Abstract

Computer-simulated synthetic populations are used by researchers and policy makers to help understand and predict the aggregate behaviour of large numbers of individuals. Research aims include explaining the structural and dynamic characteristics of populations, and the implications of these characteristics for dynamic processes such as the spread of disease, opinions and social norms. Policy makers planning for the future economic, healthcare or infrastructure needs of a population want to be able to evaluate the possible effects of their policies. In both cases, it is desirable that the structure and dynamic behaviour of synthetic populations be statistically congruent to that of real populations. Here, we present a parsimonious individual-based model for generating synthetic population dynamics that focuses on the effects that demographic change have on the structure and composition of households.

Keywords:

Demography, Synthetic Populations, Household Dynamics, Individual-Based Models

Introduction

1.1

The structure and dynamics of a population are the outcome of many events occurring to its individual members. Understanding and predicting trends in population structure and dynamics is challenging, but has the potential to be of great utility to researchers, planners and policy makers. Synthetic populations can be used as a testbed for a variety of purposes, from simulating disease outbreaks and interventions (Ajelli & Merler 2009), to evaluating the impact of land-use and transport policies (Iacono et al. 2008; Spielauer 2010). To conduct corresponding experiments in real populations may be costly, unethical or otherwise infeasible.

1.2

The key requirement for a useful synthetic population model is that it generates populations whose structure and dynamics match those of real populations (Gargiulo et al. 2010). It might seem that the method used to synthesise a model population should be relatively independent of the purpose to which that population is subsequently put. However, no model is a perfect simulacrum of reality and a model's intended purpose will influence decisions made in its design and construction. Pragmatics dictate that, for any question, an appropriate model must focus on accurately representing those aspects of a population most relevant to that question, while perhaps tolerating some deviation in those aspects judged less relevant (Taper et al. 2008). As such, an essential step in model building is the specification of which dimensions of a population are of greatest importance, and validation of model behaviour against these dimensions.

1.3

In this paper, we propose a parsimonious individual-based model of household composition and dynamics, developed for the purpose of exploring interaction between demographic processes and patterns of infection and immunity. To begin, we describe the specific questions that have motivated the development of our model, and the requirements that these questions impose. We review how these questions have been explored in a variety of different modelling paradigms. We then describe our own model, together with three case studies that demonstrate its ability to generate populations under a range of demographic scenarios relevant to infectious disease. We conclude by discussing the strengths and limitations of our model and its potential directions for its future development.

Background

Motivation

2.1

This section outlines the questions motivating the development of our model, and the requirements that these questions impose on the design of our model. The primary goal of infectious disease modelling is to understand how diseases spread and how they can be controlled, an endeavour in which modelling has long played an important role (Anderson & May 1991). A key component of infectious disease models is the representation of the population through which a disease spreads. The age structure of a population can influence patterns of susceptibility to disease (Anderson & May 1991). The social structure of a population gives rise to contact networks that affect how infection is transmitted through a population (Danon et al. 2011). Understanding the demographic processes that underlie observed populations, and how they might shape future populations, can help us explain current patterns of disease and predict how these patterns will evolve over time, insights that will aid in the design more effective strategies for disease control (John 1990).

2.2

The assumptions and approximations made when designing a model constrain the type of questions it can be used to address. One common assumption made by infectious disease models is that the composition and structure of a population is static over time. For disease outbreaks that occur over a period of weeks or months, this assumption may be appropriate, as population structure typically changes much more slowly than disease state. However, when modelling endemic diseases that may persist in a population for years or decades, this assumption becomes less appropriate, as demographic processes such as birth, death and the formation and dissolution of households will have a significant effect on population structure and composition. Households are an organisational unit of particular importance as they are a key locus of disease transmission, particularly for young infants (Jardine et al. 2010). Households are also a target for disease control strategies such as cocoon vaccination, which aims to provide a protective environment for newborns by vaccinating their parents (Coudeville et al. 2008).

2.3

In addition to the dynamics arising from individual life events, the underlying structure of global populations has undergone dramatic changes over recent centuries. The demographic transition model has been proposed to describe the long term changes to population structure associated with industrialisation, improvements in public health, education, and agriculture, and changing social values (Kirk 1996; Murphy 2011). As an example, during the twentieth century, increasing life expectancy and lower fertility rates in Australia have increased the proportion of people aged over 65 from 4% to 14% of the population, while the proportion of people aged under 15 decreased from 35% to 19%; the size of the average household has decreased from 4.5 to 2.6 people over the same period (Hayes et al. 2010; Australian Bureau of Statistics 2006b). The effects of this type of population change on patterns of disease are not yet well understood.

2.4

The questions that we would like to use models to address concern the effect of demographic changes on observed patterns of infection and immunity. What effect do shrinking household sizes have for the patterns of interaction relevant to the spread of childhood diseases? How can we assess the long-term effectiveness of vaccination strategies in a changing population? What are the implications of the rapid demographic transitions currently occurring in developing countries? A population model capable of addressing such questions must therefore capture:

• realistic patterns of household composition, in particular the household context of infants;

• the dynamic characteristics of households arising from patterns of birth, death, and household formation and dissolution;

• the differences in household dynamics across different demographic scenarios, corresponding to developed and developing countries; and

• the changes that occur to household dynamics over extended periods of time under changing demographic conditions.

2.5

This paper describes the design and validation of a synthetic population model aimed at satisfying these requirements. In the remainder of this section we briefly review how population structure, households and demography have been incorporated into existing infectious disease models.

