3rd workshop and conference on modeling infectious diseases · introductionjump markov processes...

Introduction Jump Markov Processes Some contributions to epidemiology

3rd Workshop and Conference on ModelingInfectious Diseases

Introduction to Stochastic Modelling

The Institute of Mathematical Sciences, Chennai, India

Objectives

• Recall basic concepts of infectious disease modelling.• Define the structure and formalism of stochastic models.• Understand the relationship between deterministic and

stochastic models.• Show several examples of applications of stochastic

modelling.

Contents

Introduction

Density-Dependent Jump Markov Processes

Some contributions of stochastic modelling to epidemiologyInitial extinction riskCritical community sizeMeasles and pertussis in London before/after vaccinationTwo-wave flu outbreak in Tristan da Cunha

Diseases considered

• The models are focussed toward the study of directlytransmitted, micro-parasitic infectious diseases.

• Infectious diseases (such as influenza) can be passedbetween individuals whereas non-infectious diseases(such as arthritis) develop over an individual’s lifespan.

• The infecting pathogen can be either a micro-parasite or amacro-parasite. Micro-parasitic infections develop rapidlyso that the intra-host dynamics can be ignored.

• Infectious diseases (both macro- and micro-parasitic) canalso be sub-divided into two further categories dependingon whether transmission of infection is direct (contact) orindirect (environment).

Keeling & Rohani (2008)

Diseases considered

Infection dynamics

S E I RλI ε ν

The SEIR model

S E I RλI ε ν

• λ: rate at which two individuals come into contact.• λI: per capita force of infection.• ε: inverse of the latent period.• ν: inverse of the infectious period.

The deterministic SIR model

S I RλI ν

dS(t)dt

=− λS(t)I(t)

dI(t)dt

=λS(t)I(t)− νI(t)

dR(t)dt

=νI(t)

With(S(0), I(0),R(0)

)= (S0, I0,0)

and constant population size S(t) + I(t) + R(t) = Ω.

Kermack & McKendrick (1927)

The deterministic SIR model

Two important results

• An epidemic can occur only if S0 > ν/λ, when thepopulation initially susceptible is above a critical size.

• At the end of the epidemic, it remains S∞ susceptibleindividuals, with S∞ solution of the equation:

S∞ = Ω +ν

S∞S0

). (2)

• Not all susceptibles will get infected during the epidemic.

Why using stochastic models?

1. It is the natural way to describe how an epidemic diseasespreads.

2. Some phenomena are stochastic by nature and cannot bedescribed in a deterministic setting.

3. Take into account the variability of the epidemic processwhen estimating and forecasting.

Three sources of stochasticity

1. Demographic (internal).Depends on the epidemiogical process.

2. Environmental (external).Acts on the epidemiogical process.

3. Observation.Does not change the epidemiological dynamics.

Observation stochasticity

• Diagnostic errors: false positive and false negative.• Incomplete reporting of cases: 60% for measles.• Fluctuations of the reporting rate: change in the number of

GPs in the surveillance system.• The reporting rate at time t is ρt ∼ Gamma(1/φ, ρφ).• Conditioning on ρt and the incidence in the model (Xt ), the

observed incidence Yt |ρt ,Xt ∼ Poisson(ρtXt ).• Conditioning on Xt , Yt follows a negative-binomial

distribution with:

E[Yt |Xt ] = ρXt and Var[Yt |Xt ] = ρXt + φρ2X 2t .

• This corresponds to an overdispersed observation process.

Fine & Clarkson (1982), Breto et al. (2009)

distribution with:

Environmental stochasticity

• Stochastic fluctuations of environmental factors (e.g.temperature, humidity) lead to stochastic fluctuations oftransmission parameters.

dS(t)dt

=− λ(1 + Fξ)S(t)I(t)

dI(t)dt

=λ(1 + Fξ)S(t)I(t)− νI(t)

dR(t)dt

=νI(t)

where ξ is a random variable and F is a forcing constant.

Breto et al. (2009)

Demographic stochasticity

• Results from the discrete nature of individuals in thepopulation:

(S(t), I(t),R(t)

)∈ N3.

• Mechanistic modelling of random events at the individuallevel:

• Infectious period with mean ν−1 and variance σ2.• Number of contacts ∼ Poisson process with intensity β/Ω.

• One can compute the distribution of the final size of theepidemic (S0 − S∞).

