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Transcript
Challenges of modeling the HIV
epidemic in the United-States
17-20 May 2004
DIMACS, Rutgers University
Kamal Desai MSc PhD
Dept of Infectious Disease Epidemiology
Page no./ref
© Imperial College London
‘Modeling the HIV Epidemic in the US and the Potential
Impact of HIV Vaccines and other Biomedical and
Behavioural Interventions’
collaborators from CDC
• Marta Ackers
• Hu, Dale
• Irene Hall
• John Glasser
• Peter McElroy
• Mark Messonier
• Stephanie Sansom
• Sada Soorapanth
from Futures Group
• C Bickert
• Bill McGreevey
• John Stover
from FDA
• Steven A. Anderson
from Imperial College
• Kamal Desai
• Geoff Garnett
• Marie-Claude Boily
also
• Robert Poulin
Background
• Not a single HIV epidemic in the United-States but several overlapping
ones, characterised by different modes of transmission (MSM,
heterosexual, IDU), behavioural and racial groups, sex and geography.
This heterogeneity is actually necessary to fuel the epidemic.
Background
• Not a single HIV epidemic in the United-States but several overlapping
ones, characterised by different modes of transmission (MSM,
heterosexual, IDU), behavioural and racial groups, sex and geography.
This heterogeneity is actually necessary to fuel the epidemic.
• Challenge is to develop dynamic mathematical models of HIV
transmission which encompass enough complexity (heterogeneities)
to accurately capture past, current and future HIV incidence and
prevalence trends for relevant risk groups to then be able to explore
the potential impact of HIV vaccines.
Background
• Not a single HIV epidemic in the United-States but several overlapping
ones, characterised by different modes of transmission (MSM,
heterosexual, IDU), behavioural and racial groups, sex and geography.
This heterogeneity is actually necessary to fuel the epidemic.
• Challenge is to develop dynamic mathematical models of HIV
transmission which encompass enough complexity (heterogeneities)
to accurately capture past, current and future HIV incidence and
prevalence trends for relevant risk groups to then be able to explore
the potential impact of HIV vaccines.
• Model results should make important contribution to public health
understanding of factors underlying transmission and inform the best
design of HIV vaccination strategies, useful to CDC planning and
decision-making.
Background
• Not a single HIV epidemic in the United-States but several overlapping
ones, characterised by different modes of transmission (MSM,
heterosexual, IDU), behavioural and racial groups, sex and geography.
This heterogeneity is actually necessary to fuel the epidemic.
• Challenge is to develop dynamic mathematical models of HIV
transmission which encompass enough complexity (heterogeneities)
to accurately capture past, current and future HIV incidence and
prevalence trends for relevant risk groups to then be able to explore
the potential impact of HIV vaccines.
• Model results should make important contribution to public health
understanding of factors underlying transmission and inform the best
design of HIV vaccination strategies, useful to CDC planning and
decision-making.
• CDC goal to reduce number of new HIV infections from 40,000 to
20,000 per year by 2005 (HIV/AIDS Surveillance Report to 2002 –
CDC)
Background
• We are concerned about how to best use a
HIV vaccine before it becomes available:
– How does the nature of vaccination programmes
(coverage, target groups, frequency of revaccination)
depend on the properties of the vaccine (efficacy,
duration, cross-strain reactivity, etc)?
– What is the epidemiological benefit (effectiveness)?
– What are the cost implications?
– How beneficial is vaccination when combined with other
prevention interventions?
– etc.
Current state of HIV/AIDS in United-States
HIV/AIDS diagnoses based on surveillance data from 30 states:
• Over 26,464 new diagnoses of HIV/AIDS in 2002, a 3.2% increase since 2001
• Black Americans represented 54% of all new diagnoses in 2002
• 25-34 year olds represented 28% of new diagnoses of HIV/AIDS
• By exposure category, 44% of new infections occurred in MSM, 35% through heterosexual
contact
Deaths (30 states):
• 16,371 deaths in 2002, 14% decline since 1998 (19,005 deaths)
Persons living with HIV/AIDS (30 states):
• 281,931 persons known to be living with HIV/AIDS
• 42% in age group 35-44
• by race, 50% black, 38% white, 10% hispanic
• 73% men
• of males aged >13yrs living with HIV/AIDS, 61% are MSM, 17% IDU
• of females ages > 13yrs, 72% were exposed through heterosexual contacct, 26% through
injecting drug use
UNAIDS estimates:
• 890,000 adults living with HIV/AIDS
• 180,000 women
• adult prevalence 0.6%
Current state of HIV/AIDS in United-States
Comparison of MSM in New-York City and San Francisco:
• NYC:
• SF:
• NYC:
• NYC:
333 482 MSM population (12% white, 24% black, 41%
hispanic, 23% other)
34 962 MSM (79% white, 4% black, 10% hispanic,
7% other)
• SF:
54 900 MSM living with HIV/AIDS (16.5% prevalence)
prevalence by race, 3.1% white, 18.4% black, 8.8%
hispanic
12 786 MSM lwha (36.0 % prevalence)
• NYC:
• SF:
HIV incidence 2.5% / yr in 2002
HIV incidence between 0.9% to 5.5%
Current state of HIV/AIDS in United-States
• So HIV epidemic is very heterogeneous with
respect to locale, sex, race and exposure
group.
Open questions...
• How sophisticated does a model of HIV transmission
dynamics need to be to provide truly informative data
to contribute to public health understanding and
decision-making?
• I believe we should strive for quantitative accuracy
(versus qualitative) in estimates of cases preventable
in different settings, under different vaccine properties
and intervention strategies. How feasible is this
really?
Methodology
• We are developing a stochastic dynamic
mathematical model simulating HIV infection in
heterosexual men and women and MSM men,
progression to AIDS, ARV Rx and vaccination.
• model stratified by:
– age (4 groups)
– sex (2)
– sexual activity classes (4)
– race (2)
– condom use (2)
==> implies 128 kinds of individuals.
• also migration, birth, death
birth
death
•HIV- susceptibles
•4 age groups (a=1..4)
•2 sexes (k=1..2)
•2 races (r=1..2)
•4 sexual activity class
(i=1..4)
•2 condom groups (c=1..2)
•HIV+ infecteds
•high viral load
•high transmissibility
•high CD4
•HIV+ infecteds
•low viral load
•low
transmissibility
•declining CD4
•HIV+ infecteds
•high viral load
•high transmissibility
•low CD4
•AIDS
•eligible for ARV Rx
•not sexually active
rate of
successful
ARV Rx
•HIV+ under ARV Rx
•low viral load
•low infectivity
•resumption of sexual
activity
•AIDS – ARV Rx
therapies failed
•not sexually active
 a , k , r ,i , c
X a1,k ,r,i,c


