dengueARS0

Estimating the Attack Ratioof Dengue Epidemics under

Time-varying Force ofInfection using Aggregated

Notification Data

Flavio Codeco Coelho andLuiz Max Carvalho

Motivation

Vector dynamics

Building blocks

Variable Force of Infection

Modeling Dengue

Single-strain model


Parameter estimation

Estimating S0Attack Ratio

Estimating the Attack Ratio ofDengue Epidemics under

Time-varying Force of Infection usingAggregated Notification Data

Flavio Codeco Coelho and Luiz Max Carvalho

Applied Mathematics School, Fundacao Getulio Vargas

May 1st, 2015



Notification Data


Motivation

Vector dynamics

Building blocks


Modeling Dengue

Single-strain model




Summary

MotivationVector dynamics

Building blocksVariable Force of Infection

Modeling DengueSingle-strain modelVariable Force of Infection

Parameter estimationEstimating S0

Attack Ratio



Notification Data


Motivation

Vector dynamics

Building blocks


Modeling Dengue

Single-strain model




Dengue Dynamics

I Dengue is a Multi-Strain vector-borne disease

I 4 major viral strains in circulation in Brazil

I Case-notification data is aggregated, i.e., does notdiscriminate serotype except for a handful of cases.

I It’s a Seasonal disease, but recurrence pattern is hard topredict

I Vector population dynamics plays a major role in themodulation of incidence

I Imunological structure of the population is also a keyfactor, but is mostly unknown.



Notification Data


Motivation

Vector dynamics

Building blocks


Modeling Dengue

Single-strain model




Dengue Dynamics









Notification Data


Motivation

Vector dynamics

Building blocks


Modeling Dengue

Single-strain model




Dengue Dynamics









Notification Data


Motivation

Vector dynamics

Building blocks


Modeling Dengue

Single-strain model




Dengue Dynamics









Notification Data


Motivation

Vector dynamics

Building blocks


Modeling Dengue

Single-strain model




Dengue Dynamics









Notification Data


Motivation

Vector dynamics

Building blocks


Modeling Dengue

Single-strain model




Dengue Dynamics









Notification Data


Motivation

Vector dynamics

Building blocks


Modeling Dengue

Single-strain model




4 epidemics

(a) 2010 (b) 2011

(c) 2012 (d) 2013



Notification Data


Motivation

Vector dynamics

Building blocks


Modeling Dengue

Single-strain model




Environmental determinants

Vector dynamics

I A. Aegypti population dynamics display marked seasonality

I Temperature, Humidity and rainfall are important factors

I Environmental stock of eggs

I Effects on mosquito reproduction are non-linear

I Delayed influence



Notification Data


Motivation

Vector dynamics

Building blocks


Modeling Dengue

Single-strain model





Vector dynamics





I Delayed influence



Notification Data


Motivation

Vector dynamics

Building blocks


Modeling Dengue

Single-strain model





Vector dynamics





I Delayed influence



Notification Data


Motivation

Vector dynamics

Building blocks


Modeling Dengue

Single-strain model





Vector dynamics





I Delayed influence



Notification Data


Motivation

Vector dynamics

Building blocks


Modeling Dengue

Single-strain model





Vector dynamics





I Delayed influence



Notification Data


Motivation

Vector dynamics

Building blocks


Modeling Dengue

Single-strain model




Effective Reproductive number(Rt )

The effective reproductive number can be easily estimatedfrom the incidence time-series, Yt :

Rt =

(Yt+1

Yt

)1/n

(1)

Where n is the ratio between the length of reporting intervaland the mean generation time of the disease.

Nishiura et. al. (2010)



Notification Data


Motivation

Vector dynamics

Building blocks


Modeling Dengue

Single-strain model




Rt ’s uncertainty

But what about the uncertainty about Rt1?

We explore the approach of Ederer and Mantel[4], whoseobjective is to obtain confidence intervals for the ratio of twoPoisson counts. Let Yt ∼ Poisson(λt) andYt+1 ∼ Poisson(λt+1) and define S = Yt + Yt+1. The authorsnote that by conditioning on the sum S

Yt+1|S ∼ Binomial(S , θt) (2)

θt =λt+1

λt + λt+1(3)

1Coelho, FC and Carvalho, LM (Submitted)



Notification Data


Motivation

Vector dynamics

Building blocks


Modeling Dengue

Single-strain model




Rt ’s Uncertainty

Let cα(θt) = {θ(L)t , θ

(U)t } be such that

Pr(θ(L)t < θt < θ

(U)t ) = α. Analogously, define

cα(Rt) = {R(L)t ,R

(U)t } such that Pr(R

(L)t < Rt < R

(U)t ) = α.

Ederer and Mantel (1974) [4] show that one can construct a100α% confidence interval for Rt by noting that

R(L)t =

θ(L)t

(1− θ(L)t )

and R(U)t =

θ(U)t

(1− θ(U)t )

(4)



Notification Data


Motivation

Vector dynamics

Building blocks


Modeling Dengue

Single-strain model




Rt ’s Uncertainty

Taking a Bayesian conjugate distribution approach, If wechoose a Beta conjugate prior with parameters a0 and b0 forthe Binomial likelihood in (2), the posterior distribution for θtis

p(θt |Yt+1,S) ∼ Beta(Yt+1 + a0,Yt + b0) (5)

Combining equations (4) and (5) tells us that the inducedposterior distribution of Rt is a Beta prime (or inverted Beta)with parameters a1 = Yt+1 + a0 and b1 = Yt + b0 [?]. Thedensity of the induced distribution is then

fP(Rt |a1, b1) =Γ(a1 + b1)

Γ(a1)Γ(b1)Ra1−1t (1 + Rt)

−(a1+b1) (6)

Thus, the expectation of Rt is a1/(b1 − 1) and its variance isa1(a1 + b1 − 1)/

((b1 − 2)(b1 − 1)2

).



