## Abstract

Globalization and the expansion of transport networks has transformed migration into a major policy issue because of its effects on a range of phenomena, including resource flows in economics, urbanization, as well as the epidemiology of infectious diseases. Quantifying and modeling human migration can contribute towards a better understanding of the nature of migration and help develop evidence-based interventions for disease control policy, economic development, and resource allocation. In this study we paired census microdata from 10 countries in sub-Saharan Africa with additional spatial datasets to develop models for the internal migration flows in each country, including key drivers that reflect the changing social, demographic, economic, and environmental landscapes. We assessed how well these gravity-type spatial interaction models can both explain and predict migration. Results show that the models can explain up to 87 percent of internal migration, can predict future within-country migration with correlations of up to 0.91, and can also predict migration in other countries with correlations of up to 0.72. Findings show that such models are useful tools for understanding migration as well as predicting flows in regions where data are sparse, and can contribute towards strategic economic development, planning, and disease control targeting.

## 1. Introduction

Human population movements are an important component in a wide range of diverse processes, including urban and regional development, economics, and epidemiology. The role of movement in the phenomena found in these systems has become increasingly important as recent changes in globalization and modern transportation now allow humans to move with relative ease compared to the past. In urban and regional development the flow of humans leads to the demand, creation, and maintenance of transportation capabilities ( Harris and Todaro 1970 ; Carruthers et al. 2009 ). These movements are also associated with the flow of goods, ideas, labor, and other capital that are formative for economies ( Stark and Bloom 1985 ; Fik et al. 1992 ). Finally, as humans move, they carry diseases with them, leading to the emergence, spread, and persistence of diseases such as malaria ( Martens and Hall 2000 ), HIV ( Decosas et al. 1995 ), and measles ( Pyle 1969 ), as well as evolution of drug resistant pathogens ( Lynch and Roper 2011 ). A better understanding of the driving factors behind migration and its effects in these systems can contribute towards more effective development and planning, improved resource allocation, as well as disease mitigation ( Stoddard et al. 2009 ; International Organization for Migration 2005 ).

While high income countries have a variety of rich datasets for understanding and quantifying movement, data remain sparse in resource-poor environments, such as the majority of sub-Saharan Africa ( Prothero 1977 , 1987 ; Stoddard et al. 2009 ). Datasets that can be used to quantify human movements include GPS trip logging ( Paz-Soldan et al. 2010 ), mobile phone data ( González et al. 2008 ; Wesolowski, Eagle et al. 2012 ), census data ( Tatem and Smith 2010 ) and other national level surveys ( Ghana Statistical Service 2008 ), transportation networks and their capacities and loads ( Colizza et al. 2006 ), and even data related to traceable items such as currency ( Brockmann and Theis 2008 ). These datasets are characterized by varying spatial and temporal resolution on the movement data contained within them, different access restrictions, as well as varying amounts of information about the individuals who are moving ( Pindolia et al. 2012 ). The most commonly collected and widely available data where movement is recorded are national level population and housing censuses containing migratory movements, where individuals are recorded at some location and asked where they lived at some time period previous, usually one year. These data can be used to estimate migration rates between and across countries ( Parsons et al. 2007 ; Henry 2003 ; Cohen et al. 2008 ), and have also recently been shown to be able to provide an indication of relative movements across other temporal scales ( Wesolowski, Buckee et al. 2013 ). Complete census data, however, are often difficult to obtain and are limited in the amount of information available for each individual and household due to privacy concerns ( McCaa et al. 2006 ). Increasingly, census microdata are becoming available, which address these issues by providing high quality anonymized samples of the underlying census data (ibid). Microdata contain a wealth of information about the individuals, including those who migrate, and can be used to develop more robust models of migration where data are missing.

