DDNS Discrete Dynamics in Nature and Society 1607-887X 1026-0226 Hindawi Publishing Corporation 162527 10.1155/2012/162527 162527 Research Article Stability Analysis of an HIV/AIDS Dynamics Model with Drug Resistance Li Qianqian 1 Cao Shengshan 1 Chen Xiao 1 Sun Guiquan 2 Liu Yunxi 3 Jia Zhongwei 4, 5 Raffoul Youssef 1 School of Mathematical Sciences, Ocean University of China Qingdao 266100 China ouc.edu.cn 2 Department of Mathematics North University of China Taiyuan 030051 China nuc.edu.cn 3 Department of Nosocomial Infection Management and Disease Control Chinese PLA General Hospital 28 Fuxing Road Haidian District Beijing 100853 China 4 National Institute of Drug Dependence Peking University Beijing 100191 China pku.edu.cn 5 Takemi Program in International Health Department of Global Health and Population Harvard School of Public Health 665 Huntington Avenue Boston MA 02115 USA hsph.harvard.edu 2012 19 12 2012 2012 11 08 2012 22 10 2012 2012 Copyright © 2012 Qianqian Li et al. 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.

A mathematical model of HIV/AIDS transmission incorporating treatment and drug resistance was built in this study. We firstly calculated the threshold value of the basic reproductive number (R0) by the next generation matrix and then analyzed stability of two equilibriums by constructing Lyapunov function. When R0<1, the system was globally asymptotically stable and converged to the disease-free equilibrium. Otherwise, the system had a unique endemic equilibrium which was also globally asymptotically stable. While an antiretroviral drug tried to reduce the infection rate and prolong the patients’ survival, drug resistance was neutralizing the effects of treatment in fact.

1. Introduction

It was reported that 2.7 million people were newly infected by HIV/AIDS virus and 1.8 million patients died of AIDS-related causes in 2010 worldwide. By the end of 2010, about 34 million people were living with HIV/AIDS in the world . China estimated that 2.8 million died of AIDS-related causes in 2011, and there were about 7.8 million HIV-infected people by the end of 2011 .

Since the initial infectious diseases model was presented by Anderson et al. in 1986 , various mathematical models have been developed among which the treatment has been addressed . For example, Wang and Zhou’s model tried to address HIV treatment and progression by CD4 ~+ T-cells and virus particles in microcosmic ; Blower, Boily et al., and Bachar and Dorfmayr’s works tried to investigate the effect of treatment on sexual behaviors ; Blower et al., Sharomi and Gumel and Nagelkerke et al.studied the epidemic contagion transmission in some specific regions or groups considering drug resistance .

In this work, we established a model by adding effect of drug resistance into the similar models in the literatures [611, 15, 16]. The model in  are include both virus drug resistance and drug sensitive on the treatment. However the model in  did not distinguish the stage of HIV and AIDS, our model aware of the different between HIV infections and AIDS patients. Moreover, compare to those models in [13, 14], we carefully consider some infections would exit treatment group without developing drug-resistance due to other reasons such as migration. Theoretical analysis on global stability of endemic equilibrium has then been implemented.

2. Dynamic Model

According to the progression of disease, the total populations were separated into five groups: susceptible population, early-stage HIV population, symptomatic population, AIDS patients, and those who are accepting ART; we marked them with S(t), I(t), J(t), A(t), and T(t) separately. Treatment has three outcomes: (1) a patient can respond to treatment and remain the ART; (2) exit treatment due to clinical failure, migration, or other reasons without developing drug resistance; (3) virologically fail and develop drug resistance. We use R(t) to denote patients in situation (3).

We made some assumption as below.

Infection occurred when susceptible and infected contact with each other took place.

Only people in the period of AIDS may die of AIDS disease-related, then use d denote the disease-related death rate of the AIDS. And use μ denote the mortality rate in the total population.

Although AIDS patients have the higher viral load, we assume they will not infect others because they have the obvious clinical symptoms and were accepting ART.

When in the situation (2) of treatment, we assume these people transformed into asymptomatic individuals.

