*3.7. Upper Bounds for the Cases* (*β*A, *β*H, *α*A, *α*H, *λ*) ∈ P*SP,2*×]0, 1[

For this parameter constellation, one has *φλ*(0) = 0 and *φ* 0 *λ* (0) = 0 (cf. Properties 3 (P16), (P17)). Thus, the only admissible intercept choice satisfying (47) is *r U <sup>λ</sup>* = 0 = *p U <sup>λ</sup>* − *α<sup>λ</sup>* i.e., *p U <sup>λ</sup>* = *p <sup>U</sup>* (*β*A, *<sup>β</sup>*H, *<sup>α</sup>*A, *<sup>α</sup>*H, *<sup>λ</sup>*) <sup>=</sup> *<sup>α</sup><sup>λ</sup>* <sup>=</sup> *<sup>α</sup>* <sup>&</sup>gt; <sup>0</sup> , and the minimal admissible slope which implies (35) for all *x* ∈ N<sup>0</sup> is given by *s U <sup>λ</sup>* = *φλ*(1)−*φλ*(0) <sup>1</sup>−<sup>0</sup> <sup>=</sup> *<sup>q</sup> U <sup>λ</sup>* − *β<sup>λ</sup>* = *a* (*q U λ* ) <sup>1</sup> < 0 i.e., *q U <sup>λ</sup>* = *q <sup>U</sup>* (*β*A, *<sup>β</sup>*H, *<sup>α</sup>*A, *<sup>α</sup>*H, *<sup>λ</sup>*) <sup>=</sup> (*α* + *β*A) *λ* (*α* + *β*H) <sup>1</sup>−*<sup>λ</sup>* <sup>−</sup> *<sup>α</sup>* <sup>&</sup>gt; <sup>0</sup> . Analogously to the investigation for PSP,1 in the above-mentioned Section 3.3, one can derive that *a* (*q U λ* ) *n n*∈N is strictly negative, strictly decreasing, and converges to *x* (*q U λ* ) <sup>0</sup> ∈] − *βλ*, *q U <sup>λ</sup>* − *βλ*[ as indicated in Properties 1 (P1). Moreover, in the same manner as for the case PSP,1 this leads to

**Proposition 7.** *For all* (*β*A, *β*H, *α*A, *α*H, *λ*) ∈ P*SP,2*×]0, 1[ *and all initial population sizes X*<sup>0</sup> ∈ N *there holds with p<sup>U</sup> <sup>λ</sup>* = *α*, *q U <sup>λ</sup>* = (*α* + *β*A) *λ* (*α* + *β*H) <sup>1</sup>−*<sup>λ</sup>* <sup>−</sup> *<sup>α</sup>*

(*a*) *B U <sup>λ</sup>*,*X*0,1 <sup>=</sup> exp n*<sup>q</sup> U <sup>λ</sup>* − *β<sup>λ</sup>* · *X*<sup>0</sup> o < 1,

$$\begin{aligned} (b) \qquad & \quad \text{the sequence } \left(B^{ll}\_{\lambda, \mathbf{X}\_0, \mathbb{N}}\right)\_{n \in \mathbb{N}} \text{ of upper bounds for } H\_{\lambda}(P\_{\mathcal{A}, n} || P\_{\mathcal{H}, n}) \text{ given by } \\ & \quad \quad \quad \left\{ \begin{array}{ll} (\cdot, \mathbf{u}^{\mathrm{II}}) & \sum\ (\cdot, \mathbf{u}^{\mathrm{II}}, \mathbf{u}^{\mathrm{II}}) \end{array} \right\} \end{aligned}$$

$$B\_{\lambda, X\_0, n}^{\mathcal{U}} = \exp\left\{ a\_n^{(q\_\lambda^{\mathcal{U}})} \cdot X\_0 + \sum\_{k=1}^n b\_k^{(p\_\lambda^{\mathcal{U}} \mathcal{A}\_\lambda^{\mathcal{U}})} \right\}$$

*is strictly decreasing,*

(*c*) lim*n*→<sup>∞</sup> *B U <sup>λ</sup>*,*X*0,*<sup>n</sup>* <sup>=</sup> <sup>0</sup> <sup>=</sup> lim*n*→<sup>∞</sup> *Hλ*(*P*A,*n*||*P*H,*n*) ,

$$\mu(d) \qquad \lim\_{n \to \infty} \frac{1}{n} \log \mathcal{B}\_{\lambda, X\_0, n}^{\mathrm{II}} = \left. p\_{\lambda}^{\mathrm{II}} \cdot e^{\mathbf{x}\_0^{(q\_{\lambda}^{\mathrm{II}})}} - \mathfrak{a}\_{\lambda} \right| = \left. \mathfrak{a} \left( e^{\mathbf{x}\_0^{(q\_{\lambda}^{\mathrm{II}})}} - 1 \right) < 0 \dots$$

(*e*) *the map X*<sup>0</sup> 7→ *B U λ*,*X*0,*n is strictly decreasing*.

