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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
Introduction Jump Markov Processes Some contributions to epidemiology
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.
Introduction Jump Markov Processes Some contributions to epidemiology
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
Introduction Jump Markov Processes Some contributions to epidemiology
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)
Introduction Jump Markov Processes Some contributions to epidemiology
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)
Introduction Jump Markov Processes Some contributions to epidemiology
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)
Introduction Jump Markov Processes Some contributions to epidemiology
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)
Introduction Jump Markov Processes Some contributions to epidemiology
Infection dynamics
Keeling & Rohani (2008)
Introduction Jump Markov Processes Some contributions to epidemiology
Infection dynamics
S E I RλI ε ν
Keeling & Rohani (2008)
Introduction Jump Markov Processes Some contributions to epidemiology
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.
Introduction Jump Markov Processes Some contributions to epidemiology
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)
(1)
With(S(0), I(0),R(0)
)= (S0, I0,0)
and constant population size S(t) + I(t) + R(t) = Ω.
Kermack & McKendrick (1927)
Introduction Jump Markov Processes Some contributions to epidemiology
The deterministic SIR model
Kermack & McKendrick (1927)
Introduction Jump Markov Processes Some contributions to epidemiology
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
λln(
S∞S0
). (2)
• Not all susceptibles will get infected during the epidemic.
Kermack & McKendrick (1927)
Introduction Jump Markov Processes Some contributions to epidemiology
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.
Introduction Jump Markov Processes Some contributions to epidemiology
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.
Introduction Jump Markov Processes Some contributions to epidemiology
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.
Introduction Jump Markov Processes Some contributions to epidemiology
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.
Introduction Jump Markov Processes Some contributions to epidemiology
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)
Introduction Jump Markov Processes Some contributions to epidemiology
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)
Introduction Jump Markov Processes Some contributions to epidemiology
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)
Introduction Jump Markov Processes Some contributions to epidemiology
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)
(3)
where ξ is a random variable and F is a forcing constant.
Breto et al. (2009)
Introduction Jump Markov Processes Some contributions to epidemiology
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)
Introduction Jump Markov Processes Some contributions to epidemiology
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)
Introduction Jump Markov Processes Some contributions to epidemiology
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)
Introduction Jump Markov Processes Some contributions to epidemiology
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
Introduction Jump Markov Processes Some contributions to epidemiology
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∑
m=1
am(x)dt + o(dt)
(4)
• 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)
Introduction Jump Markov Processes Some contributions to epidemiology
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∑
m=1
am(x)dt + o(dt)
(4)
• 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)
Introduction Jump Markov Processes Some contributions to epidemiology
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∑
m=1
am(x)dt + o(dt)
(4)
• 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)
Introduction Jump Markov Processes Some contributions to epidemiology
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∑
m=1
am(x)dt + o(dt)
(4)
• 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)
Introduction Jump Markov Processes Some contributions to epidemiology
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.
Andersson & Britton (2000)
Introduction Jump Markov Processes Some contributions to epidemiology
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.
Andersson & Britton (2000)
Introduction Jump Markov Processes Some contributions to epidemiology
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)
Introduction Jump Markov Processes Some contributions to epidemiology
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)
Introduction Jump Markov Processes Some contributions to epidemiology
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)
Introduction Jump Markov Processes Some contributions to epidemiology
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)
Introduction Jump Markov Processes Some contributions to epidemiology
• 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:
dΞdt
= ∂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)
Introduction Jump Markov Processes Some contributions to epidemiology
• 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:
dΞdt
= ∂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)
Introduction Jump Markov Processes Some contributions to epidemiology
• 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:
dΞdt
= ∂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)
Introduction Jump Markov Processes Some contributions to epidemiology
Monte-Carlo vs Gaussian approximation
Macroscopic value
0 5 10 15 20 25Time (days)
0
50
100
150
200
250
300
350
Infe
ctiv
es (I
)
1e-05
0.0001
0.001
0.01
0.1
195% Confidence interval
0 5 10 15 20 25Time (days)
0
50
100
150
200
250
300
350
Infe
ctiv
es (I
)
Large initial condition I0
Introduction Jump Markov Processes Some contributions to epidemiology
Monte-Carlo vs Gaussian approximation
Macroscopic value
0 5 10 15 20 25Time (days)
-100
0
100
200
300
400
Infe
ctiv
es (I
)
1e-05
0.0001
0.001
0.01
0.1
195% Confidence interval
0 5 10 15 20 25Time (days)
-100
0
100
200
300
400
Infe
ctiv
es (I
)
Small initial condition I0.
Introduction Jump Markov Processes Some contributions to epidemiology
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
Introduction Jump Markov Processes Some contributions to epidemiology
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)
Introduction Jump Markov Processes Some contributions to epidemiology
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)
Introduction Jump Markov Processes Some contributions to epidemiology
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)
Introduction Jump Markov Processes Some contributions to epidemiology
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)
Introduction Jump Markov Processes Some contributions to epidemiology
Erlang distribution
S I1... Ik R
! I! k" k" k"
time
0.0
0.1
0.2
0.3
0.4
0.5
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
kν2
Introduction Jump Markov Processes Some contributions to epidemiology
Initial extinction risk
• Pext = qI0k
• For k = 1 (exponential distribution), q1 = 1/R0.• For k = 2:
q2 =12
(1 +
4R0−
√1 +
8R0
), (11)
and q2 < q1, ∀R0 > 1.• For k ≥ 3, f (s) = s is a ≥ 4th degree polynomial equation
and we rely on numerical solutions.
