Sequential Monte Carlo Filtering Estimation of Ebola Progression in West Africa

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by  Narges Montazeri Shahtori

Sequential Monte Carlo Filtering Estimation of Ebola Progression in
West Africa
Narges Montazeri Shahtori1, , Caterina Scoglio 1, Arash Pourhabib 2, and Faryad Darabi Sahneh1
∗
approaches is that they provide an offline inference of an
Abstract—We use a multivariate formulation of sequential outbreakthatisinherentlydynamicandparametersofmodel
Monte Carlo filter that utilizes mechanistic models for Ebola
changeduringdiseaseevolution,soweneedtokeeptracking
6 virus propagation and available incidence data to simultane-
parameters when new data become available. Furthermore,
1 ously estimate the disease progression states and the model
0 parameters. This method has the advantage of performing since lots of factors such as intervention strategies could
2 the inference online as the new data becomes available and affect on parameters, we expect that the basic reproductive
estimates the evolution of basic reproductive ratio R0(t) of the ratio changes during the disease evolution. Therefore we
n Ebolaoutbreakthroughtime. Ouranalysisidentifiesapeak in
need techniques that are able to trace new data as they
a thebasicreproductiveratioclosetothetimewhenEbolacases
J become available. Towevers et al. [6] estimated the basic
were reported in Europe and the USA.

8 reproduction ratio, R0(t), by fitting exponential regression
2 I. INTRODUCTION models to small successive time intervals of the Ebola
Since December 2013, West Africa has experienced the outbreak. Therefore, they obtained an estimate of temporal
]
P largestEbola outbreakwith morethan20,000infectedcases variationsof the growthrate. Their applicationof regression
A reported [1]. Secondary infections have also been reported models ignores the systemic epidemiological information
in Spain and the United state [2]. Ebola hemorrhagic fever of Ebola progression—as reflected in the SEIR model—
.
at is considered a highly infectious and lethal disease, raising and thus are more suitable for exploratory analysis of the
t serious concerns about the public health globally. Efforts incidencedata.Amorerobustanalysisshouldtakeadvantage
s
[ to stop the spread of Ebola virus included intervention of our epidemiological knowledge of the dynamical system

strategies such as surveillance, quarantining suspected cases under study. Furthermore, the scarcity of incidence data
1
v and education of hospital workers in contact with Ebola during short time intervals impacts the stability of their
6 patients [3]. Side by side of these efforts, mathematical regression model fitting.
0 and computational epidemic models were developed and In this paper, we estimate the states of the Ebola prop-
6 implementedwiththeaimofpredictingnewlyinfectedcases agation and make inference about the basic reproductive
7
as well as evaluating mitigation strategies. ratio through time. To this end, we implement a sequential
0
The basic reproductive ratio, R , is one of the most Monte Carlo (SMC) filter, an online inference method that
. 0
1 relevantdescriptorsthathelpshealth–careauthoritiestohave allows simultaneous state and parameter estimation with
0
a quantitative understanding of the severity of the disease improvedaccuracyasnewstreamingdatabecomesavailable.
6
1 outbreak and its time projection. The most common defini- Sequential Monte Carlo filter is particularly powerful for
: tionof basic reproductivenumberis the expectednumberof inference about epidemic models which are inherently non-
v
secondaryinfectionsproducedbyatypicalsingleinfectionin linear and involve numerous uncertainties. Specifically, our
i
X a entirely susceptible population[4]. Early estimation of the SMC setting allows simultaneous estimation of the number
r basic reproductive number and other relevant descriptors is of individuals at different infection stages as well as the
a
crucial.Unfortunately,duringearlytimesavailableincidence parameters of our mechanistic epidemic model, providing
data is severely scarce, making reliable inference extremely posterior distributions of interest. In SMC, the distribution
challenging. of interest is estimated by a large number of N 1 random
≫
Researchers have attempted to analyze the recent Ebola samples termed particles conditionedon the observations.A
outbreak data and estimate the basic reproductive ratio. sampling mechanism propagates these particles [7]. After-
Althaus [5] used an offline optimization algorithm to find ward, we use the estimated values of the Ebola epidemic
parameters of the susceptible-exposed-infected-susceptible model parameters to determine the value of R0(t).
