Necessary and sufficient conditions for $R_{0}$ to be a sum of contributions of fertility loops

Abstract

Recently, de-Camino-Beck and Lewis (Bull Math Biol 69:1341–1354, 2007) have presented a method that under certain restricted conditions allows computing the basic reproduction ratio $R_0$ in a simple manner from life cycle graphs, without, however, giving an explicit indication of these conditions. In this paper, we give various sets of sufficient and generically necessary conditions. To this end, we develop a fully algebraic counterpart of their graph-reduction method which we actually found more useful in concrete applications. Both methods, if they work, give a simple algebraic formula that can be interpreted as the sum of contributions of all fertility loops. This formula can be used in e.g. pest control and conservation biology, where it can complement sensitivity and elasticity analyses. The simplest of the necessary and sufficient conditions is that, for irreducible projection matrices, all paths from birth to reproduction have to pass through a common state. This state may be visible in the state representation for the chosen sampling time, but the passing may also occur in between sampling times, like a seed stage in the case of sampling just before flowering. Note that there may be more than one birth state, like when plants in their first year can already have different sizes at the sampling time. Also the common state may occur only later in life. However, in all cases $R_0$ allows a simple interpretation as the expected number of new individuals that in the next generation enter the common state deriving from a single individual in this state. We end with pointing to some alternative algebraically simple quantities with properties similar to those of $R_{0}$ that may sometimes be used to good effect in cases where no simple formula for $R_{0}$ exists.

Appendices

Appendix A: Derivation of Eqs. (4–6)

The basic ideas of loop formulas go back to Mason (1956). Often even formulas similar to ours are attributed to this paper (e.g. Caswell 2001, p. 181). However, Mason only describes the idea of loop based calculation procedures in an electrical engineering context without giving results that are directly applicable to the population dynamical situation. The formula in Caswell (2001) also does not cover our specific case. Hence, we below give a full derivation of Eq. (4). Thus, we derive an expression for the characteristic equation of the next-generation matrix

$$\begin{aligned} 0=\det (z\mathsf{{I}}-\mathsf{F}(\mathsf{{I}}-\mathsf{S})^{-1}) \end{aligned}$$

(17)

in terms of loops in the life cycle. The largest number $z$ satisfying Eq. (17) is known as the basic reproduction ratio $R_{0}$. Multiplying both sides with $\det (\mathsf{{I}}-\mathsf{S})$ and using that for any two $n\times n$ matrices $\mathsf A$ and $\mathsf B$ holds $\det \mathsf{{A}}\det \mathsf{B}=\det (\mathsf{AB})$ we get

$$\begin{aligned} 0=\det (z(\mathsf{{I}}-\mathsf{S})-\mathsf{F})=\det (z\mathsf{{I}}-(z\mathsf{S}+\mathsf{F})). \end{aligned}$$

(18)

We continue by considering the case of an arbitrary nonnegative $n\times n$-matrix $\mathsf M$. Let $\alpha \subseteq \{1,\dots ,n\}$ be an index set and let $\mathcal P (\alpha )$ denote the power set of $\alpha$ and $\mathcal P _{p}(\alpha )$ the set of all subsets of $\alpha$ containing exactly $p$ elements. With $\mathsf{M}_{\alpha }$ we denote the sub-matrix of $\mathsf M$ with indices in $\alpha$. Then

$$\begin{aligned} \det (z\mathsf{{I}}-\mathsf{M})&= z^n-\sum _{\alpha \in \mathcal P _1}\det \mathsf{M}_{\alpha }z^{n-1}+\sum _{\alpha \in \mathcal P _2}\det \mathsf{M}_{\alpha }z^{n-2}\\&\quad -\cdots +(-1)^{n-1}\sum _{\alpha \in \mathcal P _{n-1}}\det \mathsf{M}_{\alpha }z+(-1)^n\det \mathsf{M} \end{aligned}$$