Becker (1977), Andersson & Britton (2000)
Introduction Jump Markov Processes Some contributions to epidemiology
Initial extinction risk
• Pext = qI0k
• For k = 1 (exponential distribution), q1 = 1/R0.• For k = 2:
q2 =12
(1 +
4R0−
√1 +
8R0
), (11)
and q2 < q1, ∀R0 > 1.• For k ≥ 3, f (s) = s is a ≥ 4th degree polynomial equation
and we rely on numerical solutions.
Becker (1977), Andersson & Britton (2000)
Introduction Jump Markov Processes Some contributions to epidemiology
Initial extinction risk
• Pext = qI0k
• For k = 1 (exponential distribution), q1 = 1/R0.• For k = 2:
q2 =12
(1 +
4R0−
√1 +
8R0
), (11)
and q2 < q1, ∀R0 > 1.• For k ≥ 3, f (s) = s is a ≥ 4th degree polynomial equation
and we rely on numerical solutions.
Becker (1977), Andersson & Britton (2000)
Introduction Jump Markov Processes Some contributions to epidemiology
Initial extinction risk
R0
sh
ap
e k
1
2
3
4
5
6
7
8
9
10
1 1.5 2 2.5 3 4 5 6 7 8 9 10
q
0.001
0.01
0.05
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
Pathogens with more variable infectious periods have a higherrisk of initial extinction.
Introduction Jump Markov Processes Some contributions to epidemiology
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)
Introduction Jump Markov Processes Some contributions to epidemiology
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)
Introduction Jump Markov Processes Some contributions to epidemiology
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)
Introduction Jump Markov Processes Some contributions to epidemiology
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)
Introduction Jump Markov Processes Some contributions to epidemiology
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)
Introduction Jump Markov Processes Some contributions to epidemiology
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)
Introduction Jump Markov Processes Some contributions to epidemiology
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)
Introduction Jump Markov Processes Some contributions to epidemiology
Measure of persistence• SIR with birth and death (average life expectancy = 1/µ)
S I R! "
µ
µ
• 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)
Introduction Jump Markov Processes Some contributions to epidemiology
Measure of persistence• SIR with birth and death (average life expectancy = 1/µ)
time
pre
va
len
ce
0
100
200
300
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)
• 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)
Introduction Jump Markov Processes Some contributions to epidemiology
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)
• 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)
Introduction Jump Markov Processes Some contributions to epidemiology
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)
• 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)
Introduction Jump Markov Processes Some contributions to epidemiology
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)
• 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)
Introduction Jump Markov Processes Some contributions to epidemiology
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
20
40
60
80
100
Infe
ctiv
es n
um
ber
0.0001
0.001
0.01
0.1
1
95% Confidence interval
0 5 10 15 20 25 30 35
Time (days)
0
20
40
60
80
100
Infe
ctiv
es n
um
ber
Nasell (1999, 2003, 2005)
Introduction Jump Markov Processes Some contributions to epidemiology
Critical community size
• 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)
Introduction Jump Markov Processes Some contributions to epidemiology
Critical community size
• 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)
Introduction Jump Markov Processes Some contributions to epidemiology
Critical community size
• 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)
Introduction Jump Markov Processes Some contributions to epidemiology
Critical community size
• 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)
Introduction Jump Markov Processes Some contributions to epidemiology
Introduction Jump Markov Processes Some contributions to epidemiology
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)
Introduction Jump Markov Processes Some contributions to epidemiology
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)
Introduction Jump Markov Processes Some contributions to epidemiology
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)
Introduction Jump Markov Processes Some contributions to epidemiology
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)
Introduction Jump Markov Processes Some contributions to epidemiology
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)
Introduction Jump Markov Processes Some contributions to epidemiology
Amplification of oscillations
Following vaccination, amplification increases for pertussis(diamonds) but not for measles (triangles).
Alonso et al. (2007)
Introduction Jump Markov Processes Some contributions to epidemiology
Amplification of oscillations
Following vaccination, amplification increases for pertussis(diamonds) but not for measles (triangles).
Alonso et al. (2007)
Introduction Jump Markov Processes Some contributions to epidemiology
Coherence of oscillations
Following vaccination, coherence increases for pertussis(diamonds) and decreases for measles (triangles).
Alonso et al. (2007)
Introduction Jump Markov Processes Some contributions to epidemiology
Coherence of oscillations
Following vaccination, coherence increases for pertussis(diamonds) and decreases for measles (triangles).
Alonso et al. (2007)
Introduction Jump Markov Processes Some contributions to epidemiology
Two-wave flu outbreak in Tristan da Cunha
The most isolated inhabited place in the world!
