CMMM Computational and Mathematical Methods in Medicine 1748-6718 1748-670X Hindawi Publishing Corporation 145162 10.1155/2014/145162 145162 Research Article Dynamics Analysis and Simulation of a Modified HIV Infection Model with a Saturated Infection Rate http://orcid.org/0000-0002-2178-5236 Sun Qilin 1 http://orcid.org/0000-0002-4414-3818 Min Lequan 1,2 Wei Yang 1 School of Automation and Electrical Engineering University of Science and Technology Beijing Beijing 100083 China Ustb.edu.cn 2 School of Mathematics and Physics University of Science and Technology Beijing Beijing 100083 China Ustb.edu.cn 2014 2332014 2014 17 12 2013 28 01 2014 23 3 2014 2014 Copyright © 2014 Qilin Sun and Lequan Min. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

This paper studies a modified human immunodeficiency virus (HIV) infection differential equation model with a saturated infection rate. It is proved that if the basic virus reproductive number R 0 of the model is less than one, then the infection-free equilibrium point of the model is globally asymptotically stable; if R 0 of the model is more than one, then the endemic infection equilibrium point of the model is globally asymptotically stable. Based on the clinical data from HIV drug resistance database of Stanford University, using the proposed model simulates the dynamics of the two groups of patients’ anti-HIV infection treatment. The numerical simulation results are in agreement with the evolutions of the patients’ HIV RNA levels. It can be assumed that if an HIV infected individual’s basic virus reproductive number R 0 < 1 then this person will recover automatically; if an antiretroviral therapy makes an HIV infected individual’s R 0 < 1 , this person will be cured eventually; if an antiretroviral therapy fails to suppress an HIV infected individual’s HIV RNA load to be of unpredictable level, the time that the patient’s HIV RNA level has achieved the minimum value may be the starting time that drug resistance has appeared.

1. Introduction

The human immunodeficiency virus (HIV) mainly targets a host’s CD 4 + T cells. Chronic HIV infection causes gradual depletion of the CD 4 + T cell pool and thus progressively compromises the hosts immune response to opportunistic infections, leading to Acquired Immunodeficiency Syndrome (AIDS) .

In recent years, there is much work done on HIV infection from different points of view, such as pathology , microbiology , and mathematics . Mathematical models have become essential tools to make assumptions, suggest new experiments, or help easily explain complex processes . The basic mathematical model widely used for studying the dynamics of HIV infection has the following form [4, 9]: (1) x ˙ = λ - d 1 x - k 1 x v , y ˙ = k 1 x v - d 2 y , v ˙ = a y - d 3 v , where x ( t ) , y ( t ) , and v ( t ) are the number of uninfected cells, infected cells, and free virus, respectively. Uninfected cells are produced at a constant rate λ , die at rate d 1 x , and become infected at rate k 1 x v . Infected cells are produced at rate k 1 x v and die at rate d 2 y . Free virus is produced from infected cells at rate a y and dies at rate d 3 v .

Equation (1) has a basic virus reproductive number R 0 = a k 1 λ / ( d 1 d 2 d 3 ) . According to Nowak and Bangham , R 0 is defined as the number of newly infected cells arising from any one infected cell; if R 0 is smaller than 1, then in the beginning of the infection, each virus infected cell produces on average less than one newly infected cell. Thus, the infection cannot spread, and the system returns to the uninfected state; if R 0 is larger than 1, then initially each virus infected cell produces on average more than one newly infected cell. The infected cell population will increase, whereas the uninfected cell population will decline and therefore provide less opportunity for the virus to infect new cells.

There is a discussion about the process of the HIV RNA transcribing into DNA: when an HIV enters a resting CD 4 + T cell, the HIV RNA may not be completely reverse transcribed into DNA . A proportion of resting infected cells can revert to the uninfected state before the viral genome is integrated into the genome of the lymphocyte .

Recently, some mathematical models of HIV infection have been proposed based on the assumption that a fraction of infected CD 4 + T cells return to the uninfected class . Srivastava and Chandra  have considered a model with three populations: uninfected CD 4 + T cells ( x ) , infected CD 4 + T cells ( y ) , and HIV population ( v ) . The model has the following form: (2) x ˙ = λ - d 1 x - k 1 x v + p y , y ˙ = k 1 x v - ( d 2 + p ) y , v ˙ = a y - d 3 v , where the meanings of the variables x ( t ) , y ( t ) , and v ( t ) and the parameters λ , d 1 , k 1 , d 2 , a , and d 3 are the same as those given in (1). The term p y is the rate of infected cells in the latent stage reverting to the uninfected class. Equation (2) also has a basic virus reproductive number R 0 = a k 1 λ / ( d 3 d 1 ( p + d 2 ) ) . They have proved that if R 0 1 , the infection-free steady state of (2) is globally asymptotically stable; if R 0 > 1 , the endemic steady state of (2) is globally asymptotically stable .

