + 1. 9.14: Setting up deterministic models Technical issue 2: Check for population size

You should notice that if you add the four equations of the model for measles in a closed population to each other, you obtain the result that:

S

t+1 +E

t+1 + It+1 + Rt+1 = St + Et + It + Rt

i.e. the total population size remains constant over time.

This is a useful check to see whether your equations have any errors. 9.15: Setting up deterministic models

Technical issue 3: Risks versus rates Technically, the input parameters which occur in difference equations are risks since were using e.g. the proportion of infEcted individuals who develop infectious disease in each time step, the proportion of infectious individuals who recover in each time step, etc. In practice, modellers tend to work with rates in difference equations, since these are easier to calculate and are approximately equal to the risk. For example, you may recall that risks and rates are related through the following equation:

risk = 1 erate If the rate is small (see diagram) Interaction: hyperlink: diagram Output:

End interaction. then the risk is approximately equal to the rate.

Section 10: Setting up deterministic models: Excel exercise 1 Given estimates for the model parameters, such as and the proportions of infected and infectious individuals who develop infectious disease and recover in each time

step , the model can be set up in a spreadsheet to predict the total number of individuals present in each of the compartments over time.

We shall now illustrate how this is done in further detail.

10.1: Setting up deterministic models: Excel exercise 1

Step 1: Open up the spreadsheet measles1.xls.

The model in this spreadsheet is designed to simulate the effect of introducing 1 infectious measles case into a population comprising 99,999 susceptible individuals, assuming that no individuals are born into or die from the population. You will see some: a) yellow cells which hold the size of the time step in the model and the total population size (cells F4 and F5); b) blue cells which hold the key input parameters (rows 1116) we shall discuss the derivation of these values later; c) purple cells (rows 4225594) which will (eventually) hold the equations for the number of susceptible, infected, infectious and immune individuals over time.

d) a graph (currently empty) which will show the numbers of susceptible, infectious and immune individuals over time. 10.2: Setting up deterministic models: Excel exercise 1

Look at the equations for the number of susceptible and infected individuals for day 1: you should notice that theyre expressed in terms of the number present on day 0. Step 2: Set up appropriate equations for the number of infected and immune individuals on day 1 in cells D45 and E45, assuming, for now, that no individuals are born into or die from the population. Step 3: Copy all the equations for day 1 down until day 200. Interaction: hyperlink: Hyperlink 1:

Check Excel expression for infectious individuals for day 1 Output: = D44+C44*dis_rateD44*rec_rate End Hyperlink 1 interaction.

Hyperlink 2: Check Excel expression for immune individuals for day 1 Output: = E44=D44*rec_rate

Compare your graph of the number of susceptible, infectious and immune individuals to the graph here. Interaction: hyperlink: graph here Output:

If the two differ, refer to spreadsheet measles1solna.xls to check that you have typed in the equations correctly.

The graph shows the classic epidemic curve, i.e. after a peak in the incidence of infectious cases, no further cases arise in the population.

This is attributable to the depletion of susceptible individuals: all have been infected, have developed disease and are immune to further infection. 10.3: Setting up deterministic models: Excel exercise 1 We will now alter the model to incorporate births and deaths. At present, we are assuming that the population remains constant over time, with the mortality rate per capita equalling the birth rate per capita.

Step 4: Using the contents of the cells labelled m_rate and num_births, alter the equations for the number of susceptible, infEcted, infectious and immune

individuals on day 1 to incorporate births into and deaths out of the population (assuming, for now, that individuals are born susceptible.

Step 5: Copy all the equations for day 1 down until day 25550 (i.e. 70 years). Interaction: button: Example Output: Check Excel expressions for day 1: Susceptible individuals: = B44+beta *B44*D44+num_births-B44*m_rate Infected individuals: = C44+beta *B44*D44-dis_rate*C44-C44*m_rate [Note: yes, it is C44-C44] Infectious individuals: = D44+C44*dis_rate-D44*rec_rate-D44*m_rate Immune individuals =E44+D44*rec_rate-E44*m_rate

Step 6: Select rows 18 and 37 together, click with your right mouse button and choose the unhide option. NB To select two rows together, click on the grey row header for the first row that you're interested in, and either hold down the left mouse button and drag the mouse down to the grey row header for the next row, OR hold the shift key down and click on the grey row header for the second row. You should now see the following graph Interaction: hyperlink: following graph Output:

of the numbers of susceptible and immune individuals over time. If your graph differs from this graph, check your equations against the equations in measles1solnb.xls.

