For the delayed logistic equation x n+1 = ax n (a − x n−1 ) it is well known that the nontrivial fixed point is locally stable for 1 < a ≤ 2, and unstable for a > 2. We prove that for 1 < a ≤ 2 the fixed point is globally stable, in the sense that it is locally stable and attracts all points of S, where S contains those (x 0 , x 1 ) ∈ R 2 + , for which the sequence {x n } ⊂ R + . The proof is a combination of analytical and reliable numerical methods.

We obtain radially symmetric solutions of some nonlinear (geometric) partial differential equations via a rigorous computer-assisted method. We introduce all main ideas through examples, accessible to non-experts. The proofs are obtained by solving for the coefficients of the Taylor series of the solutions in a Banach space of geometrically decaying sequences. The tool that allows us to advance from numerical simulations to mathematical proofs is the Banach contraction theorem.

Vaccines against varicella-zoster virus (VZV) are under introduction in Hungary into the routine vaccination schedule, hence it is important to understand the current transmission dynamics and to estimate the key parameters of the disease. Mathematical models can be greatly useful in advising public health policy decision making by comparing predictions for different scenarios. First we consider a simple compartmental model that includes key features of VZV such as latency and reactivation of the virus as zoster, and exogeneous boosting of immunity. After deriving the basic reproduction number R 0 , the model is analysed mathematically and the threshold dynamics is proven: if R 0 ≤ 1 then the virus will be eradicated, while if R 0 > 1 then an endemic equilibrium exists and the virus uniformly persists in the population. Then we extend the model to include seasonality, and fit it to monthly incidence data from Hungary. It is shown that besides the seasonality, the disease dynamics has intrinsic multi-annual periodicity. We also investigate the sensitivity of the model outputs to the system parameters and the underreporting ratio, and provide estimates for R 0 . MSC: 92D30

For the delayed logistic equation x n+1 = ax n (1 − x n−2 ) it is well known that the nontrivial fixed point is locally stable for 1 < a ≤ √ 5 + 1 /2, and unstable for a > √ 5 + 1 /2. We prove that for 1 < a ≤ √ 5 + 1 /2 the fixed point is globally stable, in the sense that it is locally stable and attracts all points of S, where S contains thoseThe proof is a combination of analytical and reliable numerical methods. The novelty of this article is an explicit construction of a relatively large attracting neighborhood of the nontrivial fixed point of the 3-dimensional logistic map by using center manifold techniques and the Neimark-Sacker bifurcational normal form.

No abstract