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Supplementary Material
Universal Access to HIV treatment Versus Universal ‘Test and Treat’:
transmission, drug resistance
& treatment costs
Bradley G. Wagner1, Sally Blower1#
Affiliations: 1David Geffen School of Medicine, University of California Los Angeles, Los
Angeles, California.
#
Corresponding author: [email protected]
1
1. HIV transmission model structure
To predict the epidemiological impact of implementing the universal ‘test and teat’
strategy (T&T) and achieving universal access to treatment we used a compartmental, HIV
transmission model. The model includes primary, chronic and symptomatic stages of HIV
infection, HIV treatment (initiated at different stages of infection), as well as acquired and
transmitted drug resistance. The flow diagram showing the structure of the model is given in
Figure S1B. This model is a modified version of that of Granich et al. [1]. The model is specified
by the following set of differential equations:


3
dS
S
    iI I i  iA Ai  iR Ri
 S
dt
N
i 1
dI1
dt
dI 2
dt
dI 3
dt
dA1
dt
dA2
dt
dA3
dt
dR1
dt
dR2
dt
dR3
dt
3

  iI I i  iA Ai
i 1
NS   I
1 1
  A1  ( 1   )I1
 (1)
(2)
 1 I1   2 I 2   A2  2   I 2
(3)
 2 I 2   3 I 3   A3  3   I 3
(4)


  1 I1   R     1   A1
(5)


(6)


(7)
  2 I 2   1 A1   R     2   A2
  3 I 3   2 A2   R     3   A3
3
  iR Ri
i 1
S
  R A1  ( 1   )R1
N
(8)
  R A2   1 R1  ( 2   )R2
(9)
  R A3   2 R2  ( 3   )R3
(10)
Here S represents the size of the uninfected population and N represents the total population
size. The variables I1, I2 and I3 denote the number of HIV-infected individuals in primary, chronic
and symptomatic stages of infection. A1, A2 and A3 represent the number of individuals receiving
first-line treatment regimens. Individuals initiating treatment during primary infection progress to
treatment stage A1, while those who initiate during chronic infection progress to A2. Individuals
in the symptomatic stage of infection enter the treatment stage A3. The variables R1, R2 and R3
2
denote the number of individuals infected with drug-resistant strains. For the T&T strategy, the
sum R1+ R2 +R3 gives the number of individuals receiving second-line therapy (see Table S1 for
parameterization). For achieving universal access to treatment, R3 represents the number of
individuals receiving second-line treatment regimens while R1 and R2 represent the number of
individuals with transmitted resistance who have not yet begun treatment (see Table S1 for
parameterization).
Individuals enter the sexually active population at a rate  and die at a per capita rate .
Treatment rates in primary, chronic, and symptomatic stages of infections are represented by 1,
2 and 3. Untreated individuals progress from primary to chronic infection at a rate 1 and from
chronic infection to symptomatic infection at a rate p2. Symptomatic individuals die from AIDS at
a rate 3. Individuals receiving treatment discontinue or interrupt treatment at a rate . Treated
individuals who initiate treatment during primary infection progress to the second treatment
stage at a rate 1, and those in the second stage to the third at a rate 2. Once in the third and
final treatment stage individuals die at a rate 3. Treated individuals acquire drug resistance at
an average rate R. Drug-resistant individuals progress through the stages R1, R2 and R3 at
rates 1, 2 and 3 respectively. The infectivity per unit time of individuals in stages Ii, Ai and Ri
is given, respectively, by  iI,  iA and iR. For specific parameterizations for achieving universal
access to treatment and implementing the T&T strategy for South Africa see Table S1.
2. Computation of the Control Reproduction Number (Rc)
We computed the Control Reproduction Number Rc for the transmission model in the
absence of drug resistance (i.e., for the model specified by equations 1 through 7 setting  R,
iR=0 for i=1,2,3). Rc may be computed (analytically) as the spectral radius (i.e., maximum
modulus of the eigenvalues) of the next generation matrix FV-1 [2,3], where F and V are given
by the following expressions:
 I I I A A A
2
3
1
2
3
 1
F 0 0 0
0
0
0
 M M M M M M






(11)
3
    
1
1


 1

0

V 
 1


0


0

0
0

0
  2   2
0
0

 2
  3   3
0
0
0
0
  1  
0
 2
0
 1
  2  
0
 3
0
 2



0




0


0

  3   

0
(12)
Note that FV-1 has only one nonzero eigenvalue with corresponding eigenvector
e1  [1,0,0,0,0,0]T .
The Control Reproduction Number Rc may be straightforwardly expressed in a biologically
interpretable manner as
Rc  L1  L2  L3
(13)
where L i can be interpreted as the Control Reproduction Number for a model including only
the (infected) stages I i and Ai weighted by the probability that an individual has progressed to
these stages before dying. The Li are given as follows:




1I
1A
IA
L1  r1II 

r
 1 

  1  1   
   1   
(14)




  II 
1
2I
2A
IA
L2  r 
r

r





 2
2
  1  1       2  2   
    2    
II
1




  AI 
1
2I
2A
AA
r1IA 
r

r

 2 


2 
    1       2  2   
    2    
(15)
4




 AI 
  II 
1
2
3I
3A
IA
L3  r1IA 
r
r

r

 2      3      3     
   1   
 2
   3


 
2
3
3




 II 
  II 
1
2
3I
3A
IA
r 
r
r

r

 3 

 2 
 3 
  1  1      2  2       3  3   
    3    
II
1




 AA 
  AI 
1
2
3I
3A
AA
r1IA 
r
r

r
3 
 3 



2 
   1   
    2       3  3   
    3    




 IA 
  AI 
1
2
3I
3A
AA
r 
r
r

r

 3 

 2 
 3 
  1  1        2       3  3   
    3    
II
1
(16)
where,

  k 
 1 


ri AI  
g

  i         1 g 
    i    k 0


i
i
(17)

  k 
 1 
i
i
ri IA  
g



        1 g 
i
  i  i    k 0
 i

i
i
(18)
ri II  ri AA 

g
k 0
k
i
 1 


 1  gi 



i

gi  



  i  i        i   
(19)
(20)
The gi terms give the probability that an individual in class I i begins treatment and
subsequently interrupts it (or vice versa). Similarly, the ri AI represent the sum of the probabilities
for all possible paths an individual may take from Ai to I i by successively interrupting and then
resuming treatment (and vice versa ri IA ).
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References:
1. Granich RM, Gilks CF, Dye C, De Cock KM, Williams BG (2009) Universal voluntary HIV
testing with immediate antiretroviral therapy as a strategy for elimination of HIV transmission:
a mathematical model. Lancet 373: 48-57.
2. Diekmann O, Heesterbeek JAP, Metz JAJ (1990) On the definition and the computation of the
basic reproduction ratio R0 in models for infectious diseases in heterogeneous populations. J
Math Biiol 28: 365-382.
3. van den Driessche P, Watmough J (2002) Reproduction numbers and sub-threshold endemic
equilibria for compartmental models of disease transmission. Math Biosci 180: 29-48.
4. National Department of Health (2007) The National Antenatal Sentinel HIV and Syphilis
Prevalence Survey, South Africa, 2006.
http://www.doh.gov.za/list.php?type=HIV%20and%20AIDS&year=2007. Accessed 2012 Aug
13.
5. National Department of Health (2011) The National Antenatal Sentinel HIV and Syphilis
Prevalence Survey, South Africa, 2010.
http://www.doh.gov.za/docs/reports/2011/hiv_aids_survey.pdf. Accessed 2012 Aug 13.
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