a0,k ,r ,i ,c
X a2,k ,r,i,c
 a1,k ,r ,i ,c
X a3,k ,r,i,c


 ...2
 ...3
X a4,k ,r,i,c
 
X a5,k,r,i,c
 ...4
X
h
a,k ,r ,i,c

(t)
0
a ,k ,r ,i ,c
(t )
 ......
... ,  ...
 a ,k ,r ,i ,c (t )
number of unvaccinated individuals at time t of age a,
sex k, race r, sexual activity class i, and condom class c
in HIV infection stage h (h=1 uninfected, h=2..4 are
HIV+, h=5 AIDS, h=6 HIV under ART, h=7 AIDS ART
failed) at time t
HIV infection rate of unvaccinated, specific for age a,
sex k, race r, sexual activity class i, and condom class c
at time t
rate of progression of unvaccinated between different
HIV infection states for age a, sex k, race r, sexual
activity class i, and condom class c
death rate (non-AIDS and AIDS)
entry into sexually active population
X a6,k ,r,i,c
 ...5

X a7,k,r,i,c
 
Force of infection
• The force of HIV infection for unvaccinated individual of
kind k (where k in {K}={1,...,128} ) at time t is given by:

k0 (t )  mk (t )   k , j (t )
j

h 1,2,3,4,6
 jh,k
 X hj (t ) 


 NAj (t ) 
}
Force of infection
• The force of HIV infection for unvaccinated individual of
kind k (where k in {K}={1,...,128} ) at time t is given by:

k0 (t )  mk (t )   k , j (t )
j

h 1,2,3,4,6
 jh,k
 X hj (t ) 


 NAj (t ) 
• annual rate of partner change for individual of kind k
}
Force of infection
• The force of HIV infection for unvaccinated individual of
kind k (where k in {K}={1,...,128} ) at time t is given by:

k0 (t )  mk (t )   k , j (t )
j

h 1,2,3,4,6
 jh,k
 X hj (t ) 


