  Contour integrals in the complex plane are the basis of effective numerical
methods for computing matrix functions, such as the matrix exponential and the
Mittag-Leffler function. These methods provide successful ways to solve partial
differential equations, such as convection--diffusion models. Part of the
success of these methods comes from exploiting the freedom to choose the
contour, by appealing to Cauchy's theorem. However, the pseudospectra of
non-normal matrices or operators present a challenge for these methods: if the
contour is too close to regions where the norm of the resolvent matrix is
large, then the accuracy suffers. Important applications that involve
non-normal matrices or operators include the Black--Scholes equation of
finance, and Fokker--Planck equations for stochastic models arising in biology.
Consequently, it is crucial to choose the contour carefully. As a remedy, we
discuss choosing a contour that is wider than it might otherwise have been for
a normal matrix or operator. We also suggest a semi-analytic approach to
adapting the contour, in the form of a parabolic bound that is derived by
estimating the field of values. To demonstrate the utility of the approaches
that we advocate, we study three models in biology: a monomolecular reaction, a
bimolecular reaction and a trimolecular reaction. Modelling and simulation of
these reactions is done within the framework of Markov processes. We also
consider non-Markov generalisations that have Mittag-Leffler waiting times
instead of the usual exponential waiting times of a Markov process.