The graph shows that the numbers of susceptible and immune individuals cycle over time. 10.4: Setting up deterministic models: Excel exercise 1 This cycling is attributable to the fact that susceptible individuals are being

a) added to the population as a result of new births and

b) removed from the population as they become infected and subsequently immune. The number of susceptible individuals

a) decreases when the number of susceptibles being removed (as a result of their becoming cases and subsequently immune) is larger than the number being added through new births;

b) increases when the number of susceptibles being removed is less than the number being added through new births. 10.5: Setting up deterministic models: Excel exercise 1 This cycling in the numbers of susceptible and immune individuals also leads to cycling in the infectious disease incidence.

Step 7: To see this, double click with your mouse on the righthand yaxis in the figure showing the longterm trends in the spreadsheet and change the scale to go from zero to 20. We shall discuss the relationship between the peaks in the disease incidence and the numbers of susceptible individuals, after we have considered the derivation of the input parameters in the model.

Before continuing, please save your spreadsheet as measlesfin1.xls Section 11: Input parameters for models:

is the most important parameter in infectious disease models, but at the same time, it is the most difficult to estimate.

It is defined formally as the: probability of an effective contact between two specific individuals per unit time, and is referred to by various names (which are not always consistent with its formal definition!) such as the transmission probability, transmission coefficient, transmission parameter, transmission rate, etc. An effective contact is defined as one which is sufficient to lead to infection of a susceptible individual if it occurs between a susceptible and an infectious individual.

As a result, the definition of an effective contact depends on both the method of transmission (e.g. sexual vs respiratory vs. vectorborn vs. faecaloral etc) and the infection considered. 11.1: Input parameters for models:

For tuberculosis, for example, casual contact is unlikely to lead to transmission, as shown in the Figure opposite.

The highly infectious nature of measles, on the other hand, means that contact between individuals can be less intimate than that for TB in order for transmission to occur.

Interaction: Button: References: van Geuns HA, Meijer J, Styblo K (1975) Results of contact examination in Rotterdam, 1967-1969. Bull Int Union Tuberc, 50:107-121 11.2: Input parameters for models: depends further on other factors such as age and setting.

For example, children probably contact more individuals (who are also likely to be children) than do adults; individuals living in urban areas probably have many more effective contacts than do those living in rural areas.

Question: Which of the bars in the above figure are likely to reflect the seroprevalence among females in rural areas? Interaction: Buttons: Button 1: White Bars Output: White Bars Correct. For each age group, the seropositivity is lower among individuals reflected in the white bars than for the individuals reflected in the black bars, suggesting that these individuals face a correspondingly lower risk of infection, i.e. they probably live in rural areas. Button 2: Black Bars Output: Black Bars Thats not quite right. For each age group, the seropositivity is lower among individuals reflected in the white bars than for the individuals reflected in the black bars, suggesting that these individuals face a correspondingly lower risk of infection, i.e. they probably live in rural areas. Button 3: References Output: Dowdle WR, Ferrera W, De Salles Gomes LF, King D, Kourany M, Madalengoitia J, Pearson E, Swanston WH, Tosi HC, Vilches AM. WHO collaborative study on the seroepidemiology of rubella in Carribbean and Middle and South American populations in 1968. Bull World Health Organ. 1970; 42(3):41922. 11.3: Input parameters for models: may also change over time: as a result of reduction in crowding in living conditions, Interaction: hyperlink: reduction in crowding in living conditions Output: During the second part of the 19th century, ~8% of the UK population lived in accommodation with more than 2 individuals per room, as compared with 5.5% by 1901 and ~1% by 1951. By 1991, only 0.25% of the population lived in accommodation with more than 1.5 individuals per room. End interaction.

individuals living in many Western countries in the year 1900 probably contacted more individuals than their counterparts who are alive today.

