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

Transcription

1 Influence of anti-viral drug therapy on the evolution of HIV-1 pathogens and Libin Rong Department of Mathematics Purdue University

2 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

3 Human Immunodeficiency Virus (HIV) 1. Attachement Getting in 2. Reverse Transcription From viral RNA to DNA 3. Integration transcription a. Viral DNA joins host DNA b. Making multiple viral RNAs 4. Translation Producing viral proteins 5. Viral Protease Cleaving viral proteins 6. Assembly & Budding Getting out Reverse Transcriptase Inhibitor Protease Inhibitor

4 Suscep tible α H ealthautho rities Quarantin ed Qγ uarantineα Exposλ ed()ts 2 Q E Hospita PQ lized Medica lpractition1εφ ers Presentatio Pr n δ odrom Sympto IP al ms HospitalizPNotQ ed EarlyMeEncoun ME dical ter Progr esionδ ω 1 Hospitalizδ ed3 Respira tory 1 RHP εφ Presentatio nsympto IR ms Hospita RNo lized thp Diagnosi2 δ slatemedencoun ML ical ter Rec overy δ 2 Recov ery Recove red Reδ covery Prog resion Progresio n Infe ction Pr ogresion Waning Recove ry An HIV model with treatment (Perelson and Nelson, 1999) Reverse Transcriptase Inhibitor (Entry Inhibitor?) Protease Inhibitor T(t): uninfected target T cells T *(t): infected T cells V(t): infectious viruses s, b: growth rate of healthy T cells d: per capita death rate of uninfected cells δ: per capita death rate of infected cells c: virus clearance rate k: Infection rate of an uninfected cell N: total viruses produced per infected cell r rt, r p : drug efficacy

5 Suscep tible α H ealthautho rities Quarantin ed Qγ uarantine α Expos λ ed ()ts 2 Q E Hospita PQ lized Medica lpractition 1εφ ers Presentatio Prodrom nsympto IP al ms δ HospitalizPNotQ ed EarlyMe Encoun ME dical ter Progr esion δ ω 1 Hospitalizδ ed 3 Respira tory 1 RHP εφ Presentatio nsympto IR ms Hospita RNo lized thp Diagnosi 2δ slatemed Encoun ML ical ter Rec overy δ 2 Recov ery Recove red Reδ covery Prog resion Progresio n Infe ction Pr ogresion Waning Recove ry An age-structured HIV-1 model (Nelson et al. 2004) (No treatment) T(t): uninfected target T cells at time t T *(a,t): age-density of infected T cells V(t): infectious virus s: recruitment of healthy T cells d: per capita death rate of uninfected cells δ(a): age-specific death rate of infected cells c: virus clearance rate k: Infection rate of an uninfected cell p(a): age-dependent virion production rate

6 Suscep tible α H ealthautho rities Quarantin ed Qγ uarantine α Expos λ ed ()ts 2 Q E Hospita PQ lized Medica lpractition 1εφ ers Presentatio Prodrom nsympto IP al ms δ HospitalizPNotQ ed EarlyMe Encoun ME dical ter Progr esion δ ω 1 Hospitalizδ ed 3 Respira tory 1 RHP εφ Presentatio nsympto IR ms Hospita RNo lized thp Diagnosi 2δ slatemed Encoun ML ical ter Rec overy δ 2 Recov ery Recove red Reδ covery Prog resion Progresio n Infe ction Pr ogresion Waning Recove ry Age-structured model with treatments (Feng and Rong, 2006) β(a): propotion of infected cells of infection age a in the prert phase T ( a, t) = β( a) T ( a, t): * * prert density of infected cells of age a in the prert phase (an RT inhibitor could revert it back to uninfected class) T ( a, t) = ( 1 β( a)) T ( a, t): * * postrt density of infected cells of age a progressed to the postrt phase (a protease inhibitor could hel p) r, r, r : 0 rt p e r ηt ( a, t) da: rt * prert drug efficacy of RT, protease, and entry inhibitors respectively rate at which prert infected cells become uninfected * (1 rp ) TpostRT ( a, t) da: rate at which new virion particl 0 es are produced

7 Suscep tible α H ealthautho rities Quarantin ed Qγ uarantine α Expos λ ed ()ts 2 Q E Hospita PQ lized Medica lpractition 1εφ ers Presentatio Prodrom nsympto IP al ms δ HospitalizPNotQ ed EarlyMe Encoun ME dical ter Progr esion δ ω 1 Hospitalizδ ed 3 Respira tory 1 RHP εφ Presentatio nsympto IR ms Hospita RNo lized thp Diagnosi 2δ slatemed Encoun ML ical ter Rec overy δ 2 Recov ery Recove red Reδ covery Prog resion Progresio n Infe ction Pr ogresion Waning Recove ry An age-structured model with combination therapy (I) (Including reverse transcriptase inhibitors and protease inhibitors) β(a): propotion of infected cells of age a in the prert phase η: conversion factor to non-infected cells by RT inhibitors Remark: RT inhibitors do not reduce the infection rate of target cells (kvt)