(e.g. Horn and Johnson 1985, p. 42), where $\mathcal P _k$ stands for $\mathcal P _k(\{1,\ldots ,n\})$. By multiplying both sides with $z^{-n}$ it is clear the characteristic equation can be written as

$$\begin{aligned} 1&= \sum _{\alpha \in \mathcal P _{1}}\det \mathsf{M}_{\alpha }z^{-1}-\sum _{\alpha \in \mathcal P _{2}}\det \mathsf{M}_{\alpha }z^{-2}\nonumber \\&\quad +\cdots +(-1)^{n-2}\sum _{\alpha \in \mathcal P _{n-1}}\det \mathsf{M}_{\alpha }z^{-(n-1)}+(-1)^{n-1}\det \mathsf{M}z^{-n}. \end{aligned}$$

(19)

Starting point of our further considerations is the Leibniz formula of the determinant of a $n\times n$ matrix $\mathsf M,$ $\mathrm{det} \mathsf{M}=\sum _{\sigma \in S_n}\mathrm{sgn}[\sigma ]\prod _{i=1}^{n}m_{i\sigma (i)}$, where $\sigma$ is a permutation of the set $\{1,\ldots ,n\}$ and $S_{n}$ the set of all permutations. The sign of a permutation is defined as $\text{ sgn}[\sigma ]=(-1)^{N(\sigma )}$, where $N(\sigma )$ is number of inversions in $\sigma$.

Recall from elementary group theory that any finite permutation can be decomposed into disjoint permutation cycles where a permutation cycle of length $k$ is defined as a permutation $\sigma$ of a set $\{a_{1},\dots ,a_{k}\}$ such that $\sigma (a_{i})=a_{i+1}$ and $\sigma (a_{k})=a_{1}$. The sign of a permutation cycle is given by the number of inversions. Thus, for a permutation cycle of length $k$ we have $\text{ sgn}[\sigma ]=k-1$. Let $c_{1},\ldots ,c_{j}$ denote the permutation cycles of a permutation $\sigma$ of length $n$ in cycle notation, $\sigma =c_{1}\cdots c_{j}, d(\sigma )$ the number of disjoint permutation cycles in a permutation $\sigma$ and $|c_{i}|$ the length of a permutation cycle. Then $\text{ sgn}[\sigma ]=\mathrm{sgn}[c_{1}]\cdots \text{ sgn}[c_{j}]=(-1)^{((|c_{1}|-1)+\cdots +(|c_{j}|-1))}=(-1)^{(|c_{1}|+\cdots +|c_{j}|-j)}=(-1)^{n-j}=(-1)^{n-d(\sigma )}$.

Thus, $\text{ sgn}[\sigma ]=(-1)^{n-d(\sigma )}$. The set of permutations $S_{|\alpha |}$ can be decomposed into a partition with elements $C_{j}:=\{\sigma \in S_{|\alpha |}:d(\sigma )=j\}$ and $j\in \{1,\ldots ,|\alpha |\}$. Thus, for each $i$ the set $C_{i}$ contains the permutations that consist of exactly $j$ permutation cycles. Hence, the determinant can be rewritten as

$$\begin{aligned} \det \mathsf{M}&= \sum _{j=1}^{n}\sum _{\sigma \in C_{j}}(-1)^{n-j}\prod _{i=1}^{n}m_{i\sigma (i)}\nonumber \\&= \sum _{\sigma \in C_{1}}(-1)^{n-1}\prod _{i=1}^{n}m_{i\sigma (i)}+\sum _{\sigma \in C_{2}}(-1)^{n-2}\prod _{i=1}^{n}m_{i\sigma (i)}+\cdots +\sum _{\sigma \in C_{n}}\prod _{i=1}^{n}m_{i\sigma (i)}\nonumber \\ \end{aligned}$$

(20)

Let $l\in \{1,\dots ,n\}$ and $c=(l\,\sigma (l)\,\sigma ^{2}(l)\cdots \sigma ^{|c|-1}(l))$ be a permutation cycle in cycle notation. Then $L=m_{l\,\sigma ^{|c|-1}(l)}m_{\sigma ^{|c|-1}(l)\,\sigma ^{|c|-2}(l)}\cdots m_{\sigma (l)\,l}$ is a loop of length $|\breve{L}|=|c|$. Thus, a permutation $\sigma \in C_{j}$ corresponds to the product of $j$ pairwise unconnected loops $L_{1}\cdots L_{j}$. From this follows that we can rearrange Eq. (19) in the following way:

