1. Three-Component Model
of HIV with
Sensitivity Analysis
December 10th, 2014
by
Dustin Burchett and Alyssa Mandarino
Colorado School of Mines
Abstract
HIV spread in the body can be modeled using the three-component model. The
three variables involved are uninfected T-cells, infected T-cells, and virus cells.
Six parameters are in these equations. They are λ which is the growth rate
of T-cells, µ is the T-cell death rate, k is the infection rate, δ is the infected
cell death rate, p is the virus production rate, and c is the viral clearance rate.
Sensitivity analysis was performed on the model to see which parameter of the
six, when changed, changes the virus count the most. The results of this project
show that the infection rate affects the virus count the greatest. Some current
medical treatment of HIV involve decreasing the infection.

3. 1
Human immunodeficiency virus (HIV) affects a hosts immune system. To
fully understand the model some biological background will be discussed. The
immune system distinguishes between cells and foreign invaders in an organism.
It is prompted to respond by foreign substances called antigens. Once the
immune system finds antigens in the blood they are labeled by a macrophage;
which is found in white blood cells to identify infection sites in the body. At
this point helper T-cells (a lymphocyte called Thymus cells) sends out a signal
to get the B-cell (another lymphocyte) to come and produce antibodies for the
specific antigen. Lymphocytes are white blood cells and identified as either B-
cells or T-cells. The antibodies signal to other immune system cells to come
and destroy it as well as the antigen. Eventually, the immune system rests [3].
HIV is spread through sexual contact or by contact of infected blood. HIV
cells are considered foreign and are attacked by the hosts immune system ini-
tially; however, some virus cells manage to infect T-cells. This is done by the
virus changing then integrating its DNA into the DNA of the T-cell. It enters
the cell and inputs its DNA into the nucleus. Once that is accomplished the
viral DNA then forces the T-cell to replicate the viral DNA. The viral DNA will
manufacture into more virus cells and eventually the T-cell bursts in a process
called cytolysis. When the cell bursts it spreads more virus cells into the blood
stream to infect other T-cells. What can be seen upon infection is that T-cell
count will drop significantly due to the fact that they are becoming infected
cells. As more virus cells enter the blood stream T-cells will continue to drop
until they level off at a significantly lower value than before infection.
A current model called the three-component model represents the spread of
HIV from cell to cell inside the host. The variables in this system of equations
are T, I, and V, respectively representing uninfected T-cells count, infected T-
cells count, and virus cells count. The system is:
dT
dt
= λ − µT − kTV
dI
dt
= kTV − δI
dV
dt
= pI − cV
The parameters involved are λ which is the T-cell growth rate, µ is the T-cell
death rate, k is the infection rate or the contact rate between virus cells and
T-cells, δ is the infected T-cell death rate, p is the virus production rate, and c is
the viral clearance rate or the burst rate of the infected T-cell during cytolysis.
The initial conditions used were T0 = 1000, I0 = 0, and, V0 = 50 [2].
The two equilibrium points calculated are (T1, I1, V1) = λ
µ , 0, 0 and (T2, I2, V2) =
cδ
pk , λ
δ − µc
kp , pλ
cδ − µ
k [2]. The first point, (T1, I1, V1), is known as viral extinc-
tion because the infected and virus cell counts go to zero. The second point,
(T2, I2, V2), is known as viral persistence because the virus is still present within
the host as well as infected cells. Despite the parameters being independent of

