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-Chavez2–Reference 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.