Interpreting declines in HIV prevalence: The impact of spatial
aggregation and migration on expected declines in prevalence
PT Walker*, TB Hallett, PJ White & GP Garnett
Department of Infectious Disease Epidemiology, Imperial College London, UK
*for correspondence: [email protected]
Technical Appendix
Model description
Following others (Garnett and Anderson 1993; Hallett, Aberle-Grasse et al. 2006), our model
was defined by a set of differential equations which were solved using a Runge-Kutta four
step algorithm. The population is split into three geographically independent subpopulations (denoted k), and within each sub-population into two sexes (j), each composed
of four sexual activity groups (i).
The ordinary differential equations describing the model are as follows:
dS (t ) i , j ,k
dt
dX (t ) i , j ,k
dt
dY (t ) i , j ,k
dt
dZ (t ) i , j ,k
dt
dA(t ) i , j ,k
  (t ) i , j ,k  ( (t ) i , j ,k   S (t ) i , j ,k   ) S (t ) i , j ,k   S (t ) i , j ,k
  (t ) i , j ,k S (t ) i , j ,k  ( X (t ) i , j ,k     X ) X (t ) i , j ,k   X (t ) i , j ,k
  X X (t ) i , j ,k  ( Y (t ) i , j ,k     Y )Y (t ) i , j ,k  Y (t ) i , j ,k
  Y Y (t ) i , j ,k  ( Z (t ) i , j ,k     Z ) Z (t ) i , j ,k   Z (t ) i , j ,k
  Z Z (t ) i , j ,k  ( A (t ) i , j ,k     ) A(t ) i , j ,k   A (t ) i , j ,k
dt
N (t ) i , j ,k  S (t ) i , j ,k  X (t ) i , j ,k  Y (t ) i , j ,k  Z (t ) i , j ,k  A(t ) i , j ,k
1
Here, S, X, Y, Z and A denotes the disease states individuals progress through: susceptible (S),
acute HIV-infection (X), incubating infection (Y), pre-AIDS (Z) and full-blown AIDS (A).
N (t ) i , j ,k is the total number of individuals in that group.
 (t ) i , j ,k is the force of infection for susceptible individuals in that sub-population,
1
sex and activity group.
,
1
1
 X Y
and
Z
are the average durations that individual
spend, in each disease state (acute, incubating and pre-AIDS, respectively). The net mortality
1
rate for those with full-blown AIDS is
. 
is the mean time individuals are sexually
active. (t ) i , j ,k is the rate of supply of susceptibles (through individuals ageing and starting
sexual activity) to that sexual-activity group, gender and sub-population.
i , j , k is the rate
of overall rate of in-migration to that sexual activity group, sex and sub-population, and
i , j , k is the per capita rate of out-migration from that group. The fraction of individuals in
each of the four sexual activity groups in each gender in each sub-population is: i , j ,k where

i , j ,k
1.
i
The default model parameter values describing the pathological progress between
disease states are listed in Table S1.
Descriptions of the calculation of force of infection, supply of susceptible and
migration rates follow.
Force of Infection
Individuals in the four sexual-activity groups of the gender j=1 ( ci ,1,k ) are assigned partner
change rates in a geometric series, such that individuals belonging to the sexual activity
groups with higher i indices form more sexual partnerships per year.
i 1
ci ,1, k  (c1,1, k k )(1 i k ) :
for i=2..4
2
c1,1, k is is the rate of partnership formation for those in sexual-activity group i=1);  k
the geometric scaling parameter; and,
k
is
is the power-scaling parameter.
The rates of partner change of individuals of the opposite gender (i.e. j=2) are
determined by that distribution and the number of individuals in each gender in each sexualactivity group:
ci , 2, k 
ci ,1, k ( Ni ,1, k  Ai ,1, k )
Ni , 2, k  Ai , 2, k
The partnership of individuals in the gender j=1 are distributed among the members
of the opposite gender (j=2) in the following way:
(  i ,i ) k
is the proportion of contacts
made by individuals of gender j=1 in the ith sexual activity group in the kth sub-population,
that are formed with individuals in the same sub-population of the opposite gender in the i’th
sexual activity category.


