Download Impact of emerging antiviral drug resistance on influenza

Impact of emerging antiviral drug resistance on
influenza containment and spread: influence of
subclinical infection and strategic use of a stockpile
containing one or two drugs: Appendix S1
James M. McCaw∗†, James G. Wood‡, Christopher T. McCaw∗ and Jodie McVernon∗
May 2, 2008
Contents
S1 Details of the model
S1.1 The single-drug model . . . . . . . . . . . . .
S1.1.1 ODEs . . . . . . . . . . . . . . . . . .
S1.1.2 R0 calculation . . . . . . . . . . . . .
S1.2 Multi-drug models . . . . . . . . . . . . . . .
S1.2.1 Random allocation and cycling models
S1.2.2 Treatment and Prophylaxis model . . .
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S2 Results
S2.1 Alternative scenarios . . . . . . . . . . . . . . .
S2.1.1 Extremely high fitness and lower seeding
S2.1.2 A reduced symptomatic proportion . . .
S2.2 Sensitivity Analysis . . . . . . . . . . . . . . . .
S2.2.1 Single drug model . . . . . . . . . . . .
S2.2.2 Random allocation model . . . . . . . .
S2.2.3 Treatment and Prophylaxis model . . . .
S2.2.4 Comparing models . . . . . . . . . . . .
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A Multi-drug model construction
28
A.1 Random allocation model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
A.2 Treatment and Prophylaxis model . . . . . . . . . . . . . . . . . . . . . . . . . . 39
References
44
∗
Vaccine and Immunisation Research Group, Murdoch Childrens Research Institute and Melbourne School of
Population Health, The University of Melbourne, Victoria, Australia.
†
Corresponding author. Electronic mail: [email protected], Phone: +61 3 8344 9145, Fax: +61 3 9348
1827.
‡
School of Public Health & Community Medicine, University of New South Wales and National Centre for Immunisation Research and Surveillance of Vaccine Preventable Diseases, The Childrens Hospital at Westmead and the
University of Sydney, New South Wales, Australia.
1
Abstract
We provide details of the models used in the main text, explain some of the results in
greater detail, perform a sensitivity analysis for each model and explore a number of alternate
scenarios to demonstrate the general nature of the results presented in the main text.
S1
Details of the model
The models used in this paper are extensions to the contact model described in detail in McCaw
and McVernon (2007). Here we detail how asymptomatic infections and development of antiviral
resistance is accounted for within the single-drug model. We also provide R0 calculations for the
single-drug model. The multi-drug model equations are presented for reference, while details of
their construction are left to Appendix A.
S1.1
The single-drug model
We have states Ipy with y ∈ {w, r} representing symptomatic infections for those taking prophylaxis (p) and infected with the wild-type (y = w) or resistant-type (y = r) strain. States Ayp
represent equivalent asymptomatic infections.
w
We also have the state Inp,t
representing symptomatic infections with the wild-type strain for
those not taking prophylaxis (np), but being provided with antiviral drugs as treatment (t). For
symptomatic infections with the wild-type strain that are not provided with antiviral drugs at all,
w
. Symptomatic infections with the resistant-type strain may or may not
we have the state Inp,nt
r
be provided with treatment but it has no effect so we require just one state Inp
. Asymptomatic
infections are not able to be identified and provided with antiviral drugs and so we only require
two states Aynp with y ∈ {w, r}.
The force of infection arises from each of these nine infectious classes:
w
w
λw
p = βei Ip + χAp
λw
np,nt
λw
np,t
r
=
=
λ =
(S1)
w
β Inp,nt
+ χAw
np
w
βet Inp,t
r
βφ Ipr + Inp
+χ
(S2)
(S3)
Arp + Arnp
,
(S4)
where φ is the relative transmissibility of the resistant strain relative to uncontrolled wild-type and
χ is the relative transmissibility of asymptomatic infections. Other parameters are defined as in
McCaw and McVernon (2007).
Infections arising from these nine states cause susceptible individuals (S) to become exposed
(Exy with x ∈ {p, np} and y ∈ {w, r}). Exposed individuals will become infectious after a latent
period of mean duration 1/ω days. New infections (I and A states) have contacts (Cxy states) which
may or may not be exposed. The mean infectious period is 1/γ days.
A susceptible, upon exposure, can proceed down a number of paths, dependent upon the characteristics of the infectee (wild-type, resistant-type) and whether or not they are provided with
post-exposure prophylaxis. A proportion of exposures, α, are symptomatic. Our model allows for
this proportion to depend upon the nature of the strain (w or r) and the provision of prophylaxis (p
or np).
Resistance development (“seeding”) is captured by two parameters. A proportion, ρt , of those
w
w
exposed to wild-type virus and treated (αnp
ψEnp
) convert to become symptomatic resistant-strain
2
r
infectors (Inp
). A proportion, ρp , of those exposed to wild-type virus and taking prophylaxis (Epw )
convert to become either symptomatic or asymptomatic resistant-strain infectors (Ipr or Arp ).
Figures S1 and S2 show the possible paths a susceptible can take on becoming infectious.
These diagrams are used to construct the model equations that follow.
3
S
es K w
(1 − ρp )
Epw
αpw
Ipw
(1 − αpw )
Aw
p
ρp
(1 − K w )
αr
Ipr
(1 − αr )
Arp
w
Enp
w
αnp
ψ
(1 − ρt )
w
Inp,t
ρt
(1 − ψ)
r
Inp
w
(1 − αnp
)
w
Inp,nt
Aw
np
Figure S1: Tree diagram for wild strain exposures
4
Kr
S
αr
Epr
Ipr
(1 − αr )
Arp
(1 − K r )
αr
r
Inp
r
Enp
(1 − αr )
Arnp
Figure S2: Tree diagram for resistant strain exposures
We introduce four contact states Cxy (x ∈ {p, np}, y ∈ {w, r}) which classify a contact by their
own type if they were to become infectious. We introduce:
Θw
p
=
Θw
np =
Θrp =
Θrnp =
es Cpw S
w N
Cpw + Cnp
w
Cnp
S
w N
Cpw + Cnp
Cpr
S
r
r
Cp + Cnp N
r
Cnp
S
.
r N
Cpr + Cnp
(S5)
(S6)
(S7)
(S8)
Contacts of wild-type infectives who are provided with AV drugs (Cpw ) have a reduced susceptibility accounted for by the factor es in the equation for Θw
p . All other contacts (those not on
prophylaxis and those on prophylaxis but in contact with resistant-strain infectives) remain fully
susceptible to infection.
5
S1.1.1
ODEs
The model equations are:
dS
w
r
= − λw Θw
Θrp + Θrnp
p + Θnp + λ
dt
dCpw
w
w
= κω αpw (1 − ρp ) Epw + [(1 − ρt ) ψ + (1 − ψ)] αnp
Enp
− δCpw − λw Θw
p
dt
w
dCnp
w
w
= κ (1 − ) ω αpw (1 − ρp ) Epw + [(1 − ρt ) ψ + (1 − ψ)] αnp
Enp
dt
w
w
w
+ κω 1 − αpw (1 − ρp ) Epw + 1 − αnp
Enp − δCnp
− λw Θw
np
r r
dCpr
r
w
w
= κω α Ep + Enp
+ αr ρp Epw + αnp
ρt ψEnp
− δCpr − λr Θrp
dt
r
dCnp
r
w
w
+ αr ρp Epw + αnp
ρt ψEnp
= κ (1 − ) ω αr Epr + Enp
dt
r
r
− λr Θrnp
+ (1 − αr ) ρp Epw − δCnp
+ κω (1 − αr ) Epr + Enp
(S9)
(S10)
(S11)
(S12)
(S13)
and, for x ∈ {p, np}, y ∈ {w, r}
dExy
= λy Θyx − ωExy
dt
(S14)
and, for symptomatic infectious states
dIpw
dt
w
dInp,t
dt
w
dInp,nt
dt
dIpr
dt
r
dInp
dt
= αpw (1 − ρp ) ωEpw − γI Ipw
(S15)
w
w
w
= αnp
(1 − ρt ) ψωEnp
− γI Inp,t
(S16)
w
w
w
= αnp
(1 − ψ) ωEnp
− γI Inp,nt
(S17)
= αr ρp ωEpw + αr ωEpr − γI Ipr
(S18)
w
w
r
r
ρt ψωEnp
+ αr ωEnp
− γI Inp
= αnp
(S19)
and, for asymptomatic infectious states
dAw
p
dt
dAw
np
dt
dArp
dt
dArnp
dt
= 1 − αpw (1 − ρp ) ωEpw − γA Aw
p
(S20)
w
w
= 1 − αnp
ωEnp
− γA Aw
np
(S21)
= (1 − αr ) ρp ωEpw + (1 − αr ) ωEpr − γA Arp
(S22)
r
= (1 − αr ) ωEnp
− γA Arnp
(S23)
and, for x ∈ {p, np}, y ∈ {w, r}
y
dRIx
= γI Ixy
dt
y
dRAx
= γA Ayx
dt
(S24)
(S25)
6
and finally, for the depletion of the finite stockpile of AV drugs,
w w
dO
w
w
r
r
= −κω αpw Epw + αnp
Enp
+ αr Epr + Enp
− ψω αnp
Enp + αr Enp
.
dt
S1.1.2
(S26)
R0 calculation
Define the column vector (0 is transpose)
w
w
r
r
r 0
~ = Ipw Inp,t
Inp,nt
Ipr Inp
Aw
Aw
X
p
np Ap Anp
(S27)
We may ignore the E states by factoring them into the right hand side. Defining this construction
˙
~˙ + appropriate parts of E)
~˙ allows us to write:
(Y~ ≡ X
n
o
˙
~
Y~ = [9 × 9] /γ − I~ γ X
(S28)
and R0 is the maximum eigenvalue of the 9 × 9 matrix. We construct this matrix as a linear
combination of tensor products of force of infection row vectors, λ~w and λ~r , with suitable column
vectors constructed from the differential equations, F~w and F~r :
n
o
w
w
r
r
~
~
~
~
R0 = max eigenvalue of F λ + F λ /γ ,
(S29)
where
λ~w = β ei et 1 0 0 ei χ χ 0 0
λ~r = βφ 0 0 0 1 1 0 0 χ χ ,
and
(S30)
(S31)



