epidemic models - cmap.polytechnique.fr
TRANSCRIPT
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Lajmanovich and Yorke SIS ModelAnalysis
Main resultsBack to Lajmanovich and Yorke
Epidemic Models
Based on recent work with Edouard Strickler (ArXiv, 2017)
Epidemic Models
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Lajmanovich and Yorke SIS ModelAnalysis
Main resultsBack to Lajmanovich and Yorke
Lajmanovich and Yorke SIS Model, 1976
- d groups
- In each group each individual can be infected
- 0 ≤ xi ≤ 1 = proportion of infected individuals in group i .
- Cij = rate of infection from group i to group j .
- Di cure rate in group i
dxidt
= (1− xi )(∑j
Cijxj)− Dixi
Epidemic Models
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Lajmanovich and Yorke SIS ModelAnalysis
Main resultsBack to Lajmanovich and Yorke
Lajmanovich and Yorke SIS Model, 1976
- d groups
- In each group each individual can be infected
- 0 ≤ xi ≤ 1 = proportion of infected individuals in group i .
- Cij = rate of infection from group i to group j .
- Di cure rate in group i
dxidt
= (1− xi )(∑j
Cijxj)− Dixi
Epidemic Models
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Lajmanovich and Yorke SIS ModelAnalysis
Main resultsBack to Lajmanovich and Yorke
Suppose C irreducible
A = C − diag(D)
λ(A) = largest real part of eigenvalues of A.
Theorem (Lajmanovich and Yorke 1976)
If λ(A) ≤ 0, the disease free equilibrium 0 is a global attractor
If λ(A) > 0 there exists another equilibrium x∗ >> 0 and every non
zero trajectory converges to x∗
Epidemic Models
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Lajmanovich and Yorke SIS ModelAnalysis
Main resultsBack to Lajmanovich and Yorke
Suppose C irreducible
A = C − diag(D)
λ(A) = largest real part of eigenvalues of A.
Theorem (Lajmanovich and Yorke 1976)
If λ(A) ≤ 0, the disease free equilibrium 0 is a global attractor
If λ(A) > 0 there exists another equilibrium x∗ >> 0 and every non
zero trajectory converges to x∗
Epidemic Models
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Lajmanovich and Yorke SIS ModelAnalysis
Main resultsBack to Lajmanovich and Yorke
• What if the environment �uctuates ?
• Example: Two environments
C 1 =
(1 4
0 1
), D1 =
(2
2
),
and
C 2 =
(2 0
4 2
), D2 =
(3
3
).
λ(A1) = λ(A2) = −1 < 0
⇒
The disease free equilibrium is a global attractor in each
environment
Epidemic Models
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Lajmanovich and Yorke SIS ModelAnalysis
Main resultsBack to Lajmanovich and Yorke
• What if the environment �uctuates ?
• Example: Two environments
C 1 =
(1 4
0 1
), D1 =
(2
2
),
and
C 2 =
(2 0
4 2
), D2 =
(3
3
).
λ(A1) = λ(A2) = −1 < 0
⇒
The disease free equilibrium is a global attractor in each
environment
Epidemic Models
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Lajmanovich and Yorke SIS ModelAnalysis
Main resultsBack to Lajmanovich and Yorke
• What if the environment �uctuates ?
• Example: Two environments
C 1 =
(1 4
0 1
), D1 =
(2
2
),
and
C 2 =
(2 0
4 2
), D2 =
(3
3
).
λ(A1) = λ(A2) = −1 < 0
⇒
The disease free equilibrium is a global attractor in each
environment
Epidemic Models
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Lajmanovich and Yorke SIS ModelAnalysis
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,
Figure: Phase portraits of F 1 and F 2
Epidemic Models
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Lajmanovich and Yorke SIS ModelAnalysis
Main resultsBack to Lajmanovich and Yorke
Constant switching
Q(x) =
(0 ββ 0
), β >> 1.
