  To investigate the combined effects of drug resistance, seasonality and
vector-bias, we formulate a periodic two-strain reaction-diffusion model. It is
a competitive system for resistant and sensitive strains, but the single-strain
subsystem is cooperative. We derive the basic reproduction number $\mathcal
{R}_i$ and the invasion reproduction number $\mathcal {\hat{R}}_i$ for strain
$i~(i=1,2)$, and establish the transmission dynamics in terms of these four
quantities. More precisely, (i) if $\mathcal {R}_1<1$ and $\mathcal{R}_2<1$,
then the disease is extinct; (ii) if $\mathcal {R}_1>1>\mathcal{R}_2$
($\mathcal {R}_2>1>\mathcal{R}_1$), then the sensitive (resistant) strains are
persistent, while the resistant (sensitive) strains die out; (iii) if $\mathcal
{R}_i>1$ and $\mathcal {\hat{R}}_i>1~(i=1,2)$, then two strains are coexistent
and periodic oscillation phenomenon is observed. We also study the asymptotic
behavior of the basic reproduction number with respect to small and large
diffusion coefficients. Numerically, we demonstrate the phenomena of
coexistence and competitive exclusion for two strains and explore the
influences of seasonality and vector-bias on disease spreading.

Dynamical Systems