Human migration is subject to two basic questions: why do humans migrate and how far do they migrate ( Mabogunje 1970 )? Spatial interaction theory postulates that individuals undertaking spatial interactions such as migration will seek to maximize their benefits while minimizing their costs ( Roy and Thill 2003 ). This leads to a push-pull hypothesis for human migration: some areas attract or pull migrants in where benefits are perceived to exceed costs, while other areas propel or push migrants out where costs are perceived to exceed benefits. This attractiveness or propulsiveness of a location is a combination of various characteristics as well as the spatial arrangement of locations ( Gordon and Vickerman 1982 ). In low income countries a range of push-pull factors with relationships to migration have been previously identified; this includes geographic, sociodemographic, economic, climatic and environmental factors ( Zuma et al. 2005 ; Adepoju 2004 ; Henry et al. 2004 ; De Bruijn and Van Dijk 2003 ; Perch-Nielsen et al. 2008 ; Herrin et al. 2008 ; Van der Geest et al. 2010 ). Migration between locations as a function of these factors has been modeled in the past using the gravity model ( Lewer and Van den Berg 2008 ; Karemera et al. 2000 ; Cohen et al. 2008 ; Zipf 1946 ), one of the most studied and employed spatial interaction models ( Hua and Porell 1979 ); it has been used extensively as a quantitative tool to understand the flow of goods, resources, labor, communications, as well as migration ( Long 1970 ; Shen 2004 ; De Grange et al. 2009 ; Murdock et al. 1978 ; Karemera et al. 2000 ). The gravity model as applied to migration uses population sizes at each location and the distance between them as the push-pull factors. The model can be expanded by adding the previously identified socioeconomic, demographic, and environmental characteristics of the locations that are related to migration, resulting in a series of gravity-type spatial interaction models (GTSIM). These GTSIMs can contribute to understanding migration by providing quantitative estimates of the absolute and relative importance of the location characteristics as opposed to just using population as a crude measure of attractiveness or propulsiveness. Understanding what variables are more important in these models should help formulate better generalizable predictive models. This is critical where it is uncertain what the overall effect or importance of specific drivers are, such as the recent focus on the effects of climate change on migration ( Perch-Nielsen et al. 2008 ; Raleigh et al. 2008 ). By identifying these key drivers for migration in sub-Saharan Africa, we seek to evaluate predictive capabilities of the models, and in particular if there is one particular model formulation that is more generalizable across sub-Saharan Africa in order to predict migration movements where those data are missing.

In this study, national level census microdata for 10 selected countries, shown in Fig. 1 , were combined with additional spatial datasets to create a suite of GTSIMs for the human migration data contained in the census. The ability of both these models to explain migration was assessed using goodness-of-fit measures and by evaluating coefficient estimates. Future and cross-country predictions were also analyzed to determine if these models can be used to model migration where data are scarce or missing.

Figure 1.

A. Countries used in study. The countries used in this study are labeled along with their associated census year(s). The temporal period of the movement data is shown by the shading while the spatial resolution of movement is shown by the administrative boundaries mapped within each country. B. Human movement in Ghana in 2000 during a previous five-year period. The movement contained in the 2000 census data for Ghana is shown, visualized as weighted flows between the centroids of Ghana’s regions.

Figure 1.

A. Countries used in study. The countries used in this study are labeled along with their associated census year(s). The temporal period of the movement data is shown by the shading while the spatial resolution of movement is shown by the administrative boundaries mapped within each country. B. Human movement in Ghana in 2000 during a previous five-year period. The movement contained in the 2000 census data for Ghana is shown, visualized as weighted flows between the centroids of Ghana’s regions.

## 2. Methods

### 2.1 Data

Census microdata were obtained from Integrated Public Use Microdata Series (IPUMS) International ( Minnesota Population Center 2010 ). Each set of census microdata represents a sample of a full census conducted by the national statistical agency in the country, containing complete but anonymized information about individuals and households, and is standardized for the sake of comparison between countries. Countries in this study were selected from the IPUMS database based on their location in sub-Saharan Africa and the availability of migration variables in the census microdata. Migration is captured in the census when individuals are asked where they lived previously. The temporal resolution to the migration question varied; individuals were asked where they lived at some time period previously, with one and five years being common (with one exception of 15 years). Alternatively, individuals were asked how long they had lived at their current location. Where this continuous temporal period of migration was available for a country, both one- and five-year migration flows were extracted, in order to align and compare with migration periods in census microdata that were exactly one- or five-year periods. The spatial resolution of the migration data also varied as both current and previous residences were recorded at some administrative unit level. For a complete list of countries used in this study see Fig. 1 , which also shows the spatial and temporal resolution of the migration data, along with a visualization of typical migration flow estimates contained in the microdata. The census microdata were loaded into a relational database and queried using Structured Query Language (SQL) to generate quantifications of the migrations from each administrative unit i to every other administrative unit j . This dependent variable, MIG ij , comprises observed migration flows between administrative units in each country. A summary of migration flows for each country is included in Online Appendix A .

Other demographic and socioeconomic variables known to have relationships to migration were also calculated using the census microdata (see Table 1 ) for each administrative unit. Rural to urban migration is a critical aspect of change in a number of sub-Saharan countries ( Lall et al. 2006 ; Mabogunje 1970 ). We included this factor in our models by calculating the proportion of individuals in each region living in urban areas, included in the model as URBANPROP. Similarly, literature on African migration suggests that migration is in part driven by young males looking for economic opportunities ( Henry et al. 2004 ; Konseiga 2006 ), in part because urban areas offer more opportunities for males ( Lall et al. 2006 ). Using the census microdata we calculated the corresponding variables for inclusion in the model: MALEPROP, the proportion of males in recorded in the region; MEDIANAGE, the median age of all individuals in the region; and ACTIVE, the proportion of economically active individuals between 15 and 60 years old out of the total population in that age range in the region. Basic measures of variation for all the variables are shown in Table 2 .