αpw (1 − ρp ) es K w
0
w
w 


αnp
0


 w(1 − ρt ) ψ (1 − Kw )


 αnp (1 − ψ) (1 − K ) 
0




r
w
r r




α
ρ
e
K
α
K
p
s




w
w
r
r
 and F~r =  α (1 − K )  ,
ρ
ψ
(1
−
K
)
α
(S32)
F~w = 
t
np






 1 − αw (1 − ρp ) es K w 
0
p






 1 − αw (1 − K w ) 
0
np




r
r
r
w
 (1 − α ) K


 (1 − α ) ρp es K
r
r
(1 − α ) (1 − K )
0
x
where K w and K r are the steady state proportions of Cpx / Cpx + Cnp
(x ∈ {w, r}), which, in the
w
w
x
case αp = αnp , are simply αnp (x ∈ {w, r}).
The matrix F~w λ~w + F~r λ~r can be put into block-triangular form, effectively breaking the
eignevalue calculation into two separate calculations, one from each of F~w λ~w and F~r λ~r . As each
of these matrices is, by construction, rank-1 the eigenvalues of the system are given by the scalar
product of F~x and λ~x (x ∈ {w, r}). From x = w, we obtain the reproduction number for the controlled wild strain; from x = r, the reproduction number for the reduced-fitness resistant strain:

αpw + 1 − αpw χ (1 − ρp ) es ei K w

β
w
w
+ αnp
[et (1 − ρt ) ψ + (1 − ψ)] + 1 − αnp
χ (1 − K w ) ,
R0 = × max

γ
r
r
φ (α + (1 − α ) χ)
(S33)

