    August  2017, 22(6): 2089-2120. doi: 10.3934/dcdsb.2017086

## An analysis of functional curability on HIV infection models with Michaelis-Menten-type immune response and its generalization

 Department of Mathematical Sciences, National Chengchi Uniserstiy, Taipei, 11605, Taiwan

Received  January 2016 Revised  January 2017 Published  March 2017

Let HIV infection be modeled by a dynamical system with a Michaelis-Mente-type immune response. A functional cure refers to driving the system from a stable high-viral-load state to a stable low-viral-load state. This may occur only when at least two stable equilibrium states coexist in the system. This paper analyzes how the number of biologically meaningful equilibrium states varies with system parameters. Meanwhile, it investigates how patients' profiles of immune responses determine their clinical outcomes, with focus on functional curability. The analysis provides a criterion that a functional cure is possible only if the capability of immune stimulation starts to attenuate when the density of infected cells is below a threshold. From treatment viewpoints, such a criterion is crucial because it identifies which patients cannot use a low-viral-load state as a treatment endpoint. The deriving process also provides a method to study functional curability problems with a wider class of immune response functions and functional curability problems of similar virus infections such as chronic hepatitis B virus infection.

Citation: Jeng-Huei Chen. An analysis of functional curability on HIV infection models with Michaelis-Menten-type immune response and its generalization. Discrete & Continuous Dynamical Systems - B, 2017, 22 (6) : 2089-2120. doi: 10.3934/dcdsb.2017086
##### References:

show all references

##### References: (Case 1) The graph of function $g(I)$ with $b_1 = 4$, $b_2 = 2$, $M_1 = 5$ and $M_2 = 5$ (Case 2) The graph on the left hand side is the function $g(I)$ with $b_1 = 90$, $b_2 = 80$, $M_1 = 4$ and $M_2 = 6$. The graph of $g(I)$ with $I>-M_1$ is enlarged on the right hand side. Point A is a local maximum with coordinates $(I_2^{*}, g(I_2^{*}))$ = (8.9282, 14.3078). Point B is an infection point with coordinates $(I_1^{**}, g(I_1^{**}))$ = (15.8723, 13.8299). Point C is the intersection point of $y = g(I)$ and $y=b_1-b_2$. Its coordinates is ($I_{{\mathop{\rm int}}}, g(I_{{\mathop{\rm int}}}))$ = (2.000, 10) Type 1 immune induction function (referred as a function valid in Bonhoeffer sense.) The parameters are $b_1 = 4$, $M_1 =100$, $b_2 = 2.5$, $M_2 = 120$ and $d_E = 0$ Type 2 immune induction function. The parameters are $b_1 = 2.5$, $M_1 =120$, $b_2 = 4$, $M_2 = 100$ and $d_E = 0$ Type 3 immune induction function (referred as a function valid in Adams sense.) The parameters are $b_1 = 6$, $M_1 =1.25$, $b_2 = 5$, $M_2 = 6.25$ and $d_E = 0$. The positive local maximum is located at $I^{*} = 3.55$ (point A) and the inflection point is located at $I^{**} = 6.96.