Notification Data


Motivation

Vector dynamics

Building blocks


Modeling Dengue

Single-strain model




Rt ’s Uncertainty



Notification Data


Motivation

Vector dynamics

Building blocks


Modeling Dengue

Single-strain model




Rt vs. Temperature



Notification Data


Motivation

Vector dynamics

Building blocks


Modeling Dengue

Single-strain model




Single Strain SIR

Why not multi-strain? No Multi-strain data!!

dS

dt= −β(t)SI (7)

dI

dt= β(t)SI − τ I

dR

dt= τ I

where S(t) + I (t) + R(t) = 1 ∀ t.



Notification Data


Motivation

Vector dynamics

Building blocks


Modeling Dengue

Single-strain model





From Rt , we can define a force of infection which varies withtime:

β(t) =Rt · τS

(8)

But how do we get the value of S? we need to estimate S0.



Notification Data


Motivation

Vector dynamics

Building blocks


Modeling Dengue

Single-strain model




Estimating S0

Bayesian framework:

I Define priors for S0 in the range (0,1)

I Samples from prior, calculate β(t) and run the model

I calculate Likelihood of data given current parameterization

I Determine posterior probability of parameterization

p(S0j |Yj) ∝ L(Yj|S0j ,Rt ,m, τ)π(S0j) (9)



Notification Data


Motivation

Vector dynamics

Building blocks


Modeling Dengue

Single-strain model




Estimating S0

Bayesian framework:








Notification Data


Motivation

Vector dynamics

Building blocks


Modeling Dengue

Single-strain model




Estimating S0

Bayesian framework:








Notification Data


Motivation

Vector dynamics

Building blocks


Modeling Dengue

Single-strain model




Estimating S0

Bayesian framework:








Notification Data


Motivation

Vector dynamics

Building blocks


Modeling Dengue

Single-strain model




Models vs Data

fiting the model to data (Rio de janeiro) to estimate S02.

Posterior distribution for Susceptible (S) and infectious (I)

individuous. Blue dots are data.2Coelho FC et al., 2011



Notification Data


Motivation

Vector dynamics

Building blocks


Modeling Dengue

Single-strain model




Attack Ratio

Once we have S0, we can caculate the attack ratio:

Aj =

∑Yj

S0j(10)



Notification Data


Motivation

Vector dynamics

Building blocks


Modeling Dengue

Single-strain model




Attack ratio

Table : Median attack ratio and 95% credibility intervalscalculated according to (10). Values are presented as percentageof total population. †: Year corresponds to the start of theepidemic, however the peak of cases may occur in the followingyear. ‡: Susceptible fraction. These results show considerablevariation in AR between epidemics, consistent with the accquiringand loss of serotype-specific immunity.

Year† median Attack Ratio S‡0

1996 0.39 (0.17-0.54) 0.00171(0.0012-0.0038)1997 0.87 (0.74-0.87) 0.00273(0.0027-0.0032)1998 0.5 (0.49-0.5) 0.00142(0.0014-0.0014)1999 0.11 (0.037-0.2) 0.00345(0.0018-0.01)2000 0.25 (0.24-0.27) 0.0155(0.015-0.016)2001 0.48 (0.47-0.49) 0.0495(0.048-0.051)2005 0.15 (0.1-0.21) 0.0147(0.01-0.021)2006 0.11 (0.08-0.14) 0.0281(0.022-0.037)2007 0.15 (0.15-0.15) 0.135(0.13-0.14)2008 0.14 (0.031-0.31) 0.00672(0.003-0.024)2010 0.18 (0.17-0.19) 0.0454(0.043-0.048)2011 0.086 (0.082-0.094) 0.215(0.2-0.23)2012 0.14 (0.13-0.15) 0.0621(0.058-0.068)



Notification Data


Motivation

Vector dynamics

Building blocks


Modeling Dengue

Single-strain model




References

Nishiura H, Chowell G, Heesterbeek H, Wallinga J (2010)The ideal reporting interval for an epidemic to objectivelyinterpret the epidemiological time course.J R Soc Interface 7: 297–307.

Coelho FC, Codeco CT, Gomes MG (2011) A Bayesianframework for parameter estimation in dynamical models.PLoS ONE 6: e19616.

Coelho FC, Carvalho, LM Estimating the Attack Ratio ofDengue Epidemics under Time-varying Force of Infectionusing Aggregated Notification DataArxiv. http://arxiv.org/abs/1502.01236

Ederer, F. and Mantel, N. (1974)Confidence limits on the ratio of two poisson variables.American Journal of Epidemiology volume 100:, pages165–167

http://arxiv.org/abs/1502.01236



Notification Data


Motivation

Vector dynamics

Building blocks


Modeling Dengue

Single-strain model




Thank you!

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