There is a study that the amount of sleep is associated with the probability of getting infected by the common cold. The researchers attribute the finding to the impaired functioning of the immune system, when sleeping less. This may not answer your question directly, but it is related.
Logistic regression analysis revealed that actigraphy-assessed shorter sleep duration was associated with an increased likelihood of development of a clinical cold. Specifically, those sleeping < 5 h (odds ratio [OR] = 4.50, 95% confidence interval [CI], 1.08–18.69) or sleeping between 5 to 6 h (OR = 4.24, 95% CI, 1.08–16.71) were at greater risk of developing the cold compared to those sleeping > 7 h per night; those sleeping 6.01 to 7 h were at no greater risk (OR = 1.66; 95% CI 0.40–6.95). This association was independent of prechallenge antibody levels, demographics, season of the year, body mass index, psychological variables, and health practices.
The study identifies 6h of sleep as the threshold under which catching the infection becomes more likely. To try and explain this phenomenon, the researchers cite studies that have shown that lack of sleep can disturb the effectiveness of the immune system:
Sleep, along with circadian rhythms, exerts substantial regulatory effects on the immune system.42,43 Circulating immune cells, including T and B cells, peak early in the night and then decline throughout the nocturnal hours moving out of circulation into lymphoid organs where exposure to virally infected cells occur.43–45 Studies employing experimental sleep loss also support functional changes relevant to host resistance. Sleep deprivation results in down regulation in T cell production of interleukin-219,44 and a shift away from T-helper 1 responses, marked by a reduction in the ratio of interferon-γ/IL-4 production.16 Sleep loss is associated with diminished proliferative capacity of T cells in vitro15 as well as modulation of the function of antigen presenting cells critical to virus uptake.46
Prather AA; Janicki-Deverts D; Hall MH, Cohen S. Behaviorally Assessed Sleep and Susceptibility to the Common Cold. Sleep. 2015;38(9):1353-59