$ (point B) Type 4 immune induction function. The parameters are $b_1 = 5$, $M_1 =6.25$, $b_2 = 6$, $M_2 = 1.25$ and $d_E = 0$. The positive local minimum is located at $I^{*} = 3.55$ (point A) and the inflection point is located at $I^{**} = 6.96$ (point B) A demonstration of theorem 4.8. With the immune induction function $g(I)$ valid in Bonhoeffer sense, the functions $H(I)$ and $g(I)$ intersect exactly once at $I = I_e > 0$ with $H(I_e) = g(I_e) < 0$. (They intersect at point A in the graph.) This may occur if and only if the condition $H(0) > g(0)$ holds. Equivalently, this is the condition ${R_0} > (1 + p\frac{{{c_E}}}{{{d_I}{d_E}}})$ stated in theorem 4.8. The parameters of this graph are $s_T =5$, $d_T = 0.01$, $\beta =8$, $d_I = 0.7$, $p = 0.7$, $k =5$, $d_V =13$, $c_E =1$, $b_1 =4$, $M_1 =100$, $b_2 = 2.5$, $M_2 =120$ and $d_E =1$ Part (c) of theorem 4.9. It is possible that the functions $H(I)$ and $g(I)$ intersect three times. (They intersect at points A, B and C. The upper half of the function $H(I)$ is not shown in the graph.) Each intersection point leads to one biologically meaningful equilibrium states other than $Q_1^{1}$. This may occur if $H(0) > g(0)$. Equivalently, this is the condition ${R_0} > (1 + p\frac{{{c_E}}}{{{d_I}{d_E}}})$ stated in part (c) of theorem 4.9. The parameters of this graph are $s_T =50$, $d_T = 0.01$, $\beta =8$, $d_I = 0.7$, $p = 0.7$, $k =5$, $d_V =13$, $c_E =1$, $b_1 =6$, $M_1 =1.25$, $b_2 = 5$, $M_2 = 6.25$ and $d_E =2.5$ Part (c) of theorem 4.9. It is possible that the functions $H(I)$ and $g(I)$ intersect three times. (They intersect at points A, B and C. The upper half of the function $H(I)$ is not shown in the graph.) Each intersection point leads to one biologically meaningful equilibrium states other than $Q_1^{1}$. This may occur if $H(0) > g(0)$. Equivalently, this is the condition ${R_0} > (1 + p\frac{{{c_E}}}{{{d_I}{d_E}}})$ stated in part (c) of theorem 4.9. The parameters of this graph are $s_T =50$, $d_T = 0.01$, $\beta =8$, $d_I = 0.7$, $p = 0.7$, $k =5$, $d_V =13$, $c_E =1$, $b_1 =6$, $M_1 =1.25$, $b_2 = 5$, $M_2 = 6.25$ and $d_E =2.5$
The behavior of $g(I)$ with respective to system parameters
 Type Conditions in parameters 1 (case 1) $M_1 = M_2$ and $b_1 > b_2$ (case 3) $M_1 > M_2, b_1 > b_2$ and $b_2M_1 < b_1M_2$ (case 4) $M_1 > M_2, b_1 > b_2$ and $b_2M_1 = b_1M_2$ (case 5) $M_1 < M_2, b_1 > b_2$ and $b_2M_2 < b_1M_1$ (case 6) $M_1 < M_2, b_1 > b_2$ and $b_2M_2 = b_1M_1$ 2 (case 7) $M_1 = M_2$ and $b_1 < b_2$ (case 8) $M_1 > M_2, b_1 < b_2$ and $b_1M_1 < b_2M_2$ (case 9) $M_1 > M_2, b_1 < b_2$ and $b_1M_1 = b_2M_2$ (case 10) $M_1 < M_2, b_1 < b_2$ and $b_1M_2 < b_2M_1$ (case 11) $M_1 < M_2, b_1 < b_2$ and $b_1M_2 = b_2M_1$ 3 (case 12) $M_1 < M_2, b_1 < b_2$ and $b_1M_2 > b_2M_1$ (case 13) $M_1 < M_2$ and $b_1 = b_2$ (case 2) $M_1 < M_2, b_1 > b_2$ and $b_2M_2 > b_1M_1$ 4 (case 14) $M_1 > M_2, b_1 > b_2$ and $b_2M_1 > b_1M_2$ (case 15) $M_1 > M_2$ and $b_1 = b_2$ (case 16) $M_1 > M_2, b_1 < b_2$ and $b_1M_1 > b_2M_2$