Interaction: Button: References: Hunt S. Housing-related disorders. In: Charlton J, Murphy M (eds.), The health of adult Britain, 1841-1994. Volume 1; chap. 10, 157-70. The Stationery Office; London 1997.

may also change over time as a result of changes in behaviour or interventions. During past influenza pandemics, for example, the amount of contact between individuals changed over time, e.g. with theatres and concert halls shutting and office closing hours being staggered to avoid the congregation of individuals. is difficult to measure directly for the majority of infections, excepting possibly, those involving a vector or sexual transmission. Consequently, it is typically calculated using indirect methods involving analysis of seroprevalence data and/or the basic reproduction number. 11.4: Input parameters for models: For example, for immunizing infections, and assuming that individuals mix randomly, can be formally calculated from the basic reproduction number, using the expression = R0 / (Nd),

where N is the total population size and d is the duration of infectiousness. Interaction: Hyperlink: = R0 / (Nd) For an infection for which all infected individuals develop infectious disease, the R0 reflects the number of effective contacts made by each person over a time period which is equal to the infectious period of an infectious case. R0/d reflects the number of effective contacts made by each person per unit time. Dividing this by N gives the probability of an effective contact between 2 specific individuals per unit time. Exercise:

The basic reproduction number of measles and rubella in some populations has been estimated to be 13 and 7 respectively.

The duration of infectiousness for measles is 7 days and 11 days for rubella.

Considering population of a given size, is the value for greater for measles or for rubella?

Interaction: button: Measles

Output:

Measles

Correct. In a population comprising 100 individuals, the value for is 0.019 per day for measles and 0.0064 per day for rubella. Interaction: button: Rubella

Output:

Rubella

Thats not quite right. In a population comprising 100 individuals, the value for is 0.019 per day for measles and 0.0064 per day for rubella, i.e. it is higher for measles than for rubella. 11.5: Input parameters for models: Alternatively, (and other parameters) can be obtained by fitting model predictions of disease incidence to observed data on the numbers of cases seen over time.

For more complicated assumptions about mixing e.g. assuming that contact between individuals is agedependent, is calculated separately for each age group, using an approach based on fitting to observed seroprevalence data. Unfortunately, those methods are beyond the scope of this session However, further details are provided in the following reference:

Anderson RM and May RM (1991) Infectious Diseases of Humans. Dynamics and Control, Ch 9. Oxford University Press. Section 12: Input parameters for models rate at which individuals develop infectious disease, recover, etc. An assumption which is often made in infectious disease models is that, once infected, individuals develop infectious disease at a constant rate, or that once infectious, individuals recover and become immune at a constant rate.

This assumption is convenient, since a) the rate at which something occurs is equal to 1/(average time to that event).

b) The average latent and infectious periods are usually known (at least for immunizing infections).

For example, the average infectious period for measles is 7 days; the rate at which infectious individuals recover from infectious disease is 1/7 = 0.143 per day. 12.1: Input parameters for models rate at which individuals develop infectious disease, recover, etc. The assumption that individuals develop or recover from infectious disease at a constant rate may not be realistic, since it implies that the latent or infectious periods follow the negative exponential distribution.

Interaction: hyperlink: negative exponential distribution

Output:

End interaction.

Distribution of the time interval between infection and onset of infectiousness, assuming that individuals develop infectious disease at a constant rate of 0.143/day

Days since infection

It is possible to refine this assumption e.g. to assume that the rate of infectious disease onset depends on time since infection. In practice, the level of complexity incorporated in the model usually depends on the kind of question asked.

One rule in modelling is to keep models as simple as possible, and no simpler. If model predictions fail to match the observed data, then this is a strong indicator that the model is too simplistic and some assumptions need to be changed! Section 13: Setting up deterministic models Excel exercises II We shall now return to the Excel spreadsheet that you worked with earlier (measlesfin1.xls) and incorporate these expressions for the input parameters and explore how they influence the disease incidence.

Step 1: Select rows 5 and 10 of this spreadsheet together, click with the right mouse button and choose the unhide option. You should now see some yellow cells, containing values for the average latent and infectious periods, the life expectancy and the basic reproduction number.

Step 2. Set up appropriate equations for beta, dis_rate, rec_rate and m_rate in the corresponding cells.

At this stage, your graphs should remain unchanged. If they change, check your equations against the contents of measles1solnc.xls. 13.1: Setting up deterministic models Excel exercises II The model currently incorporates the latent and infectious periods and the R0 for measles.

Step 3: Change the latent and infectious periods and the R0 to be those for rubella (10 days, 11 days and 7 respectively). Question: How does changing the input parameters in the model to reflect those of rubella affect the:

a) epidemic curve during the first 200 days following the introduction of the infectious case into the population?

b) the cycles in the numbers of immune and susceptible individuals?