7
This expression deserves some explanation. The first eigenvalue is for the wild-type strain. It’s
form is the appropriate linear combination of prophylaxis (K w ) and non-prophylaxis (1 − K w )
sectors. The prophylaxis sector is itself a linear combination of symptomatic and asymptomatic
parts, capturing the reduced susceptibility (es ) and reduced infectivity (ei ) due to prophylaxis, as
well as the leakage to the resistant sector. The non-prophylaxis sector is, again, a linear combination of symptomatic and asymptomatic parts. Furthermore, the symptomatic part is itself a linear
combination of treated (ψ) and untreated (1 − ψ) components. The treatment part captures leakage
to the resistant sector.
The second reproduction number is for the resistant strain. It is the appropriate linear combination of reproduction numbers for the symptomatic resistant-strain infections (βφ/γ) and asymptomatic resistant-strain infections (βφχ/γ). The simple form arises due to the fact that antiviral
drugs have no impact on the resistant strain.
S1.2
Multi-drug models
The multi-drug models build upon the single-drug model just detailed. More states are included to
account for the four possible strains in cicurlation (wild-type, two single-drug resistant types and
multi-drug resistant type). Parameters relating to the resistant strain (φ, ρt and ρp as well as αr )
pick up labels to tie them to the drug in use and the strain in circulation.
S1.2.1
Random allocation and cycling models
We have four strains in circulation: with wild-type virus (w label), drug 1 resistant virus (r1 label), drug 2 resistant virus (r2 label) and multiple resistant virus (r12 label). We introduce new
parameters:
• φr1 ,r2 ,r12 — the transmissibility of the (drug 1, drug 2, multiple)-resistant strain relative to
the transmissibility of the uncontrolled wild-type strain.
• ρt1,t2 — the proportion of offspring (that is, secondary infectives) of treated wild-type infectives who carry the drug 1 (drug 2) resistant strain.
• ρp1,p2 — the proportion of offspring of breakthrough (prophylaxis) wild-type infectives who
carry the drug 1 (drug 2) resistant strain.
We assume that all offspring of r1 -resistant infectives are either r1 -resistant or r12 -resistant.
Similarly, all offspring of r2 -resistant infectives are either r2 -resistant or r12 -resistant. All offspring
of r12 -resistant infectives are r12 -resistant.
The provision of AV drug 1 (drug 2) as prophylaxis to contacts of drug 2 (drug 1) resistant
infectives is assumed to have the same protective effect as provision of prophylaxis to contacts
of wild-type infectives. Prophylaxis with drug 1 (drug 2) of contacts of drug 1 (drug 2) resistant
infectives has no protective effect. Similarly, the provision of AV drug 1 (drug 2) as treatment to
drug 2 (drug 1) resistant infectives is assumed to have the same effect as provision of treatment
to wild-type infectives. Treatment with drug 1 (drug 2) has no effect on infectives carrying the
drug 1 (drug 2) resistant strain. Provision of AV drugs as prophylaxis to contacts or as treatment to
infectives carrying the multiple-resistant strain is assumed to have no effect on either infectiousness
or susceptibility.
In the random allocation model, which drug is used for treatment or prophylaxis is chosen at
random, assumed to be a function of the stockpile size of each drug. For treatment or prophylaxis,
8
drug 1 will be provided a proportion f1 (O1 , O2 ) of the time and drug 2 a proportion f2 (O1 , O2 ) of
the time. In the simplest case, f1 and f2 are given by the proportion of the stockpile that is of type
1 or 2:
f1 = O1 / (O1 + O2 )
f2 = O2 / (O1 + O2 ) .
(S34)
(S35)
In the cycling model, when using drug 1 we set f1 = 1, f2 = 0 and vice-versa when using
drug 2.
The force of infection arises from 30 infectors:
w
w
(S36)
+
χA
=
βe
I
λw
i1
p1
p1
p1
w
w
w
λp2 = βei2 Ip2 + χAp2
(S37)
w
λw
np,t1 = βet1 Inp,t1
(S38)
w
λw
np,t2 = βet2 Inp,t2
λw
np,nt
λrp11
λrp21
1
λrnp,t1
1
λrnp,t2
1
λrnp,nt
λrp12
λrp22
2
λrnp,t1
2
λrnp,t2
2
λrnp,nt
r12
λ
=
=
=
=
=
=
=
=
=
=
=
=
(S39)
w
β Inp,nt
+ χAw
np
r1
r1
βφ Ip1 + χArp11
r1
βφr1 ei2 Ip2
+ χArp21
r1
βφr1 Inp,t1
r1
βφr1 et2 Inp,t2
r1
βφr1 Inp,nt
+ χArnp1
r2
βφr2 ei1 Ip1
+ χArp12
r2
βφr2 Ip2
+ χArp22
r2
βφr2 et1 Inp,t1
r2
βφr2 Inp,t2
r2
βφr2 Inp,nt
+ χArnp2
r12
r12
r12
βφr12 Ip1
+ Ip2
+ Inp
9
(S40)
(S41)
(S42)
(S43)
(S44)
(S45)
(S46)
(S47)
(S48)
(S49)
(S50)
+χ
Arp112
+
Arp212
+
Arnp12
.
(S51)
Compared to the single-drug model, we now have twelve contact states Cxy (x ∈ {p1, p2, np},
y ∈ {w, r1 , r2 , r12 }):
Θw
p1 =
Θw
p2 =
Θw
np =
Θrp11 =
Θrp21 =
Θrnp1 =
Θrp12 =
Θrp22 =
Θrnp2 =
Θrp112 =
Θrp212 =
Θrnp12 =
w
es1 Cp1
S
w
w
w
Cp1 + Cp2 + Cnp N
w
es2 Cp2
S
w
w
w
Cp1 + Cp2 + Cnp N
w
Cnp
S
w
w
w
Cp1 + Cp2 + Cnp N
r1
Cp1
S
r1
r1
r1
Cp1 + Cp2 + Cnp N
r1
es2 Cp2
S
r1
r1
r1
Cp1 + Cp2 + Cnp N
r1
Cnp
S
r1
r1
r1
Cp1 + Cp2 + Cnp N
r2
es1 Cp1
S
r2
r2
r2
Cp1 + Cp2 + Cnp N
r2
Cp2
S
r2
r2
r2
Cp1 + Cp2 + Cnp N
r2
Cnp
S
r2
r2
r2
Cp1 + Cp2 + Cnp N
r12
Cp1
S
r12
r12
r12
Cp1 + Cp2 + Cnp N
r12
Cp2
S
r12
r12
r12
Cp1 + Cp2 + Cnp N
r12
Cnp
S
r12
r12
r12
Cp1 + Cp2 + Cnp N
(S52)
(S53)
(S54)
(S55)
(S56)
(S57)
(S58)
(S59)
(S60)
(S61)
(S62)
(S63)
.
The differential equations that describe the dynamics are:
dS
w
w
r1
= − λw Θw
Θrp11 + Θrp21 + Θrnp1
p1 + Θp2 + Θnp + λ
dt
+λr2 Θrp12 + Θrp22 + Θrnp2 + λr12 Θrp112 + Θrp212 + Θrnp12
10
(S64)
and for the wild-type contact classes
w
w
dCp1
w
w
w
= κωf1 αp1
(1 − ρp1 ) Ep1
+ αp2
(1 − ρp2 ) Ep2
dt
w
w
w
− λw Θw
− δCp1
Enp
+ [(1 − ρt1 ) f1 ψ + (1 − ρt2 ) f2 ψ + (1 − ψ)] αnp
p1 (S65)
w
w
dCp2
w
w
w
= κωf2 αp1
(1 − ρp1 ) Ep1
+ αp2
(1 − ρp2 ) Ep2
dt
w
w
w
− λw Θw
− δCp2
Enp
+ [(1 − ρt1 ) f1 ψ + (1 − ρt2 ) f2 ψ + (1 − ψ)] αnp
p2 (S66)
w
w
dCnp
w
w
w
= κ (1 − ) ω αp1
(1 − ρp1 ) Ep1
+ αp2
(1 − ρp2 ) Ep2
dt
w
w
Enp
+ [(1 − ρt1 ) f1 ψ + (1 − ρt2 ) f2 ψ + (1 − ψ)] αnp
w
w
w
w
w
w
+ κω 1 − αp1
(1 − ρp1 ) Ep1
+ 1 − αp2
(1 − ρp2 ) Ep2
+ 1 − αnp
Enp
w
− δCnp
− λw Θw
np
(S67)
and for the drug 1 resistant contact classes
r1
r1 r1
dCp1
r1
r1
r1 r1
+ [f1 ψ + (1 − ρt2 ) f2 ψ + (1 − ψ)] αnp
Enp
(1 − ρp2 ) Ep2
Ep1 + αp2
= κωf1 αp1
dt
r1
r1
w
w
w
+αp1
ρp1 Ep1
+ αnp
ρt1 f1 ψEnp
− δCp1
− λr1 Θrp11
(S68)
r1
r1 r1
dCp2
r1
r1
r1 r1
= κωf2 αp1
Ep1 + αp2
(1 − ρp2 ) Ep2
+ [f1 ψ + (1 − ρt2 ) f2 ψ + (1 − ψ)] αnp
Enp
dt
r1
r1
w
w
w
+αp1
ρp1 Ep1
+ αnp
ρt1 f1 ψEnp
− δCp2
− λr1 Θrp21
(S69)
r1
r1 r1
dCnp
r1
r1
r1 r1
= κ (1 − ) ω αp1
Ep1 + αp2
(1 − ρp2 ) Ep2
+ [f1 ψ + (1 − ρt2 ) f2 ψ + (1 − ψ)] αnp
Enp
dt
r1
w
w
w
+αp1
ρp1 Ep1
+ αnp
ρt1 f1 ψEnp
r1
r1
r1
r1
r1
r1
+ κω 1 − αp1
Ep1 + 1 − αp2
(1 − ρp2 ) Ep2
+ 1 − αnp
Enp
r1
w
r1
+ 1 − αp1
ρp1 Ep1
− δCnp
− λr1 Θrnp1
(S70)
and for the drug 2 resistant contact classes
r2
r2 r2
dCp1
r2
r2
r2 r2
= κωf1 αp2
Ep2 + αp1
(1 − ρp1 ) Ep1
+ [(1 − ρt1 ) f1 ψ + f2 ψ + (1 − ψ)] αnp
Enp
dt
r2
r2
w
w
w
+αp2
ρp2 Ep2
+ αnp
ρt2 f2 ψEnp
− δCp1
− λr2 Θrp12
(S71)
r2
r2 r2
dCp2
r2
r2
r2 r2
Enp
= κωf2 αp2
Ep2 + αp1
(1 − ρp1 ) Ep1
+ [(1 − ρt1 ) f1 ψ + f2 ψ + (1 − ψ)] αnp
dt
r2
r2
w
w
w
+ αnp
ρt2 f2 ψEnp
− δCp2
− λr2 Θrp22
(S72)
+αp2
ρp2 Ep2
r2
r2 r2
dCnp
r2
r2
r2 r2
Enp
= κ (1 − ) ω αp2
Ep2 + αp1
(1 − ρp1 ) Ep1
+ [(1 − ρt1 ) f1 ψ + f2 ψ + (1 − ψ)] αnp
dt
r2
w
w
w
+ αnp
ρt2 f2 ψEnp
+αp2
ρp2 Ep2
r2
r2
r2
r2
r2
r2
+ κω 1 − αp2
Ep2
+ 1 − αp1
(1 − ρp1 ) Ep1
+ 1 − αnp
Enp
r2
w
r2
r2 r2
+ 1 − αp2 ρp2 Ep2 − δCnp − λ Θnp
(S73)
11
and for the multi-drug resistant contact classes
r12
dCp1
r2
r1
r12
r12
r12
+ αr12 ρp1 Ep1
+ αr12 ρp2 Ep2
+ αr12 Enp
+ αr12 Ep2
= κωf1 αr12 Ep1
dt
r12
r2
r2
r1
r1
− λr12 Θrp112
− δCp1
ρt1 f1 ψEnp
+ αnp
ρt2 f2 ψEnp
+αnp
r12
dCp2
r2
r1
r12
r12
r12
+ αr12 ρp1 Ep1
+ αr12 ρp2 Ep2
+ αr12 Enp
+ αr12 Ep2
= κωf2 αr12 Ep1
dt
r12
r2
r2
r1
r1
− λr12 Θrp212
− δCp2
ρt1 f1 ψEnp
+ αnp
ρt2 f2 ψEnp
+αnp
r12
dCnp
r2
r1
r12
r12
r12
+ αr12 ρp1 Ep1
+ αr12 ρp2 Ep2
+ αr12 Enp
+ αr12 Ep2
= κ (1 − ) ω αr12 Ep1
dt
r2
r2
r1
r1
ρt1 f1 ψEnp
+ αnp
ρt2 f2 ψEnp
+αnp
r12
r12
r12
+ (1 − αr12 ) Enp
+ (1 − αr12 ) Ep2
+ κω (1 − αr12 ) Ep1
r2
r1
r12
− λr12 Θrnp12
− δCnp
+ (1 − αr12 ) ρp1 Ep1
+ (1 − αr12 ) ρp2 Ep2
(S74)
(S75)
(S76)
and, for x ∈ {p1, p2, np}, y ∈ {w, r1 , r2 , r12 },
dExy
= λy Θyx − ωExy
dt
(S77)
12
and, for symptomatic infectious states
w
dIp1
dt
w
dIp2
dt
w
dInp,t1
dt
w
dInp,t2
dt
w
dInp,nt
dt
r1
dIp1
dt
r1
dIp2
dt
r1
dInp,t1
dt
r1
dInp,t2
dt
r1
dInp,nt
dt
r2
dIp1
dt
r2
dIp2
dt
r2
dInp,t1
dt
r2
dInp,t2
dt
r2
dInp,nt
dt
r12
dIp1
dt
r12
dIp2
dt
r12
dInp
dt
w
w
w
= αp1
(1 − ρp1 ) ωEp1
− γI Ip1
(S78)
w
w
w
− γI Ip2
(1 − ρp2 ) ωEp2
= αp2
(S79)
w
w
w
= αnp
(1 − ρt1 ) f1 ψωEnp
− γI Inp,t1
(S80)
w
w
w
= αnp
(1 − ρt2 ) f2 ψωEnp
− γI Inp,t2
(S81)
w
w
w
− γI Inp,nt
(1 − ψ) ωEnp
= αnp
(S82)
r1
r1
r1
r1
w
− γI Ip1
ωEp1
ρp1 ωEp1
+ αp1
= αp1
(S83)
r1
r1
r1
− γI Ip2
(1 − ρp2 ) ωEp2
= αp2
(S84)
r1
r1
w
w
r1
− γI Inp,t1
f1 ψωEnp
= αnp
ρt1 f1 ψωEnp
+ αnp
(S85)
r1
r1
r1
= αnp
(1 − ρt2 ) f2 ψωEnp
− γI Inp,t2
(S86)
r1
r1
r1
= αnp
(1 − ψ) ωEnp
− γI Inp,nt
(S87)
r2
r2
r2
= αp1
(1 − ρp1 ) ωEp1
− γI Ip1
(S88)
r2
r2
r2
r2
w
= αp2
ρp2 ωEp2
+ αp2
ωEp2
− γI Ip2
(S89)
r2
r2
r2
= αnp
(1 − ρt1 ) f1 ψωEnp
− γI Inp,t1
(S90)
r2
w
w
r2
r2
= αnp
ρt2 f2 ψωEnp
+ αnp
f2 ψωEnp
− γI Inp,t2
(S91)
r2
r2
r2
= αnp
(1 − ψ) ωEnp
− γI Inp,nt
(S92)
r2
r12
r12
= αr12 ρp1 ωEp1
+ αr12 ωEp1
− γI Ip1
(S93)
r1
r12
r12
= αr12 ρp2 ωEp2
+ αr12 ωEp2
− γI Ip2
(S94)
r1
r1
r2
r2
r12
r12
= αnp
ρt2 f2 ψωEnp
+ αnp
ρt1 f1 ψωEnp
+ αr12 ωEnp
− γI Inp
(S95)
13
and, for asymptomatic infectious states
dAw
p1
dt
dAw
p2
dt
dAw
np
dt
dArp11
dt
dArp21
dt
dArnp1
dt
dArp12
dt
dArp22
dt
dArnp2
dt
dArp112
dt
dArp212
dt
dArnp12
dt
w
w
= 1 − αp1
(1 − ρp1 ) ωEp1
− γA Aw
p1
(S96)
w
w
− γA Aw
(1 − ρp2 ) ωEp2
= 1 − αp2
p2
(S97)
w
w
= 1 − αnp
ωEnp
− γA Aw
np
(S98)
r1
r1
r1
w
− γA Arp11
ωEp1
+ 1 − αp1
ρp1 ωEp1
= 1 − αp1
(S99)
r1
r1
− γA Arp21
(1 − ρp2 ) ωEp2
= 1 − αp2
(S100)
r1
r1
− γA Arnp1
ωEnp
= 1 − αnp
(S101)
r2
r2
− γA Arp12
(1 − ρp1 ) ωEp1
= 1 − αp1
(S102)
r2
r2
r2
w
− γA Arp22
ωEp2
ρp2 ωEp2
+ 1 − αp2
= 1 − αp2
(S103)
r2
r2
= 1 − αnp
ωEnp
− γA Arnp2
(S104)
r2
r12
= (1 − αr12 ) ρp1 ωEp1
+ (1 − αr12 ) ωEp1
− γA Arp112
(S105)
r1
r12
= (1 − αr12 ) ρp2 ωEp2
+ (1 − αr12 ) ωEp2
− γA Arp212
(S106)
r12
= (1 − αr12 ) ωEnp
− γA Arnp12
(S107)
and, for x ∈ {p1, p2, np}, y ∈ {w, r1 , r2 , r12 },
y
dRIx
= γI Ixy
dt
y
dRAx
= γA Ayx
dt
(S108)
(S109)
and finally, for the depletion of the finite stockpiles of AV drugs,
w w
dOtotal
r1 r1
r1 r1
w
w
w
w
r1 r1
Enp
+ αnp
Enp
+ αp1
Ep1 + αp2
Ep2 + αnp
= −κω αp1
Ep1 + αp2
Ep2
dt
r2 r2
r2 r2
r12
r12
r12
r2 r2
+αp1
Ep1 + αp2
+ Ep2
+ Enp
Ep2 + αnp
Enp + αr12 Ep1
w w
r1 r1
r2 r2
r12
Enp + αr12 Enp
Enp + αnp
− ψω αnp
Enp + αnp
(S110)
with
dO1
dOtotal
= f1
dt
dt
dO2
dOtotal
= f2
dt
dt
(S111)
(S112)
14
R0 is the maximum eigenvalue of a 30 × 30 matrix. As in the single-drug model, each of the
four non-zero eigenvalues corresponds to the reproduction number for one of the four strains in
circulation:

w
w
w
αp1
+ 1 − αp1
χ (1 − ρp1) es1 ei1 Kp1



w
w
w

χ (1 − ρp2 ) es2 ei2 Kp2
+ 1 − αp2
+ αp2



w

[et1 f1 (1 − ρt1) ψ+ et2 f2 (1 − ρt2 ) ψ
+ αnp

+ (1 − ψ)]


w
w
w

+
1
−
α
χ
1
−
K
−
K

p1
p2 ,
r1np
 r r1

r1
1

φ
αp1 + 1 − αp1 χ Kp1 


r1
r1
r1

χ (1 − ρp2 ) es2 ei2 Kp2
+ 1 − αp2
+ αp2

β
r1
[f1 ψ + et2 f2 (1 −ρt2 ) ψ + (1 − ψ)]
+ αnp
(S113)
R0 = × max

γ
r1
r1
r1

+ 1− αnp χ 1 − Kp1 − Kp2 ,


r2

r2
r2
r2

ρp1 ) es1 ei1 Kp1
φ
αp1 + 1 − αp1 χ (1 −



r2
r2
r2


χ Kp2
+ 1 − αp2
+ αp2


r2


[et1 f1 (1 − ρt1) ψ+ f2 ψ + (1 − ψ)]
+ αnp


r2
r2

r2

,
− Kp2
+ 1 − αnp
χ 1 − Kp1

 r12 r12
φ {α + (1 − αr12 ) χ}
S1.2.2
Treatment and Prophylaxis model
We have four strains in circulation:wild-type virus (w label), treatment-resistant virus (rt label),
prophylaxis-resistant virus (rp label) and multi-resistant virus (rtp label). We introduce new parameters:
• φrt ,rp ,rtp — the transmissibility of the (treatment, prophylaxis, multiple)-resistant strain relative to the transmissibility of the uncontrolled wild-type strain.
• ρt — the proportion of offspring (that is, secondary infectives) of treated wild-type infectives
who carry the resistant strain.
• ρp — the proportion of offspring of breakthrough (prophylaxis) wild-type infectives who
carry the resistant strain.
We assume that all offspring of rt -resistant infectives are either rt -resistant or rtp -resistant.
Similarly, all offspring of rp -resistant infectives are either rp -resistant or rtp -resistant. All offspring
of rtp -resistant infectives are rtp -resistant.
The provision of AV drugs as prophylaxis to contacts of treatment-resistant infectives is assumed to have the same protective effect as provision of prophylaxis to contacts of wild-type infectives. Treatment has no effect on infectives carrying the treatment-resistant strain. Similarly, the
provision of AV drugs as treatment to prophylaxis-resistant infectives is assumed to have the same
effect as provision of treatment to wild-type infectives. Prophylaxis of contacts of prophylaxisresistant infectives has no protective effect.
Provision of AV drugs as prophylaxis to contacts or as treatment to infectives carrying the
multiple-resistant strain is assumed to have no effect on either infectiousness or susceptibility.
15
The force of infections arises from 18 infectors:
w
w
λw
p = βei Ip + χAp
(S114)
w
λw
np,t = βet Inp,t
(S115)
w
w
λw
np,nt = β Inp,nt + χAnp
λrpp
rp
λnp,t
rp
λnp,nt
λrpt
λrnpt
rtp
λ
=
=
=
=
=
=
βφ Iprp + χArpp
rp
βφrp et Inp,t
rp
+ χArnpp
βφrp Inp,nt
βφrt ei Iprt + χArpt
rt
+ χArnpt
βφrt Inp
rtp
+χ
βφrtp Iprtp + Inp
rp
(S116)
(S117)
(S118)
(S119)
(S120)
(S121)
Arptp
+
Arnptp
.
(S122)
We have eight contact states: Cxy (x ∈ {p, np}, y ∈ {w, rt , rp , rtp }):
es Cpw S
w N
Cpw + Cnp
w
Cnp
S
= w
w N
Cp + Cnp
Θw
p =
(S123)
Θw
np
(S124)
r
Cp p
S
rp
rp
Cp + Cnp N
r
S
Cnpp
Θrnpp = rp
rp
Cp + Cnp N
es Cprt S
Θrpt = rt
rt N
Cp + Cnp
rt
Cnp
S
Θrnpt = rt
r
t
Cp + Cnp N
Θrpp =
(S125)
(S126)
(S127)
(S128)
r
Θrptp
Θrnptp
Cp tp
S
= rtp
rtp
Cp + Cnp N
r
Cnptp
S
= rtp
rtp
Cp + Cnp N
(S129)
(S130)
.
The differential equations that describe the dynamics are:
dS
w
rt
Θrpt + Θrnpt
= − λw Θw
p + Θnp + λ
dt
+λrp Θrpp + Θrnpp + λrtp Θrptp + Θrnptp
(S131)
and for the wild-type contact classes
dCpw
w
w
Enp
− δCpw − λw Θw
= κω αpw (1 − ρp ) Epw + [(1 − ρt ) ψ + (1 − ψ)] αnp
p
dt
w
dCnp
w
w
= κ (1 − ) ω αpw (1 − ρp ) Epw + [(1 − ρt ) ψ + (1 − ψ)] αnp
Enp
dt
w
w
w
+ κω 1 − αpw (1 − ρp ) Epw + 1 − αnp
Enp − δCnp
− λw Θw
np
16
(S132)
(S133)
and for the prophylaxis-strain resistant contact classes
r
dCp p
rp
+ αrp ρp Epw − δCprp − λrp Θrpp (S134)
= κω αrp Eprp + [(1 − ρt ) ψ + (1 − ψ)] αrp Enp
dt
r
dCnpp
rp
+ αrp ρp Epw
= κ (1 − ) ω αrp Eprp + [(1 − ρt ) ψ + (1 − ψ)] αrp Enp
dt
rp
+ (1 − αrp ) ρp Epw
+ κω (1 − αrp ) Eprp + (1 − αrp ) Enp
rp
− λrp Θrnpp
− δCnp
(S135)
and for the treatment-strain resistant contact classes
dCprt
w
w
rt
rt
+ αnp
ρt ψEnp
− δCprt − λrt Θrpt
Enp
= κω αprt (1 − ρp ) Eprt + αnp
dt
rt
dCnp
w
w
rt
rt
+ αnp
ρt ψEnp
Enp
= κ (1 − ) ω αprt (1 − ρp ) Eprt + αnp
dt
rt rt
rt
− λrt Θrnpt
Enp − δCnp
+ κω 1 − αprt (1 − ρp ) Eprt + 1 − αnp
(S136)
(S137)
and for the multi-drug resistant contact classes
r
dCp tp
rtp
rp
= κω αrtp Eprtp + Enp
+ αrtp ρp Eprt + αrp ρt ψEnp
− δCprtp − λrtp Θrptp
dt
r
dCnptp
rtp
rp
= κ (1 − ) ω αrtp Eprtp + Enp
+ αrtp ρp Eprt + αrp ρt ψEnp
dt
rtp
+ κω (1 − αrtp ) Eprtp + Enp
+ (1 − αrtp ) ρp Eprt
rtp
− δCnp
− λrtp Θrnptp
(S138)
(S139)
and, for x ∈ {p, np}, y ∈ {w, rp , rt , rtp },
dExy
= λy Θyx − ωExy
dt
(S140)
17
and, for symptomatic infectious states
dIpw
dt
w
dInp,t
dt
w
dInp,nt
dt
r
dIp p
dt
rp
dInp,t
dt
rp
dInp,nt
dt
dIprt
dt
rt
dInp
dt
r
dIp tp
dt
r
dInptp
dt
= αpw (1 − ρp ) ωEpw − γI Ipw
(S141)
w
w
w
= αnp
(1 − ρt ) ψωEnp
− γI Inp,t
(S142)
w
w
w
− γI Inp,nt
(1 − ψ) ωEnp
= αnp
(S143)
= αrp ρp ωEpw + αrp ωEprp − γI Iprp
(S144)
r
p
rp
− γI Inp,t
= αrp (1 − ρt ) ψωEnp
(S145)
r
p
rp
− γI Inp,nt
= αrp (1 − ψ) ωEnp
(S146)
= αprt (1 − ρp ) ωEprt − γI Iprt
(S147)
rt
rt
rt
w
w
− γI Inp
ωEnp
+ αnp
ρt ψωEnp
= αnp
(S148)
= αrtp ρp ωEprt + αrtp ωEprtp − γI Iprtp
(S149)
rp
rtp
rtp
= αrp ρt ψωEnp
+ αrtp ωEnp
− γI Inp
(S150)
and, for asymptomatic infectious states
dAw
p
dt
dAw
np
dt
r
dApp
dt
r
dAnpp
dt
dArpt
dt
dArnpt
dt
r
dAptp
dt
r
dAnptp
dt
= 1 − αpw (1 − ρp ) ωEpw − γA Aw
p
(S151)
w
w
= 1 − αnp
ωEnp
− γA Aw
np
(S152)
= (1 − αrp ) ρp ωEpw + (1 − αrp ) ωEprp − γA Arpp
(S153)
rp
= (1 − αrp ) ωEnp
− γA Arnpp
(S154)
= 1 − αprt (1 − ρp ) ωEprt − γA Arpt
(S155)
rt
rt
= 1 − αnp
ωEnp
− γA Arnpt
(S156)
= (1 − αrtp ) ρp ωEprt + (1 − αrtp ) ωEprtp − γA Arptp
(S157)
rtp
= (1 − αrtp ) ωEnp
− γA Arnptp
(S158)
and, for x ∈ {p, np}, y ∈ {w, rp , rt , rtp }
y
dRIx
= γI Ixy
dt
y
dRAx
= γA Ayx
dt
(S159)
(S160)
18
and finally, for the depletion of the finite stockpiles of AV drugs,
dOp
rp
rt
rt
w
w
+ αrp Eprp + Enp
Enp
= −κω αpw Epw + αnp
Enp
+ αprt Eprt + αnp
dt
rtp
+αrtp Eprtp + Enp
w w
dOt
rtp
rp
rt
rt
.
+ αrtp Enp
+ αrp Enp
Enp
= −ψω αnp
Enp + αnp
dt
(S161)
(S162)
When the stockpile of AV drugs for prophylaxis, Op , or treatment, Ot , runs out that form of
intervention is no longer possible. Within the simulations, we set or ψ to zero as appropriate from
that point onwards.
R0 is, again, the maximum of four eigenvalues (one for each of the strains):

