T Cell Counts and Viral Load Interactions, Epid 602, Kevin Jefferson
While reverse transcriptase inhibitors represent significant gains in understanding HIV’s affect
on the body, the exact processes by which viral load increases as CD4+ T cells decrease over the
course of an infection remain unclear. Three stages of HIV infection have been described: (1)
acute infection in which viral load peaks shortly after infection, (2) a period in which T cells
decrease while viral load increases, and (3) AIDS diagnosis, characterized by few T cells and a
high viral load (Pantaleo et al., 1993). Individuals move through these stages at different rates
but reasons for this are partially understood. Simple target-cell-limited models are within host
models of HIV. Earlier versions describe viral production as relying solely on CD4+ T cells
(Stafford et al., 2000; Gumel et al., 2001). Presumably, a decreased number of T cells would
decrease viral reproduction; thus, increased viral load despite a decrease in CD4+ t cells during
later stages of infection represents a paradox that previous models of HIV infection failed to
predict (Duffin and Tullis, 2002).
To increase the usefulness of simple target-cell-limited models beyond acute infection, Duffin
and Tullis explored and two hypotheses on why viral load increases and CD4+ T cells decrease
after acute infection (ibid). The hypotheses are: (1) the rate of clearing HIV viruses from the
body decreases over the course of an infection, and (2) HIV uses other immune cells to
reproduce as CD4+ T cells decrease. The first hypothesis would modify a mass action
compartment in the within host model to essentially decrease the death rate of HIV virions.
Although viral reproduction decreases as CD4+ T cells become depleted, this decrease in virion
death rate allows a high level of virus to be maintained within host. The second hypothesis
modifies the mass action equation by adding another niche within the host for virions; although
CD4+ T cells deplete over the course of disease, virions may use other resources (immune cells)
to replicate. The availability of this resource allows for a high viral load despite the depletion of
T Cell Counts and Viral Load Interactions, Epid 602, Kevin Jefferson
CD+ 4 T cells for viral reproduction. Both hypotheses offer insight into the paradox of high viral
loads despite low CD+ 4 T cells later in the course of infection. Studies have shown
macrophange immune cells may support high viral loads when an infection has lead to AIDS,
and that viral production in macrophanges can increase during opportunistic infections;
moreover, few researchers have examined the viral clearance rate of HIV (Igarashi et al., 2001;
Kalter et al., 1991; Moriuchi et al., 1998; Duffin and Tullis, 2002).
The authors explored their hypotheses by modifying a simple target-cell-limited model of early
HIV infection to predict subsequent phases of infection (Stafford et al., 2000). They began by
recreating a simple target-cell-limited model. The equations they used to modify the model by
introducing a decrease in viral clearance rate are as follows (Duffin and Tullis, 2002):
𝑑𝑇(𝑡)
𝑑𝑡
𝑑𝑇𝑖(𝑡)
𝑑𝑡
𝑑𝑉(𝑡)
𝑑𝑡
= 𝜆 𝑇 -𝑑𝑇 𝑇 − 𝑘𝑇 𝑇𝑉
T(0)= 𝑇0 activated T cells
= 𝑘𝑇 𝑇𝑉 -𝛿𝑇 𝑇𝑖
Ti(0)= 𝑇𝑖0 infected T cells
= 𝜋𝑇 𝑇𝑖 −𝑐𝑉
V(0)= 𝑉0 virus concentration
0.01
All values were computed per ul of blood. Normal T cells die at a rate of dT , day . The rate at
which T cells become infected, k T is
0.00027
0.39
. The death rate of infected T cells is 𝛿𝑇 , day . 𝜋𝑇 is
virusday
the rate of virus production, 850 virons per T cell daily. The rate of T cell production is given by
𝜆 𝑇 , and the viral clearance rate is given by c. Duffin and Tullis
modify 𝜆 𝑇 by introducing a decay term so it equals 𝑑 𝑇 𝑇0 (1 − 𝑓)/𝑑𝑎𝑦, rather than 𝑑 𝑇 𝑇0 . The
same rate of decay affects clearance rate in their model modification: c equals c(1-f) rather than
3/day. The authors defined the decay term f, 0.00028t, as the average daily decrease in T cell
T Cell Counts and Viral Load Interactions, Epid 602, Kevin Jefferson
production by time t. They derived f using data from Fauci et al (1996) showing an approximate
10% yearly decrease in T cells after a 40% decrease within the first year of infection.
