# Handling Hysteresis in a Referral Marketing Campaign with Self-Information. Hints from Epidemics

## Abstract

**:**

## 1. Introduction and Motivations

## 2. A Referral Marketing Model with Self-Information

## 3. The Campaign-Free and the Campaign-Standing Equilibria

**Theorem**

**1.**

**Theorem**

**2.**

**Theorem**

**3.**

## 4. Sustaining the Campaign: Forward or Backward Scenario?

**Theorem**

**4.**

**Proof.**

**Remark**

**1.**

## 5. Effects of Self-Information on the Bifurcation Thresholds

- the threshold ${\alpha}^{\ast}$ increases with increasing the information variable $\zeta $. This means that an higher information increases the threshold ${\alpha}^{\ast}$, favoring the forward regime with respect to the backward scenario. In this sense, information would act as a stabilizing mechanism;
- within the backward scenario, the saddle-node bifurcation threshold ${\sigma}_{SN}$ increases with increasing the information variable $\zeta $. This means that an higher information implies a higher value of $\sigma $ in order to stop the campaign. However, the length of the bistability range $\left[{\sigma}_{c},{\sigma}_{SN}\right]$ does not have a monotone trend as function of the information variable $\zeta $. More precisely, for intermediate values of $\zeta $, the bistability range decreases whereas it increases when the values of $\zeta $ are too small or too large. This would qualitatively mean that too much or too little information, although enlarging the chances of survival of the campaign, can have eventually a destabilizing effect on the system dynamics favoring sudden collapses in broadcasters that could lead to a sudden stop of the campaign according to a hysteretic phenomenology.

**Definition**

**1.**

## 6. Conclusions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Bifurcation diagram in the plane ($\sigma ,{b}^{\ast}$). The other parameters are $\mu =0.05$; $\rho =0.25$; $\lambda =0.02$; $p=0.7$; $q=0.8$; $a=0.5$; $k=2$; $\gamma =0.2$ so that ${\alpha}^{\ast}=1.1684$ and ${\sigma}_{c}=0.6850$. The solid lines (-) denote stability; the dashed lines (- -) denote instability. (

**Left**) Forward scenario. The case $\alpha <{\alpha}^{\ast}$, $\alpha =0.4$− At $\sigma ={\sigma}_{c}=0.6850$, system (5) exhibits a forward bifurcation. (

**Right**) Backward scenario. The case $\alpha >{\alpha}^{\ast}$, $\alpha =2$− At $\sigma ={\sigma}_{c}=0.6850$, system (5) exhibits a backward bifurcation. The value ${\sigma}_{SN}=0.9105$ is the saddle-node bifurcation threshold.

**Figure 2.**Graphical representation of an hysteresis cycle on the bifurcation diagram in the plane ($\sigma ,{b}^{\ast}$) in the case $\alpha >{\alpha}^{\ast}$, where a backward scenario is obtained. The other parameters are as in Figure 1 (right). Here ${\alpha}^{\ast}=1.1684$, ${\sigma}_{c}=0.6850$ and the value ${\sigma}_{SN}=0.9105$ is the saddle-node bifurcation threshold. The solid lines (-) denote stability; the dashed lines (- -) denote instability.

**Figure 3.**Thresholds (16) as function of the information variable $\zeta $. The other parameters are chosen as in Figure 1. (

**Top-left**) The threshold ${\alpha}^{\ast}\left(\zeta \right)$ as function of $\zeta $. (

**Top-right**) The saddle-node bifurcation threshold ${\sigma}_{SN}\left(\zeta \right)$ as function of $\zeta $. The threshold ${\sigma}_{SN}$ is feasible in the range $\left(0,{\zeta}^{\ast}\right]$, with ${\zeta}^{\ast}=0.9375$ (

**Bottom**) The length of the bistability range, i.e., ${\sigma}_{SN}-{\sigma}_{c}$, within the backward scenario as function of $\zeta $. The bistability range is increasing for $\left[0,{\zeta}_{1}\right)$ and $\left({\zeta}_{2},{\zeta}^{\ast}\right)$ and it decreases for $\left[{\zeta}_{1},{\zeta}_{2}\right]$. Here ${\zeta}^{\ast}=0.9375$; ${\zeta}_{1}=0.1135$; ${\zeta}_{2}=0.8861$.

