Transcript Bryn Mawr Friday December 12, 2003

Models of HIV Infection at the
Immune System Level
Douglas E. Norton
Villanova University
Mellon Tri-Co Faculty
Modeling Working Group
Bryn Mawr College
December 12, 2003
Villanova Summer
Research Institute
The National Health and Nutrition Examination Survey is
a survey conducted by the National Center for Health
Statistics (NCHS), Centers for Disease Control and
Prevention. This survey has been designed to collect
information about the health and diet of people in the
United States.
http://www.cdc.gov/nchs/nhanes.htm
Our immediate academic goal: to
formulate and explore some models of the
immune system and its interaction with the
human immunodeficiency virus.
We present a small number of
fundamental ideas – 6 in all – that
participants can understand at a useful level
and that we consider essentially sufficient to
work intelligently with these models.
6 Pillars of Wisdom
[William Fleischman, T. E. Lawrence with tolerance 1]
I. We will be modeling dynamical systems.
This may be a forbidding, technical
sounding term; all it means is a system with
quantities or properties we can measure that
change over time.
II. The models we will be examining are
calculus-based. They are expressed formally
as systems of differential equations, but
there is a simple, intuitive way of thinking
about them, interpreting the equations as
rate of change expressions, that enables us to
understand and then manipulate them
without distorting their meaning.
III. Although the formal solution and
analysis of the models involve deep
ideas from advanced mathematics, we
can explore a big part of the landscape
described by these models using only
(high school) algebra and geometry.
IV. Every model is a population model.
V. Complex models are built from simple
parts. They are based on useful analogies or
metaphors.
VI. Even models that are simple to the
point of caricature can be profoundly
useful tools in thinking about intricate
biological systems. Be careful not to be
dismissive of a model on account of its
simplicity.
After an introduction of the derivative as a
limit of average rates of change, we use “rate
of change” as our motivation for generating
systems of differential equations to model
biological phenomena. We move
immediately to Euler’s Method as a
computational tool, calling it simply
The Recipe.
The Recipe.
STEP 1: Compute the value of the growth
rate expression for the current time: dN/dt.
STEP 2: Since dN/dt is approximately
N/t, we can use the rate of change in step
1 to approximate the population change for a
given time change. That is, for a given
choice of t, we can use step 1 to estimate
N over the time interval t.
Specifically,
 N 
 dN 
N  
 ( t )  
(t ), or N  (growth rate) (t ).
 t 
 dt 
STEP 3: Add the approximate change N
to the population you began with to get the
approximate population at the end of the
time interval:
(current N) + N  (new N).
STEP 4: Let the new value of N be the
new choice for the current value of N and
return to step 1.
Repeat steps 1 through 3 until you reach
the final time you want.
dN  birth rate  death
dt
dN  b N  m N
dt
dN  (b  m) N
dt
dN  r N
dt
rate
1 dN  b  m
N dt
per capita
average
average
growth rate
fecundity
probabilit y
of mortality
b  f ( N )  b0  kbN
m  g ( N )  m0  kmN
b, m
bo
m0
N
EQUILIBRIUM POINT
b, m
bo
m0
N
1 dN  (b0  kbN )  (m0  kmN )
N dt
 (b0  m0 )  (kmN  kbN )
dN  N [(b0  m0 )  (km  kb) N ]
dt
(
b
0  m0 )
 (km  kb) N [
 N]
(km  kb)
dN
dt
 k N [K  N ]


dN


N
 r N 1 
K
dt




0
9.6
4.223483
1
18.3
3.564981
2
29
3.087903
3
47.2
2.571771
4
71.1
2.122624
5
119.1
1.522472
6
174.6
1.032724
7
257.3
0.460289
8
350.7
-0.10958
9
441
-0.6774
10
513.3
-1.21896
11
559.7
-1.67059
12
594.8
-2.13688
13
629.4
-2.87242
14
640.8
-3.27636
15
651.1
-3.84677
16
655.9
-4.27773
17
659.6
-4.80523
K
N
1 Ae rt
K