Subsequent literature may refine estimates for f, however the derived estimate is a good
approximation from patient data.
Recreating the simple target-cell-limited model was difficult in that Duffin and Tullis illexplained their mathematical computation of 𝑑 𝑇 𝑇0 . Nonetheless using 10 for 𝑑 𝑇 𝑇0 recreated the
simple target-cell-limited model (see figure 1). This paper runs the model for approximately one
year to show the dynamics of acute HIV infection and early chronic infection. Within this short
time frame including a decay term for T cell production did not tangibly affect the number of
uninfected T cells, infected T cells, or virions. The model in this paper includes the decay term
and created a simulink compartment to integrate decay over time t. Appendix 1 includes matlab
code and the simulink model built to recreate this model.
Figure 1:
400 days of HIV with T cell decay term
11
10
9
8
Log counts/ul
7
Red: Virus
Green: Infected T cells
Blue: Uninfected T
cells
6
5
4
3
2
1
0
0
50
100
150
200
250
300
350
400
Time (days)
Recreating the viral clearance modification done by Duffin and Tullis was easily accomplished
by changing the clearance rate parameter in the first recreated model. As that model
T Cell Counts and Viral Load Interactions, Epid 602, Kevin Jefferson
incorporated the decay term already, recreating the clearance rate modification occurred by using
the decay compartment in the clearance rate modification. Appendix 2 includes matlab code
and the simulink model built to recreate this model. Figures 2 and 3 below show the results of
this modification on the number of uninfected T cells and HIV virions over 3,000 and 4,000 days
respectively. In figure 2 T cells begin to decline precipitously around 2700 days after infection.
Figure 3 shows an exponential rise in viral load, after it has remained relatively constant, around
3700 days post infection.
Figure 2:
Clearance rate reduction model CD4+ T cells
5
4.5
4
Log (counts/ul)
3.5
3
2.5
2
1.5
1
0.5
0
0
500
1000
1500
2000
2500
3000
Time (days)
Figure 3:
Clearance rate reductionmodel-- HIV Virions
11
10
9
8
Log (Counts/ul)
7
6
5
4
3
2
1
0
0
500
1000
1500
2000
Time (days)
2500
3000
3500
4000
T Cell Counts and Viral Load Interactions, Epid 602, Kevin Jefferson
This recreated model matches the clearance rate modification by Duffin and Tullis well. To
make their clearance rate model work Duffin and Tullis assumed that HIV primarily targets T
cells, and that viral clearance decreases at the same rate that T cell concentration does (Duffin
and Tullis, 2002). Literature supports the idea that HIV changes clearance rates for other
infections, and presumably could affect HIV viral clearance rates as well (Duffin and Tullis,
2002; Matsuo et al., 2001). However, the assumptions that HIV primarily targets T cells, and
that the decrease in viral clearance rate is the same as the rate of T cell decrease may be
problematic. If HIV uses other immune cells to reproduce as T cells decrease, this model may
underestimate viral load in late stages of infection, and could possibly overestimate T cell loss.
Because little is known about the biologic mechanisms of viral load clearance, using the same
rate based on T cell concentration may simply be inaccurate. Nonetheless, the model produced
by Duffin and Tullis matches patient data very well (Duffin and Tullis, 2002).
This paper does not recreate the second modification Duffin and Tullis created. While both
modifications together offer insight into the long term within-host dynamics between HIV
virions and T cells, additional exploration of differential HIV progression rates is sorely needed.