$$\begin{aligned} 1&= \sum _{\alpha \in \mathcal P _{1}}\det \mathsf{M}_{\alpha }z^{-1}-\sum _{\alpha \in \mathcal P _{2}}\det \mathsf{M}_{\alpha }z^{-2}+\cdots +(-1)\sum _{\alpha \in \mathcal P _{n-1}}\det \mathsf{M}_{\alpha }z^{-(n-1)} \nonumber \\&\quad +(-1)^{n-1}\det \mathsf{M}z^{-n}\nonumber \\&= \sum _{\{L\in \mathcal{L }_\mathsf{M}:|\breve{L}|=1\}}\hat{L}+\sum _{\{L\in \mathcal{L }_\mathsf{M}:|\breve{L}|=2\}}\hat{L}-\sum _{\{(L,M)\in \mathcal{L }_\mathsf{M}^{2*}:|\breve{L}|+|\breve{M}|=2\}}\hat{L}\hat{M}\nonumber \\&\quad +\sum _{\{L\in \mathcal{L }_\mathsf{M}:|\breve{L}|=3\}}\hat{L}-\sum _{\{(L,M)\in \mathcal{L }_\mathsf{M}^{2*}:|\breve{L}|+|\breve{M}|=3\}}\hat{L}\hat{M}\nonumber \\&\quad +\sum _{\{(L,M,N)\in \mathcal{L }_\mathsf{M}^{3*}:|\breve{L}|+|\breve{M}|+|\breve{N}|=3\}}\hat{L}\hat{M}\hat{N}\nonumber \\&\quad \vdots \nonumber \\&\quad +\sum _{\{L\in \mathcal{L }_\mathsf{M}:|\breve{L}|=n\}}\hat{L}-\sum _{\{(L,M)\in \mathcal{L }_\mathsf{M}^{2*}:|\breve{L}|+|\breve{M}|=n\}}\hat{L}\hat{M}+\cdots +(-1)^{n-1}\nonumber \\&\quad \times \sum _{\{(L_{i},\ldots ,L_{n})\in \mathcal{L }_\mathsf{M}^{n*}:\sum _{i=1}^{n}|\breve{L}_{i}|=n\}}\prod _{i=1}^{n}\hat{L}_{i}\nonumber \\&= \sum _{\mathcal{L }_\mathsf{M}}Lz^{-|\breve{L}|}+\sum _{\mathcal{L }_\mathsf{M}^{2*}}LMz^{-(|\breve{L}|+|\breve{M}|)}-\sum _{\mathcal{L }_\mathsf{M}^{3*}}LMNz^{-(|\breve{L}|+|\breve{M}|+|\breve{N}|)}+\cdots \qquad \quad \end{aligned}$$

(21)

where $\hat{L}:=Lz^{-|\breve{L}|}$ and $\mathcal{L }_\mathsf{M}^{k*}$ denotes the $k$-fold Cartesian product over the set of loops $\mathcal{L }_\mathsf{M}$. The star indicates that $k$-tuples in which not all loops are unconnected to each other are excluded from the Cartesian product.

By replacing $\mathsf M$ with $z\mathsf{S}+\mathsf{F}$ Eq. (21) can be rewritten as

$$\begin{aligned} 0&= 1-\sum _{\mathcal{L }_\mathsf{{A}}}L\frac{z^{|L|_\mathrm{s}}}{z^{|L|}}+\sum _{\mathcal{L }_\mathsf{{A}}^{2*}}LM\frac{z^{|L|_\mathrm{s}+|M|_\mathrm{s}}}{z^{|\breve{L}|+|M|}}-\sum _{\mathcal{L }_\mathsf{A}^{3*}}LMN\frac{z^{|L|_\mathrm{s}+|M|_\mathrm{s}+|N|_\mathrm{s}}}{z^{|\breve{L}|+|\breve{M}|+|\breve{N}|}}+\cdots \nonumber \\&= 1-\sum _{\mathcal{L }_\mathsf{{A}}}Lz^{-|L|_\mathrm{f}}+\sum _{\mathcal{L }_\mathsf{{A}}^{2*}}LMz^{-(|L|_\mathrm{f}+|M|_\mathrm{f})}-\sum _{\mathcal{L }_\mathsf{{A}}^{3*}}LMNz^{-(|L|_\mathrm{f}+|M|_\mathrm{f}+|N|_\mathrm{f})}+\cdots \nonumber \\ \end{aligned}$$

(22)

where the expressions $|L|_\mathrm{f}$ and $|L|_\mathrm{s}$ denote the number of fertility parameters $f_{lk}$ and the number of transition parameters $s_{lk}$ in loop $L$, respectively. Furthermore, $|\breve{L}|$ indicates the number of i-states traversed by loop $L$ and therefore also the number of demographic parameters in $L$. Thus, $|\breve{L}|=|L|_\mathrm{f}+|L|_\mathrm{s}$.