8 Suscep tible α H ealthautho rities Quarantin ed Qγ uarantine α Expos λ ed ()ts 2 Q E Hospita PQ lized Medica lpractition 1εφ ers Presentatio Prodrom nsympto IP al ms δ HospitalizPNotQ ed EarlyMe Encoun ME dical ter Progr esion δ ω 1 Hospitalizδ ed 3 Respira tory 1 RHP εφ Presentatio nsympto IR ms Hospita RNo lized thp Diagnosi 2δ slatemed Encoun ML ical ter Rec overy δ 2 Recov ery Recove red Reδ covery Prog resion Progresio n Infe ction Pr ogresion Waning Recove ry An age-structured model with combination therapy (II) (Including entry or fusion inhibitors and protease inhibitors)

9 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 Number of uninfected cells in an infection-free population R skk s 1 k K dc d c 2 = = 2 Virus lifespan

10 Reformulation of the system (I) Let B(t)=kV(t)T(t). Solve for T * (a,t) along the characteristic line: Eliminating T*:

11 Equivalent systems (A) (B)

12 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)

13 A limiting system of (B) Steady-states Infection-free SS: Infected SS:

14 Existence of the infection steady-state E Note: E skk 2 Therefore, V > 0 if and only if skk2 > dc, that is, R > 1 ( R = ) dc

15 Stability of E (infection-free) and E (infected) Result 1: E is a global attractor if R< 1, and it is unstable if R> 1 Result 2: E exists and is locally asymptotically stable if R > 1

16 Proof of Result 1 (fluctuation method) Rewrite the V equation: where T ( ) s and R = d ksk cd 2 Therefore, if R < 1 then V = 0, or lim V( t) = 0. From B( t) = kv( t) T( t), lim B( t) = 0 t s s From the T equation we get T. Thus, T = T = d d t * () W = T + Ttotal Let then s Choose a sequence rn s.t. W( rn) W and W '( rn) 0 as n. Then W and T d s d

17 Proof of Result 2 (assume R >1) Characteristic equation at E : (1) where (Laplace transform of K i (a)) Eq. (1) is equivalent to: or If λ > 0 (real) then Kˆ ( λ) K, Kˆ ( λ) K <1. Hence, λ cannot be an eigenvalue if R > If λ is a complex eigenvalue. First show that Re( λ)<0 when R is close to 1, and then use the continuous dependence on R of the characteristic polynomial for any R > 1

18 Stability of E and E (numerical simulations) Result 1: E is a global attractor if R< 1, and it is unstable if R> 1 Result 2: exists and is locally asymptotically stable if E R > 1 R < 1 R > 1

19 Influence of drug therapy on viral fitness Suppose 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: r rt and r p: drug efficacy for the resistant strain, r i= σ iri (0< σ i <1, i = rt, p) p ( a) : viron production rate of infected cells with resistant viruses Let R and R denote the reproductive ratios for sensitive and resistant strains s r The reproductive ratio of an invading resistant strain (when the sensitive strain is at its infected equilibrium) is Available uninfected T cells =s/(r s d) The resistant strain can establish in the population if and only if R r > R s R r > Burst size 1 or

20 Optimal reproductive ratio Consider the following specific forms of parameter functions: Cost functions Type 1: Type 2: p p = σ p * * = e p * φ(1/ σ 1) *

21 Drug treatment and invasion of resistant virus Assume Type 1 cost function and all σ the same. Then this relationship holds: Consider R = R ( σ ) as a function of σ, 0 σ 1. A resistant strain with r r resistance σ can invade the sen sitive strain if and only if R r ( σ ) > R s Remark: If r 0 and r 0, then R ( σ) σr R as σ 1 rt p r s s Thus, 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., > 0 and r = 0. The case of combined therapy will be explored numerically r p rt

22 Optimal reproductive number/resistance level Let r > 0 and r = 0. Then p rt 1 2r p R r R s σ 1 1 max 2r p σ

23 Impact of treatment on invasion of resistant strains (r rt =0) R r HaL r p =0.4 R s s max s R r HcL r p =0.7 R s, R 8 rmax 6 4 R r HbL r p =0.5 R r HdL r p = s R rmax 6 R s 4 s 0.1 s max 1 s opt R rmax 6 R s s max s opt s (a) and (b): No resistant strains can invade (c) And (d): Strains with resistance levels σ max < σ < 1 can invade

24 Impact of treatment on invasion of resistant strains (r rt >0) R r r rt =0.1, R r HsL 6 5 r rt =0.1, R s r rt =0.3, R r HsL r rt =0.3, R s r rt =0.5, R r HsL 4 è ò ò è r rt =0.5, R s s The case of combination therapy with r p =0.6 fixed. Type 1 cost function is used

25 Impact of treatment on invasion of resistant strains HaL HbL 3 R s,r r r rt r p R s r rt r p 0.25 rp HcL R s <R r <1 R s <1<R r 1<R s <R r 1<R r <R s rrt Reproductive ratio vs treatment efficacy. Type 1 cost function is used

26 A different cost function f= R r f=0.9 f=2.5 Type 1 cost R s s Qualitative results are similar when Type 2 cost function is used

27 Comparison of combination therapies R r rt r e r rt r e R > 1 (a) (b) η=5 For small η, the entry inhibitor is more effective R > 1 R r rt r e r rt r e (c) (d) η=12 For large η, the RT inhibitor is more effective

28 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 r p * below which no resistant strains can invade. When r p > r p * there is a well defined range of resistance levels for which resistant strains are able to invade As the drug efficacy increases the range of invasion strains, (σ max, 1), increases the optimal resistance level, σ opt, decreases the optimal viral fitness, R r (σ opt ), decreases

29 Acknowledgements National Science Foundation DSM