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
– 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:
333 482 MSM population (12% white, 24% black, 41% hispanic, 23% other)
34 962 MSM (79% white, 4% black, 10% hispanic,
7% other)
• NYC:
• NYC:
• SF:
• NYC:
• 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)
HIV incidence 2.5% / yr in 2002
HIV incidence between 0.9% to 5.5%

Current state of HIV/AIDS in United-States

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
•HIV+ under ARV Rx
•low viral load
•low infectivity
•resumption of sexual activity rate of successful
ARV Rx
•AIDS – ARV Rx therapies failed
•not sexually active

X h

0

...
...

 
...
,
...

X 1


0
X 2

1

X 3 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


...
2
X 4
X 6


...
3
X 5

...
4


...
5
X 7

• The force of HIV infection for unvaccinated individual of kind k (where k in {K}={1,...,128} ) at time t is given by:
 k
0
( )
 k
( )
 j


 h


1,2,3,4,6
 h

 h
X ( ) j j


}

• The force of HIV infection for unvaccinated individual of kind k (where k in {K}={1,...,128} ) at time t is given by:
 k
0
( )
 k
( )
 j


 h


1,2,3,4,6
 h

 h
X ( ) j j


}
• annual rate of partner change for individual of kind k

• The force of HIV infection for unvaccinated individual of kind k (where k in {K}={1,...,128} ) at time t is given by:
 k
0
( )
 k
( )
 j


 h


1,2,3,4,6
 h

 h
X ( ) j j


}
• mixing matrix elements – probability that individual of kind k chooses a partner of kind j ( more later )

• The force of HIV infection for unvaccinated individual of kind k (where k in {K}={1,...,128} ) at time t is given by:
 k
0
( )
 k
( )
 j


 h


1,2,3,4,6
 h

 h
X ( ) j j


}
• given that individual of kind k chose a partner of kind j, this is probability that the partner is infected in HIV state h. NA j
(t) is number of sexually active individuals of kind j.

• The force of HIV infection for unvaccinated individual of kind k (where k in {K}={1,...,128} ) at time t is given by:
 k
0
( )
 k
( )
 j


 h


1,2,3,4,6
 h

 h
X ( ) j j


}
• per-partnership probability of HIV transmission from partner kind j infected in state h to susceptible individual kind k

• 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:
R ( ) event
 
0 1
• Events occur with probability:
P event
( )

R event
( ) / ( )
• where
 events


1..1792
R ( ) events

• 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
N m
 i i ij

N m
 j j ji
• where N i is the number of individuals in class i , m for class i (rate of new partner acquisition), and φ ij i is the mean activity level is the mixing matrix element between class i and j (the probability that i chooses j )
• φ ij is given by:
 ij


W N m ij j j
W N m ik k k
• where W ij are mixing preferences

• so...
N i m i
 k
W ij
W ik
N
N j m k j m k

N j m j
W
 k
W ji jk
N i
N k m i m k
• in fact this doesnt balance perfectly, the difference between the two is given by
B ij ,diff

W ji
 k
W N m ik k k

W ij
 k
W N m jk k k
• with time the W ij and m partner change (m ij change, but we would like to keep them close to values corresponding to true population mixing (W ij,target
) and rates of ij,target
). So there are two more difference equations:

W ij

W ij

W target, ij
 m i
 m i
 m target , i

• so...
N i m i
 k
W ij
W ik
N
N j m k j m k

N j m j
W
 k
W ji jk
N i
N k m i m k
• in fact this doesnt balance perfectly, the difference between the two is given by
B ij ,diff

W ji
 k
W N m ik k k

W ij
 k
W N m jk k k
128 2 equations; with 128x2+128 2 unknowns !
• with time the W ij and m partner change (m ij change, but we would like to keep them close to values corresponding to true population mixing (W ij,target
) and rates of ij,target
). So there are two more difference equations:

W ij

W ij

W target, ij
 m i
 m i
 m target , i
128 2 equations;
128 2 unknowns !
128 equations; 128 unknowns !

• ... so we need to choose W ij
F cost

W , m

 and m ij which minimise the cost function:
F
B

F

W

F
 m
• where
F
B
   
C
 
2
F

W
 
C



W ij

2
• the C
X,ij function
F
 m
 
C

  m i

2 i are weights to give preference one or the other terms of the cost
• how? Simplex minimisation algorithm, conjugate gradient algorithm
(Numerical Recipes)

• ... so we need to choose W ij
F cost

W , m

 and m ij which minimise the cost function:
F
B

F

W

F
 m
• where
F
B
   
C
 
2
F

W
 
C



W ij

2
• the C
X,ij function
F
 m
 
C

  m i

2 i are weights to give preference one or the other terms of the cost
• how? Simplex minimisation algorithm, conjugate gradient algorithm
(Numerical Recipes)
•
Seems to be working...

vaccination
(flow from X to
Y)
Y 1 vaccination
(1
 
 rs
0
...
)

(flow from X to
Y)
Y 2
(1
 
 pr
1
...
)

Y 3
(1
 
 pr
2
)

...
Y 4
(1
  pr
)

 3
...
Y 5
Y h ( ) 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 reduction in susceptibility of HIVvaccinees, specific for age a, sex k, race r, sexual activity class i, and condom class c
(1
  rp
)
 
