# American Institute of Mathematical Sciences

December  2019, 24(12): 6771-6782. doi: 10.3934/dcdsb.2019166

## Remarks on basic reproduction ratios for periodic abstract functional differential equations

 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

Received  September 2018 Revised  March 2019 Published  July 2019

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

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 $\mathcal{R}_0$.

Citation: Tianhui Yang, Lei Zhang. Remarks on basic reproduction ratios for periodic abstract functional differential equations. Discrete & Continuous Dynamical Systems - B, 2019, 24 (12) : 6771-6782. doi: 10.3934/dcdsb.2019166
##### References:

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##### References:
Comparison of results using two methods by ODEs
Comparison of results using two methods by Reaction-Diffusion systems
Comparison of results using two methods by DDEs
Comparison of results using two methods by Reaction-Diffusion systems with time-delay
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
 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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