Global Stability of a Mumps Transmission Model with Quarantine Measure.

In this paper, a model of mumps transmission with quarantine measure is proposed and then the control reproduction number ℛ c of the model is obtained. This model admits a unique endemic equilibrium P* if and only if R c > 1, while the disease-free equilibrium P 0 always exists. By using the technique of constructing Lyapunov functions and the generalized Lyapunov-LaSalle theorem, we first show that the equilibrium P 0 is globally asymptotically stable (GAS) if R c ≤ 1; second, we prove that the equilibrium P* is GAS if R c > 1. Our results reveal that mumps can be eliminated from the community for ℛ c ≤ 1 and it will be persistent for ℛ c > 1 , and quarantine measure can also effectively control the mumps transmission.

Periodic solution of a stage-structured predator-prey model incorporating prey refuge.

In this paper, we consider a delayed stage-structured predator-prey model incorporating prey refuge with Holling type II functional response. It is assumed that prey can live in two different regions. One is the prey refuge and the other is the predatory region. Moreover, in real world application, we should consider the stage-structured model. It is assumed that the prey in the predatory region can divided by two stages: Mature predators and immature predators, and the immature predators have no ability to attack prey. Based on Mawhin's coincidence degree and novel estimation techniques for a priori bounds of unknown solutions to Lu = λNu, some sufficient conditions for the existence of periodic solution is obtained. Finally, an example demonstrate the validity of our main results.