Contingency planning for a deliberate release of smallpox in Great
Britain—the role of geographical scale and contact structure:
Supplementary Material
Thomas House, Ian Hall, Leon Danon and Matt J. Keeling
This supplementary material consists of three sections: firstly, we present the formal
mathematical underpinnings of our work, which are necessary for reproduction of the
results; secondly, we present a supplementary figure of sensitivity analysis; and finally,
we present the full system of differential equations that defines our model.
1 Mathematical model
1.1 Probability of a network event
Suppose we have a network node in a state ζ, from which it ‘recovers’ at rate g, and which
‘transmits’ its state to a neighbour currently in state γ at a rate τ. We then define
p(t) := Pr(Transmission happened at some time t′ ≤ t) ,
q(t) := Pr(The system remains in the state ζ ↔ γ at time t) .
(1)
This system is then described by the equations
dq
= −(τ + g)q(t) ,
dt
dp
= τq(t) .
dt
(2)
This has the solution
q(t) = e−(τ+g)t ,
p(t) =
τ 1 − e−(τ+g)t .
τ+g
(3)
So clearly, the final probability of transmission is τ/(τ + g) and the expected number of
transmission is
τ
,
(4)
hnumber of transmissionsi = n
τ+g
for a node with n links.
1
1.2 Keeping R0, RP constant
We consider two ways to fit our network rates to reproduction numbers. For the first of
these, we define the relevant type reproduction numbers through the ‘early growth’ of the
system, defined through linearisation of the pairwise system using the following Ansatz:
˙ = K[I] ,
[I]
[A] = [A]t=0 + kA [I] ,
[AB] = [AB]t=0 + kAB [I] .
(5)
We then define
rI :=
τI kIS
,
gI
rP :=
τI kPS
,
gP
r0 := rI + rP .
(6)
These ‘early growth’ reproduction numbers can then be set to the desired values for R0 , RP.
Substituting these together with the linear Ansatz and ignoring terms of order I(t)2 and
higher allows us to fix τI , τP . It is worth noting that during the early part of an epidemic,
the growth of all disease compartments is governed by just one Malthusian parameter, and
so rI and rP should be interpreted as an natural, approximate apportionment of populationlevel disease growth to each disease class rather than a rigorous result for prodromal and
infectious reproduction numbers on networks.
An alternative approach is to hold constant the literal expected number of secondary
infections created by a single infectious individual in a fully susceptible population (the
textbook definition of R0 ), which we write as
RI :=
nτI
,
τI + gI
RP :=
nτP
,
τP + gP
R0 := RI + RP .
(7)
The reasoning behind these definitions is expounded in Section 1.1 above. We note
that contact network structure will modify (7) after the initial generation, however any
individual-level measurement in the release scenario we are considering will almost certainly still be before this modification.
1.3 Improved closure methods
The unclosed pairwise equations, generated automatically from rules using Mathematica
6.0 code, are included in section 3 below. To integrate these, approximations linking
triples and pairs are needed.
Our code relies on a novel method for closure of triples. Where the prevalence of [AB]
is being modified by the action of a C upon B, the following approximation significantly
improves the behaviour of the pairwise system during numerical integration:
!
[AB][BC]
[AB][BC][CA]
[ABC] ≈ (n − 1) (1 − φ)
.
(8)
+φ
P
n[B]
[A] a ([aB][aC]/[a])
We found that even sophisticated software such as Mathematica, using adaptive algorithms for ODE integration and a working precision of 50, gave unacceptable performance
for the sheer number and complexity of equations involved using standard closure. This
problem was avoided for the improved closure above.
1.4 Escape from control region
If we suppose that the control region exerts an external force of infection, f (t), over time,
then the probability of escape over time, h(t), is given in terms of this force of infection
by
ḣ(t) = (1 − h(t)) f (t) .
(9)
This model is generally tractable in the sense that (9) can be integrated directly to give
h(t) = 1 − e−
Rt
0
f (u)du
(10)
,
however we need a full expression for the force of infection, which is
f (t) =
nξ
(τP [P] + (1 − κ)τI [I]) .
N
(11)
1.5 Trade-off for optimisation
We start by making the following definitions for final escape probability, attack rate and
proportion vaccinated in m-ary affected regions:
(m)
(m)
h∞
:= lim h(t, m) , R(m)
∞ := lim [R]t,m , V∞ := lim [V]t,m .
t→∞
t→∞
t→∞
The expected total deaths from an outbreak in one region is then
1/M
(1)
(2)
(2)
(1
)
T := R(1)
δ
+
V
δ
+
M
1
−
−
h
R
δ
+
V
δ
,
V
∞
V
∞
∞
∞
∞
(12)
(13)
where M is the number of other regions of the same type as the initial outbreak region,
(2)
shown in Figure 3(a) of the main text. R(2)
∞ and V∞ are calculated by modifying both
Itrig and E 0 to one case, as in Table 1 of the main text. The equivalent expression for a
dispersed outbreak is
(2)
δ
+
V
δ
.
