Download Non-Linear Models Cont’d: Rumour Spread

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Growth and Decay
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Introduction
Linear Models
Non-Linear Models
Non-Linear Models Cont’d: Rumour Spread
The spread of rumours can be modelled using a modified SIR
model.
Here, as before, the population is divided into 3 groups:
1
2
3
Susceptibles (i.e. those who haven’t heard the rumour),
Influencers (those who spread the rumour to others),
Repressers (those who do not spread the rumour).
Again, symbols S, I, R denote these resp groups at time t.
This time, the following assumptions are made:
1
2
3
When an Influncer meets a Susceptable, a certain fraction β of
Susceptables are influenced and change to being Influncers,
When one Influncer meets another, a certain fraction ν of
Influncers get bored with the rumour and become Repressers,
When an Influncer meets a Represser, a certain fraction ν of
Influncers are bored and become Repressers.
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Non-Linear Models Cont’d: Rumour Spread (cont’d)
Equations are thus:
Ṡ
İ
Ṙ
=
=
=
−βIS
+βIS −νI 2 −νIR
+νI 2 +νIR
(4.35)
β is the influence rate and governs speed of rumour spread
ν is the boredom rate and governs rate of rumour quashing.
Again, S + I + R = N for constant population N, as before.
Rest of the initial conditions are
S(0) = S0 > 0, I(0) = I0 > 0, R(0) = 0.
Given these initial conditions from Eqn.(4.35b) get:
dI 
= I0 ((β + ν)S0 − ν)
dt t=0

< 0,
> 0,
if S0 < ν/(β + ν)
if S0 > ν/(β + ν)
(4.36)208 / 230
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Non-Linear Models Cont’d: Rumour Spread
Ratio ρ′ = ν/(β + ν), can be seen as relative boredom rate.
As with model for infectious disease spread, for different ρ′ ,
we have two cases:
1
2
′
From Eqn.(4.30a), dS
dt < 0, so S(t) ≤ S0 and (if S0 < ρ ) so
dI
′
′
S(t) < ρ for all t. From Eqn.(4.30b), if S < ρ then dt < 0 &
as t → ∞, rumour will be quashed.
dI
If S0 > ρ′ , then, by same reasoning dt
> 0 for all t such that
′
S(t) > ρ . This means that for some time interval t ∈ [0, t0 ),
must have I(t) > I0 . Say there is an chain reaction situation in
such cases.
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Non-Linear Models Cont’d: Rumour Spread
Again, as with SIR model of infection, consider the (S, I) phase
plane in Fig. 4.8.
Figure 4.8 : SIR Rumour Spread for ρ′ = 4/5
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Non-Linear Models Cont’d: Rumour Spread
Compared to Fig. 4.7, the trajectories are much steeper.
This is because, comparing Eqn (4.36)
dI 
= I0 ((β + ν)S0 − ν)
dt t=0

with the SIR model of Infectious Diseases Eqn(4.31):
dI 
= I0 (βS0 − ν)
dt t=0

there is now a factor of β + ν instead of β.
This has the effect of causing ’epidemic’ behaviour even for
small rate parameters.
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Non-Linear Models Cont’d: Infectious Diseases
More general model than Kermack-McKendrik or SIR (⇒
more applicable), is one where only partial immunity is
conferred.
This is known as SIRS & permits previously infected (i.e.
removed) individuals to return to susceptible pop’n at a rate
proportional to the number removed.
Mathematically SIRS can be expressed as:
dS
dt
=
dI
dt
=
dR
dt
=
+γR
−βIS
βIS −νI
(4.37)
νI −γR
Again the total population, S + I + R = N, is constant.
