Generalized Necessary and Sufficient Conditions for Annihilation of HIV-1 Virions During HAART

25 Pages·2016·0.35 MB·English

FFaayyeetttteevviillllee SSttaattee UUnniivveerrssiittyy
DDiiggiittaallCCoommmmoonnss@@FFaayyeetttteevviillllee SSttaattee UUnniivveerrssiittyy
Math and Computer Science Faculty Working
Math and Computer Science
Papers
7-7-2012
GGeenneerraalliizzeedd NNeecceessssaarryy aanndd SSuuffifficciieenntt CCoonnddiittiioonnss ffoorr AAnnnniihhiillaattiioonn
ooff HHIIVV--11 VViirriioonnss DDuurriinngg HHAAAARRTT
Frank Nani
Fayetteville State University, [email protected]
Mingxian Jin
Fayetteville State University, [email protected]
Follow this and additional works at: https://digitalcommons.uncfsu.edu/macsc_wp
RReeccoommmmeennddeedd CCiittaattiioonn
Nani, Frank and Jin, Mingxian, "Generalized Necessary and Sufficient Conditions for Annihilation of HIV-1
Virions During HAART" (2012). Math and Computer Science Faculty Working Papers. 18.
https://digitalcommons.uncfsu.edu/macsc_wp/18
This Article is brought to you for free and open access by the Math and Computer Science at
DigitalCommons@Fayetteville State University. It has been accepted for inclusion in Math and Computer Science
Faculty Working Papers by an authorized administrator of DigitalCommons@Fayetteville State University. For more
information, please contact [email protected].
Generalized Necessary and Sufficient Conditions for
Annihilation of HIV-1 Virions During HAART

Frank Nani and Mingxian Jin
Department of Mathematics and Computer Science
Fayetteville State University
Fayetteville, NC 28301, USA

