Epidemic Modeling: SIRS Models (short) - Columbia University

Epidemic Modeling: SIRS Models
Regina Dolgoarshinnykh
Columbia University
Steven P. Lalley
University of Chicago

Epidemic Models: SIR, SIRS
2

SIRS models
S t = # susceptible at time t
I t = # infected at time t
R t = # recovered (immune) at time t
N ≡ S t + I t + R t = population size
s t = S t /N, i t = I t /N
r t = R t /N = 1 − s t − i t
γ t = (s t , i t ) T 3

SIRS Model
MCs indexed by N with transition rates:
ρ (s → i) = S · θI/N = Nθsi
ρ (i → r) = ρI = Nρi
ρ (r → s) = R = Nr
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

i
i
i
N=100
N=100
1.0
1.0
0.8
0.6
0.4
0.2
population fractions
0.8
0.6
0.4
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
0.6
0.4
0.2
population fractions
0.8
0.6
0.4
0.2
0.0
0.0 0.2 0.4 0.6 0.8 1.0
s
0.0
0 10 20 30 40 50
time
N=2500
N=2500
1.0
1.0
0.8
0.6
0.4
0.2
population fractions
0.8
0.6
0.4
0.2
0.0
0.0 0.2 0.4 0.6 0.8 1.0
s
0.0
0 10 20 30 40 50
time
6

Deterministic Approximation
Fix N, h > 0
E t (s t+h ) = s t + r t h − θi t s t h + o(h)
E t (i t+h ) = i t + θi t s t h − ρi t h + o(h)
Get “mean field approximation” as h → 0
⎧
⎪⎨
⎪⎩
ds t
dt
di t
dt
= r t − θi t s t
= θi t s t − ρi t
:= F (γ t )
7

Deterministic Approximation
1
Mean field: theta=3,rho=1
0.9
0.8
0.7
0.6
i
0.5
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)
(
γ
N t
)
t≥0
- random path
Theorem 1. If γ0
N
any T > 0
→ ¯γ 0 as N → ∞ then for
lim sup |γt N − ¯γ t | = 0 a.s.
N→∞ t≤T
9

Fluctuations around (s ∞ , i ∞ )
X N t :=
⎧
⎨
⎩
x 1 t = √ N(s N t − s ∞ )
x 2 t = √ N(i N t − i ∞ )
so that
⎧
⎪⎨ s N t
⎪⎩ i N t
= s ∞ + x1 t
√
N
= i ∞ + x2 t
√
N
Theorem 2. If X0
N → D X 0 as N → ∞, then
X N ⇒ X in D R 2[0, ∞).
10

Fluctuations around (s ∞ , i ∞ )
X is generated by G
G =
2∑
i=1
µ i (x) ∂
∂x i
+ 1 2
2∑
i,j=1
σ ij
∂ 2
∂x i ∂x j
where
( )
µ1 (x)
µ 2 (x)
=
⎛
− 1+θ ⎞
1+ρ −(1+ρ)
⎛ ⎞
x 1
⎜
⎟ ⎜ ⎟
⎝
⎠ ⎝ ⎠
θ−ρ
1+ρ
0 x 2
( )
σ11 σ 12
σ 12 σ 22
=
⎛
⎜
⎝
2ρ(θ−ρ)
θ(1+ρ)
− ρ(θ−ρ)
θ(1+ρ)
− ρ(θ−ρ) ⎞
θ(1+ρ)
⎟
2ρ(θ−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 µ N satisfy LDP on X with rate
function I if
− inf I(x) ≤ lim 1
x∈F ◦ N→∞
N log µN (F )
1
≤ lim N→∞
N log µN (F ) ≤ − inf I(x)
x∈ ¯F
for F ⊂ X .
Y t = Poisson processes rate m
y N t
= N −1 Y Nt satisfy LDP with rate function
I(y) =
:=
∫ T
(ẏt )
0 ẏt log
m
∫ T
0 f(ẏ t, m) dt
− ẏ t + m dt
14

Time Changed Poisson
Processes
Y 1 (t), Y 2 (t), Y 3 (t) are rate 1 PPs
y k (t) = y N k (t) = N −1 Y k (Nt) for k = 1, 2, 3
(∫ t
) (∫ t
s t = s 0 − y 1 θs ui u du + y 3 r u du
0 0
(∫ t
) (∫ t
i t = i 0 + y 1 θs ui u du − y 2 ρi u du
0 0
)
)
.
15

Exit Path LDP
• Why standard methods don’t work
– Contraction Principle
Cont. f : X → Y & LDP for µ N on X
⇒ LDP for µ N ◦ f −1 on Y.
– Wentzell and Freidlin
• Dangers of diffusion approximations
16

Exit path LDP
Fix γ = (s t , i t ) t≥0 ∈ AC[0, T ]
Let λ, µ, ν ≥ 0 s.t.
⎧
⎪⎨
ds t
dt = ν t − λ t
⎪⎩
di t
dt = λ t − µ t
17

Exit path LDP
I(γ) = inf
λ,µ,ν
∫T
0
f(x, m) = x log
For γ ∈ AC[0, T ]
f(λ t , θs t i t ) + f(µ t , ρi t ) + f(ν t , r t )dt,
where
( x
m)
− x + m, x, m ≥ 0.
Theorem 3. SIRS processes γ N
with good rate function I(γ),
satisfy LDP
i.e.
P N (||γ − ˜γ|| T < δ) ≈ e −NI(˜γ) .
18

Exit path LDP
Lower Bound
Define measure Q ∼ λ t , µ t , ν t
P N (||γ − ˜γ|| T < δ) =
E N Q
(
I{||γ − ˜γ|| T < δ} · dP
dQ
)
Upper Bound
Exponential Approximations
Markov-type Inequality
Boundary Problem
19

Time until extinction
τ N = inf{t : i t = 0} = time to extinction
Ī = inf γ I τ (γ) = “minimal cost” of exit
In fact,for any ɛ > 0
lim
(
N→∞ PN e N(Ī−ɛ) ≤ τ N ≤ e N(Ī+ɛ)) = 1.
Conjecture.
lim
N→∞
1
N log E τ N = Ī.
20

Extensions / Future Directions
• Time to extinction
• Similar models - spatial component
• Start with small number of infected
21