Previous approaches

2.6

Methods for generating synthetic populations based on empirical data have been developed in parallel in the fields of demography, geography and social science. Despite some recent convergence, there is no single general purpose approach (Mannion et al. 2012; Birkin & Clarke 2011). There are several recent and comprehensive overviews of various methods for population projection and modelling (e.g., Stillwell & Clarke 2011; Wilson 2011; Spielauer 2010). Rather than attempting to replicate these efforts in this section, we focus specifically on approaches taken to representing populations in infectious disease models.

Mathematical models

2.7

Mathematical models of infectious disease represent populations in terms of the prevalence of infection and immunity at a given point in time. A population is divided into 'compartments' corresponding to particular disease states (e.g., susceptible, infectious, recovered). The movement of individuals between these compartments is modelled by specifying transition rates between them. The entire system can then be represented as a set of ordinary differential equations and solved (analytically or numerically) to predict future patterns of disease (Hethcote 2000; Grassly & Fraser 2008).

2.8

Demography can be included in mathematical models by further subdividing compartments according to age or sex, and by introducing additional terms for the birth and death of individuals (Anderson & May 1991). Mathematical models that incorporate household structure have also been proposed, adopting theoretical or empirical distributions for household size (e.g., Ball et al. 1997; Ball & Neal 2002; Becker et al. 2005); however, these models typically assume static populations. Glass et al. (2011) do propose a dynamic household model, however the distribution of household sizes is held fixed and the model does not include age structure. These trade-offs illustrate a fundamental limitation of mathematical approaches to capturing complex population structure. The explosion of terms that results from simultaneously incorporating age, household properties, disease state, and other factors of interest results in models that are analytically intractable.

Individual-based models

2.9

To overcome the limitations of mathematical models, individual-based models (IBMs)¹ have also been applied to the problem of understanding the dynamics of infectious disease. By explicitly modeling each individual, together with their age, location, disease status and other relevant properties, IBMs offer a great deal of flexibility in representing heterogeneous population structures (Eubank et al. 2004; Elveback et al. 1976; Ferguson et al. 2005; Longini et al. 2005). Most of these models have been aimed at capturing the dynamics of a single outbreak. Hence, their primary focus has been on the dynamics of individual activity over the course of a typical day, the resulting contact networks, and the patterns of disease transmission that these networks support. The composition and structure of the populations themselves are usually static. Compared to the proliferation of static population models, fewer IBMs have considered the relationship between the long term dynamics of population structure and the spread of infectious disease.

2.10

Those models that have considered longer time frames have adopted heuristic approaches to capturing household structure. Ajelli & Merler (2009) and Guzzetta et al. (2011) investigate long term patterns of hepatitis A and tuberculosis respectively, using simulated populations that include birth, death and household formation processes. They generate initial household distributions from Italian census and survey data using a Monte Carlo sampling method. Population demographics are updated annually, using empirical mortality rates to determine individual deaths, and allocating births according to household size and parental age. Silhol & Boëlle (2011) model the occurrence of varicella using a population model that includes realistic age and household structure and dynamics. Their approach to producing an appropriate distribution of household sizes is to generate each household as a completed unit including all children that will eventually be born there. This final household is then 'rewound', resulting in the ages of younger children becoming negative, and births occur over the course of a simulation when a child's age becomes positive. An advantage of this approach is that household structure is guaranteed to accurately reflect that of the real population; however, it does not generalise in straightforward fashion beyond available data.

2.11

While we have drawn upon aspects of these previous models in the design of the model described in Section 3, we note that they do not fulfil our requirement of capturing the changes in household dynamics arising from demographic shifts.

Microsimulation models

2.12

Microsimulation is an approach to experimenting with virtual societies by modelling the actions of individuals in a population and (potentially) the interactions between them that has its origins in economics (Orcutt 1957). These models are used to explore the potential impacts of policy decisions, where the choices made by individuals may depend on policies in a nonlinear and context-dependent fashion. Dynamic microsimulation models add in a temporal element, incorporating demographic processes and individual life courses (Mannion et al. 2012; Birkin & Clarke 2011; Spielauer 2010). Microsimulation models and IBMs share many similarities. Their differences arise primarily from having been developed and applied to different problems by different research communities. Broadly speaking, microsimulation models have been more concerned with empirical data and predictive accuracy, while IBMs tend to be more theoretically oriented. While many microsimulation models are designed for the purpose of forecasting future population trends and policy interactions, they have also been applied to the challenge of elucidating historical demographic patterns (e.g., Hammel 2005; Murphy 2011).

2.13

Given the requirement of disease models for demographically plausible populations, it seems at first surprising that, so far as we are aware, only one infectious disease model (EpiSimS, Eubank et al. 2004) has explicitly made use of a pre-existing microsimulation model (TRANSIMS, Barrett et al. 2000), albeit one developed by the same research group. Possible reasons for this include: availability of software—many microsimulation models are not released publicly; tight coupling between model design and data requirements, meaning that if the data required to initialise a population is not publicly available, or not available in the correct form for the population of interest, the model may not be suitable; limited extensibility, making it difficult to extend an existing population model to include disease dynamics; and model development effort—dynamic microsimulation models are notoriously time-consuming and expensive to build (Harding 2007).

