*3.18. Lower Bounds for the Cases* (*β*A, *β*H, *α*A, *α*H, *λ*) ∈ P*SP,3a* × (R\[0, 1])

In the current setup, where PSP,3a := n (*β*A, *β*H, *α*A, *α*H) ∈ PSP : *α*<sup>A</sup> 6= *α*H, *β*<sup>A</sup> 6= *β*H, *α*A *β*A 6= *α*H *β*H , *α*A−*α*<sup>H</sup> *β*H−*β*<sup>A</sup> ∈ ] − ∞, 0[ o (cf. (49)), we *always* have either (*α*<sup>A</sup> > *α*H) ∧ (*β*<sup>A</sup> > *β*H) or (*α*<sup>A</sup> < *α*H) ∧ (*β*<sup>A</sup> < *β*H). Furthermore, from Properties <sup>3</sup> (P16) we obtain *φλ*(0) > 0. As in the case *λ* ∈]0, 1[, the derivative *φ* 0 *λ* (0) can assume any sign on PSP,3a, take e.g., (*β*A, *β*H, *α*A, *α*H, *λ*) = (2.2, 4.5, 1, 3, 2) for *φ* 0 *λ* (0) < 0, (*β*A, *β*H, *α*A, *α*H, *λ*) = (2.25, 4.5, 1, 3, 2) for *φ* 0 *λ* (0) = 0 and (*β*A, *β*H, *α*A, *α*H, *λ*) = (2.3, 4.5, 1, 3, 2) for *φ* 0 *λ* (0) > 0 (these parameter constellations reflect "dangerous" (A) versus "highly dangerous" (H) situations within our running-example epidemiological context of Section 2.3). Nevertheless, in all three subcases one gets min*x*∈N<sup>0</sup> *φλ*(*x*) ≥ min*x*≥<sup>0</sup> *φλ*(*x*) > 0. Thus, there exist parameters *p L λ* ∈ *αλ*, *α λ* A *α* 1−*λ* H and *q L λ* ∈ *βλ*, *β λ* A *β* 1−*λ* H which satisfy (35) in particular, *p L <sup>λ</sup>* − *α<sup>λ</sup>* > 0, *q L <sup>λ</sup>* − *β<sup>λ</sup>* > 0 . We now have to look for a condition which guarantees that these parameters *additionally* fulfill (56); such a condition is clearly that both *α<sup>λ</sup>* ≥ 0 and *β<sup>λ</sup>* ≥ 0 hold, which is equivalent (cf. (57)) with

$$\lambda \in \mathcal{I}\_{\mathrm{SP,3a}}^{(\geq)} := \begin{cases} \left[ \max \left\{ \frac{-a\_{\mathcal{H}}}{a\_{\mathcal{A}} - a\_{\mathcal{H}'}} \frac{-\beta\_{\mathcal{H}}}{\beta\_{\mathcal{A}} - \beta\_{\mathcal{H}}} \right\}, \ 0 \right] \cup \left[ \cup \ \right] 1, \infty \left[ \right], & \text{if } (a\_{\mathcal{A}} > a\_{\mathcal{H}}) \wedge (\beta\_{\mathcal{A}} > \beta\_{\mathcal{H}}), \\\\ \left[ -\infty, 0 \right] \cup \left[ \text{1, } \min \left\{ \frac{a\_{\mathcal{H}}}{a\_{\mathcal{H}} - a\_{\mathcal{A}'}}, \frac{\beta\_{\mathcal{H}}}{\beta\_{\mathcal{H}} - \beta\_{\mathcal{A}}} \right\} \right], & \text{if } (a\_{\mathcal{A}} < a\_{\mathcal{H}}) \wedge (\beta\_{\mathcal{A}} < \beta\_{\mathcal{H}}); \end{cases}$$

recall that *α<sup>λ</sup>* = 0 and *β<sup>λ</sup>* = 0 cannot occur simultaneously in the current setup. If *α<sup>λ</sup>* ≤ 0 and *β<sup>λ</sup>* ≤ 0, i.e., if