1 Abstract: In this paper, the patho-physiological dynamics of Human Immuno-deficiency Virus type 1( HIV-1) induced
2 AIDS during Highly Active Anti Retroviral Therapy (HAART) is modeled by a system of non-linear deterministic
3 differential equations. The physiologically relevant and clinically plausible equations depict the dynamics of uninfected
4 CD4+ T cells (x), HIV-1 infected CD4+ T cells (x), HIV-1 virions in the blood plasma (x), HIV-1 specific CD8+ T cells
1 2 3
5 (x), and the concentration of HAART drug molecules (x). Criteria for the existence of therapeutic outcomes are
4 5
6 presented. In particular, the necessary and sufficient conditions for the annihilation of HIV-1 virions, and HIV-1 infected
7 helper T cells are clearly exhibited in terms of biological measureable model physiological parameters. Investigative
8 computer simulations are presented elucidating the patho-physiodynamics of HIV-1 induced AIDS and various
9 hypothetical patient parametric configurations. The mathematical analysis of the model equations and the computer
10 simulations are performed with regard to HAART protocols with constant continuous intravenous and transdermal drug
11 infusions.
12 Keywords: HIV-1patho-physiodynamics, mathematical modeling, HAART therapy, AIDS cure criteria,
13 Michaelis-Menten kinetics
14 AMS Subject Classification: 93A30; 93D05; 93D20; 34A34; 92C42; 92C35
15 1. Introduction
16
17 Highly Active Anti-retroviral Therapy (HAART) is currently the most therapeutically efficacious treatment protocol for
18 treating the Acquired Immunodeficiency Syndrome (AIDS). HIV-1 virions induce AIDS by orchestrating an irreversible
19 destruction of the CD4+ T cells which then paralyze the immune system of the HIV-1 positive person. The major objectives of
20 HAART therapy are the prolongation and improvement of the long-term life quality of patients; optimization of therapy such as
21 to suppress the HIV-1 viral load to below 50 copies of HIV-1 RNA; reconstitution of the patients’ immune system such that the
22 CD4+ T cells proliferate to carrying capacity; and minimization of drug toxicity. HAART treatment protocol consists of
23 nucleoside reverse transcriptase inhibitors, non-nucleoside reverse transcriptase inhibitors, protease inhibitors, anti-fungals
24 /anti-bacterials and in future, integrase inhibitors. The reverse transcriptase inhibitors prevent reverse transcription of HIV-1
25 specific DNA. The protease inhibitors are antagonistic to maturation and formation of new HIV-1 virions. The possible role of
26 integrase inhibitors is to prevent the integration of HIV-1 viral DNA into the patients’ DNA[5,12] .
27 In order to achieve the therapeutic goals of HAART, it is plausible to involve the techniques of mathematical modeling.
28 Before the advent of HAART, the primary focus of the mathematical modelers is to quantitatively analyze the observed
29 patho-physio dynamics of HIV-1 infection in the AIDS patients. These earlier research papers involve the pioneering work of
30 Perelson et al. [13], Nowark et al [11], and other contributors [16,17].
31 A recent paper by Nani and Jin [ 10] provided some physiological criteria under which HIV-1 virions in an AIDS patient
32 can be annihilated during HAART.
33 Some of the earlier mathematical modeling publications focus on single-drug AIDS therapy using Zidovudine (INN) or
34 azidothymidine (AZT) [9]. Then the advent of Active Retro-viral Therapy (ART) and the associated clinical limitation let to the
35 development of HAART treatment protocols. In spite of the initial success of HAART, there are clinically measurable and
36 observable shortcomings in the treatment of AIDS [4,]. In particular, HAART is not successful in about 40% of AIDS patients
37 because of drug-induced toxicity and complications of treatment . HAART protocols have been clinically observed to have
38 limited therapeutic efficacy due to biochemical/clinical drug resistance, short drug half-life, low bio-availability and blood
39 plasma toxicities.
40 Mathematical modeling provides a quantitative and rational approach to solve the therapeutic efficacy problems associated
41 with HAART. In particular, the models focused on finding optimal therapeutic schedules, the roles of latent viral reservoirs as
42 well as minimizing of toxic side effect[1,2,6,7,8,9,14,16,17].
43 Optimal therapies that will minimize side effects have been investigated by many authors in [8, 9, 10, 11, 13, 14, 15, 16].
44 Zaric et al. in 1998 presented a model which was focused on the simulation of protease inhibitors and role of drug resistant
45 HIV-1 virions [18]. Stengel in [14] presented a mathematical model of HIV-1 infection and HAART which demonstrated the
46 efficacy of a mathematically optimal therapy. Caetano and Yoneyama in [2] constructed a HAART model which incorporated