(14)
T d := M0 + max(M + 1 − M0 , 0) 1 − (1 − h∞ )1/(M+1−M0 ) R(2)
V
∞
∞
2 Supplementary figure
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Prodromal type reproductive ratio, RP
(d) Varying RP
Figure 1: Effects of varying network and disease parameters on cumulative escape probability.
3 Complete system of unclosed ODEs
[E]0 = E 0
[S ]0 = (N − E 0 )(1 − γ)
[S V ]0 = (N − E 0 )γ
[EE]0 =
[ES ]0 =
[ES V ]0 =
[S E]0 =
[S S ]0 =
[S S V ]0 =
[S V E]0 =
[S V S ]0 =
[S V S V ]0 =
˙ =
[E]
E 02 n
N
E 0 n(N − E 0 )(1 − γ)
N
E 0 n(N − E 0 )γ
N
E 0 n(N − E 0 )(1 − γ)
N
n(N − E 0 )2 (1 − γ)2
N
n(N − E 0 )2 (1 − γ)γ
N
E 0 n(N − E 0 )γ
N
n(N − E 0 )2 (1 − γ)γ
N
n(N − E 0 )2 γ2
N
− gE [E] − ρ[EQ] + τI [S I]
+ τP [S P] + τI [S V I] + τP [S V P]
[E˙T ] = − gE (1 − ǫ2 )[E T ] − gE ǫ2 [E T ]
+ ρ[EQ]
˙ = − gI [I] + gP (1 − θ)[P]
[I]
− ρ[IQ]
˙ = − gO (1 − ǫ1 )[O] − gO ǫ1 [O]
[O]
+ ρ[S Q] + ρ[S V Q]
˙ = gE [E] − gP (1 − θ)[P] − ρ[PQ]
[P]
− gP θ[P]Θ([I] + [R] − Itrig )
˙ = gE (1 − ǫ2 )[E T ] − gQ [Q] + ρ[IQ]
[Q]
+ ρ[PQ] + gP θ[P]Θ([I] + [R] − Itrig )
˙ = gI [I] + gQ [Q]
[R]
[S˙ ] = gO (1 − ǫ1 )[O] − τI [S I] − τP [S P]
− ρ[S Q]
[S˙V ] = − τI [S V I] − τP [S V P] − ρ[S V Q]
− νΘ([I] + [R] − Itrig )Θ([S V ])
˙ = gE ǫ2 [E T ] + gO ǫ1 [O] + νΘ([I] + [R] − Itrig )Θ([S V ])
[V]
˙ = − 2gE [EE] − 2ρ[EEQ] + 2τI [ES I]
[EE]
+ 2τP [ES P] + 2τI [ES V I] + 2τP [ES V P]
˙ T ] = − gE [EE T ] − gE (1 − ǫ2 )[EE T ]
[EE
− gE ǫ2 [EE T ] + ρ[EEQ] − ρ[E T EQ] + τI [E T S I]
+ τP [E T S P] + τI [E T S V I] + τP [E T S V P]
˙ = − gE [EI] − gI [EI] + gP (1 − θ)[EP]
[EI]
+ τI [S I] + τI [S V I] − ρ[EIQ] − ρ[IEQ]
+ τI [IS I] + τP [IS P] + τI [IS V I] + τP [IS V P]
˙ = − gE [EO] − gO (1 − ǫ1 )[EO]
[EO]
− gO ǫ1 [EO] + ρ[ES Q] + ρ[ES V Q] − ρ[OEQ]
+ τI [OS I] + τP [OS P] + τI [OS V I] + τP [OS V P]
˙ = gE [EE] − gE [EP] − gP (1 − θ)[EP]
[EP]
+ τP [S P] + τP [S V P] − ρ[EPQ] − ρ[PEQ]
+ τI [PS I] + τP [PS P] + τI [PS V I] + τP [PS V P]
− gP θ[EP]Θ([I] + [R] − Itrig )
˙ = gE (1 − ǫ2 )[EE T ] − gE [EQ] − gQ [EQ]
[EQ]
− ρ[EQ] + ρ[EIQ] + ρ[EPQ] − ρ[QEQ]
+ τI [QS I] + τP [QS P] + τI [QS V I] + τP [QS V P]