Table 1.

Models and variables. Specification for each of the gravity-type spatial interaction models fit for each country along with descriptions of the variable(s) added at each model, starting from Model 1, the standard gravity model, to Model 8, our full gravity type spatial interaction model. Suffixes I and J denote the variable at locations I and J, respectively

Model 1 ln(POPI) + ln(POPJ) + ln(DISTIJ) POPI, POPJ DISTIJ Population of origin and destination Euclidean distance between centroids of origin and destination
Model 2 Model 1 + CONTIG CONTIG Dummy variable, 1 if regions are contiguous, 0 otherwise
Model 3 Model 2 + URBANPROPI + URBANPROPJ URBANPROPI, URBANPROPJ Proportion of population living in urban areas
Model 4 Model 3 + ACTIVEI + ACTIVE J ACTIVEI, ACTIVEJ Economic activity: the proportion of economically active 15–60 year old people out of total population aged 15–60 years
Model 5 Model 4 + MALEPROPI + MALEPROPJ MALEPROPI, MALEPROPJ Male proportion: total number of men divided by the population
Model 6 Model 5 + MEDAGEI + MEDAGEJ MEDAGEI, MEDAGEJ Median age of the population
Model 7 Model 6 + DROUGHTI + DROUGHTJ DROUGHTI, DROUGHTJ Drought frequency: the number of droughts in the 25-year period before the census. Droughts as were defined as a deficit in annual rainfall of 10% or more compared to the annual mean rainfall for the 25-year period.
Model 8 Model 7 + RAINVARI + RAINVARJ RAINVARI, RAINVARJ Recent rainfall variability: the number of months in the past rainy season with less rainfall than 50% of the mean rainfall for the corresponding month during the 25-year period before the census.
Model 1 ln(POPI) + ln(POPJ) + ln(DISTIJ) POPI, POPJ DISTIJ Population of origin and destination Euclidean distance between centroids of origin and destination
Model 2 Model 1 + CONTIG CONTIG Dummy variable, 1 if regions are contiguous, 0 otherwise
Model 3 Model 2 + URBANPROPI + URBANPROPJ URBANPROPI, URBANPROPJ Proportion of population living in urban areas
Model 4 Model 3 + ACTIVEI + ACTIVE J ACTIVEI, ACTIVEJ Economic activity: the proportion of economically active 15–60 year old people out of total population aged 15–60 years
Model 5 Model 4 + MALEPROPI + MALEPROPJ MALEPROPI, MALEPROPJ Male proportion: total number of men divided by the population
Model 6 Model 5 + MEDAGEI + MEDAGEJ MEDAGEI, MEDAGEJ Median age of the population
Model 7 Model 6 + DROUGHTI + DROUGHTJ DROUGHTI, DROUGHTJ Drought frequency: the number of droughts in the 25-year period before the census. Droughts as were defined as a deficit in annual rainfall of 10% or more compared to the annual mean rainfall for the 25-year period.
Model 8 Model 7 + RAINVARI + RAINVARJ RAINVARI, RAINVARJ Recent rainfall variability: the number of months in the past rainy season with less rainfall than 50% of the mean rainfall for the corresponding month during the 25-year period before the census.
Table 2.

Basic measure of variation for variables used in regression analysis across all countries with census microdata. A total of 317 administrative units with associated demographic, socioeconomic, and climatic variables were observed across all countries. These were combined in a pairwise fashion for flow regression, along with 13,062 calculations of distance and contiguity between administrative units. For detailed distributions within countries see Online Appendix B

Variable Mean Std. Dev. Min Max
POP 317 84,066 137,088 2,642 1,156,979
DIST 13,062 3.357 2.392 0.106 15.930
CONT 13,062 0.106 0.308
URBANPROP 317 0.232 0.232
ACTIVE 317 0.606 0.146 0.172 0.884
MALEPROP 317 0.492 0.017 0.448 0.604
MEDAGE 317 17.190 2.874 13 28
DROUGHT 317 7.167 2.174 16
RAINVAR 317 0.401 0.675
Variable Mean Std. Dev. Min Max
POP 317 84,066 137,088 2,642 1,156,979
DIST 13,062 3.357 2.392 0.106 15.930
CONT 13,062 0.106 0.308
URBANPROP 317 0.232 0.232
ACTIVE 317 0.606 0.146 0.172 0.884
MALEPROP 317 0.492 0.017 0.448 0.604
MEDAGE 317 17.190 2.874 13 28
DROUGHT 317 7.167 2.174 16
RAINVAR 317 0.401 0.675