• To go further, let’s assume an exponentially distributedinfectious period: ν−2 = σ2.

• Memory-less property of the exponential distribution:(S(t), I(t),R(t)

): t ≥ 0 becomes a discrete state,

continuous time Markov process.

Andersson & Britton (2000)

Demographic stochasticity

• Results from the discrete nature of individuals in thepopulation:

(S(t), I(t),R(t)

)∈ N3.

• Mechanistic modelling of random events at the individuallevel:

• Infectious period with mean ν−1 and variance σ2.• Number of contacts ∼ Poisson process with intensity β/Ω.

• One can compute the distribution of the final size of theepidemic (S0 − S∞).

• To go further, let’s assume an exponentially distributedinfectious period: ν−2 = σ2.

• Memory-less property of the exponential distribution:(S(t), I(t),R(t)

): t ≥ 0 becomes a discrete state,

continuous time Markov process.

SIR with demographic stochasticity

S I RβI/Ω ν

Event Transition Transition probabilitywithin [t , t + dt ]

Infection (s, i , r)→ (s − 1, i + 1, r) βΩsi dt + o(dt)

Recovery (s, i , r)→ (s, i − 1, r + 1) νi dt + o(dt)

Bartlett (1949)

Contents

Introduction

Definition• Let’s take a model with d compartments and M transitions.• We note X(t) = (X1(t) . . . Xd (t))T ∈ Nd the vector of the

state of the population at time t .• We define each transition Tm,m ∈ 1, . . . ,M, by a vector

of state change: km = (k1m . . . kdm)T ∈ Zd and a jumpintensity am(x), such that:

P(X(t + dt) = x + km|X(t) = x) = am(x)dt + o(dt)

P(X(t + dt) = x|X(t) = x) = 1−M∑

am(x)dt + o(dt)

• X(t) : t ≥ 0 is a jump Markov process.• It is density-dependent if am(x) = Ωam(Ω−1x), where Ω is

the population size, assumed constant.Bartlett (1978), Ethier & Kurtz (1986)

P(X(t + dt) = x|X(t) = x) = 1−M∑

am(x)dt + o(dt)

P(X(t + dt) = x|X(t) = x) = 1−M∑

am(x)dt + o(dt)

P(X(t + dt) = x|X(t) = x) = 1−M∑

am(x)dt + o(dt)

Remarks

• Almost all Markov models used in epidemiology areDDJMPs (e.g. Bartlett’s SIR).

• Natural formalism to model demographic stochasticity.• Limited analytical results. Mainly on the final size of an

outbreak (total number of infected individuals).• The law of X(t) at every time t is intractable due to the

non-linearity of the contact process.• In order to study the dynamics of the model, mainly two

solutions:• Monte-Carlo approach: simulate many realisations of the

process to evaluate X(t) empirically.• Analytical approximations: first few moments of X(t), when

population is large.

Remarks

• Almost all Markov models used in epidemiology areDDJMPs (e.g. Bartlett’s SIR).

• Natural formalism to model demographic stochasticity.• Limited analytical results. Mainly on the final size of an

outbreak (total number of infected individuals).• The law of X(t) at every time t is intractable due to the

non-linearity of the contact process.• In order to study the dynamics of the model, mainly two

solutions:• Monte-Carlo approach: simulate many realisations of the

process to evaluate X(t) empirically.• Analytical approximations: first few moments of X(t), when

population is large.

Doob-Gillespie algorithm

• Proposed by Joseph L. Doob in 1940s and popularized byDaniel T. Gillespie in 1970s.

• Given a state x and a time t , simulate the following event:No transition occurs during the time interval [t , t + τ [ andthe transition Tµ occurs at the time t + τ .

• One can show that τ and µ are two random variables withprobability density:

p(τ) = a0(x) exp(−a0(x)τ), τ > 0, (5)

p(µ) =aµ(x)

a0(x), µ = 1, . . . ,M, (6)

where a0(x) =∑M

m=1 am(x).• Slow for large populations because E[τ ] = 1/a0(x) ∝ 1/Ω.

Doob (1942), Gillespie (1977)

Doob-Gillespie algorithm

• Proposed by Joseph L. Doob in 1940s and popularized byDaniel T. Gillespie in 1970s.

• Given a state x and a time t , simulate the following event:No transition occurs during the time interval [t , t + τ [ andthe transition Tµ occurs at the time t + τ .