N
y  ln(
)
N
dN1  r N 1 N1  N2 
1 1
12

K
K
dt
1
1




2



1

2
dN2  r N 1 N2  N
2
21
K
K
dt
2
EXAMPLE: Rabbits and sheep compete for a limited amount
of grass. We assume logistic growth for each, that rabbits
reproduce rapidly, and that the sheep can crowd out the rabbits.
Then some unrealistic coefficients (just to make this example
look friendlier) and these assumptions could give the following
model:
dx/dt = x(3 – x – 2y)
dy/dt = y(2 – x – y),
where x(t) = size of the rabbit population and y(t) = size of the
sheep population. This is an example of what the N1 – N2
system may look like, algebraically speaking, if multiplication
and constant renaming are utilized. We can explore this model
with a natural extension of our recipe to find equilibrium
points and possible extinction of one population.
EXAMPLE (a different but related example):
Three populations: grass, sheep, and wolves.
Without the sheep, the grass grows logistically (?!).
Sheep eat grass. Wolves eat sheep. Let x, y, and z
represent the sizes of the wolf, sheep, and grass
populations, respectively. Then the following
system is a possible model:
dx/dt = –x + xy
dy/dt = –y + 2yz – xz
dz/dt = 2z – z2 – yz.
ONE MORE EXAMPLE: predator-prey with child-care (!).
x1 = young prey; x2 = adult prey; y = predators.
The young are protected from predators.
Interpret the terms in this system:
dx1/dt = ax2 – bx1 – cx1
dx2/dt = bx1 – dx2 – ex2y
dy/dt = – fy + gx2y.
Dissolved Oxygen
Biodegradation kinetics and reaeration may be integrated into
a single model to predict the effect of organic pollution on
streams. The deficit equation in these natural systems must
consider the oxygen consumption due to biodegradation and
the oxygen replenishment due to reaeration. The differential
equation that describes these conditions is:
Integration of this summation yields the general equation for
the deficit in a stream (sag-curve equation or Streeter-Phelps
equation):
REM:
The Register of Ecological Models (REM) is a metadatabase for existing mathematical models in ecology.
ECOBAS:
The ECOBAS project provides a system for
documentation of mathematical descriptions of ecological
processes.
The Classical S-I-R Model
Model the spread of a disease:
N = S + I + R = total population, where
S = the number of “susceptibles,”
I = number of “infecteds,” and
R = number of “recovereds” (immune).
First version: do not include births or deaths.
Then the only changes are from S to I
(susceptibles becoming infected) and from
I to R (from infected to recovered)
To become infected, a susceptible must come
in contact with an infected, we use an analogy
with the Law of Mass Action in chemical
kinetics: the rate at which this type of
change occurs (S to I) is proportional to the
product of the sizes of the two populations
involved. For recovery, the rate is proportional
to the number of infecteds. Then with
constants  = “transmission coefficient”
and  = “recovery rate coefficient,” we have:
dS/dt = –SI
dI/dt = SI – I
dR/dt = I.
A First Modification
 = the extra per capita mortality rate due to
the disease (decreases I)
A = rate of additions (increases S)
b = per capita mortality rate (decreases all three)
 = per capita rate of loss of immunity (turns
R’s back into S’s)
a = per capita birth rate, independent of whether
S, I, or R. (If there are no additions other
than by birth, then A = a(S + I + R).)
Then the “new and improved”
model would be:
dS/dt = A – bS – SI + R
dI/dt = SI – (b +  + I
dR/dt = I – ( + b)R.
Epidemiological Models
Immune System Level
Our Beginning Model
In direct analogy with human population models, we
model the immune system players in HIV infection.
Rather than susceptible, infected, or recovered persons,
our dynamic variables now represent:
T
T*
T**
V
:
:
:
:
healthy T-cells (cells per ml)
latently infected T-cells (cells per ml)
actively infected T-cells (cells per ml)
viral particles (free virus) (particles per ml).
 T  T * T * * 
dT
  k1VT
 s  T T  rT 1 
dt
Tmax


