*3.17. Lower Bounds for the Cases* (*β*A, *β*H, *α*A, *α*H, *λ*) ∈ P*SP,2* × (R\[0, 1])

In such a constellation, where PSP,2 := { (*β*A, *β*H, *α*A, *α*H) ∈ PSP : *α*<sup>A</sup> = *α*H, *β*<sup>A</sup> 6= *β*<sup>H</sup> } (cf. (49)), one gets *φλ*(0) = 0 (cf. Properties 3 (P16)), *φ* 0 *λ* (0) = 0 (cf. (P17)). Thus, the only choice for the intercept and the slope of the linear lower bound *φ L λ* (·) for *φλ*(·), which satisfies (35) for all *x* ∈ N and (potentially) (56), is *r L <sup>λ</sup>* = 0 = *p L <sup>λ</sup>* − *α<sup>λ</sup>* (i.e., *p L <sup>λ</sup>* = *α<sup>λ</sup>* = *α* > 0) and *s L <sup>λ</sup>* = *φλ*(1)−*φλ*(0) <sup>1</sup>−<sup>0</sup> <sup>=</sup> *<sup>q</sup> L <sup>λ</sup>* − *β<sup>λ</sup>* = *a* (*q L λ* ) <sup>1</sup> > 0 (i.e., *q L <sup>λ</sup>* = (*α* + *β*A) *λ* (*α* + *β*H) <sup>1</sup>−*<sup>λ</sup>* <sup>−</sup> *<sup>α</sup>*). However, since *<sup>p</sup> L <sup>λ</sup>* = *α<sup>λ</sup>* = *α* > 0, the restriction (56) is fulfilled iff *q L <sup>λ</sup>* > 0, which is equivalent to

$$\lambda \in \mathcal{Z}\_{\text{SP},2} := \begin{cases} \left\lceil \frac{\log\left(\frac{a}{a+\beta\_{\mathcal{H}}}\right)}{\log\left(\frac{a+\beta\_{\mathcal{H}}}{a+\beta\_{\mathcal{H}}}\right)} , \mathbf{0} \right\rceil \cup \left\lceil \cup \right\rceil \mathbf{1}\_{\mathcal{H}} \left\lceil \begin{array}{c} \mathbf{1} \\ \mathbf{1} \end{array} \right\rceil, & \text{if } \beta\_{\mathcal{A}} > \beta\_{\mathcal{H}}, \\\\ \left\lceil \begin{array}{c} \mathbf{1} \\ \end{array} \right\rceil \mathbf{1}\_{\mathcal{H}} \frac{\log\left(\frac{a}{a+\beta\_{\mathcal{H}}}\right)}{\log\left(\frac{a+\beta\_{\mathcal{H}}}{a+\beta\_{\mathcal{H}}}\right)} \end{array} \tag{58}$$

Suppose that *λ* ∈ ISP,2. As we have seen above, from Properties 1 (P3a) and (P3b) one can derive that *a* (*q L λ* ) *n n*∈N is strictly positive, strictly increasing, and converges to *x* (*q L λ* ) <sup>0</sup> ∈]0, − log(*q L λ* )] iff *q L <sup>λ</sup>* ≤ min{1 , *e <sup>β</sup>λ*−1}, and otherwise it diverges to <sup>∞</sup>. Notice that both cases can occur: consider the parameter setup (*β*A, *β*H, *α*A, *α*H) = (1.5, 0.5, 0.5, 0.5) ∈ PSP,2, which leads to ISP,2 =] − 1, 0[ ∪ ]1, ∞[; within our running-example epidemiological context of Section 2.3, this corresponds to a "mild" infectious-disease-transmission situation (H) (with "low" reproduction number *β*<sup>H</sup> = 0.5 and importation mean of *α*<sup>H</sup> = 0.5), whereas (A) describes a "dangerous" situation (with supercritical *β*<sup>A</sup> = 1.5 and *α*<sup>A</sup> = 0.5). For *λ* = −0.5 ∈ ISP,2 one obtains *q L <sup>λ</sup>* ≈ 0.207 ≤ min{1 , *e <sup>β</sup>λ*−1} ≈ 0.368, whereas for *λ* = 2 ∈ ISP,2 one gets *q L <sup>λ</sup>* = 3.5 > min{1 , *e <sup>β</sup>λ*−1} <sup>=</sup> 1. Altogether, this leads to

