J. Math. Biol. (2011) 62:543–568 DOI 10.1007/s00285-010-0346-8 Mathematical Biology A reaction–diffusion malaria model with incubation period in the vector population Yijun Lou · Xiao-Qiang Zhao Received: 16 December 2009 / Revised: 1 April 2010 / Published online: 30 April 2010 © Springer-Verlag 2010 Abstract Malaria is one of the most important parasitic infections in humans and more than two billion people are at risk every year. To understand how the spatial het- erogeneity and extrinsic incubation period (EIP) of the parasite within the mosquito affect the dynamics of malaria epidemiology, we propose a nonlocal and time-delayed reaction–diffusion model. We then define the basic reproduction ratio R 0 and show that R 0 serves as a threshold parameter that predicts whether malaria will spread. Fur- thermore, a sufficient condition is obtained to guarantee that the disease will stabilize at a positive steady state eventually in the case where all the parameters are spatially independent. Numerically, we show that the use of the spatially averaged system may highly underestimate the malaria risk. The spatially heterogeneous framework in this paper can be used to design the spatial allocation of control resources. Keywords Malaria transmission · Spatial heterogeneity · Incubation period · Basic reproduction ratio · Threshold dynamics · Global attractivity Mathematics Subject Classification (2000) 35K57 · 37N25 · 92D30 Supported in part by the NSERC of Canada and the MITACS of Canada. Y. Lou (B ) · X.-Q. Zhao Department of Mathematics and Statistics, Memorial University of Newfoundland, St. John’s, NL A1C 5S7, Canada e-mail: yijunlou@hotmail.com; ylou@mun.ca X.-Q. Zhao e-mail: zhao@mun.ca 123