Journal of Applied Mathematics

A Time Scales Approach to Coinfection by Opportunistic Diseases

Marcos Marvá, Ezio Venturino, and Rafael Bravo de la Parra

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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.

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J. Appl. Math., Volume 2015 (2015), Article ID 275485, 10 pages.

First available in Project Euclid: 11 June 2015

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Marvá, Marcos; Venturino, Ezio; Bravo de la Parra, Rafael. A Time Scales Approach to Coinfection by Opportunistic Diseases. J. Appl. Math. 2015 (2015), Article ID 275485, 10 pages. doi:10.1155/2015/275485.

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  • F. E. G. Cox, “Concomitant infections, parasites and immune responses,” Parasitology, vol. 122, supplement, pp. S23–S38, 2001.
  • E. C. Griffiths, A. B. P. Pedersen, A. Fenton, and O. L. Petchey, “The nature and consequences of coinfection in humans,” Journal of Infection, vol. 63, no. 3, pp. 200–206, 2011.
  • S. Merrill, Introducing Syndemics: A Critical Systems Approach to Public and Community Health, Wiley, 2009.
  • C. Kwan and J. D. Ernst, “HIV and tuberculosis: a deadly human syndemic,” Clinical Microbiology Reviews, vol. 24, no. 2, pp. 351–376, 2011.
  • World Health Organization, WHO Report 2011: Global Tuberculosis Control, 2011.
  • L. J. Abu-Raddad, P. Patnaik, and J. G. Kublin, “Dual infection with HIV and malaria fuels the spread of both diseases in Sub-Saharan Africa,” Science, vol. 314, no. 5805, pp. 1603–1606, 2006.
  • E. Ivan, N. J. Crowther, E. Mutimura, L. O. Osuwat, S. Janssen, and M. P. Grobusch, “Helminthic infections rates and malaria in HIV-infected pregnant women on anti-retroviral therapy in Rwanda,” PLoS Neglected Tropical Diseases, vol. 7, no. 8, Article ID e2380, 2013.
  • D. A. Herring and L. Sattenspiel, “Social contexts, syndemics, and infectious disease in North Aboriginal populations,” American Journal of Human Biology, vol. 19, no. 2, pp. 190–202, 2007.
  • D. K. Eaton, R. Lowry, N. D. Brener, L. Kann, L. Romero, and H. Wechsler, “Trends in human immunodeficiency virus- and sexually transmitted disease-related risk behaviors among U.S. high school students, 1991–2009,” American Journal of Preventive Medicine, vol. 40, no. 4, pp. 427–433, 2011.
  • S. Baron, Ed., Medical Microbiology, The University of Texas Medical Branch at Galveston, Galveston, Tex, USA, 4th edition, 1996.
  • W. S. Symmers, “Opportunistic infections. The concept of `opportunistic infections',” Proceedings of the Royal Society of Medicine, vol. 58, pp. 341–346, 1965.
  • F. T. Koster, G. C. Curlin, K. N. A. Aziz, and A. Haque, “Synergistic impact of measles and diarrhoea on nutrition and mortality in Bangladesh,” Bulletin of the World Health Organization, vol. 59, no. 6, pp. 901–908, 1981.
  • M. Zlamy, S. Kofler, D. Orth et al., “The impact of Rotavirus mass vaccination on hospitalization rates, nosocomial Rotavirus gastroenteritis and secondary blood stream infections,” BMC Infectious Diseases, vol. 13, no. 1, article 112, 2013.
  • P. Auger, R. Bravo de la Parra, J.-C. Poggiale, E. Sánchez, and T. Nguyen-Huu, “Aggregation of variables and applications to population dynamics,” in Structured Population Models in Biology and Epidemiology, P. Magal and S. Ruan, Eds., vol. 1936 of Lecture Notes in Mathematics, pp. 209–263, Springer, Berlin, Germany, 2008.
  • P. Auger, J. C. Poggiale, and E. Sánchez, “A review on spatial aggregation methods involving several time scales,” Ecological Complexity, vol. 10, pp. 12–25, 2012.
  • M. Marvá, R. Bravo de la Parra, and J. C. Poggiale, “Approximate aggregation of a two time scales periodic multi-strain SIS epidemic model: A patchy environment with fast migrations,” Ecological Complexity, vol. 10, no. 1, pp. 34–41, 2012.
  • F. Brauer and C. Castillo-Chávez, Mathematical Models in Population Biology and Epidemiology, vol. 40 of Texts in Applied Mathematics, Springer, New York, NY, USA, 2001.
  • Y. Kang and C. Castillo-Chavez, “Dynamics of SI models with both horizontal and vertical transmissions as well as Allee effects,” Mathematical Biosciences, vol. 248, pp. 97–116, 2014.
  • H. McCallum, N. Barlow, and J. Hone, “How should pathogen transmission be modelled?” Trends in Ecology & Evolution, vol. 16, no. 6, pp. 295–300, 2001.
  • M. Begon, M. Bennett, R. G. Bowers, N. P. French, S. M. Hazel, and J. Turner, “A clarification of transmission terms in host-microparasite models: numbers, densities and areas,” Epidemiology & Infection, vol. 129, no. 1, pp. 147–153, 2002.
  • F. Brauer, “Compartmental models in epidemiology,” in Mathematical Epidemiology, F. Brauer, P. van den Driessche, and J. Wu, Eds., vol. 1945 of Lecture Notes in Mathematics, pp. 19–79, Springer, Berlin, Germany, 2008.
  • S. Bedhomme, P. Agnew, Y. Vital, C. Sidobre, and Y. Michalakis, “Prevalence-dependent costs of parasite virulence,” PLoS Biology, vol. 3, no. 8, article e262, 2005.
  • M. Sieber, H. Malchow, and F. M. Hilker, “Disease-induced modification of prey competition in eco-epidemiological models,” Ecological Complexity, vol. 18, pp. 74–82, 2014.
  • R. Cavoretto, S. Chaudhuri, A. de Rossi et al., “Approximation of dynamical system\textquotesingle s separatrix curves,” in International Conference on Numerical Analysis and Applied Mathematics (ICNAAM \textquotesingle 11), T. Simos, G. Psihoyios, C. Tsitouras, and Z. Anastassi, Eds., vol. 1389 of AIP Conference Proceedings, pp. 1220–1223, Halkidiki, Greece, September 2011.
  • R. Cavoretto, A. de Rossi, E. Perracchione, and E. Venturino, “Reliable approximation of separatrix manifolds in competition models with safety niches,” International Journal of Computer Mathematics, 2014. \endinput