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

From Properties 3 (P16) one gets *φλ*(0) < 0, whereas *φ* 0 *λ* (0) can assume any sign, take e.g., the parameters (*β*A, *β*H, *α*A, *α*H, *λ*) = (1.8, 0.9, 2.7, 0.7, 0.5) for *φ* 0 *λ* (0) < 0, (*β*A, *β*H, *α*A, *α*H, *λ*) = (1.8, 0.9, 2.8, 0.7, 0.5) for *φ* 0 *λ* (0) = <sup>0</sup> and (*β*A, *β*H, *α*A, *α*H, *λ*) = (1.8, 0.9, 2.9, 0.7, 0.5) for *φ* 0 *λ* (0) > 0; within our running-example epidemiological context of Section 2.3, this corresponds to a "nearly dangerous" infectious-disease-transmission situation (H) (with nearly critical reproduction number *β*<sup>H</sup> = 0.9 and importation mean of *α*<sup>H</sup> = 0.7), whereas (A) describes a "dangerous" situation (with supercritical *β*<sup>A</sup> = 1.8 and *α*<sup>A</sup> = 2.7, 2.8, 2.9). However, in all three subcases there holds max*x*∈N<sup>0</sup> *φλ*(*x*) ≤ max*x*∈[0,∞[ *<sup>φ</sup>λ*(*x*) < 0. Thus, there clearly exist parameters *<sup>p</sup> U <sup>λ</sup>* = *p <sup>U</sup>* (*β*A, *<sup>β</sup>*H, *<sup>α</sup>*A, *<sup>α</sup>*H, *<sup>λ</sup>*), *<sup>q</sup> U <sup>λ</sup>* = *q <sup>U</sup>* (*β*A, *<sup>β</sup>*H, *<sup>α</sup>*A, *<sup>α</sup>*H, *<sup>λ</sup>*) with *<sup>p</sup> U λ* ∈ - *α λ* A *α* 1−*λ* H , *α<sup>λ</sup>* - and *q U λ* ∈ - *β λ* A *β* 1−*λ* H , *β<sup>λ</sup>* - (implying (47)) such that (35) is satisfied. As explained above, we get the following

**Proposition 8.** *For all* (*β*A, *β*H, *α*A, *α*H, *λ*) ∈ P*SP,3a*×]0, 1[ *there exist parameters p U λ* , *q U <sup>λ</sup> which satisfy p U λ* ∈ - *α λ* A *α* 1−*λ* H , *α<sup>λ</sup>* - *and q U λ* ∈ - *β λ* A *β* 1−*λ* H , *β<sup>λ</sup>* - *as well as* (35) *for all x* ∈ N0*, and for all such pairs* (*p U λ* , *q U λ* ) *and all initial population sizes X*<sup>0</sup> ∈ N *there holds*

$$\mathcal{S}(\boldsymbol{a}) \qquad \mathcal{B}\_{\lambda, \mathcal{X}\_0, 1}^{\mathrm{II}} = \ \exp\left\{ \left( q\_{\lambda}^{\mathrm{II}} - \beta\_{\lambda} \right) \cdot \mathcal{X}\_0 + p\_{\lambda}^{\mathrm{II}} - \boldsymbol{a}\_{\lambda} \right\} < 1/2$$

$$\text{If } (b) \qquad \text{the sequence } \left( \mathcal{B}\_{\lambda, X\_0, n}^{\text{II}} \right)\_{n \in \mathbb{N}} \text{ of upper bounds for } \mathcal{H}\_{\lambda}(P\_{\mathcal{A}, n} || P\_{\mathcal{H}, n}) \text{ given by}$$

$$B\_{\lambda, X\_0, \eta}^{\mathcal{U}} = \exp\left\{ a\_n^{(q\_\lambda^{\mathcal{U}})} X\_0 + \sum\_{k=1}^n b\_k^{(p\_\lambda^{\mathcal{U}}, q\_\lambda^{\mathcal{U}})} \right\},$$