According to the assumptions, the flow diagram of the six subpopulations was shown in Figure 1.

Relationship between different populations.

The model was collected as the following differential equations: (2.1)S=  Λ-μS-(β1I+β2J+β3T+β4R)S,I=(β1I+β2J+β3T+β4R)S-αI-μI,J=αI-(ρ1+σ+k1)J-μJ+γT,T=σJ-(ρ2+γ+k2)T-μT,R=k1J+k2T-ρ3R-μR,A=ρ1J+ρ2T+ρ3R-(d+μ)A, where Λ is the recruitment rate of the susceptible population, β1 is the probability of transmission by an infection in the first stage, β2 is the probability of transmission by an infection in the second phase, β3=lβ2  (l<1) is the probability of transmission by a patient being treated, β4 is the probability of transmission by a drug resistance individual, α is the transfer rate constant from the asymptomatic phase I to the symptomatic phase J, σ is the proportion of treatment, γ is the exit treatment rate without developing drug resistance, k1 is probability constant of infection by transmission of drug-resistant strains, and k2 is the rate of acquiring drug resistance during treatment; ρi  (i=1,2,3) denote transfer rate constant by an infection from phase J, T, R to the AIDS cases A, respectively.

The model is established in practice; thus we assume all parameters are nonnegative.

Since the A of system (2.1) does not appear in the equations, in the following analysis, we only consider the system as follows: (2.2)S=Λ-μS-(β1I+β2J+β3T+β4R)S,I=(β1I+β2J+β3T+β4R)S-(α+μ)I,J=αI-(ρ1+σ+k1+μ)J+γT,T=σJ-(ρ2+γ+k2+μ)T,R=k1J+k2T-(ρ3+μ)R.

Theorem 2.1.

Let the initial data be S(0)=S0>0, I(0)=I0>0, J(0)=J0>0, T(0)=T0>0 and R(0)=R0>0; then the solutions of system (2.2) are all positive for all t > 0. For the model, the feasible region of system (2.2) is Ω={(S,I,J,T,R)+5:S+I+J+T+RΛ/μ,  0<SΛ/μ}, and Ω for system (2.2) is positively invariant.

Proof.

From the first equation of (2.2) (2.3)S=Λ-μS-(β1I+β2J+β3T+β4R)S, consider the following two categories.

When t0>0  and  S(t0)=0.

Equation (2.3) becomes S=Λ  (t=t0), due to Λ>0; we have S(t0)>0, that is, when t>t0,  S(t) is an increasing function about t. Therefore, we can conclude that when t is the neighborhood of t0 and t>t0,  S(t)S(t0)>0.

When t0>0  and  S(t0)>0.

Equation (2.3) can be written as (2.4)SS=ΛS-[μ+(β1I+β2J+β3T+β4R)], that is (2.5)SS-[μ+(β1I+β2J+β3T+β4R)], thus, (2.6)S(t)S(0)exp[-0t[μ+(β1I(s)+β2J(s)+β3T(s)+β4R(s))]ds]>0.

Similarly for the other equations of system (2.2) we can easily show that I(t), J(t), T(t), R(t) are increasing functions about t when t0>0 and I(t0)=0, t0>0 and J(t0)=0, t0>0 and T(t0)=0,  t0>0 and R(t0)=0, respectively; then when t>t0, X(t)X(t0)>0  (X=I,J,T,R). Otherwise, when t0>0 and X(t0)>0  (X=I,J,T,R), we have the following results corresponding to the respective hypothesis: (2.7)II-(α+μ). Thus, (2.8)I(t)I(0)exp[-(α+μ)t]>0,JJ-(ρ1+σ+k1+μ). Thus, (2.9)J(t)J(0)exp[-(ρ1+σ+k1+μ)t]>0,TT-(ρ2+γ+k2+μ). Thus, (2.10)T(t)T(0)exp[-(ρ2+γ+k2+μ)t]>0,RR-(ρ3+μ). Thus, (2.11)R(t)R(0)exp[-(ρ3+μ)t]>0. Then, we have that I(t), J(t), T(t), and R(t) are all strictly positive for t>0. Thus we can conclude that all solutions of system (2.2) remain positive for all t>0.