(The answer is on the next slide) 13.2: Setting up deterministic models Excel exercises II a) The epidemic curve for rubella

The epidemic peaks later for rubella than it does for measles, as you can see by clicking on the graph button below.

You would expect this to happen, since the latent period is longer for rubella than it is for measles. It therefore takes longer for a rubella case to appear in the population after infection than for a measles case.

You should also notice that, as for measles, all individuals are immune in the population by the end of the first 200 days following the introduction of the infectious case. Interaction: button: Graph Output:

Predictions of the daily incidence of measles and rubella following the introduction of one infectious individual into a totally susceptible population comprising 100,000 individuals 13.3: Setting up deterministic models Excel exercises II b) The cycles in the numbers of susceptible and immune individuals for rubella

The numbers of susceptible and immune individuals cycle more slowly for rubella than they do for measles.

You might have expected this to occur, since the R0 for rubella is smaller than that for measles. Since each rubella case removes fewer susceptible individuals than does each measles case, it takes longer to deplete the susceptible population for rubella than it does for measles. We shall now discuss the relationship between the R0 and these cycles in more detail.

Section 14: The relationship between the R0 and the cycles in disease incidence: Excel exercises III

From your previous epidemiological training, you will recall that the size of the net reproduction number (Rn) determines whether or not the disease incidence increases or decreases in a given population (see EC03 p11).

If Rn is bigger than one (i.e. each infectious case is leading to more than one secondary infectious case), then the disease incidence should increase.

If Rn is less than one (i.e. each infectious case is leading to less than one secondary infectious case), then the disease incidence should decrease. We will now use our spreadsheet model to explore this relationship further.

Step 1: Open up the spreadsheet measles2.xls. It is very closely related to the spreadsheet you have just been using, except that we now have some cells set up for the proportion of susceptible and immune individuals in the population for day 1.

Step 2. Set up an appropriate expression for the net reproduction number on day 1 and copy this expression down until the 18250th day. Interaction: Button: Hint: The expression for the net reproduction number on day 1 should be: =G81*R0 14.1: The relationship between the R0 and the cycles in disease incidence: Excel exercises III Step 3: Select the columns I and V together, click with your right mouse button and select the unhide option.

You should now see Figure 1

Interaction: hyperlink: Figure 1

Output:

(end interaction) which plots the Rn on the left hand yaxis and the disease incidence on the right hand yaxis in the population. The xaxis goes from the 14600th day (ie the 40th year) to the 18250th day (ie the 50th year). The gridlines on the xaxis occur at 2 year intervals.

(If Figure 1 in your spreadsheet does not resemble this Figure, you may like to check the equations in your spreadsheet against the equations in measles2ansa.xls).

Question: What is the value of the Rn when the disease incidence peaks or reaches a trough? Interaction: hotspots (set 1): Hotspot 1: ~>1 Thats not quite right. At the peak or trough of the disease incidence, Rn is approximately equal to 1. Hotspot 2: ~

Thats not quite right. When Rn~1, the disease incidence is increasing. Hotspot 3: Unchanging Thats not quite right. When Rn~>1, the disease incidence is increasing. 14.2: The relationship between the R0 and the cycles in disease incidence: Excel exercises III Step 2: Select rows 22 and 43 together, click with the right mouse button and select the unhide option.

You should now see Figure 2,

Interaction: hyperlink: Figure 2

Output:

(end interaction) which plots the proportion susceptible on the left hand yaxis and the disease incidence on the right hand yaxis.

Question. What proportion of the population is susceptible to infection when the disease incidence peaks or troughs? Interaction: button: Answer Output: At a peak or trough in disease incidence, the proportion of individuals which are susceptible is approximately 0.077. You should remember from previous sessions that, for transmission of an infectious disease to persist in a population, the proportion of the population which is immune to infection has to be below the herd immunity threshold(HIT), which is given by: HIT = 1 1/R0

Question: According to this expression, what should be the herd immunity threshold for the infection in this model? Interaction: button: Answer Output: The R0 is 13, so the herd immunity threshold is 11/R0 which is approximately equal to 92.3%.

14.3: The relationship between the R0 and the cycles in disease incidence: Excel exercises III Step 3: Select rows 49 and 70 together, click with the right mouse button and select the unhide option.

You should now see graph: Figure 3

Interaction: hyperlink: Figure 3

Output:

(end interaction) which plots the proportion immune in the population together with the disease incidence.

Question: What proportion of the population is immune when the disease incidence peaks or troughs?

