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MBW:Estimation of HIV/AIDS parameters

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Executive Summary

This paper determines HIV/AIDS disease parameters for a basic HIV model from measurements of healthy CD4+T concentrations in plasma and the viral load in plasma. First, the minimum number of necessary measurements is obtained. Next, the stages of HIV progression when parameter estimation is impossible are determined. After that, using system control techniques, an estimation algorithm of HIV parameters is created. The use of adaptive observers helps filter out the noise in measurement data and predict all six parameters reasonably well.

Project Summary

This project covers aspects of mathematical biology and biological systems, specifically within Immunology. This is an HIV/AIDS specific mathematical model. The model uses concentration of certain T cells as well as the concentration of the virus. The healthy CD4+ T cells are created at a constant rate, die depending on their concentration, and are infected at a rate proportional to both their concentration and virus concentration. The model used to effectively capture that information is:

{\dot  {x}}_{1}=s-dx_{1}-\beta x_{1}x_{3}

{\dot  {x}}_{2}=\beta x_{1}x_{3}-\mu _{1}x_{2}

{\dot  {x}}_{3}=kx_{2}-\mu _{2}x_{3}

The infected T cells reproduce and die at a concentration-proportional rate. The virus is produced by the infected T cells and also dies proportionally to the concentration.

Background Information

HIV, human immunodeficiency virus, is a virus which causes the immune system to fail, making the host vulnerable to opportunistic infections. It infects approximately 0.6% of the world's population. Since its discovery in 1981, it has taken approximately 25 million lives.

Infection Stages

The infection happens in 4 stages: incubation, acute infection, latency, and AIDS. The initial incubation period lasts two-four weeks and generally has no symptoms. During the acute infection stage, however, the patients show symptoms such as fever, lymphadenopathy, rashes, pharyngitis, and sores. The symptoms are difficult to link specifically to HIV because they are very scattered. The acute stage lasts approximately 4 weeks. During this phase, the virus replicates itself quickly. The latency stage occurs next; it is a stage when there are few to no symptoms showing. It can last anywhere between a couple weeks and 20 years. During this stage, the virus count stays relatively low. This can give the impression that the disease is under control. However, following the latency stage comes AIDS, when the T cell count drops below 200 cells/microliter. At that point, the immune system can't fight off opportunistic infections.

Virus Reproduction

Replication of the Virus inside a T Cell

The virus enters into the host cell by fusing with the cell membrane and releasing the contents of the virus capsule into the interior of the cell. Following entering the cell, the virus takes over the protein producing machinery of the cell, and uses the cell's capabilities to replicate. See the image for some of the details of replication.

The virus replicates itself very fast, 10^{9} to 10^{{10}} virions per day. The mutation rate is approximately 3*10^{{-5}} per nucleotide base, due to the errors in the reverse transcriptase process (when the RNA of the virus is used to create cDNA). The error-prone process is responsible for the high number of virus mutations created in a single host during one day. Since a single T cell may also be infected with several mutants of HIV, it becomes difficult to determine a proper course of treatment very quickly. The drug resistance caused by the frequent mutations is one of the reasons why HIV is such a difficult disease to approach.

For more information on Virus Replication please see: [1]

Mathematical Models Background

Mathematical models of HIV dynamics describe the interactions between the immune system and the virus at different stage. These models give insights to the dynamics of the disease. For more background on the dynamics of this disease please see MBW: HIV Dynamics

The basic model is what is covered in the paper. It involves 3 variables: concentration of healthy CD4+ T cells, concentration of infected CD4+ T cells, and virus concentration. The healthy CD4+ T cells are created at a constant rate, die depending on their concentration, and are infected at a rate proportional to both their concentration and virus concentration. The infected CD4+ T cells produce new virions, and die at a concentration-proportional rate. The virus is produced by the infected CD4+ T cells and dies proportionally to the concentration.

Extensions of the model can be investigated, although they are not analyzed in the paper. Some of the possible investigations include adding in a latent compartment for the virus (to potentially explain the exit from the latency stage), adding in mutations of the virus (both harmful and harmless), adding in the effect of the CD8+ T cells which are responsible for controlling virus levels, adding in effects of drugs, etc.

This paper focuses on the observability of the parameters of the model. Because it is only practical to measure the viral load and the healthy CD+ T cell concentration, the error in determining the parameters can be potentially large. The system that is used for this paper is both observable and identifiable.

