Models of HIV Infection at the Immune System Level

21st Annual Southeastern-Atlantic Regional Conference on Differential Equations

Wake Forest University November 2-3, 2001

Douglas E. NortonVillanova University

Villanova SummerResearch 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: toformulate and explore some models of theimmune system and its interaction with thehuman immunodeficiency virus.

We present a small number offundamental ideas – 6 in all – thatparticipants can understand at a useful leveland that we consider essentially sufficient towork 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, technicalsounding term; all it means is a system withquantities or properties we can measure thatchange over time.

II. The models we will be examining arecalculus-based. They are expressed formallyas systems of differential equations, butthere is a simple, intuitive way of thinkingabout them, interpreting the equations asrate of change expressions, that enables us tounderstand and then manipulate themwithout distorting their meaning.

III. Although the formal solution andanalysis of the models involve deepideas from advanced mathematics, wecan explore a big part of the landscapedescribed by these models using only(high school) algebra and geometry.

IV. Every model is a population model.

V. Complex models are built from simpleparts. They are based on useful analogies ormetaphors.

VI. Even models that are simple to thepoint of caricature can be profoundlyuseful tools in thinking about intricatebiological systems. Be careful not to bedismissive of a model on account of itssimplicity.

After an introduction of the derivative as alimit of average rates of change, we use “rateof change” as our motivation for generatingsystems of differential equations to modelbiological phenomena. We moveimmediately to Euler’s Method as acomputational 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 step1 to approximate the population change for agiven time change. That is, for a givenchoice of t, we can use step 1 to estimateN over the time interval t.

S p e c i f i c a l l y ,

S T E P 3 : A d d t h e a p p r o x i m a t e c h a n g e N t o t h e p o p u l a t i o n y o u b e g a n w i t h t o g e t t h ea p p r o x i m a t e p o p u l a t i o n a t t h e e n d o f t h et i m e i n t e r v a l :

( c u r r e n t N ) + N ( n e w N ) .

).(rate)growth (or),()( tNtdt

dNt

t

NN

STEP 4: Let the new value of N be the new choice for the current value of N andreturn to step 1.

Repeat steps 1 through 3 until you reachthe final time you want.

2

121

2

222

2

1

212

1

111

1

1

1

KN

KNNr

dtdN

KN

KNNr

dtdN

The Classical S-I-R Model

Model the spread of a disease:

N = S + I + R = total population, whereS = the number of “susceptibles,”I = number of “infecteds,” andR = 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 fromI to R (from infected to recovered)

To become infected, a susceptible must comein contact with an infected, we use an analogywith the Law of Mass Action in chemicalkinetics: the rate at which this type ofchange occurs (S to I) is proportional to theproduct of the sizes of the two populationsinvolved. For recovery, the rate is proportionalto the number of infecteds. Then withconstants = “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 tothe 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, wemodel the immune system players in HIV infection.Rather than susceptible, infected, or recovered persons,our dynamic variables now represent:

T : healthy T-cells (cells per ml)T* : latently infected T-cells (cells per ml)T** : actively infected T-cells (cells per ml)V : viral particles (free virus) (particles per ml).

VVTkTNdt

dV

TTkdt

dT

TkTVTkdt

dT

VTkT

TTTrTTs

dt

dT

Vb

b

T

T

1

2

21

1max

**

*****

***

***1

**

)(

***

*****

**)*(*

*)(***

1

*4

4

1

2

2*31

31max

MVMkdt

dM

VMkMEdt

dM

VVTkMTNdt

dV

TTkdt

dT

TkTTMkVTkdt

dT

TMkVkT

TTTrTTs

dt

dT

M

MM

VMb

b

T

T

**

)(

***

*****

**)*(*

*)(***

1

*4

4

1

2

2*31

31max

MVMkdt

dM

VMkMEdt

dM

VVTkMTNdt

dV

TTkdt

dT

TkTTMkVTkdt

dT

TMkVkT

TTTrTTs

dt

dT

M

MM

VMb

b

T

T

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 0virus)

Tf* = Latently infected CD4+ T cell population (fast-replicating 0virus)

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 0M = Uninfected macrophage population 30 mm-3

Ms* = Infected macrophage population (slow-replicating) 0Mf* = Infected macrophage population (fast-replicating) 0

dT

dtS T r

T V

C VK V K V T K M K M T

dA

dtS A r

A V

C V

dT

dtK V T T K T K M T

dT

dtK V T T K T K M T

dT

dtK T T K T A

T T TT

s f s f

A A AA

ss T s s s

f

f T f f f

ss b s s s

( ) ( * *)

** * *

** * *

* ** * * * *

11 1 2 4 1 4 2

11 2 4 1

1 2 2 4 2

2 5

dT

dtK T T K T A

dV

dtN T wII M y II M V K V T K V M

dV

dtN T w II M y II M V K V T K V M

dM

dtS M K Vs K V M

dM

dtK V M

f

f b f f f

ss b s s M s s M f f V s s

f

f b f f M s s M f f V f f

M M f

ss

* ** * * * *

* * * ( ) *

* * ( ) * ) *

( )