Understanding factors that influence HIV’s progression not only may improve HIV treatment, it
may also prove crucial for HIV prevention. Higher viral loads, as seen in later HIV stages, are
associated with more efficient disease transmission; moreover factors such as stress can
influence viral load independently of antiretroviral medication use (Antoni et al, 2006; Das, et al
2010). Thus understanding factors that influence HIV progression independently of
antiretroviral use may help bolster HIV prevention efforts.
Differences in the body’s ability to clear HIV virions offers one hypothesis with apparent
biologic plausibility as to why individuals progress through infection at different rates.
T Cell Counts and Viral Load Interactions, Epid 602, Kevin Jefferson
Allostatic stress load theory, which considers stress effects on the immune system, would
perhaps suggest stress could affect the body’s ability to clear HIV virions. Nonetheless, stress
load literature did not offer useable parameter estimates or explanations of how stress may affect
viral clearance. A more useful construct than viral clearance rate over the course of infection
proved to be viral set point. Viral set point refers to viral load level in the body after acute HIV
infection (Kelly et al, 2007). As shown in the Stafford et al simple target-cell-limited model, an
initial rise in viral load decreases T cells which then decreases viral load (Stafford et al, 2000).
The level to which viral load decreases following acute infection is the viral set point and may be
thought of as an equilibrium state (Goldstein, 2008). The body maintains this equilibrium over
the course of chronic HIV infection; this equilibrium is disturbed when HIV progresses to an
AIDS diagnosis and viral load rises (Fryer and McLean, 2011; Little et al, 1999). Individuals
with a higher viral set point cannot maintain themselves at equilibrium as well as individuals
with a lower viral set point. Thus viral set point influences HIV progression rate.
Cytotoxic T lymphocytes (CTLs) are immune cells which engulf and destroy infected T cells.
Literature suggests that the presence of many CTLs during acute infection may lower viral set
point (Fryer and McLean, 2011). Moreover, stress has been shown to impact CTL numbers
(Antoni et al, 2006; Segerstrom and Miller, 2004). These findings supported the creation of a
model modification in which CTLs decrease the number of infected T cells. The hypotheses of
this model were: (1) CTLs decrease the number of infected T cells in acute infection which
lowers viral load at set point, and (2) stress decreases the number of CTLs and may contribute to
a higher viral set point. The outcome of interest with these hypotheses was a noticeable
difference in viral set point between stress levels. To parameterize this model the following
equations were used:
T Cell Counts and Viral Load Interactions, Epid 602, Kevin Jefferson
𝑑𝑇(𝑡)
𝑑𝑡
= 𝜆 𝑇 -𝑑𝑇 𝑇 − 𝑘𝑇 𝑇𝑉
𝑑𝑇𝑖(𝑡)
𝑑𝑡
𝑑𝑉(𝑡)
𝑑𝑡
𝑑𝑋(𝑡)
𝑑𝑡
= 𝑘𝑇 𝑇𝑉 -𝑇𝑖 (𝛿𝑇 + 𝛿𝑐 𝑋)
= 𝜋𝑇 𝑇𝑖 −𝑐𝑉
𝑎𝑉𝑋
= 𝑠𝑋+𝑉 − 𝛿𝑋 𝑋
T(0)= 𝑇0 activated T cells
Ti(0)= 𝑇𝑖0 infected T cells
V(0)= 𝑉0 virus concentration
X(0)= 𝑋0 CTL concentration
X represents CTLs and 𝛿𝑐 is the rate at which CTLs destroy infected T cells. In this model
0.2/day represents 𝛿𝑐 . Infected T cells thus die off at a rate of 𝑇𝑖 (𝛿𝑇 + 𝛿𝑐 𝑋); that is they die off
at a rate influenced by HIV virions, 𝛿𝑇 , combined with the rate at which they are destroyed by
CTLs. Proliferation of X is a function of a, the default growth rate of CTLs, times the number of
virions and current CTLs, over s times the number of CTLs plus the number of virions. CTL
production occurs in response to the presence of virions, thus the growth of CTLs is a function of
CTL rate of production times viral number and the current number of CTLs. s represents stress,
it interacts with the number of CTLs to decrease the production of CTLs. The CTL number then
is a function of CTL proliferation minus the death rate of CTLs, 𝛿𝑥 , by the number of CTLs.
