Modelling the impact of vector control interventions on Anopheles
gambiae population dynamics
Supplementary Information
Michael T White1*, Jamie T Griffin1, Thomas S Churcher1, Neil M Ferguson1, María-Gloria Basáñez1, Azra
C Ghani1
1
MRC Centre for Outbreak Analysis & Modelling, Department of Infectious Disease Epidemiology,
Imperial College London, London, England
*
Corresponding author
Email addresses:
MTW: m.white08@imperial.ac.uk
JTG: jamie.griffin@imperial.ac.uk
TSC: thomas.churcher@imperial.ac.uk
NMF: neil.ferguson@imperial.ac.uk
MGB: m.basanez@imperial.ac.uk
ACG: a.ghani@imperial.ac.uk
1. Comparison of model fits
Three models for calculating the environmental carrying capacity K (t ) were tested:
i) Carrying capacity proportional to the mean rainfall during the past  days
K (t )  
1


t
t 
rain(t ) dt 
(1)
ii) Carrying capacity proportional to the linearly weighted mean rainfall during the past  days
K (t )  
2

2

t
t 
(t   t   )rain(t )dt 
(2)
iii) Carrying capacity proportional to past rainfall weighted by an exponential distribution with mean 2
K (t )  

1
 1 e
t
 t
e

0
 ( t t )
rain(t )dt 
(3)
where rain(t ) is the daily rainfall and  is a fitted scaling factor unique for each village. The three models
gave similar visual fits (Figures S1, S2 and S3). When the models were formally compared using reversible
Markov Chain Monte Carlo model comparison, the probability of each model was estimated to be
i) Carrying capacity proportional to mean past rainfall:
0.221
ii) Carrying capacity proportional to linearly weighted past rainfall: 0.303
iii) Carrying capacity proportional to exponentially weighted past rainfall: 0.476
Model (iii) was thus deemed to be the best fit. Table 1 in the main text shows the best fit parameters taken to
be the median of the fitted posterior distributions.
Figure S1: Model fits under model i): carrying capacity proportional to mean past rainfall.
Figure S2: Model fits under model (ii): carrying capacity proportional to linearly weighted past rainfall.
Figure S3: Model fits under model (iii): carrying capacity proportional to exponentially weighted past rainfall.
Figure S4: Comparison of prior (red) and posterior (blue) distributions for the 10 parameters fitted using model (iii).
2. Density-dependent larval mortality
A study to investigate the dependence of Anopheles gambiae s. l. larval mortality on larval density was undertaken
by Njunwa [1] at Ubwari Field Station in Tanzania. Batches of first instar larvae were placed in partially immersed
cages measuring 0.25m x 0.25 m x 0.2m and observed for survival. Two versions of the experiment were undertaken.
In the first, all first instars were inserted in a single batch, thus ensuring little or no variation in the age of the larvae.
In the second, first instars were inserted in daily batches thus ensuring a varying age profile of larvae. The
proportion of instars surviving was measured. These data were transformed into the daily larval mortality rates
shown in Figure S5. Although 5 data points are plotted in Figure S5A and 4 data points in Figure S4B, each of these
data points summarises information from multiple binary observations of larval survival.
Figure S5: Comparison of model fits for density-dependent larval death rates




Linear (  y  a  bx  , quadratic y  a  bx  cx 2 and logistic curves  y  a 
ba 
 were fitted to the data
1  ec ( x d ) 
assuming measurement error was normally distributed. A quadratic curve gave the best fit to the data in Figure S5A
as judged by the Akaike Information Criterion (AIC) (Table S1). All curves gave a reasonably good fit to the data in
Figure S5B with the quadratic curve giving the best statistical fit (Table S2). A potential problem when fitting models
to small data sets is the danger of over-fitting. For example the logistic curve in Figure S5A fits a 4 parameter curve
to 5 data points and the quadratic curve in Figure S5B fits a 3 parameter curve to 4 data points.
Both data sets are best fitted by quadratic curves, but they are predicted to have opposite convexities
Table S1: Results of model fitting to the single instar input data in Figure 1A. n: number of data points. k: number of model parameters.
linear
quadratic
logistic
n
5
5
5
k
2
3
4
a
0.0560
1.23 x 10-11
2.81 x 10-4
b
0.0015
0.0029
0.473
c
-4.94 x 10-6
0.027
d
92.49
AIC
-5.12
-11.00
-9.46
Table S2: Results of model fitting to the staggered instar input data in Figure 1B. n: number of data points. k: number of model parameters.
linear
quadratic
logistic
n
4
4
4
k
2
3
4
a
0.048
0.068
0.049
b
9.49 x 10-5
5.79 x 10-5
0.499
c
7.91 x 10-9
0.0011
d
2450
AIC
12.97
-0.18
25.63
An additional test to check the relationship between larval mortality and larval density was performed by adjusting
the equations for density-dependent mortality (equation (7) in the main text) so that mortality was proportional to
some power  .
  E  L  
 E ( E , L)   1  
  K  


