PERSISTENCE IN POPULATION BIOLOGY MODELS INGEMAR N ASELL This note - - PDF document

PERSISTENCE IN POPULATION BIOLOGY MODELS

INGEMAR N˚ ASELL

This note contains copies of illustrations used in the Large Deviations Conference, Ann Arbor, June 2007. Overview Quasi-stationarity and persistence The SIS model: Asymptotic approximations of QSD Uniform results The classic endemic model Birth-death process, with origin absorbing State space: S={0,1,2,. . . ,N} Transition rates: λn and µn, with λ0 = µ0 = 0 Generator: A p = (p0, p1, . . . , pN) Master equation: p′ = pA Partition the state space and condition the state probabili- ties S = S0 ∪ SQ, S0 = {0}, SQ = {1, 2, . . . , N} p = (p0, pQ), pQ = (p1, . . . , pN) p′

0 = µ1p1

p′

Q = pQAQ

qQ(t) = pQ(t)/(1 − p0(t)) QSD = stationary conditional distribution on SQ The stationary distribution of qQ is denoted q q is an eigenvector of AQ: qAQ = −µ1q1q The original process has degenerate stationary distribution: p0 = 1 The original process with pQ(0) = q pQ(t) = q exp(−µ1q1t) p0(t) = 1 − exp(−µ1q1t)

1

2 INGEMAR N˚ ASELL

Can use this case to derive ODE for pgf GQ(x) of q from PDE of pgf G(x, t) of p(t) Two auxiliary processes with nondegenerate stationary dis- tributions X(0): replace µ1 by 0 X(1): replace µn by µn−1 The stationary distributions p(0) and p(1) are known functions of λn and µn A recursion relation for q involves p(0) and p(1) Time to extinction τ P{τ < t} = p0(t) Persistence: Time to exctinction from QSD, τQ, has exponential dis- tribution with expectation EτQ = 1/µ1q1 Time to extinction from state 1, τ1, has expectation Eτ1 = 1/µ1p(0)

1

Logistic models Verhulst model has density dependence in both birth rate and death rate: λn = λ(1 − α1n/N)n, µn = µ(1 + α2n/N)n, R0 = λ/µ Special case: Logistic epidemic (aka SIS model, with recovered indi- viduals susceptible): λn = λ(1 − n/N)n, µn = µn The deterministic SIS model X′ = µ(R0 − 1 − R0X/N)X The model has a threshold at R0 = 1: X(t) → K = (1 − 1/R0)N if R0 > 1 and X(t) → 0 if R0 < 1 The counterpart to the threshold in the stochastic model is sought The SIS model Asymptotic approximations of QSD as N → ∞ show qualitatively different behaviors in three parameter regions: R0 > 1, R0 < 1, and transition region near R0 = 1, where ρ = (R0 − 1) √ N is constant Uniform approximations across the three regions are found for Eτ1, EτQ, EX(0), EX(1), EX(Q) The classic endemic model Famous model for measles since the 1950’s SIR model: recovered individuals are immune

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S′ = µN − βSI/N − µS I′ = βSI/N − (γ + µ)I α = (γ + µ)/µ, R0 = β/(γ + µ) The classic endemic model The stochastic version of the model is bivariate: {S(t), I(t)} The states (S, 0) are absorbing QSD is denoted qsi EτQ = 1/µαq·i The classic endemic model The marginal distribution q·i behaves in qualitatively different ways in three parameter regions With large α, the transition region is wide q·i is close to geometric in a large part of the transition region The classic endemic model The persistence threshold is related to the critical community size The large deviation problem of determining q·i for R0 > 1 is open Persistence Deterministic modellers use the term persistence to describe various ways in which the solution of a deterministic model can avoid getting close to zero The deterministic persistence concept is at complete odds with the stochastic one. Persistence measured by time to extinction cannot be studied in the deterministic framework References (1) I. N˚ asell: The quasi-stationary distribution of the closed endemic SIS model, Adv Appl Prob, 28, 895–932, 1996. (2) I. N˚ asell: Extinction and quasi-stationarity in the Verhulst logis- tic model, J Theor Biol, 211, 11–27, 2001. (3) www.math.kth.se/∼ingemar/forsk/verhulst/verhulst.html,

• 2006. This is an extended and updated version of (2).

(4) I. N˚ asell: On the time to extinction in recurrent epidemics, J Roy Stat Soc B, 61, 309–330, 1999. (5) I. N˚ asell, A new look at the critical community size for childhood infections, Theor Pop Biol, 67, 203–216, 2005.

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The well-known expression for expected time to extinction from state n is rewritten in terms of the stationary distributions p(0) and p(1) of the two auxiliary processes: Eτn = 1 µ1

n

• k=1

1 p(1)

k N

• j=k

p(0)

j

p(1)

1

p(0)

1

. The QSD obeys the following recursion relation. The similarity with the above expression for Eτn is noted. qn = p(0)

n n

• k=1

1 p(1)

k N

• j=k

qj p(1)

1 q1

p(0)

1

. Notation that is used to express the uniform approximation results for the SIS model is summarized as follows: f1 = max β1 ρ , 1 R0

• ,

fQ = min R0β2

1

ρ2 , 1

• ,
• ρ = ρ min

β1 ρ , 1

• ,

ρ = (R0 − 1) √ N, β1 = sgn(R0 − 1)

• 2N[log(R0 − 1) − 1 + 1/R0].