Summary

2.14

A significant impediment to modelling the demographic dynamics of populations is the limited availability of data with which to parameterise models. Hypothetically, if sufficient data were available to estimate a probability for each possible transition that an individual or household might undergo, using a demographic microsimulation model to project a population forward would be a trivial exercise. In practice, the number of possible transitions explodes combinatorially with attributes of interest, and available data will always be insufficient for this approach to be practicable.

2.15

At the other extreme, an ideal individual-based model might simulate the underlying cognitive and social behaviour of individuals in a population, perhaps using some internal representation of individual utility together with a forecasting model that predicts an individual's behaviour on the basis of its history and current context. This approach has the potential to free demographic models from the reliance on the massive quantities of data required by transition-based microsimulation models (Silverman et al. 2011). However, much rests upon the model used to represent individual decision making. Successful individual-based models of demographic processes have thus far typically been constrained to specific aspects of populations. For example, Billari et al. (2007) use an IBM of marriage based on social interaction to explore the emergence of observed trends in marriage age. Extending such an approach to all facets of human behaviour remains an open challenge.

2.16

Recently, there has been a convergence of techniques from microsimulation and individual-based modelling, as (some) IBMs encounter a need for more empirically plausible populations, and microsimulation modules begin to include more behavioural aspects (Wu & Birkin 2012). The resulting hybrid models use available demographic data to calibrate distributions of individual attributes, and stochastic behavioural models of individual decisions to generate particular events.

Our model

3.1

The basic unit of description in our model² is the individual. Individuals are characterised by their age, sex and the household to which they currently belong. A population consists of the set of currently alive individuals, structured by a network of interpersonal ties that maps couple, parent–child and household co-membership relations. The following sections describe how a population is initialised, how it is updated over time, and how it can be parameterised using available data on real populations.

Initialisation

3.2

An initial 'bootstrap' population is created by randomly generating individuals with ages drawn from a specified age distribution. These individuals are assigned to households at random according to a specified household size distribution. Households of size one or two are assigned one or two adults respectively, while households of size three or greater are assigned two adults and one or more children. This initial population structure will diverge in several ways from a real population; for example, constraints on birth interval and inter-generational age difference will not be respected. If more detailed data on household structure is available, more complex methods (Gargiulo et al. 2010; Ajelli & Merler 2009) could be used to specify a more realistic initial population structure. Alternatively, the approach we adopt here is to update the state of the population until such time as all of these bootstrap individuals and households have been replaced (i.e., for at least 100 years), at which stage internal constraints on population structure will be respected.

Updating

3.3

The state of a population is updated in discrete time steps, where each step corresponds to a specified number of days. All parameters controlling the occurrence of demographic events are specified as annual probabilities and rates, therefore these are scaled appropriately.

3.4

Five types of demographic events can occur to an individual:

• Death: Age and sex specific mortality rates are used to determine an individual's probability of death during each time unit (e.g., Figure 1(a)). If a death results in a household containing only children then the household is dissolved. Any adult children (i.e., aged 18 years or over) leave home and create new single-person households, while any children under 18 years are randomly relocated (fostered) to other households containing at least one child.

• Birth: Age (and optionally, parity³) specific fertility rates are used to determine the probability that a birth occurs to a given individual. As fertility is not an independent process, rates are transformed into relative probabilities used to designate the subset of the population from which the actual mother is then chosen (e.g., Figure 1(b)). Upon giving birth, a mother is excluded from being a candidate for future births for a number of days drawn from a truncated normal distribution, with a minimum duration of 270 days.

• Couple formation: An individual within a given age range who is currently single has a fixed probability per time unit of forming a new couple. The new partner is chosen from the pool of individuals who are currently single, of the opposite sex, and whose age differs by a normally distributed value. The newly coupled individuals move into a new household, together with any dependents (e.g., children from previous couples).

• Couple dissolution: Any currently coupled individual within a given age range has a fixed probability per time unit of dissolving the couple. Upon dissolution, one member of the couple moves into a new single person household, while the other remains in the original household together with any children.

• Leaving home: Individuals leave home automatically when they form a couple, otherwise any individual above a specified age leaves home with a fixed probability per time unit and forms a new single person household.

3.5

The model can be used to simulate populations of fixed size (i.e., with replacement level fertility), or populations that are increasing or decreasing in size. Change in population size can result from an imbalance between the number of births and deaths, or between the number of immigrants and emigrants. In our model, growth due to excess births is implemented by triggering additional birth events at each time step. Growth due to immigration is implemented by introducing new individuals and households into the population according to a specified distribution of age and household structure. Specifying the age and family composition of migrants is a complex issue, depending as it does on the circumstances of migration, country of origin, and other factors. The default assumption made by the model is that the migrant population is demographically similar to the target population. Calibrating migration to reflect the demographic characteristics of specific events would be relatively straightforward provided that suitable data were available.

Parameterisation

3.6

The input parameters that must be specified to initialise and update a population are summarised in Table 1. A general principle followed in designing the model was to parameterise the model in terms of events occurring to individuals, and to allow the sizes and types of households in a population to vary as a consequence of these individual-level events. For example, individual births were allocated to parents using age- and parity-specific fertility rates, but not to households of a specific size or composition; thus, in contrast to existing models (Silhol & Boëlle 2011; Ajelli & Merler 2009), household size distributions were not controlled, but rather result from the interaction between patterns of household formation and dissolution arising from individual-level events. Further calibration against household-level data would of course be possible, but reserving available household-level data for validation purposes enables us to better evaluate the effectiveness of our simple event-based model.