$$\lambda \in \mathcal{I}\_{\mathrm{SP,3a}}^{(<)} := \begin{cases} \begin{array}{c} \left\lfloor \begin{array}{c} -\infty \text{, } \min\left\{ \frac{-a\_{\mathcal{H}}}{a\_{\mathcal{A}} - a\_{\mathcal{H}}}; \frac{-\beta\_{\mathcal{H}}}{\beta\_{\mathcal{A}} - \beta\_{\mathcal{H}}} \right\} \right\} \end{array} \right\rfloor , & \text{if } (\mathfrak{a}\_{\mathcal{A}} > \mathfrak{a}\_{\mathcal{H}}) \wedge (\beta\_{\mathcal{A}} > \beta\_{\mathcal{H}}), \\\\ \left\lfloor \max\left\{ \frac{a\_{\mathcal{H}}}{a\_{\mathcal{H}} - a\_{\mathcal{A}}}; \frac{\beta\_{\mathcal{H}}}{\beta\_{\mathcal{H}} - \beta\_{\mathcal{A}}} \right\} \right\rangle , & \text{so} \left[ \text{, } \quad \text{if } (\mathfrak{a}\_{\mathcal{A}} < \mathfrak{a}\_{\mathcal{H}}) \wedge (\beta\_{\mathcal{A}} < \beta\_{\mathcal{H}}). \end{array} \right\} \end{cases}$$

then–due to the strict positivity of the function *ϕλ*(·) (cf. (31))–there exist parameters *p L <sup>λ</sup>* > 0 = max{0, *αλ*} and *q L <sup>λ</sup>* > 0 = max{0, *βλ*} which satisfy (56) and (34) (where the latter implies (35) and thus *p L <sup>λ</sup>* ≤ *α λ* A *α* 1−*λ* H , *q L <sup>λ</sup>* ≤ *β λ* A *β* 1−*λ* H ). With

$$\mathcal{L}\_{\text{SP,3a}} := \mathcal{L}\_{\text{SP,3a}}^{(>)} \cup \mathcal{L}\_{\text{SP,3a}}^{(<)} \tag{60}$$

and with the discussion below (56), we thus derive the following

**Proposition 12.** *For all* (*β*A, *β*H, *α*A, *α*H, *λ*) ∈ P*SP,3a* × I*SP,3a there exist parameters p L λ* , *q L <sup>λ</sup> which satisfy* max{0, *αλ*} < *p L <sup>λ</sup>* ≤ *α λ* A *α* 1−*λ* H , max{0, *βλ*} < *q L <sup>λ</sup>* ≤ *β λ* A *β* 1−*λ* H *as well as* (35) *for all x* ∈ N0*, and for all such pairs* (*p L λ* , *q L λ* ) *and all initial population sizes X*<sup>0</sup> ∈ N *one gets*

$$\begin{aligned} (a) \qquad & B^L\_{\lambda, X\_0, 1} = \tilde{B}^{(p^L\_{\lambda}, q^L\_{\lambda})}\_{\lambda, X\_0, 1} = \exp\left\{ \left( q^L\_{\lambda} - \beta\_{\lambda} \right) \cdot X\_0 + p^L\_{\lambda} - a\_{\lambda} \right\} > 1, \\ (b) \qquad & \text{then assume that } \tilde{B}^L\_{\lambda} \text{ has an eigenvalue for all } (\mathbf{D}\_{\lambda}, \|\mathbf{D}\_{\lambda}\|). \end{aligned}$$

(*b*) *the sequence B L λ*,*X*0,*n n*∈N *of lower bounds for Hλ*(*P*A,*n*||*P*H,*n*) *given by*

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

*is strictly increasing,*

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

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

(*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*.