Mathematical Model

The basic model of HIV/AIDS is as follows:

{\dot  {x}}_{1}=s-dx_{1}-\beta x_{1}x_{3}

{\dot  {x}}_{2}=\beta x_{1}x_{3}-\mu _{1}x_{2}

{\dot  {x}}_{3}=kx_{2}-\mu _{2}x_{3}

In the model, x_{1} is the concentration of healthy CD4+ T cells, x_{2} is the concentration of infected CD4+ T cells, and x_{3} is the concentration of virus in plasma. s is the rate at which new CD4+ T cells are created, d is the death rate of T cells, \beta is the infection rate, \mu _{1} is the death rate of infected T cells, \mu _{2} is the death rate of virions, and k is the rate of creation of virions.

It has been previously shown that the system is observable and identifiable. Since only the concentration of uninfected T cells and concentration of virus can be measured, the outputs are redefined as y_{1}=x_{1},y_{2}=x_{3}.

Determining Conditions for Parameter Identification

The system can be redefined as:

{\dot  {y}}_{1}=\theta _{1}+\theta _{2}y_{1}+\theta _{3}y_{1}y_{2}

{\ddot  {y}}_{2}=\theta _{4}{\dot  {y}}_{2}+\theta _{5}y_{2}+\theta _{6}y_{1}y_{2}

Where \theta values define a one-to-one map. The conditions are that \beta \neq 0 and \mu _{1}\neq \mu _{2}. From clinical data, \beta \neq 0 and \mu _{1}>\mu _{2}. Therefore,

\left[{\begin{array}{c}s\\d\\\beta \\\mu _{1}\\\mu _{2}\\k\end{array}}\right]=\left[{\begin{array}{c}\theta _{1}\\-\theta _{2}\\-\theta _{3}\\{\frac  {-\theta _{4}-{\sqrt  {\theta _{4}^{2}+4\theta _{5}}}}{2}}\\{\frac  {-\theta _{4}+{\sqrt  {\theta _{4}^{2}+4\theta _{5}}}}{2}}\\-{\frac  {\theta _{6}}{\theta _{3}}}\end{array}}\right]

Since we need 6 equations to calculate all the variables, we will need to differentiate y_{1} to the order of 3 and y_{2} to the order of 4. Therefore, at least 4 measurements of CD4+T cell count are needed and 5 measurements of the virus concentration are needed.

With the measurements labeled so that the superscript indicates the number of the measurement, and the parameter d indicating the time difference, the following system can be set up to calculate the first three theta values based on the measured concentrations.

\left[{\begin{array}{ccc}1&y_{1}^{0}&y_{1}^{0}y_{2}^{0}\\1&y_{1}^{1}&y_{1}^{1}y_{2}^{1}\\1&y_{1}^{2}&y_{1}^{2}y_{2}^{2}\end{array}}\right]\left[{\begin{array}{c}\theta _{1}\\\theta _{2}\\\theta _{3}\end{array}}\right]=\left[{\begin{array}{c}{\frac  {y_{1}^{1}-y_{1}^{0}}{d_{1}}}\\{\frac  {y_{1}^{2}-y_{1}^{1}}{d_{2}}}\\{\frac  {y_{1}^{3}-y_{1}^{2}}{d_{3}}}\end{array}}\right]

The parameters \theta _{4},\theta _{5},\theta _{6} can be similarly derived. The method will fail to predict the parameters when the viral load remains constant, so during the latency stage and during short periods after treatment, the data will not provide for useful results. The main difficulty with this method though is that it fails with noisy data. Clinical data is usually noisy, with variance of 20 for the T cell concentration and log variance of 0.2 for virion concentration. The results section shows how this system performs with parameter prediction.

Estimation of Parameters Using Adaptive Observers

To see a general process on how to estimate parameters refer to: [2]

In order to analyze noisy data, the system can be transformed into observer form by setting z to be the following:

z_{1}=x_{1}\ \ z_{2}=kx_{2}+mu_{1}x_{3}\ \ z_{3}=x_{3}

The observer form is as shown:

{\dot  {z}}_{1}=\theta _{1}+\theta _{2}y_{1}+\theta _{3}y_{1}y_{2}

{\dot  {z}}_{2}=\theta _{6}y_{1}y_{2}+\theta _{5}y_{2}

{\dot  {z}}_{3}=z_{2}+\theta _{4}y_{2}



The filtered transformation can be defined as well.

\eta _{1}=z_{1}

\eta _{2}=z_{2}-\theta _{6}\xi _{1}-\theta _{5}\xi _{2}-\theta _{4}\xi _{3}

\eta _{3}=z_{3}

{\dot  {\xi }}_{1}=-b\xi _{1}+y_{1}y_{2}

{\dot  {\xi }}_{2}=-b\xi _{2}+y_{2}

xi_{3}=-b\xi _{2}

In those expressions, b has to be greater than zero, so that the system can be transformed into the adaptive observer form and then an adaptive observer can be designed.