Interaction: button: Answer Output:

At a peak or a trough in the disease incidence, the proportion of the population which is immune is approximately 0.923, i.e. equal to the herd immunity threshold. (Note that this is equal to 1 0.077, the proportion of the population which is susceptible at this time, as we saw earlier on EC06p15c3)

End interaction.

Question: What do you notice about the proportion of the population which is immune when the disease incidence is a) declining? b) increasing? Interaction: button: Answer A Output:

When the disease incidence is declining, the proportion of the population which is immune is always above 0.923, i.e. above the herd immunity threshold. End Answer A interaction. Interaction: button: Answer B Output: When the disease incidence is increasing, the proportion of the population which is immune is always below 0.923, i.e. above the herd immunity threshold. Section 15: The relationship between the R0 and the cycles in disease incidence: summary

The spreadsheet exercise which you have just completed should help to reinforce the theory that you learnt in previous sessions: for the incidence of an infectious disease to increase or decrease, the proportion of the population which is immune must be below or above the threshold respectively, given by:

1 1/R0

This is equivalent to saying that the proportion of the population which is susceptible must be above or below the threshold 1/R0 for the disease incidence to increase or decrease respectively. The exercise also highlights the fact that cycles in the incidence of an immunizing infection occur because of cycles in the proportion of individuals who are susceptible and immune. 15.1: The relationship between the R0 and the cycles in disease incidence: summary For example, as shown in this graph, Interaction: hyperlink: graph

Output:

A summary of the process underlying the cycles in measles incidence Disease incidence Proportion susceptible A. Numbers of new births added to popn ~>number of susceptibles being removed, resulting in an increase in proportion of susceptibles B. Proportion susceptible now sufficiently large (i.e. ~> 1/Ro) for each case to start to lead to ~> 1 secondary case = ~> increase disease incidence C. Increased proportion susceptible slows and then reverses, since no. susceptible being removed ~> no. being born D. Proportion susceptible is now so low (i.e. ~< 1/Ro) that each case leads to ~< 1 secondary case and the disease incidence starts to decrease

the proportion of susceptible individuals in the population increases if the number of births entering the population exceeds the number of susceptibles being removed, as they become cases and then immune. (point A). The disease incidence starts to increase once the proportion susceptible starts to exceed 1/R0. (point B) 15.2: The relationship between the R0 and the cycles in disease incidence: summary This increase in the incidence (and hence prevalence) of infectious cases, in turn, slows down and eventually reverses the increase in the proportion of individuals

who are susceptible i.e. more susceptibles are being removed than are being added to the population. (point C)

Once the proportion of susceptible individuals has decreased to be below 1/R0, each case starts to lead to less than one secondary case and the disease incidence decreases (point D). Interaction: button: Graph Output:

A summary of the process underlying the cycles in measles incidence Disease incidence Proportion susceptible A. Numbers of new births added to popn ~> number of susceptibles being removed, resulting in an increase in proportion of susceptibles B. Proportion susceptible now sufficiently large (i.e. ~> 1/Ro) for each case to start to lead to ~> 1 secondary case = ~> increase disease incidence C. Increased proportion susceptible slows and then reverses, since number susceptibles being removed ~> no. being born D. Proportion susceptible is now so low (i.e. ~< 1/Ro) that each case leads to ~< 1 secondary case and the disease incidence starts to decrease

15.3: The relationship between the R0 and the cycles in disease incidence: summary The interepidemic period

It can be shown that the interepidemic period for an immunizing infection is related to its basic reproduction number through the following expression:

1R)'(2

0 +

DDLT

Here, L is the average life expectancy, and D and D are the durations of the latent and infectious periods respectively. is the universal constant, given approximately by 3.14. Use this formula to calculate the interepidemic period in a population in which the life expectancy is 60 years for a) measles (R0=13, D=8 days, D=7 days) b) rubella (R0=7, D=10 days, D=11 days) Interaction: button: answer: measles Output:

yearsdaysxxT 85.21039113

)87(3656014.32 ==

+

Interaction: button: answer: rubella Output:

yearsdaysxxT 76.4173917

)1011(3656014.32 ==

+

15.4: The relationship between the R0 and the cycles in disease incidence: summary You should find that the values obtained using the formula for the interepidemic period should be consistent with the values predicted by the spreadsheet model that you were working with, once the cycles have settled down.

