Dynamics of a dengue fever transmission model with crowding effect in human population and spatial variation

Tzy-Wei Hwang; Feng-Bin Wang

doi:10.3934/dcdsb.2013.18.147

Article Contents

2013, Volume 18, Issue 1: 147-161. Doi: 10.3934/dcdsb.2013.18.147

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Dynamics of a dengue fever transmission model with crowding effect in human population and spatial variation

Tzy-Wei Hwang^1, and
Feng-Bin Wang^2,

1.
Department of Mathematics, National Chung Cheng University, Min-Hsiung, Chia-Yi 621
2.
Department of Natural Science in the Center for General Education, Chang Gung University, Kwei-Shan, Taoyuan 333

Received: September 2011

Revised: April 2012

Published: January 2013

Abstract / Introduction Related Papers Cited by

Abstract

Dengue fever is a virus-caused disease in the world. Since the high infection rate of dengue fever and high death rate of its severe form dengue hemorrhagic fever, the control of the spread of the disease is an important issue in the public health. In an effort to understand the dynamics of the spread of the disease, Esteva and Vargas [2] proposed a SIR v.s. SI epidemiological model without crowding effect and spatial heterogeneity. They found a threshold parameter $R_0,$ if $R_0<1,$ then the disease will die out; if $R_0>1,$ then the disease will always exist.
To investigate how the spatial heterogeneity and crowding effect influence the dynamics of the spread of the disease, we modify the autonomous system provided in [2] to obtain a reaction-diffusion system. We first define the basic reproduction number in an abstract way and then employ the comparison theorem and the theory of uniform persistence to study the global dynamics of the modified system. Basically, we show that the basic reproduction number is a threshold parameter that predicts whether the disease will die out or persist. Further, we demonstrate the basic reproduction number in an explicit way and construct suitable Lyapunov functionals to determine the global stability for the special case where coefficients are all constant.

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Mathematics Subject Classification: 35K57, 92B05, 92D25.

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