Next, add all the the equations of system (2.2); we have (2.12)(S+I+J+T+R)=  Λ-μ(S+I+J+T+R)-(ρ1J+ρ2T+ρ3R)  Λ-μ(S+I+J+T+R).

Then, (2.13)limtsup(S+I+J+T+R)Λμ.

In a similar fashion we have SΛ-μS from the first equation of (2.2); then limtsupSΛ/μ.

Thus, the feasible solution of system remains in the region Ω, and Ω as the feasible region for system is positively invariant. In the following, the dynamics of system (2.2) will be considered in Ω.

3. The Basic Reproduction Number and the Disease-Free Equilibrium 3.1. The Basic Reproduction Number

It is easy to see that the model has a disease-free equilibrium (DFE),  P0=(Λ/μ,0,0,0,0). Following the paper , we obtain the basic reproduction number by using the next generation operator approach.

Let y=(I,J,T,R,S)T; thus we have (3.1)y=(y)-𝒱(y), where (3.2)(y)=((β1I+β2J+β3T+β4R)S0000),(3.3)𝒱(y)=((α+μ)I-αI+(ρ1+σ+k1+μ)J-γT-σJ+(ρ2+γ+k2+μ)T-k1J-k2T+(ρ3+μ)R-Λ+μS+(β1I+β2J+β3T+β4R)S). Then the derivatives of (y) and 𝒱(y) at the DFE P0-=(0,0,0,0,Λ/μ) are partitioned as (3.4)D(P0-)=(F000),D𝒱(P0-)=(V0000β1·Λμ  β2·Λμβ3·Λμβ4·Λμμ).F and V are the 4×4 matrices as follows: (3.5)F=(β1·Λμ  β2·Λμ  β3·Λμ    β4·Λμ000            000000            00            0),V=(Q1000-αQ2-  γ00-σQ300  -k1-k2  Q4), where (3.6)Q1=(α+μ),Q2=(ρ1+σ+k1+μ),Q3=(ρ2+γ+k2+μ),Q4=(ρ3+μ).

Hence the reproduction number, denoted by R0, is the spectral radius of the next generation matrix FV-1: (3.7)R0=ρ(FV-1)=R1+R2+R3+R4. Here (3.8)R1=β1Q1·Λμ,  R2=β2αQ3Q1(Q2Q3-σγ)·Λμ,R3=β3ασQ1(Q2Q3-σγ)·Λμ,R4=β4α(k1Q3+k2σ)Q1(Q2Q3-σγ)Q4·Λμ.

3.2. DFE and Stability Theorem 3.1.

The disease-free equilibrium P0 of system is globally asymptotically stable for R0<1 and unstable for R0>1.

Proof.

(1) The Jacobian matrices of system (2.2) at the DFE are (3.9)J(P0)=(-μ    -β1·Λμ    -β2·Λμ    -β3·Λμ    -β4·Λμ0000D), where (3.10)D=(β1·Λμ-Q1β2·Λμβ3·Λμβ4·Λμα-Q2  γ00σ-Q300  k1k2  -Q4).

Obviously, -D=[dij] is a 4×4 matrix with dij0, for ij,i,j=1,,4 and dii>0 for i=2,,4. If R0<1, we have R1=(β1/Q1)·(Λ/μ)<R0<1 from the expression of (3.8); thus d11=Q1-β1·(Λ/μ)=Q1(1-R1)>0.

Define the positive vector subsequently: (3.11)x=(1,αQ3Q2Q3-σγ,ασQ2Q3-σγ,α(k1Q3+k2σ)(Q2Q3-σγ)Q4)T. If R0<1, -D·x=[Q1(1-R0),0,0,0]T0.

