Epidemiological models with density-dependent demographics

Abstract

The Allee e ect is characterized by a positive relationship between population density or size and the per capita population growth rate in small populations. There are several mechanisms responsible for the Allee e ects. In light of these Allee mechanisms, we modeled both the birth and the death rates as density-dependent quadratic polynomials. This approach provides an ample opportunity for taking into account the major contributors to the Allee e ects and e ectively captures species' susceptibility variation due to the Allee e ect. We design an SI model with these demographic functions and show that the host and/or disease persistence and extinction are characterized by threshold values of the disease related parameters ( and ). For special cases of the model, veri able conditions for host population persistence (with or without infected individuals) and host extinction are derived. Numerical simulations indicate the e ects of the parameter on the host population persistence and extinction regions. Furthermore, an SEI model with frequency-dependent incidence and the same quadratic demographics is presented. This is aimed at investigating the combined impact of infectious disease and the Allee e ect at higher population levels. Indeed, the model suggests that the eventual outcome could be an inevitable population crash to extinction. The tipping point marking the unanticipated population collapse at high population level is mathematically associated with a saddle-node bifurcation. The essential mechanism of this scenario is the simultaneous population size depression and the increase of the extinction threshold owing to disease virulence and the Allee eff ect. Finally, the role of repeated exposure to mycobacteria on the transmission dynamics of bovine tuberculosis is addressed. Such exposure is found to induce the phenomenon of backward bifurcation. Two scenarios when such phenomenon does not arise are highlighted and for each case the model is proved to have a globally asymptotically stable equilibrium. The impact of vaccine is assessed via a threshold analysis approach.