Actuaciones procesales y resultados de las mismas

Gravity models are commonly used to model human spatial interaction behaviour – any movement in space that arises from a human process. They originate from the Newton's law of Universal Gravitation, which states that between two point masses there is a force of attraction with magnitude directly proportional to the product of the two masses, and inversely proportional to the square of the distance between them. It can be expressed as:

1 2 2

m m F G

F: magnitude of the gravitational force between the two point masses G: gravitational constant

m1,m2: mass of the two point masses

d: distance between the two point masses

The first conceptual application of Newton’s law to human movement was developed by Carey (Carey 1858), but it was almost a century later that a formal postulation was formulated (Stewart 1941) to quantify human movement between regions to be proportional to the populations of the origin and destination, and inversely proportional to the squared distance between the regions. The squared distance relationship was later relaxed and the exponent was calibrated from observations (Huff 1963). In the past few decades, there has been substantial development of the theoretical basis and model formulation for the family of gravity models (Sen and Smith 1995). They have been widely applied to study different types of movement patterns, and are an essential tool in trip distribution modelling and transport planning (Erlander and Stewart 1990), retail planning and modelling different types of human and commodity flows (Haynes and Fotheringham 1984; Fotheringham and O'Kelly 1988).

4.1.2 Generic model formulation

Denote Tij as the flow from origin i to destination j, O(i) as a function of propelling or outflow attribute(s) of the origin i, D(j) as a function of attracting or inflow attribute(s) of the destination j, and f(dij) as a distance deterrence function that reflects the tendency to travel relative to distance, and c as the constant. The generic form of gravity model is written as:

( ) ( ) ( )

ij ij

T cO i D j f d

Gravity models can be inflow constrained, outflow constrained, doubly constrained, or unconstrained (Haynes and Fotheringham 1984; Fotheringham and O'Kelly 1988). The constraint requires the estimated marginal flow (inflow, outflow, or both) of each study region to be equal to that of the observed marginal flow. Usually model fit improves with additional information in the constrained models compared to unconstrained models (Fotheringham and O'Kelly 1988), but the research question under study also determines the appropriate type of model to be applied besides data availability. An outflow constrained model quantifies the relative attractiveness of a region as a destination compared to all other possible destinations to travel from the origin. Similarly, an inflow constrained model quantifies the relative contribution of a region as an origin compared with other possible origins conditional upon the marginal inflow of the destinations. Denote Oi as the observed

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marginal outflow for origin i, Dj as the observed marginal outflow for destination j, Ai and Bj as the balancing factors to ensure the constraints are met. Then these models can be written as:

Outflow constrained model:

Inflow constrained model:

All the existing applications of gravity models in studying human infectious disease epidemics use the unconstrained model, and they are discussed in the following section.

4.1.3 Application in human infectious disease epidemics

With the spatial spread of human infectious disease ultimately driven by human movement, travel flows between regions modelled by gravity models have been increasingly used to establish the heterogeneous contact rate between individuals in the subpopulations, assuming the contact rate to be proportional to the flow volume. Population sizes and travel distance are the most commonly used attributes in the gravity models that parsimoniously capture the coupling between the patches and are superimposed onto metapopulation epidemic models at various geographical scales.

On the national scale, gravity models were applied in the studies of measles (Xia, Bjornstad et al. 2004) and seasonal influenza (Viboud, Bjornstad et al. 2006) epidemics, and reproduced the regional spatiotemporal patterns which could not be achieved by using a distance function alone. A gravity type of spatial coupling was also used to model the spatial spread of the 1918 influenza pandemic in the UK, but proved less satisfactory as a description of the US pandemic (Eggo, Cauchemez et al. 2010). That study also found that connectivity between cities was density dependent, that is the total connectivity of a city depends on the number of close neighbours. However another study of the epidemic patterns of measles in the coastal regions of UK (Bharti, Xia et al. 2008) using Xia et al.’s model found that the gravity model underestimated the flows to close neighbours, and that having less neighbouring areas does not make a region less connected. A less vigorous gravity model

( ) ( ) 1 ( ) ( ) for all i ij i i ij i ij j ij i j T A O D j f d A D j f d T O ( ) ( ) 1 ( ) ( ) for all j ij j ij j ij i i j ij j T B O i f d B O i f d D D T

application was used in another pandemic study (Ciofi degli Atti, Merler et al. 2008)which fitted a gravity model to Italy’s categorical work and study trip data (within municipal, outside municipal but within province, outside province but within same region, outside region). However, only the derived spatial kernel from the model was used in the model to determine the distance dependent contact rates.

