Influence of anti-viral drug therapy on the evolution of ...kuang/workshop/Feng.pdf · ASU 2/4/06...

ASU 2/4/06

Zhilan Feng

Influence of anti-viral drug therapy on the evolution of HIV-1 pathogens

Zhilan Feng and Libin Rong

Department of MathematicsPurdue University

ASU 2/4/06

Zhilan Feng

Outline

HIV-1 life cycle and Inhibitors

Age-structured models with combination therapies

Model analysis

Invasion of drug-resistant viruses

Evolution of HIV-1 pathogens

ASU 2/4/06

Zhilan Feng

Human Immunodeficiency Virus (HIV)1. Attachement

Getting in2. Reverse Transcription

From viral RNA to DNA3. Integration transcription

a. Viral DNA joins host DNAb. Making multiple viral RNAs

4. TranslationProducing viral proteins

5. Viral ProteaseCleaving viral proteins

6. Assembly & BuddingGetting out

13

4

Protease Inhibitor

25 6

Reverse Transcriptase Inhibitor

http://www.cellsalive.com/pics/hivmstr.map

ASU 2/4/06

Zhilan Feng

An HIV model with treatment (Perelson and Nelson, 1999)

H ealth Autho rities

Medica l Practition ers

Suscep tibleS

Late Med icalEncoun terML Recove redR

Quarantin edQ

Hospita lizedR No tHP

InfectionProgression

Recov ery

Prodrom alSympto msIPEarly Me dicalEncoun terME Respira torySympto msIRPresentatio nDiagnosi s Recovery

Expos edEQ uarantine

Hospitaliz edP NotQ Progression

Presentatio n WaningHospita lizedP Q

Hospitaliz edR HP

Re covery

Progression

Rec overy

Progr ession

αγ

1εφ

2δ

1 δ

δ

()tSλ2α

1 δ

2εφ

ωδ

3 δ

s, b: growth rate of healthy T cellsd: per capita death rate of uninfected cellsδ: per capita death rate of infected cellsc: virus clearance ratek: Infection rate of an uninfected cellN: total viruses produced per infected cellrrt, rp: drug efficacy

T(t): uninfected target T cells

T *(t): infected T cells

V(t): infectious viruses

Reverse Transcriptase Inhibitor(Entry Inhibitor?)

Protease Inhibitor

ASU 2/4/06

Zhilan Feng

An age-structured HIV-1 model (Nelson et al. 2004)(No treatment)



Suscep tibleS


Quarantin edQ



Recov ery





Hospitaliz edR HP

Re covery

Progression

Rec overy

Progr ession

αγ

1εφ

2δ

1 δ

δ

()tSλ2α

1 δ

2εφ

ωδ

3 δ

s: recruitment of healthy T cellsd: per capita death rate of uninfected cellsδ(a): age-specific death rate of infected cellsc: virus clearance ratek: Infection rate of an uninfected cellp(a): age-dependent virion production rate

T(t): uninfected target T cells at time t

T *(a,t): age-density of infected T cells

V(t): infectious virus

ASU 2/4/06

Zhilan Feng

Age-structured model with treatments (Feng and Rong, 2006)



Suscep tibleS


Quarantin edQ



Recov ery





Hospitaliz edR HP

Re covery

Progression

Rec overy

Progr ession

αγ

1εφ

2δ

1 δ

δ

()tSλ2α

1 δ

2εφ

ωδ

3 δ

* *

propotion of infected cells of infection age a in the preRT phase

density of infected cells of age a in the preRT phase

(an RT inhibitor cou

(a):

( , ) ( ) ( ,

ld r

) : preRTT a t a T a t

β

β=

* *

evert it back to uninfected class)

density of infected cells of age a progressed to

the postRT phase (a protease in

( , ) (

hibi

1 ( )) ( , ) :

tor could he

,

l )

,

ppostRT

rt p

T a t a T a t

r r r

β= −

*

0

*

drug efficacy of RT, protease, and entry inhibitors respectively

rate at which preRT infected cells become uninfected

:

( , ) :

(1 ) ( , ) : rate at which new virion part cl i

e

rt preRT

p postRT

r T a t da

r T a t da

η∞

−

∫

0

es are produced∞

∫

ASU 2/4/06

Zhilan Feng

An age-structured model with combination therapy (I)(Including reverse transcriptase inhibitors and protease inhibitors)