Administrative unit boundary files for each country matching the year of the census and containing the spatial regions of interest were obtained from the Global Administrative Areas Database (GADM) ( Global Administrative Areas 2012 ), the United Nations Development Programme (UNDP) ( UN Geographic Information Working Group 2012 ), and via personal communication. These shapefiles were used to calculate the DISTIJ and CONTIJ variables (see Table 1 ) which measure respectively distance and contiguity between administrative units. Distance is a standard parameter in gravity-type models, where it represents the barriers to, as well as potential costs of, migration. Euclidean distance was used between geometric centroids in this study. This method was assessed in comparison to using road distance and travel time estimation measures ( Yoshida and Deichmann 2009 ; Nelson 2008 ; Linard et al. 2012 ) as alternative distance measurements, as well as population weighted centroids as an alternative method to choosing a representative point within a region. No significant differences between the methods were found when applied in a case study for Kenya (see Online Appendix C ). Consequently, all of the GTSIMs presented here used standard centroid and Euclidean distance for distance calculations. Contiguity was included as migration flows between contiguous regions are often higher than expected in these types of migration models ( Henry 2003 ). CONTIJ equals 1 where regions have contiguous borders and 0 otherwise.

Worldwide high resolution precipitation grids obtained from the CRU TS 3.1 dataset ( New et al. 2002 ) were used to calculate possible long and short term environmental drivers of migration. These calculations are similar to variables used by other studies ( Henry 2003 ; Henry et al. 2004 ). The precipitation grids, with a 0.5 degree spatial resolution, consisted of monthly precipitation covering the period 1901 to 2009. These grids were used to calculate two precipitation-related variables for each region: DROUGHT, a long term index of rainfall variability, as well as RAINVAR, a shorter term index of rainfall variability. The drought index was calculated by first delineating the 25-year period prior to the census. In the case that migration was recorded for the one-year period previous, an individual could have moved anytime up to a year prior to the census. The 25-year period then started 26 years previous to the census and ended a year before the census. The mean yearly rainfall for each of the 25 years was calculated, along with the mean of the entire 25-year period. The drought variable was calculated as the number of years in the 25-year period that were less than 90% of the overall mean. The short term rainfall variability index was calculated by first identifying the rainy season months in the country, calculating a monthly mean over the 25-year period for the rainy season months, and then counting the number of months in the previous rainy season that were less than 50% of the 25-year mean for each rainy season month. Across all administrative units in all countries the DROUGHT variable had a mean of 7.167 (see Table 2 ), indicating that on average a little more than seven years out of the last 25 years prior to the census had an annual rainfall total that was 10% below the 25-year period’s overall annual mean rainfall. Similarly on average, roughly half of a month from the previous year had less than 50% of the monthly average over the last 25 years. For a more detailed picture of variation by country see Online Appendix A .

### 2.2 Gravity-type spatial interaction models

Spatial interaction theory postulates that individuals seek to maximize benefits while minimizing costs with regard to why they migrate and how far they migrate. The gravity model builds on this theory by explicitly stating the relationship between migration and the push-pull factors that represent benefits and costs of migration. In the modern form of the gravity model, shown in Equation 1, the flux of migration MIG ij between two places i and j is proportional to their masses and inversely proportional to distance between them, where all three terms have exponents that control the strength of interaction.

(1)
$MIGij=piαpjβdijγ$

The variables $pi$ and $pj$ refer to the population size at an origin i and a destination j , while $dij$ is the distance between origin i and destination j . The exponents $α, β,$ and γ (along with an intercept term) are unknown and estimated from the data. This form, although with known limitations ( Simini et al. 2012 ), has been used to model spatial interaction between locations extensively, including migration ( Greenwood and Hunt 2003 ). Other types of models that can be used to the same effect include more complicated aggregate models that address issues such as the spatial autocorrelation within the migration network in a country ( Chun and Griffith 2011 ) or more complicated micro approaches, such as individual choice models in migration ( Moss 1979 ). We chose to focus on gravity-type models because they provide a more general framework that can be applied where data are missing without country-specific specifications and because aggregate models will pose less of a difficulty where data are poor or missing as compared to individual choice models.

