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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

• 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

21

```
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