At a continental scale, influenza pandemic spread in Europe was modelled with an embedded gravity model of the railway transportation data between countries, with the gross domestic product (GDP) of the countries fitting the flows better than population size (Merler and Ajelli 2010). In global metapopulation models, a gravity model of local commuting patterns of 29 countries integrated with the long-range air traffic network data underlies the multi-scaled “Global Epidemic and Mobility model” (GLEaM). This model was used to simulate the spread of epidemics worldwide (Balcan, Colizza et al. 2009), study the early stage of the influenza A/H1N1 pandemic in 2009 (Colizza, Vespignani et al. 2009), and investigate the strain’s potential for seasonal transmission (Balcan, Hu et al. 2009).

The gravity models applied in all of these studies were of the general form:

j k j k jk N N T k d

where Tj→k denotes the flow from patch j to k, Nj and Nk are the population size of patch j and k, djk is the spatial distance between patches j and k, and , , and are the scaling factors of the origin and destination population and the travel distance respectively, and k is a constant. In the GLEaM, an exponential distance function f(djk)=exp(djk/r) performed better than the power law function g(djk)=djk as the kernel, and in Xia et al.’s study on measles (Xia, Bjornstad et al. 2004), the predicted number of infectives Ij in the origin patch was used instead, with the gravity model being combined with a time-series susceptible-infected-recovered (TSIR) model.

Different assumptions relating the human movement described by gravity models to disease transmission were implicitly applied in each of these studies. The gravity model is usually embedded in the force of infection term of the epidemic model, with its coupling effects with each region summed up to give the total transmission potential from the heterogeneous level of interaction between the subpopulations. As the compartmental models categorizes individuals into the states of susceptible, infective and others, the nature of gravity model is always directional – from an origin to a destination - and the flow matrices modelled are usually not symmetric. It is therefore essential to decide who travelled from their residing patch and to where, and possibly also where the contact

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takes place. A few studies (Xia, Bjornstad et al. 2004; Eggo, Cauchemez et al. 2010) used the “infector driven approach”, assuming it is the infected person that travels to other patches and infects the susceptible person there. Thus the total force of infection experienced by a susceptible is proportional to the sum of the flow from other patches to the susceptible’s patch. The opposite is the “susceptive driven approach”, assuming the susceptible individuals travel to other patches and become infected from the infectious people there (Merler and Ajelli 2010). Therefore the origin patch and the model structure was defined differently under these two approaches, though both of them assumed the movement to be unilateral in nature, with either the susceptible or the infected travelling, and those living in the destination remaining in their patches. However, these assumptions could be an oversimplified depiction of reality.

The framework of capturing “bilateral movement” in epidemic models – that allows for the possibility that the susceptible travels to the infector’s patch and becomes infected there (susceptible driven approach), and the infector travels to the susceptible’s patch and spreads the infection (infector driven approach), was proposed and developed using the spatial interaction theory (Thomas 1999). It defined the measure of regional attractiveness of region j as a destination for individuals travelling from region i as Ti→j=njexp(-rdij), where nj is the population size of j, dij the distance between regions i and j, and r is the distance decay parameter. Then the probability of a given contact by an individual in i is made in region j, denoted as pi→j = njexp(-rdij)/∑j njexp(-rdij). Conversely, the probability that an individual in region j travels to region i and makes contact there is

pj→i = niexp(-rdij)/∑i ni exp(-rdij). The sum of both terms then forms the contact probability under a bilateral movement assumption. As an alternative to using the spatial interaction model, Keeling and Rohani (Keeling and Rohani 2002) used assumptions about the leaving and returning rate of individuals from their home patches to describe the bilateral movement in a hypothetical study of a population with two patches. However, this movement assumption restricted the location of infection to be at the home patch of either the susceptible or infective.

Sattenspiel and Dietz (Sattenspiel and Dietz 1995) developed a more comprehensive framework that captures the movement of both the susceptible and infective and allows infection between them to occur in any regions they both travel to – referred as “multilateral movement” in this study - using the rates of leaving and returning from home patches. Such information was assumed to be available from other sources like travel data or survey, and the method was first applied to study the influenza epidemic in 1918-1919 in Canada (Sattenspiel and Herring 2003). The GLEaM model was built upon the work of Sattenspiel and Dietz with the commuting flow derived from a gravity model. However, it remains uncertain how the various movement assumptions affect the results of the

epidemic models, or if under certain condition the much simpler unilateral assumption would be sufficient to describe the heterogeneous contact rate.