In (2), the mass action term k 1 x v used to model infection of CD 4 + T cells by free virions is biologically problematic for several reasons. Firstly, since λ / d 1 represents the total number of CD 4 + T cells in the basic virus reproductive number R 0 = a k 1 λ / ( d 3 d 1 ( p + d 2 ) ) , this causes R 0 to depend upon the total number of CD 4 + T cells in vivo. This implies the dubious prediction that individuals with more CD 4 + T cells will be more easily infected than individuals with less CD 4 + T cells. Secondly, the rate of HIV infection is assumed to be bilinear by the mass action term k 1 x v . However, the actual incidence rate is probably not linear over the entire range of virus v ( t ) and uninfected CD 4 + T cells x ( t ) .

On biological grounds, during primary HIV infection, the rate of virus infection should be approximately proportionate to the virus load k 1 v because of a small amount of viral load with respect to a large number of CD 4 + T cells. However,since the total number of healthy CD 4 + T cells in vivo is limited, the HIV infection will approach saturation with more and more virus produced. In this case, it is more reasonable to assume that the rate of virus infection should be approximately proportionate to the number of healthy CD 4 + T cells k 1 x .

Based on the argument above, this paper describes an amended model. In this model, we use a saturated infection rate k 1 x v / ( x + v ) to replace the mass action term k 1 x v in (2). Under the formulation of this saturated infection rate, the basic virus reproductive number R 0 is independent of the total number of CD 4 + T cells. Meanwhile, the actual incidence rate is not linear over the entire range of virus v ( t ) and uninfected CD 4 + T cells x ( t ) any more. The global stabilities of the infection-free state and the endemic infection state of the modified HIV infection model have been discussed. Based on the clinical data from HIV drug resistance database of Stanford University, using the proposed model simulates the dynamics of two groups of patients’ anti-HIV infection treatment, and then make long-term predictions for the two groups’ anti-HIV infection treatment, respectively.

The rest of this paper is organized as follows. Section 2 introduces a modified model and discusses the boundedness of the solutions of the model. Sections 3 and 4 discuss the global stability of the infection-free state and the endemic infection state of the modified HIV infection model, respectively. Section 5 simulates the dynamics of two groups of patients’ anti-HIV infection treatment. Section 6 summarizes this paper.

2. Modified HIV Infection Model 2.1. The Modified HIV Infection Model

Based on (2), our modified HIV infection model has the following form: (3) x ˙ = λ - d 1 x - k 1 x v x + v + p y , y ˙ = k 1 x v x + v - ( d 2 + p ) y , v ˙ = a y - d 3 v , where the meanings of the variables x ( t ) , y ( t ) , and v ( t ) and the parameters λ , d 1 , k 1 , d 2 , p , a , and d 3 are the same as those given in (2). Equation (3) has two steady states:

The infection-free steady state (4) Q 1 = ( x 0 , 0,0 )

represents the virus infection free. Q 1 is called infection-free equilibrium point. Here, (5) x 0 = λ d 1 .

The endemic infected steady state (6) Q 2 = ( x ¯ , y ¯ , v ¯ )

represents persistent virus infection. Q 2 is called endemic infection equilibrium point. Here, (7) x ¯ = λ R 0 k 1 ( R 0 - 1 ) + d 1 R 0 - p ( R 0 - 1 ) R 0 ( d 3 / a ) , y ¯ = d 3 a ( R 0 - 1 ) x ¯ , v ¯ = a y ¯ d 3 = ( R 0 - 1 ) x ¯ .

Here, (8) R 0 = a k 1 d 3 ( d 2 + p ) .

Since the total rate of disappearance of infected cells is d 2 + p , infected cells live on average for time 1 / ( d 2 + p ) . Each infected cell produces virus at rate a . Thus, each infected cell produces on average a total of a / ( d 2 + p ) viruses. Since virus dies at rate d 3 per virion, each virus survives on average for time 1 / d 3 . During the time 1 / d 3 , each virus infects on average k 1 x 0 / ( x 0 + v 0 ) d 3 cells, where x 0 and v 0 are the preinfection target cells’ density and viruses’ density, respectively. Thus, the total number of cells infected by the a / ( d 2 + p ) viruses is a k 1 x 0 / ( x 0 + v 0 ) d 3 ( d 2 + p ) . According to (4) and (5), x 0 = λ / d 1 and v 0 = 0 at the preinfection steady state. Then one can obtain that the total number of cells infected by each infected cell is a k 1 / d 3 ( d 2 + p ) . Hence, R 0 is the basic virus reproductive number of (3) which is independent of the total number of the uninfected CD 4 + T cells.

According to (4), (6), and (7), if R 0 1 , then Q 1 is the unique infection-free equilibrium point; if R 0 > 1 , then, in addition to the infection-free equilibrium point, (3) has another equilibrium point Q 2 .

2.2. Boundedness of Solutions

It is easy to show that the solutions of (3) with initial conditions x ( 0 ) > 0 , y ( 0 ) > 0 , and v ( 0 ) > 0 have all positive components for t > 0 . Hence, one begins the analysis of (3) by observing the nonnegative octant (9) D = { ( x , y , v ) R + 3 : x 0 , y 0 , v 0 } .

