On the Hurwitz stability of quasipolynomials

Question

Suppose that $p,q\in\mathbb{R}^{+}$ and $a\in\mathbb{R}$. Consider the transcendental polynomial $$ p\lambda^2+p\lambda-a\left(e^{p\lambda}-1\right)e^{-q\lambda}=0,\;\lambda\in\mathbb{C}. $$ I would like some suggestions on beautiful and clean techniques for establishing conditions on $p,q$, and $a$ to guarantee that $\operatorname{Re}(\lambda)<0$ for this transcendental polynomial. Now, I'm familiar with the work of Pontryagin (he generalized the Hermite-Biehler theorem to so-called quasipolynomials) and others in this direction. I've even seen some people use basic ideas from the theory of complex variables (e.g. Rouche's theorem, etc) to attack this kind of problem. I'm fishing for ideas on something elegant, and rigorous. I'm open to literature suggestions as well.