**Proposition 11.** *For all* (*β*A, *β*H, *α*A, *α*H, *λ*) ∈ P*SP,2* × I*SP,2 and all initial population sizes X*<sup>0</sup> ∈ N *there holds with p<sup>L</sup> <sup>λ</sup>* = *α*<sup>A</sup> = *α*<sup>H</sup> = *α*, *q L <sup>λ</sup>* = (*α* + *β*A) *λ* (*α* + *β*H) <sup>1</sup>−*<sup>λ</sup>* <sup>−</sup> *<sup>α</sup>*

$$\mathcal{B}(a) \qquad \mathcal{B}^{L}\_{\lambda, X\_0, 1} = \underset{\lambda}{\tilde{\mathcal{B}}} \left( \underset{\lambda}{\tilde{\mathcal{B}}} \!/ \!/ \!/ \!/ \!/ \!/ \!/ \!/ \!/ \!/} \!/ \!/ \!/ \!/ \!/ \!/ \!/ \!/ \!/ \!/ \!/> \!/> \!\!/> \!\!/> \!\!/> < \!\!/> < \!\!/> < \!\!/> < \!\!/> < \!\!/> < \!\!/> < \!\!/> < \!\!/> < \!\!/> < \!\!/> < \!\!/> < \!\!/> < \!\!/> < \!\!/> < \!\!/> < \!\!\!/> < \!\!\!/> < \!\!\!/> < \!\!\!/> < \!\!\!/> < \!\!\!/> < \!\!\!/> < \!\!\!/> < \!\!\!/> < \!\!\!/> < \!\!\!\!/> < \!\!\!\!/> < \!\!\!\!\/> < \!\!\!\!\/> < \!\!\!\!\/> < \!\!\!\/> < \!\!\!\/> < \!\!\!\/> < \!\!\!\/> < \!\!\!\/> < \!\!\!\/> < \!\!\!\/> < \!\!\!\/> < \!\!\!\/> < \!\!\!\/> < \!\!\!\/> < \!\!\/> < \!\!\/> < \!\!\/> < \!\!\/> < \!\!\/> < \!\!\/> < \!\!\/> < \!\!\/> < \!\!\/> < \!\!\/> < \!\!\/> < \!\!\/> < \!\!\/$$

$$\begin{array}{ccccc} (b) & \text{the sequence } \left( \mathbf{B}\_{\lambda, \mathbf{X}\_0, \mathbb{H}}^{L} \right)\_{n \in \mathbb{N}} \text{ of lower bounds for } \mathbf{H}\_{\lambda}(\mathbf{P}\_{\mathcal{A}, \mathbb{n}} || \mathbf{P}\_{\mathcal{H}, \mathbb{n}}) \text{ given by } \\\\ & \begin{array}{ccccc} \end{array} & \begin{array}{c} \begin{array}{c} (a\_{\mathcal{A}}) \ \vdots \ \begin{array}{c} n \ \end{array} \end{array} \end{array} \end{array}$$

$$B^{L}\_{\\\lambda,\\\\\mathcal{X}\_{0},\mathcal{U}} = \ \widetilde{\mathcal{B}}^{(p^{L}\_{\lambda}q^{L}\_{\lambda})}\_{\lambda,\\\\\mathcal{X}\_{0},\mathcal{U}} = \ \exp\left\{a\_{\eta}^{(q^{L}\_{\lambda})} \cdot \mathcal{X}\_{0} + \sum\_{k=1}^{n} b\_{k}^{(p^{L}\_{\lambda},q^{L}\_{\lambda})}\right\}$$

*is strictly increasing,*

$$(\mathcal{C}) \qquad \lim\_{n \to \infty} B^{L}\_{\lambda, X\_0, n} = \infty \quad = \lim\_{n \to \infty} H\_{\lambda}(P\_{\mathcal{A}, n} || P\_{\mathcal{H}, n}) \; .$$

$$(d)\quad\lim\_{n\to\infty}\frac{1}{n}\log B\_{\lambda,X\_0,\mathbb{1}}^L = \begin{cases} \begin{array}{c} p\_\lambda^L \cdot \exp\left\{ x\_0^{(q\_\lambda^L)} \right\} - a > 0, & \text{if} \quad q\_\lambda^L \le \min\left\{ 1, e^{\theta\_\lambda - 1} \right\} \\\infty, & \text{if} \quad q\_\lambda^L > \min\left\{ 1, e^{\theta\_\lambda - 1} \right\} \end{array} \end{cases}$$

(*e*) *the map X*<sup>0</sup> 7→ *B L <sup>λ</sup>*,*X*0,*<sup>n</sup>* = *B*e (*p L λ* ,*q L λ* ) *λ*,*X*0,*n is strictly increasing*.

Nevertheless, for the remaining constellations (*β*A, *β*H, *α*A, *α*H, *λ*) ∈ PSP,2 × R\ (ISP,2 ∪ [0, 1]), all observation time horizons *n* ∈ N and all initial population sizes *X*<sup>0</sup> ∈ N one can still prove

$$1 < H\_{\lambda} \left( P\_{\mathcal{A},n} || P\_{\mathcal{H},n} \right) \qquad \text{and} \qquad \lim\_{n \to \infty} H\_{\lambda} \left( P\_{\mathcal{A},n} || P\_{\mathcal{H},n} \right) = \infty \, \, \, \, \tag{59}$$

(i.e., the achievement of the Goals (G10 ), (G20 )), which is done by a conceptually different method (without involving *p L λ* , *q L λ* ) in Appendix A.1.