+ gP θ[EP]Θ([I] + [R] − Itrig )
˙ = gI [EI] + gQ [EQ] − gE [ER] − ρ[REQ]
[ER]
+ τI [RS I] + τP [RS P] + τI [RS V I] + τP [RS V P]
˙
[ES ] = gO (1 − ǫ1 )[EO] − gE [ES ] − τI [ES I]
− τP [ES P] − ρ[ES Q] − ρ[S EQ] + τI [S S I]
+ τP [S S P] + τI [S S V I] + τP [S S V P]
ν
[ES˙ V ] = − gE [ES V ] −
Θ([I] + [R] − Itrig )Θ([S V ])[ES V ]
[S V ]
− τI [ES V I] − τP [ES V P] − ρ[ES V Q] − ρ[S V EQ]
+ τI [S V S I] + τP [S V S P] + τI [S V S V I] + τP [S V S V P]
˙ = gE ǫ2 [EE T ] + gO ǫ1 [EO] − gE [EV] − ρ[V EQ]
[EV]
+ τI [VS I] + τP [VS P] + τI [VS V I] + τP [VS V P]
ν
+
[ES V ]Θ([I] + [R] − Itrig )Θ([S V ])
[S V ]
[E T˙ E] = − gE [E T E] − gE (1 − ǫ2 )[E T E]
− gE ǫ2 [E T E] + ρ[EEQ] − ρ[E T EQ] + τI [E T S I]
+ τP [E T S P] + τI [E T S V I] + τP [E T S V P]
[E T˙E T ] = − 2gE (1 − ǫ2 )[E T E T ] − 2gE ǫ2 [E T E T ]
+ 2ρ[E T EQ]
[E˙T I] = − gI [E T I] − gE (1 − ǫ2 )[E T I]
− gE ǫ2 [E T I] + gP (1 − θ)[E T P] − ρ[E T IQ]
+ ρ[IEQ]
[E T˙ O] = − gO (1 − ǫ1 )[E T O] − gO ǫ1 [E T O]
− gE (1 − ǫ2 )[E T O] − gE ǫ2 [E T O] + ρ[E T S Q]
+ ρ[E T S V Q] + ρ[OEQ]
[E T˙ P] = gE [E T E] − gE (1 − ǫ2 )[E T P] − gE ǫ2 [E T P]
− gP (1 − θ)[E T P] − ρ[E T PQ] + ρ[PEQ]
− gP θ[E T P]Θ([I] + [R] − Itrig )
[E T˙ Q] = ρ[EQ] + gE (1 − ǫ2 )[E T E T ] − gQ [E T Q]
− gE (1 − ǫ2 )[E T Q] − gE ǫ2 [E T Q] + ρ[E T IQ]
+ ρ[E T PQ] + ρ[QEQ] + gP θ[E T P]Θ([I] + [R] − Itrig )
[E T˙ R] = gI [E T I] + gQ [E T Q] − gE (1 − ǫ2 )[E T R]
− gE ǫ2 [E T R] + ρ[REQ]
[E T˙ S ] = gO (1 − ǫ1 )[E T O] − gE (1 − ǫ2 )[E T S ]
− gE ǫ2 [E T S ] − τI [E T S I] − τP [E T S P] − ρ[E T S Q]
+ ρ[S EQ]
[E T˙S V ] = − gE (1 − ǫ2 )[E T S V ] − gE ǫ2 [E T S V ]
ν
−
Θ([I] + [R] − Itrig )Θ([S V ])[E T S V ] − τI [E T S V I]
[S V ]
− τP [E T S V P] − ρ[E T S V Q] + ρ[S V EQ]
[E T˙ V] = gE ǫ2 [E T E T ] + gO ǫ1 [E T O] − gE (1 − ǫ2 )[E T V]
ν
[E T S V ]Θ([I] + [R] − Itrig )Θ([S V ])
− gE ǫ2 [E T V] + ρ[V EQ] +
[S V ]
˙ = − gE [IE] − gI [IE] + τI [IS ]
[IE]
+ τI [IS V ] + gP (1 − θ)[PE] − ρ[EIQ]
− ρ[IEQ] + τI [IS I] + τP [IS P] + τI [IS V I]
+ τP [IS V P]
[IE˙ T ] = − gI [IE T ] − gE (1 − ǫ2 )[IE T ]