Recent work has shown how the gravity model can be improved upon by including sociodemographic and climatic variables ( Henry 2003 ). This work is extended here by pairing available census microdata with worldwide climate data as described above to generate the variables of interest (see Table 1 ) for multiple countries. A suite of gravity-type spatial interaction models were built for each country, starting with a standard gravity model consisting only of population and distance and continuing on with successive, nested models that were created by adding additional explanatory variables to the standard gravity model (see Table 1 ). Variables with multicollinearity higher than 0.8 were removed and the corresponding models were not used; subsequent models omitted the variable as well. The models were then fit to the census migration data for the time period(s) available in the census using Poisson regression, which addresses many of the problems that occur when using ordinary least squares regression ( Flowerdew and Aitkin 1982 ). The dependent variable MIG ij is comprised of the observed migration flows between the administrative units within a country. More specifically for a country with N administrative units, there are N*(N−1) observed values for MIG ij . The general model equation to be estimated is shown in Equation 2, where the predicted value of migration is assumed to follow a Poisson distribution with a mean that is logarithmically linked to a linear combination of origin and destination populations ( POPI and POPJ ), distance between populations ( DISTIJI ). The suite of models is built by successively adding spatial, demographic, socioeconomic, and environmental variables (see Table 1 ) represented in Equation 2 by the matrix X and its vector of coefficients β . Table 1 indicates how variables are added to the model at each step of the fitting procedure, by adding the observed value of a variable at both the origin (e.g. ACTIVE i ) as well as the destination (e.g. ACTIVE j ). Fitting procedures were conducted using the GLM package in the R statistical environment ( R Development Core Team 2013 ) (see Online Appendix E for all code).

(2)
$ln⁡(MIGIJ^ij)=β0+β1POPi+β2POPj+β3DISTij+β⃗[X]$

Deviance values, a goodness-of-fit statistic, are considered appropriate for evaluating the fit of Poisson regression models ( Flowerdew and Aitkin 1982 ) and used here to evaluate the GTSIMs. Deviance calculates the amount of variation in the data that is not explained by the model and is approximately distributed according to the Chi-square distribution with $n−k$ degrees of freedom, where n is the number of observations and k is the number of parameters in the model. Deviance was calculated for each of the gravity-type spatial interaction models that were fit, and Wald tests that penalized added variables were used to determine if each model’s deviance reduction compared to a null model was significant or not. Wald tests were also used to determine if sequential models significantly reduced deviance with the additional variables.

For one particular country, South Africa, census data were available for 1996, 2002, and 2007; province boundaries did not change significantly between the census years; and a common time period of five years of migration was available in all three census datasets. This allowed the prediction of future within-country migration by fitting models to the data from an earlier census and predicting the migration in a future census. Models fit to 1996 were used to predict migration in 2001 and 2007; likewise, models fit to 2001 were used to predict migration in 2007. Similar to cross-country comparisons, all predictions were evaluated for both relative and absolute predictive ability. To assess how well predicted migration lined up with actual migration, two methods were used. First, correlations between the predicted and actual migration were calculated, giving a relative measure of predictive ability or, in other words, how well the model predicted the ranked migration flows between regions. Second, the slope of a linear regression between the predicted and actual migration was calculated as an absolute measure of the predictions. While a model with perfect absolute predictive power would be ideal, one with high relative predictive power is also useful ( Wesolowski et al. 2013 ).

To examine how generalizable these models are to areas where migration data is scarce, models fit to each country were used to predict the migration in all other countries. Predictions made across countries were done within matching time periods of movement, for example, Kenya’s migration over one year was only predicted using GTSIMs fit to countries that also had the same one-year period of migration available. These cross-country predictions were assessed in a similar way to the within-country predictions for South Africa, using both relative and absolute measures of predictive capability. Various analyses were conducted to determine what patterns existed with regards to the relative and absolute predictive power of the models for cross-county predictions (see Online Appendix D ), including assessing the relative and absolute predictive abilities based on the model type, the goodness-of-fit of the predicting model within the host country, and the goodness-of-fit of the corresponding model type within the target country.

## 3. Results

### 3.1 GTSIM goodness of fits

For each country, census, and time period of migration, the successive deviance reductions for the associated set of GTSIMs ( Table 1 ) is shown in Fig. 2 , beginning with the deviance reduction of the standard gravity model (Model 1) against a null model and proceeding with each successive model all the way to the full socioeconomic and climatic model (Model 8). More detailed results, including coefficient estimates and residuals for each country and model, are included in Online Appendix B . At a maximum, the GTSIMs were able to explain up to 87.4% of the migration using Model 8 in South Africa for the 2001 census. Not all countries or models performed equally though, and Malawi’s maximum deviance reduction for the one-year and five-year migration models were just 41.14% and 38.27%, respectively. The pattern of deviance reduction was also similar for countries where both one-year and five-year migration periods were used to fit the GTSIMs (Malawi 2008; South Africa 1996 and 2007; Mali 1998; and Uganda 2002). To visualize how well the GTSIM explained the migration data, actual migrations for South Africa in 2001 are shown along with the fitted migrations from the Model 8 estimation procedure.

Figure 2.