However, if the spreadsheet model is extended to make predictions Interaction: hyperlink: predictions Output:

Predictions of the daily incidence of measles cases, following the introduction of 1 infectious case into a totally susceptible population Number of cases per day per 100,000 population Time (years) for a prolonged time period e.g. over 100 years, then the cycles eventually damp out (although they can still be seen if the yaxis scale is refined sufficiently). Since this damping out is inconsistent with what is observed in reality for immunizing infections, it suggests that other extrinsic factors must be important in sustaining these cycles.

The most likely factor is seasonal contact, which occurs as a result of children contacting each other intensively during school terms and less so during the school holidays. Interaction: button: References Output: Fine PEM and Clarkson JA (1982) Measles in England and Wales I: an analysis of factors underlying seasonal patterns. Int J Epidemiol 11(1):514.

15.5: The relationship between the R0 and the cycles in disease incidence: summary There is also a discrepancy between model predictions of disease incidence for influenza and the observed data.

For example, application of the formula Interaction: hyperlink: formula Output: For influenza, the average latent and infectious periods are both roughly 2 days, and the R0 has been estimated to be approximately 2. End interaction. for the interepidemic period suggests that the peaks in the incidence of influenza should occur roughly every 56 years, whereas, in reality, in most European countries, influenza epidemics occur almost every winter. This discrepancy suggests that the model would need to be changed to be consistent with the observed data, e.g. incorporating changes in the strain of influenza in circulation, so that individuals are not completely immune to the new strain, if its introduced into the population.

In addition, the predicted interepidemic period would change if the model were to incorporate seasonal contact between individuals or seasonality in the transmissibility of the influenza strain. Section 16: Beyond difference equations: time step sizes and differential equations

As you have just seen, difference equations describe the numbers of individuals in different categories using discrete time steps e.g. of 1, 2, 3 days etc.

Such models can be set up in any spreadsheet, although using it to make longterm predictions (e.g. for 100 years) can become difficult as the spreadsheet can become large. To reduce this problem, the size of the time step used in the model can be increased.

To see an example of how this can be achieved, open up the spreadsheet measles3.xls.

This spreadsheet is similar to the models that we have been using so far, except that and the transition, birth and mortality rates have been reexpressed to take account of the size of the time step.

Step 1: Change the size of the time step (in cell F4) to be 0.5, 2, 3, 4 and 5 days and look at predictions of the longterm disease incidence. 16.1: Beyond difference equations: time step sizes and differential equations You should notice that predictions of disease incidence based on time step sizes of 0.54 days, as shown in the graph in the spreadsheet, are very similar.

However, if the time step is 5 days then the model fails to make reliable predictions. This results from the fact that the size of is now so large that, between day 50 and 55 more individuals are predicted to become infected than were susceptible on day 50.

This, in turn, leads to negative numbers of susceptible individuals on day 55, which, in the current model, leads to negative numbers of infectious individuals and ultimately, the model fails to make sensible predictions. 16.2: Beyond difference equations: time step sizes and differential equations A way of ensuring that models make reliable longterm predictions and avoiding the problems with inaccuracies in using large step sizes is to use differential equations.

Differential equations use time step sizes which are infinitesimally small and therefore describe the transmission dynamics in continuous time. Such equations can be set up in specialist software packages (e.g. ModelMaker, Berkeley Madonna), which convert differential equations into difference equations, but then adjust for the error resulting from using discrete time steps rather than continuous time. 16.3: Beyond difference equations: time step sizes and differential equations The notation for differential equations

Interaction: hyperlink: differential equations

Output:

for the measles model we were discussing earlier differs from that for the difference equations. Interaction: hyperlink: difference equations

Output:

These differences are discussed in the optional reading section for this session. Section 17: Beyond difference equations: stochastic models

As we have seen, deterministic models describe what happens on average in a population.

As such, they are unlikely to provide a reliable answer to questions involving small populations, where chance might play an important role in the outcome. The following include possible questions which could not be adequately answered using a deterministic model:

If we introduce 1 infectious case of MRSA into a ward comprising 20 susceptible individuals, how many cases are we likely to see?

What is the probability that transmission will cease?

What is the 95% confidence interval in the number of cases that we will see?

On the other hand, these kinds of questions could be answered using a stochastic model. 17.1: Beyond difference equations: stochastic models Stochastic models incorporate chance variation e.g. in the number of individuals infected during each time step.