By sorting the terms into those containing fertility parameters and those that do not, Eq. (22) can be rearranged in the following way:

$$\begin{aligned} 0&= -\sum _{\mathcal{L }_{\mathrm{f},\mathsf{{A}}}}Lz^{-|L|_\mathrm{f}}+\sum _{\mathcal{L }_{\mathrm{f},\mathsf{A}}\overset{_{*}}{\times }\mathcal{L }_\mathsf{{A}}}LMz^{-(|L|_\mathrm{f}+|M|_\mathrm{f})}-\sum _{\mathcal{L }_{\mathrm{f},\mathsf{A}}\overset{_{*}}{\times }\mathcal{L }_\mathsf{A}\overset{_{*}}{\times }\mathcal{L }_\mathsf{{A}}}\!\!\!LMNz^{-(|L|_\mathrm{f}+|M|_\mathrm{f}+|N|_\mathrm{f})}\nonumber \\&\quad +\cdots +1-\sum _{\mathcal{L }_\mathrm{s}}L+\sum _{\mathcal{L }^{2*}_\mathrm{s}}LM-\sum _{\mathcal{L }^{3*}_\mathrm{s}}LMN+\cdots \nonumber \\&= -\sum _{\mathcal{L }_{\mathrm{f},\mathsf{{A}}}}Lz^{-|L|_\mathrm{f}}+\sum _{\mathcal{L }_{\mathrm{f},\mathsf{A}}\overset{_{*}}{\times }\mathcal{L }_\mathsf{{A}}}LMz^{-(|L|_\mathrm{f}+|M|_\mathrm{f})}\nonumber \\&\quad -\sum _{\mathcal{L }_{\mathrm{f},\mathsf{A}}\overset{_{*}}{\times }\mathcal{L }_\mathsf{A}\overset{_{*}}{\times }\mathcal{L }_\mathsf{{A}}}LMNz^{-(|L|_\mathrm{f}+|M|_\mathrm{f}+|N|_\mathrm{f})}+\cdots +\mathrm{det}(\mathsf{{I}}-\mathsf{S}). \end{aligned}$$

(23)

The last simplification follows from Eq. (21) by choosing $\mathsf{M}=\mathsf{S}$ and $z=1$. Dividing both sides by $-\det (\mathsf{{I}}-\mathsf{S})$ results in Eq. (4) in the main text.

Equation (5) in the main text can be simplified. Recall the notation introduced after Eq. (6). With this notation at hand and by using Eq. (21) the numerator of Eq. (5) can be written as

$$\begin{aligned} \sum _{\mathcal{L }_{\mathrm{f},\mathsf{{A}}}}L\left(1-\sum _{\mathcal{L }_{\mathsf{S}_{\setminus \breve{L}}}}M+\sum _{\mathcal{L }^{2*}_{\mathsf{S}_{\setminus \breve{L}}}}MN-\cdots \right)=\sum _{\mathcal{L }_{\mathrm{f},\mathsf{{A}}}}L\mathrm{det}(\mathsf{{I}}-{\mathsf{S}_{\setminus \breve{L}}}), \end{aligned}$$

resulting in Eq. (6) in the main text.

Appendix B: Proof of Lemma 2

In this appendix we prove that if all fertility loops contain only a single fertility parameter, then all fertility loops are mutually connected if and only if an i-state exists that is shared by all fertility loops. One implication is trivial. If all such loops share a common i-state, then they are mutually connected. The remainder of this appendix deals with the reverse implication.

