todd parsons - sorbonne-universite.fr

INTRODUCTION FADE-OUT PROBABILITIES APPROXIMATING THE SIR ODES PUZZLING PERSISTENCE EVOLUTIONARY EPIDEMIOLOGY

The Persistence of Emerging PathogensThe Persistence of Emerging Pathogens

Todd L. ParsonsTodd L. Parsons

CNRS & Laboratoire de Probabilites, Statistique et Modelisation (LPSM, UMR 8001)CNRS & Laboratoire de Probabilites, Statistique et Modelisation (LPSM, UMR 8001)

GdT Covid-19GdT Covid-19March 29th, 2020March 29th, 2020


OUTLINE

INTRODUCTION

FADE-OUT PROBABILITIES

APPROXIMATING THE SIR ODES

PUZZLING PERSISTENCE

EVOLUTIONARY EPIDEMIOLOGY

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THE SIR EPIDEMIC MODEL (KERMACK & MCKENDRICK, 1927)

µN// Susceptible

SN

µ��

βN IN // Infected

IN

µ��

γ // RecoveredRN

µ��

I N = S + I + RI µ = birth/immigration and death ratesI β = transmission rateI γ = recovery rate

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THE SIR EPIDEMIC MODEL, DETERMINISTIC FORMULATION

We typically analyze this via a system of differential equations:

dSN

dt= µN − β SNIN

N− µSN

dIN

dt= β

SNIN

N− γIN − µIN

dRN

dt= γIN − µRN

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THE SIR EPIDEMIC MODEL, DENSITY FORMULATION

For reasons I’ll elaborate upon, this is better expressed in terms of thedensities S = SN

N , I = INN , R = RN

N (S + I + R = 1).

dSdt

= µ− βSI − µS

dIdt

= βSI − γI − µI

dRdt

= γI − µR

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ASIDE #1: IN PRAISE OF TOY MODELS

Attributed to John von Neumann by Enrico Fermi:

“With four parameters I can fit an elephant, and with five, I can make himwiggle his trunk”

Dyson (2004) “A meeting with Enrico Fermi” Nature 427 p. 297

Even fitting a single parameter,R0, at the early stages of an epidemic is achallenge:

Park et al. (2020) “Reconciling early-outbreak estimates of the basicreproductive number and its uncertainty: framework and applications to thenovel coronavirus (SARS-CoV-2) outbreak”(doi:10.1101/2020.01.30.20019877)

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RECURRING EPIDEMICSFor a wide range of parameters, the SIR model predicts recurring epidemicsas damped cycles

S(0) = 0.99, I(0) = 0.01.

R0 = 17, γ−1 = 13 days, µ = 0.002/day, N = 100.6 / 45


ASIDE # 2: MEASLES CYCLIC EPIDEMICS

(Chen & Epureanu, 2017)

Recurrent epidemic cycles in actual populations don’t always showdamping; the reasons are a topic of ongoing research: Dushoff et. al. (2004)“Dynamical resonance can account for seasonality of influenza epidemics”PNAS 101 (48) pp. 16915-16916.

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FADE-OUT: INFLUENZA EPIDEMIC,TRISTAN DA CUNHA,1971

(Camacho & Cazelles 2013)

Other pathogens have a finite number of epidemics before stochasticityresults in extinction of the pathogen. We wish to estimate the probability thatthis occurs.

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THE BASIC REPRODUCTION RATIOThe basic reproduction number is

R0 := β︸︷︷︸transmission rate

× 1µ+ γ︸︷︷︸

duration of infection

× S0(= 1)︸︷︷︸inital proportion of susceptbles

Disease is endemic (persists indefinitely) ifR0 > 1.

(Doherty et al., 2016)

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OUTLINE

INTRODUCTION





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JOINT WITH

Ben Bolker Jonathan Dushoff David Earn

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A PERSISTENCE PUZZLE

I Provided that µ� 1, approximately e−R0 susceptible at the end of theepidemic.

I e.g. for measles, less than 1 in a billion remain susceptible at the end ofthe first outbreak⇒ suggests it should not persist.

I LargerR0 ⇒ larger, faster first epidemic⇒ persistence seems less likely.I Yet, many diseases have large values ofR0: measles (12–18), pertussis

(12–17), malaria (32–48)I How can a disease with very highR0 persist after initial invasion?

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THE SIR EPIDEMIC MODEL (KERMACK & MCKENDRICK, 1927), Redux

µN// Susceptible

SN

µ��

βN IN // Infected

IN

µ��

γ // RecoveredRN

µ��

I To properly analyze persistence and extinction, we need to return to ouroriginal, discrete and stochastic formulation.

