Influence of Intracellular Delay on the Dynamics of Hepatitis C Virus

Abstract

In this paper, we present a delay induced model for hepatitis C virus incorporating the healthy and infected hepatocytes as well as infectious and noninfectious virions. The model is mathematically analyzed and characterized, both for the steady states and the dynamical behavior of the model. It is shown that time delay does not affect the local asymptotic stability of the uninfected steady state. However, it can destabilize the endemic equilibrium, leading to Hopf bifurcation to periodic solutions with realistic data sets. The model is also validated using 12 patient data obtained from the study, conducted at the University of Sao Paulo Hospital das clinicas.

Appendix: Calculation of \(W_{20}(\theta )\) and \(W_{11}(\theta )\)

From (30) and (32), we have

$$\begin{aligned} {\dot{W}}={\dot{x}}_{t}-{\dot{z}} q-\dot{{\bar{z}}}{\bar{q}}= & {} \left\{ \begin{array}{ll} {A W- 2 \text {Re}\{\bar{q^{*}}(0)f_{0} q (\theta )\},~~~~~~ if \theta \in [-1, 0),}\\ {A W-2 \text {Re}\{\bar{q^{*}}(0)f_{0} q (0)\}+f_{0},~~if \theta =0}, \end{array}\right. \nonumber \\= & {} A W+H(z, {\bar{z}}, \theta )~~~~(say), \end{aligned}$$

(38)

where

$$\begin{aligned} H(z, {\bar{z}}, \theta )=H_{20}(\theta )\frac{z^{2}}{2}+H_{11}(\theta )z {\overline{z}} +H_{02}(\theta )\frac{{\overline{z}}^{2}}{2} +... \end{aligned}$$

(39)

Substituting the corresponding series into (38) and comparing the coefficients, we obtain

$$\begin{aligned} (A-2 \text {i} \tau _{0}\omega _{0})W_{20}(\theta ) =-H_{20}(\theta ), A W_{11}(\theta )=-H_{11}(\theta ),... \end{aligned}$$

(40)

From (38), we know that for $\theta \in [-1, 0),$

$$\begin{aligned} H(z, {\bar{z}}, \theta ) =-\bar{q^{*}}(0)f_{0} q (\theta )- q^{*}(0) \bar{f_{0}} {\bar{q}}(\theta )=-g(z, {\bar{z}})q(\theta )-{\bar{g}}(z, {\bar{z}}){\bar{q}}(\theta ). \end{aligned}$$

(41)

Comparing the corresponding coefficients with that of (39), we get,

$$\begin{aligned} H_{20}(\theta ) =-g_{20}q(\theta )-{\bar{g}}_{02}{\bar{q}}(\theta ), H_{11}(\theta ) =-g_{11}q(\theta )-{\bar{g}}_{11}{\bar{q}}(\theta ). \end{aligned}$$

(42)

From (40),(42) and the definition of A, it follows that

$$\begin{aligned} \dot{W_{20}}(\theta )=2 \text {i}\omega _{0}\tau _{k}W_{20}(\theta )+g_{20}(\theta )q(\theta )+{\bar{g}}_{02}{\bar{q}}(\theta ). \end{aligned}$$

Note that $q(\theta )=(1, a, b, c_{1})^{T}e^{\text {i} \omega _{0}\tau _{j}\theta }$, hence

$$\begin{aligned} W_{20}(\theta )=\frac{\text {i}g_{20}}{\omega _{0}\tau _{j}}q(0)e^{\text {i} \omega _{0}\tau _{j}\theta }+\frac{\text {i}{\bar{g}}_{02}}{3\omega _{0}\tau _{j}}{\bar{q}}(0)e^{-\text {i} \omega _{0}\tau _{j}\theta }+E_{1}e^{2\text {i} \omega _{0}\tau _{j}\theta }, \end{aligned}$$

(43)

Here $E_{1}=(E_{1}^{(1)}, E_{1}^{(2)}, E_{1}^{(3)}, E_{1}^{(4)})\in R^{4}$ is a constant vector. Similarly, from (40) and (43), we get

$$\begin{aligned} W_{11}(\theta )=-\frac{\text {i}g_{11}}{\omega _{0}\tau _{j}}q(0)e^{\text {i} \omega _{0}\tau _{j}\theta }+\frac{\text {i}{\bar{g}}_{11}}{3\omega _{0}\tau _{j}}{\bar{q}}(0)e^{-\text {i} \omega _{0}\tau _{j}\theta }+E_{2}, \end{aligned}$$

(44)

where $E_{2}=(E_{2}^{(1)}, E_{2}^{(2)}, E_{2}^{(3)}, E_{2}^{(4)})\in R^{4}$ is also a constant vector. Now, we have to find an appropriate constant vector $E_1$ and $E_2$ which satisfy the above conditions. From the definition of A and (40), we obtain

$$\begin{aligned} \int _{-1}^{0} d \eta (\theta ) W_{20}(\theta )= 2 \text {i}\omega _{0} \tau _{j}W_{20}(\theta )-H_{20}(\theta ), \end{aligned}$$