• One can show that τ and µ are two random variables withprobability density:

p(τ) = a0(x) exp(−a0(x)τ), τ > 0, (5)

p(µ) =aµ(x)

a0(x), µ = 1, . . . ,M, (6)

where a0(x) =∑M

m=1 am(x).• Slow for large populations because E[τ ] = 1/a0(x) ∝ 1/Ω.

Doob (1942), Gillespie (1977)

Analytical approximations• Results obtained by Thomas G. Kurtz in the 1970s for the

large population limit Ω→∞ of XΩ = XΩ(t) : t ≥ 0.• Two theorems characterise the convergence of the

sequences of Ω−1XΩ.• Law of large numbers: for large population size and large

initial conditions, Ω−1XΩ can be approximated, over abounded time interval, by the deterministic solution φ:

dφ(t)dt

= F(φ(t)),

φ(0) = φ0,(7)

where F(x) =∑M

m=1 kmam(x),x ∈ Rd .• Kermack and McKendrick’s SIR model can be retrieved as

the large population limit of Bartlett’s SIR model.Ethier & Kurtz (1986)

Analytical approximations• Results obtained by Thomas G. Kurtz in the 1970s for the

large population limit Ω→∞ of XΩ = XΩ(t) : t ≥ 0.• Two theorems characterise the convergence of the

sequences of Ω−1XΩ.• Law of large numbers: for large population size and large

initial conditions, Ω−1XΩ can be approximated, over abounded time interval, by the deterministic solution φ:

dφ(t)dt

= F(φ(t)),

φ(0) = φ0,(7)

where F(x) =∑M

m=1 kmam(x),x ∈ Rd .• Kermack and McKendrick’s SIR model can be retrieved as

the large population limit of Bartlett’s SIR model.Ethier & Kurtz (1986)

• The next step is to study the deviations of Ω−1XΩ from φ.• Kurtz’s central limit theorem establishes that these

deviations are Gaussian and of the order of 1/√

Ω.• It follows from these 2 theorems that XΩ can be

approximated in distribution by:

XΩ = Ωφ +√

ΩY, (8)

with E[Y(t)] = Φ(t ,0)Y0 and Ξ(t) = Cov[Y(t),Y(t)]solution of:

= ∂F(φ)Ξ + Ξ(∂F(φ))T + G(φ), (9)

where G(x) =∑M

m=1 kmkTmam(x).

• If Y0 = 0, XΩ is a Gaussian process with meanE[XΩ(t)] = Ωφ(t) and covariance Σ(t) = ΩΞ(t).

Ethier & Kurtz (1986)

XΩ = Ωφ +√

ΩY, (8)

= ∂F(φ)Ξ + Ξ(∂F(φ))T + G(φ), (9)

where G(x) =∑M

m=1 kmkTmam(x).

XΩ = Ωφ +√

ΩY, (8)

= ∂F(φ)Ξ + Ξ(∂F(φ))T + G(φ), (9)

where G(x) =∑M

m=1 kmkTmam(x).

Monte-Carlo vs Gaussian approximation

Macroscopic value

0 5 10 15 20 25Time (days)

0.0001

195% Confidence interval

0 5 10 15 20 25Time (days)

Large initial condition I0

Monte-Carlo vs Gaussian approximation

Macroscopic value

0 5 10 15 20 25Time (days)

0.0001

0 5 10 15 20 25Time (days)

Small initial condition I0.

Contents

Introduction

Initial extinction risk• In the stochastic SIR model, a major epidemic can still be

avoided even if R0 > 1.• The initial phase of the epidemic can be described by a

birth-death process: individuals live for a random duration(infectious period) during which they give birth (infect)according to a Poisson process with intensity β.

• The probability Pext that the birth-death process stops aftera finite number of generations is qI0 , where q is the uniqueroot in [0,1[ of s = f (s), with

f (s) =

∫ ∞0

e−βt(1−s)gI(t)dt , |s| ≤ 1, (10)

where gI is the pdf of the infectious period.• For DDJMPs, we can set gI(t ; k , ν) = Γ(k , kν), k ∈ N∗:

Becker (1977), Andersson & Britton (2000)

f (s) =

∫ ∞0

e−βt(1−s)gI(t)dt , |s| ≤ 1, (10)

f (s) =

∫ ∞0

e−βt(1−s)gI(t)dt , |s| ≤ 1, (10)

f (s) =

∫ ∞0

e−βt(1−s)gI(t)dt , |s| ≤ 1, (10)

Erlang distribution

S I1... Ik R

! I! k" k" k"

0 2 4 6 8

k = 1 k = 5

gI(t ; k , ν) =(kν)k tk−1e−kνt

(k − 1)!

