On the definition and the computation of the type-reproduction number T for structured populations in heterogeneous environments

In the context of mathematical epidemiology, the type-reproduction number (TRN) for a specific host type is interpreted as the average number of secondary cases of that type produced by the primary cases of the same host type during the entire course of infection. Here, it must be noted that T takes into account not only the secondary cases directly transmitted from the specific host but also the cases indirectly transmitted by way of other types, who were infected from the primary cases of the specific host with no intermediate cases of the target host. Roberts and Heesterbeek (Proc R Soc Lond B 270:1359–1364, 2003) have shown that T is a useful measure when a particular single host type is targeted in the disease control effort in a community with various types of host, based on the fact that the sign relation sign(R ₀ ? 1) = sign(T ? 1) holds between the basic reproduction number R ₀ and T. In fact, T can be seen as an extension of R ₀ in a sense that the threshold condition of the total population growth can be formulated by the reproduction process of the target type only. However, the original formulation is limited to populations with discrete state space in constant environments. In this paper, based on a new perspective of R ₀ in heterogeneous environments (Inaba in J Math Biol 2011), we give a general definition of the TRN for continuously structured populations in heterogeneous environments and show some examples of its computation and applications.