Table 1: Model parameters
Parameter	Description
General:
Initial population size	Number of individuals in population at start of simulation.
Population growth rate	Annual rate of change in population size due to natural increase.
Immigration rate	Annual rate of change in population size due to immigration.
Death:
Age- and sex-specific mortality probabilities	Annual probabilities of death by sex and year of age.
Birth:
Age-specific relative fertility probabilities	Relative probabilities, given the birth of a child, that the mother is of a specified age.
Parity-specific relative fertility probabilities	(optional) Relative probability, given the birth of a child, that it is to a woman with a specified number of previous children.
Birth gap (mean and SD)	Parameters governing the minimum inter-birth interval.
Couple formation:
Couple formation parameters	Age range that a currently single individual is eligible to form a couple, and annual probability that this will occur.
Partner age difference (mean and SD)	Parameters governing the sex-dependent age difference between partners during couple formation.
Couple dissolution:
Couple dissolution parameters	Age range that a currently coupled individual is eligible dissolve that couple, and annual probability that this will occur.
Leaving home:
Leaving home parameters	Minimum age at which an individual currently living with a parent/guardian will form a new single-person household, and annual probability that this will occur.

3.7

Mortality data is often available in the form of life tables, a demographic tool that shows, for each age, the probability of an individual of that age dying in the next year. Fertility rates by age are also available for most countries. These fertility rates can be used in two different ways. One option is to use them to determine the probability that a woman of a particular age will give birth in a given year. The number of births that occur in a year will then emerge from the aggregate application of these probabilities across the female population. The alternative (which we adopted here) is to specify the number of births that occur in a year, and then use age-specific fertility rates to determine the ages of the women to which those births occur. Effectively, rather than asking “what is the probability of a woman of age x giving birth this year?”, we ask “given that a birth occurred, what is the probability that the mother was aged x?” An advantage of the latter approach was that it enabled us to control the total size of a population. For example, population size could be held constant by triggering sufficient births each year to replace the individuals dying in that year, or constrained to grow (or shrink) at a given rate.

3.8

Couple formation and dissolution, and the departure of children from their parents' household are all governed by simple models that assume a constant probability of a particular event occurring per unit of time. Thus, these probabilities are independent of age, duration of relationship, previous marital status, and other potentially relevant factors. This approach is considerably simpler than that taken by dynamic microsimulation models. These more complex models rely heavily on the availability of appropriate statistical data in order to parameterise the effects that diverse factors have on, for example, marriage and divorce. Our primary motivation for adopting this simpler approach was to be able to deal with the availability of data across a variety of different countries and historic time periods.

3.9

The specific parameter values, data sources, and estimation processes used to specify the probability of individual events are detailed at the beginning of each case study in the following section.

Results

4.1

As described above, the primary motivation for the development of our model was to use it as a demographically plausible testbed in which to explore interactions between population dynamics and patterns of disease. In turn, this aim imposed requirements on model behaviour, as described in Section 2.1. In this section, we use three demographic scenarios to evaluate the extent to which our model meets these requirements. The first two scenarios consider populations corresponding to those of a developed and a developing country. The third scenario considers a population undergoing a demographic transition from high to low fertility.

4.2

Quantitative comparison between model output and empirical data is an important component of validation. However, suitable data for comparison is not always available, and we also rely on comparison of more qualitative aspects of model behaviour to build confidence in the validity of a model (Korb et al. 2013; Grimm et al. 2005).

Case study 1: A developed country population

4.3

Our first case study explored the ability of our model to capture patterns of household composition and dynamics comparable to those exhibited by Australian population at the beginning of the 21st century. Real populations are ever-changing: fertility rates that deviate from replacement and immigration act to increase or decrease the size of a population, while changes in longevity reshape its age structure. For this case study, we explore a simpler scenario in which population size, mortality and fertility rates and other event probabilities are fixed, and there is no migration. The resulting steady state population has a constant size and, over time, approaches a stable age structure. While such a scenario is obviously of only limited use for forecasting purposes, it does have the significant benefit of providing a stable background for theoretical investigation of interactions between the dynamics of demographic and epidemic processes (e.g., as described in Glass et al. 2011). In particular, it allows us to evaluate the model from the perspective of the first two requirements described in Section 2.1. A more realistic scenario with time-varying demographic parameters is explored in Case Study 3.

4.4

The initial population was parameterised using recent Australian data on age structure and households (de Vaus 2004; Australian Bureau of Statistics 2006b). Values for other parameter were estimated on the basis of census data reported in de Vaus (2004). Couple formation parameters were calibrated against data on the percentage of individuals by age who had never married nor cohabited. Couple dissolution parameters were calibrated against data on percentage of marriages surviving by duration. Leaving home parameters were calibrated against data on the number of individuals by age living at home, accounting for those who had left to marry or cohabit. The population was simulated for two hundred years, to ensure that a steady state had been achieved, before composition and dynamics were analysed.

Table 1: Case Study 1: Model parameters
Parameter	Value / Data source
Initial population size	20,000
Population growth rate	0%
Immigration rate	0%
Mortality probabilities	Australia, 2006, by year of age (Australian Bureau of Statistics 2007) (see Figure 1(a))
Fertility probabilities	Australia, 2006, by year of age (Australian Bureau of Statistics 2006a) (see Figure 1(b))
Birth gap	mean: 365 days; SD: 90 days
Couple formation parameters	age range: 18–60 years; annual probability: 7.5%
Partner age difference	mean: 2 years; SD: 2 years)
Couple dissolution parameters	age range: 18–60; annual probability: 1.5%
Leaving home parameters	minimum age: 18; annual probability: 0.8%

Figure 1: Australian census data used to parameterise the model in Case Study 1. (a) Mortality rates by age and sex; (b) Relative probability of a birth occurring to a woman by age. Data sources as in Table 2.

