Acute hepatitis B virus infection model within the host incorporating immune cells and cytokine responses

Abstract

We formulate and analyze a within-host hepatitis B viral mathematical model for hepatitis B in the acute phase of infection. The model incorporates hepatocytes, hepatitis B virus, immune system cells and cytokine dynamics using a system of ordinary differential equations. We use the model to demonstrate the trends of the hepatitis B infection qualitatively without the effects of immune cells and cytokines. Using these trends, we tested the effects of incorporating the immune cells only and immune cells with cytokine responses at low and high inhibitions on the hepatitis B virus infection. Our results showed that it is impossible to have the immune cells work independently from cytokines when there is an acute hepatitis B virus infection. Therefore, our results suggest that incorporating immune cells and cytokine dynamics in the acute hepatitis B virus infection stage delays infection in the hepatocytes and excluding such dynamics speeds up infection during this phase. Results from this study are useful in developing strategies for control of hepatocellular carcinoma which is caused by hepatitis B virus infection.

Appendix

$$\begin{aligned} a_{0} & = \frac{\alpha _2 T_4^0}{\mu _4},\quad a_1=\frac{a_0K_4-c_4\mu _4c_2}{K_4+c_4\mu _4},\quad a_2=\frac{a_1\delta N}{\mu _v},\\ a_3 & = \frac{K_{10}}{\mu _{10}},\\ a_4 & = a_3a_2,\quad a_5=\frac{\gamma _{\mathrm{c}} T_8^0}{\mu _{\mathrm{c}}},\quad a_6=\frac{K_{\gamma 1}}{\mu _\gamma },\quad a_7=\frac{\beta \delta N K}{r},\\ a_8 & = \frac{\alpha _1 T_4^0}{\mu _1},\\ a_9 & = ca_2+ca_4,\quad a_{10}=\frac{K_12}{\mu _{12}},\\ a_{11} & = a_5K_{12}+c_{\mathrm{c}} K_{12}+c_{\mathrm{c}} c_{I2}\mu _{12},\\ a_{12} & = a_5K_{12},\quad a_{13}=c_{I2}c_{\mathrm{c}}\mu _{12}c_1,\quad a_{14}=a_6a_5K_{12},\\ a_{15} & = a_{11}a_6+a_{14},\\ a_{16} & = a_6a_{13}+a_{14}c_1,\quad a_{17}=a_{11}a_6+a_{14}+c_\gamma a_{11},\\ a_{18} & = a_6a_{13}+c_\gamma a_{13}+c_\gamma c_1 a_{11},\\ a_{19} & = a_{14}c_1+c_\gamma c_1a_3, \quad a_{20}=a_{15}\delta N a_{10}a_8a_9,\\ a_{21} & = a_{16}\delta N a_{10}a_9a_8,\\ a_{22} & = a_{15}a_8cc_2\delta N a_{10},\quad a_{23}=a_{16}a_8cc_2\delta N a_{10},\\ a_{24} & = K_{12}+c_{I2},\\ a_{25} & = a_{24}a_4a_{17}\mu _v,\quad a_{26}=a_{18}a_{24}a_4\mu _v,\\ a_{27} & = a_{17}c_{I2}c_1a_4\mu _v,\\ a_{28} & = a_{24}a_4\mu _v a_{19},\quad a_{29}=a_{18}a_4c_{I2}c_1a_4\mu _v,\\ a_{30} & = a_{19}a_4c_{I2}c_1a_4\mu _v,\\ a_{31} & = a_{26}+a_{27},\quad a_{32}=a_{28}+a_{29},\quad a_{33}=a_{31}-a_{22},\\ a_{34} & = a_{21}+a_{23}+a_{32},\\ a_{35} & = a_5K_{12}\gamma _{\mathrm{c}}, \quad a_{36}=K_{12}+c_{I2}\mu _{12},\quad a_{37}=c_{I2}\mu _{12}c_1,\\ a_{38} & = a_9a_{35}a_{15},\\ a_{39} & = a_{35}a_{15}cc_2,\quad a_{40}=a_{35}a_{16}a_9,\quad a_{41}=a_{35}cc_2a_{16},\\ a_{42} & = a_{17}a_{36}a_4,\\ a_{43} & = a_{36}a_{18}+a_{37}a_4a_{17},\quad a_{44}=a_{36}a_4a_{19}+a_{37}a_4a_{18},\\ a_{45} & = a_{19}a_{37}a_4,\\ a_{46} & = a_{38}+a_{42}\delta +a_{42}\mu ,\quad a_{47}=a_{43}\mu +a_{43}\delta ,\\ a_{48} & = a_{44}\mu +a_{44}\delta ,\\ a_{49} & = a_{45}\mu +a_{45}\delta , \quad a_{50}=\beta \delta N r \mu _v a_{42}+\beta ^2\delta ^2 N^2 K a_4^2,\\ a_{51} & = \beta ^2 \delta ^2 N^2 K a_4+\beta \delta N r \mu _v a_4,\\ a_{52} & = \beta \delta N r \mu _v a_{44}+\beta ^2 \delta ^2 N^2 K a_{44},\\ a_{53} & = \beta \delta N r \mu _v a_{45}+\beta ^2 \delta ^2 N^2 K a_{45}, \quad a_{54}=\beta \delta N K r \mu _v a_{42},\\ a_{55} & = \beta \delta N K r \mu _v a_4,\\ a_{56} & = \beta \delta N K r \mu _v a_{45},\quad a_{57}=r \mu _v^2 a_{46},\quad a_{58}=r \mu _v^2 a_{47}, \\ a_{59} & = r\mu _v^2 a_{49}, \end{aligned}$$