1 
AppendixS1 Implications ofmultiplesources ofheterogeneityfor controlsuccess 
In themain paper weexploretheimplications for controlof key hosts thatexhibitonly one
formof asymmetry (either super-abundant, super-infected or super-shedder).  In reality 
however, hostspecies arelikely to show asymmetry in severalof theseaspects (seeempirical
datain main paper).  Hereweexploretheimplications of such mixed mechanisms for control
success.  Specifically, weassumethatspecies iis akey hostin thesensethatitcontributes a
significantproportion, T, to overalltransmission, butitdoes so through theequal
combination of two sources of asymmetry.  Thatis, hostspecies iis akey hosteither dueto a
combination of being super-abundantand super-infected, super-abundantand super-shedding 
or super-infected and super-shedding;in allcases both mechanisms areassumed to contribute
equally to theoveralldegreeof asymmetry of thathost (seebelow).  As in themain paper, for 
each key hostscenario weexplored theeffectof controlthatremoves acertain number of 
individuals of species i(Ci) under two controlpossibilities:(1) Untargeted control, whereCi 
individuals areremoved regardless of infection status and (2) Targeted control, whereonly 
infected individuals areremoved.  Again, wequantified theimpact of controlas the
proportion of theparasite's initialinfectious pool remaining after either targeted (ξT) or 
untargeted (ξU) control(Eqns 2 and 3, main paper). 
 
Assuming mechanisms ofasymmetryareequal
Fromthemain paper weknow theoverallcontribution of hostspecies ito parasite
transmission is given by:
!! =
!!
!!!
!!!
!
!
= !        
If weconsider thecasewherespecies iis akey hostthrough two mechanisms (e.g. !!
! and !!
!  
>> 1, !!
!  = 1, wherex, yand z areeach oneof thethreemechanisms asymmetry), this 
becomes: 
 
2 
!!
!!!
!
!
= ! , or  !!
! =
!"
!!
! .      
Hence, multiplecombinations of !!
!  and !!
! (any pair of mechanisms) willallow hostspecies 
ito contributeaproportion T to overallparasitetransmission.  For this reason, in what
follows weassumeboth mechanisms contributeequally to transmission, such that:
!!
! = !!
! = !"                           Eq S1, 
which provides asingle, uniquevaluefor themagnitudeof each asymmetry. 
 
Super-abundantand super-infected host 
Here!!
!=1, so λ! =
!!!!!!!
!!!!!
.  Furthermore, since!!
! = !" (fromEq S1), p! =
!" !!!!!
!!!
.  
Hence, fromEqs 2 and 3 in themain paper: 
ξ! = 1−
!!
!!!!!
and 
ξ! = 1−
!! !"
!!!
.                                
 
Super-abundantand super-shedding host
Here!!
!=1, so p! =
!!!!!
!!!
.  Furthermore, since!!
! = !" (fromEq S1), λ! =
!" !!!!!!!
!!!!!
.  
Hence
ξ! = 1−
!! !"
!!!!!
and 
ξ! = 1−
!! !"
!!!
.                                
  
 
3 
Super-infected and super-shedding host
Here!!
! = !", soλ! =
!" !!!!!!!
!!!!!
, and so:
ξ! = 1−
!! !"
!!!!!
Furthermore, since!!
!!!
! = !", p!λ! =
!" !!!!!!!
!!!
.  Hence:
ξ! = 1−
!!!"
!!!
. 
 
Results 
FigureS1 illustrates theeffects of (a) untargeted and (b) targeted controlunder thesevarious 
scenarios (which can becompared to Fig.1, which assumed only singlemechanisms of 
asymmetry).  In generaltheefficacy of agiven controleffortunder multiplesources of 
heterogeneity (Fig.S1) lies between thetwo extremes of therelevantsingle-sources of 
heterogeneity (Fig.1).  For example, untargeted controlis moreeffective(fewer individuals 
need to betreated) for amixed super-abundantand super-infected key host(Fig.S1a, purple
line) than an equivalentpurely super-abundantkey host(Fig.1a, black line).  However, the
samemixed key hostis harder to controlthan theequivalentpurely super-infected key host
(Fig.1a, red line).  Specifically, moving fromoneextremesinglemechanism(e.g. apure
super-abundanthost) to theequalmixed case(e.g., an equally super-abundantand super-
infected host) to theother extremesinglemechanism(e.g. apuresuper-infected host) alters 
theefficacy of controlby afactor !" each step(TableS1).