Basic Reproduction Ratios for Periodic Abstract Functional Differential Equations (with Application to a Spatial Model for Lyme Disease)

• Published in: Journal of dynamics and differential equations Vol. 31; no. 3; pp. 1247 - 1278
• Main Authors: Liang, Xing, Zhang, Lei, Zhao, Xiao-Qiang
• Format: Journal Article
• Language: English
• Published: New York Springer US 01-09-2019; Springer Nature B.V
• Subjects: Applications of Mathematics; Differential equations; Lyme disease; Mathematics; Mathematics and Statistics; Ordinary Differential Equations; Partial Differential Equations; Threshold dynamics; Lyme disease; 92D30; Abstract functional differential system; 37B55; Periodic solution; 35K57; Basic reproduction ratio; 34K20
• ISSN: 1040-7294; 1572-9222
• DOI: 10.1007/s10884-017-9601-7

In this paper, we develop the theory of basic reproduction ratios
R
0
for abstract functional differential systems in a time-periodic environment. It is proved that
R
0
-
1
has the same sign as the exponential growth bound of an associated linear system. Then we apply it to a time-periodic Lyme disease model with time-delay and obtain a threshold type result on its global dynamics in terms of
R
0
.