For life cycles with one or two fertility loops the statement is trivial. For life cycles with more than two fertility loops we prove the statement by induction. Assume that three mutually connected fertility loops $L_{1}, L_{2}$ and $L_{3}$ exist. If one of them is a self loop the statement is obviously true. Assume none of them is a self loop. We then prove the statement by contradiction. Thus, assume $\{L_{1}\}\cap \{L_{2}\}\cap \{L_{3}\}=\emptyset$. Let $a, b, c$ denote i-states in the life cycle such that $a\in \{L_{1}\}\cap \{L_{3}\}, b\in \{L_{1}\}\cap \{L_{2}\}$ and $c\in \{L_{2}\}\cap \{L_{3}\}$ with $a\ne b, a\ne c$ and $b\ne c$. A path is defined as a sequence of demographic parameters $s_{lk}, f_{lk}$ that lead from one i-state to another i-state without passing through any i-state more than once. The path transmission $P_{vu}$ equals the product of the demographic parameters along the path leading from $u$ to $v$. Here we use the terms path and path transmission synonymously and often denote a path with its transmission. Loops in a life cycle can be written as a concatenation of paths, e.g. $L=P_{uv}P_{vu}$ or $M=P_{wv}P_{vu}P_{uw}$. Thus, the life cycle under consideration has the structure shown in Fig. 3 where the arrows correspond to paths and not necessarily to single demographic parameters as in Fig. 1 in the main part. Thus, next to the loops $L_{1}=P_{ab}P_{ba}, L_{2}=P_{bc}P_{cb}$ and $L_{3}=P_{ac}P_{ca}$ two more loops exist. These are $L_{4}:=P_{ab}P_{bc}P_{ca}$ and $L_{5}:=P_{ac}P_{cb}P_{ba}$. Each path $P_{ab},P_{ba},P_{bc},P_{cb},P_{ac},P_{ca}$ contains either exactly one fertility parameter $f_{lk}$ or no fertility parameter. We indicate the presence of a fertility parameter in a path with the subscript f, e.g. $P_{ab,\mathrm f}$. Then the three fertility parameters can be distributed over the six different paths in $2^{3}$ different ways with resulting loops $L_{4}$ and $L_{5}$ as follows:

$$\begin{aligned}&\{P_{ba,\mathrm f},P_{cb,\mathrm f},P_{ac,\mathrm f}\}\Longleftrightarrow L_{4}=P_{ba,\mathrm f}P_{cb,\mathrm f}P_{ac,\mathrm f}\wedge L_{5}=P_{ca}P_{bc}P_{ab}\\&\{P_{ba,\mathrm f},P_{cb,\mathrm f},P_{ca,\mathrm f}\}\Longleftrightarrow L_{4}=P_{ba,\mathrm f}P_{cb,\mathrm f}P_{ac} \wedge L_{5}=P_{ca,\mathrm f}P_{bc}P_{ab}\\&\{P_{ba,\mathrm f},P_{bc,\mathrm f},P_{ac,\mathrm f}\}\Longleftrightarrow L_{4}=P_{ba,\mathrm f}P_{cb}P_{ac,\mathrm f} \wedge L_{5}=P_{ca}P_{bc,\mathrm f}P_{ab}\\&\{P_{ab,\mathrm f},P_{cb,\mathrm f},P_{ac,\mathrm f}\}\Longleftrightarrow L_{4}=P_{ba}P_{cb,\mathrm f}P_{ac,\mathrm f} \wedge L_{5}=P_{ca}P_{bc}P_{ab,\mathrm f}\\&\{P_{ab,\mathrm f},P_{bc,\mathrm f},P_{ac,\mathrm f}\}\Longleftrightarrow L_{4}=P_{ba}P_{cb}P_{ac,\mathrm f} \wedge L_{5}=P_{ca}P_{bc,\mathrm f}P_{ab,\mathrm f}\\&\{P_{ba,\mathrm f},P_{bc,\mathrm f},P_{ca,\mathrm f}\}\Longleftrightarrow L_{4}=P_{ba,\mathrm f}P_{cb}P_{ac}\wedge L_{5}=P_{ca,\mathrm f}P_{bc,\mathrm f}P_{ab}\\&\{P_{ab,\mathrm f},P_{cb,\mathrm f},P_{ca,\mathrm f}\}\Longleftrightarrow L_{4}=P_{ba}P_{cb,\mathrm f}P_{ac} \wedge L_{5}=P_{ca,\mathrm f}P_{bc}P_{ab,\mathrm f}\\&\{P_{ab,\mathrm f},P_{bc,\mathrm f},P_{ca,\mathrm f}\}\Longleftrightarrow L_{4}=P_{ba}P_{cb}P_{ac} \wedge L_{5}=P_{ca,\mathrm f}P_{bc,\mathrm f}P_{ab,\mathrm f} \end{aligned}$$

In each case a loop with more than one fertility parameter exists, contradicting the assumption that no such loops exist. Hence, the loops $L_{1}, L_{2}$ and $L_{3}$ have to have at least one i-state in common.