{\dot  {\eta }}_{1}=k_{1}\eta _{1}+\theta _{1}+\theta _{2}y_{1}+\theta _{3}y_{1}y_{2}-k_{1}y_{1}

{\dot  {\eta }}_{2}=k_{2}\eta _{3}+b[\theta _{6}\xi _{1}+\theta _{5}\xi _{2}+\theta _{4}(\xi _{3}+y_{2})]-k_{2}y_{2}

{\dot  {\eta }}_{3}=k_{3}\eta _{3}+\eta _{2}+[\theta _{6}\xi _{1}+\theta _{5}\xi _{2}+\theta _{4}(\xi _{3}-y_{2})]-k_{3}y_{2}

\left[{\begin{array}{c}\theta _{1}\\\theta _{2}\\\theta _{3}\end{array}}\right]=\Gamma _{1}\left[{\begin{array}{c}1\\y_{1}\\y_{1}y_{2}\end{array}}\right](y_{1}-\eta _{1})

\left[{\begin{array}{c}\theta _{4}\\\theta _{5}\\\theta _{6}\end{array}}\right]=\Gamma _{2}\left[{\begin{array}{c}y_{2}+\xi _{3}\\\xi _{2}\\\xi _{1}\end{array}}\right](y_{2}-\eta _{3})

The \Gamma _{1},\Gamma _{2} are symmetric positive definite matrices, </math>k_1</math> is a negative constant, k_{2}=-b,k_{3}=-b-\lambda .


The simulation was set up in Matlab Simbiology package, in order to obtain data for parameter estimation. The parameters were as follows: s=7,5=.007,beta=.00000042163,\mu _{1}=.0999,\mu _{2}=0.2,k=90.67. The time is on the x axis and is in days.

Simbiology Setup

The HIV progression can be seen for all three variables both from the paper and the Simbiology run. They are very similar.

Simulation done in the paper Results from Simbiology Simulation


The parameter estimation done on s, d and \beta shows that without the adaptive observer technique, the results tend to be very noisy.




When the adaptive observer technique is used, the parameters s, d and \beta look much smoother.


The \mu estimations are still not very smooth because the virus and infected cells have a much higher turnover (a factor of 10 compared to d for the infected cell death and a factor of over 100 for the virus death).


The parameters can be determined using the adaptive observer method. They are most easily determined during the early infection stage when there is sufficient change in viral load. Therefore the main question to be asked is how likely is it that the disease can be monitored at the early stage? (See MBW:When to Initiate HIV Therapy for more information on HIV at an early stage) In any case, by disturbing the steady state of the disease with drugs, the parameters can be calculated as well.

By monitoring the parameters for the disease in one patient, the drug resistance development could be potentially monitored. Potentially, a study of how the parameters \beta and k change with resistance could be done.


The determination of parameters for the HIV/AIDS model was studied. It was determined that at least 4 measurements of CD4+ T cell load and 5 measurements of viral load need to be obtained in order to estimate model parameters. The model does not allow for estimation of HIV/AIDS parameters during the latency stage. Using adaptive observers, the parameters can be calculated while the data is obtained, and all 6 parameters needed converge at close to the correct value when the model is simulated. For a more detailed description of what each these parameters mean see MBW:HIV_Dynamics.

Recent Papers

Over the years there have been about 41 different papers citing the original paper, "Estimation of HIV/AIDS Parameters". One specific paper touches the topic of the estimation of parameters and extends upon the research that was already done. In this paper the concept of algebraic identifiability of a three dimensional HIV/AIDS dynamic model. Identifiability is defined as a condition that a model must satisfy in order to be statistically inferred. The researchers proposed a new method of finding the parameters of the model by using the multiple time point method. Previously, the high-order derivative method was used in order to find the missing parameters. The paper uses data and the new method to compute the new parameters and presents the new method.


Xiaohua Xia. "Estimation of HIV/AIDS Parameters." Automatica 39 (2003) 1983-1988.

Xia, X. Moog, C.H. (2003) "Identifiability of nonlinear systems with application to HIV/AIDS models." IEEE Transactions on Automatic Control, 48, 330-336.

Wu, H. Zhu, H. Miao, H. Perelson, A. (2008) "Parameter Identifiability and Estimation of HIV/AIDS Dynamic Models." Bulletin of Mathematical Biology, 70, 785-799. [3]

HIV Wikipedia Page