Epidemic Models
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Lajmanovich and Yorke SIS ModelAnalysis
Main resultsBack to Lajmanovich and Yorke
,
Figure: Random Switching may reverses the trend
Epidemic Models
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Lajmanovich and Yorke SIS ModelAnalysis
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• More surprising !
A0 =
−1 0 0
10 −1 0
0 0 −10
, A1 =
−10 0 10
0 −10 0
0 10 −1
.
D0 =
11
11
20
, D1 =
20
20
11
C i = Ai + D i .
F 0,1 = the LY vector �eld on [0, 1]3 induced by (C 0,1,D0,1).
Epidemic Models
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Lajmanovich and Yorke SIS ModelAnalysis
Main resultsBack to Lajmanovich and Yorke
F t = (1− t)F 0 + tF 1
= LY vector �eld induced by
C t = (1− t)C 0 + tC 1,Dt = (1− t)D0 + tD1
• For all 0 ≤ t ≤ 1, the disease free equilibrium is a global
attractor of F t
• Still, a random switching between the dynamics leads to the
persistence of the disease.
Epidemic Models
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Lajmanovich and Yorke SIS ModelAnalysis
Main resultsBack to Lajmanovich and Yorke
F t = (1− t)F 0 + tF 1
= LY vector �eld induced by
C t = (1− t)C 0 + tC 1,Dt = (1− t)D0 + tD1
• For all 0 ≤ t ≤ 1, the disease free equilibrium is a global
attractor of F t
• Still, a random switching between the dynamics leads to the
persistence of the disease.
Epidemic Models
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Lajmanovich and Yorke SIS ModelAnalysis
Main resultsBack to Lajmanovich and Yorke
F t = (1− t)F 0 + tF 1
= LY vector �eld induced by
C t = (1− t)C 0 + tC 1,Dt = (1− t)D0 + tD1
• For all 0 ≤ t ≤ 1, the disease free equilibrium is a global
attractor of F t
• Still, a random switching between the dynamics leads to the
persistence of the disease.
Epidemic Models
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Lajmanovich and Yorke SIS ModelAnalysis
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Figure: Simulation of Xt for β = 10.
Epidemic Models
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Lajmanovich and Yorke SIS ModelAnalysis
Main resultsBack to Lajmanovich and Yorke
Figure: Simulation of ‖Xt‖ for β = 10.
Epidemic Models
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Lajmanovich and Yorke SIS ModelAnalysis
Main resultsBack to Lajmanovich and Yorke
Growth rates
Analysis
• First step :
Generalize !
•E = {1, . . . ,m},
•F 1, . . . ,Fm smooth vector �elds on Rd ,
F 1(0) = . . . = Fm(0) = 0.
• 0 ∈ M ⊂ Rd = compact positively invariant set under each Φi ,
X = F It (X ),
(It) jump process controled by X
P(It+s = j |It = i ,Xt = x ,Ft) = Qij(x)t + o(s).
Epidemic Models
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Lajmanovich and Yorke SIS ModelAnalysis
Main resultsBack to Lajmanovich and Yorke
Growth rates
Analysis
• First step : Generalize !
•E = {1, . . . ,m},
•F 1, . . . ,Fm smooth vector �elds on Rd ,
F 1(0) = . . . = Fm(0) = 0.
• 0 ∈ M ⊂ Rd = compact positively invariant set under each Φi ,
X = F It (X ),
(It) jump process controled by X
P(It+s = j |It = i ,Xt = x ,Ft) = Qij(x)t + o(s).
Epidemic Models
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Lajmanovich and Yorke SIS ModelAnalysis
Main resultsBack to Lajmanovich and Yorke
Growth rates
Analysis
• First step : Generalize !
•E = {1, . . . ,m},
•F 1, . . . ,Fm smooth vector �elds on Rd ,
F 1(0) = . . . = Fm(0) = 0.
• 0 ∈ M ⊂ Rd = compact positively invariant set under each Φi ,
X = F It (X ),
(It) jump process controled by X
P(It+s = j |It = i ,Xt = x ,Ft) = Qij(x)t + o(s).