Suscep tibleS


Quarantin edQ



Recov ery





Hospitaliz edR HP

Re covery

Progression

Rec overy

Progr ession

αγ

1εφ

2δ

1 δ

δ

()tSλ2α

1 δ

2εφ

ωδ

3 δ

β(a): propotion of infected cells of age a in the preRT phase

η: conversion factor to non-infected cells by RT inhibitorsRemark: RT inhibitors do not reduce the infection rate of target cells (kVT)

ASU 2/4/06

Zhilan Feng

An age-structured model with combination therapy (II)(Including entry or fusion inhibitors and protease inhibitors)



Suscep tibleS


Quarantin edQ



Recov ery





Hospitaliz edR HP

Re covery

Progression

Rec overy

Progr ession

αγ

1εφ

2δ

1 δ

δ

()tSλ2α

1 δ

2εφ

ωδ

3 δ

ASU 2/4/06

Zhilan Feng

Notation

Age specific survival probability of infected cells:

Effective viral production rate of a productively infected cell of age a:

Total amount of infectious virion particles produced by one infected cell in its lifespan:

The reproductive number

22

1sk skdc d c

⎛ ⎞⎛ ⎞= = ⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠

KR K

Number of uninfected cells in an infection-free population

Virus lifespan

ASU 2/4/06

Zhilan Feng

Reformulation of the system (I)

Eliminating T*:

Let B(t)=kV(t)T(t). Solve for T*(a,t) along the characteristic line:

ASU 2/4/06

Zhilan Feng

Equivalent systems

(A)

(B)

ASU 2/4/06

Zhilan Feng

Existence of (positive) solutions

(A)

System (A) can be written as

with

Existence of solutions follows from Theorem 1.1 in Gripenberg et al. (1990)

ASU 2/4/06

Zhilan Feng

A limiting system of (B)

Steady-states

Infection-free SS:

Infected SS:

ASU 2/4/06

Zhilan Feng

Existence of the infection steady-state E ◊

E ◊

Note:

22Therefore, 0 if and only if , that is, ) (1

skV

dcsk dc◊ > > =>K

KRR

ASU 2/4/06

Zhilan Feng

Stability of (infection-free) and (infected)E E ◊

is a attractor if , and it is unstable if

exists an

g

d is locally asymptotically stable

Result 1:

Resul if

t 2:

lobal 1 1

1

E

E ◊

>

>

R < R

R

ASU 2/4/06

Zhilan Feng

Proof of Result 1 (fluctuation method)

Rewrite the V equation:

where

2

( )

and kssTd cd

∞∗⎛ ⎞

≤ =⎜ ⎟⎜ ⎟⎝ ⎠

KR

Therefore, if 1 then 0, or lim ( ) 0. From ( ) ( ) ( ), lim ( ) 0

From the equation we get . Thus,

t tV V t B t kV t T t B t

s sT T T Td d

∞

→∞ →∞

∞∞ ∞

< = = = =

≥ = =

R

Choose a sequence s.t. ( ) and '( ) 0 as . Then and n n ns sr W r W W r n W Td d

∞ ∞ ∞→ → → ∞ ≤ ≤

* Let th n e( ) totalW T T= +∗

ASU 2/4/06

Zhilan Feng

Proof of Result 2 (assume R >1)

2 12 10 cannot be an eigenvalue if 1

is a co

ˆ ˆ If (real) then ( ) , ( ) <1. Hence,

If . First show that when is closmplex eigenval e to 1, and thue Re( )<0 en

us

e the continuous d p

e

K Kλ λλ λ

λ λ

•

•

≤ ≤> >K K

R

R

endence on of the characteristic polynomial for any 1>R R

Eq. (1) is equivalent to:

Characteristic equation at :E (1)

(Laplace transform of Ki(a))where

or

ASU 2/4/06

Zhilan Feng

Stability of and (numerical simulations)E E ◊

is a global attractor if , and it is unstable if

exists and is locally asymptotically stable if

Result 1:

Result 2:1

1 1

E

E ◊

>

>

R < R

R

R > 1R < 1

ASU 2/4/06

Zhilan Feng

Influence of drug therapy on viral fitnessSuppose that the drug-sensitive strain of HIV-1 infection is at the infected steady state and that a small number of drug resistant viruses have been introduced into the population. Assume that all parameters are the same for both strains except:

pand : drug efficacy for the resistant strain, (0< <1, , )

: viron production rate of infected cells with resistant

vi

=

u es( s) rrt i i i i i rr r

p

t pr r

a

σσ =

Let and denote the reproductive ratios for sensitive and resistant strainss rR R

The reproductive ratio of an invading resistant strain (when the sensitive strain is at its infected equilibrium) is