 Type Conditions in parameters 1 (case 1) $M_1 = M_2$ and $b_1 > b_2$ (case 3) $M_1 > M_2, b_1 > b_2$ and $b_2M_1 < b_1M_2$ (case 4) $M_1 > M_2, b_1 > b_2$ and $b_2M_1 = b_1M_2$ (case 5) $M_1 < M_2, b_1 > b_2$ and $b_2M_2 < b_1M_1$ (case 6) $M_1 < M_2, b_1 > b_2$ and $b_2M_2 = b_1M_1$ 2 (case 7) $M_1 = M_2$ and $b_1 < b_2$ (case 8) $M_1 > M_2, b_1 < b_2$ and $b_1M_1 < b_2M_2$ (case 9) $M_1 > M_2, b_1 < b_2$ and $b_1M_1 = b_2M_2$ (case 10) $M_1 < M_2, b_1 < b_2$ and $b_1M_2 < b_2M_1$ (case 11) $M_1 < M_2, b_1 < b_2$ and $b_1M_2 = b_2M_1$ 3 (case 12) $M_1 < M_2, b_1 < b_2$ and $b_1M_2 > b_2M_1$ (case 13) $M_1 < M_2$ and $b_1 = b_2$ (case 2) $M_1 < M_2, b_1 > b_2$ and $b_2M_2 > b_1M_1$ 4 (case 14) $M_1 > M_2, b_1 > b_2$ and $b_2M_1 > b_1M_2$ (case 15) $M_1 > M_2$ and $b_1 = b_2$ (case 16) $M_1 > M_2, b_1 < b_2$ and $b_1M_1 > b_2M_2$
The chosen values of system parameters
 Parameters Value Parameters Value $s_T$ $10^{4} \frac{{cells}}{{ml \cdot day}}$ $c_E$ $1 \frac {cells} {ml \cdot day}$ $d_T$ $0.01 \frac{1}{day}$ $b_1$ $0.3 \frac {1}{day}$ $\beta$ $8 \times 10^{-7} \frac{ml}{{virions \cdot day}}$ $M_1$ $100 \frac {cells}{ml}$ $d_I$ $0.7 \frac {1}{day}$ $b_2$ $0.25 \frac {1}{day}$ $p$ $10^{-5} \frac{ml}{{cells \cdot day}}$ $M_2$ $500 \frac {cells}{ml}$ $k$ $100 \frac {virions}{day}$ $d_E$ $0.1 \frac {1} {day}$ $d_V$ $13 \frac {1}{day}$
 Parameters Value Parameters Value $s_T$ $10^{4} \frac{{cells}}{{ml \cdot day}}$ $c_E$ $1 \frac {cells} {ml \cdot day}$ $d_T$ $0.01 \frac{1}{day}$ $b_1$ $0.3 \frac {1}{day}$ $\beta$ $8 \times 10^{-7} \frac{ml}{{virions \cdot day}}$ $M_1$ $100 \frac {cells}{ml}$ $d_I$ $0.7 \frac {1}{day}$ $b_2$ $0.25 \frac {1}{day}$ $p$ $10^{-5} \frac{ml}{{cells \cdot day}}$ $M_2$ $500 \frac {cells}{ml}$ $k$ $100 \frac {virions}{day}$ $d_E$ $0.1 \frac {1} {day}$ $d_V$ $13 \frac {1}{day}$
Summarized numerical results for $c_E=0$
 Case Varied parameters $R_0$ Immunity Outcomes (a) $\beta$ = $1.6 \times 10^{-7}$, $k =20$, $c_E = 0$ 0.35 Adams Virus eradication $Q_1^{0}$ (b) $c_E = 0$ 8.7912 Adams A functional cure ($Q_2^{0}$, $Q_3^{0}$) (c) $b_1 = 0.4$, $M_2 = 120$ $c_E =0$ 8.7912 Bonhoeffer Elite controller $Q_3^{0}$ (d) $M_2 = 110$, $c_E = 0$ 8.7912 Bonhoeffer High viral load $Q_2^{0}$ (e) $d_E = 0.2$, $c_E = 0$ 8.7912 Adams High viral load $Q_2^{0}$
 Case Varied parameters $R_0$ Immunity Outcomes (a) $\beta$ = $1.6 \times 10^{-7}$, $k =20$, $c_E = 0$ 0.35 Adams Virus eradication $Q_1^{0}$ (b) $c_E = 0$ 8.7912 Adams A functional cure ($Q_2^{0}$, $Q_3^{0}$) (c) $b_1 = 0.4$, $M_2 = 120$ $c_E =0$ 8.7912 Bonhoeffer Elite controller $Q_3^{0}$ (d) $M_2 = 110$, $c_E = 0$ 8.7912 Bonhoeffer High viral load $Q_2^{0}$ (e) $d_E = 0.2$, $c_E = 0$ 8.7912 Adams High viral load $Q_2^{0}$
Summarized numerical results for case $c_E \neq 0$
 Case Varied parameters $R_0$ Immunity Outcomes (a) $\beta$ = $1.6 \times 10^{-7}$ $k = 20$ 0.35 Adams Virus eradication $Q_1^{1}$ (b) No changes 8.7912 Adams A functional cure ($Q_2^{1}$, $Q_4^{1}$) (c) $b_1 = 0.4$, $M_2 = 120$ 8.7912 Bonhoeffer Elite controller $Q_4^{1}$ (d) $\beta$ = $1.6 \times 10^{-7}$ $b_1 = 0.4$, $k = 20$ $M_2 = 120$ 0.35 Bonhoeffer Virus eradication $Q_1^{1}$ (e) $M_1 =1 \times 10^{-4}$ $M_2 = 5 \times 10^{4}$ 8.7912 Adams High viral load $Q_4^{1}$