*is strictly decreasing,*

$$(c) \qquad \lim\_{n \to \infty} B\_{\lambda, X\_0, n}^{II} = \begin{array}{c} 0 \ = \lim\_{n \to \infty} H\_{\lambda}(P\_{\mathcal{A}, n} || P\_{\mathcal{H}, n}) \ . \end{array}$$

$$\mu(d) = \lim\_{n \to \infty} \frac{1}{n} \log \mathcal{B}\_{\lambda, X\_0, \mathfrak{n}}^{\mathcal{U}} = \left. p\_{\lambda}^{\mathcal{U}} \cdot e^{\chi\_0^{(q\_{\lambda}^{\mathcal{U}})}} - \mathfrak{a}\_{\lambda} \right. \\ < \left. 0 \right.$$

(*e*) *the map X*<sup>0</sup> 7→ *B U λ*,*X*0,*n is strictly decreasing*.

Notice that all parts of this proposition also hold true for parameter pairs (*p U λ* , *q U λ* ) satisfying (35) and additionally either *p U <sup>λ</sup>* = *αλ*, *q U <sup>λ</sup>* < *β<sup>λ</sup>* or *p U <sup>λ</sup>* < *αλ*, *q U <sup>λ</sup>* = *βλ*.

Let us briefly illuminate the above-mentioned possible parameter choices, where we begin with the case of *φ* 0 *λ* (0) ≤ 0, which corresponds to *λβ*<sup>A</sup> (*α*A/*α*H) *<sup>λ</sup>*−<sup>1</sup> + (<sup>1</sup> <sup>−</sup> *<sup>λ</sup>*)*β*<sup>H</sup> (*α*A/*α*H) *<sup>λ</sup>* <sup>−</sup> *<sup>β</sup><sup>λ</sup>* <sup>≤</sup> <sup>0</sup> (cf. (P17)); then, the function *φλ*(·) is strictly negative, strictly decreasing, and–due to (P19)–strictly concave (and thus, the assumption *<sup>α</sup>*H−*α*<sup>A</sup> *β*A−*β*<sup>H</sup> < 0 is superfluous here). One pragmatic but yet reasonable parameter

choice is the following: take any intercept *p U λ* ∈ [*α λ* A *α* 1−*λ* H , *αλ*] such that (*p U <sup>λ</sup>* − *αλ*) + 2(*φλ*(1) − (*p U λ* − *αλ*)) ≥ *φλ*(2) i.e., 2 (*α*<sup>A</sup> + *β*A) *λ* (*α*<sup>H</sup> + *β*H) <sup>1</sup>−*<sup>λ</sup>* <sup>−</sup> *<sup>p</sup> U <sup>λ</sup>* + *α<sup>λ</sup>* ≥ (*α*<sup>A</sup> + 2*β*A) *λ* (*α*<sup>H</sup> + 2*β*H) 1−*λ* and *q U λ* := *φλ*(1) − (*p U <sup>λ</sup>* − *αλ*) + *β<sup>λ</sup>* = (*α*<sup>A</sup> + *β*A) *λ* (*α*<sup>H</sup> + *β*H) <sup>1</sup>−*<sup>λ</sup>* <sup>−</sup> *<sup>p</sup> U λ* , which corresponds to a linear function *φ U <sup>λ</sup>* which is (i) nonpositive on N<sup>0</sup> and strictly negative on N, and (ii) larger than or equal to *φ<sup>λ</sup>* on N0, strictly larger than *φ<sup>λ</sup>* on N\{1, 2}, and equal to *φ<sup>λ</sup>* at the point *x* = 1 ("discrete tangent or secant line through *x* = 1"). One can easily see that (due to the restriction (34)) not all *p U λ* ∈ [*α λ* A *α* 1−*λ* H , *αλ*] might qualify for the current purpose. For the particular choice *p U <sup>λ</sup>* = *α λ* A *α* 1−*λ* H and *q U <sup>λ</sup>* = (*α*<sup>A</sup> + *β*A) *λ* (*α*<sup>H</sup> + *β*H) <sup>1</sup>−*<sup>λ</sup>* <sup>−</sup> *<sup>α</sup> λ* A *α* 1−*λ* H one obtains *r U <sup>λ</sup>* = *p U <sup>λ</sup>* − *α<sup>λ</sup>* = *b* (*p U λ* ,*q U λ* ) <sup>1</sup> < 0 (cf. Lemma A1) and *s U <sup>λ</sup>* = *q U <sup>λ</sup>* − *β<sup>λ</sup>* = *φλ*(1) − *φλ*(0) = *a* (*q U λ* ) <sup>1</sup> < 0 (secant line through *φλ*(0) and *φλ*(1)).

