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Attach importance to the procedure of deriving reproduction numbers from compartmental models: Letter to the editor in response to ‘Seasonality of the transmissibility of hand, foot and mouth disease: a modelling study in Xiamen City, China

Published online by Cambridge University Press:  02 March 2020

Shi Zhao*
Affiliation:
JC School of Public Health and Primary Care, Chinese University of Hong Kong, Hong Kong, China Shenzhen Research Institute of Chinese University of Hong Kong, Shenzhen, China
Jinjun Ran
Affiliation:
School of Public Health, Li Ka Shing Faculty of Medicine, University of Hong Kong, Hong Kong, China
Guangpu Yang
Affiliation:
Department of Orthopaedics and Traumatology, Chinese University of Hong Kong, Hong Kong, China SH Ho Scoliosis Research Lab, Joint Scoliosis Research Center of Chinese University of Hong Kong and Nanjing University, Hong Kong, China
Peihua Cao
Affiliation:
Clinical Research Centre, Zhujiang Hospital, Southern Medical University, Guangzhou, China
*
Author for correspondence: Shi Zhao, E-mail: zhaoshi.cmsa@gmail.com
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Abstract

Type
Letter to the Editor
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Copyright
Copyright © The Author(s), 2020. Published by Cambridge University Press

To the Editor

We read with interests Huang et al.'s article ‘Seasonality of the transmissibility of hand, foot and mouth disease: a modelling study in Xiamen City, China’ [Reference Huang1]. Huang et al. modelled the long-term transmission dynamics of hand, foot and mouth disease (HFMD) in Xiamen, China. However, we suspected that there was an inconsistency between the model formulation and the effective reproduction number, R eff, in [Reference Huang1]. As such, we highlighted the importance of following the next generation matrix (NGM) analytic procedure to find reproduction numbers in compartmental models.

The NGM approach is widely adopted to derive the reproduction numbers in modelling infectious disease epidemiology [Reference Brauer and Castillo-Chavez2Reference Van den Driessche and Watmough9]. With this approach, we re-analysed the ‘SEIAR’ model in [Reference Huang1], and found the effective reproduction number, denoted by r eff in this letter to distinguish from R eff in [Reference Huang1], in Eqn (1).

(1)$$r_{{\rm eff}} = \beta S\cdot \displaystyle{\omega \over {\omega + dr}}\cdot \left[{\displaystyle{\,p \over {\gamma + dr + f}} + \displaystyle{{\lpar {1-p} \rpar k} \over {{\gamma \,}^{\prime} + dr}}} \right].$$

Here, all notations have the same definitions as in Huang et al. [Reference Huang1]. We found that r eff was different from that in Huang et al. [Reference Huang1]. For a sufficiently small value of the human natural death rate (dr), we have an approximated version of r eff as in Eqn (2), and it has an analytic form closer to the R eff in [Reference Huang1].

(2)$$\mathop {\lim }\limits_{dr\to 0^ + } r_{{\rm eff}} = \beta S\cdot \left[{\displaystyle{\,p \over {\gamma + f}} + \displaystyle{{\lpar {1-p} \rpar k} \over {{\gamma \,}^{\prime}}}} \right].$$

It was notable that the denominator of the first fractional term in the brackets, i.e. [⋅], has an additional term ‘f’ in Eqn (2), compared to the R eff in [Reference Huang1]. This difference implied that the HFMD-induced fatality rate (f) was neglected in Huang et al. [Reference Huang1]. Furthermore, according to Table 1 in [Reference Huang1], the value of f was set to be (0.03% =) 0.0003 that appears relatively small compared to the main ‘removing’ rates (γ and γ′) of HFMD infections. This setting suggested that the R eff in [Reference Huang1] could be treated as a reasonable simplification from its theoretical version. Hence, this inconsistency between the model formulation and the reproduction number was unlikely to affect the main conclusions.

In conclusion, we call for caution in deriving and simplifying reproduction numbers from compartmental models.

Data

No data or material was used in this work.

Acknowledgements

The authors would like to acknowledge anonymous colleagues for helpful comments, and Ying Huo for proofreading.

Authors’ contributions

All authors conceived the study, carried out the analysis, discussed the results, drafted the first manuscript, critically read and revised the manuscript, and gave final approval for publication.

Financial support

PC was supported by the National Natural Science Foundation of China (Grant Number: 81903406) and the China Postdoctoral Science Foundation (Grant Number: 2017M620378).

Conflict of interest

The authors declared no competing interests.

Ethical standards

The ethical approval or individual consent was not applicable.

Consent for publication

Not applicable.

Disclaimer

The funding agencies had no role in the design and conduct of the study; collection, management, analysis and interpretation of the data; preparation, review or approval of the manuscript; or decision to submit the manuscript for publication.

References

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