Then begin to show that all the eigenvalues of -D are nonzero: (3.12)det(-D)=Q1Q4(Q2Q3-σγ)-β1(Q2Q3-σγ)Q4  -α(β2Q3+β3σ)Q4-α(β2Q3+β3σ)Q4. Simplify the above expression through substituting formula (3.8) into (3.12): (3.13)det(-D)=Q1Q4(Q2Q3-σγ)(1-R0). Here Q2Q3-σγ=(ρ1+σ+k1+μ)(ρ2+γ+k2+μ)-σγ>0; thus, det(-D)>0; namely, -D has non zero eigenvalue for R0<1.

In conclusion, if R0<1,  -D is an irreducible matrix with dii>0 and dij0(ij), there exists a positive vector x such that -D·x0. Hence, the real part of each nonzero eigenvalue of -D is positive according to the M-matrix theory; that is, each eigenvalue of D has negative real part.

Through the structure of the Jacobian matrix J(P0), it can be seen that the eigenvalues of J(P0) consist of -μ and all eigenvalues of D. Hence, all eigenvalues of J(P0) have negative real part for R0<1; thus, disease-free equilibrium P0 is locally asymptotically stable.

(2) Let U1=m1I+m2J+m3T+m4R, where mi(i=14) are positive constants as follows: (3.14)m1=β1Q1+β2αQ3Q1(Q2Q3-σγ)+β3ασQ1(Q2Q3-σγ)+β4α(k1Q3+k2σ)Q1(Q2Q3-σγ)Q4,m2=β2Q3Q2Q3-σγ+β3σQ2Q3-σγ+β4(k1Q3+k2σ)(Q2Q3-σγ)Q4,m3=β2γQ2Q3-σγ+β3Q2Q2Q3-σγ+β4(k1γ+k2Q2)(Q2Q3-σγ)Q4,m4=β4Q4. When R0<1, the time derivative of U1 is (3.15)U1|(2.2)=m1I+m2J+m3T+m4R=(m1β1S-m1Q1+m2α)I+(m1β2S-m2Q2+m3σ+m4k1)J+(m1β3S+m2γ-m3Q3+m4k2)T+(m1β4S-m4Q4)R=(m1S-1)(β1I+β2J+β3T+β4R). Let M=max(β1/m1,β2/m2,β3/m3,β4/m4); due to limtsupSΛ/μ, M  is the finite number, then (3.16)U1(m1Λμ-1)(β1I+β2J+β3T+β4R)=(R0-1)(β1I+β2J+β3T+β4R)M(R0-1)(m1I+m2J+m3T+m4R). Solve U1M(R0-1)U1, have U1U1(0)eM(R0-1)t, that is, limtU1(t)=0, thus when t, (I,J,T,R)(0,0,0,0). When R0<1, U10, and equalities hold if and only if S=Λ/μ,I=J=T=R=0, that is U1=0 if and only if t. We can conclude that the solutions of system (2.2) are all in Ψ={(S,I,J,T,R):S=Λ/μ,I=J=T=R=0} and the only invariant set in Ψ is P0 by the LaSalle’s invariance principle. Thus the solutions of system (2.2) are limits to the endemic equilibrium P0 when R0<1. Combine locally asymptotically stable of P0 with convergence properties of the P0, we conclude that P0 of system is globally asymptotically stable for R0<1.

(3) If R0>1, detD=Q1Q4(Q2Q3-σγ)(1-R0)<0; thus D has eigenvalue with positive real part, otherwise, detD>0; this is contradiction. Hence, P0 is unstable for R0>1.

4. Endemic Equilibrium and Stability

Equating each equation in system (2.2) to zero and solving this equilibrium equations, system has the unique positive equilibrium P*=(S*,I*,J*,T*,R*) for R0>1, here (4.1)S*=Λμ·1R0,I*=ΛQ1(1-1R0),  J*=αQ3Q2Q3-σγI*T*=ασQ2Q3-σγI*,R*=α(k1Q3+k2σ)(Q2Q3-σγ)Q4I*.

Theorem 4.1.

Endemic equilibrium P* of system is globally asymptotically stable for R0>1.

Proof.

