The SEIR model with nonlinear incidence rates in epidemiology is studied. Global stability of the endemic equilibrium is proved using a general criterion for the orbital stability of periodic orbits associated with higher-dimensional nonlinear autonomous systems as well as the theory of competitive systems of differential equations.
In this paper we establish some new oscillation criteria for the third-order nonlinear dynamic equation (c(t) (a(t)x ∆ (t)) ∆ γ) ∆ + f (t, x(t)) = 0, t ∈ [a, ∞) T on time scales, where γ ≥ 1 is a quotient of odd integers. Our results not only unify the oscillation theory for third-order nonlinear differential and difference equations but also are new for… (More)
For the Nicholson's blowflies equation with a distributed delay ˙ N (t) = −δN (t) + p Z t h(t) N (s)e −aN (s) dsR(t, s), t ≥ 0, we obtain existence, positiveness and permanence results for solutions with positive initial conditions. We prove that all nonoscillatory about the positive equilibrium N * solutions tend to N *. In the case δ < p < δe there are no… (More)
Host community composition and biodiversity can limit and regulate tick abundance which can have profound impacts on the incidence and severity of tick-borne diseases. Our understanding of the relationship between host community composition and tick abundance is still very limited. Here, we present a novel mathematical model of a stage-structured tick… (More)