According to the first two equations of (3), one can get (10) x ˙ + y ˙ = λ - d 1 x - d 2 y λ - d ( x + y ) , d = min ( d 1 , d 2 ) , and then (11) x + y λ d . So x ( t ) and y ( t ) are bounded. From the last equation of (3), it follows that (12) v ˙ = a y - d 3 v a λ d - d 3 v , and then (13) v a λ d d 3 . So v ( t ) are bounded. Hence there is a bounded subset of D : (14) Ω = { ( x , y , v ) R + 3 : 0 x + y λ d , 0 v a λ d d 3 } such that any solution trajectory ( x ( t ) , y ( t ) , v ( t ) ) of (3) with initial value ( x ( 0 ) , y ( 0 ) , v ( 0 ) ) in Ω will keep in the subset Ω .

According to (7), x ¯ > 0 , y ¯ > 0 , and v ¯ > 0 . One can get that the endemic infection equilibrium point Q 2 exists in the interior of Ω : (15) Ω 0 = { a λ d d 3 ( x , y , v ) Ω : 0 < x , 0 < y , 0 < x + y < λ d , 0 < v < a λ d d 3 } . Therefore, the stability of the endemic infection equilibrium point Q 2 only needs to be discussed in Ω 0 .

3. Stability of the Infection-Free Equilibrium Point <inline-formula> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M122"> <mml:mrow> <mml:msub> <mml:mrow> <mml:mi>Q</mml:mi></mml:mrow> <mml:mrow> <mml:mn>1</mml:mn></mml:mrow> </mml:msub></mml:mrow> </mml:math></inline-formula>

In this section, we discuss locally asymptotical stability and globally asymptotical stability of the infection-free equilibrium point Q 1 of (3).

3.1. Locally Asymptotical Stability of the Infection-Free<bold /> Equilibrium Point <inline-formula> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M124"> <mml:mrow> <mml:msub> <mml:mrow> <mml:mi>Q</mml:mi></mml:mrow> <mml:mrow> <mml:mn>1</mml:mn></mml:mrow> </mml:msub></mml:mrow> </mml:math></inline-formula> Theorem 1.

If R 0 = a k 1 / ( d 3 ( d 2 + p ) ) < 1 , then the infection-free equilibrium point Q 1 of (3) is locally asymptotically stable. If R 0 > 1 , then the infection-free equilibrium point Q 1 is unstable.

Proof.

The Jacobi matrix of (3) at an arbitrary point is given by (16) J ( x , y , v ) = [ - d 1 - a 1 p - a 2 a 1 - p - d 2 a 2 0 a - d 3 ] , where a 1 = k 1 v 2 / ( x + v ) 2 and a 2 = k 1 x 2 / ( x + v ) 2 .

Substituting the equilibrium point Q 1 into matrix (16) gives (17) J Q 1 = [ - d 1 p - k 1 0 - p - d 2 k 1 0 a - d 3 ] . The corresponding eigenequation is (18) | λ E - J Q 1 | = | λ + d 1 - p k 1 0 λ + p + d 2 - k 1 0 - a λ + d 3 | = 0 . Solving (19) | λ E - J Q 1 | = ( λ + d 1 ) [ ( λ + p + d 2 ) ( λ + d 3 ) - a k 1 ] = 0 gives (20) λ 1 = - d 1 < 0 , (21) ( λ + d 2 + p ) ( λ + d 3 ) - a k 1 k 3 = 0 . Equation (21) can be written as (22) λ 2 + ( d 2 + p + d 3 ) λ + d 3 ( p + d 2 ) - a k 1 = 0 .

Solving equation (22) gives (23) λ 2 = - ( d 2 + p + d 3 ) - ( d 2 + p + d 3 ) 2 - 4 [ d 3 ( p + d 2 ) - a k 1 ] 2 = - ( d 2 + p + d 3 ) - ( d 2 + p + d 3 ) 2 - 4 d 3 ( p + d 2 ) ( 1 - R 0 ) 2 , λ 3 = - ( d 2 + p + d 3 ) + ( d 2 + p + d 3 ) 2 - 4 [ d 3 ( p + d 2 ) - a k 1 ] 2 = - ( d 2 + p + d 3 ) + ( d 2 + p + d 3 ) 2 - 4 d 3 ( p + d 2 ) ( 1 - R 0 ) 2 .

If R 0 < 1 , then λ 2 < 0 and λ 3 < 0 . Hence the infection-free equilibrium point Q 1 is locally asymptotically stable. If R 0 > 1 , then λ 3 > 0 such that the infection-free equilibrium point Q 1 is unstable.

3.2. Globally Asymptotical Stability of the Infection-Free<bold /> Equilibrium Point <inline-formula> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M146"> <mml:mrow> <mml:msub> <mml:mrow> <mml:mi>Q</mml:mi></mml:mrow> <mml:mrow> <mml:mn>1</mml:mn></mml:mrow> </mml:msub></mml:mrow> </mml:math></inline-formula> Theorem 2.