(SEIR) epidemicmodelthat fits best to collected Ebola data Compared with existing studies on the recent Ebola epi-
during a fixed time period. The major shortcoming of such demics in West Africa, our approach has the advantage of
performing the inference online as the new data becomes
This material is based upon work supported by the National Science available and estimates the evolution of basic reproductive
Foundation underGrantNo.SCH:1513639.
1Shahtori,Sahneh,andScolgioarewiththeDepartmentofElectricaland rate R0(t) of the Ebola outbreak through time. Interestingly,
ComputerEngineering,KansasStateUniversity,Manhattan,KS,66506.E- our analysis identifies a peak in the basic reproductive ratio
mail:{nargesmsh92,caterina,faryad}@ksu.edu. close to the time when cases were reported in Europe and
2Pourbabib is with the School of Industrial Engineering and Man-
the USA.
agement, Oklahoma State University, Stillwater, OK 74078. E-mail:
[e-mail]. The remainder of this paper is organized as follows.
SectionIIpresentsbasictoolsusedinsequentialMonteCarlo definitionofR inmathematicalepidemiologyistheaverage
0
filter and discusses the problem background in epidemic numberofexpected new infectionsoverall possible infected
modeling. Section III outlines our modified SEIR model for types[12].A widely usedtechniqueforfindingR , which is
0
Ebola and explains the particle setup and data, and Section basedonthisdefinition,isthenextgenerationmatrixmethod
IV presents main results of this study. Section V concludes [12].ApplyingthistechniquetotheSEIRmodelabovefinds
the paper by suggestions for future research. R = b c [10].
0 g
II. BACKGROUND B. Sequential Monte Carlo Filter
A. Epidemic Modeling Sequential Monte Carlo (SMC) — or particle filter —
refers to a class of statistical techniques that estimate un-
Mathematical models of infectious disease offer a mech-
known parameters, namely states in this context, as new
anistic frameworkto describe and study the spread of infec-
streaming noisy observations becomes available [13]. In
tions within human and animal populations, providing deep
SMC,weiterativelysamplefromtheposteriordistributionof
insightintotheirdynamicsandsuggestingpracticalstrategies
parametersuntiltheparametersconvergetostationaryvalues
to reduce the severity of epidemics [8]. Here, we intro-
[14]. This iterative sampling is updated using a stream of
duce a brief backgrounddiscussing the susceptible-exposed-
data,andassuch,itenablesustomodifyourbestguessesfor
infected-recovered (SEIR) model which is compatible with
the states accordingto actualobservations.In the following,
our understanding of Ebola virus epidemiology.
weexplainthedynamicstate–spacemodelandestimationof
In the SEIR model, individuals are assumed to be in
posterior PDF briefly for the particle filters algorithm.
one of these compartments: susceptible, exposed, infected
a) Dynamicstate–spacemodel: Thestate–spacemodel
and removed/recovered. In this model, when susceptible
assumes the Markov property, i.e.,
individuals have contact with an infected person, they enter
intotheexposedcompartment(E)withrateb I.Homogeneity
Pr(x x )=Pr(x x ). (2)
k 0:k 1 k k 1
of the population and how people have contact with each | − | −
others in host population is represented by a percentage and describesthe distribution of the system state in the next
factor c. After the incubation period of disease, mean of step, as well as the observation, given the current state of
1/l ,theyenterintotheinfectedcompartment(I).Infectious the system. More rigorously, a state–space model is defined
individualsmovetotherecovered/removedcompartment(R) as [13], [15]:
at rate g [9]. The compartmental SEIR model is [9]
x f(x x ,q ),
k k k 1
∼ | − (3)
y f(y x ,q ),
dS = b cSI, k∼ k| k
dt − where y represents kth observation, x represents states
k k
dE =b cSI l E, corresponding to the kth observation, and q represents pa-
dt − rametersofthemodel.Therefore,y dependsonlyonx and
(1) k k
dI =l E g I, q and xk depends on xk 1 and q .