 ( Ni , 2, k  Ai , 2, k )ci , 2, k 
( i ,i  ) k   k i ,i   (1   k )

(
N

A
)
c
i , 2 , k
i , 2 , k 
  i , 2, k
 i

 k is the proportion of contacts which are formed “assortatively”, i.e. between individuals in
equivalent sexual activity classes; and ci, 2,k is the rate of partnership change for individuals
in the opposite gender in the i’th sexual activity-group and N i, 2 ,k is the number of
individuals in that group.  i ,i is a matrix such that:  i ,i =1 if i  i' ; and  i ,i =0 otherwise.
Note that all partnerships are formed between individuals in the same sub-population.
The force of infection is then calculated by multiplying the rates of sexual partner
change for individuals in each activity group, gender and sub-population by the prevalence
of HIV infection (weighted by stage of infection) among their sexual partners and the chance
of transmission per partnership:
 (t )i, j, k
X
Y
Z


 i' X i', j , k  i' Yi', j , k  i' Z i', j , k ) 

 ci, j, k  (i,i  ) k 

N

A
i 


i ', j , k
i ', j , k



3
Here,
 iX
,
 iY and  iZ
are the probabilities of infection per partnership, in
partnerships with individuals with acute HIV-infection, incubating HIV infection and preAIDS, respectively in the ith sexually activity group.
Supply of Susceptibles
Three alternative scenarios can be used in the model to determine the rate of supply of
susceptible to each group,
(t )i , j , k
(see text for more details).
(1) The distribution of risk of individuals starting sex remains constant (“constant”):
 i , j , k (t )   j ,k (t )i , j ,k (0)
Where
i , j ,k (0) is the initial proportion of individuals of sex j in the kth population who
are in the ith sexual activity class and
 j ,k (t ) is the overall rate of individuals starting sex
at time t.
This assumption was used in model simulation, unless it is specified otherwise.
(2) The distribution of risk is continuously adjusted to compensate for any changes in the
distribution of risk in the whole population due to AIDS-mortality (“replace”):
 i , j ,k (t )  N i , j ,k  Ai , j ,k  i , j ,k 
i , j ,k (0)[ j ,k   ( N i , j ,k  Ai , j ,k  i , j ,k )],
i 1
where i , j , k is the difference between the total rate of out-migration and in-migration from
the sub-population k of individuals in that gender and activity group.
(3) The pattern of risk behaviour in those starting sex at time t corresponds to the existing
pattern of risk within the population (“current”).
4
 i , j ,k (t )   j ,k (t )
N i , j ,k (t )
N
i , j ,k
(t )
.
i
The overall rate of re-supply to all sexual activity groups (  j , k ) can be calculated in two
alternative ways:
(1) Constant per-capita growth rate.
 j ,k   (b   ) N i , j ,k
i
This assumption means that in the absence of AIDS, the population grows at a rate equal to
b, and that with AIDS mortality the population grows at a slower rate. The value of b is
selected so that the rate of population growth is never negative. This assumption was used
in model simulation, unless it is specified otherwise.
(2) Constant population growth rate.
 j ,k   ((b   ) N i , j ,k  Ai , j ,k )
i
With this assumption, the population grows at rate b throughout the epidemic, irrespective
of the level of AIDS mortality. In effect, this leads to supply of individuals being increased
when AIDS mortality increases.
Migration
As described above,
istate
, j , k is the per capita rate of out-migration of individuals from that
group with the HIV-status. This is calculated as the product of the average (overall) rate of
migration, multiplied by indicator variables that allow the actual rate out out-migration to
vary according to gender, HIV-status and sexual-activity group:
state gender
 statei , j ,k  irisk
, j ,k i, j ,k i, j ,k 
state
gender
Here,  is the overall rate and  irisk
modify the rate according to risk, j , k ,  i , j , k and  i , j ,k
group, HIV-status and gender, respectively. The  irisk
, j , k modifier variable is calculated as:
5