Notice that the assertions (a) to (e) of Proposition 12 hold true for parameter pairs (*p L λ* , *q L λ* ) *whenever* they satisfy (35) and (56); in particular, we may allow either *p L <sup>λ</sup>* = max{0, *αλ*} or *q L <sup>λ</sup>* = max{0, *βλ*}. Let us furthermore mention that in part (d) both asymptotical behaviours can occur: consider e.g., the parameter setup (*β*A, *β*H, *α*A, *α*H) = (0.3, 0.2, 4, 3) ∈ PSP,3a, leading to ]1, ∞[ ( I (≥) SP,3a ( ISP,3a. For *λ* = 2 ∈ ISP,3a, the parameters *p L λ* :<sup>=</sup> *<sup>p</sup>*e*<sup>λ</sup>* :<sup>=</sup> 5.25, *<sup>q</sup> L λ* :<sup>=</sup> *<sup>q</sup>*e*<sup>λ</sup>* :<sup>=</sup> 0.45 (corresponding to the asymptote *<sup>φ</sup>*e*λ*(·), cf. (P20)) fulfill (35), (56) and additionally *q L <sup>λ</sup>* = 0.45 < min{1,*e <sup>β</sup>λ*−1} ≈ 0.549. Analogously, in the setup (*β*A, *β*H, *α*A, *α*H, *λ*) = (3, 2, 4, 3, 2) ∈ PSP,3a × ISP,3a, the choices *p L λ* :<sup>=</sup> *<sup>p</sup>*e*<sup>λ</sup>* :<sup>=</sup> 5.25, *<sup>q</sup> L λ* :<sup>=</sup> *<sup>q</sup>*e*<sup>λ</sup>* :<sup>=</sup> 4.5 satisfy (35), (56) and there holds *q L <sup>λ</sup>* = 4.5 > min{1,*e <sup>β</sup>λ*−1} <sup>=</sup> 1.

For the remaining two cases (*α<sup>λ</sup>* ≤ 0) ∧ (*β<sup>λ</sup>* > 0) (e.g., (*β*A, *β*H, *α*A, *α*H, *λ*) = (6, 5, 3, 2, −3)) and (*α<sup>λ</sup>* > <sup>0</sup>) ∧ (*β<sup>λ</sup>* ≤ <sup>0</sup>) (e.g., (*β*A, *β*H, *α*A, *α*H, *λ*) = (3, 2, 6, 5, −3)), one has to proceed differently. Indeed, for all parameter constellations (*β*A, *β*H, *α*A, *α*H, *λ*) ∈ PSP,3a × R\ (ISP,3a ∪ [0, 1]), all observation time horizons *n* ∈ N and all initial population sizes *X*<sup>0</sup> ∈ N one can still prove

$$1 < H\_{\lambda} \left( P\_{\mathcal{A}, \mathfrak{n}} || P\_{\mathcal{H}, \mathfrak{n}} \right) \quad \text{and} \quad \lim\_{\mathfrak{n} \to \infty} H\_{\lambda} \left( P\_{\mathcal{A}, \mathfrak{n}} || P\_{\mathcal{H}, \mathfrak{n}} \right) = \infty \,, \tag{61}$$

which is done in Appendix A.1, using a similar method as in the proof of assertion (59).

*3.19. Lower Bounds for the Cases* (*β*A, *β*H, *α*A, *α*H, *λ*) ∈ P*SP,3b* × (R\[0, 1])

Within such a constellation, where PSP,3b := n (*β*A, *β*H, *α*A, *α*H) ∈ PSP : *α*<sup>A</sup> 6= *α*H, *β*<sup>A</sup> 6= *β*H, *α*A *β*A 6= *α*H *β*H , *α*A−*α*<sup>H</sup> *β*H−*β*<sup>A</sup> ∈ ]0, ∞[\N o (cf. (49)), one *always* has either (*α*<sup>A</sup> < *α*H) ∧ (*β*<sup>A</sup> > *β*H) or (*α*<sup>A</sup> > *α*H) ∧ (*β*<sup>A</sup> < *β*H). Moreover, from Properties 3 (P15) one can see that *φλ*(*x*) = 0 for *x* = *x* ∗ = *α*H−*α*<sup>A</sup> *β*A−*β*<sup>H</sup> > 0. However, *x* <sup>∗</sup> ∈/ N0, which implies *φλ*(*x*) > 0 for all *x* on the relevant subdomain N0. Again, we incorporate (57) and consider the set of all *λ* ∈ R\[0, 1] such that *α<sup>λ</sup>* ≥ 0 and *β<sup>λ</sup>* ≥ 0 (where *α<sup>λ</sup>* = 0 ∧ *β<sup>λ</sup>* = 0 cannot appear), i.e.,

