Deterministic epidemic models with explicit household structure
Introduction
Mathematical modelling is now seen as a key tool in planning for a range of epidemics [1], [2] and such models can play an important role in determining an effective control policy [3], [4]. Understanding emergent infectious diseases in humans is viewed with increasing importance. Our recent experience with the rapid spread of SARS [5], the perceived threat of bioterrorism [6] and concerns over influenza pandemics [7] have all highlighted our vulnerability to (re)emerging infections. For all these examples, mathematical modelling has been used to develop an understanding of the relevant epidemiology, as well as to quantify the likely effects of different intervention strategies [2], [8], [9], [10], [11].
The basis of most epidemiological models is the compartmentalisation of individuals into a range of classes dependent on their infection history. As such the simple SIR (Susceptible–Infectious–Recovered) model, which ignores births and deaths, is amongst the most studied [12], [13]. Despite its simplicity, this model is generally quite effective at describing the dynamics of a range of infections in many populations [14]. The model is specified by the parameter , together with initial conditions and the natural time-scale of the infection. These quantities then uniquely determine early growth, peak location and final size of the epidemic.
While such simplicity has its advantages, it precludes capturing the full complexity of population heterogeneities and targeted intervention strategies, both of which have important implications in preparation for potential influenza pandemics or bioterrorism. Faced with these very real threats to human health it is natural to construct parameter-rich simulation models, which aim to capture the full complexity of pathogen transmission and human interaction. While such models have been highly successful [9], [11], [10] and are a key requisite for control planning [1], [15], [16], their inherent complexity precludes an intuitive understanding of the wealth of interacting component parameters and variables. A complementary approach, and one adopted here, is to make more modest extensions of the SIR model that focus on specific aspects of the complex problem whilst maintaining a relatively small parameter set and deterministic equations.
Here we focus on the transmission dynamics within and between households, essentially constructing a metapopulation model [17], in which contacts that can lead to disease transmission within households are considered to be much stronger than those that can lead to transmission between [18], [19].
There has been much recent interest in household modelling [11], [20], [21]. Formal models dealing with SIS-type dynamics in household-structured populations have appeared in [22], [23], [24], and for SIR-type dynamics in [25], [26], [27]. Recent work linking more formal results on household models with applied questions, making use of different modelling techniques from ours, has also recently appeared [28], [29].
The present work is unique in presenting analysis of a simple ODE model for transmission of disease that is suited to the analysis of both threshold behaviour and interventions that are affected by transient features of the system. We pay particular attention to robustness of assumptions under reparameterisation of the model, and also consider from the start the need to fit model parameters dynamically to available epidemiological data. In addition to this, we are able to derive analytic expressions for small households that support the robustness of many of our conclusions across the full range of parameter values.
Our model consists of a (relatively) large but simply described set of differential equations, where the stochastic nature of transmission is captured by modelling all possible household configurations. These household models are parameterised to fit a fixed early growth rate, replicating the fitting procedure that would occur from early epidemic data. Finally, we consider control by prophylactic vaccination, responsive vaccination or antiviral treatment [11], [30] and focus on whether such control should be targeted towards entire households or individuals.
Section snippets
‘Household gas’ models
Although the SIR model is well known and its dynamics have already been studied in considerable detail [12], [13], we define it here both for completeness and to introduce the basic parameters. We consider a closed population, without births or deaths, and adopt a frequentist approach, meaning that are the proportion of the population that are susceptible, infectious and recovered, respectively. The deterministic dynamics of the model arewhere is the
Parameter fitting
For the simple SIR epidemic of Eq. (2.1), assuming that the initial proportion of the population infected is sufficiently small to ignore, we have only two parameters: the recovery rate sets the basic time-scale of the dynamics, while determines all other aspects, including the early epidemic growth rate, the final size of the epidemic and the levels of control required for eradication. It is generally assumed that , or more precisely the average infectious period , can be
Epidemic dynamics
We now use each fitting method in turn, together with numerical integrations of the household model (2.4), to investigate the implications of household structure for both the shape of the infection curves and summary statistics of the epidemic.
Fig. 1 (top row) shows the changing prevalence of infection through the epidemic for ‘pure’ households of size n. All epidemics are constrained to have the same early growth by matching , and we additionally fit the initial, very small, level of
Modelling control measures
In this section we consider modelling three possible control methods: prophylactic vaccination, responsive vaccination and distribution of antivirals. These methods operate in very different ways: prophylactic vaccination suppresses infection by reducing the initial number of available susceptibles in the population; responsive vaccination removes susceptibles at a certain rate during an epidemic; and antivirals are administered to infected individuals in order to reduce their subsequent
Discussion
In this paper, we have developed a relatively simple model for the transmission of an infectious agent through a population of well mixed households. To make a fair comparison as household size is varied, we fix the early epidemic growth rate—essentially all models would agree with the case-report data gathered during the first few generations of an epidemic. We feel that this agreement is vital if we are to compare models reliably. However, even with this constraint there is still a free
Acknowledgments
This work was funded by EU Grant INFTRANS (FP6 STREP; Contract No. 513715). We give thanks to members of the Infectious Disease Epidemiology Research Interest Group at Warwick, to Joshua Ross, and to other members of the INFTRANS consortium for their helpful comments on this work.
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