− gE ǫ2 [IE T ] + gP (1 − θ)[PE T ] − ρ[E T IQ]
+ ρ[IEQ]
˙ = − 2gI [II] + gP (1 − θ)[IP]
[II]
+ gP (1 − θ)[PI] − 2ρ[IIQ]
˙ = − gI [IO] − gO (1 − ǫ1 )[IO]
[IO]
− gO ǫ1 [IO] + gP (1 − θ)[PO] + ρ[IS Q]
+ ρ[IS V Q] − ρ[OIQ]
˙ = gE [IE] − gI [IP] − gP (1 − θ)[IP]
[IP]
+ gP (1 − θ)[PP] − ρ[IPQ] − ρ[PIQ]
− gP θ[IP]Θ([I] + [R] − Itrig )
˙ = gE (1 − ǫ2 )[IE T ] − gI [IQ] − gQ [IQ]
[IQ]
− ρ[IQ] + gP (1 − θ)[PQ] + ρ[IIQ]
+ ρ[IPQ] − ρ[QIQ] + gP θ[IP]Θ([I] + [R] − Itrig )
˙ = gI [II] + gQ [IQ] − gI [IR] + gP (1 − θ)[PR]
[IR]
− ρ[RIQ]
˙ ] = gO (1 − ǫ1 )[IO] − gI [IS ] − τI [IS ]
[IS
+ gP (1 − θ)[PS ] − τI [IS I] − τP [IS P]
− ρ[IS Q] − ρ[S IQ]
ν
[IS˙ V ] = − gI [IS V ] − τI [IS V ] −
Θ([I] + [R] − Itrig )Θ([S V ])[IS V ]
[S V ]
+ gP (1 − θ)[PS V ] − τI [IS V I] − τP [IS V P]
− ρ[IS V Q] − ρ[S V IQ]
˙ = gE ǫ2 [IE T ] + gO ǫ1 [IO] − gI [IV] + gP (1 − θ)[PV]
[IV]
ν
− ρ[V IQ] +
[IS V ]Θ([I] + [R] − Itrig )Θ([S V ])
[S V ]
˙ = − gE [OE] − gO (1 − ǫ1 )[OE]
[OE]
− gO ǫ1 [OE] + ρ[ES Q] + ρ[ES V Q] − ρ[OEQ]
+ τI [OS I] + τP [OS P] + τI [OS V I] + τP [OS V P]
˙ T ] = − gO (1 − ǫ1 )[OE T ] − gO ǫ1 [OE T ]
[OE
− gE (1 − ǫ2 )[OE T ] − gE ǫ2 [OE T ] + ρ[E T S Q]
+ ρ[E T S V Q] + ρ[OEQ]
˙ = − gI [OI] − gO (1 − ǫ1 )[OI]
[OI]
− gO ǫ1 [OI] + gP (1 − θ)[OP] + ρ[IS Q]
+ ρ[IS V Q] − ρ[OIQ]
˙ = − 2gO (1 − ǫ1 )[OO] − 2gO ǫ1 [OO]
[OO]
+ 2ρ[OS Q] + 2ρ[OS V Q]
˙ = gE [OE] − gO (1 − ǫ1 )[OP] − gO ǫ1 [OP]
[OP]
− gP (1 − θ)[OP] − ρ[OPQ] + ρ[PS Q]
+ ρ[PS V Q] − gP θ[OP]Θ([I] + [R] − Itrig )
˙ = gE (1 − ǫ2 )[OE T ] − gQ [OQ] − gO (1 − ǫ1 )[OQ]
[OQ]
− gO ǫ1 [OQ] + ρ[S Q] + ρ[S V Q] + ρ[OIQ]
+ ρ[OPQ] + ρ[QS Q] + ρ[QS V Q] + gP θ[OP]Θ([I] + [R] − Itrig )
˙ = gI [OI] + gQ [OQ] − gO (1 − ǫ1 )[OR]
[OR]
− gO ǫ1 [OR] + ρ[RS Q] + ρ[RS V Q]
˙ ] = gO (1 − ǫ1 )[OO] − gO (1 − ǫ1 )[OS ]
[OS
− gO ǫ1 [OS ] − τI [OS I] − τP [OS P] − ρ[OS Q]
+ ρ[S S Q] + ρ[S S V Q]
[OS˙ V ] = − gO (1 − ǫ1 )[OS V ] − gO ǫ1 [OS V ]
ν
−
Θ([I] + [R] − Itrig )Θ([S V ])[OS V ] − τI [OS V I]
[S V ]
− τP [OS V P] − ρ[OS V Q] + ρ[S V S Q] + ρ[S V S V Q]
˙ = gE ǫ2 [OE T ] + gO ǫ1 [OO] − gO (1 − ǫ1 )[OV]
[OV]
ν
− gO ǫ1 [OV] + ρ[VS Q] + ρ[VS V Q] +