 NAj (t ) 
• mixing matrix elements – probability that individual of
kind k chooses a partner of kind j (more later)
}
Force of infection
• The force of HIV infection for unvaccinated individual of
kind k (where k in {K}={1,...,128} ) at time t is given by:

k0 (t )  mk (t )   k , j (t )
j

h 1,2,3,4,6
 jh,k
 X hj (t ) 


 NAj (t ) 
}
• given that individual of kind k chose a partner of kind j,
this is probability that the partner is infected in HIV state
h. NAj(t) is number of sexually active individuals of kind j.
Force of infection
• The force of HIV infection for unvaccinated individual of
kind k (where k in {K}={1,...,128} ) at time t is given by:

k0 (t )  mk (t )   k , j (t )
j

h 1,2,3,4,6
 jh,k
 X hj (t ) 


 NAj (t ) 
}
• per-partnership probability of HIV transmission from
partner kind j infected in state h to susceptible individual
kind k
Methodology
•
•
We have 14 possible events (flows) and 128 kinds of individuals  1792
possible events in absence of vaccination, with varying probability of occuring
(excluding ageing which is not stochastic and migration)
An event vector corresponding to the probabilities of the 1792 events is
constructed. For example, the infection event of susceptible individual of age
a, sex k, race r, sexual activity i, condom c is:
Revent (t )  
0
a ,k ,r ,i ,c
•
(t ) X
1
a ,k ,r ,i ,c
Events occur with probability:
Pevent (t )  Revent (t ) / S (t )
•
where
S (t ) 

events 1..1792
Revents (t )
(t )
Methodology
• The event chosen at each step of the
sequence is determined by a random number
generator (see Numerical Recipes)
• The time between any two events is
exponentially distributed with mean 1/S(t)
Balancing mixing equations
•
•
In order for the model to remain valid, the mixing must balance as the
simulation is going on. That is, the number of partnerships between
individuals of kind i and kind j must be equal to the number of partnerships
between individuals j and i.
the mixing balance equation is
Ni miij  N j m j ji
•
•
where Ni is the number of individuals in class i, mi is the mean activity level
for class i (rate of new partner acquisition), and φij is the mixing matrix
element between class i and j (the probability that i chooses j)
φij is given by:
ij 
Wij N j m j
W
kK
•
where Wij are mixing preferences
ik
N k mk
Balancing mixing equations
•
so...
N i mi
Wij N j m j
W
ik
N k mk
k
•
 N jmj
W ji N i mi
W
jk
N k mk
k
in fact this doesnt balance perfectly, the difference between the two is given
by
Bij ,diff  W ji Wik N k mk  Wij W jk N k mk
k
•
k
with time the Wij and mij change, but we would like to keep them close to
values corresponding to true population mixing (Wij,target) and rates of
partner change (mij,target). So there are two more difference equations:
Wij  Wij  Wtarget,ij
mi  mi  mtarget,i
Balancing mixing equations
•
so...
N i mi
Wij N j m j
W
ik
N k mk
k
•
 N jmj
W ji N i mi
W
jk
N k mk
k
in fact this doesnt balance perfectly, the difference between the two is given
by
Bij ,diff  W ji Wik N k mk  Wij W jk N k m
k
•
k
1282 equations;
2
k with 128x2+128
unknowns !
with time the Wij and mij change, but we would like to keep them close to
values corresponding to true population mixing (Wij,target) and rates of
partner change (mij,target). So there are two more difference equations:
Wij  Wij  Wtarget,ij
mi  mi  mtarget,i
1282 equations;
1282 unknowns !
128 equations; 128
unknowns !
Balancing mixing equations
•
... so we need to choose Wij and mij which minimise the cost function:
Fcost W, m  FB  FW  Fm
•
where
FB  W, m    CB ,ij  Bij 
2
i, j
FW  W    CW ,ij  Wij 
2
i, j
Fm  m    Cm,i  mi 
•
•
i
2
the CX,ij are weights to give preference one or the other terms of the cost
function
how? Simplex minimisation algorithm, conjugate gradient algorithm
(Numerical Recipes)
Balancing mixing equations
•
... so we need to choose Wij and mij which minimise the cost function:
Fcost W, m  FB  FW  Fm
•
where
FB  W, m    CB ,ij  Bij 
2
i, j
FW  W    CW ,ij  Wij 
2
i, j
Fm  m    Cm,i  mi 
•
•
•
i
2
the CX,ij are weights to give preference one or the other terms of the cost
function
how? Simplex minimisation algorithm, conjugate gradient algorithm
(Numerical Recipes)
Seems to be working...
What about vaccination?
vaccination
(flow from X to
Y)
vaccination
(flow from X to
Y)
(1   rs ) 