I N = S + I + RI µ = birth/immigration and death ratesI β = transmission rateI γ = recovery rate

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ASIDE # 3: CONTINUOUS-TIME MARKOV CHAINS (CTMCS)

I A Markov process is a ’memoryless’ stochastic process: looselyspeaking, a process satisfies the Markov property if one can makepredictions for the future of the process based solely on its present statejust as well as one could knowing the process’s full history.

I i.e., conditional on the present state of the system, its future and past areindependent

I A Markov process X(t) is described by it’s initial state X(0) (which maybe a random variable) and it’s jump (or transiition) rates qx,y:

P (∆X(t + ∆t) = y|X(t) = x) = qx,y∆t + o(∆t).

I j = y− x is a jump; sometimes, I’ll use J for the set of allowable jumps,i.e., those j for which qx,x+j 6= 0.

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AN INDIVIDUAL-BASED STOCHASTIC FORMULATION

I We work with a continuous-time Markov chain with transitionsJump Rate(SN, IN)→ (SN + 1, IN) µN(SN, IN)→ (SN − 1, IN) µSN

(SN, IN)→ (SN − 1, IN + 1) βN SNIN

(SN, IN)→ (SN, IN − 1) (γ + µ)IN

I The corresponding master equation is, unfortunately, analyticallyintractable.

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ASIDE #4: DENSITY DEPENDENT POPULATION PROCESSES (KURTZ,1970)

I The SIR model is included in a broad class of models, that includeschemical reaction equations and many models of biological interest.

I Let {λj(x)}j∈J be a collection of non-negative functions defined on asubset E ⊆ Rd

+. Let E(n) be the set of (rescaled) lattice points in E:

E(n) := E ∩ 1nZd,

and assume that x ∈ E(n) and λj(x) > 0 imply x + n−1j ∈ E(n).I The density dependent family corresponding to the λj(x) is a sequence{X(n)} of jump Markov processes such that X(n) has state space E(n) andintensities

q(n)x,y = nλn(y−x)(x), x, y ∈ E(n).

I Can also allow rates that depend on n, λ(n)n(y−x)(x), provided

λ(n)n(y−x)(x)→ λn(y−x)(x) as n→∞.

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ASIDE #4: LAW OF LARGE NUMBERS (KURTZ, 1970)

I Let {λ(n)j (x)}j∈J be as above and let {X(n)} be the corresponding

density-dependent family.I Assume that there exist functions {λj(x)}j∈J such that

limn→∞

∑j∈ZK

‖j‖ supx∈K|λ(n)

j (x)− λj(x)| = 0 and∑j∈J

‖j‖ supx∈K

λj(x) <∞

for all compact sets K ⊂ E.

I Let F(x) =∑

j∈J jλj(x). Suppose X(n)(0)→ x0 and let X(t, x0) satisfy

ddt

X = F(X).

with X(0, x0) = x0

I Then, for any fixed T > 0,

limn→∞

supt≤T|X(n)

(t)− X(t, x0)| = 0 a.s.

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ASIDE #4: CENTRAL LIMIT THEOREM (KURTZ, 1971)

I Assume in addition that

limn→∞

√n∑j∈J

‖j‖ supx∈K|λ(n)

j (x)− λj(x)| = 0 and∑j∈J

‖j‖2 supx∈K

λj(x) <∞.

I Let V(n) =√

n(X(n) − X) and suppose that V(n)(0)→ V(0).I Then, V(n) ⇒ V in DE[0,∞), where V is an Ornstein-Uhlenbeck process

with (Ito) SDE:

dV(t) = J(X(t, x0))V(t) dt +∑j∈J

j√λj(X(t, x0)) dBj(t)

where J is the Jacobian of F (previous slide) and the Bj are independentBrownian motions.

I Intuitively: X(n)= X(t, x0) + 1√

n V(t) + lower order terms.

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LAW OF LARGE NUMBERS FOR THE SIR MODEL

I On finite time intervals, the densities for the Markov chain SN(t)N , IN(t)

Nconverge to the solution of the Kermack-McKendrick ordinarydifferential equations:

dSdt

= µ− βSI − µS

dIdt

= βSI − γI − µI

almost surely as N →∞.I Initial conditions

S(0) = limN→∞

SN(t)N

and I(0) = limN→∞

IN(t)N

.