Epidemic Models
![Page 21: Epidemic Models - cmap.polytechnique.fr](https://reader031.vdocuments.mx/reader031/viewer/2022012016/61da7accd52b671725486673/html5/thumbnails/21.jpg)
Lajmanovich and Yorke SIS ModelAnalysis
Main resultsBack to Lajmanovich and Yorke
Growth rates
Analysis
• First step : Generalize !
•E = {1, . . . ,m},
•F 1, . . . ,Fm smooth vector �elds on Rd ,
F 1(0) = . . . = Fm(0) = 0.
• 0 ∈ M ⊂ Rd = compact positively invariant set under each Φi ,
X = F It (X ),
(It) jump process controled by X
P(It+s = j |It = i ,Xt = x ,Ft) = Qij(x)t + o(s).
Epidemic Models
![Page 22: Epidemic Models - cmap.polytechnique.fr](https://reader031.vdocuments.mx/reader031/viewer/2022012016/61da7accd52b671725486673/html5/thumbnails/22.jpg)
Lajmanovich and Yorke SIS ModelAnalysis
Main resultsBack to Lajmanovich and Yorke
Growth rates
• Extinction set: M0 = {0} × E .
On M0 the dynamics is trivial and doesn't convey any information!
⇒ Has'minskii trick (1960, for linear SDEs)
Replace {0Rd} by {0}×Sn−1
Epidemic Models
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Lajmanovich and Yorke SIS ModelAnalysis
Main resultsBack to Lajmanovich and Yorke
Growth rates
• Extinction set: M0 = {0} × E .
On M0 the dynamics is trivial and doesn't convey any information!
⇒ Has'minskii trick (1960, for linear SDEs)
Replace {0Rd} by {0}×Sn−1
Epidemic Models
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Lajmanovich and Yorke SIS ModelAnalysis
Main resultsBack to Lajmanovich and Yorke
Growth rates
• Extinction set: M0 = {0} × E .
On M0 the dynamics is trivial and doesn't convey any information!
⇒ Has'minskii trick (1960, for linear SDEs)
Replace {0Rd} by {0}×Sn−1
Epidemic Models
![Page 25: Epidemic Models - cmap.polytechnique.fr](https://reader031.vdocuments.mx/reader031/viewer/2022012016/61da7accd52b671725486673/html5/thumbnails/25.jpg)
Lajmanovich and Yorke SIS ModelAnalysis
Main resultsBack to Lajmanovich and Yorke
Growth rates
• Extinction set: M0 = {0} × E .
On M0 the dynamics is trivial and doesn't convey any information!
⇒ Has'minskii trick (1960, for linear SDEs)
Replace {0Rd} by {0}×Sn−1
Epidemic Models
![Page 26: Epidemic Models - cmap.polytechnique.fr](https://reader031.vdocuments.mx/reader031/viewer/2022012016/61da7accd52b671725486673/html5/thumbnails/26.jpg)
Lajmanovich and Yorke SIS ModelAnalysis
Main resultsBack to Lajmanovich and Yorke
Growth rates
θt =Xt
‖Xt‖, ρt = ‖Xt‖
⇒dθ
dt= F It (ρ, θ)− 〈θ, F It (ρ, θ)〉θ.
dρ
dt= ρ〈F It (ρ, θ), θ〉
where
F j(ρ, θ) =
F j(ρθ)
ρif ρ > 0
DF j(0)θ if ρ = 0
Epidemic Models
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Lajmanovich and Yorke SIS ModelAnalysis
Main resultsBack to Lajmanovich and Yorke
Growth rates
θt =Xt
‖Xt‖, ρt = ‖Xt‖
⇒dθ
dt= F It (ρ, θ)− 〈θ, F It (ρ, θ)〉θ.
