Global Stability of a Mumps Transmission Model with Quarantine Measure
Yu-Zhen Bai1, Xiao-Jing Wang1, Song-Bai Guo1,2
1School of Science, Beijing University of Civil Engineering and Architecture, Beijing, 102616 China.
Abstract:
In this paper, a model of mumps transmission with quarantine measure is proposed and then the control reproduction number of the model is obtained. This model admits a unique endemic equilibrium P* if and only if R > 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 ≤ 1; second, we prove that the equilibrium P* is GAS if R > 1. Our results reveal that mumps can be eliminated from the community for and it will be persistent for , and quarantine measure can also effectively control the mumps transmission.
Insights
This study models mumps transmission, finding that a control reproduction number (R0) below 1 eliminates the disease. Quarantine measures effectively control mumps spread when R0 is greater than 1.
Area of Science:
- Epidemiology
- Mathematical Biology
- Public Health
Background:
- Mumps transmission dynamics require understanding for effective control.
- Quarantine is a key public health intervention for infectious diseases.
Purpose of the Study:
- To develop and analyze a mathematical model for mumps transmission incorporating quarantine measures.
- To determine the conditions for mumps elimination and persistence based on the control reproduction number (R0).
Main Methods:
- Mathematical modeling of disease transmission.
- Analysis of model equilibria (disease-free and endemic).
- Application of Lyapunov functions and the Lyapunov-LaSalle theorem for stability analysis.
Main Results:
- The model demonstrates a unique endemic equilibrium exists if R0 > 1.
- The disease-free equilibrium is globally asymptotically stable (GAS) when R0 ≤ 1.
- The endemic equilibrium is GAS when R0 > 1, indicating disease persistence.
- Quarantine measures are shown to be effective in controlling mumps transmission.
Conclusions:
- Mumps can be eliminated from a community when R0 ≤ 1.
- Mumps will persist in the community when R0 > 1.
- Quarantine interventions play a significant role in managing and reducing mumps transmission dynamics.
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