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MULTI-SCALE ANALYSIS

I However, if we assume that IN(t)� N, then

limN→∞

IN(0)

N= 0.

I The law of large numbers doesn’t “see” the infected individuals.I If SN(0) = Θ(N), then S(0) > 0, and we get a much simpler

approximation for the density of susceptibles

dSdt

= µ(1− S),

I This has solution S(t) = 1 + e−µt(S(0)− 1).I We need to characterize the number infected.

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(BRANCHING) BIRTH AND DEATH PROCESSES

I Time-dependent individual infection rate β SN(t)N ≈ βS(t).

I Recovery and/or death rate γ + µ.I The number of infecteds is approximately birth (infections) and death

(or recovery) process.I This approximation holds exactly when the pathogen can go extinct:

when first appearing, and at the end of a major outbreak.21 / 45


ASIDE # 5: MONOTONE COUPLING

LemmaFor ε > 0 sufficiently small, we can simultaneously construct SN(t), IN(t), and andsupercritical time-inhomogeneous Markov birth-and-death processes Z±ε(t) withtransition rates

P {Z±ε(t + ∆t) = Z±ε(t) + 1} = β (S(t)± ε) Z±ε(t)∆t + o(∆t)P {Z±ε(t + ∆t) = Z±ε(t)− 1} = (µ+ γ)Z±ε(t)∆t + o(∆t)

such thatZ−ε(t) ≤ IN(t) ≤ Z+ε(t)

for all t such that IN(t) ≤ εN. Thus,

P{

supt

IN(t) ≥ εN}≥ P

{sup

tZ−ε(t) ≥ εN

}P{IN(t) = 0} ≥ P{Z+ε(t) = 0}

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ASIDE # 5: MONOTONE COUPLING (SKETCH PROOF)I Order individuals so that Z−ε(t) ⊆ IN(t) ⊆ Z+ε(t).I Generate potential births and deaths at the fastest possible rate for all

processes:β(S(t) + ε)Z+ε(t) and (γ + µ)Z+ε(t)

I Each potential birth is always a birth for Z+ε(t), and is a birth for IN(t)and Z−ε(t) with probabilities

SN(t)IN(t)N(S(t) + ε)Z+ε(t)

and(S(t)− ε)Z−ε(t)(S(t) + ε)Z+ε(t)

Add a “phantom individual” for each birth not occurring in IN(t).I Each potential death is always a death for Z+ε(t), and is a birth for IN(t)

and Z−ε(t) with probabilities

IN(t)Z+ε(t)

andZ−ε(t)Z+ε(t)

Kill an individual in Z−ε(t), IN(t) \ Z−ε(t) or Z+ε(t) \ IN(t) accordingly.I Refine Kurtz to show that either Z−ε(t) > εN or Z+ε(t) = 0 before

SN(t) 6∈ [N(S(t) + ε),N(S(t) + ε)] with high probability.23 / 45


BIRTH AND DEATH PROCESSES: EXTINCTION PROBABILITIES

Theorem (Kendall, 1948)Let Z(t) be a continuous time Markov birth-and-death process with transitions andrates given by

P {Z(t + ∆t) = Z(t) + 1} = b(t)Zt∆t + o(∆t)P {Z(t + ∆t) = Z(t)− 1} = d(t)Zt∆t + o(∆t)

Then, the probability of extinction in finite time, q, is∫∞0 e

∫ s0 d(u)−b(u) dud(s) ds

1 +∫∞

0 e∫ s

0 d(u)−b(u) dud(s) ds,

which is equal to 1 if and only if the integral∫∞

0 e∫ s

0 d(u)−b(u) dud(s) ds diverges. Ifthe process does not go extinct, it grows indefinitely.

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OUTBREAK AND PERSISTENCE PROBABILITIES

I In particular, the probability of persistence starting from(S(0) = S0, IN(0) = I0) is asymptotically

P {IN(t) = 0} =

( ∫ t0 e−

∫ τ0 βS(u)−γ−µ du(γ + µ) dτ

1 +∫ t

0 e−∫ τ

0 βS(u)−γ−µ du(γ + µ) dτ

)I0

S(t) = 1 + e−µt(S0 − 1)

I At the beginning of the outbreak S(t) ≈ S0 ≈ 1 and one infected.I Simplifies to give outbreak probability ≈ 1− 1

R0.

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EXACT EXPRESSIONS

In particular, setting κ = βµ

(1− S0) and θ = β−γµ− 1, the probability of

burnout is ((γ + µ)eκκ−θγ(θ, κ)

1 + (γ + µ)eκκ−θγ(θ, κ)

)I0

.

