Evaluating the introduction of vaginal microbicides
I- only model
• Thus, the model
I-only model
• Thus, the model
dI
dt =
µ µ ⇥
⇥ I
Λ=birth rate µ=background death rate β=infection rate γ=disease death rate
I-only model
• Thus, the model
will always overestimate the epidemic
dI
dt =
µ µ ⇥
⇥ I
Λ=birth rate µ=background death rate β=infection rate γ=disease death rate
I-only model
• Thus, the model
will always overestimate the epidemic
• This model also has eradication threshold
dI
dt =
µ µ ⇥
⇥ I
Λ=birth rate µ=background death rate β=infection rate γ=disease death rate
I-only model
• Thus, the model
will always overestimate the epidemic
• This model also has eradication threshold
dI
dt =
µ µ ⇥
⇥ I
R0,I =
µ(µ + ⇥)
Λ=birth rate µ=background death rate β=infection rate γ=disease death rate
I-only model
• Thus, the model
will always overestimate the epidemic
• This model also has eradication threshold
• It follows that there will be eradication in the linear model if and only if there is eradication in the SI model.
dI
dt =
µ µ ⇥
⇥ I
R0,I =
µ(µ + ⇥)
Λ=birth rate µ=background death rate β=infection rate γ=disease death rate
BMC Public Health 2009, 9(Suppl 1):S15 http://www.biomedcentral.com/1471-2458/9/S1/S15
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(page number not for citation purposes)
will always overestimate the epidemic. We shall refer to this as the I-only model.
This model has only the trivial equilibrium I = 0. If the trivial equilibrium is stable, then all trajectories will approach it. If the trivial equilibrium is unstable, then solutions will increase without bound.
Although the total population without infection is not constant, it should nevertheless be noted that the I-only model has the eradication threshold
which is the same eradication threshold as the SI model (as expected). It follows that there will be eradication in the I-only model if and only if there is eradication in the SI model.
To illustrate, we simulated two cases: R0,SI < 1 (Figure 1) and R0,SI> 1 (Figure 2). Parameters used in the simulations
were ! = 20 people·years-1, " = years-1, # = years
-1, with $ = 0.00007 people-1years-1 (Figure 1) and $ = 0.0002 people-1years-1 (Figure 2). Despite the fact that the transient dynamics are vastly different in the two models, the linear approximation has the same eradication thresh-old as the more accurate SI model and, furthermore, always overestimates the epidemic. It follows that, for eradication purposes, the simple, linear model should determine whether our control methods will be sufficient.
We now extend the SI model to a metapopulation model with p regions. For 1 % i % p, we can write
Si = susceptible individuals in the ith region;
Ii = infected individuals in the ith region;
!i = the rate of appearance of new susceptible individ-uals in the ith region;
"i = background death rate in the ith region;
#i = death rate due to disease in the ith region;
nij = migration rate of susceptible individuals from jth region to ith region;
dI
dt = − − I
$
"Λ " #
Ro I,
( ) ,
= +
$
" " #
Λ (1)
701 1
10
When R0 < 1, both models lead to eradication Figure 1
When R0 < 1, both models lead to eradication. When R0 < 1, both models lead to eradication. The SI model (solid curve) is always below the I-only model (dashed curve). In this case, R0 = 0.8575.
0 50 100 150 200 250 300
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
time (years)
population (millions)
Eradication (R
0=0.8575)
I-only model
SI model
BMC Public Health 2009, 9(Suppl 1):S15 http://www.biomedcentral.com/1471-2458/9/S1/S15
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(page number not for citation purposes)
mij = migration rate of infected individuals from jth region to ith region.
With these assumptions, we have nii ! mii ! 0. Then, for i = 1,..., p,
If the system (2) is well-posed, we can find a bound for the population of susceptibles in each patch. We can then extend the same idea to obtain the linear system of I-equa-tions only. (See Appendix for details.)
From Theorem 1 in the Appendix, we have
Thus, the total population of susceptible individuals in the ith region is limited. So we can write
"i = # × (total population in the ith region without infection)
di = $i + %i,
where di represents the total death rate in the ith region.
Thus, for i = 1,..., p, the equations of the I-only model are
This can be rewritten as
where I(t) = (I1(t), I2(t),..., Ip(t))T and
Analysis
A two-region example
The simplest nontrivial version of model (2) is the case when p = 2, which can be presented as follows:
dSi
dt S I S n S n S
dIi
dt S I
i i i i i ij j ji i
j p j
p
i i i
= − − + −
= − +
=
=
∑
∑
Λ # $
# $ %
1 1
( ii i ij j ji i
j p j
I p m I m I
) + − .
