Stability Analysis of an HIV / AIDS Dynamics Model with Drug Resistance

1 School of Mathematical Sciences, Ocean University of China, Qingdao 266100, China 2 Department of Mathematics, North University of China, Taiyuan 030051, China 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 5 Takemi Program in International Health, Department of Global Health and Population, Harvard School of Public Health, 665 Huntington Avenue, Boston, MA 02115, USA


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 1 .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 2 .
Since the initial infectious diseases model was presented by Anderson et al. in 1986 3-5 , various mathematical models have been developed among which the treatment has been addressed 6-16 .For example, Wang and Zhou's model tried to address HIV treatment and progression by CD4 ∼ T-cells and virus particles in microcosmic 8 ; Blower, Boily et al., and Bachar and Dorfmayr's works tried to investigate the effect of treatment on sexual behaviors 9-11 ; Blower et al., Sharomi and Gumel and Nagelkerke et al. studied the epidemic contagion transmission in some specific regions or groups considering drug resistance 12-14 .In this work, we established a model by adding effect of drug resistance into the similar models in the literatures 6-11, 15, 16 .The model in 12-14 are include both virus drug resistance and drug sensitive on the treatment.However the model in 12 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.

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.
1 Infection occurred when susceptible and infected contact with each other took place.
2 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.
3 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.
4 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.
The model was collected as the following differential equations: where Λ is the recruitment rate of the susceptible population, β 1 is the probability of transmission by an infection in the first stage, β 2 is the probability of transmission by an infection in the second phase, β 3 lβ 2 l < 1 is the probability of transmission by a patient being treated, β 4 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, k 1 is probability constant of infection by transmission of drug-resistant strains, and k 2 is the rate of acquiring drug resistance during treatment; ρ i 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:
Proof.From the first equation of 2.2 consider the following two categories.
Equation 2.3 becomes S Λ t t 0 , due to Λ > 0; we have S t 0 > 0, that is, when t > t 0 , S t is an increasing function about t.Therefore, we can conclude that when tis the neighborhood of t 0 and t > t 0 , S t ≥ S t 0 > 0.

Equation 2.3 can be written as
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 t 0 > 0 and I t 0 0, t 0 > 0 and J t 0 0, t 0 > 0 and T t 0 0, t 0 > 0 and R t 0 0, respectively; then when t > t 0 , X t ≥ X t 0 > 0 X I, J, T, R .Otherwise, when t 0 > 0 and X t 0 > 0 X I, J, T, R , we have the following results corresponding to the respective hypothesis: Thus,

2.10
Thus, 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 In a similar fashion we have S ≤ Λ − μS from the first equation of 2.2 ; then lim t → ∞ sup S ≤ Λ/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.

The Basic Reproduction Number
It is easy to see that the model has a disease-free equilibrium DFE , P 0 Λ/μ, 0, 0, 0, 0 .Following the paper 17 , we obtain the basic reproduction number by using the next generation operator approach.
Let y I, J, T, R, S T ; thus we have where Then the derivatives of F y and V y at the DFE P 0 0, 0, 0, 0, Λ/μ are partitioned as F and V are the 4 × 4 matrices as follows: where

3.6
Hence the reproduction number, denoted by R 0 , is the spectral radius of the next generation matrix FV −1 :

DFE and Stability
Theorem 3.1.The disease-free equilibrium P 0 of system is globally asymptotically stable for R 0 < 1 and unstable for R 0 > 1.
Proof. 1 The Jacobian matrices of system 2.2 at the DFE are where Obviously, −D d ij is a 4 × 4 matrix with d ij ≤ 0, for i / j, i, j 1, . . ., 4 and Define the positive vector subsequently: Then begin to show that all the eigenvalues of −D are nonzero:

3.12
Simplify the above expression through substituting formula 3.8 into 3.12 : Here In conclusion, if R 0 < 1, −D is an irreducible matrix with d ii > 0 and d ij ≤ 0 i / j , there exists a positive vector x such that −D • x ≥ 0. 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 P 0 , it can be seen that the eigenvalues of J P 0 consist of −μ and all eigenvalues of D. Hence, all eigenvalues of J P 0 have negative real part for R 0 < 1; thus, disease-free equilibrium P 0 is locally asymptotically stable.
2 Let U 1 m 1 I m 2 J m 3 T m 4 R, where m i i 1 • • • 4 are positive constants as follows:

3.14
When R 0 < 1, the time derivative of U 1 is

3.16
Solve I, J, T, R → 0, 0, 0, 0 .When R 0 < 1, U 1 ≤ 0, and equalities hold if and only if S Λ/μ, I J T R 0, that is U 1 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 P 0 by the LaSalle's invariance principle.Thus the solutions of system 2.2 are limits to the endemic equilibrium P 0 when R 0 < 1. Combine locally asymptotically stable of P 0 with convergence properties of the P 0 , we conclude that P 0 of system is globally asymptotically stable for R 0 < 1.
thus D has eigenvalue with positive real part, otherwise, det D > 0; this is contradiction.Hence, P 0 is unstable for R 0 > 1.

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 R 0 > 1, here
Proof.Equating each equation in system 2.2 to zero, the equilibrium equations as follows are useful: where Q i is defined in 3.6 .Setting x S, I, J, T, R ∈ Ω ⊂ R 5 , construct a Lyapunov function where x * P * S * , I * , J * , T * , R * and A i > 0 is constant.There, U 2 x ≥ 0 for x ∈ IntΩ, and U 2 x 0 ⇔ x x * .Computing the time derivative of U 2 , we have

4.4
Using 2.2 and 4.10 , we obtain

4.5
Similarly, we obtain

4.6
Substituting formula 4.5 and 4.6 into 4.4 and arranging the equation we have where

4.8
In 4.7 , Q 2 consists of all constant terms, Q 3 contains all linear terms of I, J, T , R, and Q 4 contains all negative nonlinear.In order to determine the coefficient A i of U 2 , let Q 3 ≡ 0 in Ω ; then the coefficients of state variables I, J, T , R are equal to zero, that is: 4.9 Solving 4.9 , and using the expression of R 0 and S * , we have 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
Using the arithmetic mean geometric to get that 2 − x − 1/x is less than or equal to zero, substituting A i , and simplifying the other expressions in 4. 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 R 0 > 1.

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 R 0 .If control R 0 < 1, the disease can be eliminated from population.R052 RCT study indicated that treatment can prevent in HIV transmission 18 ; this sounds that increasing proportion of treated population is helpful to control HIV epidemic overall.In our study, ART is clearly affecting R 0 in the HIV procession.However, we cannot yet give this positive result based on the formula of R 0 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 k 1 , k 2 and treatment exiting γ are the feasible measures to reduce R 0 .Improving treatment standard and patients' compliance are the feasible and effective measures to reduce k 1 , k 2 .Certainly, new effective antiretroviral drugs might be the real determinants.R 0 is also linked with drug resistance by the transmission rate β 4 of drug resistance individual and removing rate ρ 3 from the population R. When the other parameters keep constant, R 0 is positive with β 4 and negative with ρ 3 .The value of R 0 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 k 1 , k 2 can be prevented by improving treatment quality, and the transmission coefficient parameter β 4 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.
11after 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 U 2 ≤ 0, 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.