47 the roles of latently infected CD4+ T cells, and discussed how the reverse transcriptase and protease inhibitors affected HIV-1
48 dynamics during HAART, using the LQR, Scheme. In a future paper, we will use Pontryagin’s Minimal Principle to construct
49 admissible optimal therapies such as to minimize the toxicity of the drug but maximize the therapeutic efficacy of HAART.
50 In this paper, an elaborate mathematical model will be constructed which will incorporate physiologically plausible effects
51 such as Michaelis-Menten kinetics, role of HIV-1 latent viral reservoirs, continuous transdermal drug delivery, and the implicit
52 lymphocyte proliferation induction by the CD4+ T cells. The activation and proliferation is accomplished by a paracrine and
53 autocrine processes which are mediated by the cytokine interleukin-2, secreted by the CD4+ T cells. Several authors
54 investigated the consequences of structured long-term and short-term treatment interruptions during HAART [1, 2, 4, 8]. The
55 current model will discuss these consequences by means of simulations.
56 The current paper will be divided into seven sections. The first section gives the introduction into HAART therapy and
57 provides the basis for current research. This is followed by presentation and discussion of the model parameters in Section 2. In
58 Section 3 the mathematical model of HAART therapy will be constructed. Also the necessary and sufficient criteria for
59 annihilation of HIV-1 virions during HAART will be presented in sections 4, 5, and 6. In Section 7, clinically plausible
60 computer simulations will be exhibited. Section 5 will be the summary and discussion of the basic results of the paper.
61
62 2. Parameters
63 In this section, the physiological variable and parameters of the HAART model equations will be defined and explained. It
64 must be emphasized that some of these parameters are biologically measurable or can be estimated using clinical techniques. In
65 clinical experience, these parameters are different from patient to patient depending on their patho-physiological conditions.
66 A list of model parameters, constants, and variables is shown as follows.
67 x : the number density of non-HIV-1-infected CD4+ helper T-lymphocytes per unit volume
1
68 x : the number density of HIV-1 infected CD4+ helper T-lymphocytes per unit volume
2
69 x : the number density of HIV-1 virions in the blood plasma per unit volume
3
70 x : the number density of HIV-1 specific CD8+ cytotoxic T-lymphocytes per unit volume
4
71 x : the concentration of drug molecules of the HAART treatment protocol
5
72 S : rate of supply of un-infected CD4+ T -lymphocytes
1 4
73 S : rate of supply of latently infected CD4+ T -lymphocytes
2 4
74 S : rate of supply of HIV-1 virions from macrophage, monocytes, microglial cells and other lymphoid tissue different from
3
75 T -lymphocytes
4
76 S : rate of supply of CD8+ T lymphocytes from the thymus
4 8
77 D: rate of HAART drug infusion by transdermal delivery
78 a, b: constant associated with activation of lymphocytes by cytokine interleukin-2 (IL-2) (i =1, 2, 3, 4)
i i
79 c: rate of HAART drug degradation and excretion
80 α: constant associated with HIV-1 infection of CD4+ T helper cells (i =1, 2, 3)
i 4
81 β the number of HIV-1 virions produced per day by replication and budding in CD4+ T helper cells
1: 4
82 β: rate constant associated with replication and “budding” of HIV-1 in syncytia CD4+ T helper cells per day per microliter (µl)
2 4
83 and released into the blood plasma
84 β the number of HIV-1 virions produced per day by replication and “budding” in non-syncytia CD4+ T helper cells and
3: 4
85 released into the blood plasma
86 η: constant depicting the rate of which HIV-1 virions incapacitate the CD8+ T cytotoxic cells (i =1, 2)
i 8
87 (σ, λ): Michaelis-Menten nonlinear metabolic rate constants associated with HAART drug elimination
0 0
88 (σ, λ): Michaelis-Menten nonlinear metabolic rate constants associated with HAART drug pharmacokinetics (i =2, 3)
i i
2
89 ξ: cytotoxic coefficient where 0 ≤ ξ ≤ 1 (i = 2, 3)
i i
90 q: constant depicting competition between infected and un-infected CD4+ T helper cells (i =1, 2)
i 4
91 k: constant depicting degradation, loss of clonogenicity or “death” (i =1, 2, 3, 4)
i
92 k : rate constant depicting linear drug elimination pharmacokinetics
5
93 e : constant depicting death or degradation or removal by apoptosis (programmed cell death) (i =1, 2, 3, 4)
i0
94 K: constant associated with the killing rate of infected CD4+ T cells by CD8+ T cytotoxic lymphocytes (i =1, 2)
i 4 8
95 All the parameters are positive
96
97 3. Model Equations
98 3.1. Description of the Model Equations
99 The HIV-1 patho-physiological dynamics during HAART therapy can be modeled using the following system of
100 non-linear ordinary differential equations:
101