dt − b) Estimationofpo−steriorPDF: Giventheobservation
dR
=g I. data y1:k up to time k, the ultimate goal is to define the
dt posteriordistribution,p(x ,q y ),whichdescribesthehidden
k k
InthiscompartmentalSEIR,thesizeofhostpopulationis state xk and parameters q o|f the dynamical system. The
assumed to remain constant throughout the evolution time, estimation of posterior probability density function (PDF)
i.e., P=S+E+I+R, and demographiceffects are ignored. based on Bayes’ theorem is [13]:
An important mathematical tool in the study of epi-
x ,q y (cid:181) f(y x ,q )p (x ,q )
demics is the basic reproductive ratio. Usually the basic k k| 1:k k| k 0 k (4)
(cid:181) f(y x ,q )p (x ,q y ).
reproductive number is defined as the expected number of k k 0 k 1:k 1
| | −
secondary individuals produced by a typical single infected
The sequentialMonteCarlo filter approximatesthe poste-
individual during its infectious period [10]. Thus, R is a
0 rior PDF as [13]:
dimensionless value that represents the average number of
J
additional susceptible people to whom an infected person Pr(x ,q y ) (cid:229) w(i)1 . (5)
passes the disease before he/she recovers[11]. For instance, k k| 1:k ≈i=1 k {(xk,qk)=(x(ki),qk(i))}
if an infectious person passes the disease on three others on
(i)
average, during his/her infectious lifetime, then R =3>1, where 1 is the indicator function, x is a particle and
0 . k
indicating that the number of new infectious individuals w(i) is its{}weight. The approximationis more accurate if the
k
would increase with each generation, so we can expect to number of samples, i.e., particles, is large. Particle’ weights
experience an epidemic. Conversely, if R0 <1 the disease are normalized thus [16]
will die out [11]. Thus, the basic reproductive ratio is a
J
threshold condition for epidemics as R0 =1 separates the Pr(x ,q y )dx (cid:229) w(i). (6)
increments or decrements of new infected [11]. A common Z k k| 1:k k≈ k
i=1
Among particle filter techniques are the bootstrap filter, eliminated.Aftereachround,thesurvivingparticlesproduce
auxiliary particle filter, and kernel density particle filter. In new particles. The main problem when bootstrap filter is
this paper, we use the latter, namely kernel density particle employed is that w(i) might become very small for some
k
filter,duetoitsflexibilityinmodelingofnon-linearprocesses particles and affect the accuracy of the particle filter. A
as explained in Section III-B. properwaytodecreasethenumberofparticlesistoselectthe
importanceprobabilityfunctionclosetotheoptimalone[15].
III. SMC FOREBOLA EPIDEMICS
Auxiliaryparticlefilteraimsispredictingwhichparticleswill
In this paper,we use the discrete-time SEIR modelwhich
haveasmallweightandminimizesthoseparticleswithsmall
is compatible with the epidemiology of the Ebola virus.
weights. With respect to these two filters, the kernelparticle
filter not only minimizes those kinds of particles with small
A. Modeling of Ebola
weights, but also estimates the unknownparameters[17]. In
The state variablesS, E, I, and R denotethe fractionof
t t t t this method, the main goal is to reduce the mean integrated
peoplewho are susceptible, exposed,infectedand recovered
square error of the kernel approximation and the posterior
or removed, at time step t, respectively which t step is
PDF in each step.
one day. For our analysis, we use the discrete–time form
Using the kernel density, we approximate and reproduce
of equation 1 with stochastic fluctuation and the following
parameters of the system in addition to the posterior proba-
assumptions to modify the original SEIR model 1.
bility density function, p(x ,q y ), of x . Here, for each
Assumption 1: Since the populationis much greaterthan tk k| 1:k tk
particle i, q (i) is the value of parametersat time t which we
the number of infected cases of Ebola, we assume S 1. k
≃ only have k observations up to time t.