Definitions of the functions H1, H0, H that are needed to express the uniform results for the SIS model are as follows: H1(y) = Φ(y) φ(y) , φ(y) = 1 √ 2π exp(−y2/2), Φ(y) = y

−∞

φ(t)dt, H(y) = 1 y + 1/H(y) y

−1/H(y)

H1(t)dt, Ha(y) = − log |y| − 1 2y2 + 3 4y4 − 5 2y6, H0(y) =

• Ha(y),

if y ≤ −3, Ha(−3) + y

−3 H1(t)dt,

if y > −3.

PERSISTENCE IN POPULATION BIOLOGY MODELS 5

Uniform approximations for the expected times to extinction from the state 1 and from the quasi-stationary distribution for the SIS model: Eτ1 ≈ 1 µf1

• log

√ N + H0( ρ)

• ,

EτQ ≈ 1 µfQH( ρ) √ N. Uniform approximations for expectations of the stationary distribu- tions of the two auxiliary processes and of the qsd for the SIS model: EX(1) ∼ min 1 R0 , 1 1 + ρH1(ρ) H1(ρ) √ N, EX(0) ≈ min 1 R0 , R0

• H1(ρ)

log √ N + H0(ρ) √ N, EX(Q) ≈ min 1 R0 , 1 H1(ρ) − H1(−1/H(ρ)) 1 + ρH(ρ) √ N. Approximations of EτQ for the SIS model in different parameter re- gions: EτQ ≈ 1 µfQH( ρ) √ N, EτQ ≈ 1 µ R0β2

1

ρ2 H(β1) √ N, R0 ≥ 1, EτQ ≈ 1 µ

• 2π

N R0 (R0 − 1)2 exp(β2

1/2),

R0 > 1, EτQ ≈ 1 µH(ρ) √ N, R0 ≤ 1, EτQ ≈ 1 µ 1 1 − R0 , R0 < 1, EτQ ≈ 1 µH(ρ) √ N, ρ = O(1).

Department of Mathematics, The Royal Institute of Technology, S-100 44 Stockholm, Sweden E-mail address: ingemar@math.kth.se

6 INGEMAR N˚ ASELL

10 20 30 40 50 0.01 0.02 0.03 0.04 0.05 10 20 30 40 50 500 1000 1500 2000 2500

SIS model: N=100, R0=1.5, µ=1 n QSD Eτn

10 20 30 40 50 0.01 0.02 0.03 0.04 0.05 10 20 30 40 50 40 80 120

SIS model: N=100, R0=1.3, µ=1 n QSD Eτn

10 20 30 40 0.02 0.04 0.06 0.08 10 20 30 40 10 20 30

SIS model: N=100, R0=1.1, µ=1 n QSD Eτn

5 10 15 20 25 30 0.05 0.1 0.15 0.2 5 10 15 20 25 30 5 10 15

SIS model: N=100, R0=0.9, µ=1 n QSD Eτn

5 10 15 20 0.1 0.2 0.3 0.4 5 10 15 20 2 4 6 8

SIS model: N=100, R0=0.7, µ=1 n Eτn QSD

Figure 1. The quasi-stationary distribution and the ex- pected time to extinction from the state n are shown for the SIS model for several values of R0.

PERSISTENCE IN POPULATION BIOLOGY MODELS 7

10 20 30 40 50 60 0.01 0.02 0.03 0.04 0.05

SIS model: N=100, R0=1.5, µ=1 n QSD P0 P1

10 20 30 40 50 0.01 0.02 0.03 0.04 0.05

SIS model: N=100, R0=1.3, µ=1 n P1 P0 QSD

10 20 30 40 0.02 0.04 0.06 0.08

SIS model: N=100, R0=1.1, µ=1 n P0 QSD P1

10 20 30 0.05 0.1 0.15 0.2

SIS model: N=100, R0=0.9, µ=1 n P0 P1 QSD

5 10 15 20 0.1 0.2 0.3 0.4

SIS model: N=100, R0=0.7, µ=1 n P0 QSD P1

Figure 2. The quasi-stationary distribution and the stationary distributions of the two auxiliary processes are shown for the SIS model for several values of R0.

8 INGEMAR N˚ ASELL

0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 5 10 15 20 25 30

SIS model: N=100 R0 EX0 EXQ EX1

Figure 3. Uniform approximations of the expectations

• f the QSD and of the stationary distributions of the

two auxiliary processes for the SIS model are shown as functions of R0.

0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 10 10

1

10

2

10

3

10

4

10

5

SIS model: N=100, µ=1 R0 Eτ1 EτQ

Figure 4. Uniform approximations of the expectations

• f the time to extinction from the state 1 and from the

quasi-stationary distribution for the SIS model are shown as functions of R0.