[OS V ]Θ([I] + [R] − Itrig )Θ([S V ])
[S V ]
˙ = gE [EE] − gE [PE] − gP (1 − θ)[PE]
[PE]
+ τP [PS ] + τP [PS V ] − ρ[EPQ] − ρ[PEQ]
+ τI [PS I] + τP [PS P] + τI [PS V I] + τP [PS V P]
− gP θ[PE]Θ([I] + [R] − Itrig )
˙ T ] = gE [EE T ] − gE (1 − ǫ2 )[PE T ] − gE ǫ2 [PE T ]
[PE
− gP (1 − θ)[PE T ] − ρ[E T PQ] + ρ[PEQ]
− gP θ[PE T ]Θ([I] + [R] − Itrig )
˙ = gE [EI] − gI [PI] − gP (1 − θ)[PI]
[PI]
+ gP (1 − θ)[PP] − ρ[IPQ] − ρ[PIQ]
− gP θ[PI]Θ([I] + [R] − Itrig )
˙ = gE [EO] − gO (1 − ǫ1 )[PO] − gO ǫ1 [PO]
[PO]
− gP (1 − θ)[PO] − ρ[OPQ] + ρ[PS Q]
+ ρ[PS V Q] − gP θ[PO]Θ([I] + [R] − Itrig )
˙ = gE [EP] + gE [PE] − 2gP (1 − θ)[PP]
[PP]
− 2ρ[PPQ] − 2gP θ[PP]Θ([I] + [R] − Itrig )
˙ = gE [EQ] + gE (1 − ǫ2 )[PE T ] − gQ [PQ]
[PQ]
− ρ[PQ] − gP (1 − θ)[PQ] + ρ[PIQ]
+ ρ[PPQ] − ρ[QPQ] + gP θ[PP]Θ([I] + [R] − Itrig )
− gP θ[PQ]Θ([I] + [R] − Itrig )
˙ = gE [ER] + gI [PI] + gQ [PQ] − gP (1 − θ)[PR]
[PR]
− ρ[RPQ] − gP θ[PR]Θ([I] + [R] − Itrig )
˙ ] = gE [ES ] + gO (1 − ǫ1 )[PO] − τP [PS ]
[PS
− gP (1 − θ)[PS ] − τI [PS I] − τP [PS P]
− ρ[PS Q] − ρ[S PQ] − gP θ[PS ]Θ([I] + [R] − Itrig )
[PS˙ V ] = gE [ES V ] − τP [PS V ] − gP (1 − θ)[PS V ]
− τI [PS V I] − τP [PS V P] − ρ[PS V Q] − ρ[S V PQ]
ν
− gP θ[PS V ]Θ([I] + [R] − Itrig ) −
[PS V ]Θ([I] + [R] − Itrig )Θ([S V ])
[S V ]
˙ = gE [EV] + gE ǫ2 [PE T ] + gO ǫ1 [PO] − gP (1 − θ)[PV]
[PV]
ν
− ρ[V PQ] − gP θ[PV]Θ([I] + [R] − Itrig ) +
[PS V ]Θ([I] + [R] − Itrig )Θ([S V ])
[S V ]
˙ = gE (1 − ǫ2 )[E T E] − gE [QE] − gQ [QE]
[QE]
− ρ[QE] + ρ[EIQ] + ρ[EPQ] − ρ[QEQ]
+ τI [QS I] + τP [QS P] + τI [QS V I] + τP [QS V P]
+ gP θ[PE]Θ([I] + [R] − Itrig )
˙ T ] = gE (1 − ǫ2 )[E T E T ] + ρ[QE] − gQ [QE T ]
[QE
− gE (1 − ǫ2 )[QE T ] − gE ǫ2 [QE T ] + ρ[E T IQ]
+ ρ[E T PQ] + ρ[QEQ] + gP θ[PE T ]Θ([I] + [R] − Itrig )
˙ = gE (1 − ǫ2 )[E T I] − gI [QI] − gQ [QI]
[QI]
− ρ[QI] + gP (1 − θ)[QP] + ρ[IIQ]
+ ρ[IPQ] − ρ[QIQ] + gP θ[PI]Θ([I] + [R] − Itrig )
˙ = gE (1 − ǫ2 )[E T O] − gQ [QO] − gO (1 − ǫ1 )[QO]
[QO]
− gO ǫ1 [QO] + ρ[QS ] + ρ[QS V ] + ρ[OIQ]
+ ρ[OPQ] + ρ[QS Q] + ρ[QS V Q] + gP θ[PO]Θ([I] + [R] − Itrig )
˙ = gE (1 − ǫ2 )[E T P] + gE [QE] − gQ [QP]