If R 0 < 1 , then the infection-free equilibrium point Q 1 of (3) is globally asymptotically stable in Ω .

Proof.

Define a global Lyapunov function by (24) V 1 ( x , y , v ) = y + ( d 2 + p ) v a . The derivative of V 1 ( x , y , v ) along the positive solutions of (3) is (25) V ˙ 1 = y ˙ + ( d 2 + p ) v ˙ a = k 1 x v x + v - d 2 y - p y + ( d 2 + p ) y - d 3 ( d 2 + p ) v a k 1 v - d 3 ( d 2 + p ) v a = [ a k 1 d 3 ( d 2 + p ) - 1 ] d 3 ( d 2 + p ) a v = ( R 0 - 1 ) d 3 ( d 2 + p ) a v .

If R 0 < 1 , then V ˙ 1 0 holds in Ω . Moreover, V ˙ 1 = 0 if and only if v = 0 . Hence, the largest compact invariant set in Ω is (26) E 1 = { ( x , y , v ) Ω V ˙ 1 = 0 } = { ( x , y , v ) Ω v = 0 } . According to the LaSalle’s invariance principle, lim t + v ( t ) = 0 . Then one can get limit equations: (27) x ˙ = λ - d 1 x + p y , y ˙ = - d 2 y - p y .

Define a global Lyapunov function by (28) V 2 ( x , y ) = x - x 0 - x 0 ln x x 0 + y , where (29) λ = d 1 x 0 , Q 1 = ( x 0 , 0,0 ) . The derivative of V 2 ( x , y ) along the positive solutions of (27) is (30) V ˙ 2 = x ˙ + y ˙ - x 0 x x ˙ = λ - d 1 x + p y - d 2 y - p y - x 0 x ( λ - d 1 x + p y ) . Since λ = d 1 x 0 , (31) V ˙ 2 = d 1 x 0 - d 1 x - x 0 x d 1 x 0 + d 1 x 0 - x 0 x p y - d 2 y = d 1 x 0 [ 2 - x x 0 - x 0 x ] - ( x 0 x p + d 2 ) y .

Since the arithmetic mean is greater than or equal to the geometric mean, we obtain 2 - ( x / x 0 ) - ( x 0 / x ) 0 .

Therefore, V ˙ 2 0 holds in E 1 , and V ˙ 2 = 0 if and only if x = x 0 and y = 0 . There is the largest compact invariant set in E 1 : (32) E 2 = { ( x , y , v ) E 1 V ˙ 2 = 0 } = { ( x , y , v ) E 1 x = x 0 , y = 0 } = { Q 1 } .

Hence if R 0 < 1 , all solution paths in Ω approach the infection-free equilibrium point Q 1 .

4. Stability of the Endemic Infection Equilibrium Point <inline-formula> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M179"> <mml:mrow> <mml:msub> <mml:mrow> <mml:mi>Q</mml:mi></mml:mrow> <mml:mrow> <mml:mn>2</mml:mn></mml:mrow> </mml:msub></mml:mrow> </mml:math></inline-formula>

In this section, we analyze local asymptotical stability and global asymptotical stability of the endemic infection equilibrium point Q 2 of (3).

4.1. Locally Asymptotical Stability of the Endemic Infection Equilibrium Point <inline-formula> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M181"> <mml:mrow> <mml:msub> <mml:mrow> <mml:mi>Q</mml:mi></mml:mrow> <mml:mrow> <mml:mn>2</mml:mn></mml:mrow> </mml:msub></mml:mrow> </mml:math></inline-formula> Theorem 3.

If R 0 > 1 , then the endemic infection equilibrium point Q 2 of (3) is locally asymptotically stable.

Proof.

Put the equilibrium point Q 2 into matrix (16); then one obtains (33) J Q 2 = [ - d 1 - k 1 v ¯ 2 ( x ¯ + v ¯ ) 2 p - k 1 x ¯ 2 ( x ¯ + v ¯ ) 2 k 1 v ¯ 2 ( x ¯ + v ¯ ) 2 - p - d 2 k 1 x ¯ 2 ( x ¯ + v ¯ ) 2 0 a - d 3 ] , | λ E - J Q 2 | = | λ + d 1 + k 1 v ¯ 2 ( x ¯ + v ¯ ) 2 - p k 1 x ¯ 2 ( x ¯ + v ¯ ) 2 - k 1 v ¯ 2 ( x ¯ + v ¯ ) 2 λ + p + d 2 - k 1 x ¯ 2 ( x ¯ + v ¯ ) 2 0 - a λ + d 3 | = 0 .