Explanatory ability of the models. The full suite of gravity-type spatial interaction models ( Table 1 ) was fit to each census dataset. The first model fit for each country is the standard gravity model (shown in black). Successive models are shown in their respective shading. Analysis of variation was used to determine how much each successive model additionally reduced deviance, the remaining unexplained variance in the movement according to the models. Note that not all models were used for each country, due to multicollinearity and a lack of variability in the underlying variable (see Online Appendix B ).

Figure 2.

Explanatory ability of the models. The full suite of gravity-type spatial interaction models ( Table 1 ) was fit to each census dataset. The first model fit for each country is the standard gravity model (shown in black). Successive models are shown in their respective shading. Analysis of variation was used to determine how much each successive model additionally reduced deviance, the remaining unexplained variance in the movement according to the models. Note that not all models were used for each country, due to multicollinearity and a lack of variability in the underlying variable (see Online Appendix B ).

For each set of models, the standard gravity model’s deviance reduction (Model 1, depending only on population and distances and shown in black in Fig. 2 ) always provided the greatest deviance reduction out of all the models, with an average reduction of 51.85%, even when the overall deviance for the full model was low (e.g. Malawi’s five-year migration models, where the full model’s deviance reduction was only 38.27% but the gravity model accounted for 72.38% of the total reduction). Model 3, which included contiguity between regions and the urban proportion variable, tended to give the next largest deviance reductions (shown in Fig. 2 ). In 14 of the 17 sets of models this urban proportion variable accounted for the next greatest deviance reduction after the standard gravity model. While environmental variables were always significant, their contribution to reducing deviance in the models was relatively low compared to other variables. These results are similar to a recent study investigating the link between climate and conflict ( O’Loughlin and Witmer 2012 ): while climatic variability was significant, its influence was modest compared to other model variables.

Investigating coefficient estimates agrees for the most part with expectations about the impact of the variables on migration. Fig. 3 presents a summary of coefficient estimates for models across all countries and periods of migration where the deviance reduction in the model was 50% or greater, in order to investigate coefficient behavior where the models explained a majority of the variation in movement. Node characteristics generally have a higher effect in the destination node than the origin node. This reflects the hypothesis that individuals move based on their expectation of increased benefits in the destination and the relative gains compared to the origin ( Todaro 1980 ). The range of population and distance coefficients generally agreed with values that have been published elsewhere ( Anderson 2010 ; Long 1970 ), with the destination population clustering higher than the origin population. Estimates for contiguity showcase that migration was generally higher between adjacent units than predicted simply by the friction of distance. Urban proportion, economic activity, and male proportions had the largest effects. This agrees with the importance of the rural-urban aspects of migration and that urban destinations attract more than they propel ( Agesa and Kim 2001 ). The wide range in coefficient estimates for MALEPROP combined with its low range of variability indicates unstable estimation, however. This could also be related to the changes in migration patterns and increasing role of young females in migration ( Adepoju 2004 ).

Figure 3.

Summary of coefficients. Violin plots for coefficient estimates for all models that had a deviance reduction of at least 50%. The underlying coefficient estimates are shown as points.

Figure 3.

Summary of coefficients. Violin plots for coefficient estimates for all models that had a deviance reduction of at least 50%. The underlying coefficient estimates are shown as points.

Figure 4.

Five-year movements between provinces in South Africa in 2001 for: A. Actual census microdata B. Fitted movements from Model 8, a gravity-type spatial interaction model for the same year and time period. Movements are shown as edges between province centroids with the weight reflecting number of individuals moving.

Figure 4.

Five-year movements between provinces in South Africa in 2001 for: A. Actual census microdata B. Fitted movements from Model 8, a gravity-type spatial interaction model for the same year and time period. Movements are shown as edges between province centroids with the weight reflecting number of individuals moving.

### 3.2 Within-country (across time) predictions

Models fit to the South Africa data for 1996 were used to predict migration in the 2001 (shown visually in Fig. 4 ) and 2007 census; models fit to South Africa for 2001 were used to predict migration in the 2007 census. The relative and absolute predictive ability of the models for these predicted migrations are shown in Fig. 5 . Note because of multicollinearity in the 1996 census data, the GTSIMs in this case did not include Models 4, 5, or 6; because of lack of variability in its underlying RAINVAR variable, Model 8 was also excluded, giving a total of four models to generate predictions from. At a maximum, the relative predictive power peaked at a correlation of 0.90 between the actual and predicted flows using 1996’s Model 7 when predicting 2001 migration. In this case Model 7 consists of the gravity model augmented with contiguity of provinces, urban proportion, and the drought variable. However, Model 3, the simpler model that excluded the drought variable, still achieved a correlation of 0.89. This pattern was consistent for all sets of predictions, where Model 3 gave a large increase in the correlation of the flows as a result of adding the urban proportion of each province. For absolute predictive power, the best result was using 1996’s Model 3 when predicting migration in 2001, where the slope of the regression between predicted and actual was 0.96. The slopes of predictions varied more widely for 2007 migration predictions than the correlations, and both the 1996 and 2001 models overestimated the predicted migration by a factor of 2 to 3.5 for 2007 actual migration. This was in part expected as the 2007 microcensus dataset was only a 2% sample as compared to 10% for both 2001 and 2007.