x& = S +a x2e−b1x1 −αx x −q x x −k x −e
1 1 1 1 1 1 3 1 1 2 1 1 10

102 x&2 = S2 +a2x1x2e−b2x1 +α2x1x3 −q2x1x2 −k2x2 −β1x3
 ξσ x x
103  − K x x −e − 2 2 2 5
1 2 4 20 λ + x
104  2 5
x& = S +βx x +βx −αx x −ηx x −k x −e
105  3 3 2 2 3 3 3 3 1 3 1 3 4 3 3 30
 ξσ x x
 − 3 3 3 5
106 λ + x
 3 5
107 x& = S +a x x e−b4x1 − K x x −ηx x −k x −e
 4 4 4 1 4 2 2 4 2 3 4 4 4 40
 σ x σ x x σ x x
108 x& = Df (t)− 0 5 − 2 2 5 − 3 3 5 −k x
109  5 λ + x λ + x λ + x 5 5
 0 5 2 5 3 5
110   1 for constant continuous input
f(t)= 
111112   sin nt for periodic input

x (t ) = x for i ={1,2,3,4,5} (3.1)
i 0 i0
113 The model includes the following clinical improvements:
114 (i) The drug delivery uses transdermal, stealth-liposome encapsulated drug delivery, instead of the matrix tablet form because
115 of improved therapeutic efficacy and reduced gastro-intestinal toxicity [6]. It is also assumed that elastic liposomes are
116 formulated and selectively targeted such as to reduce toxicity to non-HIV-1-infected CD4+ T cells (x ) and CD8+ cytotoxic
1
117 T cells (x ).
4
118 (ii) The HAART drug is such that each renal excretion and body clearance rate follows Michaelis-Menten kinetics.
119 (iii) g (x , x ) = a x x e −bjx1 for j=(1, 2, 4)
1 j j 1 j
120 This function depicts the process of lymphocyte activation which is mediated by x (CD 4+) T helper cells. These cells
1
121 secrete a cytokine called interleukin-2.
122 (iv) The periodic input function f(t)= can be depicted by the following plot:
123
3
124
125
126 3.2. Boundedness and Invariance of Non-negativity of Solutions
127 In this subsection theoretical conditions will be constructed under which solutions to the HAART mathematical model
128 equations are well-posed, ultimately bounded, and exhibit invariance of non-negativity for all t ∈[t ,T] ⊂ ℜ = [0,∞). In this
0 +
129 case, t , and T are defined respectively as times at which HAART therapy begins and terminates.
0
130 Theorem 3.1 Consider
131 (i) Ω = {(x ,x ,x ,x ,x )∈ ℜ5 0 ≤ x ≤ Φ i =1 , 2,3, 4, 5}
1 2 3 4 5 + i i
132 where Φ = sup x
i i
t∈[t0,T]
133 Let
 S + C − e 
134 Φ = max (x , i i i0 ), i =1, 2,3, 4
i  i0 k 
135 (ii) i
D
136 Φ5 = max{ x50,δ}
137 Then there exists a T >0 such that for T < t < ∞, all solutions to the HAART model equations (3.1) with initial values x ∈ℜ5
0 0 i0 +
138 ={x∈ℜ | x≥0, i=(1,2,3,4,5)} are ultimately bounded, dissipative, and will eventually enter the non-negatively invariant region
i i
139 Ω. In particular, the solutions are trapped in the region Ω for all t > T ⊂ ℜ
0 +
140 Proof. Using the result from Nani and Jin in [10], let
141 [ ]
C = sup a x x e−bjx1 for j ={1,2,4}
142 j t∈[t0,T] j 1 j (3.2)
C = sup [β x x +β x ]
3 2 2 3 3 3
t∈[t0,T]
143 Define
 1 
144 δ = sup   for i ={ 0,2,3} (3.3)
i t∈[t0,T]λi + x5 
145 And set
146 δ = δ +δ +δ (3.4)
0 2 3
147 The Kamke comparison theorem , cf. [10] can be used to establish the following inequalities.
148  S + C − e
149 xi ≤ i kii i0 +γie−kit for i = {1,2,34} (3.5)
 D
150 x ≤ +γ e−δt where δ = δ +δ +δ
5 δ 5 0 2 3