4. 2
each other, it was calculated that the parameters must follow the relationship
c < kpλ
δµ to maintain positive equilibrium points. If this condition is not met the
system levels off to the viral extinction point. This model accommodates this
scenario.
The goal of this project was to determine which of the parameters is the most
sensitive, in turn this could help with treatments to target a certain aspect of
the infection process. The effect on the virus count was observed. To test which
parameter effects the system the greatest sensitivity analysis was conducted.
To quantitatively show how parameters affect the system the time it took for
the virus count to reach one of two equilibria was recorded for a multitude of
parameter values. An = 0.1 was picked so when the virus count was within
of its equilibrium point, which was determined by the parameter values, the
time value was recorded and called t ,i; with i representing an index.
To make sure that most parameter combinations were observed, a random set
of parameters were chosen. Four of the five parameters were then held constant
and t ,i was recorded for each individual value, over the range of the remaining
parameter not being held constant (look at Table1 to see all parameter ranges).
Only five parameters are being manipulated because c always equals 3. This
process was done 1000 times checking the range of the same parameter each
time. A histogram of t ,i was plotted.
Now to test the sensitivity of all the parameters, the ranges were normalized
to [-1, 1]. This allows the parameters to be studied without units. This was a
transformation represented by zi = τ(xi) with xi being the original range value
randomly selected, zi is the normalized value, and τ is the transformation. Least
squares was then used to fit the linear model t ,i ≈ ˆu0 + ˆu1z1 + · · · + ˆu5z5. The
ˆu values are weights to the linear model and those were plotted against the
parameters. This gives sensitivity of each parameter to the system. The weight
lie on the [-1, 1] range therefore a parameter with weight close to zero does not
affect the system greatly; however, a weigh near -1 or 1 has a significant effect
on the system. The sufficient summary plot plots t ,i versus ˆui. This shows us
the relationship between the normalized random parameters and t ,i. In other
words, how changes in the parameters effect t ,i.
In terms of this model, it was shown that k was the only parameter that
affects the system significantly. This can be observed. Looking at Graph4 of
the weights, ˆu, versus the parameters it shows that ˆu = −1 at k. µ, k, δ, and
p do not change the system. λ shows some sensitivity because the ˆu at λ does
not equal exactly 0. It is close enough to 0 that the effect cause by changes in λ
can be ignored and the effects of k can be focused on. Essentially, the sufficient
summary plot, Graph5, is t versus k. When comparing the change λ has on the
system in Graph2 and the change k has on the system in Graph1 it is clear that
k changes the system more than λ. This means that if effective HIV treatments
are to be implemented they should limit the contact between virus cells and
uninfected T-cells.
For current treatments today, it is extremely common to take more than
one medication which stops HIV production at different stages of development.
Each medication is put into a class describing at what point the medication stops

5. 3
the virus replication. Nucleoside/Nucleotide Reverse Transcriptase Inhibitors
(NRTIs) is a class that involves blocking the use of the enzyme necessary to make
copies of the viral DNA. The Non-Nucleoside Reverse Transcriptase Inhibitors
(NNRTIs) is very similar to NRTIs but it just prevents the enzyme from working
properly. Protease Inhibitors (PIs) prevents the viral DNA that has already
been replicated from being split into smaller pieces that then become virus
cells[3]. It can be noticed, that all of these classes described involve the HIV
already infecting the T-cell. There is one class called Entry/Fusion Inhibitors
that prevent the HIV cells from attaching to healthy T-cells receptors. These
medications can target both virus cells as well as T-cells[1]. The sensitivity
analysis on this model supports the idea of expanding treatment in Entry/Fusion
Inhibitors. Unfortunately, only two treatment drugs are in this class and two
more are being tested for use[1]. Of the 30 drugs in the market this is not
significant.
Using the previously derived three-component model, this project argues
that the most sensitive parameter in this system of equation representing HIV
within the body is the infection rate of virus cells on healthy T-cells. Since
this parameter is mathematically the most sensitive this means that if scientists
wish to decrease the virus count they should look in the direction of medication
that changes the contact rate of T-cells and HIV cells.

6. Growth Media
Parameters Biological Process Minimum Mean Value Maximum Units
λ T-cell growth rate 0.043 0.1089 0.2 µL−1
day−1
µ T-cell death rate 0.0043 0.01089 0.02 day−1
k Infection rate 0.00019 0.001179 0.0048 µL−1
day−1
δ Infected T-cell death rate 0.13 0.3660 0.8 day−1
p Virus production rate 98 1427 7100 day−1
c Viral clearance rate 3 3 3 day−1
Table 1: range of parameters

7. Figure 1: Plot of T, I , and V with k being the variable that changes
Figure 2: Plot of T, I , and V with λ being the variable that changes

8. Figure 3: Histogram of λ
Figure 4: Showing that k is the most sensitive parameter

9. Figure 5: Sufficient summary plot

10. Bibliography
[1] ”Entry Inhibitors (including Fusion Inhibitors).” Entry Inhibitors (in-
cluding Fusion Inhibitors). N.p., 16 Sept. 2011. Web. 09 Dec.
2014.<http://www.aidsmeds.com/archive/EIs_1627.shtml>
[2] Jones, Eric, Peter Roemer, Mrinal Raghupathi, and Stephen
Pankavich. Analysis and the Simulation of the Three-Component
Model of HIV Dynamics (n.d.): 90-105. Web. 12 Nov.
2014.<https://www.siam.org/students/siuro/vol7/S01269.pdf>
[3] ”Stages of HIV Infection.” Stages of HIV Infection. N.p., n.d. Web. 19 Oct.
2014.<http://www.aids.gov/hiv-aids-basics/just-diagnosed-with-
iv-aids/hiv-in-your-body/stages-of-hiv/>
Thanks to Stephen Pankavich for help with the three-component model as
well as Paul Constantine for help with the sensitivity analysis.