 Case Varied parameters $R_0$ Immunity Outcomes (a) $\beta$ = $1.6 \times 10^{-7}$ $k = 20$ 0.35 Adams Virus eradication $Q_1^{1}$ (b) No changes 8.7912 Adams A functional cure ($Q_2^{1}$, $Q_4^{1}$) (c) $b_1 = 0.4$, $M_2 = 120$ 8.7912 Bonhoeffer Elite controller $Q_4^{1}$ (d) $\beta$ = $1.6 \times 10^{-7}$ $b_1 = 0.4$, $k = 20$ $M_2 = 120$ 0.35 Bonhoeffer Virus eradication $Q_1^{1}$ (e) $M_1 =1 \times 10^{-4}$ $M_2 = 5 \times 10^{4}$ 8.7912 Adams High viral load $Q_4^{1}$
Functional curability under different conditions
 $g(I)$ valid in Bonhoeffer sense $c_E =0$ and $R_0 < 1$ not possible $c_E = 0$ and $R_0 > 1$ inconclusive $c_E > 0$ and ${R_0} < (1 + p\frac{{{c_E}}}{{{d_I}{d_E}}})$ not possible $c_E > 0$ and ${R_0} > (1 + p\frac{{{c_E}}}{{{d_I}{d_E}}})$ not possible $g(I)$ valid in Adams sense $c_E =0$ and $R_0 < 1$ not possible $c_E = 0$ and $R_0 > 1$ possible $c_E > 0$ and ${R_0} < (1 + p\frac{{{c_E}}}{{{d_I}{d_E}}})$ inconclusive (with immunity counter-induced) $c_E > 0$ and ${R_0} > (1 + p\frac{{{c_E}}}{{{d_I}{d_E}}})$ possible with the criterion satisfied
 $g(I)$ valid in Bonhoeffer sense $c_E =0$ and $R_0 < 1$ not possible $c_E = 0$ and $R_0 > 1$ inconclusive $c_E > 0$ and ${R_0} < (1 + p\frac{{{c_E}}}{{{d_I}{d_E}}})$ not possible $c_E > 0$ and ${R_0} > (1 + p\frac{{{c_E}}}{{{d_I}{d_E}}})$ not possible $g(I)$ valid in Adams sense $c_E =0$ and $R_0 < 1$ not possible $c_E = 0$ and $R_0 > 1$ possible $c_E > 0$ and ${R_0} < (1 + p\frac{{{c_E}}}{{{d_I}{d_E}}})$ inconclusive (with immunity counter-induced) $c_E > 0$ and ${R_0} > (1 + p\frac{{{c_E}}}{{{d_I}{d_E}}})$ possible with the criterion satisfied
  Steffen Eikenberry, Sarah Hews, John D. Nagy, Yang Kuang. The dynamics of a delay model of hepatitis B virus infection with logistic hepatocyte growth. Mathematical Biosciences & Engineering, 2009, 6 (2) : 283-299. doi: 10.3934/mbe.2009.6.283  Bao-Zhu Guo, Li-Ming Cai. A note for the global stability of a delay differential equation of hepatitis B virus infection. Mathematical Biosciences & Engineering, 2011, 8 (3) : 689-694. doi: 10.3934/mbe.2011.8.689  Hui li, Manjun Ma. Global dynamics of a virus infection model with repulsive effect. Discrete & Continuous Dynamical Systems - B, 2019, 24 (9) : 4783-4797. doi: 10.3934/dcdsb.2019030  Zhaohui Yuan, Xingfu Zou. Global threshold dynamics in an HIV virus model with nonlinear infection rate and distributed invasion and production delays. Mathematical Biosciences & Engineering, 2013, 10 (2) : 483-498. doi: 10.3934/mbe.2013.10.483  Ariel D. Weinberger, Alan S. Perelson. Persistence and emergence of X4 virus in HIV infection. Mathematical Biosciences & Engineering, 2011, 8 (2) : 605-626. doi: 10.3934/mbe.2011.8.605  Nicola Bellomo, Youshan Tao. Stabilization in a chemotaxis model for virus infection. Discrete & Continuous Dynamical Systems - S, 2020, 13 (2) : 105-117. doi: 10.3934/dcdss.2020006  Hui Li, Manjun Ma. Corrigendum on “H. Li and M. Ma, global dynamics of a virus infection model with repulsive