I have studied Marchuk's immune system models (which were of the above form) and some authors use the linearization argument to examine the stability of equilibria. However they did not refer to any formal argument.
Check Jack K. Hale's "Functional Differential Equations". If you just need a reference for a paper, that is more than enough. If you want to understand the whole thing, it is quite hard. Ask again if you need more help, but... It's a Russian theory, you know, so it requieres Russian trining!
See pages 54-55 in the recent book of Hal Smith "An introduction to delay differential equations with applications to the life sciences". Texts in Applied Mathematics, 57. Springer, New York, 2011. xii+172 pp. ISBN: 978-1-4419-7645-1
A very good textbook is "Stability of Functional Differential Equations" by V.B. Kolmanovskii and V. R. Nosov, Academic Press, 1986.
A detailed description, with proofs, of the behaviour near a hyperbolic equilibrium can be found in the book: Delay Equations, Functional-, Complex-, and Nonlinear Analysis, O. Diekmann, S.A. van Gils, S.M. Verduyn Lunel and H.-O. Walther authors, Applied Mathematical Sciences 110, Springer Verlag, 1995.
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