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’

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’ [1]. 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 [1]. 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 [2–9]. With this approach, we re-analysed the ‘SEIAR’ model in [1], and found the effective reproduction number, denoted by r _eff in this letter to distinguish from R _eff in [1], 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. [1]. We found that r _eff was different from that in Huang et al. [1]. 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 [1].

(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 [1]. This difference implied that the HFMD-induced fatality rate (f) was neglected in Huang et al. [1]. Furthermore, according to Table 1 in [1], 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 [1] 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.