Where γ(θ, κ) is the incomplete γ-function:

γ(θ, κ) =

∫ θ

0e−xxκ−1 dx

= e−κκθ∞∑

n=0

κn

θ(n+1) ,

where θ(n) = θ(θ + 1) · · · (θ + n− 1) is the Pochhammer symbol and the sumconverges for all κ > 0.

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PERSISTENCE PROBABILITIES

I To apply this to the probability of post-epidemic burnout, we need tochoose an initial condition (S0, I0) that reflect the end of the initialepidemic.

I The law of large numbers tells us that the Kermack-McKendrick ODEsdescribe the process up to corrections of o(N):

SN(t) = NS(t) + o(N) IN(t) = NI(t) + o(N).

I If Imin is the minimum value attained by the ODE, and 1� δ ≥ Imin, takeI0 = bδNc.

I There is then a tδ and Sδ such that (S(tδ), I(tδ)) = (Sδ, δ).

I By the Law of Large Numbers,(

SN(tδ)N , IN(tδ)

N

)→ (S0, δ) a.s. as N →∞.

I Take (S0, I0) = (Sδ, δ).

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OUTLINE

INTRODUCTION





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SIR PHASE PORTRAIT

R0 ≈ 2, γ−1 = 13 days, µ = 0.001/day, N = 106.

S(0) = 1− 1N , I(0) = 1

N .

I Unfortunately, if µ > 0, the SIR ODEs are analytically intractable, so weneed to either numerically evaluate or approximate (tδ, Sδ).

I Use the host mortality rate µ (e.g. ≈ 5× 10−5/day for humans) as asmall parameter for asymptotic expansions.

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OUTER APPROXIMATION

I For µ� 1, we have

dIdS

=βSI − γI − µIµ(1− S)− βSI

≈ −1 +γ

β

1S,

which may be solved (Kermack & McKendrick, 1927) to yield

I = I0 + (S0 − S) +1R0

ln

(SS0

).

I For large values of N, we can approximate (S0, I0) = (1, 0).I This gives a good approximation of the true trajectory where dS

dt < 0;I Recognizing that maxima and minima of I(t) occur when S(t) = 1

R0, we

see that Imax ≈(

1− 1R0

)+ 1R0

ln(

1R0

).

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OUTER APPROXIMATION

R0 ≈ 2, γ−1 = 13 days, µ = 0.001/day, N = 106.

S(0) = 1− 1N , I(0) = 1

N .

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BOUNDARY LAYER APPROXIMATION

I To obtain the behaviour near the boundary, we assume I(t) = µι(t).

dιdS

=1µ

(βS− γ − µ)ι

(1− S− βSι)≈ 1µ

(βS− γ)ι

(1− S).

I Solve this to obtain “boundary-layer” solution

I(S) ≈√εIb

(1− 1R0

)(1− Sb

1− S

)R0−1ε

eR0(Sb−S)

ε .

for ε = µγ+µ

, where (Sb, Ib) are the point of entry into the boundary(somewhat arbitrary).

I In practice, we take Ib = µβ

(R0 − 1) (the proportion at the endemicequilibrium), while to first approximation Sb solves

(1− Sb) +1R0

ln (Sb) =µ

β(R0 − 1)

I Obtain analytical expressions for Sδ and the minimum infected, Imin.

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BOUNDARY LAYER APPROXIMATION

R0 ≈ 2, γ−1 = 13 days, µ = 0.001/day, N = 106.

S(0) = 1− 1N , I(0) = 1

N .

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OUTLINE

INTRODUCTION





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PROBABIILTY OF PERSISTENCE (µ ≈ 0.001/DAY)

Curves correspond to mean infectious period in days.The probability of surviving the initial epidemic is maximized atR0 ≈ 1 andat large values ofR0, and for longer infectious periods.

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R0 vs. Imin (µ ≈ 0.001/DAY)

Imin is minimized when 2 ≤ R0 ≤ 4.

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PIECING TOGETHER THE PUZZLE

Recall our approximation to the fade-out probability:

P {IN(t) = 0} =

( ∫ t0 e−

∫ τ0 βS(u)−γ−µ du(γ + µ) dτ

1 +∫ t

0 e−∫ τ

0 βS(u)−γ−µ du(γ + µ) dτ

)I0

I It depends not on the current density of susceptibles, but on the futuredensity.I With demography, the susceptibles recover exponentially fast.