=
=
∑
∑
1 1(2)
S ti( ) ≤ N t( ) ≤ L*, for i =1, , .! p
dIi
dt i iI d Ii i m Iij j m Iji i
j p
j p
= − + −
=
=
∑
∑
"
1 1
. (3)
d
dt I( )t = KI( ),t
K =
− +
− +
=
=
∑
∑
"
"
1 1 1 1 12 1
21 2 2 1 2 2
1 2
( )
( )
d m m m
m d m m
m m
j j p
p
j j p
p
p p
"
"
# # $ #
"
" "p p jp
j
d p m
− +
∑ =
( )
.
1
When R0 > 1, both models lead to endemic disease Figure 2
When R0 > 1, both models lead to endemic disease. When R0 > 1, both models lead to endemic disease. The I-only model (dashed curve) always overestimates the SI model (solid curve). In this case, R0 = 2.45.
0 100 200 300 400 500
0 50 100 150 200 250 300
time (years)
population (millions)
0 50 100
0 5 10
Persistence (R
0=2.45)
SI model I-only model
• For i=1,...,p,
A patch model with p regions
• For i=1,...,p,
A patch model with p regions
Λ=birth rate µ=background death rate β=infection rate γ=disease death rate nik=migration rate (susceptibles) mik=mmigration rate (infectives)
dSi
dt = i iSiIi µiSi +
p
j=1
nijSj
p
j=1
njiSi
dIi
dt = iSiIi (µI + ⇥i)Ii +
p
j=1
mijIj
p
j=1
mjiIi
• For i=1,...,p,
• We can bound
A patch model with p regions
Λ=birth rate µ=background death rate β=infection rate γ=disease death rate nik=migration rate (susceptibles) mik=mmigration rate (infectives)
dSi
dt = i iSiIi µiSi +
p
j=1
nijSj
p
j=1
njiSi
dIi
dt = iSiIi (µI + ⇥i)Ii +
p
j=1
mijIj
p
j=1
mjiIi
• For i=1,...,p,
• We can bound
A patch model with p regions
Λ=birth rate µ=background death rate β=infection rate γ=disease death rate nik=migration rate (susceptibles) mik=mmigration rate (infectives)
dSi
dt = i iSiIi µiSi +
p
j=1
nijSj
p
j=1
njiSi
dIi
dt = iSiIi (µI + ⇥i)Ii +
p
j=1
mijIj
p
j=1
mjiIi
Si(t) L , for i = 1, . . . , p
• For i=1,...,p,
• We can bound where
A patch model with p regions
Λ=birth rate µ=background death rate β=infection rate γ=disease death rate nik=migration rate (susceptibles) mik=mmigration rate (infectives)
dSi
dt = i iSiIi µiSi +
p
j=1
nijSj
p
j=1
njiSi
dIi
dt = iSiIi (µI + ⇥i)Ii +
p
j=1
mijIj
p
j=1
mjiIi
Si(t) L , for i = 1, . . . , p
• For i=1,...,p,
• We can bound where
A patch model with p regions
Λ=birth rate µ=background death rate β=infection rate γ=disease death rate nik=migration rate (susceptibles) mik=mmigration rate (infectives)
dSi
dt = i iSiIi µiSi +
p
j=1
nijSj
p
j=1
njiSi
dIi
dt = iSiIi (µI + ⇥i)Ii +
p
j=1
mijIj
p
j=1
mjiIi
Si(t) L , for i = 1, . . . , p
L = 1 + · · · + p
min{µ1, · · · , µp} .
The linear model is an overestimate
• The linear model has the same eradication threshold as the more accurate SI model
The linear model is an overestimate
• The linear model has the same eradication threshold as the more accurate SI model
• The linear model always overestimates the epidemic
The linear model is an overestimate
• The linear model has the same eradication threshold as the more accurate SI model
• The linear model always overestimates the epidemic
• For eradication purposes, the linear model should
determine whether our control methods will be sufficient.
A two-region example: the flow chart
Λ=birth rate µ=background death rate β=infection rate γ=disease death rate nik=migration rate (susceptibles) mik=mmigration rate (infectives)