dρ
dt= ρ〈F It (ρ, θ), θ〉
where
F j(ρ, θ) =
F j(ρθ)
ρif ρ > 0
DF j(0)θ if ρ = 0
Epidemic Models
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Lajmanovich and Yorke SIS ModelAnalysis
Main resultsBack to Lajmanovich and Yorke
Growth rates
• New state space:
M = Sd−1 × R+ × E
M0 = Sd−1 × {0} × E ≈ Sd−1 × E
Epidemic Models
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Lajmanovich and Yorke SIS ModelAnalysis
Main resultsBack to Lajmanovich and Yorke
Growth rates
• On M0 the dynamics is a PDMP
dΘ
dt= G Jt (θ).
dρ
dt= 0,
J jump process with rate matrix Q(0).
where
G i (θ) = Aiθ − 〈Aiθ, θ〉θ.
Ai = DF i (0)
• It also induces a PDMP on Pd−1 × E , where
Pd−1 = Sd−1/x ∼ −x
is the projective space
Epidemic Models
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Lajmanovich and Yorke SIS ModelAnalysis
Main resultsBack to Lajmanovich and Yorke
Growth rates
• On M0 the dynamics is a PDMP
dΘ
dt= G Jt (θ).
dρ
dt= 0,
J jump process with rate matrix Q(0).
where
G i (θ) = Aiθ − 〈Aiθ, θ〉θ.
Ai = DF i (0)
• It also induces a PDMP on Pd−1 × E , where
Pd−1 = Sd−1/x ∼ −x
is the projective space
Epidemic Models
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Lajmanovich and Yorke SIS ModelAnalysis
Main resultsBack to Lajmanovich and Yorke
Growth rates
• On M0 the dynamics is a PDMP
dΘ
dt= G Jt (θ).
dρ
dt= 0,
J jump process with rate matrix Q(0).
where
G i (θ) = Aiθ − 〈Aiθ, θ〉θ.
Ai = DF i (0)
• It also induces a PDMP on Pd−1 × E , where
Pd−1 = Sd−1/x ∼ −x
is the projective space
Epidemic Models
![Page 32: Epidemic Models - cmap.polytechnique.fr](https://reader031.vdocuments.mx/reader031/viewer/2022012016/61da7accd52b671725486673/html5/thumbnails/32.jpg)
Lajmanovich and Yorke SIS ModelAnalysis
Main resultsBack to Lajmanovich and Yorke
Growth rates
• On M0 the dynamics is a PDMP
dΘ
dt= G Jt (θ).
dρ
dt= 0,
J jump process with rate matrix Q(0).
where
G i (θ) = Aiθ − 〈Aiθ, θ〉θ.
Ai = DF i (0)
• It also induces a PDMP on Pd−1 × E , where
Pd−1 = Sd−1/x ∼ −x
is the projective space
Epidemic Models
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Lajmanovich and Yorke SIS ModelAnalysis
Main resultsBack to Lajmanovich and Yorke
Growth rates
Growth rates
V (θ, ρ, i) = − log(ρ)
⇒ H(θ, ρ, θ, i) = −〈F i (ρ, θ), θ〉
On M0 = Sd−1 × {0} × E
H(θ, 0, i) = −〈Aiθ, θ〉
Growth rate : For each µ ergodic for (Θ, J)
Λ(µ) = sup
∫〈Aiθ, θ〉µ(dθdi)
= −µH
Maximal growth rates :
Λ− = minµ
Λ(µ) ≤ Λ+ = supµ
Λ(µ).
Epidemic Models
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Lajmanovich and Yorke SIS ModelAnalysis
Main resultsBack to Lajmanovich and Yorke
Growth rates
Growth rates
V (θ, ρ, i) = − log(ρ)
⇒ H(θ, ρ, θ, i) = −〈F i (ρ, θ), θ〉On M0 = Sd−1 × {0} × E
H(θ, 0, i) = −〈Aiθ, θ〉
Growth rate : For each µ ergodic for (Θ, J)
Λ(µ) = sup
∫〈Aiθ, θ〉µ(dθdi) = −µH
Maximal growth rates :
Λ− = minµ
Λ(µ) ≤ Λ+ = supµ
Λ(µ).