(45)

and

$$\begin{aligned} \int _{-1}^{0} d \eta (\theta ) W_{11}(\theta )= -H_{11}(0), \end{aligned}$$

(46)

where $\eta (\theta )=\eta (0, \theta ).$ Using (38), we have

$$\begin{aligned} H_{20}(0)=-g_{20}q(0)-{\bar{g}}_{20}{\bar{q}}(0)+2 \tau _{j}\left( \begin{array}{c} -(1+a)\frac{r}{T_{max}}-(1-c\eta _{1})\alpha b e^{-2 \text {i} \omega _{0}\tau _{j}} \\ (1-c\eta _{1})\alpha b e^{-2 \text {i} \omega _{0}\tau _{j}} \\ 0 \\ 0 \\ \end{array} \right) \end{aligned}$$

(47)

and

$$\begin{aligned} H_{11}(0)=-g_{11}q(0)-{\bar{g}}_{11}{\bar{q}}(0)+2 \tau _{j}\left( \begin{array}{c} -(1+Re(a))\frac{r}{T_{max}}-(1-c\eta _{1})\alpha Re(b) \\ (1-c\eta _{1})\alpha Re(b) \\ 0 \\ 0 \\ \end{array} \right) \end{aligned}$$

(48)

Substituting (43) and (47) into (45), we obtain

$\left( 2 \text {i}\omega _{0} \tau _{j} I-\int _{-1}^{0}e^{2 \text {i} \omega _{0}\tau _{j}\theta } d \eta (\theta )\right) E_{1}=2 \tau _{j}\left( \begin{array}{c} -(1+a)\frac{r}{T_{max}}-(1-c\eta _{1})\alpha b e^{-2 \text {i} \omega _{0}\tau _{j}} \\ (1-c\eta _{1})\alpha b e^{-2 \text {i} \omega _{0}\tau _{j}} \\ 0 \\ 0 \\ \end{array} \right)$

which leads to

$$\begin{aligned}&\left( \begin{array}{cccc} i \omega _{0}-F&{} \frac{r T^{*}}{T_{max}}&{}(1-c \eta _{1})\alpha T^{*}&{}0\\ -(1-c \eta _{1})\alpha V_{I}^{*}e^{-i \omega _{0}\tau _{j}} &{}i \omega _{0}+d_{2} &{} -(1-c \eta _{1})\alpha T^{*}e^{-i \omega _{0}\tau _{j}}&{}0 \\ 0 &{} -\left( 1-\frac{\eta _{r}+\eta _{1}}{2}\right) \beta &{} i \omega _{0}+d_{3} &{} 0 \\ 0 &{} -\left( \frac{\eta _{r}+\eta _{1}}{2}\right) \beta &{} 0 &{} i \omega _{0}+d_{3}\\ \end{array} \right) \left( \begin{array}{c} E_{1}^{(1)}\\ E_{1}^{(2)} \\ E_{1}^{(3)} \\ E_{1}^{(4)} \\ \end{array}\right) \\&\quad =2\left( \begin{array}{c} -(1+a)\frac{r}{T_{max}}-(1-c\eta _{1})\alpha b e^{-2 \text {i} \omega _{0}\tau _{j}} \\ (1-c\eta _{1})\alpha b e^{-2 \text {i} \omega _{0}\tau _{j}} \\ 0 \\ 0 \\ \end{array} \right) \end{aligned}$$

From above, we can easily calculate a constant vector $E_{1}=(E_{1}^{(1)}, E_{1}^{(2)}, E_{1}^{(3)}, E_{1}^{(4)})\in R^{4}$. Similarly, substituting (44) and (48) into (46), we can get

$$\begin{aligned}&\left( \begin{array}{cccc} F&{} -\frac{r T^{*}}{T_{max}}&{}-(1-c \eta _{1})\alpha T^{*}&{}0\\ -(1-c \eta _{1})\alpha V_{I}^{*} &{}-d_{2} &{} -(1-c \eta _{1})\alpha T^{*}&{}0 \\ 0 &{} \left( 1-\frac{\eta _{r}+\eta _{1}}{2}\right) \beta &{} -d_{3} &{} 0 \\ 0 &{} \left( \frac{\eta _{r}+\eta _{1}}{2}\right) \beta &{} 0 &{} -d_{3}\\ \end{array} \right) \left( \begin{array}{c} E_{2}^{(1)}\\ E_{2}^{(2)} \\ E_{2}^{(3)} \\ E_{2}^{(4)} \\ \end{array}\right) \\&\quad =2 \left( \begin{array}{c} -(1+Re(a))\frac{r}{T_{max}}-(1-c\eta _{1})\alpha Re(b) \\ (1-c\eta _{1})\alpha Re(b) \\ 0 \\ 0 \\ \end{array} \right) \end{aligned}$$

In the similar manner, we can calculate the constant vector $E_{2}=(E_{2}^{(1)}, E_{2}^{(2)}, E_{2}^{(3)}, E_{2}^{(4)})\in R^{4}.$