Dynamics of SI epidemic with a demographic Allee effect

Abstract

In this paper, we present an extended SI model of Hilker et al. (2009). In the presented model the birth rate and the death rate are both modeled as quadratic polynomials. This approach provides ample opportunity for taking into account the major contributors to an Allee effect and effectively captures species' differential susceptibility to the Allee effects. It is shown that, the behaviors (persistence or extinction) of the model solutions are characterized by the two essential threshold parameters λ 0 and λ 1 of the transmissibility λ and a threshold quantity µ * of the disease pathogenicity µ. If λ < λ 0 , the model is bistable and a disease cannot invade from arbitrarily small introductions into the host population at the carrying capacity, while it persists when λ > λ 0 and µ < µ * . When λ > λ 1 and µ > µ * , the disease derives the host population to extinction with origin as the only global attractor. For the special cases of the model, verifiable conditions for host population persistence (with or without infected individuals) and host extinction are derived. Interestingly, we show that if the values of the parameters α and β of the extended model are restricted, then the two models are similar. Numerical simulations show how the parameter β affects the dynamics of the model with respect to the host population persistence and extinction.

Pattern Dynamics of Nonlocal Delay SI Epidemic Model with the Growth of the Susceptible following Logistic Mode

“…e nonspatial system (6) gives rise to a Hopf bifurcation at endemic steady state E * 1 if only and if the condition…”

“…It is obvious that a 2 (0) > 0 and a 0 (0) > 0 due to the existence of endemic equilibrium E * 1 (i.e., (H1)). According to the theorem of [38], when conditions a 2 (0) > 0, a 0 (0) > 0, and a 2 (0)a 1 (0) − a 0 (0) � 0 hold, a Hopf bifurcation occurs for nonspatial system (6).…”

“…Next, we find the conditions under which system (4) generates Turing instability near endemic steady state E * 1 : E * 1 is locally asymptotic stability for system (6), but E * 1 loses stability for system (6) with diffusion (i.e., system (4)). Furthermore, we already know a 2 (k) > 0 for any k, while the signs of a 0 (k) and a 2 (k)a 1 (k) − a 0 (k) are uncertain.…”

“…Research on infectious disease models can be traced back to the pioneering work of Kermack and Mckendrick [1,2], and the traditional model is usually described by ordinary differential equations as a result of the spatially homogeneous assumption [3][4][5][6][7][8][9][10]. Considering the instantaneous change, researchers have established epidemic models via impulsive differential equations [11][12][13][14][15].…”