Gravity coupled epidemic models can either have the parameters of the gravity model derived independently from the origin-destination travel data (Viboud, Bjornstad et al. 2006; Balcan, Colizza et al. 2009; Merler and Ajelli 2010), or simultaneously estimated with the epidemic model parameters as an “endogenous system” (Xia, Bjornstad et al. 2004; Eggo, Cauchemez et al. 2010). Although at the expense of heightened technical challenges and computational complexity, the latter does not rely on the availability of the relevant travel flow data, and can serve as a means to validate the general assumption of gravity coupling. However, when a standard gravity relationship is not adequate to describe the flow dynamics, for instance, if long distance flow beyond several hundred kilometres exhibits different gravity behaviour than the short distance flow (Viboud, Bjornstad et al. 2006; Balcan, Colizza et al. 2009), gravity models fitted as a standalone statistical model allows model adequacy to be diagnosed on a finer level.

The interpretation of the gravity model parameters was also different under the two methods. In an “endogenous” model, it is the contact rate between individuals in different subpopulations that is directly modelled by the gravity component, whereas the exogenous gravity model represents the factors driving the flow, and the flow is used as the surrogate of the heterogeneous contact rate, assuming they are related proportionally. It remains uncertain how valid this assumption is. Contact surveys (Mossong, Hens et al. 2008) have shown that the number and type of contacts made in different settings vary (e.g. school and work). Therefore contacts generated related to a work trip may be different from that of a study trip.

Independently fitted gravity models have fairly consistent estimates of the origin and destination population exponents ( , ) – (0.3, 0.64) for US work flow of less than 119km (Viboud, Bjornstad et al. 2006), (0.28, 0.66) for work and study flows of Italy (Ciofi degli Atti, Merler et al. 2008), (0.46, 0.64) for commuting patterns of less 300 km in 29 countries. Long distance flow generally had lower and values (Viboud, Bjornstad et al. 2006; Balcan, Colizza et al. 2009). Endogenous estimation resulted in the population exponents to be insignificant for US, and at (0, 0.40) for UK (Eggo, Cauchemez et al. 2010). Fixing =1 resulted in =1.5 when the number of infective was directly used as the origin attribute (Xia, Bjornstad et al. 2004). This unique finding of the / ratio greater than 1 opposed to other studies could be partly attributed to the size of infective group (measles predominantly affecting children). In general, trip flows are more sensitive to the size of the destination population than the origin population with a higher population exponent in most cases, though the increment in both the propulsiveness and attractiveness of the regions decreases with

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greater population size. Therefore small populations are disproportionally more important than large populations.

The spatial kernels of the gravity model from the above studies were reproduced based on the kernel parameters available in the studies (Figure 4.1). All studies except Eggo et al. apparently used the non-normalised form of the spatial kernel but with a constant term in the gravity model formulation. Therefore, although model constant or the kernel normalisation factors were not available, reproduced kernels for the same country or those with similar geographic scale can provide general ideas about the shape of the kernels. Eggo et al. normalised the origin population and the spatial distance together for each country, therefore the US kernel before normalisation is not comparable to the UK kernel before normalization.The US short distance work flow kernel was found to be almost identical to Italy’s work and study flow kernel, while the two UK kernels for year 1918 and 1940-60s’ were close to each other. Balcan et al’s common short distance kernel for 29 countries based on contemporary data was much slower decaying than the others. Spatial kernels implicitly capture the transportation infrastructure of the country under study during the study period, and are also affected by the spatial scale of the study units. The common kernel of Balcan et al. was based on commuting data including small countries like Hong Kong with maximum travel distance less than 40 km and large countries like US (>8000 times of the physical size of Hong Kong), each of different geographic division – from 18 districts in Hong Kong, 3141 districts in US, to 36602 districts in France. Therefore the derived kernel has to be applied with caution. The study also found that an exponential kernel fitted the data better than a power law kernel, while all other studies simply assumed the kernel to be distributed as a 1-parameter power law function.

Figure 4.1 Spatial kernels for gravity models from previous epidemic studies reproduced, with the country and type of

flow and the first author of the studies specified.

0.0000001 0.000001 0.00001 0.0001 0.001 0.01 0.1 1 1 10 100 sp atial k e rn e l fun ction distance (km)

US work flow (<119 km) - Viboud US - Eggo (before normalization) UK - Eggo (before normalization) UK - Xia

Italy - Ciofi

Besides being country specific, the optimal type of flow to be fitted by gravity models for studying epidemic spread should also be disease specific, determined by the dominant population subgroup for disease transmission (Viboud, Bjornstad et al. 2006). Existing gravity models fitted specifically to Hong Kong do not fit into the context of this study, neither were the spatial kernel parameters given (Wong, Tong et al. 1999).

4.2 Alternative methods to capture spatial heterogeneity in contact rates