They are therefore analogous to the ReedFrost mechanistic model which you saw in one of the earlier slides

In the same way that the outcome from one simulation run of that model was not interpretable, stochastic models have to be used to run many simulations in order for the findings to be useful. This is illustrated in the spreadsheet stoch1.xls which shows an example of a stochastic model.

This model describes the transmission dynamics of an infection following the introduction of 1 infectious case into a population of 10 susceptible individuals.

The status of each individual at the end of each time step (taken to be one serial interval) is described in the orange cells and is determined by the size of the random number drawn in the adjacent pink cell.

If this random number is less than the risk of infection, then the person becomes a case; otherwise that person remains susceptible. 17.2: Beyond difference equations: stochastic models You will notice that pressing the F9 (or any other) key leads to a new set of random numbers and hence to new predictions of the numbers of cases in the population.

If a stochastic model is run many (e.g. 1000) times, then the average result from all of these runs should be very similar to the outcome from a deterministic model. Section 18: Final conclusions: how complex do models have to be? As you have seen in the Excel exercises, the simple models of the transmission of an immunizing infection such as measles were able to recreate the cycles in incidence which are seen in reality.

These models also help to provide insight into how the basic reproduction number influences these cycles and they can be extended to explore the effect of vaccinating infants on the overall incidence. However, this model as it stands, has limited use, if, for example, we would like to know the numbers of cases in various age groups, or the effect of agedependent mixing on the impact of an intervention.

Additional complexities can be built in to examine these issues (e.g. stratifying the population by age and incorporating an agedependent parameter.

The methods for doing this are, unfortunately, beyond the scope of this session.

This then leads on to the question of how complex does a model need to be? 18.1: Final conclusions: how complex do models have to be? The development of fast computers has meant that it is now possible to have models which are as realistic as possible.

An example of such a model is the ONCHOSIMmodel, which was developed during the 1990s and described the transmission dynamics of onchocerciasis explicitly for every individual and every parasite in a given village population. Interaction: Button: References Output: Palasier AP, von Oortmarssen G, Habbema JDF, Remme J, Alley ES. ONCHOSIM: a model and computer simulation program for the transmission and control of onchocerciasis. Computer Methods and Programs in Biomedicine, 1990; 31;4356. A more recent example is the model (published in 2005) describing the transmission dynamics of pandemic influenza in Thailand at the individual level, also accounting for movement patterns of individuals to work and elsewhere.

Interaction: Button: References

Output: Ferguson NM, Cummings DA, Cauchemez S, Fraser C, Riley S, Meeyai A, Iamsirlthaworn S, Burke DS. Strategies for containing and emerging influenza pandemic in Southeast Asia. Nature. 2005 Sep 8;437(7056):20914.

End interaction.

The main drawback of complex models is that, because they incorporate many different parameters and assumptions, the individual effect of a particular factor on the model prediction becomes difficult to understand. 18.2: Final conclusions: how complex do models have to be? An alternative approach is that reflected in a quote by Einstein, namely that:

a model should be as simple as possible and no simpler i.e. the complexity of a model should depend on the question being answered Ultimately, models are often useful for informing policy and providing insight into the epidemiology of an infection.

However, they should only be used as a guide: the input parameters and assumptions should be questioned before the output is accepted Section 19: Summary This is the end of EC06. When you are happy with the material covered here please move on to session EC07. The main points of this session will appear below as you click on the relevant title. Applications of models

There are two main areas of application of modelling:

a) Predicting the level of treatment or vaccination coverage for control and the future numbers of cases

b) Providing insight into the natural history of an infection 19.1: Summary Models are classified into those which are deterministic and those which are stochastic.

Deterministic models describe what happens on average in a population. Stochastic models incorporate chance variation in e.g. the number of individuals infected and might be used to address questions such as:

what is the probability of an outbreak following the introduction of one case into a given population? 19.2: Summary Deterministic models can be set up using either difference equations of differential equations.

Difference equations describe the transmission dynamics of an infection using discrete time steps of e.g. 1, 2 days etc.

Differential equations describe the transmission dynamics using continuous time. Interaction: tabs: Tab 1: Difference Equations An example of a model for an immunizing infection set up using difference equations:

Tab 2: Differential Equations An example of a model for an immunizing infection set up using differential equations:

19.3: Summary The most important parameter in infectious disease models is , defined as the probability of an effective contact between two specific individuals per unit time.