$$\lambda \in \mathcal{I}\_{\text{SP,3b}}^{(\geq)} := \begin{cases} \left[\frac{-\beta\_{\text{H}}}{\beta\_{\text{A}} - \beta\_{\text{H}}}, 0\right] \cup \left[\begin{array}{c} 1, \frac{a\_{\text{H}}}{a\_{\text{H}} - a\_{\text{A}}} \end{array}\right], & \text{if } (a\_{\text{A}} < a\_{\text{H}}) \wedge (\beta\_{\text{A}} > \beta\_{\text{H}}), \\\\ \left[\frac{-a\_{\text{H}}}{a\_{\text{A}} - a\_{\text{H}}}, 0\right] \cup \left[\begin{array}{c} 1, \frac{\beta\_{\text{H}}}{\beta\_{\text{H}} - \beta\_{\text{A}}} \end{array}\right], & \text{if } (a\_{\text{A}} > a\_{\text{H}}) \wedge (\beta\_{\text{A}} < \beta\_{\text{H}}). \end{cases} \tag{62}$$

As above in Section 3.18, if *<sup>λ</sup>* ∈ I(≥) SP,3b then there exist parameters *p L λ* ∈ *αλ*, *α λ* A *α* 1−*λ* H , *q L λ* ∈ *βλ*, *β λ* A *β* 1−*λ* H (which thus fulfill (56)) such that (35) is satisfied for all *x* ∈ N0. Hence, for all *λ* ∈ ISP,3b := I (≥) SP,3b, all assertions (a) to (e) of Proposition 12 hold true. Notice that for the current setup PSP,3b one cannot have *α<sup>λ</sup>* ≤ 0 and *β<sup>λ</sup>* ≤ 0 simultaneously. Furthermore, in each of the two remaining cases (*α<sup>λ</sup>* < 0) ∧ (*β<sup>λ</sup>* > 0) respectively (*α<sup>λ</sup>* > 0) ∧ (*β<sup>λ</sup>* < 0) it can happen that there do not exist parameters *p L λ* , *q L <sup>λ</sup>* > 0 which satisfy both (35) and (56). However, as in the case PSP,3a above, for all *λ* ∈ I / SP,3b we prove in Appendix A.1 (by a method without *p L λ* , *q L λ* ) that for all observation times *n* ∈ N and all initial population sizes *X*<sup>0</sup> ∈ N there holds

$$1 < H\_{\lambda} \left( P\_{\mathcal{A},n} || P\_{\mathcal{H},n} \right) \qquad \text{and} \qquad \lim\_{n \to \infty} H\_{\lambda} \left( P\_{\mathcal{A},n} || P\_{\mathcal{H},n} \right) = \infty \,. \tag{63}$$

*3.20. Lower Bounds for the Cases* (*β*A, *β*H, *α*A, *α*H, *λ*) ∈ P*SP,3c* × (R\[0, 1])