[QP]
− ρ[QP] − gP (1 − θ)[QP] + ρ[PIQ]
+ ρ[PPQ] − ρ[QPQ] + gP θ[PP]Θ([I] + [R] − Itrig )
− gP θ[QP]Θ([I] + [R] − Itrig )
˙ = gE (1 − ǫ2 )[E T Q] + ρ[IQ] + ρ[PQ]
[QQ]
+ gE (1 − ǫ2 )[QE T ] + ρ[QI] + ρ[QP]
− 2gQ [QQ] + 2ρ[QIQ] + 2ρ[QPQ] + gP θ[PQ]Θ([I] + [R] − Itrig )
+ gP θ[QP]Θ([I] + [R] − Itrig )
˙ = gE (1 − ǫ2 )[E T R] + gI [QI] + gQ [QQ]
[QR]
− gQ [QR] + ρ[RIQ] + ρ[RPQ] + gP θ[PR]Θ([I] + [R] − Itrig )
˙ ] = gE (1 − ǫ2 )[E T S ] + gO (1 − ǫ1 )[QO]
[QS
− gQ [QS ] − ρ[QS ] − τI [QS I] − τP [QS P]
− ρ[QS Q] + ρ[S IQ] + ρ[S PQ] + gP θ[PS ]Θ([I] + [R] − Itrig )
[QS˙ V ] = gE (1 − ǫ2 )[E T S V ] − gQ [QS V ] − ρ[QS V ]
− τI [QS V I] − τP [QS V P] − ρ[QS V Q] + ρ[S V IQ]
ν
[QS V ]Θ([I] + [R] − Itrig )Θ([S V ])
+ ρ[S V PQ] + gP θ[PS V ]Θ([I] + [R] − Itrig ) −
[S V ]
˙ = gE (1 − ǫ2 )[E T V] + gE ǫ2 [QE T ] + gO ǫ1 [QO]
[QV]
− gQ [QV] + ρ[V IQ] + ρ[V PQ] + gP θ[PV]Θ([I] + [R] − Itrig )
ν
[QS V ]Θ([I] + [R] − Itrig )Θ([S V ])
+
[S V ]
˙ = gI [IE] + gQ [QE] − gE [RE] − ρ[REQ]
[RE]
+ τI [RS I] + τP [RS P] + τI [RS V I] + τP [RS V P]
˙ T ] = gI [IE T ] + gQ [QE T ] − gE (1 − ǫ2 )[RE T ]
[RE
− gE ǫ2 [RE T ] + ρ[REQ]
˙ = gI [II] + gQ [QI] − gI [RI] + gP (1 − θ)[RP]
[RI]
− ρ[RIQ]
˙ = gI [IO] + gQ [QO] − gO (1 − ǫ1 )[RO]
[RO]
− gO ǫ1 [RO] + ρ[RS Q] + ρ[RS V Q]
˙ = gI [IP] + gQ [QP] + gE [RE] − gP (1 − θ)[RP]
[RP]
− ρ[RPQ] − gP θ[RP]Θ([I] + [R] − Itrig )
˙ = gI [IQ] + gQ [QQ] + gE (1 − ǫ2 )[RE T ]
[RQ]
− gQ [RQ] + ρ[RIQ] + ρ[RPQ] + gP θ[RP]Θ([I] + [R] − Itrig )
˙ = gI [IR] + gQ [QR] + gI [RI] + gQ [RQ]
[RR]
˙ ] = gI [IS ] + gQ [QS ] + gO (1 − ǫ1 )[RO]
[RS
− τI [RS I] − τP [RS P] − ρ[RS Q]
[RS˙ V ] = gI [IS V ] + gQ [QS V ] − τI [RS V I] − τP [RS V P]
ν
− ρ[RS V Q] −
[RS V ]Θ([I] + [R] − Itrig )Θ([S V ])
[S V ]
˙ = gI [IV] + gQ [QV] + gE ǫ2 [RE T ] + gO ǫ1 [RO]
[RV]
ν
[RS V ]Θ([I] + [R] − Itrig )Θ([S V ])
+
[S V ]
[S˙E] = gO (1 − ǫ1 )[OE] − gE [S E] − τI [ES I]
− τP [ES P] − ρ[ES Q] − ρ[S EQ] + τI [S S I]
+ τP [S S P] + τI [S S V I] + τP [S S V P]
˙ T ] = gO (1 − ǫ1 )[OE T ] − gE (1 − ǫ2 )[S E T ]
[S E
− gE ǫ2 [S E T ] − τI [E T S I] − τP [E T S P] − ρ[E T S Q]