Consistent with the Duffin and Tullis models, all values were computed per ul of blood.
The equation for X was adapted from Fryer and McLean (2011) in which the number of unique
CTLs responses was studied. A cumulative density function allowed for the parameterization of
X above (Fryer and McLean, 2011). The death rate of CTLs, 𝛿𝑥 , was estimated from the same
study to be 0.02/day. The default growth rate of CTLs, a, was derived from this paper by
dividing the total daily body production of CTLs by the average number of uls in an adult male.
The average ul of adult males was used rather than the average ul of adult females as the
T Cell Counts and Viral Load Interactions, Epid 602, Kevin Jefferson
majority of people living with HIV in the United States are male. This value was 0.00092593.
Initial CTL concentration, 𝑋0, was also parameterized as 0.00092593. This value was used with
the assumption that CTL production when HIV is first introduced to the body is merely the
natural CTL growth rate. The model was run with a decay term in the production of T cells, but
without the Duffin and Tullis clearance rate modification.
The feasibility of adding CTLs was first explored by holding CTL number constant, arbitrarily
set at 5, and introducing 𝛿𝑐 to the infected T cell compartment. 𝛿𝑐 was inflated to .5. The
simulation was run over 350 days to cover acute infection and the beginning of chronic infection.
Appendix 3 contains the simulink model used for test. Figure 4 below shows that adding 𝛿𝑐 will
decrease viral load in comparison with the earlier model (see figure 1).
Figure 4:
HIV over 350 days with T cell production decay, and constant CTL
10
Red: Virus
Blue: Uninfected T cells
Green: Infected T cells
9
8
Log Counts/ul
7
6
5
4
3
2
1
0
0
50
100
150
200
250
300
350
Time (days)
Next the CTL number was allowed to vary as a function of stress and the model was run (see
appendix 4 for the simulink model). Figures 5-7 show results of the simulation with stress at
lowest at 500,000, increased to 1,000,000, and highest at 5,000,000. As can be seen lower stress
T Cell Counts and Viral Load Interactions, Epid 602, Kevin Jefferson
levels produced stronger CTL responses and stress level had an inverse relationship with viral set
point.
Figure 5:
HIV over 350 days with CTL as a function of stress, s=500,000
11
10
Red: Virus
Blue: Uninfected T cells
Yellow: CTLs
Green: Infected T cells
9
8
Log Count/ul
7
6
5
4
3
2
1
0
0
50
100
150
200
250
300
350
Time (days0
Figure 6:
HIV over 350 days with CTL as function of stress, s=1,000,000
11
10
Red: Virus
Blue: Uninfected T cells
Yellow: CTLs
Green: Infected T cells
9
8
Log Counts/ul
7
6
5
4
3
2
1
0
0
50
100
150
200
250
300
350
Time (days)
Figure 7:
HIV over 350 days with CTL as function of stress, s=5,000,00
11
10
Red: Virus
Blue: Uninfected T cells
Yellow: CTLs
Green: Infected T cells
9
8
Log Counts/ul
7
6
5
4
3
2
1
0
0
50
100
150
200
Time (days)
250
300
350
T Cell Counts and Viral Load Interactions, Epid 602, Kevin Jefferson
These model results suggest additional research ought to be done on the potential casual
relationship between stress level in acute infection and HIV progression. While sufficient
literature suggests the modification of CTL as a function of stress is plausible biologically,
undoubtedly such a relationship would be far more complex than it is portrayed in this model.
For instance, it is possible stress minimizes the effective killing rate (𝛿𝑐 ) of CTLs, or it may have
a more direct relationship to viral clearance rate as parameterized by c in the Duffin and Tullis
paper. It is also possible that a biologic or environmental construct other than stress is actually
influential in the model to shape CTL number. Additional research on stress and viral setpoint
should help clarify these questions and would be a very worthwhile endeavor.