For an immunizing infection in a randomly mixing population, it can be calculated using the expression:

R0/(ND)

where N is the population size and D is the duration of infectiousness. For simple models, the transition rates, (e.g. rates of infectious disease onset, the recovery rate) can be calculated using the expressions: 1/(average latent period)

or 1/(average duration of infectiousness)

assuming that these rates are constant over time. 19.4: Summary The level of complexity incorporated in a model depends on the question being addressed.

For example, a simple model of the transmission dynamics of measles following the introduction of 1 infectious case into a totally susceptible population can provide useful insight into the factors underlying the cycles in incidence e.g. the basic reproduction number.

On the other hand, such a model would need to be extended to make predictions about the numbers of cases in specific age groups, or the effect of nonrandom mixing on the impact of control strategies. Section 20: Optional reading: writing down differential equations The first distinction between difference and differential equations lies in the notation.

For example, as shown in the diagram opposite, the number of susceptible individuals at time t is denoted by the symbol S(t) rather than St, reflecting the fact that were considering events occurring in continuous time, rather than taking discrete time steps. Difference equation model:

Differential equation model:

20.1: Optional reading: writing down differential equations When describing the transmission using continuous time, it is usually not possible to express the number of susceptibles at a given time in terms of that at a previous time.

Instead, we use the notation shown opposite we shall now describe the interpretation of these equations.

20.2: Optional reading: writing down differential equations Technically, differential equations describe the rate of change of a given quantity.

The rate of change in the number of individuals in a given category is just given by the difference between the number of individuals entering the category and the number of individuals moving out per unit time or

+ the number entering the category per unit time the number leaving the category per unit time

Returning to the above model and considering the susceptible category, for example, as noone enters the susceptible class, the rate of change in the number of susceptible individuals is given by the expression: the number of susceptible individuals who become newly infected per unit time Using mathematical notation, this would be written as: dS(t)

= -S(t)I(t)

dt 20.3: Optional reading: writing down differential equations

Considering the infected category in the same model, newly infected individuals enter this category and diseased individuals exit this category.

The rate of change in the number of infected individuals is given by the expression:

+ the number of susceptible individuals who become newly infected per unit time

the number of infected individuals who develop infectious disease per unit time.

Using mathematical notation, the rate of change in the number of infected individuals would be written as:

dE(t)

= -S(t)I(t)-fE(t) dt

20.4: Optional reading: writing down differential equations

Exercise:

Considering the above model, write down the differential equations for the rate of change in the number of Interaction: Button: (a)infectious individuals Output:

A total of fE(t) Infection individuals enter this category per unit time and rI(t) exit this category per unit time. The rate of change in the number of infectious individuals is therefore given by:

)()()( trltfEdt

tdl=

Interaction: Button: (b) immune individuals Output: A total of rI(t) Infectious individuals enter this category per unit time and no individuals exit this category per unit time. The rate of change in the number of immune individuals is therefore given by:

)()( trldt

tdR=

Exercise: Write down the differential equations corresponding to the following model diagram, assuming that no individuals are born into or die from the population.

Interaction: hotspot: Answer

3.1: What are models?3.2: What are models?3.3: What are models?5.1: Applications of modelling predicting the critical levels of treatment or vaccination coverage5.2: Applications of modelling predicting the critical levels of treatment or vaccination coverage5.3: Applications of modelling predicting the critical levels of treatment or vaccination coverage5.4: Applications of modelling predicting the critical levels of treatment or vaccination coverage5.5: Applications of modelling predicting the critical levels of treatment or vaccination coverage5.6: Applications of modelling predicting the critical levels of treatment or vaccination coverage5.7: Applications of modelling predicting the critical levels of treatment or vaccination coverage6.1: Applications of modelling predicting the impact of control strategies6.2: Applications of modelling predicting the impact of control strategies6.3: Applications of modelling predicting the impact of control strategies6.4: Applications of modelling predicting the impact of control strategies6.5: Applications of modelling predicting the impact of control strategies6.6: Applications of modelling predicting the impact of control strategies9.1: Setting up deterministic models9.2: Setting up deterministic models9.3: Setting up deterministic models9.4: Setting up deterministic models9.5: Setting up deterministic models9.6: Setting up deterministic models9.7: Setting up deterministic models9.8: Setting up deterministic models9.9: Setting up deterministic models9.10: Setting up deterministic models9.11: Setting up deterministic models9.12: Setting up deterministic models9.