*

2 5

11 3 1

1 2 3 2

3 1 3 2

3 1

1

1

M s s

f

f M f f

M K M A

dM

dtK V M M K M A

*

*

* *

** *

6

3 2 6

Parameters Working values ST = source term for uninfected T4 cells 10 d-1 mm-3

SA = source term for killer cells 15 d-1 mm-3

rT = maximal proliferation of CD4+ T cell population 0.02 d-1

rA= maximal proliferation of CD8+ T cell population 0.03 d-1

CT = half-saturation constant of proliferation process (helper) 100 mm-3

CA = half-saturation constant of proliferation process (killer) 100 mm-3

K11= rate T4 cell becomes infected by Vs 1.8 x 10-5 mm3 d-1

K12 = rate T4 cell becomes infected by Vf 2 x 10-4 mm3 d-1

K2 = rate T* converts to actively infected 0.003 mm3 d-1

K31 = rate macrophage becomes infected by Vs 8 x 10-6 mm3 d-1

K32 = rate macrophage becomes infected by Vf 5 x 10-6 mm3 d-1

K41 = rate Ms* infects T4 cells 1 x 10-7 mm3 d-1

K42 = rate Mf* infects T4 cells 1 x 10-7 mm3 d-1

K5 = rate CD8+ cells kill T** cells 7.4 x 10-4 mm3 d-1

K6 = rate CD8+ cells kill M* 7.4 x 10-4 mm3 d-1

T = death rate of uninfected T4 cells 0.02 d-1

A = death rate of CD8+ cells 0.02 d-1

v = death rate of virus 0.4 d-1

M = death rate of uninfected macrophage 0.005 d-1

M* = death rate of infected macrophage 0.005 d-1

bs = death rate of infected Ts** cells 0.24 d-1

bf = death rate of infected Tf** cells 0.3 d-1

EM = equilibrium for macrophage 30 mm-3

Ns = number of free virus produced by Ts** cells 1000Nf = number of free virus produced by Tf** cells 1000

Ms = rate of free virus produced by infected Ms* 300 d-1

Mf = rate of free virus produced by infected Mf* 300 d-1

y = fast replicating mutate to slow replicating 0.95w = slow replicating mutate to fast replicating 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 .

= s o u r c e / n a t u r a l d e a t h / i m m u n e r e s p o n s e g r o w t h / i n f e c t i o n / “ i n f e c t i o n ” / d e a t h b y T B / i n f e c t i o n

2 .

= s o u r c e / n a t u r a l d e a t h / b u r s t / s o u r c e b y m a c r o p h a g e / d e a t h b y T B

3 .

= s o u r c e / n a t u r a l d e a t h / b u r s t / s o u r c e b y m a c r o p h a g e / d e a t h b y T B

TVkTTkTMkTVkTVC

TVTrTs

dt

dTsbf

b

bTTT 117

*41

*8

*4

**1

*

fbfbfTff TTkTMkTTTVk

dt

dT

*8

*4

**11

*

1 sbsbsTss TTkTMkTTTVk

dt

dT

4 .

= s o u r c e / d e a t h b y T - c e l l / n a t u r a l d e a t h / e n g u l f e d / s o u r c e b y m a c r o p h a g e / b u r s t s i z e

5 .

= s o u r c e / d e a t h b y T - c e l l / e n g u l f e d

6 .

= s o u r c e / n a t u r a l d e a t h / d e a t h b y H I V / s t i m u l a t i o n / r e c r u i t m e n t / d e a t h b y H I V

*8

*31

** TTkNzMzMVkVTVkTzN

dt

dVbMffvfb

f

ssMs MVkTVkMz

dt

dV1 311

**1

sbMMfMM MVkMTrMVrMVkMsdt

dM1 3

123

7 .

= s o u r c e / n a t u r a l d e a t h

8 .

= s o u r c e / n a t u r a l d e a t h

9 .

= s o u r c e / n a t u r a l d e a t h / d e a t h b y i m m u n e s y s t e m

*3

*

* fMff MMVk

dt

dM

*13

*

* sMss MMVk

dt

dM

MkTkTTTTrdt

dTbbTbbT

b

bb 65)(

1 0 .

1 1 .

1 2 .

sf VVV

***sf TTT

***sf MMM

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 = CLTl

L(t) – CPTlP(t)

• dTL/dt = CPTP(t) – CLTL(t) – kTL(t) + ųaTaL(t)

• dTlL/dt = pkTL(t) – CLTl

L(t) + CPTlP(t) – ųlTl

L(t) – slTlL(t) + siTi

L(t)

• dTaL/dt = rkTL(t) – ųaTa

L(t)

• dTiL/dt = qkTL(t) – ųiTi

L(t) + slTlL(t) – siTi

L(t)

Modifications

• dTp/dt = CLTL(t) – CPTP(t) +

s*(1-(Tp(t)+Tlp (t)+TL (t)+Tl

L (t)+TaL (t)+Ti

L (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) – ųaTa

L(t)

• dTlL/dt = pkV(t)TL(t) – CLTl

L(t) + CPTlP(t) – ųlTl

L(t) – slTlL(t) + siTi

L(t)

• dTiL/dt = qkV(t)TL(t) – ųiTi

L(t) + slTlL(t) – siTi

L(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