ABSTRACT
Using the technique of edge-based compartmental modelling (EBCM) for the spread of susceptible-infected-recovered (SIR) diseases in networks, in a recent paper (PloS One, 8(2013), e69162), Miller and Volz established an SIR disease network model with heterogeneous infectiousness and susceptibility. The authors provided a numerical example to demonstrate its validity but they did not perform any mathematical analysis of the model. In this paper, we resolve this problem. Using the nature of irreducible cooperative system in the theory of monotonic dynamical system, we prove that the dynamics of the model are completely determined by a critical value ρ0: When ρ0 > 0, the disease persists in a globally stable outbreak equilibrium; while when ρ0 < 0, the disease dies out in the population and the disease free equilibrium is globally stable.
Subject(s)
Communicable Diseases , Epidemics , Communicable Diseases/epidemiology , Disease Outbreaks , Disease Susceptibility , Humans , Models, BiologicalABSTRACT
Huang et al. [1] recently developed a toxin-dependent predator-prey model and analyzed its global dynamics. Their results showed that environmental toxins may influence both predators and prey and induce bistable situation, and intermediate toxin concentrations may affect predators disproportionately through biomagnification. Environmental noises can change the dynamical behaviors of the toxin-based predator-prey model. In this paper, by formulating a stochastically forced predator-prey model with environmental toxins, we study the dynamical phenomenon of noise-induced transitions from coexistence to prey-only extirpation in the bistable zone. Numerical simulations based on the technique of stochastic sensitivity functions are provided for constructing the confidence ellipse and estimating the threshold value of the noise intensity of state switching. Meanwhile, we construct the confidence band and study the configurational arrangement of the stochastic cycle.