Therefore,theequationforevolutionofE(t)in(1)simplifies
We choose reported Ebola data in Guinea, one of the
to:
E =E +b cI l E. (7) threemajorWestAfricacountriesthatexperiencedtheEbola
t+1 t t t t
− outbreak and we analyze the cumulative cases and death
Assumption 2: We assume that ct, representing how the counts.Toupdateparametersandstatesusingkernelparticle
populationismixed,isdynamicduetodifferentintervention filter, we need data. However, the number of cases and
strategies to prevent the spread of Ebola such as social deathinGuineaarereportedatirregularintervals.Toaddress
distancing and quarantining. Specifically, we assume ct is this issue, we implement particle filter only for those time
decreasing at rate a , i.e., intervals that we have information and use the last updated
parameters and states that are generated by particle filter to
c =c a c. (8)
t+1 t− t update states for those intervals that we do not have any
This is a simplified assumption to account for different information.
intervention strategies. We assumed that when the control 1) Evolution Setup: According to our Ebola model in 9,
measured are introduced and information regarding Ebola wecanwritethestate–spacemodelrequiredforSMCmethod
disease is disseminated, the transmission rate decays expo- as
nentially. xt+1 xt NW (g(xt,q ),Q(q )), (10)
| ∼
According to above assumptions and modifications to the
where x =[c,E,I,R,D]T is the state and
SEIR model 1, we propose the following set of stochastic t t t t t t
difference equations as our base epidemic model for the ct a ct
Ebola spread. Et+b −ctIt l Et
g(x,q )= I +l E −g I ,
ct+1=ct−a ct+x a , t  t Rt+tg−It t 
Et+1=Et+b ctIt−l Et+x l −x b ,  j It+jg It 
It+1=It+l Et g It x l +x g, (9)  a 0 0 0 0 (11)
− −
DRtt++11==Rj tR+t+g1It=−jxRgt+jg It−x g +x j . Q(q )= P12000 l −+0l b l −+lg g g−0g gj−0gj .
In the above equations, x c where c a ,b ,l ,g ,j is 0 0 −gj gj gj 2
∈{ }
a random component, with zero mean and given variance  − 
√c /P,wherePisthepopulationsize.Thevarianceofnoises whereQ(q )isthecovariancematrixofxt=[ct,Et,It,Rt,Dt]T
are due to stochasticity of the underlyingprocess[13].Each according to the state-space model (9) with stochastic fluc-
of these component are assumed to be uncorrelated. tuations. Here, NW (m ,S ) represents the truncated normal
distribution where W = (c,E,I,R,D):c,E,I,R,D
t t t t t t t t t t
B. Filtering Setup { ≥
0,E +I +R 1 .
t t t
≤ }
The technique of bootstrap filter and auxiliary particle 2) Observation Setup: Observations are positive values
filter are employed to generate the kernel density particle and in each reported day, WHO reports cumulative cases
filter method. In bootstrap filter, probability density p(x) is and death. Using SEIR model, we only estimate number
estimatedbyasetofparticlesandateachroundtheirweights of infected and recovered individuals at each time step.
are computed and those particles with small weights, are Therefore, to configure SMC observation setup based on
reported data, we need to estimate cumulative cases and In above, a=1−h2 and h=1−(3f2f−1)2. These two quan-
death. To this end, using states I and R in equation 9, we tities control the smoothness of kernel density estimation,
model the observationsY as follows [13]: whilef (0,1)isadiscountfactorwhichreducesthechance
∈
offailureinthefilter.Readersareencouragedtoreferto[13]
Y N (m Y,S Y), (12) for more details. Typically, f is assumed to be a number
∼
between .95 and .99.