risk
i , j ,k

RRirisk
,k

 Ni, j ,k 
 risk j
i  RRi,k
N i , j ,k 



i
j
where RRirisk
is the rate of individuals in the ith activity group in the kth sub-population
,k
migrating relative to that of a individuals in the lowest sexual activity class in the same
state
gender
population ( RR1risk
,k  1 by definition). The variables  i , j , k and  i , j , k are calculated in an
analogous way.
The population was divided into two geographically independent populations and the
potential effects of migration between these sub-populations was then explored, with the
net number of migrants leaving one population equal to the net number of migrants
entering the other:
 S i , j ,k  iS, j ,k ' Si , j ,k '
 X i , j ,k  iX, j ,k ' X i , j ,k '
Y i , j ,k  iY, j ,k 'Yi , j ,k '
 Z i , j ,k  iZ, j ,k ' Z i , j ,k '
 Ai , j ,k  iA, j ,k ' Ai , j ,k '
(For ease of reading, we have not written that each of these terms is also a function of time,
t.)
Uncertainty Analysis (Figure 4 in the main text)
The uncertainty analysis was conducted by generating 150 unique sets of parameters
determining the timing, number and characteristics of the migration, by randomly drawing
parameters from distributions designed to describe the uncertainty in each of the
parameters. A standard stratified sampling technique (Blower and Dowlatabadi 1994) was
used to ensure that values from all regions of the probability distributions were used. The
chosen distributions for these parameters are listed in Table S2.
The model was run with each of these sets of parameters and the decline in the
model prevalence was measured and compared to the decline when no migration was
simulated.
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Analysis of model sensitivity to assumption about risk behaviour of AIDS patients
In order to relax the assumption that AIDS patients are not sexually active the force of
infection and contact rate parameters had to be modified thus:
 (t )i, j, k
X
Y
Z
A


 i' X i', j , k  i' Yi', j , k  i' Z i', j , k  i' Ai', j , k ) 

 ci, j, k  (i,i ) k 

N
i 


i ', j , k



where the proportion of the contacts made by individuals of the ith sexual activity category
which are made with individuals of the opposite gender in the i’th is modified thus:


 N i , 2, k ci , 2, k 
( i ,i  ) k   k i ,i   (1   k )

N
c
  i , 2 , k i , 2 , k 
 i

with partnerships balanced across genders thus:
ci , 2, k 
ci ,1, k Ni ,1, k
N i , 2, k
.
Different assumptions about the transmissibility of HIV from a partnership with an individual
with AIDS can then be tested by varying the
i'A parameter.
References
Blower, S. M. and H. Dowlatabadi (1994). "Sensitivity and uncertainty analysis of complex
models of disease transmission: an HIV model, as an example." International
Statistical Review 62(2): 229-243.
Garnett, G. P. and R. M. Anderson (1993). "Factors controlling the spread of HIV in
heterosexual communities in developing countries: patterns of mixing between
different age and sexual activity classes." Philos Trans R Soc Lond B Biol Sci
342(1300): 137-59.
Hallett, T. B., J. Aberle-Grasse, et al. (2006). "Declines in HIV prevalence can be associated
with changing sexual behaviour in Uganda, urban Kenya, Zimbabwe, and urban
Haiti." Sex Transm Infect 82(suppl_1): i1-8.
Pettifor, A. E., M. G. Hudgens, et al. (2007). "Highly efficient HIV transmission to young
women in South Africa." Aids 21(7): 861-5.
Wawer, M. J., R. H. Gray, et al. (2005). "Rates of HIV-1 transmission per coital act, by stage of
HIV-1 infection, in Rakai, Uganda." J Infect Dis 191(9): 1403-9.
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Table S1: Default values for parameters describing progression between disease states and
associated references.
Parameter
 iY
Description
Initial base case value (in specific
category)
Transmission probability per partnership from individuals
0.030 (i=1)
with incubating HIV infection. (Based on recent observational
0.040 (i=2)
data from South Africa (Pettifor, Hudgens et al. 2007)). The
0.075 (i=3)
higher transmission probability from individuals in higher-risk
0.090 (i=4)
groups may represent the possible influence of co-factor
sexually transmitted infections that enhance HIV
transmission.
 iX ,  iZ
Transmission probability for acute and pre-AIDS infectious
stage (per partnership). (Based on observation data from
10 Y
Uganda (Wawer, Gray et al. 2005))
Average time sexually active (years)
40
X
Rate of transition from acute to incubating infection stage
4
Y
Rate of transition from incubating infection stage to pre-AIDS
Z
Rate of transition from pre-AIDS stage to AIDS (year-1)
2
1
Average survival time following onset of AIDS. (years)
0.5
Partner change rate of individuals in the sexually activity
0.8
1

