Basic Reproduction Ratios for Periodic Compartmental Models with Time Delay

• Published in: Journal of dynamics and differential equations Vol. 29; no. 1; pp. 67 - 82
• Main Author: Zhao, Xiao-Qiang
• Format: Journal Article
• Language: English
• Published: New York Springer US 01-03-2017; Springer Nature B.V
• Subjects: Applications of Mathematics; Environment models; Linear systems; Mathematics; Mathematics and Statistics; Ordinary Differential Equations; Partial Differential Equations; Time lag; Threshold dynamics; Compartmental models; 92D30; 37B55; Periodicity; Time delay; Reproduction ratio; 34K20
• ISSN: 1040-7294; 1572-9222
• DOI: 10.1007/s10884-015-9425-2

In this paper, we establish the theory of basic reproduction ratio
R
0
for a large class of time-delayed compartmental population models in a periodic environment. It is proved that
R
0
serves as a threshold value for the stability of the zero solution of the associated periodic linear systems. As an illustrative example, we also apply the developed theory to a periodic SEIR model with an incubation period and obtain a threshold result on its global dynamics in terms of
R
0
.