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

Jeng-Huei Chen

doi:10.3934/dcdsb.2017086

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2017, Volume 22, Issue 6: 2089-2120. Doi: 10.3934/dcdsb.2017086

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

Jeng-Huei Chen

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

Author Bio: E-mail address: jhchen@nccu.edu.tw

Received: December 31, 2015

Revised: December 31, 2016

Published: May 2017

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Abstract

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.

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Mathematics Subject Classification: Primary:37N25;Secondary:92D25.

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Figure 1. (Case 1) The graph of function $g(I)$ with $b_1 = 4$, $b_2 = 2$, $M_1 = 5$ and $M_2 = 5$

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Figure 2. (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)

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Figure 3. 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 $d_E = 0$

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Figure 4. Type 2 immune induction function. The parameters are $b_1 = 2.5$, $M_1 =120$, $b_2 = 4$, $M_2 = 100$ and $d_E = 0$

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Figure 5. 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)

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Figure 6. 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)

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Figure 7. The graph of the function $z(I)$ in lemma 4.6 with $a=5$ and $b=10$

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Figure 8. 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 $d_E =1$

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Figure 9. 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 $d_E =2.5$

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Figure 10. 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 $d_E =2.5$

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Table 1. 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 $b_1M_1 > b_2M_2$

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Table 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 $	$13 \frac {1}{day}$

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Table 3. 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 $Q_2^{0}$

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Table 4. 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 $Q_4^{1}$

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Table 5. 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

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