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: λ n and µ n , with λ 0 = µ 0 = 0 Generator: A p = ( p 0 , p 1 , . . . , p N ) Master equation: p ′ = pA Partition the state space and condition the state probabili- ties S = S 0 ∪ S Q , S 0 = { 0 } , S Q = { 1 , 2 , . . . , N } p = ( p 0 , p Q ) , p Q = ( p 1 , . . . , p N ) p ′ 0 = µ 1 p 1 p ′ Q = p Q A Q q Q ( t ) = p Q ( t ) / (1 − p 0 ( t )) QSD = stationary conditional distribution on S Q The stationary distribution of q Q is denoted q q is an eigenvector of A Q : qA Q = − µ 1 q 1 q The original process has degenerate stationary distribution: p 0 = 1 The original process with p Q (0) = q p Q ( t ) = q exp( − µ 1 q 1 t ) p 0 ( t ) = 1 − exp( − µ 1 q 1 t ) 1

INGEMAR N˚ 2 ASELL Can use this case to derive ODE for pgf G Q ( x ) of q from PDE of pgf G ( x, t ) of p ( t ) Two auxiliary processes with nondegenerate stationary dis- tributions X (0) : replace µ 1 by 0 X (1) : replace µ n by µ n − 1 The stationary distributions p (0) and p (1) are known functions of λ n and µ n A recursion relation for q involves p (0) and p (1) Time to extinction τ P { τ < t } = p 0 ( t ) Persistence: Time to exctinction from QSD, τ Q , has exponential dis- tribution with expectation E τ Q = 1 /µ 1 q 1 Time to extinction from state 1, τ 1 , has expectation E τ 1 = 1 /µ 1 p (0) 1 Logistic models Verhulst model has density dependence in both birth rate and death rate: λ n = λ (1 − α 1 n/N ) n , µ n = µ (1 + α 2 n/N ) n , R 0 = λ/µ Special case: Logistic epidemic (aka SIS model, with recovered indi- viduals susceptible): λ n = λ (1 − n/N ) n , µ n = µn The deterministic SIS model X ′ = µ ( R 0 − 1 − R 0 X/N ) X The model has a threshold at R 0 = 1: X ( t ) → K = (1 − 1 /R 0 ) N if R 0 > 1 and X ( t ) → 0 if R 0 < 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: R 0 > 1, R 0 < 1, and √ transition region near R 0 = 1, where ρ = ( R 0 − 1) N is constant Uniform approximations across the three regions are found for E τ 1 , E τ Q , E X (0) , E X (1) , E X ( Q ) The classic endemic model Famous model for measles since the 1950’s SIR model: recovered individuals are immune

PERSISTENCE IN POPULATION BIOLOGY MODELS 3 S ′ = µN − βSI/N − µS I ′ = βSI/N − ( γ + µ ) I α = ( γ + µ ) /µ, R 0 = 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 q si E τ Q = 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 R 0 > 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.

