Among those that were well-fed, some were found to be carrying more parasites than others, and so were more prone to spreading the disease.
Researchers said their study highlights the need to consider the impact of environmental conditions, such as nutrition, on individuals in large populations at risk of disease.
Dr Pedro Vale, from the University of Edinburgh's School of Biological Sciences, led the study. "If we have an idea of which individuals transmit a lot of disease, we will be better able to stop its spread," he said.
"We know that contact between individuals is important; but now we know that, for some animals at least, nutrition may also play an important role in the spread of disease."
The research, published in Biology Letters, was supported by the Wellcome Trust and the Centre National de la Recherche Scientifique in France.