T Cell Counts and Viral Load Interactions, Epid 602, Kevin Jefferson
References:
Antoni, M., Carrico, A., Duran, R., Spitzer, S., Pendo, F., et al (2007). Randomized Clinical
Trial of Cognitive Behavioral Stress Management on Human Immunodeficiency Virus Viral
Load in Gay Men Treated With Highly Active Antiretroviral Therapy. Psychosomatic Medicine,
68, 143–151. doi: 0033-3174/06/6801-0143.
Bouhdoud, L. (2000). T-Cell Receptor-Mediated Anergy of a Human Immunodeficiency Virus
(HIV) gp120-Specific CD4+ Cytotoxic T-Cell Clone, Induced by a Natural HIV Type 1 Variant
Peptide. Journal of Virology, 74(5), 2121-2130.
Charlebois, E. D., Das, M., Porco, T. C., & Havlir, D. V. (2011). The Effect of Expanded
Antiretroviral Treatment Strategies on the HIV Epidemic Among Men Who Have Sex With Men
in San Francisco. Clinical Infectious Diseases, 52(8), 1046-1049. doi:10.1093/cid/cir085
Das M, Chu PL, Santos G-M, Scheer S, Vittinghoff E, et al. (2010) Decreases in Community
Viral Load Are Accompanied by Reductions in New HIV Infections in San Francisco. PLoS
ONE 5(6): e11068. doi:10.1371/journal.pone.0011068
Duffin, R. P., & Tullis, R. H. (2002). Mathematical Models of the Complete Course of HIV
Infection and AIDS. Computational and Mathematical Methods in Medicine, 4(4), 215-221.
Fauci, A. S., Pantaleo, G., Stanley, S., & Weissman, D. (1996). Immunopathogenic mechanisms
of HIV infection. Annals of internal medicine, 124(7), 654-663.
Finzi, D., Blankson, J., Siliciano, J. D., Margolick, J. B., Chadwick, K., Pierson, T., Smith, K., et
al. (1999). Latent infection of CD4 + T cells provides a mechanism for lifelong persistence of
HIV-1, even in patients on effective combination therapy. Nature Medicine, 5(5), 512-517.
Fryer, H., McLean, A. (2011). Using Mathematical Models to Explore the Role
of Cytotoxic T Lymphocytes in HIV Infection. In C. Molina-Par´ıs and G. Lythe (Eds.),
Mathematical Models and Immune Cell Biology (pp. 363-382). doi: 10.1007/978-1-4419-7725-0
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Ganusov,V., Goonetilleke, N., Liu, M., Ferrari, G., Shaw, G., et al. (2011). Fitness costs and
diversity of CTL response determine the rate of CTL escape during the acute and chronic phases
of HIV infection. J. Virol. 2011 August 10; [e-pub ahead of print]. doi:10.1128/JVI.00655-11
Goldstein, D. (2008) Stress, Neurotransmitters, and Hormones. Annals of the New York
Academy of Science, 1148, 223-231. doi: 10.1196/annals1410.061.
T Cell Counts and Viral Load Interactions, Epid 602, Kevin Jefferson
Gumel, A. B., Loewen, T. D., Shivakumar, P. N., Sahai, B. M., Yu, P., & Garba, M. L. (2001).
Numerical modelling of the perturbation of HIV-1 during combination anti-retroviral therapy.
Computers in biology and medicine, 31(5), 287-301.
Igarashi, T., Brown, C.R., Endo, Y., Buckler-White, A., Plishka, R., Bischofberger, N., Hirsh, V.
and Martin, M.A. (2001) "Macrophange are the principal reservoir and sustain high virus loads
in rhesus macques after the depletion of CD4+ T cells by a highy pathogenic simian
immunodeficiency virus/HIV type 1 chimera (SHIV): implications for HIV-1 infections of
humans", Proceedings of the National Academy of Sciences of the United States of America
98(2), 658-663.