Equating each equation in system (2.2) to zero, the equilibrium equations as follows are useful: (4.2)Λ=μS*+(β1I*+β2J*+β3T*+β4R*)S*,Q1I*=(β1I*+β2J*+β3T*+β4R*)S*,Q2J*=αI*+γT*,Q3T*=σJ*,Q4R*=k1J*+k2T*, where Qi is defined in (3.6).

Setting x=(S,I,J,T,R)Ω5+, construct a Lyapunov function (4.3)U2=U2(x)=(S-S*-S*lnSS*)+A1(I-I*-I*lnII*)+A2(J-J*-J*lnJJ*)+A3(T-T*-T*lnTT*)+A4(R-R*-R*lnRR*), where  x*=P*=(S*,I*,J*,T*,R*) and Ai>0 is constant. There,  U2(x)0 for xIntΩ, and U2(x)=0x=x*.

Computing the time derivative of U2, we have (4.4)U2=S(1-S*S)+A1I(1-I*I)+A2J(1-J*J)+A3T(1-T*T)+A4R(1-R*R). Using (2.2) and (4.10), we obtain (4.5)S(1-S*S)=(1-S*S)[Λ-μS-(β1I+β2J+β3T+β4R)S]=(1-S*S)[μS*+(β1I*+β2J*+β3T*+β4R*)S*-μS-(β1I+β2J+β3T+β4R)S]=μS*(2-SS*-S*S)-(β1I+β2J+β3T+β4R)S+(β1I+β2J+β3T+β4R)S*+(β1I*+β2J*+β3T*+β4R*)S*-(β1I*+β2J*+β3T*+β4R*)S*2S. Similarly, we obtain (4.6)A1I(1-  I*I)=A1[I*I(β1I+β2J+β3T+β4R)S+(β1I*+β2J*+β3T*+β4R*)S*  A1I(1-  I*I)=A1A1-Q1I-(β1I+β2J+β3T+β4R)SI*I],A2J(1-  J*J)=A2[αI-Q2J+γT+αI*+γT*-αIJ*J-γTJ*J],A3T(1-T*T)=A3[σJ-Q3T+σJ*-σJT*T],A4R(1-R*R)=A4[k1J+k2T-Q4R+Q4R*+k1J*+k2T*-k1JR*R-k2TR*R]. Substituting formula (4.5) and (4.6) into (4.4) and arranging the equation we have (4.7)U2=Q0+Q1+Q2+Q3+Q4, where (4.8)Q0=μS*(2-SS*-S*S),Q1=(A1-1)(β1I+β2J+β3T+β4R)S,Q2=(A1+1)(β1I*+β2J*+β3T*+β4R*)S*+A2αI*+A2γT*+A3σJ*+A4k1J*+A4k2T*,Q3=(β1I+β2J+β3T+β4R)S*+A2αI+A2γT+A3σJ+A4k1J+A4k2T-A1Q1I-A2Q2J-A3Q3T-A4Q4R,Q4=-(β1I*+β2J*+β3T*+β4R*)S*2S-A1(β1I+β2J+β3T+β4R)I*I-A2αIJ*J-A2γTJ*J-A3σJT*T-A4k1JR*R-A4k2TR*R. In (4.7), Q2 consists of all constant terms, Q3 contains all linear terms of I, J, T, R, and Q4 contains all negative nonlinear.

In order to determine the coefficient Ai of U2, let Q30 (in Ω); then the coefficients of state variables I, J, T, R are equal to zero, that is: (4.9)β1S*-A1Q1+A2α=0β2S*-A2Q2+A3σ+A4k1=0β3S*+A2γ-A3Q3+A4k2=0β4S*-A4Q4=0. Solving (4.9), and using the expression of R0 and S*, we have (4.10)A1=1,A2=1Q2Q3-σγ(β2Q3+β3σ+β4(k1Q3+k2σ)Q4)S*,A3=1Q2Q3-σγ(β2γ+β3Q2+β4(k1γ+k2Q2)Q4)S*,A4=β4Q4S*, then,  Q1=0.

