A two-strain reaction–diffusion malaria model with seasonality and vector-bias

Abstract

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 sensitive and resistant 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 \(i=1,2\), and establish the transmission dynamics in terms of these four quantities. More precisely, (i) if \(\mathcal {R}_1\le 1\) and \(\mathcal {R}_2\le 1\), then the disease is extinct; (ii) if \(\mathcal {R}_1>1\ge \mathcal {R}_2\) (\(\mathcal {R}_2>1\ge \mathcal {R}_1\)), then the sensitive (resistant) strains are persistent, while the resistant (sensitive) strains die out; (iii) if \(\hat{\mathcal {R}}_1>1\) and \(\mathcal {\hat{R}}_2>1\), then two strains are coexistent and periodic oscillation phenomenon is observed. We also study the asymptotic behavior of the basic reproduction number \(\mathcal {R}_0=\max \{\mathcal {R}_1, \mathcal {R}_2\}\) for our model regarding small and large diffusion coefficients. Numerically, we demonstrate the outcome of competition for two strains in different cases.