I It strongly depends on the number of infected individualsI Small or large values of R0 maximize the number of susceptibles remaining

at the end of the first epidemic.I Caveat: this is still insufficient to explain the persistence of e.g. measles.

Other (extrinsic) mechanisms have been suggested:I Multiple introductions into the host population.I Spatial spread among cities promoting global persistence in spite of local

fadeouts.I Invasion with low R0 followed by evolution to higher R0.

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OUTLINE

INTRODUCTION





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JOINT WITH

Troy DayQueen’s University

Sylvain GandonCNRS Montpellier

Amaury LambertSorbonne Universite& College de France

I Day et al. (2020) “The Price equation and evolutionary epidemiology”Philos. Trans. Royal Soc. B. 375 (1797) 20190357

I Parsons et. al (2018) “Pathogen evolution in finite populations: slow andsteady spreads the best” J. Royal Soc. Interface 15 (47) 20180135

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VIRULENCE

I The term virulence appears hazily throughout the epidemiologicalliterature, often used to describe distinct aspects of a pathogen’sdisease-producing capacity:I infectivity: the ability to colonise and to invade a host, andI the severity of the disease produced.

I This is in part because there is evidence that these two quantities arepositively associated: avenues of increasing transmission often involvemore severe effects in hosts.

I We distinguish between transmissibility (β) from the virulence (α),which we define as the increase in the mortality rate in infected overnon-infected hosts.

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ATTENUATION OF VIRULENCE

Ongoing evidence that new host-pathogen associations tend to be morevirulent (Read, 1994):I As early as 1881, Pasteur observed the attenuation of anthrax bascillus.I Myxomatosis, introduced into Australia in 1950 to control exploding

rabbit populations initially had mortality rates of 99.8%, killing 85% ofrabbits within 6–10 days of infection.I By 1957, mortality rates had declined to 50%, with those dying surviving 3–4

weeks after infection.I By infecting Australian and European rabbits with both current and

originally introduced strains, Fenner & Marshall (1957) showed that thiswas due to both evolving resistance in rabbits and reduction inmyxomatosis virulence.

I SIV (ancient) is less virulent than HIV (1970’s)I SARS-CoV-2 appears to be less virulent than SARS-CoV-1

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THE “CONVENTIONAL WISDOM”

“[without] the early appearance and dominance of strains of virus whichcaused a lower mortality [. . . ] rabbits would have been eradicated or greatlyreduced in numbers, and the rabbit itself would have disappeared from suchlocalities”

Fenner & Ratcliffe. Myxomatosis. Cambridge University Press, 1965.

“The ‘conventional wisdom’ that successful parasites have to become be-nign is not based on exact evolutionary thinking. Rather than minimizingvirulence, selection will work to increase a parasite’s reproductive rate.”

Nowak & May (1994) Proc. R. Soc. Lond. B 255 (4): 81–89

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SIR MODEL WITH DEMOGRAPHY

InfectedsI1

δ+α1+γ1

""λ // Susceptibles

S

β1 I1

;;

β2 I2 ##

δ // RemovedR

InfectedsI2

δ+α2+γ2

<<

λ immigration rateδ base mortality rateβi infectivity, strain iαi virulence, strain iγi recovery rate, strain i

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EQUILIBRIUM INVASION

If strain one is already at its endemic steady state when strain two arrives,then it’s probability of invading is{

1− R(0,1)R(0,2)

ifR(0,2) > R(0,1), and

0 otherwise,

This is consistent withR0 optimization.

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OUT-OF-EQUILIBRIUM INVASIONIf we start away from equilibrium, however, virulence matters:I Let I?1 be the endemic density of strain one.I Suppose due to some (small) perturbation, it is instead at density I1 6= I?1

when strain two appears.I Strain two now has invasion probability(

1−R(0,1)

R(0,2)

)(1−R(0,1)(δ + α2 + γ2)

λR(0,1) − δ(I1 − I?1 )

)+O

((1−R(0,1)

R(0,2)

)3)

I RecallR(0,2) = β2δ+α2+γ2

. Any given value ofR(0,2) can be achievedinfinitely many ways.I If I1 > I?1 , among all those ways, those with the lowest virulence have the

greatest invasion probability.I If I1 < I?1 , those with the highest virulence have the greatest invasion

probability.

I So, not only the ratio matters, but how it is achieved.I Work in progress confirms that less virulent strains are more likely to

escape fade-out.

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todd parsons - sorbonne-universite.fr

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