0
E


EL 
 L ( E , L)   L0 1   
 

K

 

(4)
Fitting this model of larval mortality gave   1.10 (95% CrI, 1.00 - 1.21) which is compatible with a linear
relationship between larval mortality and density.
3. Equilibrium solution of density-dependent model
The population dynamics of Anopheles gambiae mosquitoes can be described by the following set of ordinary
differential equations
dE
E
 EL
  M   E0 1 
E 
dt
K 
dE

dL E
EL
L


  L0 1  
L
dt d E
K 
dL

(5)
dP L
P

 P P 
dt d L
dP
dM 1 P

 M M
dt
2 dP
These equations can be solved at equilibrium to estimate the number of adult mosquitoes and larvae observed in
the environment. Setting the equations in (5) equal to zero and solving, we get:
EK

  
1
 0  0  1
   1   L d E  L d L 
 

1
 0  0  1
   1   L d E  L d L 
dp
 

1
1
PK
 0  0  1
d L 1  d P  P   (  1)   L d E  L d L

LK
M K
1
(6)
 

1
1
1
 0  0  1
2d L  M 1  d P  P   (  1)   L d E  L d L

2


L0 d E
1  L0 d E
1  L0 d E
0
0








1

d






1

d








where
L E 
L E 
2  E0 d L
4  E0 d L
2E0 M d L 1  d P P 


By rearranging the equation for M in (6) we can derive an expression for the larval carrying capacity in terms of the
number of observed mosquitoes at equilibrium:
K  M eq
2d L M 1  d P  P   (  1)
 

1
 0  0  1
 L d E L d L 
(7)
And hence the aquatic populations can be represented in terms of the population of adult mosquitoes:
E  2 M d L 1  d P  P  M
L  2 M d L 1  d P  P  M
P  2d P  M M
M M
(8)
4. Interventions
3.1. LLINs
We adapt an existing model [3,4] of the effects of LLINs and IRS on the Anopheline vector to our model. We describe
it briefly here, but for more details see Griffin et al [4].
Figure S4: Flow chart of mosquito life cycle based on the diagram from Le Menach et al [3]. Note the addition of an extra class of human for
those people not sleeping under bed nets.
The human feeding rate a needs to be broken down into two terms, the average time to complete one feeding cycle
  1/ f , and the proportion of bites taken on humans Q
a  fQ
(9)
From the flow chart in Figure S6, the probability that a surviving mosquito succeeds in feeding during a single
attempt can be calculated to be
w  1  Q0  Q0 (1   LLIN )  Q0  LLIN (1   )  Q0  LLIN s
w  1  Q0  LLIN (1  s)
(10)
where  LLIN is the coverage of bed nets,  is the proportion of people in bed when they are bitten, and s is the
probability of a mosquito feeding successfully on a person sleeping under a bed net. The probability of a mosquito
resetting and beginning a new search is
z  Q0  LLIN r
(11)
where r is the probability of a mosquito being repelled by a bed net. At zero LLIN coverage the length of a feeding
cycle is given by
1
  1 (0)   2
f
(12)
where  1 (0) is the time spent searching for a blood meal, and  2 is the time spent resting. Increased LLIN coverage
will cause the mosquito to spend a longer time foraging for a blood meal. And hence at LLIN coverage  LLIN
 1 (  LLIN )   1 (0)  z 1 (  LLIN ) 
 1 (0)
(13)
1 z
And therefore the length of a feeding cycle at coverage  LLIN is
1
f (  LLIN )

 1 (0)
1 z
 2
(14)
Increased LLIN coverage will also reduce the probability of a mosquito surviving the foraging stage p1 .
p1 (  LLIN )  p1 (0)[ w  zp1 (  LLIN )] 
p1 (0) w
1  zp1 (0)
(15)
Thus the probability of a mosquito surviving one day is given by
p (  LLIN )   p1 (  LLIN ) p2 
f (  LLIN )
 p (0) wp2 
 1