where
Y = log(yI,k) ,m = bI(It+Rt)zI ,S s I 0 . C. Data Explanation
(cid:20)log(yD,k)(cid:21) Y (cid:20) bD(Dt)zD (cid:21) Y ∼(cid:20)0 s D(cid:21)
(13) EpidemicCurvesandItsApproximation(X)
4000
Xh: Approximation of X
Index D and I in observation matrix Y, represent total I I
3500 X : Cumulative number of infectious
numberofdeadandinfectedindividualsattimet,asreported I
byWHO.Inmatrixm Y,Irepresentstheestimationofnumber 3000 XhR: Approximation of XR
X : Cumulative number of death
of infected individuals (state I) and R represents estimation R
of state R; individuals who are dead or recovered. z s,s s n 2500
o
and bs for both infected and dead are typically unknown, ulati 2000
butwecanpredictthesevaluesbylinearregressionmethods p
o
[16]. In particular, bs are multiplicative constants and z s P 1500
are power-law exponents which can be calculated based on
1000
the significantof dispersedof data points.Since fluctuations
exist in the data, z s are not estimated precisely. The scaling
500
1
law for S , standard deviation, is s (cid:181) where P is the
Y .
√P 0
Feb14 Apr14 May14 Jul14 Sep14 Oct14 Dec14 Feb15 Mar15 May15
population size [16].
Time(Days)
3) Kernel density particle filter: Model (12) and (13)
define the likelihood of observations, y , given x and q Fig. 1:Cumulativecases anddeathdata andtheir estimation
— p(y x ,q ) — as a log normal distrkibution witkth mean by the particle filter in Guinea, reported by [WHO]
m , whki|chtkis two-by-one vector and the standard deviation,
Y
S , which is a two-by-two diagonal matrix with diagonal The Ebola virus, commonly known as Ebola, causes a
Y
element equal to s and s . Initially, particle i which serious illness which is fatal if untreated in most cases [2].
I D
i 1,...,J is sampled from an initial probability density It is transmitted via direct contact with blood, secretions,
fu∈nc{tion(PD}F) p(x ).Fortimestepk=1,weightsareequal organs, or other bodily fluids of infected individuals [18].
0
to 1/J for all particles, and q and x are generated by The incubation period, the time interval from infection with
0 0
randomsamplingfromspecificdistributions.Supposek+1th Ebola virus to the onset of symptoms, is between two
observation becomes available. The following steps present to twenty one days [2]. First symptoms of Ebola include
algorithm which applied kernel particle filter, to update and the sudden onset of fever, fatigue, muscle pain, headache,
estimate p(x ,q y ) for k>1 [13]. and sore throat [2]. Since approximately December 2013,
tk+1 | 1:k+1 West Africa has been affected by this virus. However, The
1) Calculate m(i) : m(i) =aq (i)+(1 a)q¯.
k+1 k+1 k − k World Health Organization (WHO) declared the epidemic
(i)
2) Compute the expectation of xtk+1: to be a public health emergency of international concern
m (i) =E(x q (i),x(i)), for all i 1,2,..,J . on August 8, 2014 [2]. We analyze the cumulative case
k+1 tk+1| k tk ∈{ }
3) Compute auxiliary weights and normalize them: and death counts in Guinea, one of the three West Africa
g(i) =w(i)p(y m (i) ,m(i) ) , g(i) = g(ki+)1 . Countries that experienced the Ebola outbreak. Cumulative
k+1 k k+1| k+1 k+1 k+1 (cid:229)J g(l) cases are classified into three categories: confirmed, prob-
l=1 k+1 able, suspected cases. Similarly, we have three cases for
4) Sampling: Select an index j randomlyand afterwards
death counts. Confirmed cases are those individuals who
(j) (1) (J)
sample x with its weights g ,...,g .
k { k+1 k+1} are diagnosed by polymerase chain reaction (PCR) method.