Solving the eigenequation of the matrix above, here is (34) λ 3    + [ d 1 + d 2 + d 3 + p + k 1 v ¯ 2 ( x ¯ + v ¯ ) 2 ] a 1 λ 2 + [ d 3 ( d 2 + p ) + ( d 1 + k 1 v ¯ 2 ( x ¯ + v ¯ ) 2 ) ( d 2 + p + d 3 ) - p k 1 v ¯ 2 ( x ¯ + v ¯ ) 2 - a k 1 x ¯ 2 ( x ¯ + v ¯ ) 2 ] a 2 λ + d 3 ( d 2 + p ) ( d 1 + k 1 v ¯ 2 ( x ¯ + v ¯ ) 2 ) + a k 1 2 x ¯ 2 v ¯ 2 ( x ¯ + v ¯ ) 2 - p d 3 k 1 v ¯ 2 ( x ¯ + v ¯ ) 2 - ( d 1 + k 1 v ¯ 2 ( x ¯ + v ¯ ) 2 ) a k 1 x ¯ 2 ( x ¯ + v ¯ ) 2 a 3 = 0 . If R 0 > 1 , one obtains that (35) a 1 = d 1 + d 2 + d 3 + p + k 1 v ¯ 2 ( x ¯ + v ¯ ) 2 > 0 , a 2 = d 3 ( d 2 + p ) + ( d 1 + k 1 v ¯ 2 ( x ¯ + v ¯ ) 2 ) ( d 2 + p + d 3 ) - p k 1 v ¯ 2 ( x ¯ + v ¯ ) 2 - a k 1 x ¯ 2 ( x ¯ + v ¯ ) 2 = a k 1 R 0 + d 1 ( d 2 + d 3 + p ) + k 1 v ¯ 2 ( x ¯ + v ¯ ) 2 ( d 2 + d 3 ) - a k 1 x ¯ 2 ( x ¯ + v ¯ ) 2 = a k 1 R 0 - a k 1 x ¯ 2 R 0 2 x ¯ 2 + d 1 ( d 2 + d 3 + p ) + k 1 v ¯ 2 ( x ¯ + v ¯ ) 2 ( d 2 + d 3 ) = a k 1 ( R 0 - 1 ) R 0 2 + d 1 ( d 2 + d 3 + p ) + k 1 v ¯ 2 ( x ¯ + v ¯ ) 2 ( d 2 + d 3 ) > 0 , a 3 = d 3 ( d 2 + p ) ( d 1 + k 1 v ¯ 2 ( x ¯ + v ¯ ) 2 ) + a k 1 2 x ¯ 2 v ¯ 2 ( x ¯ + v ¯ ) 2 - p d 3 k 1 v ¯ 2 ( x ¯ + v ¯ ) 2 - ( d 1 + k 1 v ¯ 2 ( x ¯ + v ¯ ) 2 ) a k 1 x ¯ 2 ( x ¯ + v ¯ ) 2 = d 1 d 3 ( d 2 + p ) - a d 1 k 1 x ¯ 2 ( x ¯ + v ¯ ) 2 + d 2 d 3 k 1 v ¯ 2 ( x ¯ + v ¯ ) 2 = d 1 a k 1 ( R 0 - 1 ) R 0 2 + d 2 d 3 k 1 v ¯ 2 ( x ¯ + v ¯ ) 2 > 0 , a 1 a 2 - a 3 = ( d 2 + d 3 + p + k 1 v ¯ 2 ( x ¯ + v ¯ ) 2 ) × a k 1 ( R 0 - 1 ) R 0 2 + d 1 ( d 2 + d 3 + p ) × ( d 2 + d 3 + p + d 1 + k 1 v ¯ 2 ( x ¯ + v ¯ ) 2 ) + d 2 k 1 v ¯ 2 ( x ¯ + v ¯ ) 2 ( d 2 + d 3 + p + d 1 + k 1 v ¯ 2 ( x ¯ + v ¯ ) 2 ) + d 3 k 1 v ¯ 2 ( x ¯ + v ¯ ) 2 ( d 3 + p + d 1 + k 1 v ¯ 2 ( x ¯ + v ¯ ) 2 ) > 0 .

Hence all inequalities of the Routh-Hurwitz criterion are satisfied. Therefore, the endemic infection equilibrium point Q 2 is locally asymptotically stable.

4.2. Globally Asymptotical Stability of the Endemic Infection Equilibrium Point <inline-formula> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M190"> <mml:mrow> <mml:msub> <mml:mrow> <mml:mi>Q</mml:mi></mml:mrow> <mml:mrow> <mml:mn>2</mml:mn></mml:mrow> </mml:msub></mml:mrow> </mml:math></inline-formula>

In this subsection, we firstly introduce a lemma outlined by Li and Wang , and then using this lemma discusses the globally asymptotical stability of the endemic infection equilibrium point Q 2 of (3).

The lemma is briefly summarized as follows.

Let x f ( x ) R n be a C 1 function for x in an open set Γ R n . Consider the differential system (36) x ˙ = f ( x ) . Denote by x ( t , x 0 ) the solution to (36) such that x ( 0 , x 0 ) = x 0 . Let x ~ be an equilibrium point of (36). Li and Wang  made the following two basic assumptions:

there exists a compact absorbing set K Γ ;

equation (36) has a unique equilibrium x ~ in Γ .

Li and Wang (see Theorem  2.5 in ) have given the following lemma.