 1  zp1 (0) 
f (  LLIN )
(16)
where p2 is the probability of surviving resting. The mosquito mortality can then be calculated as
M (  LLIN )   log p(  LLIN )
(17)
3.2. IRS
The IRS model is implemented in the same manner as the LLIN model except for a few changes to mosquito
behaviour. When a mosquito searches for a blood meal in a sprayed house it will be repelled with probability
rIRS and attempt to take a blood meal with probability 1  rIRS . If a mosquito takes a blood meal it will rest indoors
on a sprayed wall with probability  . This is the endophily, the proportion of mosquitoes resting indoors after
feeding, and can be estimated from studies of vector behaviour [5,6]. The probability that a mosquito is repelled,
dies or is successful during a feeding attempt is thus given by
r  rIRS
s  (1  rIRS )(1  )
(18)
d  (1  rIRS )
Table S3: List of parameters estimated from the literature and quantities estimated from the model. All parameters are taken from Griffin et
al [4]. Parameters are for Anopheles gambiae s. s.
Parameter
Description
Frequency of feeding = inverse of gonotrophic cycle
Time spent seeking blood meal during gonotrophic cycle
Value
1/3 day-1
0.69 days
Time spent resting during gonotrophic cycle
2.31 days
Human blood index – proportion of blood meals on humans
0.92
Daily mosquito mortality at zero bed net coverage
0.096 day-1
Probability of surviving blood meal seeking at zero net coverage
0.91
Probability of surviving resting period at zero net coverage
0.74
LLIN
Proportion of bites taken on humans when in bed
0.89
d LLIN
Probability that a mosquito is killed by LLIN
0.41
sLLIN
rLLIN
Probability that a mosquito feeds successfully with ITNs
0.03
Probability that a mosquito is repelled by ITNs
0.56
IRS

Proportion of bites taken on humans while indoors
0.97
Endophily: proportion of mosquitoes resting indoors after feeding
Probability that a mosquito is repelled by lambdacyhalothrin
0.86
0.207
LLIN/IRS coverage
-
Probability of mosquito successfully surviving feeding attempt
-
Probability of mosquito repeating
-
Daily survival probability
-
Daily mosquito mortality
-
f
1
2
Q0
M (0)
p1  e  M (0)1
p2  e
  M (0) 2
rIRS
 LLIN/IRS
w(  LLIN )
z (  LLIN )
p(  LLIN )
M (  LLIN )
3.3. Larvicide
Breeding sites that have been larvicided with Bacillus thuringiensis israelensis (BTI) experience an 87% reduction in
the number of An. gambiae pupae [1]. We assume the larvicide works by increasing the death rate to both larvae
and pupae by some factor  . Having selected a value for  we can investigate the consequences of a larviciding
program with varying coverage. The environment will now consist of a collection of breeding sites with larvicide, a
collection of breeding sites without larvicide and a shared population of adult mosquitoes. The dynamics can be
described by the following set of 7 coupled differential equations. In principle these equations can be solved at
equilibrium and an expression for the number of adult mosquitoes derived in terms of the larvicide coverage, but in
practice it is easier to simulate the whole system and estimate M .

dE
E
EL 
 (1  c)  M 
  E0 1 
E
dt
dE
 (1  c) K 

dL E L
EL 

   L0 1  
L
dt d E d L
(1  c) K 

dP L P


 P P
dt d L d P
dEBTI
E
 E  LBTI 
 c M  BTI   E0 1  BTI
 EBTI
dt
dE
cK


(19)
dLBTI EBTI LBTI
E  LBTI 



  L0 1   BTI
 LBTI
dt
dE
dL
cK


dPBTI LBTI PBTI


  P PBTI
dt
dL
dP
dM 1 P  PBTI

 M M
dt
2 dP
3.4. Pupacide
We assume that in breeding sites receiving a sufficient quantity of a pupacide based on the juvenile hormone
analogue Pyriproxyfen (PPF), there is a 95% reduction in the number of pupae that successfully emerge as adults. As
PPF has been considered suitable for auto-dissemination from adult mosquito resting sites to breeding sites by
female mosquitoes during oviposition, it is possible that low levels of coverage of resting sites could result in high
levels of coverage of breeding sites [7,8]. Here we only consider coverage of breeding sites. If PPF causes a fraction
FPPF of pupae to die before emergence and a proportion c of breeding sites are covered, then the equations are the
same as before except with the following adjustment:
dE
E
 EL
  M   E0 1 
E 
dt
K 
dE

dL E
EL
L


  L0 1  
L
dt d E
K 
dL

dP L
P

 P P 
dt d L
dP
(20)
dM
1 P
 1  cFPPF 
 M M
dt
2 dP
Equation (20) can be solved at equilibrium in the same manner as equation (5) and the reduction in adult mosquito
density as a function of coverage can be estimated.
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