Introduction Jump Markov Processes Some contributions to epidemiology
Two-wave flu outbreak in Tristan da Cunha
• Small population ≈300individuals
• Fully isolated• Free social mixing
Population almost naive toinfluenza (Tristan escapedthe 1918 Spanish flu!)
Introduction Jump Markov Processes Some contributions to epidemiology
Two-wave flu outbreak in Tristan da Cunha
• 1971: a boat returning from Cape-Town landed 2individuals with flu symptoms
• A welcome home (homogeneous mixing?) dance wasgiven
Introduction Jump Markov Processes Some contributions to epidemiology
1971 influenza epidemic on Tristan da Cunha
time
nu
mb
er
of
ca
se
s
0
10
20
30
40
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)
Introduction Jump Markov Processes Some contributions to epidemiology
1971 influenza epidemic on Tristan da Cunha
time
nu
mb
er
of
ca
se
s
0
10
20
30
40
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)
Introduction Jump Markov Processes Some contributions to epidemiology
1971 influenza epidemic on Tristan da Cunha
time
nu
mb
er
of
ca
se
s
0
10
20
30
40
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)
Introduction Jump Markov Processes Some contributions to epidemiology
1971 influenza epidemic on Tristan da Cunha
time
nu
mb
er
of ca
se
s
0
10
20
30
40
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)
Introduction Jump Markov Processes Some contributions to epidemiology
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
Introduction Jump Markov Processes Some contributions to epidemiology
Mechanistic modeling of reinfection hypotheses
S E I C Hλ ε ν γ
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
Introduction Jump Markov Processes Some contributions to epidemiology
Mechanistic modeling of reinfection hypotheses
S E I C H
I ECH
λ ε ν γ
σλενγ
Tmut
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
Introduction Jump Markov Processes Some contributions to epidemiology
Mechanistic modeling of reinfection hypotheses
S E I C H
I ECH
λ1 ε ν γ
λ2ενγ
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
Introduction Jump Markov Processes Some contributions to epidemiology
Mechanistic modeling of reinfection hypotheses
S E I C Hλ ε ν αγ
(1− α)γ
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
Introduction Jump Markov Processes Some contributions to epidemiology
Mechanistic modeling of reinfection hypotheses
S E I C Hλ ε ν γ
σλ
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
Introduction Jump Markov Processes Some contributions to epidemiology
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
Introduction Jump Markov Processes Some contributions to epidemiology
Mechanistic modeling of reinfection hypotheses
S E I C Hλ ε ν αγ
(1− α)γ
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
Introduction Jump Markov Processes Some contributions to epidemiology
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
Introduction Jump Markov Processes Some contributions to epidemiology
Mechanistic modeling of reinfection hypotheses
S E I C W Hλ ε ν αγ
(1− α)γ
ω
λ
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
Introduction Jump Markov Processes Some contributions to epidemiology
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∏
t=1
∑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
Introduction Jump Markov Processes Some contributions to epidemiology
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∏
t=1
∑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
Introduction Jump Markov Processes Some contributions to epidemiology
Parameter inference and model selection
Model
5
10
15
20
25
30
0.5
0.6
0.7
0.8
0.9
1.0
R0 = β ν
Prob. of long−term immunity: α
WoN 2Vi Mut InH PPI
0.5
1.0
1.5
2.0
2.5
3.0
0.65
0.70
0.75
0.80
0.85
0.90
Latent period (days): 1 ε
Partial protection: 1 − σ
WoN 2Vi Mut InH PPI
1
2
3
4
5
6
7
0.55
0.60
0.65
0.70
0.75
0.80
Infective period (days): 1 ν
Reporting rate: ρ
WoN 2Vi Mut InH PPI
5
10
15
20
−118
−117
−116
−115
−114
−113
Removed period (days): 1 γ
Maximized log−likelihood
WoN 2Vi Mut InH PPI
0
1
2
3
4
5
6
0
2
4
6
8
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
Camacho et al. (2011) Proc. Roy. Soc. B
Introduction Jump Markov Processes Some contributions to epidemiology
Parameter inference and model selection
days post-infection
pro
po
rtio
n o
f in
div
idu
als
0.0
0.2
0.4
0.6
0.8
1.0
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
Camacho et al. (2011) Proc. Roy. Soc. B
Introduction Jump Markov Processes Some contributions to epidemiology
The role of demographic stochasticity
time
0
10
20
30
40
50
60
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
Incid
en
ce
Extin
ction
The inter-wave extinction probability varies from onemodel to another
Camacho et al. (2011) Proc. Roy. Soc. B
Introduction Jump Markov Processes Some contributions to epidemiology
The role of demographic stochasticity
!"#$
%&'()("*"!+
,-.
,-/
,-0
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
Camacho et al. (2011) Proc. Roy. Soc. B
Introduction Jump Markov Processes Some contributions to epidemiology
The role of demographic stochasticity
!"#$
%&'()("*"!+
,-.
,-/
,-0
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
Camacho et al. (2011) Proc. Roy. Soc. B
Introduction Jump Markov Processes Some contributions to epidemiology
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.