Multiparametric bifurcations of an epidemiological model with strong Allee effect

Abstract

In this paper we completely study bifurcations of an epidemic model with five parameters introduced by Hilker et al. (Am Nat 173:72-88, 2009), which describes the joint interplay of a strong Allee effect and infectious diseases in a single population. Existence of multiple positive equilibria and all kinds of bifurcation are examined as well as related dynamical behavior. It is shown that the model undergoes a series of bifurcations such as saddle-node bifurcation, pitchfork bifurcation, Bogdanov-Takens bifurcation, degenerate Hopf bifurcation of codimension two and degenerate elliptic type Bogdanov-Takens bifurcation of codimension three. Respective bifurcation surfaces in five-dimensional parameter spaces and related dynamical behavior are obtained. These theoretical conclusions confirm their numerical simulations and conjectures by Hilker et al., and reveal some new bifurcation phenomena which are not observed in Hilker et al. (Am Nat 173:72-88, 2009). The rich and complicated dynamics exhibit that the model is very sensitive to parameter perturbations, which has important implications for disease control of endangered species.

Dynamics of SI epidemic with a demographic Allee effect

“…This is because it is algebraically impossible to find explicit necessary and sufficient conditions for the possible number of endemic equilibria of system (4) depending on all the values of the model parameters (see the comment in Cai et al (2013)). According to the first case of Theorem 3, it is possible for model (4) to have a unique endemic equilibrium.…”

“…Detailed analysis of this case can be found in (Cai et al, 2013;Friedman and Yakubu, 2012a;Hilker et al, 2009). …”

“…They also showed that additional deaths due to the disease infections increase the Allee threshold of the host population. Cai et al (2013) used the same SI model of Hilker et al (2009) to analytically study bifurcations and dynamical behaviors of the model. These researchers found that their qualitative conclusions support the numerical bifurcation analysis and conjunctures reported in Hilker et al (2009).…”

“…In addition, some new bifurcations phenomena are explored such as pitchfork bifurcation, Bogdanov-Takens (BT) bifurcation of codimension two, degenerate Hopf bifurcation and degenerate BT bifurcation of codimension three in elliptic case Cai et al (2013). These bifurcations exhibit complicated dynamical behaviors of the model in Cai et al (2013) such as multiple attractors, homoclinic loop, and limit cycles, whose respective biological consequences are disease persistence, host population extinction and disease cycles.…”

Codimension two and three bifurcations of a predator–prey system with group defense and prey refuge

“…Notice that the coefficients of the terms x 2 and xy in system (10) are not zero if β = 2 √ 2/K, hence, the equilibrium (0, 0) of system (10) is a cusp of codimension 2, as used in [1,15].…”