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

A simple epidemiological model for populations in the wild with Allee effects and disease-modified fitness

Pages: 89 - 130, Volume 19, Issue 1, January 2014      doi:10.3934/dcdsb.2014.19.89

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Yun Kang - Science and Mathematics Faculty, School of Letters and Sciences, Arizona State University, Mesa, AZ 85212, United States (email)
Carlos Castillo-Chávez - Mathematical, Computational & Modeling Science Center, Arizona State University, Tempe, AZ 85287-1904, United States (email)

Abstract: The focus here is on the study disease dynamics under the assumption that a critical mass of susceptible individuals is required to guarantee the population's survival. Specifically, the emphasis is on the study of the role of an Allee effect on a Susceptible-Infectious (SI) model where the possibility that susceptible and infected individuals reproduce, with the S-class being the best fit. It is further assumed that infected individuals loose some of their ability to compete for resources, the cost imposed by the disease. These features are set in motion as simple model as possible. They turn out to lead to a rich set of dynamical outcomes. This toy model supports the possibility of multi-stability (hysteresis), saddle node and Hopf bifurcations, and catastrophic events (disease-induced extinction). The analyses provide a full picture of the system under disease-free dynamics including disease-induced extinction and proceed to identify required conditions for disease persistence. We conclude that increases in (i) the maximum birth rate of a species, or (ii) in the relative reproductive ability of infected individuals, or (iii) in the competitive ability of a infected individuals at low density levels, or in (iv) the per-capita death rate (including disease-induced) of infected individuals, can stabilize the system (resulting in disease persistence). We further conclude that increases in (a) the Allee effect threshold, or (b) in disease transmission rates, or in (c) the competitive ability of infected individuals at high density levels, can destabilize the system, possibly leading to the eventual collapse of the population. The results obtained from the analyses of this toy model highlight the significant role that factors like an Allee effect may play on the survival and persistence of animal populations. Scientists involved in biological conservation and pest management or interested in finding sustainability solutions, may find these results of this study compelling enough to suggest additional focused research on the role of disease in the regulation and persistence of animal populations. The risk faced by endangered species may turn out to be a lot higher than initially thought.

Keywords:  Allee effects, infectious disease, reduced reproduction, multiple interior equilibria, bifurcation, catastrophe, mathematical biology, conservation biology, sustainability.
Mathematics Subject Classification:  Primary: 35B32, 37C75; Secondary: 92B05, 92D30, 92D40.

Received: April 2012;      Revised: September 2013;      Available Online: December 2013.