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Basic reproduction ratios for periodic and time-delayed compartmental models with impulses
Zhenguo Bai1, Xiao-Qiang Zhao2
1School of Mathematics and Statistics, Xidian University, Xi'an, 710071, China. zgbai@xidian.edu.cn.
Journal of Mathematical Biology
|November 27, 2019
Summary
This study develops the theory of the basic reproduction number for impulsive, time-delayed models. Findings suggest the basic reproduction number may overestimate virus spread risk in averaged delayed impulsive systems.
Area of Science:
- Mathematical epidemiology
- Dynamical systems theory
- Impulsive differential equations
Background:
- The basic reproduction number (R0) is crucial for analyzing infectious disease dynamics.
- Existing R0 theory is limited for periodic and time-delayed impulsive models.
- Impulsive models are relevant for interventions like vaccination or treatment.
Purpose of the Study:
- To extend the theory of the basic reproduction number (R0) to impulsive, time-delayed models.
- To establish R0 as a threshold parameter for the stability of associated linear systems.
- To analyze the global dynamics of a time-delayed computer virus model with impulse treatment.
Main Methods:
- Development of R0 theory for a class of impulsive differential equations.
- Analysis of the stability of the zero solution for a linearized impulsive system.
- Application of the developed R0 theory to a specific computer virus model with time delays and impulse control.
Main Results:
- Introduced a novel definition and framework for R0 in impulsive, time-delayed systems.
- Demonstrated that the new R0 is a critical threshold parameter for system stability.
- Derived a threshold criterion for the global dynamics of the computer virus model based on R0.
Conclusions:
- The developed R0 theory provides a robust tool for analyzing complex epidemiological models.
- The basic reproduction number for time-averaged delayed impulsive systems may overestimate virus spread risk.
- Further research is needed to refine R0 calculations for time-delayed, impulsive systems.
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