+ ρ[S EQ]
[S˙I] = gO (1 − ǫ1 )[OI] − gI [S I] − τI [S I]
+ gP (1 − θ)[S P] − τI [IS I] − τP [IS P]
− ρ[IS Q] − ρ[S IQ]
[S ˙O] = gO (1 − ǫ1 )[OO] − gO (1 − ǫ1 )[S O]
− gO ǫ1 [S O] − τI [OS I] − τP [OS P] − ρ[OS Q]
+ ρ[S S Q] + ρ[S S V Q]
[S˙P] = gO (1 − ǫ1 )[OP] + gE [S E] − τP [S P]
− gP (1 − θ)[S P] − τI [PS I] − τP [PS P]
− ρ[PS Q] − ρ[S PQ] − gP θ[S P]Θ([I] + [R] − Itrig )
[S ˙Q] = gO (1 − ǫ1 )[OQ] + gE (1 − ǫ2 )[S E T ]
− gQ [S Q] − ρ[S Q] − τI [QS I] − τP [QS P]
− ρ[QS Q] + ρ[S IQ] + ρ[S PQ] + gP θ[S P]Θ([I] + [R] − Itrig )
[S˙R] = gO (1 − ǫ1 )[OR] + gI [S I] + gQ [S Q]
− τI [RS I] − τP [RS P] − ρ[RS Q]
[S˙S ] = gO (1 − ǫ1 )[OS ] + gO (1 − ǫ1 )[S O]
− 2τI [S S I] − 2τP [S S P] − 2ρ[S S Q]
[S S˙ V ] = gO (1 − ǫ1 )[OS V ] − τI [S S V I] − τP [S S V P]
− ρ[S S V Q] − τI [S V S I] − τP [S V S P] − ρ[S V S Q]
ν
−
[S S V ]Θ([I] + [R] − Itrig )Θ([S V ])
[S V ]
[S˙V] = gO (1 − ǫ1 )[OV] + gE ǫ2 [S E T ] + gO ǫ1 [S O]
ν
[S S V ]Θ([I] + [R] − Itrig )Θ([S V ])
− τI [VS I] − τP [VS P] − ρ[VS Q] +
[S V ]
ν
Θ([I] + [R] − Itrig )Θ([S V ])[S V E]
[S V˙ E] = − gE [S V E] −
[S V ]
− τI [ES V I] − τP [ES V P] − ρ[ES V Q] − ρ[S V EQ]
+ τI [S V S I] + τP [S V S P] + τI [S V S V I] + τP [S V S V P]
[S V˙E T ] = − gE (1 − ǫ2 )[S V E T ] − gE ǫ2 [S V E T ]
ν
−
Θ([I] + [R] − Itrig )Θ([S V ])[S V E T ] − τI [E T S V I]
[S V ]
− τP [E T S V P] − ρ[E T S V Q] + ρ[S V EQ]
ν
Θ([I] + [R] − Itrig )Θ([S V ])[S V I]
[S V ]
+ gP (1 − θ)[S V P] − τI [IS V I] − τP [IS V P]
− ρ[IS V Q] − ρ[S V IQ]
[S ˙V I] = − gI [S V I] − τI [S V I] −
[S V˙ O] = − gO (1 − ǫ1 )[S V O] − gO ǫ1 [S V O]
ν
Θ([I] + [R] − Itrig )Θ([S V ])[S V O] − τI [OS V I]
−
[S V ]
− τP [OS V P] − ρ[OS V Q] + ρ[S V S Q] + ρ[S V S V Q]
[S V˙ P] = gE [S V E] − τP [S V P] − gP (1 − θ)[S V P]
− τI [PS V I] − τP [PS V P] − ρ[PS V Q] − ρ[S V PQ]
ν
− gP θ[S V P]Θ([I] + [R] − Itrig ) −
[S V P]Θ([I] + [R] − Itrig )Θ([S V ])
[S V ]
[S V˙ Q] = gE (1 − ǫ2 )[S V E T ] − gQ [S V Q] − ρ[S V Q]
− τI [QS V I] − τP [QS V P] − ρ[QS V Q] + ρ[S V IQ]
ν
[S V Q]Θ([I] + [R] − Itrig )Θ([S V ])
+ ρ[S V PQ] + gP θ[S V P]Θ([I] + [R] − Itrig ) −
[S V ]
