Modelling the Spread of Wolbachia in Spatially Heterogeneous Environments

Hancock, Penelope A., and H. Charles J. Godfray. “Modelling the Spread of Wolbachia in Spatially Heterogeneous Environments.” Journal of The Royal Society Interface 9, no. 76 (November 7, 2012): 3045–54. doi:10.1098/rsif.2012.0253.

Hancock and Godfray (2012) developed an age-structured, density-dependent metapopulation model of mosquito populations, in which to explore the spread of Wolbachia across a heterogeneous landscape. Wolbachia is a mosquito endosymbiont that causes cytoplasmic incompatibility between infected males and uninfected females, leading to higher mortality in eggs resulting from these matings compared to those of positive male-positive female matings. In addition, it can also incur fitness costs amongst adults. The dynamics of Wolbachia in mosquito populations are clearly complex, as they have opposing effects at different life-stages, and studying their spread in space requires an equally complex model.

The authors build upon past reaction-diffusion models by incorporating density-dependent larval competition at the subpopulation level into their metapopulation model, and explore how differing the correlation between patch quality effects the speed of the wave of a newly introduced Wolbachia infection. Density-dependence is introduced at the larval stage through an exponential term in larval mortality as a function of density, with stronger density dependence inducing higher mortality. They incorporate the effect of Wolbachia through traits at the adult stage, with Wolbachia reducing fecundity and longevity. Wolbachia is spread vertically (from mother to offspring), however incompatible matings result in uninfected larvae, of which a proportion die due to cytoplasmic incompatibility. Matings are random, and dependent on the frequency of infected individuals in the subpopulation. Adults migrate among patches following a linear dispersal kernel, and each simulation introduces Wolbachia infected adults at one “end” of the metapopulation, allowing for a continuous introduction in that patch until the infection frequency exceeds a threshold, p*, at which point Wolbachia can spread to other patches. To explore the effect of spatial heterogeneity in patch quality, they define patches as good and bad quality, which lead to decreased and increased larval mortality, respectively. The environment then varies periodically in space, switching from runs of poor quality to good quality and back. Spatial correlation is then introduced into the model, allowing patches to be more like those near them in quality.

There were many results from the exploration of a range of parameter spaces, however there were two important conclusions regarding i) density-dependence and ii) spatial heterogeneity:

i) In a homogeneous environment, weak density-dependance slows the spread of Wolbachia. Weak density-dependence allows for higher immigration of infected adults into uninfected patches, which then cause increased rates of cytoplasmic incompatibility, and therefore higher rates of larval mortality. Because density-dependence is weak, lower densities does not translate to lower mortality in larvae, and fewer infected adults emerge in the patch.

ii) Spatial heterogeneity reduces the speed of the wave of Wolbachia in periodically-varying environments, and even more so in spatially correlated environments. Low quality environments produce few infected adults, which slows the spread of Wolbachia. However, high quality environments are difficult for Wolbachia to become established in because they produce uninfected adults that then migrate to infected lower quality environments. Both of these mechanisms make it more difficult for a patch to overcome p*, the infection frequency necessary for establishment.

Hancock and Godfray’s study incorporates additional biological realism missing from prior models, greatly improving our understanding of Wolbachia dynamics in the field. They model the larval and adult density (both infected and uninfected) at each patch, in order to understand the movement of Wolbachia between patches, which I believe to be an excellent application of multi-scale modeling. Movement between patches was limited to linear dispersal in this study, and a further study in two-dimensions would add additional realism to their model. Additionally, because of the phenomenon in which high quality, uninfected patches slow the wave, it would be interesting to explore how multiple introductions of Wolbachia targeted at high quality patches throughout the landscape would affect overall prevalence. And finally, the most obvious next step would be to expand this modeling framework to include mosquito-borne disease dynamics, as Wolbachia is most known for its ability to inhibit the transmission of arboviruses.