Age distribution

4.5

Figure 2(a) shows the simulated age structure of the final population for ten independent runs, together with the starting age distribution (based on 2006 census data). In the absence of migration, the age structure of a population is determined solely by births and deaths. When age-specific rates of fertility and mortality remain constant over a long period of time, and there is no migration, the resulting population is called 'stable'. Stable populations are characterised by constant age structure and fixed growth rates. A stable population with a growth rate of zero is called 'stationary'. Stable and stationary population structures are largely theoretical constructs that are rarely observed in real populations. Over the last century, life expectancy in Australia has increased steadily, while fertility and migration have fluctuated. One interpretation of the final age structures shown in Figure 2(a) is that they represent what Australia's population could look like after a century with replacement-level fertility, no migration, and constant age-specific birth and death rates.

Figure 2: (a) The final age distributions of ten simulated populations (grey), compared to the empirical age distribution used to initialise the population (black). (b) The final household size distribution (white), averaged over the ten simulated populations (error bars indicate standard deviation), compared to empirical data (black).

Household size distribution

4.6

Household sizes and compositions were allowed to vary in response to the individual level processes of leaving home, forming and dissolving couples, birth and death. The final household size distribution after 200 years (mean and standard deviation across 10 independent runs) is shown in Figure 2(b). Some of the variation between simulated and empirical age structure can also be observed in the final household size distribution. Older individuals, who are over-represented in the simulated population, are more likely to live in households of size one and two, which are also over-represented in the simulated population.

4.7

As individual households are formed and dissolved, the number of households of a particular size increases or decreases; however, because we are holding the demographic rates fixed, the shape of the household size distribution remains relatively constant, with some stochastic fluctuation (Figure 3(a)).

Figure 3: The evolution of (a) household size distribution and (b) family household type distribution over 100 years under fixed demographic conditions, for a single simulation run.

4.8

To test how the final household size distribution depended upon the the household size distribution that was used to create the initial bootstrap population (as described in Section 3.3) we also ran simulations in which the population was initialised with uniformly distributed household sizes. By year 200 of these simulations, the distribution of household sizes once again approximated the empirical distribution. Thus, we are confident that the household size distribution is an emergent property of the underlying individual-level demographic processes, rather than a simple consequence of the initial conditions.

Household type distribution

4.9

Beyond looking simply at household size, we also investigated the types of household that individuals tended to belong to at different stages of their lives. Figure 4 shows the proportion of individuals that living in various types of household situation (couple with children, couple without children, single parent, lone person, and living with parents), broken down by age category (mean and standard deviation across 10 independent runs). The simulation model produces a plausible representation of household type prevalence observed in the Australian population. While there is some variation, general trends across age groups are well captured; for example, the proportion of individuals living in households without children initially increases as individuals form couples, then decreases as these couples have children, before finally increasing as these children leave the family household.

Figure 4: Type of household by age group at the end of a simulation run (white; error bars indicate standard deviation), compared with empirical data (de Vaus 2004) (black).

4.10

Figure 3(b) indicates that, as with household size distributions, the proportion of family households fluctuates stochastically over the time period reported, but is generally stable.

Changes in household structure

4.11

The distribution of household sizes (Section 4.1.2) and the prevalence of household types (Section 4.1.3) both remain stable over the course of a particular simulation run. However, this stability hides the fact that the type of household that any one individual belongs to changes multiple times over the course of their life (see Section 4.1.5). An individual's household type can change when they leave home, when they form or dissolve a couple, when their first child is born or their last child leaves home, or when another member of their household dies. While longitudinal data is not typically captured in a census, the Household, Income and Labour Dynamics in Australia (HILDA) Survey (Wilkins et al. 2011) reports statistics on the proportion of individuals who change household type over a five year period (Table 3). Collating output from multiple simulation runs reveals a similar pattern of transitions between household types (Table 4).

Table 3: Changes in household structure, 2003 to 2008 (HILDA) (%)
	Couple only	Couple with children	Single with children	Single person
Couple only	75.3	14.0	0.8	8.4
Couple with children	10.1	76.8	6.0	4.9
Single with children	6.4	18.0	59.5	13.0
Single person	11.1	8.8	3.5	74.9

Table 4: Changes in household structure, mean and standard deviation over 10 runs (%)
	Couple only	Couple with children	Single with children	Single person
Couple only	78.2 (0.6)	11.3 (0.4)	0.6 (0.1)	9.8 (0.3)
Couple with children	14.8 (1.3)	77.4 (1.8)	5.2 (0.8)	2.6 (0.2)
Single with children	9.5 (0.6)	41.6 (3.0)	42.6 (2.4)	6.3 (0.9)
Single person	8.8 (0.7)	11.6 (0.9)	1.2 (0.3)	78.3 (1.0)

4.12

Given that model behaviour was not explicitly calibrated against these transition rates, there is a remarkable level of agreement between the dynamics of real and simulated households. The primary point of disagreement is the transition from 'single with children' to 'couple with children', which is over-represented in simulated populations, suggesting a possible direction for future refinement.