w
w
χ (1 − ρp ) es ei K w
+
1
−
α
α

p
p


w
w

+ αnp
[et (1 − ρt ) ψ + (1 − ψ)] + 1 − αnp
χ (1 − K w ) ,




 φrp {(αrp + (1 − αrp ) χ) K rp
β
+ (αrp [et (1
− ρt ) ψ + (1 − ψ)] + (1 − αrp ) χ) (1 − K rp )} ,
R0 = × max

γ

φrt αprt + 1 − αprt χ (1 − ρp) es ei K rt 


rt
rt

+ αnp
+ 1 − αnp
χ (1 − K rt ) ,


 rtp rtp
rtp
φ {α + (1 − α ) χ}
(S163)
S2
Results
We consider a number of different scenarios to demonstrate the general nature of the results presented in the main text. We also provide a sensitivity analysis for the single-drug, random allocation
and treatment and prophylaxis models.
S2.1
Alternative scenarios
In this section we examine two plausible alternative scenarios where it would be expected that a
successful intervention is possible:
1. The resistant strain has a higher relative fitness than φ = 0.8, but a lower seeding rate. We
examine a 90% fit resistant strain (φ = 0.9) with a seeding rate of 10 or 100 times lower
(ρt = 10−2 or ρt = 10−3 ). We keep ρp = ρt /10.
2. The symptomatic proportion is somewhat lower. With a fixed clinical attack rate of 40%
this implies a higher R0 . We examine a scenario with α = 0.6 and an increased (but still
realistic) intervention of providing prophylaxis to 40% of contacts (up from 30% in the main
text) and treatment to 50% of symptomatic cases (up from 40% in the main text).
S2.1.1
Extremely high fitness and lower seeding
w
We keep the symptomatic proportion unchanged at αnp
= 0.7, and the intervention unchanged at
= 0.3, ψ = 0.4. We restrict ourselves to a 90–10 stockpile. Figures S3– S5 are the equivalent of
Figures 5–7 in the main text. While each strategy is less capable of controlling the epidemic compared to the main text, the treatment and prophylaxis strategy remains the most effective choice. In
fact, especially in the case of lower seeding, it is more clear cut that treatment and prophylaxis is
19
a better strategy compared to either single-drug usage or a random allocation strategy. The reason
is clear: with higher fitness (φ = 0.9) it becomes even more important to delay emergence of the
multi-drug resistant strain.
5
5
x 10
7
Number of symptomatic infections
Number of symptomatic infections
7
6
5
4
3
2
1
0
0
0.5
1
Time (years)
1.5
x 10
6
5
4
3
2
1
0
0
2
0.5
(a)
1
Time (years)
1.5
2
(b)
Figure S3: a. ρt = 0.01 (one order of magnitude less than usual) b. ρt = 0.001 (two orders of
magnitude less than usual).
6
x 10
Cumulative number of symptomatic infections
Cumulative number of symptomatic infections
6
8
7
6
5
4
3
2
1
0
0
0.5
1
Time (years)
1.5
2
(a)
8
x 10
7
6
5
4
3
2
1
0
0
0.5
1
Time (years)
1.5
2
(b)
Figure S4: a. ρt = 0.01 (one order of magnitude less than usual) b. ρt = 0.001 (two orders of
magnitude less than usual).
20
1
0.8
0.8
0.6
0.6
Proportion
Proportion
1
0.4
0.2
0.4
0.2
0
0
0.5
1
Time (years)
1.5
0
0
2
0.5
1
Time (years)
(a)
1.5
2
(b)
Figure S5: a. ρt = 0.01 (one order of magnitude less than usual) b. ρt = 0.001 (two orders of
magnitude less than usual).
S2.1.2
A reduced symptomatic proportion
Consider a scenario where α = 0.6 (R0 = 1.x). Without an increase in the intervention the
impact of treatment and/or prophylaxis is minimal (see Figure 2b in the main text). Here, we
examine the ability of an increased intervention to delay the onset of the epidemic. Because the
baseline epidemic duration is shorter, an increased intervention can not produce delays to median
infection as great as for α = 0.7. However, a similar multiplicative increase in the time to median
infection can be achieved and the different strategies (single-drug, random allocation, treatment and
prophylaxis) have the same qualitative impacts. Figure S6 shows the epidemic curves, cumulative
curves and cumulative resistant proportion for a 90–10 stockpile scenario.
5
6
8
6
4
2
0
0
0.5
1
Time (years)
1.5
2
8
x 10
1
7
0.8
6
5
Proportion
x 10
Cumulative number of symptomatic infections
Number of symptomatic infections
10
4
3
0.6
0.4
2
0.2
1
0
0
(a)
0.5
1
Time (years)
(b)
1.5
2
0
0
0.5
1
Time (years)
1.5
2
(c)
w
Figure S6: αnp
= 0.6, = 0.4, ψ = 0.5. a. Epidemic curves. b. Cumulative curves. c Cumulative
resistance proportion.
We explore this scenario (α = 0.6, = 0.4, ψ = 0.5) a little more below when performing a
sensitivity analysis for multi-drug strategies.
S2.2
Sensitivity Analysis
Throughout the main text we assumed that:
21
1. antiviral drugs had a fixed efficacy. For treatment, we set et = 0.7 (relative infectiousness).
For prophylaxis, we set es = 0.5 (relative susceptibility) and ei = 0.4 (relative infectiousness
if nevertheless infected).
2. resistant strain seeding was either “high” (ρt = 10−1 , ρp = 10−2 ) or “low” (ρt = 10−3 ,
ρp = 10−4 ).
3. relative transmissibility of asymptomatic infections was 50% (χ = 0.5).
Here, we examine sensitivity to these parameter choices. We also compare conclusions regarding
optimal strategy across these same parameter sensitivity analyses. Throughout, we restrict ourselves to a high-fitness (φ = 0.8) resistant strain scenario. In the multi-drug strategy section we
further restrict the analysis to high seeding only (ρt = 10−1 , ρp = 10−2 ).
w
We perform this analysis with the symptomatic proportion fixed as in the main text (αnp
= 0.7).
The intervention also remains as in the main text ( = 0.3, ψ = 0.4).
S2.2.1
Single drug model
1
1.6
Time of median infection (years)
Cumulative proportion resistant at expiry of stockpile
With a single drug for treatment and prophylaxis, it is implausible that et and ei vary independently.
Thus, we tie et to ei : as ei varies from 0.4 to 1, et varies from 0.7 to 1. Figure S7 shows the singledrug model’s sensitivity to variation in antiviral drug efficacy.
0.8
0.6
0.4
0.2
0
0.4
0.5
0.6
0.7
ei
0.8
0.9
1.4
1.2
1
0.8
0.6
0.4
0.2
0.4
1
(a)
0.5
0.6
0.7
ei
0.8
0.9
1
(b)
Figure S7: Impact of simultaneously varying ei ∈ [0.4, 1], et ∈ [0.7, 1] and es ∈ [0.5, 1], for high
fitness (φ = 0.8, solid line) and low fitness (φ = 0.3, dashed line). The seeding rate is kept fixed at
ρt = 10−1 , ρp = 10−2 . a. The unfit resistant strain (dashed line) is unable to establish itself in the
population. For the fit resistant strain (solid line) the proportion of resistant infections at stockpile
expiry increases as ability to reduce transmission increases (ei , et and es getting smaller). b. The
unfit resistant strain (dashed line) always leads to a longer time to median infection than the fit
resistant strain (solid line). For the unfit strain, improving AV transmission reduction (ei getting
smaller) increases the time to median infection. For the fit resistant strain, as the AV intervention
becomes effective enough to lead to a resistant-strain dominated epidemic we find the time to
median infection flattens out. The properties of the AV intervention have little influence on an
epidemic dominated by resistant virus.
22
1
1.4
Time of median infection (years)
Cumulative proportion resistant at expiry of stockpile
Figure S8 shows the single-drug model’s sensitivity to variation in the seeding rate of the resistant strain. We always keep ρp = ρt /10.
0.8
0.6
0.4
0.2
0 −4
10
−3
10
−2
ρt
10
1.3
1.2
1.1
1
0.9
0.8
0.7 −4
10
−1
10
(a)
−3
10
−2
ρt
10
−1
10
(b)
Figure S8: Impact of varying ρt and ρp (logscale) for φ above the threshold (solid line) and φ
below the threshold (dashed line). a. Above the threshold, increasing the seeding rate increases the
proportion resistant. Below the threshold, the proportion resistant is essentially independent of ρt
and ρp . b. Time to median infection decreases as seeding rate increases above the threshold and
increases above the threshold (the “immunising” effect).
Figure S9 shows the single-drug model’s sensitivity to variaiton in the assumed relative transmissibility of asymptomatic infections.
23
2
Time of median infection (years)
Cumulative proportion resistant at expiry of stockpile
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
0.2
0.4
χ
0.6
0.8
1
1.8
1.6
1.4
1.2
1
0.8
0
(a)
0.2
0.4
χ
0.6
0.8
1
(b)
Figure S9: Impact of varying χ. Colour is seeding rate (ρt = 0.1, ρp = 0.01 (blue), ρt = 0.01,
ρp = 0.001 (red), ρt = 0.001, ρp = 0.0001 (green)). Solid lines are high fitness (φ = 0.8), dashed
lines are low fitness (φ = 0.3). a. For low fitness strains there is no ability to generate significant
proportions of resistance. For high fitness strains, and for high seeding (blue) we have a resistantstrain dominated epidemic. Therefore, relative transmissibility of asymptomatics is not important.
For lower seeding (red and green), the epidemic is a mixture of wild-type and resistant-type strains
and thus, as χ increases and the wild-type is more able to continue transmitting, the proportion
resistant is reduced. b. For transmissible resistant strains (solid lines) we generally have significant
proportions resistant so the time to median infection is fairly independent of χ. For untransmissible
resistant strains (dotted lines) the time to median infection is reduced as asymptomatics are more
capable of transmitting.
S2.2.2
Random allocation model
In a random allocation strategy, both drugs 1 and 2 are used for treatment and prophylaxis. Therefore, for each drug we must tie ei , et and es as in the single drug strategy. However, we can vary
the efficacy of the two drugs. For a fixed fitness and seeding (φ = 0.8, ρt = 10−1 , ρp = 10−2 ),
we vary ei1 and ei2 independently. et1 and es1 are locked to ei1 and et2 and es2 are locked to ei2 as