Burst sizeAvailable uninfected T cells =s/(Rsd)

The resistant strain can establish in the population if and only if 1 or r

r s

◊ >

>

R R R

ASU 2/4/06

Zhilan Feng

Optimal reproductive ratioConsider the following specific forms of parameter functions:

Cost functions * *

* (1/ 1) *

Type 1:

Type 2:

p

p

p

e pφ σ

σ− −

=

=

ASU 2/4/06

Zhilan Feng

Drug treatment and invasion of resistant virusAssume Type 1 cost function and all σ the same. Then this relationship holds:

Consider as a function of , 0 1. A resistant strain with resistance can invade the sen

Remark:

sitive strain if and only if

If 0 and 0, then ( ) as 1

Th

= ( )(

us

)

,rt p r s s

r r

r s

r r

σσ

σ σσ

σ σ σ

≤ ≤

≈ ≈ ≈ ≤ ≤•

>

R RR R

R R R

drug treatments act as a selection force for resistant strains Will derive analytic understanding for the case when only a single-drug

therapy with a protease inhibitor is considered, i.e., 0pr•

> and 0.

The case of combined therapy will be explored numericall

yrtr =

•

ASU 2/4/06

Zhilan Feng

Optimal reproductive number/resistance level

Let 0 and 0. Thenp rtr r> =

12 pr

max1

2 prσ 1

σ

Rs

Rr

ASU 2/4/06

Zhilan Feng

Impact of treatment on invasion of resistant strains (rrt=0)

1sopt0.1 smaxs

Rs 4

6Rr max

Rr HcL rp=0.7

0.1 0.8soptsmax 1s

Rs

4Rr max

6

Rr HdL rp=0.8

0.5 1 smaxs

4

6

8Rs

Rr HaL rp=0.4

0.5 1 1.5s

Rs,

4

6

8Rr max

Rr HbL rp=0.5

(a) and (b): No resistant strains can invade

(c) And (d): Strains with resistance levels σmax< σ < 1 can invade

ASU 2/4/06

Zhilan Feng

Impact of treatment on invasion of resistant strains (rrt>0)

0.2 0.4 0.6 0.8 1s

3

4

5

6

Rr

èè

òò

÷÷

rrt=0.5, Rs

rrt=0.5, RrHsLrrt=0.3, Rs

rrt=0.3, RrHsLrrt=0.1, Rs

rrt=0.1, RrHsL

The case of combination therapy with rp=0.6 fixed. Type 1 cost function is used

ASU 2/4/06

Zhilan Feng

Impact of treatment on invasion of resistant strains

0 0.2 0.4 0.6 0.8 1rrt

0

0.2

0.4

0.6

0.8

1

r p

HcL

Rs<1<Rr

Rs<Rr<1

1<Rr<Rs

1<Rs<Rr

HaL

00.25

0.50.75

1rrt 0

0.250.5

0.751

rp0123

Rs, Rr

00.25

0.50.75rrt

HbL

00.25

0.50.75

1rrt 0

0.250.5

0.751

rp0123

Rs

00.25

0.50.75rrt

Reproductive ratio vs treatment efficacy. Type 1 cost function is used

ASU 2/4/06

Zhilan Feng

A different cost function

0.2 0.4 0.6 0.8 1 1.2s

1

2

3

4

5

6

Rr

Rs

Type 1 cost

f=2.5

f=0.9

f=0.5

Qualitative results are similar when Type 2 cost function is used

ASU 2/4/06

Zhilan Feng

Comparison of combination therapies

R > 1

R

R > 1

R

(a) (b) η=5For small η, the entry inhibitor is more effective

rrtre

rrtre

rrtre

rrtre

(c) (d) η=12For large η, the RT inhibitor is more effective

ASU 2/4/06

Zhilan Feng

Conclusion

Age-structured model allows us to incorporate distinct features of RT inhibitors and entry inhibitors which may have very different impact on viral persistence

There exists a threshold drug efficacy rp* below which no resistant strains

can invade. When rp> rp* there is a well defined range of resistance levels for which resistant strains are able to invade

As the drug efficacy increasesthe range of invasion strains, (σmax, 1), increases

the optimal resistance level, σopt, decreases

the optimal viral fitness, R r(σopt), decreases

ASU 2/4/06

Zhilan Feng

Acknowledgements

National Science Foundation DSM-0314575

Influence of anti-viral drug therapy on the evolution of ...kuang/workshop/Feng.pdf · ASU 2/4/06...

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