# A Mathematical Model for Early HBV and -HDV Kinetics during Anti-HDV Treatment

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## Abstract

**:**

## 1. Introduction

## 2. Background

_{1}(t) is the number of HBV-infected cells at time t, y

_{2}(t) is the number of HDV-infected cells at time t, y

_{3}(t) is the number of infected cells with both HBV and HDV at time t, v

_{1}(t) is the HBV viral load at time t, and v

_{2}(t) is the HDV viral load at time t. Both the variables and all of the parameters of the model are shown in Table 1.

## 3. Materials and Methods

_{0}, constant, i.e., ignoring the dynamics of infected cells, such as infection of susceptible cells and/or infected cell proliferation and/or death (assuming a long-infected cell life-span as previously performed [16,17]). Treatment is assumed to solely block HDV production rate p

_{1}by (1 − ε), where 0 ≤ ε ≤ 1 represents its efficacy. To account for the biphasic HDV RNA decay in the absence of infected cell dynamics, we assume that treatment has additional time-dependent inhibitory effects on HDV production and we model it by decreasing p

_{1}further by e

^{−gt}, where g ≥ 0 is a constant and t is the time in days post-treatment initiation, as previously performed [16,18].

_{1}, at time t = 0, D = D

_{0}, one plugs in the initial condition and C

_{1}is obtained:

## 4. Results

## 5. Discussion and Conclusions

^{−gt}), which represents the time-dependent blocking of viral production that explains the second-phase decline observed with LNF. The HBV equation is inversely dependent on HDV as per the empirical results, and includes patient-specific variables. The clearance rate of HDV and HBV from circulation is considered similar to that in [11]. We assume that the number of infected cells is constant, a feature that we intend to relax in future models.

## Supplementary Materials

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**A schematic diagram of our new model. The figure relates to the phase where HDV declines and HBV increases.

**Table 1.**The variables and the parameters in the model of de Sousa and Cunha [12].

x(t) | the number of uninfected cells at time t |

y_{1}(t) | the number of HBV-infected cells at time t |

y_{2}(t) | the number of HDV-infected cells at time t |

y_{3}(t) | the number of infected cells with both HBV and HDV at time t |

v_{1}(t) | the HBV viral load at time t |

v_{2}(t) | the HDV viral load at time t |

λ | production rate of uninfected cells (day^{−1}) |

δ | death rate of uninfected cells (day^{−1}) |

b_{1} | infection rate of HBV-infected cells (day^{−1}) |

b_{2} | infection rate of HDV-infected cells (day^{−1}) |

d_{1} | death rate of HBV-infected cells (day^{−1}) |

d_{2} | death rate of HDV-infected cells (day^{−1}) |

u_{1} | clearance rate of HBV virions (day^{−1}) |

u_{2} | clearance rate of HDV virions (day^{−1}) |

k_{1} | production rate of HBV virions (day^{−1}) |

k_{2} | production rate of HDV virions (day^{−1}) |

k_{3} | production rate of HBV virions (?) |

η | therapy efficacy of inhibiting new virus infections as a result of virus clearance |

ε | therapy efficacy of inhibiting viral production from infected cells |

**Table 2.**The variables and the parameters in the model of Packer et al. [6].

x(t) | uninfected cells |

y(t) | HBV-only-infected cells |

z(t) | HDV-only-infected cells |

w(t) | HBV-HDV-coinfected cells |

r | maximum proliferation rate (day^{−1}) |

K | homeostatic liver size (number of cells) |

α | infected cells death rate (day^{−1}) |

c | HBV inhibition coefficient |

σ | HBV infection rate (day^{−1}) |

δ | HDV infection rate (day^{−1}) |

D | HDV viral load (IU/mL) |

D_{0} | HDV viral load before treatment (IU/mL) |

B | HBV viral load (IU/mL) |

B_{0} | HBV viral load before treatment (IU/mL) |

I_{0} | Number of HBV-HDV-coinfected cells |

p_{1} | Production rate constant of HDV |

p_{2} | Production rate constant of HBV |

g | Additional treatment inhibitory effect in blocking HDV production (day^{−1}) |

ε | Treatment efficacy in blocking viral production (between 0 and 1) |

n | HBV exponent that governs the rate increase in HBV |

c | HDV and HBV clearance constant from blood (day^{−1}) |

**Table 4.**Model parameter values for Patient 1 and Patient 2 of [7].

B_{0} | D_{0} | g | ε | n | τ | c | |
---|---|---|---|---|---|---|---|

Patient 1 | 29,512 | 5.5 × 10^{5} | 0.042 | 0.73 | 2.4 | 0.1 | 0.51 |

Patient 2 | 148 | 2.2 × 10^{5} | 0.023 | 0.82 | 3.8 | 0.1 | 0.51 |

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**MDPI and ACS Style**

Zakh, R.; Churkin, A.; Bietsch, W.; Lachiany, M.; Cotler, S.J.; Ploss, A.; Dahari, H.; Barash, D. A Mathematical Model for Early HBV and -HDV Kinetics during Anti-HDV Treatment. *Mathematics* **2021**, *9*, 3323.
https://doi.org/10.3390/math9243323

**AMA Style**

Zakh R, Churkin A, Bietsch W, Lachiany M, Cotler SJ, Ploss A, Dahari H, Barash D. A Mathematical Model for Early HBV and -HDV Kinetics during Anti-HDV Treatment. *Mathematics*. 2021; 9(24):3323.
https://doi.org/10.3390/math9243323

**Chicago/Turabian Style**

Zakh, Rami, Alexander Churkin, William Bietsch, Menachem Lachiany, Scott J. Cotler, Alexander Ploss, Harel Dahari, and Danny Barash. 2021. "A Mathematical Model for Early HBV and -HDV Kinetics during Anti-HDV Treatment" *Mathematics* 9, no. 24: 3323.
https://doi.org/10.3390/math9243323