Fig. 3

Life cycle containing the three mutually connected loops $L_{1}=P_{ab}P_{ba}, L_{2}=P_{bc}P_{cb}$ and $L_{3}=P_{ca}P_{ac}$. For these tree loops no shared i-state exists. This life cycle contains two more loops: $L_{4}=P_{ab}P_{bc}P_{ca}$ and $L_{5}=P_{ac}P_{cb}P_{ba}$

Next consider a life cycle containing the fertility loops $L_{1},\ldots , L_{n},L_{n+1}$ which all contain only a single fertility parameter. Assume that the loops $L_{1},\ldots , L_{n}$ have a least one i-state in common: $\alpha :=\breve{L}_{1}\cap \ldots \cap \breve{L}_{n}\ne \emptyset$. Furthermore, assume that (i) the loop $L_{n+1}$ is connected to all other loops but (ii) not via one of the i-states that is shared by those loops: $\breve{L}_{n+1}\cap \breve{L}_{j}\ne \emptyset$ for all $L_{j}\in \{L_{1},\ldots , L_{n}\}$ and $\breve{L}_{n+1}\cap \alpha =\emptyset$. It follows that two loops $L_{l},L_{k}\in \{L_{1},\ldots , L_{n}\}$ exist such that $L_{n+1}$ intersects with $L_{l}$ at an i-state that is not passed by $L_{k}$ and that $L_{n+1}$ intersects with $L_{k}$ at an i-state that is not passed by $L_{l}$: $\beta :=(\breve{L}_{n+1}\cap \breve{L}_{l}){\setminus }\breve{L}_{k}\ne \emptyset$ and $\gamma :=(\breve{L}_{n+1}\cap \breve{L}_{k}){\setminus }\breve{L}_{l}\ne \emptyset$. If such loops would not exist, then (i) and (ii) could not be fulfilled simultaneously. Let $a,b$ and $c$ denote three i-states such that $a\in \alpha , b\in \beta$ and $c\in \gamma$. Then the life cycle under consideration contains three loops that are connected as shown in Fig. 3. With the above argument follows that then a loop with more than one fertility parameter has to exist, contradicting our assumption that no such loop exists. Thus, $\breve{L}_{n+1}\cap \alpha \ne \emptyset$.

Appendix C: Proof of Lemma 4

Next we prove Lemma 4. Let $\mathsf A$ be an irreducible population projection matrix. Let us call a path through which a newborn has to pass till it reaches a state where it can reproduce a maturation path. In particular, in case of a self-loop $f_{kk}$ the maturation path has length zero. Assume that (i) all fertility loops in the life cycle given by $\mathsf A$ contain exactly one fertility parameter and (ii) an i-state exists that is passed by all fertility loops. Note that for each maturation path a fertility parameter $f_{lk}$ exists that closes the maturation path into a fertility loop. This follows from (i) in combination with the fact that any two i-states are connected by a path (since $\mathsf A$ is assumed irreducible). Then it follows from (ii) that an i-state exists that is passed by all maturation paths. The reverse implication follows immediately since a fertility loop with more than one fertility parameter cannot exist as it would have to pass more than once through that specific i-state.

Appendix D: Proof of Proposition 6

In this Appendix we prove that if all fertility loops contain only a single fertility parameter and share a common i-state, then $R_{0}$ gives the expected number of offspring surviving until they enter the common i-state that are born over the remaining life time of an individual that has just entered the common i-state. In other words, $R_{0}$ projects the population size in the common i-state from one generation to the next.

