Big city, small world: density, contact rates, and transmission of dengue across Pakistan

Abstract

Macroscopic descriptions of populations commonly assume that encounters between individuals are well mixed; i.e. each individual has an equal chance of coming into contact with any other individual. Relaxing this assumption can be challenging though, due to the difficulty of acquiring detailed knowledge about the non-random nature of encounters. Here, we fitted a mathematical model of dengue virus transmission to spatial time-series data from Pakistan and compared maximum-likelihood estimates of 'mixing parameters' when disaggregating data across an urban-rural gradient. We show that dynamics across this gradient are subject not only to differing transmission intensities but also to differing strengths of nonlinearity due to differences in mixing. Accounting for differences in mobility by incorporating two fine-scale, density-dependent covariate layers eliminates differences in mixing but results in a doubling of the estimated transmission potential of the large urban district of Lahore. We furthermore show that neglecting spatial variation in mixing can lead to substantial underestimates of the level of effort needed to control a pathogen with vaccines or other interventions. We complement this analysis with estimates of the relationships between dengue transmission intensity and other putative environmental drivers thereof.

Keywords: dengue; epidemiology; heterogeneity; mixing; spatial accessibility; spatial dynamics.

Figure 1.

Covariates used in this study to derive environmental drivers of transmission. Aedes aegypti probability of occurrence (a); A. albopictus probability of occurrence (b); urbanicity (c); weighted urban accessibility (d); population density and study area (e); precipitation (f); EVI mean (g).

Figure 2.

Model outputs using a backwards model selection procedure in the model using climatological variables (a, i–iv), and including the density-dependent variables (b, i–iv). Every subplot shows the predictions of the model for the indicated parameter carrying across the indicated range and every other parameter set to their mean. Figure (b, iv) shows the differences in the transmission coefficient from Lahore (green) and all other districts (red).

Figure 3.

Average distribution of R₀ with green representing Lahore versus red all other districts (a), their geographical distribution (b) and over time in which the green line again is representing Lahore versus the red line representing all other districts (c). Outputs shown here correspond to the model without including density-related variables.

Figure 4.

Ratio of betas (R₀) assuming equal force of infection and a difference in α₂–α₁ of 0.2, 0.15 (green), and 0.1, from top to bottom. The straight line indicates a ratio of 1. Order of magnitude of the infectious population refers to number of infectious people in powers of 10; i.e. the range is 10–100 000.

Figure 5.

Critical proportion of the population to control in population 2 as a function of R₀ in population 1, the order of magnitude of the infectious numbers in each population, and a difference in α₂–α₁ of 0.1, 0.15 (green) and 0.2. The straight line indicates the critical proportion assuming the α in each population are equal. Order of magnitude of the infectious population refers to number of infectious people in powers of 10; i.e. the range is 10–100 000.