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Non-Linear Models Cont’d: Infectious Diseases
SIRS can be analysed using standard methods, steady states
can be found
Ṡ = 0
İ = 0
Ṙ = 0
⇒
⇒
⇒
βIS
γ
βIS
νI
γ
=
=
=
R
νI
R
(4.38)
These yield 2 steady states:
1
first is trivial
S̄1 = N,
Ī1 = 0
R̄1 = 0,
i.e. all pop’n healthy but susceptible & disease eradicated;
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A further steady state
2
second found from inserting S = ν/β (from Eqn.(4.38b)) &
R = νI/γ (from Eqn.(4.38c)) into S + I + R = N to give:
S̄2 =
ν
,
β
Ī2 =
γ βN − ν
,
β γ+ν
R̄2 =
ν βN − ν
β γ+ν
(4.39)
(S̄2 , Ī2 , R̄2 ) is only meaningful if all values are +ive.
i.e.
β
νN
> 1 (i.e. +ive numerators for Ī2 , R̄2 ).
This threshold effect corresponds to the minimum population
necessary for a disease to become endemic.
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Have seen ρ = ν/β (relative removal rate) above;
Now define its reciprocal β/ν as follows:
as removal rate from infective class is ν (with units 1/time),
average period of infectivity is obviously 1/ν.
as β is fraction of contacts (between I and S) that result in
infections, then β × 1/ν gives pop’n fraction that comes into
contact with infective during infectious period.
Hence σ = βN/ν is defined as disease’s infectious contact
number or intrinsic reproducive rate sometimes denoted R0 .
So, from above, and usefully enough, the disease will become
endemic in the population if σ > 1.
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Can see this effect quite clearly in phase-plane. By using
R = N − S − I, can write Eqn.s(4.37a,b) as:
dS
dt
dI
dt
=
=
−βIS + γ(N − S − I)
βIS − νI
(4.40)
In Fig 4.9(a) pop’n cannot sustain disease & it dies out;
In Fig 4.9(b), get steady-state
S̄2 =
ν
,
β
Ī2 =
γ βN − ν
.
β γ+ν
Jacobian corresponding to Eqn.(4.40) at (S̄2 , Ī2 ) is
A(S̄2 , Ī2 )
=
!
−(β Ī2 + γ) −(β S̄2 + γ)
−β Ī2
β S̄2 − ν
"
(4.41)
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(a) (S, I) for σ < 1
(b) (S, I) for σ > 1
Figure 4.9 : SIRS Phase-Plane Plots
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Non-Linear Models Cont’d: Infectious Diseases
As seen above, conditions of stability are for the trace of the
Jacobian to be always -ive and its determinant to be always
+ive.
It is left as an exercise to show that the stability of the
steady-states (S̄2 , Ī2 ), (S̄2 , Ī2 ) is assured when the threshold
condition σ > 1 is satisfied.
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Non-Linear Models Cont’d: Infectious Diseases
The SIS Model
A special case of SIR where infection does not confer any long
lasting immunity.
Such infections (e.g. tuberculosis, meningitis, & infections
leading to the common cold) do not have a recovered state &
individuals become susceptible again after infection. The
equations are thus:
dS
dt
=
−βIS +νI
dI
dt
=
βIS −νI
(4.42)
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Non-Linear Models Cont’d: Infectious Diseases
From this can see that for a total population, N, it holds that
dN
dS
dI
=
+
=0
dt
dt
dt
i.e. S + I = N for constant (initial) population N.
Expressing I in terms of S in eqn.(4.42), can be seen that:
dI
= (βN − ν)I − βI 2
dt
This is a form of the logistic growth equation with
r = βN − ν and K = N − βν so that we have two cases:
1
for
2
for
β
νN
β
νN
> 1, limt→∞ I(t) =
βN−ν
β
& disease will spread,
≤ 1, limt→∞ I(t) = 0 & disease will die out.
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A plot of the former SIS model case is shown in Fig 4.10.
Population
Time, t
Figure 4.10 : SIS Model for ρ = 100: Susceptable, Infected
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Eradication & Control for the Models above
For SIR model, for eqn(4.39) S0 ≈ N, the total pop’n.
Hence, from eqn(4.39) & eqns(4.42,4.39) for the SIS & SIRS
models, resp that infectious contact number or intrinsic
reproductive rate,
σ or R0 = βN/ν
is highly important.
It is fraction of population that comes into contact with an
infective individual during the period of infectiousness.