Family life cycle

4.13

The family life cycle is a demographic pattern that captures the life experience of a large proportion of the developed world's past, current, and most probably future population:“most persons will grow up, establish families, rear and launch their children, experience an empty nest period and eventually reach the end of their life.” (Glick 1989, p. 123). Figure 5 illustrates two different views of the family life cycle from the perspective of an individual, showing the age distribution at which various significant events occur. Exploring the sequences of events that constitute a simulated individual's family life cycle at both an individual and aggregate level provides a straightforward way to verify that these life courses appear plausible.

Figure 5: The age distribution of (a) major life events (marriage, birth of first child and death) and (b) birth of first and subsequent children over the total sample population.

4.14

The approach taken in Figure 5 to exploring the distribution of significant events over an individual's life span can also be applied to households. An issue that arises is how best to define a household's 'age'. The approach taken here is to define the creation of a household as occurring when either (a) an individual leaves her parents' household to form a new single person household; (b) two individuals leave their parents' households to form a new couple household; or (c) an individual leaves his spouse's household after divorce and forms a new single person household. Other inter-household movements do not result in the creation of a new household; for example, the formation of a couple where at least one individual currently resides in a single person household. In these cases, one household merges with another, but no new household (i.e., of age zero) is created. A household is dissolved when the last individual in that household dies or leaves; for example, if two individuals, each living in a single person household, form a couple, then one of these households will continue, and the other will be dissolved.

4.15

Assigning households an age as described above then allows us to look at the distribution of particular individual or household level events by household age. Figure 6 represents the family life-cycle in terms of household size: households typically begin their life with one or two individuals (e.g., a new couple who have just left their own parents' households). Household size increases steadily, peaking between twenty and twenty-five years (when our example couple are in their forties and have had all the children that they will have). Thereafter, household size declines back to a mean of two by around forty-five years (with our example couple being 'empty nesters' in their sixties). The period of peak size represents the stage when most or all of the children that will be born into that household have been born, but none or few of them have yet left home.

Figure 6: Household size by age. Bold line indicates mean household size. Boxplots show median size (red bars), inter-quartile range (box), minimum/maximum within 1.5 of the inter-quartile range (whiskers) and outliers. The sample population of households used was the set of households that were both created and dissolved during the timeframe of the simulation run; that is, households created at the beginning and households still existing at the end of the run were excluded.

Household composition

4.16

We simplify the issue of describing household composition by dividing individuals into discrete age categories relevant for disease transmission: infants under five years; school-children between five and seventeen years; adults from eighteen to sixty-four years; and elderly individuals sixty-five years and over. More or different categories could be chosen depending upon the age groups that are of most interest in a particular application of the model. Figures 7(a) and 7(b) show the distribution of individual ages by household age and household size respectively in a single population at a given point in time. These figures match observed trends of the household locations of different types of individuals (de Vaus 2004). Elderly individuals tend to be found primarily in smaller and older households. Children are found primarily in households of intermediate age and intermediate-to-large size.

Figure 7: Age distribution of individuals given (a) household age and (b) household size. Individuals are grouped according to age categories: < 5 years; 5-17 years; 18-64 years; 65 years.

4.17

Figure 7(b) depicts how individuals of various ages are allocated across households of different sizes, but provides no information on the co-occurrence of individuals from different age groups in households. Figure 8 shows a novel approach to visualising the types of households that appear in a population and their relative frequency. As above, this figure represents a snapshot of a population at a given point in time, rather than an aggregation across time. Each cluster of n circles represents a unique type of household containing n individuals, with the count of households of that type appended below. Each circle is coloured according to the age category of the individual in that household type, and circle size reflects household type frequency, with larger circles indicating more common household types.⁴ Household types are further arranged by household size (increasing left to right) and frequency (more frequent household types appear at the top). The representation shown in Figure 8 provides a convenient overview of the diverse household structures that arise in simulated populations. At the same time, it is straightforward to gain an impression of which household types are most likely to contain infants and their frequency in the population.

Figure 8: A chart of household types appearing in a population at a given point in time. Each circle represents an individual (coloured by age category). Each cluster of circles represents a household type. The frequency with which a particular household type occurs in a population is written beneath, and represented visually by size (larger circles indicate more common household types).

Case study 2: A developing country population

4.18

The third requirement of our model was that it be able to produce plausible patterns of household dynamics across a variety of demographic scenarios. In particular, we are interested in exploring the dynamics of infection and immunity in a developing country setting. As discussed above, the demographic transition model describes a country's transition from a phase in which both fertility and mortality are high, through a phase where mortality falls, but fertility remains high, to a phase where both fertility and mortality have fallen. During the first and last phases, fertility and mortality are balanced, and population size is stable. By contrast, in the middle phase, fertility exceeds mortality, and population size increases. We chose Zambia as a country representative of this middle phase of demographic transition: both mortality and fertility rates are higher than Australia's, and population growth of approximately 2.5% per annum is almost entirely due to natural increase (rather than immigration). Compared to Australia, less data is available on the Zambian population, restricting the amount of validation possible. Here, we focus on age and household size distribution.

4.19

We simulated an initial population of 500 individuals, growing at an annual rate of 2.5% (giving a final population size of approximately 70,000 after 200 years). Age-specific mortality rates were obtained from (Lopez et al. 2000), and age-specific fertility rates were obtained from the UNdata website (data.un.org). Insufficient resources were available to estimate precise values for couple formation and dissolution parameters. We therefore estimated values based on comparison with values used in Case Study 1. Namely, we specified an earlier age for couple eligibility, a lower rate of divorce and a lower rate of leaving home as a single individual (Republic of Zambia Central Statistical Office 2000). As with the first case study, we model the hypothetical scenario in which demographic rates are constant over time.

