# A New Model for the Dynamics of Hepatitis C Infection: Derivation, Analysis and Implications

## Abstract

**:**

## 1. Introduction

- Globally, an estimated 71 million people were living with chronic HCV infection;
- An estimated 1.75 million new HCV infections occurred worldwide, while 399,000 people died from end-stage HCV infection and 843,000 were cured;
- 20% of HCV-infected persons (14 million) have been diagnosed, and of these, 7.4% (1.1 million) had started treatment;
- HCV infection affects all regions with the highest reported prevalence in the Eastern Mediterranean and European Regions.

## 2. A Review of Existing Models of HCV Infection

#### 2.1. The Neumann Model

#### 2.2. The First Dahari Model

#### 2.3. The Second Dahari Model

#### 2.4. Other Models

## 3. A New Model of HCV Infection

#### 3.1. Cell Regeneration

#### 3.2. Stem Cells

#### 3.3. Infection

#### 3.4. The Revised Model

#### 3.5. Spontaneous Clearance

#### 3.6. Non-Dimensionalisation

## 4. Validation of the New HCV Model

**Theorem**

**1.**

**Proof.**

**Theorem**

**2.**

**Proof.**

**Theorem**

**3.**

**Proof.**

## 5. Analysis of the New HCV Model

#### 5.1. Assumptions

#### 5.2. Steady States ($S=0$)

#### 5.3. Bifurcations $(S=0)$

#### 5.3.1. Bifurcation on the Uninfected Branch

**Lemma**

**1.**

**Proof.**

**Lemma**

**2.**

**Proof.**

#### 5.3.2. Bifurcation on the Pure Infection Branch

**Lemma**

**3.**

**Proof.**

**Lemma**

**4.**

**Proof.**

- if ${c}_{4}<0$, then $x\left(\u03f5\right)$ has a positive slope, and valid solutions exist locally only for $\u03f5\ge {\u03f5}_{2}$;
- if ${c}_{4}>0$, then $x\left(\u03f5\right)$ has a negative slope, and valid solutions exist locally only for $\u03f5\le {\u03f5}_{2}$.

#### 5.3.3. The Infected Steady State Branch ($S=0$)

**Theorem**

**4.**

**Remark**

**1.**

**Proof.**

#### 5.4. Steady States ($S>0$)

#### 5.4.1. Bifurcation on the Uninfected Branch

#### 5.4.2. Bifurcation on the Pure Infection Branch

- if ${c}_{4}<0$, then no limit points exist;
- if ${c}_{4}>0$, then there are two limit points at $x=\pm \sqrt{S{y}_{p}/{c}_{4}}$.

#### 5.4.3. Infected Branch of Solutions

#### 5.4.4. The Case of ${\u03f5}_{2}<0$

## 6. Stability

## 7. Comparison with the Neumann/Dahari Models

- For the Neumann/Dahari models, treatment will only be effective once the treatment factor $\u03f5$ exceeds a critical value determined by the bifurcation point, regardless of the viral load when treatment commences. For our model, if the viral load is close to the infected steady state before treatment starts, then similarly, the treatment factor $\u03f5$ must exceed the critical value ${\u03f5}_{LP}$ determined by the limit point for the treatment to be effective. However, if the infection is caught and treated in the early stages, while the viral load is still relatively low, then our model predicts that a lower drug dose, with a corresponding smaller value of the treatment parameter $\u03f5$, will be effective.
- As mentioned previously, once treatment is stopped, the prediction of the Neumann/Dahari models is that the infection will take hold again unless the infected hepatocytes and virus have been completed eliminated during treatment. The prediction from our model, if the bifurcation point on the uninfected solution branch occurs at a negative value of $\u03f5$ (${\u03f5}_{0}<0$), is that the body will be able to eliminate a small amount of infected hepatocytes and virus cells without further treatment once their levels have been reduced sufficiently. On the other hand, if the bifurcation point on the uninfected branch occurs at a positive value of $\u03f5$ (${\u03f5}_{0}>0$), then our model predicts in this case that the infection will take hold again on cessation of treatment unless the infected hepatocytes and virus cells have been completely eliminated during treatment. However, our model also predicts that continuing with a low level of drug treatment, corresponding to a small value of $\u03f5$, will stop the infection recurring in this case.
- The Neumann/Dahari models suggest that treatment will only be effective if the treatment parameter $\u03f5$ is greater than the critical value during the whole period of treatment, which is the way that patients are generally treated in practice. Our model suggests that the drug dose could be reduced as treatment progresses and that this will still be effective, provided that it is not reduced too far too quickly. If this is indeed the case, it could save some of the costs of treatment, and a lower drug dosage may also mean a reduction in side effects, which would benefit the patient.