Lemma 4 (see [<xref ref-type="bibr" rid="B18">18</xref>]).

Assume that

assumptions ( H 1 ) and ( H 2 ) hold;

equation (36) satisfies the Poincaré-Bendixson Property;

for each periodic solution x = p ( t ) to (36) with p ( 0 ) Γ , the linear system (the second additive compound system) (37) w ˙ ( t ) = f [ 2 ] x ( P ( t ) ) w ( t )

is asymptotically stable, where f [ 2 ] / x is the second additive compound matrix of the Jacobian matrix f / x ;

( - 1 ) n det ( ( f / x ) ( x ~ ) ) > 0 .

Then the unique equilibrium x ~ is globally asymptotically stable in Γ .

Now one uses Lemma 4 to show the following.

Theorem 5.

If R 0 > 1 , then the endemic infection equilibrium point Q 2 of (3) is globally asymptotically stable in Ω 0 , where Ω 0 is defined by (15).

Proof.

Based on Lemma 4, the proof of Theorem 5 has been implemented via the following four steps.

( 1 ) For epidemic models and many other biological models where the feasible region is a bounded cone, ( H 1 ) is equivalent to the uniform persistence of the system . By (15), Ω 0 is bounded, so it only needs to show the uniform persistence of (3). According to Proposition  3.3 in , the necessary and sufficient condition for the uniform persistence of (3) is equivalent to the equilibrium point Q 1 being unstable. Theorem 1 has shown that Q 1 is unstable if R 0 > 1 . Therefore, (3) is uniformly persistent if R 0 > 1 so that ( H 1 ) holds if R 0 > 1 .

Meanwhile, Q 1 = ( x 0 , 0,0 ) by (4), so Q 1 does not exist in Ω 0 . Hence, Q 2 is the unique equilibrium point of (3) in Ω 0 so that ( H 2 ) holds.

The results above verify the condition ( 1 ) of Lemma 4.

( 2 ) The Jacobian matrix of (3) is (38) J ( x , y , v ) = [ - d 1 - a 1 p - a 2 a 1 - p - d 2 a 2 0 a - d 3 ] , where a 1 = ( k 1 v 2 / ( x + v ) 2 ) and a 2 = ( k 1 x 2 / ( x + v ) 2 ) .

If H = diag ( 1 , - 1,1 ) , then (39) H J H = [ - d 1 - k 1 v 2 x + v - p - k 1 x 2 ( x + v ) 2 - k 1 v 2 ( x + v ) 2 - p - d 2 - k 1 x 2 ( x + v ) 2 0 - a - d 3 ] , and one can obtain that H J H has nonpositive off-diagonal elements in Ω 0 . Therefore (3) is competitive in Ω 0 . It is known that 3-dimensional competitive systems have the Poincaré-Bendixson Property . Hence, (3) satisfies the Poincaré-Bendixson Property. This verifies condition ( 2 ) of Lemma 4.

( 3 ) Let P ( t ) = ( x ( t ) , y ( t ) , v ( t ) ) be a periodic solution in Ω 0 .

According to , if B = ( b i j ) is a 3 × 3 matrix, then the second additive compound B [ 2 ] of B is (40) B [ 2 ] = [ b 11 + b 22 b 23 - b 13 b 32 b 11 + b 33 b 12 - b 31 b 21 b 22 + b 33 ] . The Jacobian matrix of (3) is (41) J ( x , y , v ) = [ - d 1 - a 1 p - a 2 a 1 - p - d 2 a 2 0 a - d 3 ] , where a 1 = k 1 v 2 / ( x + v ) 2 and a 2 = k 1 x 2 / ( x + v ) 2 .

And then the second additive compound matrix of the Jacobian matrix of (3) is given by (42) J [ 2 ] = [ - d 1 - k 1 v 2 ( x + v ) 2 - p - d 2 k 1 x 2 ( x + v ) 2 k 1 x 2 ( x + v ) 2 a - d 1 - k 1 v 2 ( x + v ) 2 - d 3 p 0 k 1 v 2 ( x + v ) 2 - p - d 2 - d 3 ] , and the second additive compound system of (3) along the periodic solution P ( t ) = ( x ( t ) , y ( t ) , v ( t ) ) is (43) w ˙ 1 = ( - d 1 - k 1 v 2 ( x + v ) 2 - p - d 2 ) w 1 + k 1 x 2 ( x + v ) 2 w 2 + k 1 x 2 ( x + v ) 2 w 3 , w ˙ 2 = a w 1 + ( - d 1 - k 1 v 2 ( x + v ) 2 - d 3 ) w 2 + p w 3 , w ˙ 3 = k 1 v 2 ( x + v ) 2 w 2 + ( - p - d 2 - d 3 ) w 3 . Define a global Lyapunov function by (44) V ( w 1 , w 2 , w 3 , P ) = ( w 1 , y ( t ) v ( t ) w 2 , y ( t ) v ( t ) w 3 ) , where · is the norm in set D defined by (45) ( w 1 , w 2 , w 3 ) = sup { | w 1 | , | w 2 | + | w 3 | } . Suppose that the solution P ( t ) is periodic of least period ω > 0 and that P ( 0 ) Ω 0 . According to , (3) is uniformly persistent, if there exists a positive constant μ such that (46) lim inf t x ( t ) μ , lim inf t y ( t ) μ , lim inf t v ( t ) μ . Step ( 1 ) has shown that (3) is uniformly persistent if R 0 > 1 . Hence, there always exists a positive constant μ which satisfies (46). The orbit of P ( t ) remains at a positive distance from the boundary of Ω by the uniform persistence, and one can obtain that (47) y ( t ) μ , v ( t ) μ for    large    enough    t .