Epidemic Models
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Lajmanovich and Yorke SIS ModelAnalysis
Main resultsBack to Lajmanovich and Yorke
Growth rates
Growth rates
V (θ, ρ, i) = − log(ρ)
⇒ H(θ, ρ, θ, i) = −〈F i (ρ, θ), θ〉On M0 = Sd−1 × {0} × E
H(θ, 0, i) = −〈Aiθ, θ〉
Growth rate : For each µ ergodic for (Θ, J)
Λ(µ) = sup
∫〈Aiθ, θ〉µ(dθdi) = −µH
Maximal growth rates :
Λ− = minµ
Λ(µ) ≤ Λ+ = supµ
Λ(µ).
Epidemic Models
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Lajmanovich and Yorke SIS ModelAnalysis
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Growth rates
Link with Lyapunov expoents
Proposition (BS 16)
Λ+ coincides with the top-Lyapounov exponent given by the
Multiplicative Ergodic Theorem of the skew product system
dY
dt= AJtY
and Λ− is another exponent.
Remark
If there exists for (Θ, J) an accessible point (on Pd−1) at which the
weak bracket condition holds. Then
Λ+ = Λ−.
Epidemic Models
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Lajmanovich and Yorke SIS ModelAnalysis
Main resultsBack to Lajmanovich and Yorke
Growth rates
Link with Lyapunov expoents
Proposition (BS 16)
Λ+ coincides with the top-Lyapounov exponent given by the
Multiplicative Ergodic Theorem of the skew product system
dY
dt= AJtY
and Λ− is another exponent.
Remark
If there exists for (Θ, J) an accessible point (on Pd−1) at which the
weak bracket condition holds. Then
Λ+ = Λ−.
Epidemic Models
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Growth rates
Example in dimension 2
Corollary (BS 16)
Suppose d = 2 and that either
(a) One matrix Ai has no real eigenvalues; or
(b) Two matrices Ai ,Aj have no common eigenvectors
Then Λ+ = Λ−.
Epidemic Models
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Lajmanovich and Yorke SIS ModelAnalysis
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Growth rates
Metzler matrices
Proposition (BS, 16)
Assume the matrices are Metzler and at least one convex
combination is irreducible. Then Λ+ = Λ−
Epidemic Models
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Lajmanovich and Yorke SIS ModelAnalysis
Main resultsBack to Lajmanovich and Yorke
ExtinctionPersistence
Main results
Epidemic Models
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ExtinctionPersistence
Extinction : Λ+ < 0
Assume Λ+ < 0.
Theorem (BS, 16)
There exists a neighborhood U of 0 and η > 0 such that for all
x ∈ U and i ∈ E
Px ,i (lim supt→∞
1
tlog(‖Xt‖) ≤ Λ+) ≥ η.
If furthermore 0 is accessible then for all x ∈ M and i ∈ E
Px ,i (lim supt→∞
1
tlog(‖Xt‖) ≤ Λ+) = 1.
Epidemic Models
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ExtinctionPersistence
Persistence : Λ− > 0.
Assume Λ− > 0.
Theorem (BS, 16)
∀x ∈ M∗ = M \ {0}, Px ,i almost surely, every limit point Π of (Πt)belongs to Pinv and
Π({0} × E ) = 0.
More precisely, ∃θ,K > 0 (independent of Π) such that∫1
‖x‖θdΠ ≤ K .
Epidemic Models
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Lajmanovich and Yorke SIS ModelAnalysis
Main resultsBack to Lajmanovich and Yorke
ExtinctionPersistence
Persistence : Λ− > 0.
Assume Λ− > 0.
Theorem (BS, 16)
∀x ∈ M∗ = M \ {0}, Px ,i almost surely, every limit point Π of (Πt)belongs to Pinv and
Π({0} × E ) = 0.
More precisely, ∃θ,K > 0 (independent of Π) such that∫1
‖x‖θdΠ ≤ K .