[S V˙ R] = gI [S V I] + gQ [S V Q] − τI [RS V I] − τP [RS V P]
ν
− ρ[RS V Q] −
[S V R]Θ([I] + [R] − Itrig )Θ([S V ])
[S V ]
[S V˙ S ] = gO (1 − ǫ1 )[S V O] − τI [S S V I] − τP [S S V P]
− ρ[S S V Q] − τI [S V S I] − τP [S V S P] − ρ[S V S Q]
ν
[S V S ]Θ([I] + [R] − Itrig )Θ([S V ])
−
[S V ]
[S V˙S V ] = − 2τI [S V S V I] − 2τP [S V S V P] − 2ρ[S V S V Q]
2ν[S V S V ]Θ([I] + [R] − Itrig )Θ([S V ])
−
[S V ]
[S V˙ V] = gE ǫ2 [S V E T ] + gO ǫ1 [S V O] − τI [VS V I] − τP [VS V P]
ν
[S V S V ]Θ([I] + [R] − Itrig )Θ([S V ])
− ρ[VS V Q] +
[S V ]
ν
[S V V]Θ([I] + [R] − Itrig )Θ([S V ])
−
[S V ]
[V˙E] = gE ǫ2 [E T E] + gO ǫ1 [OE] − gE [V E] − ρ[V EQ]
+ τI [VS I] + τP [VS P] + τI [VS V I] + τP [VS V P]
ν
+
[S V E]Θ([I] + [R] − Itrig )Θ([S V ])
[S V ]
˙ T ] = gE ǫ2 [E T E T ] + gO ǫ1 [OE T ] − gE (1 − ǫ2 )[V E T ]
[V E
ν
[S V E T ]Θ([I] + [R] − Itrig )Θ([S V ])
− gE ǫ2 [V E T ] + ρ[V EQ] +
[S V ]
[V˙I] = gE ǫ2 [E T I] + gO ǫ1 [OI] − gI [V I] + gP (1 − θ)[V P]
ν
− ρ[V IQ] +
[S V I]Θ([I] + [R] − Itrig )Θ([S V ])
[S V ]
˙ = gE ǫ2 [E T O] + gO ǫ1 [OO] − gO (1 − ǫ1 )[VO]
[VO]
ν
[S V O]Θ([I] + [R] − Itrig )Θ([S V ])
− gO ǫ1 [VO] + ρ[VS Q] + ρ[VS V Q] +
[S V ]
[V˙P] = gE ǫ2 [E T P] + gO ǫ1 [OP] + gE [V E] − gP (1 − θ)[V P]
ν
− ρ[V PQ] − gP θ[V P]Θ([I] + [R] − Itrig ) +
[S V P]Θ([I] + [R] − Itrig )Θ([S V ])
[S V ]
[V˙Q] = gE ǫ2 [E T Q] + gO ǫ1 [OQ] + gE (1 − ǫ2 )[V E T ]
− gQ [V Q] + ρ[V IQ] + ρ[V PQ] + gP θ[V P]Θ([I] + [R] − Itrig )
ν
[S V Q]Θ([I] + [R] − Itrig )Θ([S V ])
+
[S V ]
˙ = gE ǫ2 [E T R] + gO ǫ1 [OR] + gI [V I] + gQ [V Q]
[VR]
ν
[S V R]Θ([I] + [R] − Itrig )Θ([S V ])
+
[S V ]
˙ ] = gE ǫ2 [E T S ] + gO ǫ1 [OS ] + gO (1 − ǫ1 )[VO]
[VS
ν
− τI [VS I] − τP [VS P] − ρ[VS Q] +
[S V S ]Θ([I] + [R] − Itrig )Θ([S V ])
[S V ]
[VS˙ V ] = gE ǫ2 [E T S V ] + gO ǫ1 [OS V ] − τI [VS V I] − τP [VS V P]
ν
[S V S V ]Θ([I] + [R] − Itrig )Θ([S V ])
− ρ[VS V Q] +
[S V ]
ν
[VS V ]Θ([I] + [R] − Itrig )Θ([S V ])
−
[S V ]
˙ = gE ǫ2 [E T V] + gO ǫ1 [OV] + gE ǫ2 [V E T ] + gO ǫ1 [VO]
[VV]
ν
ν
+
[S V V]Θ([I] + [R] − Itrig )Θ([S V ]) +
[VS V ]Θ([I] + [R] − Itrig )Θ([S V ])
[S V ]
[S V ]