  Subdiffusion in a system in which mobile particles $A$ can chemically react
with static particles $B$ according to the rule $A+B\rightarrow B$ is
considered within a persistent random walk model. This model, which assumes a
correlation between successive steps of particles, provides hyperbolic Cattaneo
normal diffusion or fractional subdiffusion equations for a system without
chemical reactions. Starting with the difference equation, which describes a
persistent random walk in a system with chemical reactions, using the
generating function method and the continuous time random walk formalism, we
will derive the Cattaneo--type subdiffusion differential equation with
fractional time derivatives in which the chemical reactions mentioned above are
taken into account. We will also find its solution over a long time limit.
Based on the obtained results, we will find the Cattaneo--type
subdiffusion--reaction equation in the case in which mobile particles of
species $A$ and $B$ can chemically react according to a more complicated rule.