Figure 5.

Model predictive abilities within-country for South Africa for five-year movements. In the first two panels, models fit to the 1996 South Africa census were used to predict movements in the 2001 and 2007 census. In the third panel, fit to the 2001 South Africa census were used to predict movements in the 2007 census. All models and movements were on a five-year time period, available in all census years. The correlation between predicted and actual movements determines how well the models predict relative movements between provinces (i.e. the rank of flows), while the slope of the regression between predicted and actual flows gives an absolute measure of accuracy. Models not shown were excluded due to multicollinearity or a lack of variance in the underlying variable (see Section 2, Methods).

Figure 5.

Model predictive abilities within-country for South Africa for five-year movements. In the first two panels, models fit to the 1996 South Africa census were used to predict movements in the 2001 and 2007 census. In the third panel, fit to the 2001 South Africa census were used to predict movements in the 2007 census. All models and movements were on a five-year time period, available in all census years. The correlation between predicted and actual movements determines how well the models predict relative movements between provinces (i.e. the rank of flows), while the slope of the regression between predicted and actual flows gives an absolute measure of accuracy. Models not shown were excluded due to multicollinearity or a lack of variance in the underlying variable (see Section 2, Methods).

### 3.3 Cross-country comparisons

For cross-country predictions, the models fit within each country were used to predict the migration flows in all other countries, and the correlation and slopes of the predicted versus actual flows were also calculated (for detailed results consult Online Appendix B ). Fig. 6 highlights the relative predictive ability of GTSIMs within East and West African countries. Each country’s boxplot represents the correlations between its actual migration and the predicted migration from models fit to other countries with the same time period of migration. Similar to Fig. 2 , there are trends where some countries’ migration seem to be predicted well, while others are not (e.g. Malawi’s correlations for predicted migration are never higher than 0.28). Because of the poor explanatory power that GTSIMs had within Malawi (see Fig. 2 ), this indicated that the ability of any given GTSIM to predict migration is related to how well the model type fits within the country and its own migration data to begin with, or more generally, how well does the migration within the country correspond to a spatial interaction model with the selected variables. More specifically, poor prediction seemed to correspond to lower levels of urbanization overall, such as Malawi, where URBANPROP was at most 0.6 and mostly rural otherwise (see Online Appendix A ). Fig. 7 confirms this by showing the goodness-of-fit for both the model being used to predict migration, and the corresponding model type’s goodness-of-fit within the country being predicted. The correlation (size) changes little along the horizontal axis, the gradient in size is apparent along the vertical axis. Simply put, where migration is not explained well by any type of gravity model, none of the GTSIMs were able to predict migration well.

Figure 6.

Cross-country predictions. Correlation between predicted flows and actual flows for each country’s predicted movement. Top row shows countries with one year of movement while bottom row shows countries with five years of movement. Countries were limited to East and West Africa for brevity.

Figure 6.

Cross-country predictions. Correlation between predicted flows and actual flows for each country’s predicted movement. Top row shows countries with one year of movement while bottom row shows countries with five years of movement. Countries were limited to East and West Africa for brevity.

Figure 7.

Model type and predictions. For each prediction of movement using a GTSIM, the deviance reduction for the predicting model is shown against the deviance reduction of the same model type when fit within the target country. Shading corresponds to the model type (see Table 1 ), while size corresponds to the correlation between the predicted and actual flows.

Figure 7.

Model type and predictions. For each prediction of movement using a GTSIM, the deviance reduction for the predicting model is shown against the deviance reduction of the same model type when fit within the target country. Shading corresponds to the model type (see Table 1 ), while size corresponds to the correlation between the predicted and actual flows.

## 4. Discussion

There is an increasing need for a quantitative understanding of human population movements, particularly migration. This need stems from a range of fields in which studies need to incorporate human movements such as migration in their methodologies, for instance, anticipating the development of services for large influxes of migration from rural to urban regions ( Scott 2009 ) or estimating continent-wide movement rates for complex disease simulators ( Griffin et al. 2010 ). These studies are impacted by the lack of either reliable migration data in low income settings or models with known uncertainties from which to estimate migration.

