  We prove Li-Yau and Harnack inequalities for systems of linear
reaction-diffusion equations. By introducing an additional discrete spatial
variable, the system is rewritten as a scalar diffusion equation with an
operator sum. For such operators in a mixed continuous and discrete setting, we
introduce the hybrid curvature-dimension condition $CD_{hyb} (\kappa,d)$, which
is a combination of the Bakry-\'Emery condition $CD(\kappa,d)$ and one of its
discrete analogues, the condition $CD_\Upsilon (\kappa,d)$. We establish a
hybrid tensorisation principle and prove that under $CD_{hyb} (0,d)$ with
$d<\infty$ a differential Harnack estimate of Li-Yau type holds, from which a
Harnack inequality can be deduced by an integration argument.
