Remarks on basic reproduction ratios for periodic abstract functional differential equations

Tianhui Yang; Lei Zhang

doi:10.3934/dcdsb.2019166

Article Contents

2019, Volume 24, Issue 12: 6771-6782. Doi: 10.3934/dcdsb.2019166

This issue Previous Article Semidefinite approximations of invariant measures for polynomial systems Next Article Bifurcation analysis of an enzyme-catalyzed reaction system with branched sink

Remarks on basic reproduction ratios for periodic abstract functional differential equations

Tianhui Yang^1, and
Lei Zhang^2, ,

1.
School of Mathematical Sciences, University of Science and Technology of China, Hefei, Anhui 230026, China
2.
Department of Mathematics, Harbin Institute of Technology in Weihai, Weihai, Shandong 264209, China

^* Corresponding author
^* Corresponding author

Received: September 2018

Revised: March 2019

Early access: July 2019

Published: September 2019

Our research were supported by National Natural Science Foundation of China (11571334 and 11801232) and Natural Science Foundation of Shandong Province (ZR2019QA006)

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Abstract

In this paper, we extend the theory of basic reproduction ratios $ \mathcal{R}_0 $ in [Liang, Zhang, Zhao, JDDE], which concerns with abstract functional differential systems in a time-periodic environment. We prove the threshold dynamics, that is, the sign of $ \mathcal{R}_0-1 $ determines the dynamics of the associated linear system. We also propose a direct and efficient numerical method to calculate $ \mathcal{R}_0 $.

Keywords:

Basic reproduction ratio,
abstract functional differential system,
time-periodic,
threshold dynamics,
numerical method.

Mathematics Subject Classification: Primary: 34K20, 35K57; Secondary: 37B55.

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Figure 1. Comparison of results using two methods by ODEs

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Figure 2. Comparison of results using two methods by Reaction-Diffusion systems

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Figure 3. Comparison of results using two methods by DDEs

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Figure 4. Comparison of results using two methods by Reaction-Diffusion systems with time-delay

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Table 1. Mean values and relative errors under different partitions

m	Mean numerical value	Relative error(%)
500	1.7599	0.5681
1000	1.7550	0.2838
2000	1.7525	0.1419
8000	1.7506	0.0351

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