in the single drug strategy sensitivity analysis. Figure S10 shows the random allocation model’s
sensitivity to variation in antiviral drug efficacy for a 90–10 stockpile split. Figure S11 shows the
result for a 50–50 stockpile split.
For the 90–10 split, we see, unsurprisingly, little sensitivity to the properties of drug 2. For the
50–50 split the system is symmetric in drug 1 and drug 2.
24
0.8
0.7
0.5
0.6
0.5
0
0.4
0.4
0.6
Time to median infection (years)
Cumulative proporiton resistant at expiry
0.9
1
0.8
1
0.7
0.65
0.5
0.6
0.55
0
0.4
0.5
0.4
0.8
1
0.8
ei1
1
0.4
0.8
0.2
0.6
0.45
0.6
0.3
ei2
0.75
ei2
0.6
1
(a)
0.35
0.8
ei1
1
0.4
(b)
0.9
1
0.8
0.7
0.5
0.6
0.5
0
0.4
0.4
0.6
0.3
0.4
0.8
ei2
0.6
1
0.2
0.8
ei1
1
Time to median infection (years)
Cumulative proporiton resistant at expiry
Figure S10: Random allocation sensitivity to ei1 and ei2 . et1 and et2 vary in the interval [0.7, 1] and
es1 and es2 vary in the interval [0.5, 1] as ei1 and ei2 vary in the interval [0.4, 1]. The stockpile split
is 90–10. a. Cumulative proportion resistant at expiry of stockpile. b. Time to median infection.
1
0.8
0.7
0.5
0.6
0
0.4
0.5
0.6
ei2
(a)
0.4
0.8
0.4
0.6
1
0.8
ei1
1
(b)
Figure S11: Details as in Figure S11 but for a 50–50 stockpile split. a. Cumulative proportion
resistant at expiry of stockpile. b. Time to median infection.
Figures S12 and S13 show the random allocation model’s sensitivity to variation in the assumed relative transmissibility of asymptomatic infections with a 90–10 and 50–50 stockpile split
w
respectively. We show two scenarios: as in the main text (αnp
= 0.7, = 0.3, ψ = 0.4, blue) and
w
the alternative scenario presented above in Section S2.1.2 (αnp = 0.6, = 0.4, ψ = 0.5, red).
For the 90-10 stockpile split, most resistance (that is, more than half) is single-drug resistance.
The results are largely insensitive to χ While the proportion resistant at stockpile expiry is similar
for the α = 0.6 scenario (compared to the α = 0.7 scenario), the time to median infection is much
shorter. This reflects the naturally faster dynamics due to the higher R0 .
For a 50–50 stockpile, fewer of the infections are resistant in total compared to in the 90–10
case, but of those that are, multi-strain resistance is dominant.
25
0.9
Time of median infection (years)
Cumulative proporiton resistant at expiry
1
0.8
0.6
0.4
0.2
0.85
0.8
0.75
0.7
0.65
0.6
0.55
0.5
0
0
0.2
0.4
χ
0.6
0.8
0.45
0
1
0.2
0.4
(a)
χ
0.6
0.8
1
(b)
w
= 0.7, = 0.3, ψ = 0.4). The alternative
Figure S12: The main text scenario is in blue (αnp
w
scenario is in red (αnp = 0.6, = 0.4, ψ = 0.5). The stockpile split is 90–10. a. Solid lines
are total proportion resistant. Dashed lines are the proportion of all infections that are multi-drug
resistant. b. Solid lines are the time of median infection.
0.95
0.9
Time of median infection (years)
Cumulative proporiton resistant at expiry
1
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0
0.9
0.85
0.8
0.75
0.7
0.65
0.6
0.55
0.2
0.4
χ
0.6
0.8
0.5
0
1
(a)
0.2
0.4
χ
0.6
0.8
1
(b)
Figure S13: Details as in Figure S12 but for a 50–50 stockpile.
S2.2.3
Treatment and Prophylaxis model
In a treatment and prophylaxis strategy we have one drug used solely for treatment and the other
solely for prophylaxis. It follows that we examine sensitivity to antiviral drug efficacy in the
(ei , et )-plane. We tie es to ei as in the single-drug sensitivity analysis. Figure S14 shows the
proportion resistant at stockpile expiry and the time of median infection as a function of ei and et .
26
1
0.7
0.6
0.5
0.5
0
0.7
0.4
0.3
0.8
0.4
0.9
1
1
1.5
0.9
1
0.8
0.5
0.7
0
0.7
0.6
0.4
0.9
0.8
ei
1
0.5
0.8
0.2
0.6
et
Time of median infection (years)
Cumulative proporiton resistant at expiry
0.8
0.4
0.6
1
et
0.8
ei
1
(a)
(b)
Figure S14: Treatment and prophylaxis drugs strategy’s sensitivity to ei and et . The susceptibility
factor is fixed at es = 0.5. a. Cumulative proportion resistant at expiry of stockpile. b. Time to
median infection.
Figure S15 shows the variation in output as a function of relative transmissibility of asymptomatic infections, for both the main text scenario (α = 0.7, blue) and the alternative scenario
considered earlier in this document (α = 0.6, red).
1.1
Time of median infection (years)
Cumulative proporiton resistant at expiry
1
0.8
0.6
0.4
0.2
0
0
0.2
0.4
χ
0.6
0.8
1
0.9
0.8
0.7
0.6
0.5
0.4
0
1
(a)
0.2
0.4
χ
0.6
0.8
1
(b)
Figure S15: Details as in Figure S12 but for the treatment and prophylaxis strategy.
S2.2.4
Comparing models
We have explored the sensitivity of each model to key parameter assignments. Here, as in the main
text, we make a comparison between models.
A comparison of Figures S12 and S15 demonstrates that across the range of χ values considered, the treatment and prophylaxis strategy provides better outcomes, in terms of both proportion
of infections resistant and time of median infection, than does the random allocation strategy. The
single-drug strategy result (Figure S9, solid blue line) is inferior to both multi-drug strategies.
27
Time of median infection (years)
Cumulative proporiton resistant at expiry
To make a comparison in terms of sensitivity to antiviral drug effectiveness is more difficult.
If there is a large asymmetry in the effectiveness of the two drugs, the treatment and prophylaxis
strategy can yield a poorer result compared to the random allocation strategy.
Consider a situation where the 10% stockpile (drug 2) turns out to be ineffective (et → 1,
ei2 → 1, et2 → 1). For the random allocation strategy, this is of little consequence. The 90%
stockpile (drug 1) is used for both treatment and prophylaxis and the result is essentially that given
by a single-drug strategy. For the treatment and prophylaxis strategy however, we essentially have
a prophylaxis only single-drug strategy. Our previous work (McCaw and McVernon, 2007) has
shown that a “synergistic” effect is observed when effective treatment (low et ) is introduced in a
population receiving prophylaxis. Thereby, for efficacious drug 1 (low ei , ei1 and et1 ) and poor
drug 2 (high et , ei2 and et2 ) in Figure S16b we find that a random allocation strategy provides a
longer time to median infection than does a treatment and prophylaxis strategy.
1
0.5
0
Increasing drug 1
effectiveness
1.5
1
0.5
0
Increasing drug 2
effectiveness
(a)
Increasing drug 1
effectiveness
Increasing drug 2
effectiveness
(b)
Figure S16: For a 90–10 stockpile split, we compare the random allocation (blue) and treatment and prophylaxis (red) models. The two surfaces are those already presented in Figures S10
and S14. For the blue surface, ei1 and et1 vary on the drug 1 axis, and ei2 and et2 on the drug 2 axis.
For the red surface, ei varies on the drug 1 axis and et varies on the drug 2 axis. The impact on susceptibility of each drug is kept fixed throughout at es = 0.5 for both strategies. If we also tie es to
ei (for drug 1 in the treatment and prophylaxis model, and for both drugs in the random allocation
model) we find a qualitatively similar result. a. Cumulative proportion resistant at stockpile expiry.
The random allocation surface lies above the treatment and prophylaxis surface for most parameter values. b. Time of median infection. For the most part, the treatment and prophylaxis strategy
provides a longer time to median infection. Only when drug 2 (the 10% drug) is ineffective and
drug 1 is effective does the random allocation model provide a longer time to median infection.
A
Multi-drug model construction
This technical appendix provides the diagrams and R0 calculations for the multi-drug models.
A.1
Random allocation model
The possible flows are shown in the following figures.
28
S
w
es1 Kp1
w
αp1
(1 − ρp1 )
w
Ep1
w
Ip1
w
(1 − αp1
)
Aw
p1
ρp1
r1
αp1
r1
Ip1
w
es2 Kp2
r1
)
(1 − αp1
Arp11
w
αp2
(1 − ρp2 )
w
Ep2
w
Ip2
w
(1 − αp2
)
Aw
p2
ρp2
w
w
(1 − Kp1
− Kp2
)
r2
αp2
r2
Ip2
r2
(1 − αp2
)
Arp22
See next figure
Figure S17: Tree diagram for wild strain exposures (prophylaxis part)
29
From above figure (S)
w
w
(1 − Kp1
− Kp2
)
w
Enp
w
αnp
(1 − ρt1 )
f1 ψ
w
Inp,t1
ρt1
r1
Inp,t1
f2 ψ
(1 − ρt2 )
w
Inp,t2
ρt2
w
)
(1 − αnp
r2
Inp,t2
(1 − ψ)
w
Inp,nt
Aw
np
Figure S18: Tree diagram for wild strain exposures (treatment part)
30
S
r1
Kp1
r1
αp1
r1
Ep1
r1
Ip1
r1
)
(1 − αp1
Arp11
r1
es2 Kp2
r1
αp2
(1 − ρp2 )
r1
Ep2
r1
Ip2
r1
(1 − αp2
)
Arp21
ρp2
αr12
(1 −
r1
Kp1
−
r12
Ip2
r1
Kp2
)
(1 − αr12 )
See next figure
Arp212
Figure S19: Tree diagram for drug 1-resistant strain exposures (prophylaxis part)
31
From above figure (S)
r1
r1
(1 − Kp1
)
− Kp2
r1
Enp
r1
αnp
f1 ψ
r1
Inp,t1
f2 ψ
(1 − ρt2 )
r1
Inp,t2
ρt2
r1
(1 − αnp
)
r12
Inp
(1 − ψ)
r1
Inp,nt
Arnp1
Figure S20: Tree diagram for drug 1-resistant strain exposures (treatment part)
32
S
r2
es1 Kp1
r2
Ep1
r2
αp1
(1 − ρp1 )
r2
Ip1
r2
)
(1 − αp1
Arp12
ρp1
αr12
r12
Ip1
r2
Kp2
(1 − αr12 )
Arp112
r2
αp2
r2
Ep2
r2
Ip2
r2
(1 − αp2
)
r2
r2
(1 − Kp1
− Kp2
)
Arp22
See next figure
Figure S21: Tree diagram for drug 2-resistant strain exposures (prophylaxis part)
33
From above figure (S)
r2
r2
(1 − Kp1
)
− Kp2
r2
Enp
r2
αnp
(1 − ρt1 )
f1 ψ
r2
Inp,t1
ρt1
r12
Inp
f2 ψ
r2
Inp,t2
(1 − ψ)
r2
(1 − αnp
)
r2
Inp,nt
Arnp2
Figure S22: Tree diagram for drug 2-resistant strain exposures (treatment part)
34
S
r12
Kp1
αr12
r12
Ep1
r12
Ip1
(1 − αr12 )
r12
Kp2
Arp112
αr12
r12
Ep2
r12
Ip2
(1 − αr12 )
(1 −
r12
Kp1
−
Arp212
r12
Kp2
)
αr12
r12
Enp
r12
Inp
(1 − αr12 )
Arnp12
Figure S23: Tree diagram for multi-resistant strain exposures
35
We have four force-of-infection row vectors (1 × 30):