Consider a life cycle with $n$ i-states, $m\le n$ of which are birth states. We number the i-states such that the birth states have indices in $\{1,\dots ,m\}$. Then the rows $m+1$ till $n$ of $\mathsf{G}=\mathsf{F}(\mathsf{{I}}-\mathsf{S})^{-1}$ contain only zeros:

$$\begin{aligned} \mathsf{G}=\mathsf{F}(\mathsf{{I}}-\mathsf{S})^{-1}= \left(\begin{array}{c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c} g_{11}&\cdots&g_{1m}&g_{1,m+1}&\cdots&g_{1n}\\ \vdots&\ddots&\vdots&\vdots&\cdots&\vdots \\ g_{m1}&\cdots&g_{mm}&g_{m,m+1}&\cdots&g_{mn}\\&\,&\,&0&\,&\\ \end{array}\right) \end{aligned}$$

(24)

Remember that $g_{lk}$ gives the expected number of offspring in birth state $l$ that are born to an individual in i-state $k$.

The eigenvalues of $\mathsf G$ are given by the eigenvalues of the $m\times m$ upper left block of $\mathsf G$, which we denote by $\mathsf{G}_{11}$. We define $\mathsf{B}=[b_{lk}]:=(\mathsf{{I}}-\mathsf{S})^{-1}$ and let $q$ be an i-state that is passed by all fertility loops. Then

$$\begin{aligned} g_{lk}=\sum _{j=1}^{n}f_{lj}b_{jk}=\sum _{j=1}^{n}f_{lj}b_{jq}b_{qk}=b_{qk}\sum _{j=1}^{n}f_{lj}b_{jq}. \end{aligned}$$

Thus, all rows of $\mathsf{G}_{11}$ are linear dependent and therefore $\text{ rank}\,\mathsf{G}_{11}=1$. In particular,

$$\begin{aligned} R_0=\sum _{l=1}^mg_{ll}=\sum _{l=1}^{m}b_{ql}\sum _{j=1}^{m}f_{lj}b_{jq} \end{aligned}$$

(25)

which proves the statement.

Appendix E: Derivation of Eq. (12)

Assume that $\mathsf A$ can be written as in Eq. (11). Then $\mathsf G$ has the form as given in Eq. (24). The dominant eigenvalue of $\mathsf G, R_{0}$, is given by the dominant eigenvalue of the upper left $m\times m$-matrix on the right-hand side of Eq. (24), denoted with $\mathsf{G}_{11}$. It is clear that

$$\begin{aligned} \mathsf{G}_{11}=\mathsf{F}_{12}(\mathsf{{I}}-\mathsf{S}_{22})^{-1}\mathsf{S}_{21}(\mathsf{{I}}-\mathsf{S}_{11})^{-1}=\mathsf{F}_{12}(\mathsf{{I}}-\mathsf{S}_{22})^{-1}{\varvec{u}}_{2}{\varvec{u}}_{1}^\mathrm{T}(\mathsf{{I}}-\mathsf{S}_{11})^{-1}. \end{aligned}$$

Let $\varvec{v}$ be the dominant right eigenvector of $\mathsf{G}_{11}$. If we multiply both sides of the eigenvalue equation for $\mathsf{G}_{11}$ with ${\varvec{u}}_{1}^\mathrm{T}(\mathsf{{I}}-\mathsf{S}_{11})^{-1}$, we obtain

$$\begin{aligned} {\varvec{u}}_{1}^\mathrm{T}(\mathsf{{I}}-\mathsf{S}_{11})^{-1}\mathsf{G}_{11}{\varvec{v}}&= {\varvec{u}}_{1}^\mathrm{T}(\mathsf{{I}}-\mathsf{S}_{11})^{-1}\mathsf{F}_{12}(\mathsf{{I}}-\mathsf{S}_{22})^{-1}{\varvec{u}}_{2}{\varvec{u}}_{1}^\mathrm{T}(\mathsf{{I}}-\mathsf{S}_{11})^{-1}{\varvec{v}}\\&= {\varvec{u}}_{1}^\mathrm{T}(\mathsf{{I}}-\mathsf{S}_{11})^{-1}{\varvec{v}}R_{0} \end{aligned}$$

from which Eq. (12) follows immediately.