## 8. Description of Observed Viral Load Profiles

#### 8.1. Sustained Virologic Response

#### 8.2. Relapse

#### 8.3. Partial Virologic Response

#### 8.4. Breakthrough

#### 8.5. Null Response

#### 8.6. Biphasic and Triphasic Decline

#### 8.7. Initial Increase in Viral Load

#### 8.8. Direct Acting Antiviral Agents

- (i)
- It has been noted that the first phase decline when treating with DAAs is both longer and faster than when using IFN and RBV. Since the rate of the first phase decline is essentially given by the parameter C (or c for the original equations), it has been suggested that both $\u03f5$ and c should be increased in the models for treatment with DAAs [25]. However, we claim that an increase in $\u03f5$ alone is sufficient to produce a longer and faster first phase. To see this, we consider the decline in viral load during the first phase that is given in (77). For this solution, we find that the initial rate of decline is:$$\begin{array}{c}\hfill {\left(\right)}_{\frac{d\left({log}_{10}z\right)}{d\tau}}\tau =0\\ =& -\left(\right)open="("\; close=")">1-\frac{{z}_{p}\left({\u03f5}^{*}\right)}{{z}_{p}\left(0\right)}C{log}_{10}e\hfill \end{array}& =& -{\u03f5}^{*}C{log}_{10}e\hfill $$
- (ii)
- The second observation made for DAAs is that the second phase is also faster than that for treatment with IFN and RBV. We have seen in Section 8.6 that for our model, the rate of decay in the second phase is proportional to ${e}^{-{\lambda}_{0}t}$ where $-{\lambda}_{0}<0$ is the solution of the characteristic equation (78) that is closest to zero. With our assumption that $\eta =1-\alpha \u03f5$ is constant, (78) becomes:$$p\left(\lambda \left(\u03f5\right)\right)=\lambda {\left(\u03f5\right)}^{2}+(\eta B+C+D)\lambda \left(\u03f5\right)-\eta BP(1-\u03f5)+D(\eta B+C)=0$$$${\lambda}^{\prime}\left(\u03f5\right)=-\frac{\eta BP}{(\eta B+C+D+2\lambda (\u03f5\left)\right)}$$$${\lambda}^{\prime}\left({\u03f5}^{*}\right)=-\frac{\eta BP}{(\eta B+C+D-2{\lambda}_{0})}$$$$p\left(\right)open="("\; close=")">-\frac{1}{2}(\eta B+C+D)0$$$$-{\lambda}_{1}<-\frac{1}{2}(\eta B+C+D)<-{\lambda}_{0}$$

## 9. Data Fitting

- In the PVR, breakthrough and triphasic cases, we have $I\left(t\right)<V\left(t\right)$ for all t, but in the null response case, $I\left(t\right)$ is significantly higher than $V\left(t\right)$.
- The initial viral load is highest in the PVR case, and we predicted in Section 8.3 that PVR would be associated with a high initial viral load.
- In the null response case, it is interesting to observe that the fitted viral load V and infected hepatocyte concentration I both reduce towards zero, but very slowly. The start of the decline in these variables can be observed from around 300 days in Figure 12c. At 1000 days after the start of treatment, the predicted values are ${log}_{10}I\left(1000\right)=5.1202$ and ${log}_{10}V\left(1000\right)=2.8515$.
- In Section 8.6, we stated that triphasic decline might be expected when the patient has been infected for a long time before treatment, which means that the viral load will be high while the healthy hepatocyte concentration will be very low. This is precisely the situation observed in Figure 12d.
- In Section 8.7, we showed that an initial increase in viral load at the start of treatment is possible, and we see this in the null response case.
- We saw in Section 3.1 that the regeneration rate for a healthy liver is $1.15\times {10}^{-2}$ day${}^{-1}$ for females and $1.11\times {10}^{-2}$ day${}^{-1}$ for males. This is effectively the parameter ${r}_{T}+d$ in our new model. The value of this parameter when fitted to data is a little lower than this for the PVR and breakthrough cases and is slightly higher in the null response and triphasic cases.
- The condition (33) for the solution to be bounded for all $t\ge 0$ in terms of the parameters we are using here is given by:$$min(s,{p}^{*})(1+R)-({r}_{T}+d)(1+D(1+R))<0$$

## 10. Conclusions

- If the infection is caught and treated in the early stages, then our model predicts that a lower drug dose may be effective in eliminating the infection.
- If the viral load relapses on cessation of treatment, then continuing with a low level of drug treatment may keep the viral load low.
- The infected branch from the bifurcation on the uninfected branch to the limit point has a positive slope, and this suggests that the drug dose could be reduced as treatment progresses, which could save some of the costs of treatment and give a reduction in side effects for the patient.

## Acknowledgments

## Conflicts of Interest

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**Figure 2.**The bifurcation from the uninfected steady state. The bifurcation point occurs for (

**a**) $\u03f5<0$, (

**b**) $\u03f5=0$, (

**c**) $\u03f5>0$. Stable steady states are indicated by solid lines and unstable steady states by dashed lines.

**Figure 3.**The invariant region in the positive octant bounded by the cylinder $\mathcal{C}$ (see Theorem 1) and the plane (34).