..
reduction in rate of progression between different HIV infection states due to vaccination
(1
  tr
)
  ij h 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)

...
4
Y 6 vaccination
(flow from X to
Y)
(1
  pr
)


5
...
Y 7
NOTE1: flow from Y to X (not show) depends on assumptions of vaccine waning

Vaccinated individuals can acquire infection
μ X μ Y
Λ (1-p VE take
)
σ Y
Susceptible
(X)
Infected
(Y)
β X(Y/N)+ β R
I
R sex
X (W/N)
Λ p VE take
1/D vp
Clinical AIDS
Case (A)
R s
R sex
V(Y/N)+ β R
I
R s
R sex
2 V(W/N)
Vaccinated
Vaccinated but
(V)
Infected (W)
I u
/I v
σ W
μ V
R s
=Reduced susceptibility
R
I
=Reduced infectivity
μ W
I u
/I v
= Slower disease progression
Department of Infectious Disease Epidemiology, Faculty of Medicine, Imperial College London
( μ + α )A

Susceptibles
Vaccinated susceptibles
Infected unvaccinateds
Infected vaccinateds
AIDS cases dX / dt
 
( 1
 p )
 uX dV / dt
  p
 uV
 
X (
Y
N
  rqV (
Y
N

 rs
W
N
)
 
V sr
W
N
)
 
V dY / dt
 
X (
Y
N
 rs
W
N
)
 
Y
 
Y dW dA /
/ dt dt



 rqV
Y

Y
(

N
W


(
W rs

N

)


)

W
A
 
W
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
(y v
*>y*, N v
*<N*)
Vaccination & good outcome of increased HIV-1 infection but increased population size
(y v
*>y*, N v
*>N*)
Vaccination & good outcome of decreased HIV-1 infection
& increased population size
(y v
*<y*, N v
*>N*)
Trivial state
No persistence of HIV-1 infection
(R o
<1, y*=0)
Vaccination & eradication of HIV-1 infection
(y v
*=0)
No vaccination, p=0
Endemic persistence of HIV-1 infection
(R o
>1, y*>0)
Department of Infectious Disease Epidemiology, Faculty of Medicine, Imperial College London

p c

[ 1

VE
1
/ take
R
[
0
1
][

1
= critical fraction of each cohort immunised

R / R ]
0 v 0
Basic reproductive rate among vaccinees
Ro v

•p c
•R
0
= basic reproductive number for unvaccinateds
L

/

D
R s vp

R
I
]

R sex
2 
I v
•R
0v
= basic reproductive numbers for vaccinateds
•L =sexual life expectancy,
•D vp
= duration of vaccine protection
•VE take
=% of protected people/ vaccinated
•R s
= Reduced susceptibility/ protected (1-VE )

For prevalence to decrease & population size to increase
R
0 v

R
0 which can be expressed as:
1

R s

R
I
 2
R sex

I v
/ I u
Reduced susceptibility
Reduced infectivity
Increase in infectiousness period
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 (R
I
) and reduced susceptibility (Rs) of vaccinated individuals, as a function of the ratio of the incubation period of AIDS in unvaccinateds and vaccinateds, I u
/I v
, and the increase in risk behavior increases in vaccinated individuals, Rr (r=1 denotes no change).
1

R s

R
I
 2
R sex

I v
/ I u
Vaccination & good outcome of decreased HIV-1 infection
& increased population size
(y v
*<y*, N v
*>N*)
1
0.6
/ I v 0.2
1
Vaccination & perverse outcome of increased HIV-1 infection
& decreased population size
(y v
*>y*, N v
*<N*)
Vaccination & good outcome of increased HIV-1 infection but increased population size
(y v
*>y*, N v
*>N*)
1.2
1.4
1.8
1
2
0.2
0.1
0
0.6
0.5
0.4
0.3
0.9
0.8
0.7
R
I
Rs
Reduced infectivity
Reduced susceptibility
1.6
Increase in risk R sex
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
)
R
0
=2.0, R
0v
=1.0, R r
=1.0 (no increased risk behavior), VE take
=100%, D vp
=10.0yrs.
y* v
/y* u
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0 y v
*
   p

(

(


 

)( 1

  qr

)
 p
0
0.5
0.375
0.25
0.125
R s
R
I where R s
= 0.5
Reducced susceptibility
Reduced infectivity

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 R s
R
I
, which denotes the product of the proportional reduction in infectiousness times the reduction in susceptibility.
R
0
=2.0, R
0v
=1.0, R r
=1.0 (no increased risk behavior), VE take
=1.0, D vp
=10.0yrs.
1
0.8
y* v
/y* u
0.6
0.4
0.2
0 y v
*
   p

(

(


 

)( 1
 
)

 qr

) p
0
0.5
0.375
0.25
0.125
R
I
R s where
R
I
= 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*
R
0
=R
0v
, r=1.0 (no increased risk behavior), VE take
=100%, D vp
=10.
1.6
Vaccination & good outcome of increased HIV-1 infection but increased population size
(y v
*>y*, N v
*>N*) qsr
2 
I u
I v
 qr ( R
0

1 )( 1
 
)
1.5
1.4
Population ratio N* v
/N*
1.3
1.2
1.1
1 p
0.66
0.33
0.50
R
I
R s where R s
= 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?