Table 5: Case Study 2: Model parameters
Parameter	Value / Data source
Initial population size	500
Population growth rate	2.5%
Immigration rate	0%
Mortality probabilities	Zambia, 2000, 5-year age groups Lopez et al. (2000)
Fertility probabilities	Zambia, 2000, by year of age (data.un.org)
Birth gap	mean: 270 days, SD: 0 days (i.e., uniform)
Couple formation parameters	age range: 15–60 years; annual probability: 8%
Partner age difference	mean: 2 years; SD: 2 years
Couple dissolution parameters	age range: 18–60 years; annual probability: 0.1%
Leaving home parameters	minimum age: 18 years; 0.5%

4.20

As depicted in Figure 9(a), the age structure of the Zambian population is captured with a high degree of accuracy. Household size distribution (Figure 9(b)) is reproduced less accurately: small household sizes (1–4 people) are over-represented, while very large households (> 8) people are under-represented. One possible explanation is that our parameter values were poorly chosen. However, we also note that Zambia has a reasonably high level of multi-generational and multi-nucleus households (Republic of Zambia Central Statistical Office 2000). Neither of these are currently represented in our model, which may explain some of the discrepancy. Comparing Figures 10 and 11 with the corresponding figures from the first case study (Figures 7 and 8) demonstrates the considerable differences in patterns of household composition that exist between countries with different demographic properties.

Figure 9: (a) The final age distributions of ten simulated populations (grey), compared to the empirical age distribution used to initialise the population (black). (b) The final household size distribution (white), averaged over the ten simulated populations (error bars indicate standard deviation), compared to empirical data (black).

Figure 10: Age distribution of individuals given (a) household age and (b) household size. Individuals are grouped according to age categories: < 5 years; 5-17 years; 18-64 years; 65 years.

Figure 11: Household composition for Zambia.

Case Study 3: A population undergoing demographic change

4.21

The two previous case studies depict stable demographic scenarios: there is no change to fertility and mortality rates over the simulated period. The final requirement of our model was that it produce household dynamics of a population across a period of demographic change. As discussed in Section 2, patterns of infection and immunity emerge over long time-frames, during which time population structure is neither static nor changing in a uniform fashion. Rather, underlying demographic rates change as life expectancy increases due to improvements in health and medicine, and fertility patterns change in response to availability of birth control and shifting social norms. As a test case, we parameterised our model with 100 years of Australian census data, covering the period from 1910-2010. During this period, Australia's population increased from almost 4.5 million to over 22 million. While natural increase (i.e., resulting from birth rates being higher than death rates) accounts for around two-thirds of this growth, immigration has also played a significant role. At the conclusion of World War II, Australia initiated a large immigration programme in an effort to boost population numbers (Department of Immigration and Multicultural Affairs 2001). To approximate this demographic history, we used a starting population size of 500 individuals, growing at a rate of 2.5% per year during the initial 100 year initialisation period, and at a decreasing rate during the following 100 years. We used an immigration rate of 0% prior to 1950 and 1% per year after 1950. The resulting final populations contained approximately 30,000 people after 200 years of simulation. We estimated time-varying couple formation and dissolution rates on the basis of historical trends of increasing marriage age, and increasing divorce rate (de Vaus 2004).

Table 6: Case Study 6: Model parameters
Parameter	Value / Data source
Initial population size	500
Population growth rate	2.5% decreasing to 0.5% over 100 years
Immigration rate	0% until 1950, 1% thereafter
Mortality probabilities	Australia, 1910-2008, varying frequency, by year of age (Australian Bureau of Statistics 2008)
Fertility probabilities	Australia, 1910-2008, varying frequency, 5 year age groups (Australian Bureau of Statistics 2008)
Birth gap	mean: 365 days; SD: 90 days
Couple formation parameters	age range: (15 increasing to 18 over 100 years)–60; annual probability: 7.5%
Partner age difference	mean: 2 years; SD: 2 years
Couple dissolution parameters	age range: 18–60; annual probability: 0.1% increasing to 1.5% over 100 years
Leaving home parameters	minimum age: 18; annual probability: 0.8%

4.22

The key demographic trends that were reproduced in the simulated populations were the effects that time-varying demographic rates have on households. Figure 12(a) shows how average household size decreases at a rate comparable to that observed in the empirical data. Figure 12(b) shows how the (rescaled) number of households changes over the same time period, again compared to the empirical trend.

4.23

Figure 13 shows two further perspectives on how households evolve over the course of the simulation: the prevalence of households of size one and two increases relative to larger households (Figure 13(a)), and the prevalence of households containing couples without children increases (Figure 13(b)). Only limited data is available to validate these trends across the full time period; however, rates of change across the most recent decades are in agreement with census data. Empirical values for two time points (1981 and 2001) are overlaid on the plot showing the evolution of household size distribution over time (Figure 13(a)). Even after seventy years of simulated population evolution, the model reproduces comparable rates of increase and decrease in the occurrence of households of a given size. With respect to the distribution of household types (Figure 13(b)), in our ten simulated populations, the proportion of family households containing a couple with children decreased by an average of 23% (SD 1.5%) over the penultimate two decades, while the proportion containing only a couple increased by an average 28% (SD 3.8%). The corresponding changes in the Australian population between 1981 and 2001 were a 20% decrease and 28% increase respectively (de Vaus 2004).