VarI(t ; k , ν) =1

Initial extinction risk

• Pext = qI0k

• For k = 1 (exponential distribution), q1 = 1/R0.• For k = 2:

q2 =12

4R0−

√1 +

), (11)

and q2 < q1, ∀R0 > 1.• For k ≥ 3, f (s) = s is a ≥ 4th degree polynomial equation

and we rely on numerical solutions.

• Pext = qI0k

q2 =12

4R0−

√1 +

), (11)

• Pext = qI0k

q2 =12

4R0−

√1 +

), (11)

1 1.5 2 2.5 3 4 5 6 7 8 9 10

Pathogens with more variable infectious periods have a higherrisk of initial extinction.

Critical community size

• Minimum size of a closed human population within which apathogen can persist indefinitely.

• The value of the CCS depends on the pathogen.• The CCS of measles is between 100,000 and 200,000 ind.• There is a lack of such data for other diseases.• How can we estimate the CCS for these diseases?• One solution is to use stochastic modelling since the CCS

is linked to the concept of risk of extinction.

Bartlett (1957), Choisy et al. (2007)

Critical community size• Minimum size of a closed human population within which a

pathogen can persist indefinitely.• The value of the CCS depends on the pathogen.• The CCS of measles is between 100,000 and 200,000 ind.

• There is a lack of such data for other diseases.• How can we estimate the CCS for these diseases?• One solution is to use stochastic modelling since the CCS

is linked to the concept of risk of extinction.Bartlett (1957), Choisy et al. (2007)

Measure of persistence• SIR with birth and death (average life expectancy = 1/µ)

S I R! "

• We consider the expected time to extinction E[TΩ] as ameasure of persistence: TΩ = inft ≥ 0 : IΩ(t) = 0.

• We start from the quasi-stationary distribution, which is thelimit of the distribution conditioned on non-extinction:

π(Ω)x = lim

t→∞p(XΩ(t) = x|XΩ(t) /∈ A). (12)

• It follows that TΩ is exponentially distributed with mean:

E[TΩ] =1

(ν + µ)π(Ω).,1

, (13)

where π(Ω).,1 =

∑Ωs=0 π

(Ω)s,1 is the marginal probability of

having a single infectious individual at quasi-stationarity.Darroch & Seneta (1967), Andersson & Britton (2000)

Measure of persistence• SIR with birth and death (average life expectancy = 1/µ)

0 50 100 150 200 250 300

epidemic fadeout endemic fadeout deterministic

• We consider the expected time to extinction E[TΩ] as ameasure of persistence: TΩ = inft ≥ 0 : IΩ(t) = 0.

• We start from the quasi-stationary distribution, which is thelimit of the distribution conditioned on non-extinction:

π(Ω)x = lim

t→∞p(XΩ(t) = x|XΩ(t) /∈ A). (12)

E[TΩ] =1

(ν + µ)π(Ω).,1

, (13)

where π(Ω).,1 =

∑Ωs=0 π

Measure of persistence• SIR with birth and death (average life expectancy = 1/µ)• We consider the expected time to extinction E[TΩ] as a

measure of persistence: TΩ = inft ≥ 0 : IΩ(t) = 0.• We start from the quasi-stationary distribution, which is the

limit of the distribution conditioned on non-extinction:

π(Ω)x = lim

t→∞p(XΩ(t) = x|XΩ(t) /∈ A). (12)

E[TΩ] =1

(ν + µ)π(Ω).,1

, (13)

where π(Ω).,1 =

∑Ωs=0 π

π(Ω)x = lim

t→∞p(XΩ(t) = x|XΩ(t) /∈ A). (12)

E[TΩ] =1

(ν + µ)π(Ω).,1

, (13)

where π(Ω).,1 =

∑Ωs=0 π

π(Ω)x = lim

t→∞p(XΩ(t) = x|XΩ(t) /∈ A). (12)

E[TΩ] =1

(ν + µ)π(Ω).,1

, (13)

where π(Ω).,1 =

∑Ωs=0 π

Quasi-stationary distribution• In practice, it is not possible to compute π(Ω)

s,i analytically.• Heuristic: use the endemic equilibrium of the Gaussian

approximation.