Since v < a λ / ( d d 3 ) by (15), (48) V ( w 1 , w 2 , w 3 , P ) μ d d 3 a λ ( w 1 , w 2 , w 3 ) , for all ( w 1 , w 2 , w 3 ) R 3 .

Along a solution ( w 1 , w 2 , w 3 ) of (43), V ( w 1 , w 2 , w 3 , P ) becomes (49) V ( w 1 , w 2 , w 3 , P ) = sup { | w 1 | , y ( t ) v ( t ) ( | w 2 | + | w 3 | ) } . The right-hand derivative of V ( t ) along the positive solution of (43) is (50) D + | w 1 | ( - d 1 - k 1 v 2 ( x + v ) 2 - p - d 2 ) | w 1 | + k 1 x 2 ( x + v ) 2 ( | w 2 | + | w 3 | ) , D + | w 2 | a | w 1 | + ( - d 1 - k 1 v 2 ( x + v ) 2 - d 3 ) | w 2 | + p | w 3 | D + | w 3 | k 1 v 2 ( x + v ) 2 | w 2 | + ( - p - d 2 - d 3 ) | w 3 | . Therefore (51) D + y v ( | w 2 | + | w 3 | ) = y ˙ v - y v ˙ v 2 ( | w 2 | + | w 3 | ) + y v D + ( | w 2 | + | w 3 | ) y v ( y ˙ y - v ˙ v ) ( | w 2 | + | w 3 | ) + y v ( a | w 1 | + ( - d 1 - d 3 ) | w 2 | + ( - d 2 - d 3 ) | w 3 | ) a y v | w 1 | + y v ( | w 2 | + | w 3 | ) × ( y ˙ y - v ˙ v - d 3 - min ( d 1 , d 2 ) ) , (52) D + V ( t ) sup { g 1 ( t ) , g 2 ( t ) } V ( t ) , where (53) g 1 ( t ) = - d 1 - p - d 2 - k 1 v 2 ( x + v ) 2 + k 1 v x 2 y ( x + v ) 2 y ˙ y - d 1 , g 2 ( t ) = a y v + y ˙ y - v ˙ v - d 3 - min ( d 1 , d 2 ) = y ˙ y - min ( d 1 , d 2 ) . Denote d = min ( d 1 , d 2 ) , and then (54) sup { g 1 ( t ) , g 2 ( t ) } y ˙ y - d . By (52) and Gronwall’s inequality, one obtains (55) V ( t ) V ( 0 ) y ( t ) e - d t V ( 0 ) e - d t λ d . V ( t ) 0 when t , and then ( w 1 , w 2 , w 3 ) 0 when t by (48). The second additive compound system is asymptotically stable. This verifies the condition ( 3 ) of Lemma 4.