dT *
 k1VT  T T * k 2T *
dt
dT * *
 k 2T *  bT * *
dt
dV
 NbT * *  k1VT  V V
dt
 T T*T**
dT

 s TT rT
1
(k1V k3M*)T


dt
Tmax


dT*
(k1VTk3M*T) T*T*k2T*
dt
dT**
k2T*bT**
dt
dV
 NbT**MM*k1VTVV
dt
dM
 M(EM M) k4VM
dt
dM*
k4VMM*M*
dt
 T T*T**
dT

 s TT rT
1
(k1V k3M*)T


dt
Tmax


dT*
(k1VTk3M*T) T*T*k2T*
dt
dT**
k2T*bT**
dt
dV
 NbT**MM*k1VTVV
dt
dM
 M(EM M) k4VM
dt
dM*
k4VMM*M*
dt
Dependent Variables
Values
T = uninfected CD4+ T cell population
1000 mm-3
A = CD8+ T cell population
500 mm-3
Ts* = Latently infected CD4+ T cell population (slow-replicating
0
virus)
Tf* = Latently infected CD4+ T cell population (fast-replicating
0
virus)
Ts** = Actively infected CD4+ T cell population (slow-replicating 0
virus)
Tf** = Latently infected CD4+ T cell population (fast-replicating
0
virus)
Vs = Slow replicating infectious HIV population
.001 mm-3
Vf = Fast replication infectious HIV population
0
M = Uninfected macrophage population
30 mm-3
Ms* = Infected macrophage population (slow-replicating)
0
Mf* = Infected macrophage population (fast-replicating)
0
dT
TV
 ST   T T  rT
 ( K11 Vs  K12V f )T  ( K41 M s *  K42 M f *)T
dt
CT  V
dA
AV
 S A   A A  rA
dt
CA  V
dTs *
 K11Vs T   T Ts *  K2 Ts *  K41 M s * T
dt
dTf *
 K12V f T   T Tf *  K2 Tf *  K42 M f * T
dt
dTs **
 K2 Ts *   bs Ts **  K5Ts ** A
dt
dTf **
 K2 Tf *   bf Tf **  K5Tf ** A
dt
dVs
 N s  bs Ts **  wII Ms M s *  (1  y ) II Mf M f *   V V  K11Vs T  K31Vs M
dt
dV f
 N f  bf Tf **  (1  w) II Ms M s *  y ) II Mf M f *   V V  K12V f T  K32V f M
dt
dM
 S M   M M  ( K31Vs  K32V f ) M
dt
dM s *
 K31Vs M   M * M s *  K6 M s * A
dt
dM f *
 K32V f M   M * M f *  K6 M f * A
dt
Parameters
ST = source term for uninfected T4 cells
SA = source term for killer cells
rT = maximal proliferation of CD4+ T cell population
rA= maximal proliferation of CD8+ T cell population
CT = half-saturation constant of proliferation process (helper)
CA = half-saturation constant of proliferation process (killer)
K11= rate T4 cell becomes infected by Vs
K12 = rate T4 cell becomes infected by Vf
K2 = rate T* converts to actively infected
K31 = rate macrophage becomes infected by Vs
K32 = rate macrophage becomes infected by Vf
K41 = rate Ms* infects T4 cells
K42 = rate Mf* infects T4 cells
K5 = rate CD8+ cells kill T** cells
K6 = rate CD8+ cells kill M*
T = death rate of uninfected T4 cells
A = death rate of CD8+ cells
v = death rate of virus
M = death rate of uninfected macrophage
M* = death rate of infected macrophage
bs = death rate of infected Ts** cells
bf = death rate of infected Tf** cells
EM = equilibrium for macrophage
Ns = number of free virus produced by Ts** cells
Nf = number of free virus produced by Tf** cells
Ms = rate of free virus produced by infected Ms*
Mf = rate of free virus produced by infected Mf*
y = fast replicating mutate to slow replicating
w = slow replicating mutate to fast replicating
Working values
10 d-1 mm-3
15 d-1 mm-3
0.02 d-1
0.03 d-1
100 mm-3
100 mm-3
1.8 x 10-5 mm3 d-1
2 x 10-4 mm3 d-1
0.003 mm3 d-1
8 x 10-6 mm3 d-1
5 x 10-6 mm3 d-1
1 x 10-7 mm3 d-1
1 x 10-7 mm3 d-1
7.4 x 10-4 mm3 d-1
7.4 x 10-4 mm3 d-1
0.02 d-1
0.02 d-1
0.4 d-1
0.005 d-1
0.005 d-1
0.24 d-1
0.3 d-1
30 mm-3
1000
1000
300 d-1
300 d-1
0.95
0.8
T = healthy T-cells (initial value = 1000 to 2000 cells/ml)