0
...
Ya1,k,r,i,c
Ya2,k,r,i,c

1
...
(1   pr ) 
(1   pr ) 
(1   pr ) 
Ya3,k,r,i,c
 ...2
Ya4,k,r,i,c
 ...3
Ya5,k,r,i,c
 ...4
h
a,k ,r,i,c
Y
(t)
number of vaccinated individuals at time t of age a, sex
k, race r, sexual activity class i, and condom class c in
HIV infection stage h (h=1 uninfected, h=2..4 are HIV+,
h=5 AIDS, h=6 HIV under ART, h=7 AIDS ART failed)
(1   rs )  
0
a ,k ,r ,i ,c
(1   rp )   ..
(1  tr )  
h
ij
(t )
reduction in susceptibility of HIVvaccinees, specific for age a, sex k, race
r, sexual activity class i, and condom
class c
Ya7,k,r,i,c
Ya6,k,r,i,c
(1   pr ) 
vaccination
(flow from X to
Y)
 ...5
reduction in rate of progression between
different HIV infection states due to
vaccination
reduction in transmissibility from a vaccinated
HIV + individual of kind i and infection state h
to a partner of kind j. (This term incorporated
inside lambda)
NOTE1: flow from Y to X (not show)
depends on assumptions of vaccine
waning
Results from a simpler model
work by :
Roy M Anderson
Marie-Claude Boily
Mathew Hansen
(NOTE: this was not presented at DIMACS)
Simple model - Imperfect vaccine
Vaccinated individuals can acquire infection
μX
μY
Λ(1-p VEtake)
Susceptible
(X)
Infected
(Y)
βX(Y/N)+β RI Rsex X (W/N)
1/Dvp
Λ p VEtake
Rs RsexV(Y/N)+β RI Rs Rsex2 V(W/N)
Vaccinated
(V)
μV
Rs =Reduced
susceptibility
Vaccinated
but
Infected (W)
RI =Reduced
infectivity
μW
σY
(μ+α)A
Clinical AIDS
Case (A)
Iu/Iv σ W
Iu/Iv= Slower
disease
progression
Department of Infectious Disease Epidemiology, Faculty of Medicine, Imperial College London
Imperfect vaccine
Susceptibles
Vaccinated
susceptibles
Infected
unvaccinateds
Infected
vaccinateds
AIDS cases
W
Y
dX / dt   (1  p )  uX  X (  rs )  V
N
N
W
Y
dV / dt  p  uV  rqV (  sr )  V
N
N
W
Y
dY / dt  X (  rs )  Y  Y
N
N
W
Y
dW / dt  rqV (  rs )  W  W
N
N
dA / dt  Y  W  (    ) A
Department of Infectious Disease Epidemiology, Faculty of Medicine, Imperial College London
Six possible scenarios depending on different
epidemiological and vaccine property related parameter
combinations
Vaccination & perverse outcome
of increased HIV-1 infection
& decreased population size
(yv*>y*, Nv*<N*)
Vaccination & good
outcome
of increased HIV-1
infection
but increased population
size
(yv*>y*, Nv*>N*)
Vaccination & good outcome
of decreased HIV-1 infection
& increased population size
(yv*<y*, Nv*>N*)
Trivial state
No persistence
of HIV-1 infection
(Ro<1, y*=0)
Vaccination & eradication
of HIV-1 infection
(yv*=0)
No vaccination, p=0
Endemic persistence
of HIV-1 infection
(Ro>1, y*>0)
Department of Infectious Disease Epidemiology, Faculty of Medicine, Imperial College London
Eradication fraction – imperfect
vaccines
pc 
[1  1 / R0 ][1  L / Dvp ]
VEtake[1  R0v / R0 ]
Basic reproductive rate among vaccine
Rov    Rs  RI  Rsex  I v
2
•pc = critical fraction of each cohort immunised
•R0 = basic reproductive number for unvaccinateds
•R0v = basic reproductive numbers for vaccinateds
•L =sexual life expectancy,
•Dvp = duration of vaccine protection
•VEtake =% of protected people/ vaccinated
•Rs= Reduced
susceptibility/
protected
) Imperial College London
Department
of Infectious Disease
Epidemiology,(1-VE
Faculty of
Medicine,
degree
Conditions for beneficial
outcome
For prevalence to decrease & population size to increase
R0 v  R0
which can be expressed as:
1  Rs  RI  R  I v / I u
2
sex
Reduced susceptibility
Increase in infectiousness period
