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Journal of Applied Mathematics
Volume 2015, Article ID 275485, 10 pages
Research Article

A Time Scales Approach to Coinfection by Opportunistic Diseases

1Departamento de Física y Matemáticas, Universidad de Alcalá, 28871 Alcalá de Henares, Spain
2Dipartimento di Matematica “Giuseppe Peano”, Università di Torino, Via Carlo Alberto 10, 10123 Torino, Italy

Received 23 October 2014; Accepted 25 March 2015

Academic Editor: Winston Garira

Copyright © 2015 Marcos Marvá 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.


Traditional biomedical approaches treat diseases in isolation, but the importance of synergistic disease interactions is now recognized. As a first step we present and analyze a simple coinfection model for two diseases simultaneously affecting a population. The host population is affected by the primary disease, a long-term infection whose dynamics is described by a SIS model with demography, which facilitates individuals acquiring a second disease, secondary (or opportunistic) disease. The secondary disease is instead a short-term infection affecting only the primary infected individuals. Its dynamics is also represented by a SIS model with no demography. To distinguish between short- and long-term infection the complete model is written as a two-time-scale system. The primary disease acts at the slow time scale while the secondary disease does at the fast one, allowing a dimension reduction of the system and making its analysis tractable. We show that an opportunistic disease outbreak might change drastically the outcome of the primary epidemic process, although it does among the outcomes allowed by the primary disease. We have found situations in which either acting on the opportunistic disease transmission or recovery rates or controlling the susceptible and infected population size allows eradicating/promoting disease endemicity.