For the remaining case *φ* 0 *λ* (0) > 0, which corresponds to *λβ*<sup>A</sup> (*α*A/*α*H) *<sup>λ</sup>*−<sup>1</sup> + (<sup>1</sup> <sup>−</sup> *λ*)*β*<sup>H</sup> (*α*A/*α*H) *<sup>λ</sup>* <sup>−</sup> *<sup>β</sup><sup>λ</sup>* <sup>&</sup>gt; 0, the function *<sup>φ</sup>λ*(·) is strictly negative, strictly concave and hump-shaped (cf. (P18)). For the derivation of the parameter choices, we employ *<sup>x</sup>*max :<sup>=</sup> argmax*x*∈]0,∞[ *φλ*(*x*) which is the unique solution of

$$\lambda \beta\_{\mathcal{A}} \left[ \left( \frac{f\_{\mathcal{A}}(\mathbf{x})}{f\_{\mathcal{H}}(\mathbf{x})} \right)^{\lambda - 1} - 1 \right] + (1 - \lambda) \beta\_{\mathcal{H}} \left[ \left( \frac{f\_{\mathcal{A}}(\mathbf{x})}{f\_{\mathcal{H}}(\mathbf{x})} \right)^{\lambda} - 1 \right] = 0, \qquad \mathbf{x} \in ]0, \infty[ , \tag{50}$$

(cf. (P17), (P19)); notice that *x* = *x* ∗ := *α*H−*α*<sup>A</sup> *β*A−*β*<sup>H</sup> ∈ ]0, ∞[ formally satisfies the Equation (50) but does not qualify because of the current restriction *x* ∗ < 0.

Let us first inspect the case *φλ*(b*x*maxc) > *φλ*(b*x*maxc + 1), where b*x*c denotes the integer part of *x*. Consider the subcase *φλ*(b*x*maxc) + b*x*maxc (*φλ*(b*x*maxc) − *φλ*(b*x*maxc + 1)) ≤ 0, which means that the secant line through *φλ*(b*x*maxc) and *φλ*(b*x*maxc + 1) possesses a non-positive intercept. In this situation it is reasonable to choose as *intercept* any *p U <sup>λ</sup>* − *α<sup>λ</sup>* = *b* (*p U λ* ,*q U λ* ) <sup>1</sup> = *r U λ* ∈ [*φλ*(b*x*maxc), *φλ*(b*x*maxc) + b*x*maxc (*φλ*(b*x*maxc) − *φλ*(b*x*maxc + 1))], and as corresponding *slope q U <sup>λ</sup>* − *α<sup>λ</sup>* = *a* (*q U λ* ) <sup>1</sup> = *s U <sup>λ</sup>* = *φλ*(b*x*maxc)−*r U λ* (b*x*maxc)−<sup>0</sup> <sup>≤</sup> 0. A larger intercept would lead to a linear function *<sup>φ</sup> U λ* for which (35) is not valid at b*x*maxc + 1. In the other subcase *φλ*(b*x*maxc) + *x*max (*φλ*(b*x*maxc) − *φλ*(b*x*maxc + 1)) > 0, one can choose any intercept *p U <sup>λ</sup>* − *α<sup>λ</sup>* = *b* (*p U λ* ,*q U λ* ) <sup>1</sup> = *r U λ* ∈ [*φλ*(b*x*maxc), 0] and as corresponding slope *q U <sup>λ</sup>* − *α<sup>λ</sup>* = *a* (*q U λ* ) <sup>1</sup> = *s U <sup>λ</sup>* = *φλ*(b*x*maxc)−*r U λ* (b*x*maxc)−<sup>0</sup> <sup>≤</sup> <sup>0</sup> (notice that the corresponding line *<sup>φ</sup> U λ* is on ]b*x*maxc, ∞[ strictly larger than the secant line through *φλ*(b*x*maxc) and *φλ*(b*x*maxc + 1)).