 


ei1
0
0
0
 ei2 
 0 
 0 
0





 

 et1 
 0 
 0 
0





 

 0 
 0 
0
 et2 





 

 1 
 0 
 0 
0





 

 0 
 1 
 0 
0





 

 ei2 
 0 
0
 0 





 

 0 
 1 
 0 
0





 

 0 
 et2 
 0 
0





 

 0 
0
 0 
 1 





 

 0 
 0 
 ei1 
0





 

 0 
 0 
 1 
0





 

0
 0 
 0 
 et1 





 

 0 
 0 
 1 
0





 






 

0
0
0  ~r1 0
r1  0  ~r2 0
r2  1 
r12  0 
r
w
~
~

12
λ =β
,
λ
=
βφ
,
λ
=
βφ
,
λ
=
βφ
 0 

 0 
1
 0 



 

 0 
 0 
 0 
1





 

 0 
 0 
 0 
1



 



ei1 χ
 0 
 0 
0





 

ei2 χ
 0 
 0 
0





 

 χ 
 0 
 0 
0


 




 0 
 χ 
 0 
0





 

 0 
ei2 χ
 0 
0





 

 0 
 χ 
 0 
0
 






 0 
 0 
ei1 χ
0





 

 0 
 0 
 χ 
0





 

 0 
 χ 
0
 0 





 

 0 
 0 
 0 
χ





 