151  and γ ∈ R = (0,∞) and i = {1,2,3,4,5}
i +

152 In particular, the following results can be obtained.
S + C − e
lim sup x (t) ≤ i i i0 , i = {1,2,3,4}
t k
i 4

D
lim sup x (t) ≤
5 δ
153
154 (3.6)
155
156 Thus, Ω is non-negatively invariant and the system is ultimately bounded, dissipative, with the bounds defined by the following
157 equations.
S +C −e
158 sup x (t) = max {x , i i i0}, i ={1,2,3,4}
t∈[t0,T] i t∈[tL,tP] i0 ki
159 159D (3.7)
sup x (t) = max {x , }
160 t∈[t0,T] 5 t∈[t0,T] 50 δ
161 This completes the proof. 
162 4. The Rest Points and Computation of the Jacobian Matrices
163 4.1 The list of rest points or physiological outcomes of HAART
164 In this section, the possible patho-physiological outcomes from constant continuous transdermal HAART therapy are
165 listed and analyzed.
166 The physiological outcomes or the steady states during constant continuous transdermal HAART therapy occur when
167 x& = 0 for i=1,2,3,4,5 and f(t) ≡1
i
168 In particular, the clinically relevant and physiological plausible therapeutic outcomes include the following:
169 E = [0, 0, 0, 0, x ]
1 5
170 E = [0, 0, 0, x , x ]
2 4 5
171 E = [x , 0, 0, x , x ]
3 1 4 5
172 E = [x , 0, 0, 0, x ]
4 1 5
173 E = [0, x , x , 0, x ]
5 2 3 5
174 E = [0, 0, x , 0, x ]
6 3 5
175 E = [0, x , 0, 0, x ]
7 2 5
176 E = [x , x , 0, 0, x ]
8 1 2 5
177 E = [x , x , 0, x , x ]
9 1 2 4 5
178 E = [x , x , x , x , x ]
10 1 2 3 4 5
179 There are some other steady states which are not listed because they are less clinically interesting.
180 The clinically desirable steady states for a HIV-1 AIDS patient are E and E . The steady state E depicts a person who is
3 4 10
181 living with AIDS. In this case, the model exhibits persistence and the viral titer is not sufficient to annihilate the immune system.
182 The steady states E and E represent scenarios of therapeutic failure because the CD4+ T cells (x ) are obliterated by the
1 2 1
183 cytotoxicity of the HAART protocol. On the other hand, E represents the scenario in which HIV-1 virions (x ), HIV-1 infected
5 3
184 CD4+ T cells (x ), and the HAART drug (x ) eliminate the uninfected CD4+ T cells (x ), and HIV-1 specific CD8+ T cells (x ).
2 5 1 4
185 This is also an example of therapeutic failure for HAART protocol. In E and E , the uninfected CD4+ T cells are obliterated by
6 7
186 the HAART protocol and consequently are not clinically desirable. The steady states E and E represent curious scenarios