**Figure 4.**Sensitivity indices of the different thresholds ${\alpha}^{\ast}$, ${\sigma}_{c}$ and ${\sigma}^{SN}$ as function of the information variable $\zeta $. The other parameters are chosen as in Figure 1. (

**Top-left**) Plot of the sensitivity ${\varphi}_{\zeta}^{{\alpha}^{\ast}}$ versus the information variable $\zeta $; (

**Top-right**) Plot of the sensitivity ${\varphi}_{\zeta}^{{\sigma}_{c}}$ versus the information variable $\zeta $; (

**Bottom**) Plot of the sensitivity ${\varphi}_{\zeta}^{{\sigma}^{SN}}$ versus the information variable $\zeta $.

**Figure 5.**Impact of the customer satisfaction parameter q on the referral campaign in the bistability region, $\sigma \in [{\sigma}_{c},{\sigma}_{SN}]$, for different levels of self-information. Initial conditions are chosen in the neighbouring of the campaign-standing equilibrium. The other parameters are as in Figure 1. (

**Top-left**) Low level of the self-information parameter, i.e., $\zeta =0.08$ ($k=2;\gamma =0.04$) and $\sigma =0.4$. (

**Top-right**) Intermediate level of the self-information parameter, i.e., $\zeta =0.5$ ($k=2;\gamma =0.25$) and $\sigma =0.7$. (

**Bottom**) High level of the self-information parameter, i.e., $\zeta =0.9$ ($k=2;\gamma =0.45$) and $\sigma =0.9$.

**Table 1.**Sensitivity indices of the thresholds ${\alpha}^{\ast}$, ${\sigma}_{c}$ and ${\sigma}^{SN}$ for three different levels of information: low, intermediate and high. The numerical values of the system parameters used for the computations are: $\mu =0.05$; $\rho =0.25$; $\lambda =0.02$; $p=0.7$; $q=0.8$. Here ${\zeta}^{\ast}=0.9375$; ${\zeta}_{1}=0.1135$; ${\zeta}_{2}=0.8861$.

Low Information | Intermediate Information | High Information |
---|---|---|

$\mathbf{0}<\mathit{\zeta}<{\mathit{\zeta}}_{\mathbf{1}}$ | ${\mathit{\zeta}}_{\mathbf{1}}<\mathit{\zeta}<{\mathit{\zeta}}_{\mathbf{2}}$ | ${\mathit{\zeta}}_{\mathbf{2}}<\mathit{\zeta}<{\mathit{\zeta}}^{\ast}$ |

$\zeta =0.08$ | $\zeta =0.5$ | $\zeta =0.9$ |

${\varphi}_{\zeta}^{{\alpha}^{\ast}}=0.2154$ | ${\varphi}_{\zeta}^{{\alpha}^{\ast}}=0.6380$ | ${\varphi}_{\zeta}^{{\alpha}^{\ast}}=0.7614$ |

${\varphi}_{\zeta}^{{\sigma}_{c}}=0.38$ | ${\varphi}_{\zeta}^{{\sigma}_{c}}=0.74$ | ${\varphi}_{\zeta}^{{\sigma}_{c}}=0.8404$ |

${\varphi}_{\zeta}^{{\sigma}_{SN}}=0.1734$ | ${\varphi}_{\zeta}^{{\sigma}_{SN}}=0.3450$ | ${\varphi}_{\zeta}^{{\sigma}_{SN}}=0.9713$ |

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Lacitignola, D. Handling Hysteresis in a Referral Marketing Campaign with Self-Information. Hints from Epidemics. *Mathematics* **2021**, *9*, 680.
https://doi.org/10.3390/math9060680

**AMA Style**

Lacitignola D. Handling Hysteresis in a Referral Marketing Campaign with Self-Information. Hints from Epidemics. *Mathematics*. 2021; 9(6):680.
https://doi.org/10.3390/math9060680

**Chicago/Turabian Style**

Lacitignola, Deborah. 2021. "Handling Hysteresis in a Referral Marketing Campaign with Self-Information. Hints from Epidemics" *Mathematics* 9, no. 6: 680.
https://doi.org/10.3390/math9060680