If *φλ*(b*x*maxc) ≤ *φλ*(b*x*maxc + 1), one can proceed as above by substituting the crucial pair of points (b*x*maxc, b*x*maxc + 1) with (b*x*maxc + 1, b*x*maxc + 2) and examining the analogous two subcases.

## *3.9. Upper Bounds for the Cases* (*β*A, *β*H, *α*A, *α*H, *λ*) ∈ P*SP,3b*×]0, 1[

The only difference to the preceding Section 3.8 is that–due to Properties 3 (P15)–the maximum value of *φλ*(·) now achieves 0, at the positive *non-integer* point *x*max = *x* ∗ = *α*H−*α*<sup>A</sup> *β*A−*β*<sup>H</sup> ∈ ]0, ∞[\N (take e.g., (*β*A, *β*H, *α*A, *α*H, *λ*) = (1.8, 0.9, 1.1, 3.0, 0.5) as an example, which within our running-example epidemiological context of Section 2.3 corresponds to a "nearly dangerous" infectious-disease-transmission situation (H) (with nearly critical reproduction number *β*<sup>H</sup> = 0.9 and importation mean of *α*<sup>H</sup> = 3), whereas (A) describes a "dangerous" situation (with supercritical *β*<sup>A</sup> = 1.8 and *α*<sup>A</sup> = 1.1)); this implies that *φλ*(*x*) < 0 for all *x* on the relevant subdomain N0. Due to (P16), (P17) and (P19) one gets automatically *λβ*<sup>A</sup> (*α*A/*α*H) *<sup>λ</sup>*−<sup>1</sup> + (<sup>1</sup> <sup>−</sup> *<sup>λ</sup>*)*β*<sup>H</sup> (*α*A/*α*H) *<sup>λ</sup>* <sup>−</sup> *<sup>β</sup><sup>λ</sup>* <sup>&</sup>gt; <sup>0</sup> for all *λ* ∈]0, 1[. Analogously to Section 3.8, there exist parameter *p U λ* ∈ [*α λ* A *α* 1−*λ* H , *αλ*] and *q U λ* ∈ [*β λ* A *β* 1−*λ* H , *βλ*] such that (47) and (35) are satisfied. Thus, all the assertions (a) to (e) of Proposition 8 also hold true for the current parameter constellations.

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

The only difference to the preceding Section 3.9 is that the maximum value of *φλ*(·) now achieves 0 at the *integer* point *x*max = *x* ∗ = *α*H−*α*<sup>A</sup> *β*A−*β*<sup>H</sup> ∈ N (take e.g., (*β*A, *β*H, *α*A, *α*H, *λ*) = (1.8, 0.9, 1.2, 3.0, 0.5) as an example). Accordingly, there do not exist parameters *p U λ* , *q U λ* , such that (35) and (47) are satisfied simultaneously. The only parameter pair that ensures exp <sup>n</sup> *a* (*q U λ* ) *<sup>n</sup>* · *X*<sup>0</sup> + ∑ *n k*=1 *b* (*p U λ* ,*q U λ* ) *k* o ≤ 1 for all *n* ∈ N and all *X*<sup>0</sup> ∈ N without further investigations, leads to the choices *p U <sup>λ</sup>* = *α<sup>λ</sup>* as well as *q U <sup>λ</sup>* = *βλ*. Consequently, *B U <sup>λ</sup>*,*X*0,*<sup>n</sup>* ≡ 1, which coincides with the general upper bound (9), but violates the above-mentioned desired Goal (G1). However, there might exist parameters *p U <sup>λ</sup>* < *αλ*, *q U <sup>λ</sup>* > *β<sup>λ</sup>* or *p U <sup>λ</sup>* > *αλ*, *q U <sup>λ</sup>* < *βλ*, such that at least the parts (c) and (d) of Proposition 8 are satisfied. Nevertheless, by using a conceptually different method we can prove

*<sup>H</sup>λ*(*P*A,*n*||*P*H,*n*) <sup>&</sup>lt; <sup>1</sup> <sup>∀</sup> *<sup>n</sup>* <sup>∈</sup> <sup>N</sup>\{1} as well as the convergence lim*n*→<sup>∞</sup> *Hλ*(*P*A,*n*||*P*H,*n*) = 0 (51)

which will be used for the study of complete asymptotical distinguishability (entire separation) below. This proof is provided in Appendix A.1.
