Discrete and Continuous Dynamical Systems - Series B (DCDS-B)

A periodic Ross-Macdonald model in a patchy environment

Pages: 3133 - 3145, Volume 19, Issue 10, December 2014      doi:10.3934/dcdsb.2014.19.3133

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Daozhou Gao - Francis I. Proctor Foundation for Research in Ophthalmology, University of California, San Francisco, San Francisco, CA 94143, United States (email)
Yijun Lou - Department of Applied Mathematics, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong, China (email)
Shigui Ruan - Department of Mathematics, University of Miami, Coral Gables, FL 33124, United States (email)

Abstract: Based on the classical Ross-Macdonald model, in this paper we propose a periodic malaria model to incorporate the effects of temporal and spatial heterogeneity on disease transmission. The temporal heterogeneity is described by assuming that some model coefficients are time-periodic, while the spatial heterogeneity is modeled by using a multi-patch structure and assuming that individuals travel among patches. We calculate the basic reproduction number $\mathcal{R}_0$ and show that either the disease-free periodic solution is globally asymptotically stable if $\mathcal{R}_0\le 1$ or the positive periodic solution is globally asymptotically stable if $\mathcal{R}_0>1$. Numerical simulations are conducted to confirm the analytical results and explore the effect of travel control on the disease prevalence.

Keywords:  Malaria, patch model, seasonality, threshold dynamics, basic reproduction number.
Mathematics Subject Classification:  Primary: 92D30; Secondary: 34D23.

Received: July 2013;      Revised: January 2014;      Available Online: October 2014.