T* = all infected T-cells (initial value = 0)
Tf* = T-cell infected with fast replicating HIV (initial value = 0)
Ts* = T-cell infected with slow replicating HIV (initial value = T)
V = both strands of HIV (initial value = 0.001 virions/ml)
Vf = fast replicating HIV (initial value = 0)
Vs = slow replicating HIV (initial value = V)
M = macrophage uninfected by HIV (initial value = 85 or 100)
M* = macrophages infected with either strand of HIV (initial value = 0)
Mf* = macrophage infected with fast replicating virus
Ms* = macrophage infected with slow replicating virus
Tb = tuberculosis cell (initial value = 1 mm-3)
sT = source of healthy T-cells = 10 (cells/ml/day)
sM = source of uninfected macrophages =
T = natural healthy T-cell death rate = 0.02 (1/day)
b = lysis rate of an infected T-cell = 0.24 (1/day)
v = natural death rate of HIV particles = 2.4 (1/day)
M = natural death rate of macrophages = 0.005 or 0.003 (1/day)
Tb = natural death rate of Tb cells
k1 = rate at which a fast replicating virus infects a healthy T-cell = 0.000024
(1/cells*day/ml)
k3 = rate at which fast replicating virus is engulfed by macrophage = 0.000002
k4 = rate at which infected macrophage “infects” healthy T-cell =
k5 = rate at which a healthy T-cell kills a Tb cell = 0.5 (ml/day)
k6 = rate at which a macrophage kills a Tb cell =
k7= = rate at which a Tb cell kills a healthy T-cell =
k8 = rate at which a Tb cell kills a T-cell infected by fast-replicating HIV particles =
k11 = rate at which a slow replicating virus infects a healthy T-cell
k13 = rate at which a slow replicating virus is engulfed by a macrophage =
rT = coefficient for immune response initiated by emergence of foreign invaders = 0.02
rTb = coefficient for immune response inititated by appearance of Tb = 1
K = carrying capacity of Tb = 1000
z = effect of AZT on burst size of virion particles
N = burst size of virion particles from infected T-cells =
 = burst size of virion particles from infected macrophages = from 100-1000
 = percent fast replicating viruses coming from an infected macrophage
1 -  = percent of slow replicating viruses coming from an infected macrophage
C = something = 1000 per cubic mm
1.
 V  Tb 
dT
*
 sT   T T  rT T 
  k1V f T  k 4 M T  k 7 Tb T  k11V s T
dt
 C  V  Tb  
= source / natural death / immune response growth / infection / “infection” / death by TB /infection
2. dT *
f
dt
 k1V f T   T T f*   b T f*  k 4 M *T  k 8Tb T f*
= source / natural death / burst / source by macrophage / death by TB
*
3. dTs
 k11V s T   T Ts*   b Ts*  1   k 4 M *T  k 8Tb Ts*
dt
= source / natural death / burst / source by macrophage / death by TB
4. dV f
 zN  b T *  k1V f T   vV f  k 3 MV f  zM * M *  zN k 8Tb T *
dt
= source / death by T-cell / natural death / engulfed / source by macrophage / burst size
5. dV s
 1   zM * M *  k11V s T  k13 MV s
dt
= source / death by T-cell / engulfed
6. dM  s   M  k MV  r 2 MV  r 1 MT  k MV
M
M
3
f
M
M
b
13
s
dt
= source / natural death / death by HIV / stimulation / recruitment / death by HIV
*
dM
f
7.
 k 3 MV f   M * M *f
dt
= source / natural death
*
8. dM s  k MV   * M *
13
s
s
M
dt
= source / natural death
9. dTb  r T (  T )   T  T k T  k M 
Tb b
b
Tb b
b
5
6
dt
= source / natural death / death by immune system
10. V  V f  Vs
*
*
*
T