Epidemic Models
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Lajmanovich and Yorke SIS ModelAnalysis
Main resultsBack to Lajmanovich and Yorke
ExtinctionPersistence
Persistence : Λ− > 0.
Assume Λ− > 0.
τ = inf{t ≥ 0 : ‖Xt‖ ≥ ε}.
Theorem (BS, 16)
∃b > 1, c > 0 such that ∀x ∈ M∗
Ex(bτ ) ≤ c(1 + ‖x‖−θ).
Epidemic Models
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Lajmanovich and Yorke SIS ModelAnalysis
Main resultsBack to Lajmanovich and Yorke
ExtinctionPersistence
Persistence : Λ− > 0.
Assume Λ− > 0. In addition assume ∃p ∈ M∗ accessible from M∗
Theorem (BS, 16)
Weak Bracket condition at p ⇒
Pinv ∩ P(M∗ × E ) = {Π}
and ∀x ∈ M∗
limt→∞
Πt = Π
Px ,i almost surely.
Epidemic Models
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Lajmanovich and Yorke SIS ModelAnalysis
Main resultsBack to Lajmanovich and Yorke
ExtinctionPersistence
Persistence : Λ− > 0.
Assume Λ− > 0. In addition assume ∃p ∈ M∗ accessible from M∗
Theorem (BS, 16)
Strong Bracket condition at p ⇒
∀x ∈ M∗ ‖ Px ,i (Zt ∈ ·)− Π(·) ‖≤ const(1+ ‖ x ‖−θ)e−κt
for some κ, θ > 0.
Epidemic Models
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Lajmanovich and Yorke SIS ModelAnalysis
Main resultsBack to Lajmanovich and Yorke
Back to Lajmanovich and Yorke
• Second step :
Generalize again !
Call a vector �eld F : [0, 1]d 7→ Rd epidemic provided
• F (0) = 0, xi = 1⇒ Fi (x) < 0
• DF (x) is Metzler (o� diagonal entries ≥ 0) irreducible
• F is (strongly) sub homogeneous on ]0, 1[d : F (tx) << tF (x) for
t ≥ 1.
Remark
LY vector �eld is epidemic
Epidemic Models
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Lajmanovich and Yorke SIS ModelAnalysis
Main resultsBack to Lajmanovich and Yorke
Back to Lajmanovich and Yorke
• Second step : Generalize again !
Call a vector �eld F : [0, 1]d 7→ Rd epidemic provided
• F (0) = 0, xi = 1⇒ Fi (x) < 0
• DF (x) is Metzler (o� diagonal entries ≥ 0) irreducible
• F is (strongly) sub homogeneous on ]0, 1[d : F (tx) << tF (x) for
t ≥ 1.
Remark
LY vector �eld is epidemic
Epidemic Models
![Page 49: Epidemic Models - cmap.polytechnique.fr](https://reader031.vdocuments.mx/reader031/viewer/2022012016/61da7accd52b671725486673/html5/thumbnails/49.jpg)
Lajmanovich and Yorke SIS ModelAnalysis
Main resultsBack to Lajmanovich and Yorke
Back to Lajmanovich and Yorke
• Second step : Generalize again !
Call a vector �eld F : [0, 1]d 7→ Rd epidemic provided
• F (0) = 0, xi = 1⇒ Fi (x) < 0
• DF (x) is Metzler (o� diagonal entries ≥ 0) irreducible
• F is (strongly) sub homogeneous on ]0, 1[d : F (tx) << tF (x) for
t ≥ 1.
Remark
LY vector �eld is epidemic
Epidemic Models
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Lajmanovich and Yorke SIS ModelAnalysis
Main resultsBack to Lajmanovich and Yorke
Back to Lajmanovich and Yorke
• Second step : Generalize again !
Call a vector �eld F : [0, 1]d 7→ Rd epidemic provided
• F (0) = 0, xi = 1⇒ Fi (x) < 0
• DF (x) is Metzler (o� diagonal entries ≥ 0) irreducible
• F is (strongly) sub homogeneous on ]0, 1[d : F (tx) << tF (x) for
t ≥ 1.