Since in this subcase one has PSP,3c := n (*β*A, *β*H, *α*A, *α*H) ∈ PSP : *α*<sup>A</sup> 6= *α*H, *β*<sup>A</sup> 6= *β*H, *α*A *β*A 6= *α*H *β*H , *α*A−*α*<sup>H</sup> *β*H−*β*<sup>A</sup> ∈ N o (cf. (49)) and thus *φλ*(*x* ∗ ) = 0 for *x* <sup>∗</sup> ∈ N, there do not exist parameters *p L λ* , *q L λ* such that (35) and (56) are satisfied. The only parameter pair that ensures exp <sup>n</sup> *a* (*q L λ* ) *<sup>n</sup>* · *X*<sup>0</sup> + ∑ *n k*=1 *b* (*p L λ* ,*q L λ* ) *k* o ≥ 1 for all *n* ∈ N and all *X*<sup>0</sup> ∈ N within our proposed method, is the choice *p L <sup>λ</sup>* = *αλ*, *q L <sup>λ</sup>* = *βλ*. Consequently, *B L <sup>λ</sup>*,*X*0,*<sup>n</sup>* ≡ 1, which coincides with the general lower bound (11) but violates the above-mentioned desired Goal (G10 ). However, in some constellations there exist *nonnegative* parameters *p L <sup>λ</sup>* < *αλ*, *q L <sup>λ</sup>* > *β<sup>λ</sup>* or *p L <sup>λ</sup>* > *αλ*, *q L <sup>λ</sup>* < *βλ*, such that at least the parts (c) and (d) of Proposition 12 are satisfied. As in Section 3.19 above, by using a conceptually different method (without *p L λ* , *q L λ* ) we prove in Appendix A.1 that for all *λ* ∈ R\[0, 1], all observation times *n* ∈ N and all initial population sizes *X*<sup>0</sup> ∈ N there holds

$$1 < H\_{\lambda} \left( P\_{\mathcal{A}, \mathfrak{n}} || P\_{\mathfrak{H}, \mathfrak{n}} \right) \qquad \text{and} \qquad \lim\_{\mathfrak{n} \to \infty} H\_{\lambda} \left( P\_{\mathcal{A}, \mathfrak{n}} || P\_{\mathfrak{H}, \mathfrak{n}} \right) = \infty \,. \tag{64}$$

*3.21. Lower Bounds for the Cases* (*β*A, *β*H, *α*A, *α*H, *λ*) ∈ P*SP,4a* × (R\[0, 1])

In the current setup, where PSP,4a := { (*β*A, *β*H, *α*A, *α*H) ∈ PSP : *α*<sup>A</sup> 6= *α*<sup>H</sup> > 0, *β*<sup>A</sup> = *β*<sup>H</sup> ∈ ]0, 1[ } (cf. (49)), the function *φλ*(·) is strictly positive and strictly decreasing, with lim*x*→<sup>∞</sup> *φλ*(*x*) = lim*x*→<sup>∞</sup> *φ* 0 *λ* (*x*) = 0. The only choice of parameters *p L λ* , *q L <sup>λ</sup>* which fulfill (35) and exp <sup>n</sup> *a* (*q L λ* ) *<sup>n</sup>* · *X*<sup>0</sup> + ∑ *n k*=1 *b* (*p L λ* ,*q L λ* ) *k* o ≥ 1 for all *n* ∈ N and all *X*<sup>0</sup> ∈ N, is the choice *p L <sup>λ</sup>* = *α<sup>λ</sup>* as well as *q L <sup>λ</sup>* = *β<sup>λ</sup>* = *β*•, where *β*• stands for both (equal) *β*<sup>H</sup> and *β*A. Of course, this leads to *B L <sup>λ</sup>*,*X*0,*<sup>n</sup>* ≡ 1, which is consistent with the general lower bound (11), but violates the above-mentioned desired Goal (G10 ). Nevertheless, in Appendix A.1 we prove the following

**Proposition 13.** *For all* (*β*A, *β*H, *α*A, *α*H, *λ*) ∈ P*SP,4a* × R\[0, 1] *there exist parameters p L <sup>λ</sup>* > *α<sup>λ</sup> (not necessarily satisfying p L <sup>λ</sup>* ≥ 0*) and* 0 < *q L <sup>λ</sup>* < *β<sup>λ</sup>* = *β*• < min{1,*e <sup>β</sup>*•−1} <sup>=</sup> *<sup>e</sup> β*•−1 *such that* (35) *holds for all x* ∈ [0, ∞[ *and such that for all initial population sizes X*<sup>0</sup> ∈ N *the parts (c) and (d) of Proposition 12 hold true.*
