Global Stability of a Mumps Transmission Model with Quarantine Measure

Abstract

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 ${ℛ}_c\le 1$ and it will be persistent for ${ℛ}_c>1$ , and quarantine measure can also effectively control the mumps transmission.

Keywords: control reproduction number; global stability; mumps transmission model; quarantine measure.