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Epidemic Modeling: SIRS Models
Regina Dolgoarshinnykh
Columbia University
Steven P. Lalley
University of Chicago
Epidemic Models: SIR, SIRS
2
SIRS models
St = # susceptible at time t
It = # infected at time t
Rt = # recovered (immune) at time t
N ≡ St + It + Rt = population size
st = St/N, it = It/N
rt = Rt/N = 1 − st − it
γt = (st, it)T
3
SIRS Model
MCs indexed by N with transition rates:
ρ (s → i) = S · θI/N = N θsi
ρ (i → r) = ρI = N ρi
ρ (r → s) = R = N r
Questions:
• Establishment: Will the infection spread?
• Spread: How does it develop with time?
• Persistance: When does it disappear?
4
Overview
• Limiting ODEs
• Fluctuations about (s∞, i∞)
• Exit path LDP
• Time until infection “dies out”
5
N=100 N=100
1.0 1.0
0.8 0.8
population fractions
0.6 0.6
i
0.4 0.4
0.2 0.2
0.0 0.0
0.0 0.2 0.4 0.6 0.8 1.0 0 10 20 30 40 50
s time
N=400 N=400
1.0 1.0
0.8 0.8
population fractions
0.6 0.6
i
0.4 0.4
0.2 0.2
0.0 0.0
0 10 20 30 40 50
0.0 0.2 0.4 0.6 0.8 1.0
s time
N=2500 N=2500
1.0 1.0
0.8 0.8
population fractions
0.6 0.6
i
0.4 0.4
0.2 0.2
0.0 0.0
0 10 20 30 40 50
0.0 0.2 0.4 0.6 0.8 1.0
s time
6
Deterministic Approximation
Fix N , h > 0
Et(st+h) = st + rth − θitsth + o(h)
Et(it+h) = it + θitsth − ρith + o(h)
Get “mean field approximation” as h → 0
dst
= rt − θitst
dt
:= F (γt)
dit
= θitst − ρit
dt
7
Deterministic Approximation
Mean field: theta=3,rho=1
1
0.9
0.8
0.7
0.6
0.5
i
0.4
0.3
0.2
0.1
0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
s
8
Deterministic Approximation
γ
(¯t)t≥0 - solution of mean path ODE,
˙
i.e. γ = F (γ)
N
γt - random path
t≥0
N γ
Theorem 1. If γ0 → ¯0 as N → ∞ then for
any T > 0
N γ
lim sup |γt − ¯t| = 0 a.s.
N →∞ t≤T
9
Fluctuations around (s∞, i∞)
√
1 = N (sN − s )
xt t ∞
N
Xt :=
2 √
xt = N (iN − i∞)
t
so that
N
x1
st = s∞ + √ t
N
N x2
i = i∞ + √ t
t
N
N
Theorem 2. If X0 →D X0 as N → ∞, then
X N ⇒ X in DR2 [0, ∞).
10
Fluctuations around (s∞, i∞)
X is generated by G
2
∂ 1 2 ∂2
G= µi(x) + σij
i=1 ∂xi 2 i,j=1 ∂xi∂xj
where
1+θ
− 1+ρ −(1+ρ)
µ1(x) x1
=
µ2(x)
θ−ρ x2
1+ρ 0
2ρ(θ−ρ) ρ(θ−ρ)
− θ(1+ρ)
σ11 σ12 θ(1+ρ)
= .
σ12 σ22
ρ(θ−ρ) 2ρ(θ−ρ)
− θ(1+ρ) θ(1+ρ)
11
Fluctuations around (s∞, i∞)
100
80
60
40
20
x2
0
−20
−40
−60
−80
−100
−80 −60 −40 −20 0 20 40 60 80
x1
12
Time to Extinction
For all N , infection dies out with prob.1.
How long until this happens?
• If Y ∼ Geometric(q) then E(Y ) = 1
q
• Connection to “most likely” path
• Large Deviations for exit paths (LDP).
13
Large Deviations Principle
Def. Family µN satisfy LDP on X with rate
function I if
1
− inf ◦ I(x) ≤ limN →∞ log µN (F )
x∈F N
1
≤ limN →∞ log µN (F ) ≤ − inf I(x)
N x∈F¯
for F ⊂ X .
Yt = Poisson processes rate m
N
yt = N −1YN t satisfy LDP with rate function
T ˙
yt
I(y) = ˙
yt log ˙
− yt + m dt
0 m
T
:= ˙
f (yt, m) dt
0
14
Time Changed Poisson
Processes
Y1(t), Y2(t), Y3(t) are rate 1 PPs
N
yk (t) = yk (t) = N −1Yk (N t) for k = 1, 2, 3
t t
st = s0 − y1 θsuiu du + y3 ru du
0 0
t t
it = i0 + y1 θsuiu du − y2 ρiu du .
0 0
15
Exit Path LDP
• Why standard methods don’t work
– Contraction Principle
Cont. f : X → Y & LDP for µN on X
⇒ LDP for µN ◦ f −1 on Y.
– Wentzell and Freidlin
• Dangers of diffusion approximations
16
Exit path LDP
Fix γ = (st, it)t≥0 ∈ AC[0, T ]
Let λ, µ, ν ≥ 0 s.t.
dst
dt = νt − λt
di
t
= λt − µt
dt
17
Exit path LDP
For γ ∈ AC[0, T ]
T
I(γ) = inf f (λt, θstit) + f (µt, ρit) + f (νt, rt)dt,
λ,µ,ν
0
where
x
f (x, m) = x log − x + m, x, m ≥ 0.
m
Theorem 3. SIRS processes γ N satisfy LDP
with good rate function I(γ),
i.e.
γ
PN (||γ − ˜||T < δ ) ≈ e−N I(˜).
γ
18
Exit path LDP
Lower Bound
Define measure Q ∼ λt, µt, νt
PN (||γ − ˜||T < δ ) =
γ
dP
EN
Q γ
I{||γ − ˜||T < δ} ·
dQ
Upper Bound
Exponential Approximations
Markov-type Inequality
Boundary Problem
19
Time until extinction
τ N = inf{t : it = 0} = time to extinction
¯
I = inf γ Iτ (γ) = “minimal cost” of exit
In fact,for any >0
¯ ¯
lim PN eN (I − ) ≤ τ N ≤ eN (I + ) = 1.
N →∞
Conjecture.
1
lim log E τ N = I .
¯
N →∞ N
20
Extensions / Future Directions
• Time to extinction
• Similar models - spatial component
• Start with small number of infected
21