8 9
187 because the HIV-1 virions (x ) are eliminated from blood plasma but unfortunately the HIV-1 infected CD4+ cells (x ) remain
3 2
188 and will constitute a reservoir from which the HIV-1 virions will burst and repopulate the blood plasma and re-infect other
189 lymphoid organs.
190 The clinically desirable steady states E and E as well as undesirable states E will be discussed in this section. The
3 4 5
191 mathematical techniques used include the Hartman-Grobman theorem, non-linear dynamic systems theory, and the principles
192 of linearized stability.
5
193
194 4.2 Computation of the Jacobian Matrices
195 Using the Hartman-Grobman theorem, it is possible to investigate the physiological stability of HIV-1 AIDS disease
196 dynamics associated with the model equations, in the neighborhood of the physiological outcomes (steady states).
197 The Jacobian matrix of linearization near any physiological outcome is denoted symbolically by
[ ] { }
198 J E := a ∈M (ℜ) k =3,4,5
k ij 5×5 5×5 (4.1)
199 In particular, the a entries are defined as follows:
ij
200 a := a x (2−b x )e−b1x1 −αx −q x −k
11 1 1 1 1 1 3 1 2 1
201 a := −q x
12 1 1
202 a := −αx
13 1 1
203 a := 0
14
a := 0
204 15
205 a21 := a2x2(1−b2x1)e−b2x1 −q2x2
ξσ x
206 a := a x e−b2x1 −q x −k −K x − 2 2 5
22 2 1 2 1 2 1 4 λ + x
207 2 5
a :=α x −β
23 2 1 1
208
a := −K x
24 1 2
209 ξλσ x
a := − 2 2 2 2
210 25 (λ + x )2
2 5
211 a := −α x
31 3 3
212 a32 :=β2x3 (4.2)
ξσ x
213 a :=β x +β −α x −ηx −k − 3 3 5
33 2 2 3 3 1 1 4 3 λ + x
214 3 5
a := −ηx
34 1 3
215
ξσλ
a := − 3 3 3
216 35 (λ + x )2
3 5
217 a := a x (1−b x )e−b4x1
41 4 4 4 1
218 a := −K x
42 2 4
219 a := −η x
43 2 4
220 a := a x e−b4x1 −K x −η x −k
44 4 1 2 2 2 3 4
221 a45 := 0
a := 0
222 51
σ x
223 a := − 2 5
52 λ + x
2 5
224
σ x
a := − 3 5
225 53 λ + x
3 5
226 a := 0
54
227 σλ σλx σλx
a := − 0 0 − 2 2 2 − 3 3 3
228 55 (λ0 + x5)2 (λ2 + x5)2 (λ3 + x5)2
229 The Jacobian matrices for the steady states E , E , E are respectively listed as follows:
3 4 5
6
J{E[0,0,0,0,x ]}=
1 5
−k 0 0 0 0 
1
 ξσx 
 0 −k − 2 2 5 −β 0 0 
230 2 λ +x 1
 2 5 
 ξσx ξσλ 
0 0 β −k − 3 3 5 0 − 3 3 3
 3 3 λ +x (λ +x )2
 3 5 3 5 
0 0 0 −k 0
 4 
 σx σx σλ 
0 − 2 5 − 3 5 0 − 0 0
 λ2 +x5 λ3+x5 (λ0 +x5)2 (4.3)
J{E [0,0,0,x ,x ]}=
2 4 5
−k 0 0 0 0 
1
 ξσx 
231  0 −k2−K1x4−λ22+2x55 −β1 0 0 
 ξσx ξσλ 
0 0 β −ηx −k − 3 3 5 0 − 3 3 3
 3 1 4 3 λ+x (λ+x )2
 3 5 3 5 
a4x4 −K2x4 −η2x4 −k4 0 
 σx σx σλ 
0 − 2 5 − 3 5 0 − 0 0
 