Remark
LY vector �eld is epidemic
Epidemic Models
![Page 51: Epidemic Models - cmap.polytechnique.fr](https://reader031.vdocuments.mx/reader031/viewer/2022012016/61da7accd52b671725486673/html5/thumbnails/51.jpg)
Lajmanovich and Yorke SIS ModelAnalysis
Main resultsBack to Lajmanovich and Yorke
Back to Lajmanovich and Yorke
• Second step : Generalize again !
Call a vector �eld F : [0, 1]d 7→ Rd epidemic provided
• F (0) = 0, xi = 1⇒ Fi (x) < 0
• DF (x) is Metzler (o� diagonal entries ≥ 0) irreducible
• F is (strongly) sub homogeneous on ]0, 1[d : F (tx) << tF (x) for
t ≥ 1.
Remark
LY vector �eld is epidemic
Epidemic Models
![Page 52: Epidemic Models - cmap.polytechnique.fr](https://reader031.vdocuments.mx/reader031/viewer/2022012016/61da7accd52b671725486673/html5/thumbnails/52.jpg)
Lajmanovich and Yorke SIS ModelAnalysis
Main resultsBack to Lajmanovich and Yorke
Theorem (Hirsch, 1994)
If F is epidemic, conclusions of Lajmanovich and Yorke theorem
hold true
Theorem (BS, 16)
Suppose the F i , i = 1 . . .m are epidemic. Then
(a) Λ+ = Λ− = Λ
(b) Λ < 0⇒ almost sure Extinction
(c) Λ > 0⇒ Persistence
Epidemic Models
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Lajmanovich and Yorke SIS ModelAnalysis
Main resultsBack to Lajmanovich and Yorke
Theorem (Hirsch, 1994)
If F is epidemic, conclusions of Lajmanovich and Yorke theorem
hold true
Theorem (BS, 16)
Suppose the F i , i = 1 . . .m are epidemic. Then
(a) Λ+ = Λ− = Λ
(b) Λ < 0⇒ almost sure Extinction
(c) Λ > 0⇒ Persistence
Epidemic Models
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Lajmanovich and Yorke SIS ModelAnalysis
Main resultsBack to Lajmanovich and Yorke
Back to the "surprising" example
A0 =
−1 0 0
10 −1 0
0 0 −10
, A1 =
−10 0 10
0 −10 0
0 10 −1
.
D0 =
11
11
20
, D1 =
20
20
11
C i = Ai + D i .
F 0,1 = the LY vector �eld on [0, 1]3 induced by (C 0,1,D0,1).
Epidemic Models
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Lajmanovich and Yorke SIS ModelAnalysis
Main resultsBack to Lajmanovich and Yorke
• : Jump rate β
(i.e Switching from one environment to the other
between times t and t + s, s << 1, occurs with probability ≈ βs.)
Figure: β 7→ Λ(β).
Epidemic Models
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Lajmanovich and Yorke SIS ModelAnalysis
Main resultsBack to Lajmanovich and Yorke
• : Jump rate β (i.e Switching from one environment to the other
between times t and t + s, s << 1, occurs with probability ≈ βs.)
Figure: β 7→ Λ(β).
Epidemic Models
![Page 57: Epidemic Models - cmap.polytechnique.fr](https://reader031.vdocuments.mx/reader031/viewer/2022012016/61da7accd52b671725486673/html5/thumbnails/57.jpg)
Lajmanovich and Yorke SIS ModelAnalysis
Main resultsBack to Lajmanovich and Yorke
• : Jump rate β (i.e Switching from one environment to the other
between times t and t + s, s << 1, occurs with probability ≈ βs.)
Figure: β 7→ Λ(β).
Epidemic Models
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Lajmanovich and Yorke SIS ModelAnalysis
Main resultsBack to Lajmanovich and Yorke
Figure: Simulation of ‖Xt‖ for β = 10.
Epidemic Models