( 4 ) The Jacobi matrix of (3) at the endemic infection equilibrium Q 2 is (56) J Q 2 = [ - d 1 - k 1 v ¯ 2 ( x ¯ + v ¯ ) 2 p - k 1 x ¯ 2 ( x ¯ + v ¯ ) 2 k 1 v ¯ 2 ( x ¯ + v ¯ ) 2 - p - d 2 k 1 x ¯ 2 ( x ¯ + v ¯ ) 2 0 a - d 3 ] , and then (57) det ( J Q 2 ) = | - d 1 - k 1 v ¯ 2 ( x ¯ + v ¯ ) 2 p - k 1 x ¯ 2 ( x ¯ + v ¯ ) 2 k 1 v ¯ 2 ( x ¯ + v ¯ ) 2 - p - d 2 k 1 x ¯ 2 ( x ¯ + v ¯ ) 2 0 a - d 3 | = - [ d 1 + k 1 v ¯ 2 ( x ¯ + v ¯ ) 2 ] ( d 2 + p ) d 3 - a k 1 2 x ¯ 2 v ¯ 2 ( x ¯ + v ¯ ) 4 + d 3 p k 1 v ¯ 2 ( x ¯ + v ¯ ) 2 + a k 1 x ¯ 2 ( x ¯ + v ¯ ) 2 [ d 1 + k 1 v ¯ 2 ( x ¯ + v ¯ ) 2 ] = - d 1 ( d 2 + p ) d 3 - k 1 v ¯ 2 ( x ¯ + v ¯ ) 2 ( d 2 + p ) d 3 - a k 1 2 x ¯ 2 v ¯ 2 ( x ¯ + v ¯ ) 4 + d 3 p k 1 v ¯ 2 ( x ¯ + v ¯ ) 2 + a d 1 k 1 x ¯ 2 ( x ¯ + v ¯ ) 2 + a k 1 2 x ¯ 2 v ¯ 2 ( x ¯ + v ¯ ) 4 = d 1 [ a k 1 x ¯ 2 ( x ¯ + v ¯ ) 2 - ( d 2 + p ) d 3 ] - d 2 d 3 k 1 v ¯ 2 ( x ¯ + v ¯ ) 2 - p d 3 k 1 v ¯ 2 ( x ¯ + v ¯ ) 2 + d 3 p k 1 v ¯ 2 ( x ¯ + v ¯ ) 2 = a d 1 k 1 [ ( x ¯ x ¯ + v ¯ ) 2 - 1 R 0 ] - d 2 d 3 k 1 v ¯ 2 ( x ¯ + v ¯ ) 2 . According to (7), v ¯ = ( R 0 - 1 ) x ¯ , and then one can obtain (58) det ( J Q 2 ) = a d 1 k 1 [ 1 R 0 2 - 1 R 0 ] - d 2 d 3 k 1 v ¯ 2 ( x ¯ + v ¯ ) 2 = a d 1 k 1 ( 1 - R 0 ) R 0 2 - d 2 d 3 k 1 v ¯ 2 ( x ¯ + v ¯ ) 2 . Since J Q 2 is a 3 × 3 matrix, one gets n = 3 . Then (59) ( - 1 ) 3 det ( J Q 2 ) = - a d 1 k 1 ( 1 - R 0 ) R 0 2 + d 2 d 3 k 1 v ¯ 2 ( x ¯ + v ¯ ) 2 . If R 0 > 1 , then ( - 1 ) 3 det ( J Q 2 ) > 0 holds in Ω 0 . This verifies condition ( 4 ) of Lemma 4.

Hence, if R 0 > 1 , then the endemic infection equilibrium point Q 2 is globally asymptotically stable in Ω 0 by Lemma 4.

5. Numerical Simulation

In the first subsection, we determine some parameter values of an anti-HIV infection treatment model based on (3). In the second subsection, using the anti-HIV infection treatment model simulates the dynamics of the Group I’s anti-HIV infection treatment. In the third subsection, using the anti-HIV infection treatment model simulates the dynamics of the Group II’s anti-HIV infection treatment. In the fourth subsection, we make long-term predictions for the two groups’ anti-HIV infection treatment, respectively.

5.1. Modeling

Baxter et al.  have reported a randomized study of antiretroviral management based on plasma genotypic antiretroviral resistance testing in HIV patients failing therapy, which was enrolled from 14 units of the Terry Beirn Community Programs for Clinical Research on AIDS and the Walter Reed Army Medical Center (see the HIV drug resistance database of Stanford University ). These patients were failing virologically on a combination antiretroviral regimen containing protease inhibitors (PI) and nucleoside reverse transcriptase inhibitors (NRTI) . The patients were seen at 4, 8, and 12 weeks. At each follow-up visit, changes in antiretroviral treatment were recorded and the tested items included patients’ plasma CD 4 + T cells counts and plasma HIV-1 RNA levels by the Chiron 2.0 bDNA assay .

In the following subsections, we select, from [24, 25], two group patients’ mean uninfected CD 4 + T cells counts and mean HIV RNA levels to simulate and make long-term predictions for the patients’ treatment outcomes. Group I consists of 15 patients. Group II consists of 13 patients. The two groups of patients received the same PI: ritonavir (RTV) and saquinavir (SQV). Additionally, Group I received NRTI: strvudine (D4T). Group II received NRTI: strvudine (D4T) and dideoxyinosine (DDI) [24, 25].

Based on (3), the anti-HIV infection treatment model has the form (60) x ˙ = λ - d 1 x - ( 1 - m ) k 1 v x x + v + p y , y ˙ = ( 1 - m ) k 1 v x x + v - d 2 y - p y , v ˙ = ( 1 - n ) a y - d 3 v , where m , n    ( 0 m , n 1 ) are the efficacy variables of the treatment.

The infection-free equilibrium point Q 1 of (60) is the same as that defined by (4): (61) Q 1 = ( λ d 1 , 0,0 ) .

The endemic infection equilibrium point Q 2 of (60) is given by (62) Q 2 = ( x ¯ , y ¯ , v ¯ ) , where, (63) x ¯ = λ R 0 ( 1 - m ) k 1 ( R 0 - 1 ) + d 1 R 0 - p ( R 0 - 1 ) R 0 ( d 3 / ( 1 - n ) a ) , y ¯ = d 3 ( 1 - n ) a ( R 0 - 1 ) x ¯ , v ¯ = ( 1 - n ) a y ¯ d 3 = ( R 0 - 1 ) x ¯ , and R 0 is the basic virus reproductive number of (60): (64) R 0 = ( 1 - n ) (