5) wRheperreodNuwce(mth,se)piasraamtreutnecrast:eqdk(+ni)o1r∼maNlwdi(smtr(ki+jb)u1,tViokqn+.1), Othnosetheindoitvhiedruahlasndth,astuhspavecetesdymanpdtomprsoboafblEebocalasesbudteintoties
6) Sample the x(i) : x(i) p(x(i) q (i) ,x(j)), for all i not confirmed if they are actually infected [1]. We should
tk+1 tk+1 ∼ tk+1| k+1 tk ∈
1,2,..,J . mention that the cases were reported at irregular intervals.
7) {Recompu}te the expectation of x(i) : Thisdata has beencollected fromreportsof WHO available
tk+1
m (i) =E(x(i) q (i),x(j)), for all i 1,2,..,J . at http://www.healthmap.org/ebola/.
k+1 tk+1| k tk ∈{ }
8) Compute weights and normalize them again:
IV. RESULTS
w(i) = p(yk+1|xt(ki)+1,qk(+i)1) , w(i) = w(ki+)1 . We estimate the states of Ebola propagation from March
k+1 p(yk+1|mk(+j)1,m(k+j)1) k+1 (cid:229)J w(kl+)1 23, 2014 to April 30, 2015 having data reported only in
l=1
T =170 days. Guinea population in 2015 was estimated to Evolution of Basic Reproductive Ratio (R0)
1.6
be around 12,500,000,however, the population in danger to
be infected by the Ebola virus was estimated to be roughly
about P=1,000,000. t() 1.4
R0
We estimate parameters by sampling from a log-normal
o
distribution using J=5,000, number of particles. The sen- ati 1.2
r
sitivity to changes in discount factor is chosen as f =0.95 on
and initial weights are all set equal to w0 =J−1. To run ductii 1
the particle filter, the initial state, x0, and initial parameters, pro
q 0, are generated randomly. Specification of initial priors ere 0.8
distributions are reported in Table I and II . ctiv
e
ff
E 0.6
TABLE I: Priors specification for parameters
Parameters Priors 0.4
mitigation rate(a ) U(.0059,.00593) Feb14 Apr14 May14 Jul14 Sep14 Oct14 Dec14 Feb15 Mar15 May15
transmissionrate(b ) U(.259,.379) Time(Days)
latency rate(l ) Beta(78,577) (a) Changes in basic reproductive ratio over disease evolution
recovery/remove rate(g) Beta(21,246)
fatality rate(j ) Beta(37,15) EvolutionofEbolaDiseaseParameters
0.2
βc 0.1
0
Basedoncollecteddataandexpertopinion,somemeasure- Feb14 Apr14 May14 Jul14 Sep14 Oct14 Dec14 Feb15 Mar15 May15
ments for g , l , and j are available. Therefore, we specify 0.15
λ
beta distributions for each of parameters, using Beta buster
0.1
andBetaSlicer[19],[20].Basedoncollectedinformation,the Feb14 Apr14 May14 Jul14 Sep14 Oct14 Dec14 Feb15 Mar15 May15
averageincubationperiod,l 1islessthan21dayswith95%
− 0.15
confidenceintervalandmeanaround8days[2].Fatalityrate, γ 0.1
0.05
j , is less than 80% with 95% confidence intervaland mode Feb14 Apr14 May14 Jul14 Sep14 Oct14 Dec14 Feb15 Mar15 May15
71% [2], [21], [22]. The average duration of illness onset to 1
death or recovery, g 1, is around 12 days [21], [22]. Since ϕ 0.8
− 0.6
we do not have enough information about transmission and Feb14 Apr14 May14 Jul14 Sep14 Oct14 Dec14 Feb15 Mar15 May15
mitigation rates, b and c, we assume uniform distributions. Time(Days)
Since we do not have enough information about initial (b) Parameters changes over Ebola disease evolution
states,weuseuniformdistributions.Forobservationconstant
in SMC filter, we assume that b =.88 and b =.54 and
I D
Fig. 2: Variation in R and parametersduringdisease evolu-
standarddeviation,S isatwobytwodiagonalmatrixwhose 0
Y tion
diagonal elements are s =.00125 and s =.00085. Power
I D
–lawconstantz ,forinfectedindividuals,is.88andfordead
individualsis .68. Fig. 1 shows cumulative cases and deaths
[23], R (t) is estimated as a single value with confidence
data and their estimation by the particle filter in Guinea. In 0
interval, while in [6] for the reproduction ratio, R (t), is
our model, the basic reproductive ratio is equal to R (t)= 0
0
cbg 1. The result indicates that transmission rate, latency computed by fitting exponential growth curves to small
−
successive time intervals of the Ebola outbreak. Instead,
rate and recovery rate are not constant during the disease
our method finds the basic reproductive ratio, R (t), as a
evolution. Therefore, as demonstrated in Fig. 2a, R (t) is 0
0
continuous function of time during the Ebola evolution and
not constant neither.
