A two-strain malaria transmission model with seasonality and incubation period

Abstract

Malaria is one of the most common mosquito-borne diseases in the world. To understand the joint effects of the vector-bias, seasonality, spatial heterogeneity multi-strain and the extrinsic incubation period of the parasite on the dynamics of malaria, we formulate a time-periodic two-strain malaria reaction–diffusion model with delay and nonlocal terms. We then consider threshold conditions that determine whether malaria will spread. More specifically, the basic reproduction number $\mathcal {R}_{i}$ for single strain-i and invasion number $\hat{\mathcal {R}}_{i}$ for each strain-$i~(i=1,2)$ are derived. Our results imply that if $\max \{\mathcal {R}_{1},\mathcal {R}_{2}\}<1$, then the disease-free periodic solution is globally attractive; if $\mathcal {R}_{i}>1>\mathcal {R}_{j}~(i,j=1,2, i\ne j)$, then competitive exclusion, where the jth strain dies out and the ith strain persists; if $\min \{\hat{\mathcal {R}}_{1},\hat{\mathcal {R}}_{2},\mathcal {R}_1,\mathcal {R}_2\}>1$, then the disease persists. Our numerical simulations show that spatially heterogeneous infection can increase the basic reproduction number and the omission of the vector-bias effect will underestimate the infection risk of the disease.