Figure 12: (a) Average household size over 100 years. (b) Number of households over 100 years (Number of households in 1910 = 100). Both figures show the output from 10 simulation runs compared with historical Australian census data (bold).

Figure 13: (a) Evolution of household size distribution over 100 years. (b) Evolution of family household type distribution over 100 years. Family households, in our model, include any household containing at least two people (i.e., single person households are excluded). Each figure shows the output from a single representative simulation run.

Evaluation

5.1

Many different forces act upon populations to produce their characteristic age and household structure. Over the last century, improvements to healthcare, government policies around migration and fertility, changing social norms, as well as unpredictable events such as wars and natural disasters have all played a role. Capturing all of these complex and interacting forces in a model is challenging, and we have favoured a general and flexible approach to mechanism design. That said, we believe that, for our research agenda, the model described in Section 3 represents a good balance between simplicity and plausibility. Currently, the data sources required to parameterise the model are relatively modest, hence it can be used to model populations for which only limited data are available (e.g., the international and historical populations considered in the second and third case studies) and hypothetical scenarios. Undoubtedly a closer fit to empirical data could be achieved, but this would inevitably come at the cost of introducing further data dependence into the model.

5.2

In its current form, our model meets the four requirements set out in Section 2.1; namely, it can produce realistic patterns of household composition and household dynamics, while being flexible enough to capture a variety of demographic scenarios and transitions. In endeavouring to design a parsimonious model, we have made several simplifying approximations. We briefly discuss refinements to be considered in future model development: First the absence of extended family households, as noted above, is an omission that must be addressed in order to more accurately capture household composition in certain populations. The cultural and economic conditions under which extended family households arise vary by country and across time (Hammel & Laslett 1974). As with the model described by Murphy (2011), our model contains a great deal of information on kinship relationships between individuals that could be used to construct more complex household structures. From an epidemiological perspective, multi-generational may have implications for patterns of social contact and disease transmission (Mossong et al. 2008). Second, the model does not currently distinguish between marriage and cohabitation. We judged that, from the perspective of household structures relevant to infectious disease transmission, marriage and cohabitation are effectively indistinguishable; however, households in which partners are married as opposed to cohabiting do appear to have different characteristics (such as duration of couple relationship) that may influence household dynamics (de Vaus 2004). Finally, a more realistic model of immigration is certainly possible. Specifically, immigrant populations are likely to have different demographic characteristics to the populations they join (Haug et al. 2002), and their arrival is therefore likely to influence age and household structure in non-trivial ways. Iannelli & Manfredi (2007) have shown how changes to the age structure of a population can have implications for disease dynamics.

5.3

To conclude, mathematical and network models that work with stylised population structures will continue to provide important insights into the dynamics and control of infectious diseases. However, many open questions in infectious disease epidemiology concern the roles played by specific types of population heterogeneity. To answer these questions, more realistic population models, such as that described here, will prove invaluable. As discussed at the beginning of this paper, synthetic population models find application in many domains, and there is no reason why the utility of our model could not extend beyond the domain of infectious diseases. For example, a similar approach is being used to explore issues around the provision of social care in ageing populations (Silverman et al. 2012), another issue for which changing patterns of household demography have important implications. However, the requirements guiding the development of our model were based on our particular research agenda, and it is critical to ensure that, for other applications, model behaviour is suited to the problem at hand.

Appendix

1. Create initial population, according to initial age distribution and household size distribution.
2. Scale all annual probabilities and rates to give rates per time step.
3. At each time step, each individual has the possibility of experiencing a life event as follows (individual attributes that affect event probabilities are listed in parentheses):
   1. Test for death (age, sex). If death occurs, the following occurs:
      1. The dead individual is removed from the population. If this results in a household containing orphaned children, any children who are old enough (e.g., > 18 years) leave home, as per below, while any younger children are randomly allocated to another family household containing at least one child.
      2. The birth of a replacement individual is triggered, and a mother is chosen (age, sex, parity, time since last birth).
   2. Test for couple formation (age, sex). If couple formation is to occur, select a partner from the population with an appropriate age difference and update their households as follows:
      1. If both individuals live at home, they move into a newly created household.
      2. If either (or both) of the individuals have their own household, the other partner (together with any dependents) joins them in this household.
   3. Test for leaving home (age). If an individual leaves home, they form a single person household.
   4. Test for couple separation (age). If a couple separate, one individual remains in their former household, together with any dependents, while the other individual leaves to form a new single person household.
4. Calculate the number of additional births due to natural increase. Mothers are chosen for each of these new individuals as above.
5. Calculate the number of new arrivals due to immigration. Immigrants arrive as a household unit, with the size of the household and the age of its occupants drawn from the current population age and household size distributions.
6. Repeat from Step 3.

Notes

¹ The terms 'agent-based models' and 'individual-based models' are frequently used interchangeably to describe models in which each unit in a population is explicitly represented. Where a distinction is made, the term agents is used to refer to entities whose behaviour is determined on the basis of cognitive functions, while the behaviour of individuals is governed by less complex behavioural rules (Parrott et al. 2011). We have chosen to use 'individual-based models' throughout.

² A concise outline of the model is provided in the Appendix. The model is implemented in Python and source code is available from http://github.com/nlgn/sim-demog.

³ Parity refers to the number of children born to a particular woman. For example, a woman who has given birth to two previous children has a parity of two.

⁴ Representing household frequency by size helps to counteract the visual bias that otherwise results from the dominance of larger but less common household types.

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