-1
(year )
0.2
-1
stage (year )

c1,1, k




i , j , k
group, i=1. (By default, the same value is used for individuals
in both genders and all sub-populations).
Geometric scaling parameter for the partner change rate for
7
other sexual activity groups.
Power scaling parameter for the partner change rate for
0
other sexual activity groups.
Rate of migration.
0
Pattern of partnership formation (fraction partnerships
0.7
allocated between individuals in the same activity group).
Fraction of individuals assigned to each sexual-activity group
0.001 (i=4)
at start of epidemic.
0.029 (i=3)
0.400 (i=2)
0.570 (i=1)
8
Table S2: Ranges and marginal distribution (representing possible range of values) describing
the timing, number and characteristic of migration.
Variable
Timing of change in migration rate (years after peak
Range
Marginal distribution
(-3.5,3.5)
Uniform
(2,8)
Uniform
Size of donor population relative to recipient
(1,4)
Uniform
Rate of migration (percent of overall population per
(0.5,7.5)
Uniform
2
( ,1.5) , i=2
3
Uniform on logarithmic scale
prevalence in recipient population)
Growth rate of donor population (percent of overall
population per year)
year)
Ratio between proportion of individuals in ‘donor’
population in the
ith
sexual activity, relative to
proportion of individuals in the ‘recipient’
population in the ith sexual activity group.
1
( ,5) , i=3
5
(

(proportion of contacts made non-randomly
1
,20) , i=4
20
(0.1-0.9)
Uniform
2
( ,1.5)
3
Uniform on logarithmic scale
1
( , 4)
4
Uniform on logarithmic scale
1
( , 4)
4
Uniform on logarithmic scale
with members of the opposite sex in the equivalent
sexual activity category in the donor population)
Relative rate of migration for gender k=1 (vs k=2)
individuals.
Relative rate of migration for infected (vs not
infected) individuals.
Relative rate of migration for individuals in higher
sexual activity groups (vs. next-lowest group). (e.g. a
value of 2 implies that relative chances of migration
would be:
i=1; relative rate=1 (definition)
i=2; relative rate=2
i=3; relative rate=4
i=4; relative rate=8).
9
Figure S1
Figure S1: Sensitivity analysis using alternative assumptions on mean survival with HIV
infection. As in Figure 1, black lines shows the effect of alternative type of recruitment to
risk groups as the epidemic matures (Thick Lines- “constant” re-supply assumption; thin
lines- “current” re-supply; dotted lines- “replaced pattern”) using the default survival
assumption (mean survival with HIV 6.25 years). The red lines show the same analysis but
assuming mean survival with HIV is 11 years (all progression rate parameters reduced by
factor 0.59).
10
Figure S2
Figure S2: Simulated prevalence curves using different assumptions about the relative rate
of transmission from individuals with full-blown AIDS. Individuals with AIDS are: not
infectious (thick line); as infectious as those with latent infection (thin line); twice as
infectious as those with latent infection (dashed lines); five-time as infectious as those with
latent infection (dashed and dashed lines); or, ten-times as infectious as those with latent
infection (equivalent to pre-AIDS state).
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