 0 
 0 
 0 
χ
0
0
0
χ
36
(S164)
We construct column vectors F~w , F~r1 , F~r2 and F~r12 as follows:




w
w
(1 − ρp1 ) es1 Kp1
αp1
0
w
w




αp2
(1 − ρp2 ) es2 Kp2
0






αw (1 − ρt1 ) f1 ψ 1 − K w − K w 
0
p1
p2 


 np


αw (1 − ρ ) f ψ 1 − K w − K w 
0
t2
2


 np

p1
p2



 αw (1 − ψ) 1 − K w − K w
0




p2
p1
np
r
r
r1




w
αp11 Kp11
ρp1 es1 Kp1
αp1




r1
r1




(1
−
ρ
)
e
K
α
0
p2
s2 p2 



p2
r1
r1



 αw ρ f ψ 1 − K w − K w
r1
−
K
f
ψ
1
−
K
α




p2
p1
np 1
p2
p1
np t1 1




r
r
r
0
αnp1 (1 − ρt2 ) f2 ψ 1 − Kp11 − Kp21 



 αr1 (1 − ψ) 1 − K r1 − K r1


0




p2
p1
np




0
0








r2
w
0
αp2 ρp2 es2 Kp2








0
0








w
w
w
0
− Kp2
ρt2 f2 ψ 1 − Kp1



 αnp




0
0

 ~r1 

,
F
=
F~w = 



0
0








r
r12
1
α ρp2 es2 Kp2
0








r1
r1
r1
α
ρ
f
ψ
1
−
K
−
K
0




p1
p2
np t2 2




w
w
0




1 − αp1 (1 − ρp1 ) es1 Kp1




w
w



0
1 − αp2 (1 − ρp2 ) es2 Kp2 




w
w
w




0 1 − αnp
1− Kp1
− Kp2




r1
r1
r1
w




1 − αp1 Kp1
1 − αp1 ρp1 es1 Kp1




r1
r1



1 − αp2 (1 − ρp2 ) es2 Kp2 
0




r1
r1
r1




1 − αnp
1 − Kp1
− Kp2
0








0
0




r2
w




0
1
−
α
ρ
e
K
p2
s2
p2
p2








0
0








0
0




r1
r
12




(1 − α ) ρp2 es2 Kp2
0
0
0
(S165)
37
and



0
0




0
0








0
0








0
0








0
0








0
0








0
0








0
0








0
0








0
0




r2
r2




αp1 (1 − ρp1 ) es1 Kp1
0






r2
r2


αp2 Kp2
0




αr2 (1 − ρ ) f ψ 1 − K r2 − K r2 


0
 np
t1
1
p1 p2 




r2
r2
r2


0
αnp f2 ψ 1 − Kp1 − Kp2 





r2
r2
r2


0
−
K
(1
−
ψ)
1
−
K
α


p2
p1
np
 (S166)
F~r2 = 
 , F~r12 = 
r
r2
r
12
r12
12


α
K
α ρp1 es1 Kp1


p1




r12 r12


α Kp2
0






r
r
r
r
r2
 αr12 1 − Kp112 − Kp212

ρt1 f1 ψ 1 − Kp12 − Kp22

 αnp






0
0








0


0






0


0






0


0








0
0








0
0




r2
r2




0
1 − αp1
(1 − ρp1 ) es1 Kp1




r2
r2




0
1− αp2 Kp2




r
r


r
2
2


0
1 − αnp2 1 − Kp1 − Kp2




r12
r12
r2


r12


(1
−
α
)
K
(1
−
α
)
ρ
e
K
p1 s1 p1
p1




r
r
12




(1 − α 12 ) Kp2
0
r12
r12
r12
(1 − α ) 1 − Kp1 − Kp2
0

R0 is the maximum eigenvalue of a 30 × 30 matrix. As in the single-drug model we can
put the matrix into block-triangular form where each block is constructed from F~x and λ~x (x ∈
38
{w, r1 , r2 , r12 }). There are four non-zero eigenvalues:

w
w
w
αp1
+ 1 − αp1
χ (1 − ρp1) es1 ei1 Kp1



w
w
w

+ αp2 + 1 − αp2 χ (1 − ρp2 ) es2 ei2 Kp2



w

+ αnp
[et1 f1 (1 − ρt1) ψ+ et2 f2 (1 − ρt2 ) ψ

+ (1 − ψ)]


w
w
w

+ 1− αnp χ 1 − Kp1 − Kp2 ,

r1


r1
r1
r1

+
1
−
α
α
φ

p1 χ Kp1
p1


r1
r1
r1

χ (1 − ρp2 ) es2 ei2 Kp2
+ 1 − αp2
+ αp2

β
r1
[f1 ψ + et2 f2 (1 −ρt2 ) ψ + (1 − ψ)]
+ αnp
R0 = × max

γ
r1
r1
r1

,
− Kp2
+ 1− αnp
χ 1 − Kp1



r2
r2
r2
r2

φ
αp1 + 1 − αp1 χ (1 −


ρp1 ) re2s1 ei1 Kp1

r2
r2


+ αp2 + 1 − αp2 χ Kp2


r2


[et1 f1 (1 − ρt1) ψ+ f2 ψ + (1 − ψ)]
+ αnp


r2
r2

r2

,
− Kp2
+ 1 − αnp
χ 1 − Kp1

 r12 r12
r12
φ {α + (1 − α ) χ}
A.2
(S167)
Treatment and Prophylaxis model
The possible flows are shown in the following figures.
We define the force of infection row vectors (1 × 18) (note the transpose):
 
 
 
 
ei
0
0
0
 et 
0
 0 
0
 
 
 
 
 1 
0
 0 
0
 
 
 
 
1
 0 
0
 0 
 
 
 
 
 0 
et 
 0 
0
 
 
 
 
 0 
1
 0 
0
 
 
 
 
 ei 
0
 0 
0
 
 
 
 
 0 
0
 1 
0
 
 
 
 
 0 
0
 0 
1
0
0
0
0
r
r
r
p
t
tp
r
r
r
w
  ~t
 ~p
  ~tp
 
λ~ = β 
 0  , λ = βφ  0  , λ = βφ  0  , λ = βφ  1 
 
 
 
 
ei χ
0
 0 
0
 
 
 
 
χ
0
 0 
0
 
 
 
 
0
 0 
χ
 0 
 
 
 
 
 0 
χ
 0 
0
 
 
 
 
 0 
0
ei χ
0
 
 
 
 
 0 
0
χ
0
 
 
 
 
 0 
0
 0 
χ
0
0
0
χ
39
(S168)
S
es K w
(1 − ρp )
Epw
αpw
Ipw
(1 − αpw )
Aw
p
ρp
(1 − K w )
αrp
r
Ip p
(1 − αrp )
r
App
w
Enp
w
αnp
ψ
(1 − ρt )
w
Inp,t
ρt
(1 − ψ)
rt
Inp
w
)
(1 − αnp
w
Inp,nt
Aw
np
Figure S24: Tree diagram for wild strain exposures
40
K rp
S
αrp
r
Ep p
r
Ip p
(1 − αrp )
r
App
(1 − K rp )
r
Enpp
αrp
ψ
(1 − ρt )
r
p
Inp,t
ρt
(1 − ψ)
r
Inptp
(1 − αrp )
r
p
Inp,nt
r
Anpp
Figure S25: Tree diagram for prophylaxis-resistant strain exposures
41
S
es K rt
Eprt
αprt
(1 − ρp )
Iprt
(1 − αprt )
Arpt
ρp
(1 − K rt )
αrtp
r
Ip tp
(1 − αrtp )
r
Aptp
rt
αnp
rt
Enp
rt
Inp
rt
(1 − αnp
)
Arnpt
Figure S26: Tree diagram for treatment-resistant strain exposures
42
K rtp
S
αrtp
r
Eptp
r
Ip tp
(1 − αrtp )
r
Aptp
(1 − K rtp )
αrtp
r
r
Enptp
Inptp
(1 − αrtp )
r
Anptp
Figure S27: Tree diagram for multi-resistant strain exposures
We construct column vectors F~w , F~rp , F~rt and F~rtp as follows:




αpw (1 − ρp ) es K w
0
w 
w


αnp
0


 w(1 − ρt ) ψ (1 − Kw )


 αnp (1 − ψ) (1 − K ) 
0




rp
w
rp rp




α ρp es K
α K


 r


α p (1 − ρt ) ψ (1 − K rp )

0


 r


 α p (1 − ψ) (1 − K rp ) 

0








0
0




w
w


 αnp ρt ψ (1 − K ) 
0








0
0
r
w
~
~




p
,
F
=
F =

 αrp ρt ψ (1 − K rp ) 
0






 1 − αw (1 − ρp ) es K w 
0
p






 1 − αw (1 − K w ) 
0
np




r
r
p
p


 (1 − αrp ) ρp es K w 
(1
−
α
)
K




rp
rp




0

 (1 − α ) (1 − K ) 





0
0








0
0








0
0
0
0
43
(S169)
and
















F~rt = 


















0
0



0
0






0
0






0
0






0
0






0
0



rt
rt



αp (1 − ρp ) es K
0



rt
rt



αnp (1 − K )
0



rtp rtp



αrtp ρp es K rt
α
K
 , F~rtp = 

r

 αrtp (1 − K tp ) 
0






0
0






0
0






0
0






0
0



rt
rt 


1 − αp (1 − ρp ) es K 
0


rt
rt



0
1 − αnp (1 − K ) 


 (1 − αrtp ) K rtp 
(1 − αrtp ) ρp es K rt 
(1 − αrtp ) (1 − K rtp )
0
(S170)
x
(x ∈ {w, rp , rt , rtp }).
where K w , K rp , K rt and K rtp are the steady state proportions of Cpx / Cpx + Cnp
R0 is the maximum eigenvalue of the 18×18 matrix F~w λ~w + F~rp λ~rp + F~rt λ~rt + F~rtp λ~rtp /γ.
We put the matrix into block-triangular form and obtain four non-zero eigenvalues:

w
w
χ (1 − ρp ) es ei K w
+
1
−
α
α

p
p


w
w

+
α
[e
(1
−
ρ
)
ψ
+
(1
−
ψ)]
+
1
−
α
χ (1 − K w ) ,

t
t
np
np


β
{αrp (K rp + [et (1 −ρt ) ψ + (1 − ψ)] (1 − K rp )) + (1 − αrp ) χ} ,
φrp R0 = × max
φrt αprt + 1 − αprt χ (1 − ρp) es ei K rt 
γ


rt
rt

+ 1 − αnp
χ (1 − K rt ) ,
+ αnp


 rtp rtp
φ {α + (1 − αrtp ) χ}
(S171)
References
J. M. McCaw and J. McVernon. Prophylaxis or treatment? Optimal use of an antiviral stockpile
during an influenza pandemic. Mathematical BioSciences, 209(2):336–360, 2007. 2, 28
44