Kalter, D., Nakamura, M., Turpin, J., Baca, L., Hoover, D., et al. (1991). Enhanced HIV
Replication in Macrophage Colony-Stimulating Factor-Treated Monocytes. Journal of
Immunology, 146(1), 298-306.
Kelley CF et al. The relation between symptoms, viral load, and viral load set point in primary
HIV infection. J Acquir Immune Defic Syndr 2007 May 17; [e-pub ahead of print].
Little, S., McLean, A., Spina, C., Richman, D., & Havlir, D. (1999). Viral Dynamics of Acute
HIV-1 Infection. Journal of Experimental Medicine, 190 (6), 841-850.
doi: 10.1084/jem.190.6.841
Matsuo, K., Honda, M., Shiraki, K., & Niimura, M. (2001). Prolonged herpes zoster in a patient
infected with the human immunodeficiency virus. JOURNAL OF DERMATOLOGY, 28(12),
728-733.
McEwen, B., Stellar, E., (1993-09-27). Stress and the Individual- Mechanisms Leading to
Disease. Archives of internal medicine (1960), 153(18), 2093-2101.
Moriuchi, M., Moriuchi, H., Turner, W., & Fauci, A. S. (1998). Exposure to bacterial products
renders macrophages highly susceptible to T-tropic HIV-1. The Journal of clinical investigation,
102(8), 1540-1550.
Pantaleo, G., Graziosi, C., & Fauci, A. S. (1993). New concepts in the immunopathogenesis of
human immunodeficiency virus infection. The New England journal of medicine, 328(5), 327.
Perelson, Alan S, Neumann, A. U., Markowitz, M., Leonard, J. M., & Ho, D. D. (1996). HIV-1
Dynamics in Vivo: Virion Clearance Rate, Infected Cell Life-Span, and Viral Generation Time.
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Meta-Analytic Study of 30 Years of Inquiry. Pysch. Bulletin, 130(40, 601-630.
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Stafford, M. A., Corey, L., Cao, Y., Daar, E. S., Ho, D. D., & Perelson, A. S. (2000). Modeling
plasma virus concentration during primary HIV infection. Journal of theoretical biology, 203(3),
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Zhang, L., Ramratnam, B., Tenner-Racz, K., He, Y., Vesanen, M., Lewin, S., Talal, A., et al.
(1999). Quantifying Residual HIV-1 Replication in Patients Receiving Combination
Antiretroviral Therapy. New England Journal of Medicine, 340(21), 1605-1613.
T Cell Counts and Viral Load Interactions, Epid 602, Kevin Jefferson
Appendix 1: A Simple Model of HIV (Stafford et al., 2000) with decay of T cell production
1
la*(1-u(4))-d*u(1)-k*u(1)*u(3)
Mux
Fcn
Mux
k*u(1)*u(3)-de*u(2)
Integrator
Integrator1
k=0.00027;
d=0.01;
de=0.39;
p=850;
c=3;
To=10;
Io=0;
Vo=0.000001;
> plot (tout, log(T));
>> hold on;
>> plot (tout, log(I), 'g');
>> hold on;
T o Workspace1
1
V
s
Fcn2
la=10;
I
s
p*u(2)-c*u(3)
Function
T o Workspace
1
Fcn1
.00028
T
s
Integrator2
1
s
Integrator3
T o Workspace2
f
T o Workspace3
T Cell Counts and Viral Load Interactions, Epid 602, Kevin Jefferson
>> plot (tout, log(V), 'r');
400 days of HIV with T cell decay term
11
10
9
8
Log counts/ul
7
Red: Virus