Appendix F: Relationship between $R_{0}, Q_\mathsf{{A}}$ and $Q_\mathsf{S}$

Assume all fertility loops contain only a single fertility parameter and an i-state exists, overt or hidden, that is passed by all fertility loops. Then, with Theorem 9 we know that $R_{0}$ is given by Eq. (5). Recall that $Q_\mathsf{S}:=-\det (\mathsf{{I}}-\mathsf{S})$ and $Q_\mathsf{{A}}:=-\det (\mathsf{{I}}-\mathsf{{A}})$. Thus, Eq. (5) can be rewritten as

$$\begin{aligned} -Q_\mathsf{S}R_{0}=\sum _{\mathcal{L }_{\mathrm{f},\mathsf{A}}}L-\sum _{\mathcal{L }_{\mathrm{f},\mathsf{A}}\overset{_{*}}{\times }\mathcal{L }_\mathsf{S}}LM+\sum _{\mathcal{L }_{\mathrm{f},\mathsf{A}}\overset{_{*}}{\times }\mathcal{L }_\mathsf{S}\overset{_{*}}{\times }\mathcal{L }_\mathsf{S}}LMN-\cdots . \end{aligned}$$

(26)

From the results of Appendix A, in particular Eq. (), it is clear that

$$\begin{aligned} -\det (\mathsf{{I}}-\mathsf{{A}})=\sum _{\mathcal{L }_{\mathrm{f},\mathsf{A}}}L-\sum _{\mathcal{L }_{\mathrm{f},\mathsf{A}}\overset{_{*}}{\times }\mathcal{L }_\mathsf{S}}LM+\sum _{\mathcal{L }_{\mathrm{f},\mathsf{A}}\overset{_{*}}{\times }\mathcal{L }_\mathsf{S}\overset{_{*}}{\times }\mathcal{L }_\mathsf{S}}LMN-\cdots -\det (\mathsf{I}-\mathsf{S}). \end{aligned}$$

Thus, the right-hand side of Eq. (26) equals $Q_\mathsf{A}-Q_\mathsf{S}$ so that Eq. (26) can be rewritten as

$$\begin{aligned} Q_\mathsf{{A}}=(-Q_\mathsf{S})(R_{0}-1). \end{aligned}$$

Note that for purely age-dependent models $-Q_\mathsf{S}=1$.

That $Q_\mathsf{{A}}$ is affine in the entries of $\mathsf A$ follows from the fact that this is a property of determinants. Now assume that there exists another function $\mathsf{{A}} \mapsto Q^{\prime }_\mathsf{{A}}$ that is both locally sign equivalent to $R_{0}-1$ and affine in the entries of $\mathsf A$. In that case $Q_\mathsf{{A}}$ and $Q^{\prime }_\mathsf{{A}}$ have the same sign for $\mathsf A$ in an open set of nonnegative matrices containing the set of matrices for which $R_{0}=1$. Consider a matrix such that $R_{0}=1$. Now concentrate on two arbitrary matrix entries that are not in the same column or row of $\mathsf A$, referred to as $x$ and $y$, while keeping the other matrix entries fixed. Then there exist an interval of $x$-values such that there exists a nonlinear function $f$ such that $R_0=1$, and hence $Q_\mathsf{{A}}=0=Q^{\prime }_\mathsf{{A}}$, for $y=f(x)$. Then, by the affineness in the matrix entries, there exist constants $a, b, c, d, a^{\prime }, b^{\prime }, c^{\prime }, d^{\prime }$ such that $Q_\mathsf{{A}}=xf(x)a+xb+f(x)c+d=0$ and $Q^{\prime }_\mathsf{{A}}=xf(x)a^{\prime }+xb^{\prime }+f(x)c^{\prime }+d^{\prime }=0$. Hence $f(x)=(bx+d)/(ax+c)=(b^{\prime }x+d^{\prime })/(a^{\prime }x+c^{\prime })$. Generically in the remaining matrix entries the right hand equality holds good if and only if there exists a constant $k$ (still depending on those other entries) such that $a^{\prime }=ka, b^{\prime }=kb, c^{\prime }=kc$ and $d^{\prime }=kd$. (The exceptions occur when $bc-ad=0$.) From this and the local sign equivalence of $Q_\mathsf{{A}}$ and $Q^{\prime }_\mathsf{{A}}$ we can conclude that $Q^{\prime }_\mathsf{{A}}=k(\mathsf{{A}})Q_\mathsf{{A}}$ with $k(\mathsf{{A}})>0$. From the fact that $Q^{\prime }$ is affine in the entries of $\mathsf A$ we can moreover conclude that $k$ is constant.