Reproduction Number Versus Turnover Number in Structured Discrete-Time Population Models

Abstract

The analysis of the discrete-time dynamics of structured iteroparous populations involves a basic yearly turnover operator $B = A + H$ with a structural transition operator A and a mating and fertility operator H. A and H map a normal complete cone $X_+$ of an ordered normed vector space X into itself and are (positively) homogenous and continuous on $X_+$, A is additive and H is order-preserving. Assume that $\textbf{r}(A) < 1$ for the spectral radius of A. Let $H R_1$ with $R_1 = \sum _{j=0}^\infty A^j$ be the next generation operator and ${\mathcal T}= \textbf{r}(B)$, the spectral radius of B, be the (basic) turnover number and ${\mathcal R}= \textbf{r}( H R_1)$ be the (basic) reproduction number. We explore conditions for a turnover/reproduction trichotomy, namely one (and only one) of the following three possibilities to hold:       (i) $1 < {\mathcal T}\le {\mathcal R}$,    (ii) $1 = {\mathcal T}= {\mathcal R}$,    (iii) $1 > {\mathcal T}\ge {\mathcal R}$. In some cases, one may also like to consider the lower reproduction number ${\mathcal R}_\diamond = \lim _{\lambda \rightarrow 1+} \textbf{r}(H R_\lambda )$, $R_\lambda = \sum _{j=0}^\infty \lambda ^{-(n+1)} A^n$. ${\mathcal R}_\diamond$ is also useful to study the case $\textbf{r}(A) =1$ to explore conditions for the dichotomy       $1 = {\mathcal T}\ge {\mathcal R}_\diamond$    or    $1 < {\mathcal T}\le {\mathcal R}_\diamond \le \infty$.

Dedicated to Odo Diekmann on the occasion of his 75th birthday