≡ mean number of secondary cases one infected case will
cause in a pop’n with no immunity & without interventions to
control the infection.
Useful because it helps determine if an infectious disease will
spread thro a pop’n.
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See from βN/ν that can reduce infectious disease spread by:
1
2
3
↑ ν, removal rate of infectives. Seen in the UK
Foot-and-Mouth epidemic by slaughtering those infected
cattle.
β ↓, infectious contact rate btw susceptibles and infectives.
Disinfection & movement controls in Foot-and-Mouth ↓ β.
Decrease the effective number of N which has the effect of
↓ S. Again for the Foot-and-Mouth example, slaughtering
potential contacts surrounding infected farms was employed.
Vaccination of susceptibles for the epidemic, which became
increasingly politically controversial.
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Non-Linear Models Cont’d: Infectious Diseases
Immunizing an entire population from a disease is impractical
due to matters of cost & the logistics of administering the
vaccine to (potentially) hundreds of thousands.
Costs:
direct costs of producing & administering the dosage and
indirect costs of providing information to the public and
making sure that everyone that is vaccinated has been.
Thus would like to be able to provide safety from disease at
the lowest possible cost.
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Actually only have to immunize a fraction of population in
order to give the entire population herd immunity.
Specifically need to reduce the effective value of N so that
that the disease disappears of its own accord.
In terms of SIR model need to move enough people such that
(from eqn(4.31)), βS0 − ν < 0 so that the rate of increase of
dI
is negative.
infectives dt
i.e. need to lower epidemic threshold below one.
Or to write it more directly the fraction of people that needs
to be immunized is such that S0 < ν/β i.e. 1 − ν/β percent
of the susceptible pool need to be immunized.
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This makes intuitive sense since if ν is small that means it
takes longer to recover from infection & an infective person
has more time to infect people.
Thus as ν ↓, 1 − ν/β ↑; need to inoculate a larger fraction of
the population.
As β ↑, each infected person contacts more people in a given
period and 1 − ν/β ↑.
Thus again need to inoculate a larger fraction of population.
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R0 & 1 − 1/R0 shown in % Table 4.2 for common diseases.
Disease
Measles
Pertussis
Diphtheria
Smallpox
Polio
Rubella
HIV/AIDS
SARS
Influenza (1918)
Cholera
Transmission
Airborne
Airborne droplet
Saliva
Social contact
Fecal-oral route
Airborne droplet
Sexual contact
Airborne droplet
Airborne droplet
Fecal-oral route
R0
12 to 18
12 to 17
6 to 7
5 to 7
5 to 7
5 to 7
2 to 5
2 to 5
2 to 3
2.9
1 − R10 %
92 to 94.5
92 to 94
84
80 to 85
80 to 85
80 to 85
50 to 80
50 to 80
50 to 80
65.5
Table 4.2 : Values for R0 for Several Common Infectious Diseases
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Non-Linear Models Cont’d: The Chemostat Revisited
The Chemostat Revisited
Returning to chemostat above, can look at phase-plane plot.
Derived the equations:
dn
nc
= f (n, c) = α1
dτ
1+c
#
$
−n
(4.43)
and
dc
nc
= g(n, c) = −
dτ
1+c
#
$
− c + α2
(4.44)
containing (dimensionless) parameters:
α1 =
VKmax
C0
and α2 =
F
Kn
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Eqns(4.43, 4.44) also contain dimensionless time, bacterial
population & nutrient concentrations respectively:
tF
NαVKmax
C
, n =
, c =
V
FKn
Kn
τ =
The phase-plane plot for nVc is shown in Fig 4.11. It will be
seen from the figure that when α1 = 3, α2 = 1 there are two
equilibrium points:
(n̄1 , c̄1 ) =
#
#
α1 α2 −
1
1
,
α1 − 1
α1 − 1
$
$
3 1
= ( , )
2 2
and
(n̄2 , c̄2 ) = (0, α2 ) = (0, 1)
as predicted.
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Figure 4.11 : Chemostat Phase-Plane Plot, α1 = 3, α2 = 1
230 / 230