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Discrete Dynamics in Nature and Society
Volume 2014, Article ID 101965, 8 pages
http://dx.doi.org/10.1155/2014/101965
Research Article

Codynamics of Four Variables Involved in Dengue Transmission and Its Control by Community Intervention: A System of Four Difference Equations

1Department of Population and International Health, Harvard School of Public Health, 665 Huntington Avenue, Boston, MA 02115, USA
2Department of Mathematical Sciences, Bentley University, 175 Forest Street, Waltham, MA 02452, USA

Received 29 June 2014; Revised 22 October 2014; Accepted 23 October 2014; Published 17 November 2014

Academic Editor: Beatrice Paternoster

Copyright © 2014 T. Awerbuch-Friedlander et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

In the case of Dengue transmission and control, the interaction of nature and society is captured by a system of difference equations. For the purpose of studying the dynamics of these interactions, four variables involved in a Dengue epidemic, proportion of infected people (), number of mosquitoes involved in transmission (), mosquito habitats (), and population awareness (), are linked in a system of difference equations: , , , and , . The constraints have socioecological meaning. The initial conditions are such that , the parameters , and the parameters ,  ,  , and are positive. The paper is concerned with the analysis of solutions of the above system for . We studied the global asymptotic stability of the degenerate equilibrium. We also propose extensions of the above model and some open problems. We explored the role of memory in community awareness by numerical simulations. When the memory parameter is large, the proportion of infected people decreases and stabilizes at zero. Below a critical point we observe periodic oscillations.