effect, Discrete and Continuous Dynamical Systems, Series B, 24(9) 4783-4797, 2019”. Discrete & Continuous Dynamical Systems - B, 2020, 25 (12) : 4925-4925. doi: 10.3934/dcdsb.2020299  Yuri Nechepurenko, Michael Khristichenko, Dmitry Grebennikov, Gennady Bocharov. Bistability analysis of virus infection models with time delays. Discrete & Continuous Dynamical Systems - S, 2020, 13 (9) : 2385-2401. doi: 10.3934/dcdss.2020166  Suxia Zhang, Xiaxia Xu. A mathematical model for hepatitis B with infection-age structure. Discrete & Continuous Dynamical Systems - B, 2016, 21 (4) : 1329-1346. doi: 10.3934/dcdsb.2016.21.1329  Suxia Zhang, Hongbin Guo, Robert Smith?. Dynamical analysis for a hepatitis B transmission model with immigration and infection age. Mathematical Biosciences & Engineering, 2018, 15 (6) : 1291-1313. doi: 10.3934/mbe.2018060  Yun Tian, Yu Bai, Pei Yu. Impact of delay on HIV-1 dynamics of fighting a virus with another virus. Mathematical Biosciences & Engineering, 2014, 11 (5) : 1181-1198. doi: 10.3934/mbe.2014.11.1181  Tao Feng, Zhipeng Qiu, Xinzhu Meng. Dynamics of a stochastic hepatitis C virus system with host immunity. Discrete & Continuous Dynamical Systems - B, 2019, 24 (12) : 6367-6385. doi: 10.3934/dcdsb.2019143  Tin Phan, Bruce Pell, Amy E. Kendig, Elizabeth T. Borer, Yang Kuang. Rich dynamics of a simple delay host-pathogen model of cell-to-cell infection for plant virus. Discrete & Continuous Dynamical Systems - B, 2021, 26 (1) : 515-539. doi: 10.3934/dcdsb.2020261  Zhikun She, Xin Jiang. Threshold dynamics of a general delayed within-host viral infection model with humoral immunity and two modes of virus transmission. Discrete & Continuous Dynamical Systems - B, 2021, 26 (7) : 3835-3861. doi: 10.3934/dcdsb.2020259  Yu Yang, Shigui Ruan, Dongmei Xiao. Global stability of an age-structured virus dynamics model with Beddington-DeAngelis infection function. Mathematical Biosciences & Engineering, 2015, 12 (4) : 859-877. doi: 10.3934/mbe.2015.12.859  Elamin H. Elbasha. Model for hepatitis C virus transmissions. Mathematical Biosciences & Engineering, 2013, 10 (4) : 1045-1065. doi: 10.3934/mbe.2013.10.1045  Praveen Kumar Gupta, Ajoy Dutta. Numerical solution with analysis of HIV/AIDS dynamics model with effect of fusion and cure rate. Numerical Algebra, Control & Optimization, 2019, 9 (4) : 393-399. doi: 10.3934/naco.2019038  Cameron Browne. Immune response in virus model structured by cell infection-age. Mathematical Biosciences & Engineering, 2016, 13 (5) : 887-909. doi: 10.3934/mbe.2016022  Hossein Mohebbi, Azim Aminataei, Cameron J. Browne, Mohammad Reza Razvan. Hopf bifurcation of an age-structured virus infection model. Discrete & Continuous Dynamical Systems - B, 2018, 23 (2) : 861-885. doi: 10.3934/dcdsb.2018046  Hongying Shu, Lin Wang. Global stability and backward bifurcation of a general viral infection model with virus-driven proliferation of target cells. Discrete & Continuous Dynamical Systems - B, 2014, 19 (6) : 1749-1768. doi: 10.3934/dcdsb.2014.19.1749

2020 Impact Factor: 1.327