Macroscopic value

0 5 10 15 20 25 30 35

Time (days)

0.0001

0 5 10 15 20 25 30 35

Time (days)

Nasell (1999, 2003, 2005)

• Bartlett (1960): the size of the community for whichmeasles is as likely as not to fade out after a majorepidemic.

• I. Nasell (2005): The critical community size is that value ofΩ for which the median time to extinction equals onequasi-period T0:

T0 = E[TΩ] log 2 (14)

• For measles (R0 = 14, ν−1 = 1 week, µ−1 = 70 years) thequasi-period is T0 = 2.06 years.

• Finally, using the Gaussian approximation to estimateE[TΩ] in (13), one obtains ΩCCS ≈ 115,000 ind.

Bartlett (1960), Nasell (2005)

T0 = E[TΩ] log 2 (14)

Measles and pertussis in London before/aftervaccination

• The biannual cycle of measles becomes irregular aftervaccination.

• The multi-annual cycle of pertussis becomes more regular.• Although both incidences decrease, the amplitude of

pertussis epidemics is similar to that in the pre-vaccine era.• How pathogens with similar R0 (≈ 17) and infectious

periods (around 2 weeks) can have such differentdynamics and transitions following vaccination?

Keeling (2001), Rohani et al. (1999, 2002)

Deterministic vs Stochastic paradigms

• Deterministic paradigm: complex interactions betweennon-linearity and seasonal forcing lead to amplifiedoscillations (resonance) and very rich dynamics.

• In this paradigm, stochasticity only plays a passive role:switch between attractors.

• Stochastic paradigm: resonance depends on the capacityof demographic stochasticity to amplify oscillations (activerole).

• Gaussian approximation allows us to derive two quantities:• Amplification: system’s capacity to sustain and amplify the

stochastic fluctuations.• Coherence: measures the concentration of spectral power

around the dominant endogenous frequency.

Earn (2000), Dushoff (2004), Alonso et al. (2007)

Deterministic vs Stochastic paradigms

• Deterministic paradigm: complex interactions betweennon-linearity and seasonal forcing lead to amplifiedoscillations (resonance) and very rich dynamics.

• In this paradigm, stochasticity only plays a passive role:switch between attractors.

• Stochastic paradigm: resonance depends on the capacityof demographic stochasticity to amplify oscillations (activerole).

• Gaussian approximation allows us to derive two quantities:• Amplification: system’s capacity to sustain and amplify the

stochastic fluctuations.• Coherence: measures the concentration of spectral power

around the dominant endogenous frequency.

Earn (2000), Dushoff (2004), Alonso et al. (2007)

Amplification of oscillations

Following vaccination, amplification increases for pertussis(diamonds) but not for measles (triangles).

Alonso et al. (2007)

Amplification of oscillations

Following vaccination, amplification increases for pertussis(diamonds) but not for measles (triangles).

Coherence of oscillations

Following vaccination, coherence increases for pertussis(diamonds) and decreases for measles (triangles).

Coherence of oscillations

Following vaccination, coherence increases for pertussis(diamonds) and decreases for measles (triangles).

Two-wave flu outbreak in Tristan da Cunha

The most isolated inhabited place in the world!

• Small population ≈300individuals

• Fully isolated• Free social mixing

Population almost naive toinfluenza (Tristan escapedthe 1918 Spanish flu!)

• 1971: a boat returning from Cape-Town landed 2individuals with flu symptoms

• A welcome home (homogeneous mixing?) dance wasgiven

1971 influenza epidemic on Tristan da Cunha

14/08 21/08 28/08 04/09 11/09 18/09 25/09 02/10 09/10

• 273 (96%) of 284 islanders were infected• 92 (32%) were rapidly reinfected during the 2nd wave• Unexplained: lack of empirical data• 312 (85%) of 365 cases are known with daily accuracyMantle and Tyrrell (1973)

14/08 21/08 28/08 04/09 11/09 18/09 25/09 02/10 09/10

Objectives

• Disentangling between 5 biological hypotheses that couldexplain these cases of rapid reinfection

• Accounting for both demographic stochasticity in theepidemic dynamics and under-reporting in theobservations

Mantle and Tyrrell (1973)

Mechanistic modeling of reinfection hypotheses

Primary immune response to influenzaMut the virus Mutated during the first wave2Vi 2 Viruses since the beginning of the epidemicInH Intra-Host reinfectionPPI Partially Protective Immunity

