Research Article | Open Access
Stability Analysis of an HIV/AIDS Dynamics Model with Drug Resistance
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 () by the next generation matrix and then analyzed stability of two equilibriums by constructing Lyapunov function. When , 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.
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 [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 ; 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  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 , , , , and 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 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 denote the disease-related death rate of the AIDS. And use 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 is the recruitment rate of the susceptible population, is the probability of transmission by an infection in the first stage, is the probability of transmission by an infection in the second phase, is the probability of transmission by a patient being treated, is the probability of transmission by a drug resistance individual, is the transfer rate constant from the asymptomatic phase to the symptomatic phase , is the proportion of treatment, is the exit treatment rate without developing drug resistance, is probability constant of infection by transmission of drug-resistant strains, and is the rate of acquiring drug resistance during treatment; denote transfer rate constant by an infection from phase , , to the AIDS cases , respectively.
The model is established in practice; thus we assume all parameters are nonnegative.
Since the of system (2.1) does not appear in the equations, in the following analysis, we only consider the system as follows:
Theorem 2.1. Let the initial data be , , , and ; 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 , and for system (2.2) is positively invariant.
Proof. From the first equation of (2.2)
consider the following two categories.(1) When . Equation (2.3) becomes , due to ; we have , that is, when , is an increasing function about . Therefore, we can conclude that when is the neighborhood of and ,.(2)When . Equation (2.3) can be written as
Similarly for the other equations of system (2.2) we can easily show that , , , are increasing functions about when and , and , and and , respectively; then when , . Otherwise, when and , we have the following results corresponding to the respective hypothesis: Thus, Thus, Thus, Thus, Then, we have that , , , and are all strictly positive for . Thus we can conclude that all solutions of system (2.2) remain positive for all .
Next, add all the the equations of system (2.2); we have
In a similar fashion we have from the first equation of (2.2); then .
Thus, the feasible solution of system remains in the region , and as the feasible region for system is positively invariant. In the following, the dynamics of system (2.2) will be considered in .
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), . Following the paper , we obtain the basic reproduction number by using the next generation operator approach.
Let ; thus we have where Then the derivatives of and at the DFE are partitioned as and are the matrices as follows: where
Hence the reproduction number, denoted by , is the spectral radius of the next generation matrix : Here
3.2. DFE and Stability
Theorem 3.1. The disease-free equilibrium of system is globally asymptotically stable for and unstable for .
Proof. (1) The Jacobian matrices of system (2.2) at the DFE are
Obviously, is a matrix with , for and for . If , we have from the expression of (3.8); thus .
Define the positive vector subsequently: If , .
Then begin to show that all the eigenvalues of are nonzero: Simplify the above expression through substituting formula (3.8) into (3.12): Here ; thus, ; namely, has non zero eigenvalue for .
In conclusion, if is an irreducible matrix with and , there exists a positive vector such that . Hence, the real part of each nonzero eigenvalue of is positive according to the M-matrix theory; that is, each eigenvalue of has negative real part.
Through the structure of the Jacobian matrix , it can be seen that the eigenvalues of consist of and all eigenvalues of . Hence, all eigenvalues of have negative real part for ; thus, disease-free equilibrium is locally asymptotically stable.
(2) Let , where are positive constants as follows: When , the time derivative of is Let ; due to , is the finite number, then Solve , have , that is, , thus when , . When , , and equalities hold if and only if , that is if and only if . We can conclude that the solutions of system (2.2) are all in and the only invariant set in is by the LaSalle’s invariance principle. Thus the solutions of system (2.2) are limits to the endemic equilibrium when . Combine locally asymptotically stable of with convergence properties of the , we conclude that of system is globally asymptotically stable for .
(3) If , ; thus has eigenvalue with positive real part, otherwise, ; this is contradiction. Hence, is unstable for .
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 for , here
Theorem 4.1. Endemic equilibrium of system is globally asymptotically stable for .
Proof. Equating each equation in system (2.2) to zero, the equilibrium equations as follows are useful:
where is defined in (3.6).
Setting , construct a Lyapunov function where and is constant. There, for , and .
Computing the time derivative of , we have Using (2.2) and (4.10), we obtain Similarly, we obtain Substituting formula (4.5) and (4.6) into (4.4) and arranging the equation we have where In (4.7), consists of all constant terms, contains all linear terms of , , , , and contains all negative nonlinear.
In order to determine the coefficient of , let (in ); then the coefficients of state variables , , , are equal to zero, that is: Solving (4.9), and using the expression of and , we have then,.
Let, , , , ; substituting these expressions into (4.9), and then substituting the changing expression into (4.7) Using the arithmetic mean geometric to get that is less than or equal to zero, substituting , and simplifying the other expressions in (4.11) after tedious algebraic manipulations, we can get the other expressions such as , , and is less than or equal to zero; this indicates that , and equalities hold if and only if . Furthermore, , ; then substituting , , , , into the first equation of system (2.2), and in contrast to the first equality of (4.2), we have .
In conclusion, the limit sets of solutions in are all in , and the only invariant set in is by the LaSalle’s invariance principle. Thus the solutions of system (2.2) in are limits to the endemic equilibrium , and is globally asymptotically stable for .
For public health view, to bring HIV/AIDS into control, the prerequisite is reducing the threshold value of basic reproductive number . If control , 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 in the HIV procession. However, we cannot yet give this positive result based on the formula of and 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 , and treatment exiting are the feasible measures to reduce . Improving treatment standard and patients’ compliance are the feasible and effective measures to reduce , . Certainly, new effective antiretroviral drugs might be the real determinants.
is also linked with drug resistance by the transmission rate of drug resistance individual and removing rate from the population . When the other parameters keep constant, is positive with and negative with. The value of 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 , can be prevented by improving treatment quality, and the transmission coefficient parameter 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.
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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