 λ2+x5 λ3+x5 (λ0+x5)2 (4.4)
J{E [x ,0,0,x ,x ]}=
3 1 4 5
a x (2−bx )e−b1x1 −k −q x −αx 0 0 
 1 1 1 1 1 1 1 ξσx 1 1 
232  0 a xe−b2x1 −q x −k −K x − 2 2 5 αx −β 0 0 
 2 1 2 1 2 1 4 λ +x 2 1 1 
2 5
 ξσx ξσλ 
 0 0 β −αx −ηx −k − 3 3 5 0 − 3 3 3 
3 3 1 1 4 3 λ+x (λ+x )2
 3 5 3 5 
 a x (1−b x )e−b4x1 −K x −ηx a xe−b4x1 −k 0 
4 4 4 1 2 4 2 4 4 1 4
 σx σx σλ 
 0 − 2 5 − 3 5 0 − 0 0 
 λ2+x5 λ3+x5 (λ0+x5)2
233 (4.5)
J{E [x,0,0,0,x ]}=
4 1 5
ax(2−bx)e−b1x1 −k −qx −αx 0 0 
 1 1 1 1 1 1 1 ξσx 1 1 
234  0 a xe−b2x1 −q x −k − 2 2 5 αx −β 0 0 
 2 1 2 1 2 λ+x 2 1 1 
2 5
 ξσx ξσλ 
 0 0 β −αx −k − 3 3 5 0 − 3 3 3 
3 3 1 3 λ+x (λ+x )2
 3 5 3 5 
 0 0 0 a xe−b4x1 −k 0 
4 1 4
 σx σx σλ 
 0 − 2 5 − 3 5 0 − 0 0 
 λ2+x5 λ3+x5 (λ0+x5)2 (4.6)
235
J{E [0,x ,x ,0,x ]}=
5 2 3 5
−q x −k −αx 0 0 0 0 
1 2 1 1 3
 ξσx ξλσx 
 a x −q x −k − 2 2 5 −β −K x − 2 2 2 2 
2 2 2 2 2 λ +x 1 1 2 (λ +x )2
 2 5 2 5 
 −αx βx βx +β −k −ξ3σ3x5 −ηx − ξ3σ3λ3 
 3 3 2 3 2 2 3 3 λ+x 1 3 (λ+x )2 
 3 5 3 5 
0 0 0 −K x −ηx −k 0
 2 2 2 3 4 
 σx σx σλ σλx σλx 
0 − 2 5 − 3 5 0 − 0 0 − 2 2 2 − 3 3 3
 λ +x λ+x (λ +x )2 (λ +x )2 (λ+x )2
2 5 3 5 0 5 2 5 3 5
7
236
(4.7)
J{E [0,x ,0,0,x ]}=
7 2 5
−q x −k 0 0 0 0 
1 2 1
 ξσx ξλσx 
a x −q x −k − 2 2 5 −β −K x − 2 2 2 2 
237  2 2 2 2 2 λ2 +x5 1 1 2 (λ2 +x5)2 
 ξσx ξσλ 
0 0 βx +β −k − 3 3 5 0 − 3 3 3
 2 2 3 3 λ +x (λ +x )2 
 3 5 3 5 
0 0 0 −K x −k 0
 2 2 4 
 σx σx σλ σλx 
0 − 2 5 − 3 5 0 − 0 0 − 2 2 2
 λ2 +x5 λ3+x5 (λ0 +x5)2 (λ2 +x5)2 (4.8)
J{E[x,x ,0,x ,x ]}=
9 1 2 4 5
ax(2−bx)e−b1x1+qx −k −qx −αx 0 0 
238  1 1 1 1 1 2 1 1 1 ξσx 1 1 ξσλx 
 a x (1−bx)e−b2x1+q x a xe−b2x1−q x −k −Kx − 2 2 5 αx −β −K x − 2 2 2 2 
 2 2 2 1 2 2 2 1 2 1 2 1 4 λ+x 2 1 1 1 2 (λ+x )2 
2 5 2 5
 ξσx ξσλ 
 0 0 β2x2+β3−α3x1−η1x4−k3−λ3+3x5 0 −(λ3+3x3)2 
 3 5 3 5 
 a x (1−bx)e−b4x1 −K x −ηx a xe−b4x1+K x −k 0 
4 4 4 1 2 4 2 4 4 1 2 2 4
 σx σx σλ σλx 
 0 − 2 5 − 3 5 0 − 0 0 − 2 2 2 
 λ2+x5 λ3+x5 (λ0+x5)2 (λ2+x5)2
239
240 (4.9)
241
242 5. Necessary Criteria for Various Therapeutic Outcomes of AIDS during HAART
243
244 In this section, the necessary mathematical criteria for all therapeutic outcomes during HAART are computed and
245 presented in the form of theorems.
246 Theorem 5.1. Suppose