WoN Window of reinfection or No humoral response

Camacho et al. (2011) Proc. Roy. Soc. B

S E I C Hλ ε ν γ

S E I C H

λ ε ν γ

σλενγ

S E I C H

λ1 ε ν γ

λ2ενγ

S E I C Hλ ε ν αγ

(1− α)γ

S E I C Hλ ε ν γ

S E I C Hλ ε ν αγ

(1− α)γ

S E I C W Hλ ε ν αγ

(1− α)γ

Confrontation with data• The unobserved Markov process X(t) : t ≥ 0 depends

on an unknown parameter vector θ• Data consist in a collection of T = 59 discrete

observations y1:T = (y1,y2, ...,yT ).• Poisson observation process h(yt |xt , θ) with underreportingρ < 1.

• The likelihood is given by the identity:

L(y1:T |θ) =T∏

∑xt∈C

h(yt |xt ,θ)∑

xt−1∈C

p(xt |xt−1,θ)P(xt−1|y1:t−1,θ),

but is intractable• Sequential Monte-Carlo: only requires to be able to sample

from p(xt |xt−1,θ) (Doob-Gillespie algo)• Iterated Filtering (Ionides et al., 2006): find θ which

maximizes the likelihood

Confrontation with data• The unobserved Markov process X(t) : t ≥ 0 depends

on an unknown parameter vector θ• Data consist in a collection of T = 59 discrete

observations y1:T = (y1,y2, ...,yT ).• Poisson observation process h(yt |xt , θ) with underreportingρ < 1.

• The likelihood is given by the identity:

L(y1:T |θ) =T∏

∑xt∈C

h(yt |xt ,θ)∑

xt−1∈C

p(xt |xt−1,θ)P(xt−1|y1:t−1,θ),

but is intractable• Sequential Monte-Carlo: only requires to be able to sample

from p(xt |xt−1,θ) (Doob-Gillespie algo)• Iterated Filtering (Ionides et al., 2006): find θ which

maximizes the likelihood

Parameter inference and model selection

R0 = β ν

Prob. of long−term immunity: α

WoN 2Vi Mut InH PPI

Latent period (days): 1 ε

Partial protection: 1 − σ

WoN 2Vi Mut InH PPI

Infective period (days): 1 ν

Reporting rate: ρ

WoN 2Vi Mut InH PPI

−118

−117

−116

−115

−114

−113

Removed period (days): 1 γ

Maximized log−likelihood

WoN 2Vi Mut InH PPI

Reinfection window (days): 1 τ

∆AICc

WoN 2Vi Mut InH PPI

• High estimates for R0 ≈ 10• We used the corrected Akaike Information Criterion (AICc)

to select the best model• The best model corresponds to the Window of reinfection

or No humoral response hypothesis

Parameter inference and model selection

days post-infection

0 10 20 30 40 50 60

infected protected by CTLs protected by antibodies unprotected

Immuno-dynamicsThe probability to remain unprotected peaks after 3 weeks dueto the window of reinfection and then decreases to ≈ 20% dueto the lack of protective antibodies

The role of demographic stochasticity

0.10.20.30.40.50.60.7

Window or Nothing

14/08 21/08 28/08 04/09 11/09 18/09 25/09 02/10 09/10

Mutation

14/08 21/08 28/08 04/09 11/09 18/09 25/09 02/10 09/10

The inter-wave extinction probability varies from onemodel to another

%&'()("*"!+

1/2,3 .12,3 .32,3 ,/2,4 112,4 132,4 .52,4 ,.21, ,421,

Best model

• The most likely mechanism is robust to inter-waveextinctions

• and prevents a third wave to occur

%&'()("*"!+

1/2,3 .12,3 .32,3 ,/2,4 112,4 132,4 .52,4 ,.21, ,421,

Best model

• The most likely mechanism is robust to inter-waveextinctions

• and prevents a third wave to occur

Conclusion

• Propagation of pathogens in a population is inherentlystochastic.

• Stochastic models require Monte-Carlo simulations→intensive as Ω→∞.

• Deterministic models arise as the large population limit(Ω→∞) of the stochastic model.

• One can approximate the stochastic process by aGaussian process.

• No closed form for the likelihood for stochastic model→numerical methods, Monte-Carlo approach

• With increasing computer resources, more and morestochastic modelling & fitting to data.

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