Reduced infectivity
Increase in risky sex
Department of Infectious Disease Epidemiology, Faculty of Medicine, Imperial College London
Boundary relationship between the product of the degree of reduced infectiouness (RI)
and reduced susceptibility (Rs) of vaccinated individuals, as a function of the ratio of the
incubation period of AIDS in unvaccinateds and vaccinateds, Iu/Iv, and the increase in risk
behavior increases in vaccinated individuals, Rr (r=1 denotes no change).
Vaccination & perverse outcome
of increased HIV-1 infection
& decreased population size
(yv*>y*, Nv*<N*)
Vaccination & good
outcome
of increased HIV-1
infection
but increased population
size
(yv*>y*, Nv*>N*)
2
1  Rs  RI  Rsex
 Iv / Iu
Vaccination & good outcome
of decreased HIV-1 infection
& increased population size
(yv*<y*, Nv*>N*)
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
1
RI Rs
Reduced infectivity
0.2
Ratio I / I
u v
Reduced susceptibility
0.1
0.6
0
0.2
1
1.2
1.4
1.6
1.8
2
Increase in risk Rsex
Department of Infectious Disease Epidemiology, Faculty of Medicine, Imperial College London
Equilibrium relationship between the ratio of HIV-1 prevalence
after cohort vaccination divided by that before (y*v/y*u)
R0=2.0, R0v=1.0, Rr=1.0 (no increased risk behavior), VEtake=100%, Dvp=10.0yrs.
1
0.9
0.8
0.7
0.6
y* v /y* u 0.5
0.4
0.3
0.2
0.5
0.375
0.25
0.125
0.1
0.9
0.6
0.45
0.3
0.75
yv *    p
0.15
0
0
0
Rs RI where Rs = 0.5
Reducced susceptibility
(    )(1   )
p
Reduced infectivity
of Infectious
Disease Epidemiology, Faculty of Medicine, Imperial College London
( Department
   qr
)
Equilibrium relationship between the ratio of HIV-1 prevalence after cohort
vaccination divided by that before (y*v/y*u), with the fraction p of each cohort
vaccinated and the ratio Rs RI, which denotes the product of the proportional
reduction in infectiousness times the reduction in susceptibility.
R0=2.0, R0v=1.0, Rr=1.0 (no increased risk behavior), VEtake=1.0, Dvp=10.0yrs.
1
0.8
0.6
y* v /y* u
0.4
0.2
0.5
0.375
0.25
0
0.9
0.6
0.75
p
0.45
0.3
(    )(1   )
yv *    p
(     qr )
0.125
0.15
0
0
RI Rs where RI
= 0.5
Reducced infectivity
Reduced susceptibility
Department of Infectious Disease Epidemiology, Faculty of Medicine, Imperial College London
Ratio of equilibrium population size after and before cohort
vaccination, N*v/N*
R0=R0v, r=1.0 (no increased risk behavior), VEtake=100%, Dvp=10.
Vaccination & good
outcome
of increased HIV-1
infection
but increased population
size
(yv*>y*, Nv*>N*)
1.6
1.5
qsr 2 
Iu
 qr ( R0  1)(1   )
Iv
1.4
Population
1.3
ratio N*v/N*
1.2
1.1
0.9
0.75
0.50
0.6
0.3
p
0.45
0.66
0.15
0
1
0.33
RIqs
Rs
where Rs= 0.666
Department of Infectious Disease Epidemiology, Faculty of Medicine, Imperial College London
Issues for discussion
• How sophisticated does a model of HIV transmission dynamics need
to be to provide truly informative data to contribute to public health
understanding and decision-making?
• What are others’ experiences simulating vaccination strategies for
highly stratified model populations (age, risk groups, etc)?
• Is it ambitious to try to ‘fit’ model prevalence and incidence to each
strata of individual in a chosen setting (say New-York City)? Better to
work with fewer ‘kinds’ of individuals?
• This model has sequential partnership formation only. What would be
the effect of concurrent partnerships to validity of model outputs?