Let  S/S*=x, I/I*=y, J/J*=z, T/T*=u, R/R*=v; substituting these expressions into (4.9), and then substituting the changing expression into (4.7) (4.11)U2=μS*(2-x-1x)+2β1S*I*+2β2S*J*+2β3S*T*+2β4S*R*+A2αI*+A2γT*+A3σJ*+A4k1J*+A4k2T*-β1S*I*1x-β2S*J*1x-β3S*T*1x-β4S*R*1x-β1S*I*x-β2S*J*xzy-β3S*T*xuy-β4S*R*xvy-A2αI*yz-A2γT*uz-A3σJ*zu-A4k1J*zv-A4k2T*uv=μS*(2-x-1x)+β1S*I*(2-x-1x)+β2S*J*(2-xzy-1x)+β3S*T*(2-xuy-1x)+β4S*R*(2-xvy-1x)+A2αI*(1-yz)+A2γT*(1-uz)+A3σJ*(1-zu)+A4k1J*(1-zv)+A4k2T*(1-uv). Using the arithmetic mean geometric to get that 2-x-1/x is less than or equal to zero, substituting Ai, and simplifying the other expressions in (4.11) after tedious algebraic manipulations, we can get the other expressions such as (2-xz/y-1/x), (2-xu/y-1/x), and (2-xv/y-1/x) is less than or equal to zero; this indicates that U20, and equalities hold if and only if x=1,y=z=u=v. Furthermore, S=S*, I/I*=J/J*=T/T*=R/R*=a; then substituting S=S*, I=aI*, J=aJ*, T=aT*, R=aR* into the first equation of system (2.2), and in contrast to the first equality of (4.2), we have a=1.

In conclusion, the limit sets of solutions in Ω are all in Γ={(S,I,J,T,R):S=S*,I=I*,J=J*,T=T*,R=R*}, and the only invariant set in Γ is P* by the LaSalle’s invariance principle. Thus the solutions of system (2.2) in Ω are limits to the endemic equilibrium P*, and P* is globally asymptotically stable for R0>1.

5. Discussion

This paper is an extended model about the works in [15, 16] by adding treatment and drug resistance in the whole transmission as well as considering the reasons of treatment exiting.

For public health view, to bring HIV/AIDS into control, the prerequisite is reducing the threshold value of basic reproductive number R0. If control R0<1, the disease can be eliminated from population. R052 RCT study indicated that treatment can prevent in HIV transmission ; this sounds that increasing proportion of treated population is helpful to control HIV epidemic overall. In our study, ART is clearly affecting R0   in the HIV procession. However, we cannot yet give this positive result based on the formula of R0 and σ in our work. That is because the treatment might also induce drug resistance which neutralizes the effect of treatment. ART might produce a more complicated HIV progress. However, decreasing acquiring drug-resistant rate k1, k2 and treatment exiting γ are the feasible measures to reduce R0. Improving treatment standard and patients’ compliance are the feasible and effective measures to reduce k1, k2. Certainly, new effective antiretroviral drugs might be the real determinants.

R 0 is also linked with drug resistance by the transmission rate β4  of drug resistance individual and removing rate ρ3 from the population R. When the other parameters keep constant, R0 is positive with β4 and negative with  ρ3. The value of R0 will increase if more patients enter this kind of population; that means the drug resistance can fuel HIV epidemic. However, the acquiring drug-resistant rate k1, k2 can be prevented by improving treatment quality, and the transmission coefficient parameter β4 can be reduced by decreasing contacts between these patients and other people at public health level. Generally, early finding by routine screening for drug resistance is an important way to find them.

Limitations in our model exist. We ignored the changing drug when treatment failed in practice. A refinement of the model can be done in future. Additionally, we did do simulation with actual data, which are ongoing under further study.

Acknowledgments

This study was supported by Chinese government grants administered under the Twelfth Five-Year Plan (2012ZX10001-002), the Beijing Municipal Commission of Education (KM201010025010), and National Nature Science Foundation (30973981).

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