Green: Infected T cells
Blue: Uninfected T
cells
6
5
4
3
2
1
0
0
50
100
150
200
250
300
350
Time (days)
Appendix 2: Introducing Duffin and Tullis Clearance Rate modification:
la*(1-u(4))-d*u(1)-k*u(1)*u(3)
Fcn
1
T
s
Integrator
T o Workspace
Mux
k*u(1)*u(3)-de*u(2)
Mux
Fcn1
p*u(2)-c*(1-u(4))*u(3)
Fcn2
.00028
Function
1
s
Integrator1
1
s
Integrator2
1
s
Integrator3
I
T o Workspace1
V
T o Workspace2
f
T o Workspace3
400
T Cell Counts and Viral Load Interactions, Epid 602, Kevin Jefferson
plot (tout, log(T));
Clearance rate reduction model CD4+ T cells
5
4.5
4
Log (counts/ul)
3.5
3
2.5
2
1.5
1
0.5
0
0
500
1000
1500
2000
2500
3000
Time (days)
plot (tout, log(V), 'r');
Clearance rate reductionmodel-- HIV Virions
11
10
9
8
Log (Counts/ul)
7
6
5
4
3
2
1
0
0
500
1000
1500
2000
2500
Time (days)
Appendix 3: Introducing CTL death rate without CTL number variation
dc=.50;
3000
3500
4000
T Cell Counts and Viral Load Interactions, Epid 602, Kevin Jefferson
1
la*(1 -u(4)) -d*u(1) -k*u(1)*u(3)
Mux
Fcn
Mux
k*u(1)*u(3) - u(2)*(de + dc*5)
Integrator
I
s
Integrator1
T o Workspace1
1
p*u(2)-c*u(3)
V
s
Fcn2
Function
T o Workspace
1
Fcn1
.00028
T
s
Integrator2
T o Workspace2
1
f
s
Integrator3
T o Workspace3
HIV over 350 days with T cell production decay, and constant CTL
10
Red: Virus
Blue: Uninfected T cells
Green: Infected T cells
9
8
Log Counts/ul
7
6
5
4
3
2
1
0
0
50
100
150
200
Time (days)
Appendix 4: Allowing CTL to vary as a function of stress
>> dc=.2;
250
300
350
T Cell Counts and Viral Load Interactions, Epid 602, Kevin Jefferson
>> la=10;
k=0.00027;
d=0.01;
de=0.39;
p=850;
c=3;
To=10;
Io=0;
Vo=0.000001;
>> dx=.02;
>> Xo=.00092593;
a=.0009259259259;
la*(1 -u(4)) -d*u(1) -k*u(1)*u(3)
Fcn
k*u(1)*u(3) - u(2)*(de + dc*u(5) )
Fcn1
Mux
Mux
p*u(2)-c*u(3)
Fcn2
.00028
Function
a*u(3)*u(5)*u(6)- dx*u(5)
Function1
1/(s*u(5)+u(3))
Function2
1
T
s
Integrator
T o Workspace
1
I
s
Integrator1
1
s
Integrator2
1
s
Integrator3
1
s
Integrator4
1
s
Integrator5
T o Workspace1
V
T o Workspace2
f
T o Workspace3
X
T o Workspace4
r
T o Workspace5
T Cell Counts and Viral Load Interactions, Epid 602, Kevin Jefferson
With >> s=500,00
HIV over 350 days with CTL as a function of stress, s=500,000
11
10
Red: Virus
Blue: Uninfected T cells
Yellow: CTLs
Green: Infected T cells
9
8
Log Count/ul
7
6
5
4
3
2
1
0
0
50
100
150
200
250
300
350
Time (days0
With >> s=1,000,000
HIV over 350 days with CTL as function of stress, s=1,000,000
11
10
Red: Virus
Blue: Uninfected T cells
Yellow: CTLs
Green: Infected T cells
9
8
Log Counts/ul
7
6
5
4
3
2
1
0
0
50
100
150
200
Time (days)
With >> s=5,000,000
250
300
350
T Cell Counts and Viral Load Interactions, Epid 602, Kevin Jefferson
HIV over 350 days with CTL as function of stress, s=5,000,00
11
10
Red: Virus
Blue: Uninfected T cells
Yellow: CTLs
Green: Infected T cells
9
8
Log Counts/ul
7
6
5
4
3
2
1
0
0
50
100
150
200
Time (days)
250
300
350