S1 + a1xˆ12e−b1xˆ1 − k1xˆ1 − e10 = 0
247 (i) S − e = 0 (5.1)
 2 20
S − e = 0
3 30

S4 + a4xˆ1xˆ4e−b4xˆ1 − k4xˆ4 − e40 = 0
 σ xˆ
D − 0 5 − k xˆ = 0
 λ0 + xˆ5 5 5
a x (2 − b x )e−b1x1 − k < 0
1 1 1 1 1

ξσ x
a x e−b2x1 − q x − k − 2 2 5 < 0
248 (ii)  2 1 2 1 2 λ + x (5.2)
2 5

ξσ x
β −α x − k − 3 3 5 < 0
 3 3 1 3 λ + x
 3 5
a x e−b4x1 − k < 0
4 1 4
249 The HAART therapeutic outcome E [x , 0, 0, 0, x ] exists and it is a local attractor.
4 1 5
8
250
251
252
253 Proof. Consider the model equations (3.1). Then criterion (5.1) is a necessary condition for the existence of E [x , 0, 0, 0, x ].
4 1 5
254 Now the Jacobian matrix of linearization of the model equations in the neighborhood of E [x , 0, 0, 0, x ] is such that the
4 1 5
255 eigenvalues are given by the following expressions.
256 λ = a x (2−b x )e−b1x1 −k
 1 1 1 1 1 1
257 λ =a xe−b2x1 −q x −k −ξ2σ2x5
258  2 2 1 2 1 2 λ2 +x5
 ξσ x
259 λ =β −α x −k − 3 3 5 (5.3)
3 3 3 1 3 λ + x
260  3 5
λ =a xe−b4 x1 −k
 4 4 1 4
261
 σλ
λ =− 0 0
262  5 (λ0 +x5)2
263 The eigenvalues have negative real parts when criterion (4.2) holds. Thus, two criteria (5.1) and (5.2) constitute the necessary
264 conditions for the local existence of E [x , 0, 0, 0, x ]. In particular, the principles of linearized stability of dynamical systems
4 1 5
265 can be used to imply that the rest point E [x , 0, 0, 0, x ] is locally asymptotically stable and hence a local attractor. 
4 1 5
266 Clinical Implication 5.1. The criteria (5.1) and (5.2) guarantee a temporary cure for the AIDS patient. There will be a finite
267 time interval during which the HIV-1 virions will be annihilated from the patient’s blood plasma. This will however be
268 short-lived because the rest point E [x , 0, 0, 0, x ] may become unstable and the criteria for temporal cure are violated. It is
4 1 5
269 possible for therapeutic criteria to be derived to maintain the patient to be permanently free of AIDS, which we shall discuss in
270 Theorem 5.5.
271
272 Theorem 5.2. Suppose
S − e = 0
 1 10
S − k x − β x − e − ξ2σ2x2x5 = 0
273 (i)  2 2 2 1 3 20 λ + x (5.4)
 2 5
 ξσ x x
S + β x x + β x − k x − e − 3 3 3 5 = 0
3 2 2 3 3 3 3 3 30 λ + x
 3 5
S − e = 0
 4 40
 σ x σ x x σ x x
D − 0 5 − 2 2 5 − 3 3 5 − k x = 0
 λ0 + x5 λ2 + x5 λ3 + x5 5 5
274 (ii) Let σ(J{E [0,x ,x ,0,x ]})be the eigen-spectrum of
5 2 3 5
275 a a a a a
11 12 13 14 15
276 a a a a a
21 22 23 24 25
277 J{E [0,x ,x ,0,x ]}:= a a a a a
5 2 3 5 31 32 33 34 35
278 a a a a a (5.5)
41 42 43 44 45
279 a a a a a
51 52 53 54 55
280 such that
281 σ(J{E [0,x ,x ,0,x ]})={λ|λ3+aλ2 +a λ+a =0,i=1,2,3}∪{λ,λ} (5.6)
5 2 3 5 i 1 2 3 4 5
282 where
9