T

T
f
s
11.
*
*
*
M

M

M
12.
f
s
Goals
• Accurately implement the current models
• Modify existing equations to make them
more mathematically accurate and
biologically realistic
• Create equations to model the viral load,
number of HIV strains, and the immune
response
• Model the effects of the number of viral
strains on the progression of the virus
Original System of Equations
• dTp/dt = CLTL(t) – CPTP(t)
• dTlp/dt = CLTlL(t) – CPTlP(t)
• dTL/dt = CPTP(t) – CLTL(t) – kTL(t) + ųaTaL(t)
• dTlL/dt = pkTL(t) – CLTlL(t) + CPTlP(t) – ųlTlL(t) – slTlL(t) + siTiL(t)
• dTaL/dt = rkTL(t) – ųaTaL(t)
• dTiL/dt = qkTL(t) – ųiTiL(t) + slTlL(t) – siTiL(t)
Modifications
• dTp/dt = CLTL(t) – CPTP(t) +
s*(1-(Tp(t)+Tlp (t)+TL (t)+TlL (t)+TaL (t)+TiL (t))/Smax) ųu*Tp(t)
• dTL/dt = CPTP(t) – CLTL(t) – kTL(t) + ųaTaL(t) – ųu* TL(t)
• dV/dt = bTil(t) - cV(t) - KR(t)
• dS/dt = un*(q*k* TL(t) + Sl * TlL(t))
• dR/dt = [g* V(t) * R(t) * (1- R(t) / Rmax)]/ floor S(t)
Future Modifications
• dTL/dt = CPTP(t) – CLTL(t) – kV(t)TL(t) + ųaTaL(t) –
muU*Tp(t)
• dTaL/dt = rkV(t)TL(t) – ųaTaL(t)
• dTlL/dt = pkV(t)TL(t) – CLTlL(t) + CPTlP(t) – ųlTlL(t) – slTlL(t) + siTiL(t)
• dTiL/dt = qkV(t)TL(t) – ųiTiL(t) + slTlL(t) – siTiL(t)
• dS/dt = un*(q*k*V(t)*Tl(t) + Sl * Tll(t))
Uninfected blood CD4+ cells over 10
years
Before
After
Incorrect display of uninfected T
cells
• The cell count does not get low enough to
induce AIDS
Uninfected CD4+ cells
in blood
Uninfected CD4+
cells in lymph
Latently infected CD4+ cells in blood
over 10 years
Before
After
Uninfected CD4+ cells in lymph over 10
years
Before
After
Latently (red), abortively (green), and actively
(yellow) infected CD4+ cells in the lymph over
10 years
Before
After
Viral Load over 1 year
(in powers of 10)
Viral Load over 10 years
(in powers of 10)
Number of Virus Strains over 10
years