INGEMAR N˚ 4 ASELL 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: n N � � p (1) E τ n = 1 1 p (0) 1 . j p (1) p (0) µ 1 1 k =1 k j = k The QSD obeys the following recursion relation. The similarity with the above expression for E τ n is noted. n N � � p (1) 1 1 q 1 q n = p (0) q j . n p (1) p (0) k =1 k j = k 1 Notation that is used to express the uniform approximation results for the SIS model is summarized as follows: � β 1 � ρ , 1 f 1 = max , R 0 � R 0 β 2 � 1 f Q = min ρ 2 , 1 , � β 1 � ρ = ρ min � ρ , 1 , √ ρ = ( R 0 − 1) N, � β 1 = sgn( R 0 − 1) 2 N [log( R 0 − 1) − 1 + 1 /R 0 ] . Definitions of the functions H 1 , H 0 , H that are needed to express the uniform results for the SIS model are as follows: H 1 ( y ) = Φ( y ) φ ( y ) , 1 2 π exp( − y 2 / 2) , φ ( y ) = √ � y Φ( y ) = φ ( t ) dt, −∞ � y 1 H ( y ) = H 1 ( t ) dt, y + 1 /H ( y ) − 1 /H ( y ) H a ( y ) = − log | y | − 1 2 y 2 + 3 4 y 4 − 5 2 y 6 , � H a ( y ) , if y ≤ − 3, H 0 ( y ) = � y H a ( − 3) + − 3 H 1 ( 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: � � √ E τ 1 ≈ 1 µf 1 log N + H 0 ( � ρ ) , √ E τ Q ≈ 1 µf Q H ( � ρ ) N. Uniform approximations for expectations of the stationary distribu- tions of the two auxiliary processes and of the qsd for the SIS model: � 1 � 1 + ρH 1 ( ρ ) √ E X (1) ∼ min , 1 N, R 0 H 1 ( ρ ) � 1 � √ H 1 ( ρ ) E X (0) ≈ min , R 0 √ N, R 0 log N + H 0 ( ρ ) � 1 � H 1 ( ρ ) − H 1 ( − 1 /H ( ρ )) √ E X ( Q ) ≈ min , 1 N. R 0 1 + ρH ( ρ ) Approximations of E τ Q for the SIS model in different parameter re- gions: √ E τ Q ≈ 1 µf Q H ( � ρ ) N, √ R 0 β 2 E τ Q ≈ 1 1 ρ 2 H ( β 1 ) N, R 0 ≥ 1 , µ � E τ Q ≈ 1 2 π R 0 ( R 0 − 1) 2 exp( β 2 1 / 2) , R 0 > 1 , µ N √ E τ Q ≈ 1 µH ( ρ ) N, R 0 ≤ 1 , E τ Q ≈ 1 1 , R 0 < 1 , µ 1 − R 0 √ E τ Q ≈ 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

INGEMAR N˚ 6 ASELL SIS model: N=100, R 0 =1.5, µ =1 SIS model: N=100, R 0 =1.3, µ =1 0.05 2500 0.05 120 E τ n E τ n 0.04 2000 0.04 80 0.03 1500 0.03 0.02 1000 0.02 40 QSD 0.01 500 0.01 QSD 0 0 0 0 0 0 10 10 20 20 30 30 40 40 50 50 0 0 10 10 20 20 30 30 40 40 50 50 n n SIS model: N=100, R 0 =1.1, µ =1 SIS model: N=100, R 0 =0.9, µ =1 0.08 30 0.2 15 E τ n E τ n QSD QSD 0.06 0.15 20 10 0.04 0.1 10 5 0.02 0.05 0 0 0 0 0 0 10 10 20 20 30 30 40 40 0 0 5 5 10 10 15 15 20 20 25 25 30 30 n n SIS model: N=100, R 0 =0.7, µ =1 0.4 8 E τ n QSD 0.3 6 0.2 4 0.1 2 0 0 0 0 5 5 10 10 15 15 20 20 n 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 R 0 .

PERSISTENCE IN POPULATION BIOLOGY MODELS 7 SIS model: N=100, R 0 =1.5, µ =1 SIS model: N=100, R 0 =1.3, µ =1 0.05 0.05 P1 P1 QSD P0 0.04 0.04 0.03 0.03 0.02 0.02 0.01 0.01 QSD P0 0 0 0 10 20 30 40 50 60 0 10 20 30 40 50 n n SIS model: N=100, R 0 =1.1, µ =1 SIS model: N=100, R 0 =0.9, µ =1 0.08 0.2 P0 P0 QSD 0.06 0.15 P1 QSD 0.04 0.1 0.02 0.05 P1 0 0 0 10 20 30 40 0 10 20 30 n n SIS model: N=100, R 0 =0.7, µ =1 0.4 P0 0.3 0.2 QSD P1 0.1 0 0 5 10 15 20 n 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 R 0 .

INGEMAR N˚ 8 ASELL SIS model: N=100 30 25 20 15 10 EX1 EXQ EX0 5 0 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 R 0 Figure 3. Uniform approximations of the expectations of the QSD and of the stationary distributions of the two auxiliary processes for the SIS model are shown as functions of R 0 . SIS model: N=100, µ =1 5 10 4 10 3 10 2 10 E τ Q E τ 1 1 10 0 10 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 R 0 Figure 4. Uniform approximations of the expectations of the time to extinction from the state 1 and from the quasi-stationary distribution for the SIS model are shown as functions of R 0 .