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

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 72,i and the invasion reproduction number circumflex expressionccent 72,i for strain i = 1, 2, and establish the transmission dynamics in terms of these four quantities. More precisely, (i) if 72,1 < 1 and 72,2 < 1, then the disease is extinct; (ii) if 72,1 > 1 > 72,2 (72,2 > 1 > 72,1), then the sensitive (resistant) strains are persistent, while the resistant (sensitive) strains die out; (iii) if circumflex expressionccent 72,1 > 1 and circumflex expressionccent 72,2 > 1, then two strains are coexistent and periodic oscillation phenomenon is observed. We also study the asymptotic behavior of the basic reproduction number 72,0 = maxt72,1, 72,2} for our model regarding small and large diffusion coefficients. Numerically, we demonstrate the outcome of competition for two strains in different cases.