Here we have not only calculated quantitative migration estimates at a regional scale within multiple countries in Africa, but also developed models of migration, paying attention to their goodness-of-fit, parameter estimate, uncertainty and prediction power. The modeling approach used in this study highlights the potential for pairing census microdata and other spatial datasets to generate migration models. Broadly, the results indicate that the gravity-type models of migration are useful tools in understanding how humans move, and these models can be applied to other census and demographic datasets to estimate migration with known boundaries in their accuracy, which will be especially useful in countries where migration data are missing or poor, but census and demographic data are available. Model performance for predicting migration in other countries varied greatly in absolute terms. However, the ability of these models to predict the relative migration, as indicated by the correlation of predicted flows, is still promising for applications where absolute predictions are not required. This may include for example the identification of regional clusters that are connected by migration ( Tatem and Smith 2010 ) or the allocation of scarce resources such as development funds ( Lucas 2006 ). The models performed reasonably well for predictions of absolute migration for the special case of within-country future movement, highlighting their potential for within-country planning that requires estimates of absolute volume of movement, for example in forecasting around transportation network development ( Carruthers et al. 2009 ).

There are, however, uncertainties and limitations that must be acknowledged in the application of these models both in the methods used and the complexity of migration itself. While goodness-of-fit was high for the majority of the GTSIMs, for some countries (e.g. Malawi) the models fit poorly, suggesting that in these cases, the true drivers or structure of a country’s migration patterns are not completely captured by these models. Qualitative explorations of migration within these countries could help identify why the models fail to fit well (e.g. countries where circulatory migration is more prominent and not captured in a census) or whether other potential drivers should be incorporated, such as the impact of conflict events ( Lozano-Gracia et al. 2009 ). Other limitations arise from the underlying data used for the models and the interaction with the spatial resolution of migration flows used in an aggregate model. The administrative unit at which migration was captured varied between the countries in the study; there is no guarantee that it is the best scale at which to analyze and model migration. The migration captured is itself of the permanent type and circulatory movements are missing. This limitation is inherent to the publicly-available census data and can only be overcome by using alternative data sources. Furthermore, the explanatory variables were extracted at the same scale, which could mean drivers are not captured at a scale at which they influence migration—for instance, climate variables as calculated would not capture extreme local-scale weather events. Finally, where explanatory variables were themselves based on models, such as CRU precipitation grids, it’s likely that their uncertainty is higher in sub-Saharan Africa when compared to North America or Europe, given the lower density of weather stations that go into generating the precipitation estimates ( New et al. 2002 ).

Future directions include extending these models to the whole of Africa, as well as exploring migration microdata for countries in Asia and South America. Alternate datasets will be explored for further validation and census microdata will be assessed for its potential to model inter-country migration, which also play a role in spreading processes such as disease ( Wongsrichanalai and Sirichaisinthop 2001 ), as well as its potential to develop micro models of movement for individuals based on sociodemographic characteristics. The motivation for this study itself is malaria strategic planning; migration plays an important component in the malaria system where infected humans can import malaria, including drug resistant strains ( Lynch and Roper 2011 ), into an area and then contribute to future transmission, compromising control and elimination efforts ( Prothero 1961 ; Cohen 2012 ). By combining the migration estimates and models showcased here with risk maps of malaria, evidence-based recommendations can be formulated for these decisions, which will either contribute to the success of elimination efforts or direct resources better used elsewhere in favor of continued control and suppression.

## Supplementary data

Supplementary data is available at Migration Studies online:

Online Appendix A : Census microdata and migration flows

Online Appendix B : Statistical details

Online Appendix C : Distance calculations and comparisons

Online Appendix D : Cross-country predictions

Online Appendix E : Source code

## Funding

This work was supported partially for AJG by funding from the National Science Foundation under Grant No. 0801544 in the Quantitative Spatial Ecology, Evolution, and Environment Program at the University of Florida. AJT was supported by funding from NIH/NIAID (U19AI089674), the Bill & Melinda Gates Foundation (49446;1032350), and the RAPIDD program of the Science and Technology Directorate, Department of Homeland Security, and the Fogarty International Center, National Institutes of Health.

## Acknowledgements

The authors thank Ian T. Kracalik, Dr Timothy Fik, and Jake M. Ferguson for their valuable input. The authors thank the national statistical agencies who partner with IPUMS to provide census microdata: Ghana Statistical Services, Ghana; National Statistics Directorate, Guinea; National Bureau of Statistics, Kenya; National Statistical Office, Malawi; National Directorate of Statistics and Informatics, Mali; National Agency of Statistics and Demography, Senegal; Statistics Sierra Leone, Sierra Leone; Statistics South Africa, South Africa; Central Bureau of Statistics, Sudan; National Bureau of Statistics, Tanzania; Bureau of Statistics, Uganda. This study is part of a wider initiative in human movement data and modeling, the Human Mobility Mapping Project < http://www.thummp.org >, an open access initiative to provide data and models at various temporal and spatial scales.

Conflict of interest statement . None declared.

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