**Figure 4.**Solutions near the bifurcation involving the infected (red) and pure infection (blue) branches. Solid lines indicate stable solutions, while dashed lines represent unstable solutions.

**Top**: ${c}_{4}>0$.

**Middle**: ${c}_{4}<0$, ${c}_{6}<0$.

**Bottom**: ${c}_{4}<0$, ${c}_{6}>0$.

**Figure 5.**The pure infection (blue) and infected (red) steady state branches. Note that the dashed-dotted lines indicate invalid solutions.

**Figure 6.**Unfolding of the transcritical bifurcation given by (72) in the $(\u03f5,x)$ plane for ${c}_{4}>0$ (top) and ${c}_{4}<0$ (bottom).

**Figure 7.**Steady state solutions where the bifurcation between the infected and pure infection branches has been unfolded as $S>0$, assuming that ${\u03f5}_{2}>0$. The dashed-dotted lines are invalid solutions as $x<0$.

**Figure 8.**The steady state solutions when (73) holds and $S>0$. The dashed-dotted line consists of invalid solutions as $x<0$.

**Figure 9.**The bifurcation diagram, where solid lines indicate stable solutions and dashed lines indicate unstable solutions. Note that the vertical scale is either (

**a**) z or (

**b**) ${log}_{10}z$.

**Figure 10.**A typical bifurcation diagram for the Neumann/Dahari models, where solid lines indicate stable solutions and dashed lines indicate unstable solutions. Note that the vertical scale is either (

**a**) z or (

**b**) ${log}_{10}z$.

**Figure 11.**The decline in viral load given by (77) with ${z}_{p}\left(0\right)=1$, $C=1$ and ${z}_{p}\left({\u03f5}^{*}\right)=0.5$ (blue) or ${z}_{p}\left({\u03f5}^{*}\right)=0.2$ (red).

**Figure 12.**Plots of the viral load V (blue), healthy hepatocytes T (green) and infected hepatocytes I (red) against time fitted to the viral load datasets for (

**a**) partial virologic response (PVR), (

**b**) breakthrough, (

**c**) null response, (

**d**) triphasic.

PVR | Breakthrough | Null Response | Triphasic | |
---|---|---|---|---|

s (day${}^{-1}$) | $1.1178\times {10}^{-1}$ | $1.5104\times {10}^{-4}$ | $4.6260\times {10}^{-3}$ | $3.1259\times {10}^{-3}$ |

${r}_{T}{T}_{\mathrm{max}}$ (IU/ml/day) | $1.0645\times {10}^{4}$ | $2.8556\times {10}^{4}$ | $1.2920\times {10}^{6}$ | $1.1149\times {10}^{2}$ |

${r}_{T}+d$ (day${}^{-1}$) | $1.9927\times {10}^{-3}$ | $2.9890\times {10}^{-3}$ | $3.8518\times {10}^{-2}$ | $1.7882\times {10}^{-2}$ |

R | $3.0078\times {10}^{1}$ | $1.1686\times {10}^{3}$ | $2.6011$ | $2.0350\times {10}^{-1}$ |

D | $5.8954\times {10}^{1}$ | $5.7302\times {10}^{2}$ | $1.1064$ | $1.0962\times {10}^{1}$ |

${\beta}^{*}$ (ml/IU/day) | $8.3376\times {10}^{-9}$ | $7.1149\times {10}^{-9}$ | $1.9493\times {10}^{-7}$ | $3.3281\times {10}^{-8}$ |

${p}^{*}$ (day${}^{-1}$) | $2.0396\times {10}^{2}$ | $9.4025\times {10}^{1}$ | $3.4868\times {10}^{-2}$ | $1.1646\times {10}^{3}$ |

c (day${}^{-1}$) | $1.7908\times {10}^{1}$ | $3.3659$ | $2.7784\times {10}^{-4}$ | $1.4294$ |

$T\left(0\right)$ (IU/ml) | $3.3246$ | $8.2935\times {10}^{6}$ | $1.7755\times {10}^{4}$ | $1.9948$ |

$I\left(0\right)$ (IU/ml) | $4.1752\times {10}^{5}$ | $9.5880\times {10}^{3}$ | $1.4523\times {10}^{6}$ | $1.0355\times {10}^{2}$ |

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Aston, P.J.
A New Model for the Dynamics of Hepatitis C Infection: Derivation, Analysis and Implications. *Viruses* **2018**, *10*, 195.
https://doi.org/10.3390/v10040195

**AMA Style**

Aston PJ.
A New Model for the Dynamics of Hepatitis C Infection: Derivation, Analysis and Implications. *Viruses*. 2018; 10(4):195.
https://doi.org/10.3390/v10040195

**Chicago/Turabian Style**

Aston, Philip J.
2018. "A New Model for the Dynamics of Hepatitis C Infection: Derivation, Analysis and Implications" *Viruses* 10, no. 4: 195.
https://doi.org/10.3390/v10040195