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Modelling changes in HIV prevalence among women attending antenatal
clinics in Uganda
Brian Williams
0
5
10
15
20
1970 1990 2010 2030
Pre
vale
nce
(%)
ANC women
Behaviour change in Uganda
= birth rate
N = S + I
= rate at which new infections occur
= mortality
S I
I N SI /N I
S
The basic model
0
20
40
60
80
100
1970 1990 2010 2030
Pre
vale
nce
(%)
012345678910
Inci
denc
e/M
orta
lity
(%/y
r)
ANC women in Uganda
R0 = 3.3
0
0
-1=70%
R
R
= birth rate
N = S + I
= infection rate
I = Weibull mortality
S I
I N SI /N I
S
0.0
0.2
0.4
0.6
0.8
1.0
0 10 20 30Time (years)
P(s
urv
ivin
g)
Normal (Weibull 2)
Exponential(Weibull 1)
0
20
40
60
80
1970 1990 2010 2030
Pre
vale
nce
(%)
0
2
4
6
8
10
12
Inci
denc
e/M
orta
lity
(%/y
r)
ANC women in Uganda
= birth rate
N = population = e–P
I = Weibull mort.
~
~
S I
I N S I /N I S
0.0
0.2
0.4
0.6
0.8
1.0
0 10 20 30Prevalence (%)
Re
lativ
e t
ran
smis
sio
n
.
–Pe
Heterogeneity in sexual behaviour
0
5
10
15
1970 1990 2010 2030
Pre
vale
nce
(%)
0
1
2
Inci
denc
e/M
orta
lity
(%/y
r)
ANC women in Uganda
0.0
0.2
0.4
0.6
0.8
1.0
1985 1990 1995 2000Year
Re
lativ
e t
ran
smis
sio
n
.
~
S I
I N SI /N I
S ~
= birth rate
N = population = C(t)
I = mortality
~
~
C(t)
Including control
0
5
10
15
1970 1990 2010 2030
Pre
vale
nce
(%)
0
1
2
Inci
denc
e/M
orta
lity
(%/y
r)
ANC women in Uganda
~
S I
I N SI /N I
S *
= birth rate
N = population = e
I = mortality
~
* –M
0.0
0.2
0.4
0.6
0.8
1.0
0 2 4Annual mortality (%)
Re
lativ
e t
ran
smis
sio
n
. –Me
Mortality leads to behaviour change
0
5
10
15
1970 1990 2010 2030
Pre
vale
nce
(%)
0
1
2
Inci
denc
e/M
orta
lity
(%/y
r)
ANC women in Uganda
Nairobi
6 yr
Nunn P et al. Tuberculosis control in the era of HIV. Nat Rev Immunol. 2005 Oct;5(10):819-26.
9.4
1.11.11.0
5.9
2.2
0
2
4
6
8
10
1991-1994 1995-1997 1998-1999
Ann
ual in
ciden
ce (
%)
.HI V- HI V+
TB incidence among gold miners in SACorbett EL Stable incidence rates of tuberculosis (TB) among human immunodeficiency virus (HIV)-negative South African gold miners during a decade of epidemic HIV-associated TB. J Infect Dis. 2003;188: 1156-63.
SS+ Tuberculosis
Prevalence Incidence Disease Duration
(%) (%/yr) (yr)
HIV+ 0.44 (0.02-1.05) 2.87 (1.94-4.25) 0.15 (0.05-0.48)
HIV- 0.55 (0.14–0.95) 0.48 (0.27-0.84) 1.15 (0.48-1.13) DDR = 0.13 (0.09–0.20)
Gold miners in South Africa
We define disease duration as prevalence divided by incidence
Repeat the model 4 times, once for each stage of HIV. Use time series of HIV prevalence to determine incidence. Incidence gives rate at which people enter first stage; overall (Weibull) survival determines rate at which people move to next stage.
TB-HIV model
Williams BG et al. The impact of HIV/AIDS on the control of tuberculosis in India. PNAS 2005 102: 9619-9624.
Impact of interventions on TB cases in KenyaT
B in
cide
nce/
100k
/yr
800
600
400
200
0
.Baseline
ARV 80%
TLTI (6 m)
TLTI (life)
ARV 100%
TB detect.
TB cure
HIV incid
Base line:CDR = 50%CR = 70%Interventions:1% increase
1980 2000 2020 2040 Year
Currie, C. et al. Cost, affordability and cost-effectiveness of strategies to control tuberculosis in countries with high HIV prevalence. BMC, 2005. 5: 130.
Per
cent
Per
cent
HIV
pos
itive
HIV
neg
ativ
e
Williams BG et al. HIV Infection, Antiretroviral Therapy, and CD4+ Cell Count Distributions in African Populations. J Infect Dis, 2006 194: 1450-8.
1,000
2,000
10 20
Time to death (yrs)
Initi
al C
D4/
L
Time to death (yrs)
1,000
2,000
10 20
Initi
al C
D4/
L
Model 1
CD4 decline independent of starting value
Survival determined by pre-infection CD4
Model 2
Survival independent of starting value
CD4 decline determine entirely by starting value and survival distribution
Spatial Epidemiology of HIV
Doubling time = 1 yearLife expectancy = 10 yearsNumber of partners = 4
Proportion of random partners chosen at random = 0 (left hand set) or 10% (right hand set) in the following slides.
Note that in this model migrants have exactly the same sexual behaviour and individual risk as non-migrants.
1. Can we combine spatial/network models with our more conventional continuous time models of HIV?
2. Can we get a better understanding of the host-viral interaction?
3. What are the population level implications of 2?
4. Do we have enough data to explore fully the joint dynamics of TB and HIV?
Questions for all of us
Advice to young epidemiologists
Never make a calculation until you know the answer. Make an estimate before every calculation,
try a simple biological argument (R0, generation time, selection, survival, control). Guess the answer to every puzzle. Courage: no one else needs to know what the guess is. Therefore, make it quickly, by instinct. A right guess reinforces this instinct. A wrong guess brings the refreshment of surprise. In either case, life as an epidemiologist, however long, is more fun.
Plagiarised from E.F. Taylor and J.A. Wheeler Space-time Physics 1963