foreachtimeaprobabilityfunctionisrepresented.Usingthis
The maximum value of the basic reproductive number is
method, we can also see that not only R (t) changes over
1.51 on March 2014 and it decreases until September 2014 0
time, but also parameters such as b , l and g change.
which R 1. Afterward, a pick is occurred on October
0
∼
2014 and after that it decreases. We can see in Fig. 2b that
V. DISCUSSION AND CONCLUSION
transmission rates change during the disease evolution. In
Our analysis identifies a correlation between R (t) tem-
0
poral variations and important events in the 2014 Ebola
TABLE II: Priors specification for states
outbreak. For instance, a reduction of R (t) can be seen
0
States Priors around the time WHO first announced the Ebola outbreak
c U(.36,.40) in Guinea. This reduction can be explained by taking into
E U(.000128,.000141)
accountthe introductionof some initialmedicalsupportand
I U(.000050,.000061)
R U(.000042,.000058) public awareness. Conversely, a peak of R0(t) corresponds
D U(.000029,.000030) to the first Ebola cases in Europe and the USA.
Further improvement of our method would involve ac- [17] C. Chang and R. Ansari, “Kernel particle filter for visual tracking,”
counting for intermittent measurements and heterogeneous Signalprocessingletters, IEEE,vol.12,no.3,pp.242–245, 2005.
[18] P.E.LekoneandB.F.Finkensta¨dt,“Statisticalinferenceinastochastic
variance in the particle filtering scheme. Furthermore, our
epidemicseirmodelwithcontrolintervention:Ebolaasacasestudy,”
experiencewith numericalsimulationsshowedfairly consis- Biometrics, vol.62,no.4,pp.1170–1177, 2006.
tent outcomes. However, a more objective quantification of [19] G.JonesandW.O.Johnson,“Priorelicitation:Interactivespreadsheet
graphics with sliders can be fun, and informative,” The American
involved uncertainties can be a great addition to the current
Statistician, vol.68,no.1,pp.42–51,2014.
work. At the end, we would like to reiterate that we did [20] S. Chun-Lung, “Betabuster Program, Department of Medicine and
not account for any spatial dependencies in our analysis. A Epidemiology, University ofCalifornia, Davis.”
[21] G.Chowell andH.Nishiura, “Transmissiondynamics andcontrol of
spatial implementation of the particle filter can account for
ebolavirusdisease(evd):areview,”BMCmedicine,vol.12,no.1,p.
several countries in West Africa all at once. 196,2014.
[22] W. E. R. Team et al., “Ebola virus disease in west africathe first 9
ACKNOWLEDGMENT monthsoftheepidemicandforwardprojections,”NEnglJMed,vol.
371,no.16,pp.1481–95, 2014.
The authors would like to thank Joan Saldan˜a for fruitful [23] M. F. Gomes, A. P. y Piontti, L. Rossi, D. Chao, I. Longini, M. E.
discussions regarding basic reproductive ratio in realistic Halloran, and A. Vespignani, “Assessing the international spreading
risk associated with the 2014 West African Ebola outbreak,” PLoS
epidemics,NoraBelloforfruitfuldiscussionsandsuggestion
currents,vol.6,2014.
regardingthepriorsspecifications,andAlaFardfororganiz-
ing Ebola incidence data for our analysis.
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