Published online 15 February 2012 A nonlinear programming …faculty.bscb.cornell.edu › ~hooker...

J. R. Soc. Interface (2012) 9, 1983–1997

*Author for c

doi:10.1098/rsif.2011.0829Published online 15 February 2012

Received 28 NAccepted 26 J

A nonlinear programming approachfor estimation of transmission

parameters in childhood infectiousdisease using a continuous time model

Daniel P. Word1, Derek A. T. Cummings2, Donald S. Burke3,Sopon Iamsirithaworn4 and Carl D. Laird1,*

1Artie McFerrin Department of Chemical Engineering, Texas A&M University,College Station, TX 77843, USA

2Faculty of Johns Hopkins Bloomberg School of Public Health, 615 N. Wolfe Street,Baltimore, MD 21205, USA

3Graduate School of Public Health, University of Pittsburgh, 130 DeSoto Street,A624 Crabtree Hall, Pittsburgh, PA 15261, USA

4Bureau of Epidemiology, Ministry of Public Health, Thailand

Mathematical models can enhance our understanding of childhood infectious disease dynamics,but these models depend on appropriate parameter values that are often unknown and must beestimated from disease case data. In this paper, we develop a framework for efficient estimationof childhood infectious disease models with seasonal transmission parameters using continuousdifferential equations containing model and measurement noise. The problem is formulatedusing the simultaneous approach where all state variables are discretized, and the discretizeddifferential equations are included as constraints, giving a large-scale algebraic nonlinear pro-gramming problem that is solved using a nonlinear primal–dual interior-point solver. Thetechnique is demonstrated using measles case data from three different locations having differ-ent school holiday schedules, and our estimates of the seasonality of the transmission parametershow strong correlation to school term holidays. Our approach gives dramatic efficiency gains,showing a 40–400-fold reduction in solution time over other published methods. While ourapproach has an increased susceptibility to bias over techniques that integrate over theentire unknown state-space, a detailed simulation study shows no evidence of bias. Further-more, the computational efficiency of our approach allows for investigation of a large modelspace compared with more computationally intensive approaches.

Keywords: nonlinear optimization; measles; infectious diseases; mathematicalprogramming; Gauss–Lobatto collocation

1. INTRODUCTION

The development of reliable, mechanistic models for thespread of infectious diseases remains the subject ofextensive research. Such models are desirable for scien-tists to enhance the identification and understanding offactors that affect infectious disease dynamics, and forpublic health officials who would find a long-term,quantitative, dynamic model valuable for predictingdisease outbreak risk and performing response plan-ning. In addition, reliable models can be used toquantify the effectiveness of previous response tacticsand predict the benefit of future planned responses.To ensure the reliability of these models, they must beable to describe past system behaviour.

Owing to the availability of case data, measles hasbeen widely studied using mathematical modelling.

orrespondence ([email protected]).

ovember 2011anuary 2012 1983

The reporting interval for measles case data differs byorigin, and is commonly weekly, monthly or quarterly.The number of cases that are reported can be signifi-cantly lower than the actual number of cases, and thelevel of under-reporting can differ widely over longtime horizons, even in the same location [1]. Thechallenges inherent in the data have led to a numberof proposed modelling and parameter estimationapproaches. While the spread of measles is a continuousprocess, discrete time generation-based models havebeen formulated in addition to continuous timemodels, and, while spread of measles is inherently sto-chastic, this characteristic is included in some modelsand ignored in others. An examination of the datashows strong seasonality in the reported cases, leadingmany models to include a seasonally varying trans-mission parameter. Identifying correlations betweenpotential system inputs and transmission dynamics isimportant for understanding factors that affect disease

This journal is q 2012 The Royal Society

mailto:[email protected]

1984 Seasonal parameters in diverse settings D. P. Word et al.

dynamics. This is especially apparent in long-termmodels where we expect social structure and environ-mental factors to have changed significantly over thetime horizon studied. A better understanding of thesefactors is important for improving public health policyand aiding public health officials to establish appropriatecontrol strategies.

In this paper, we develop a framework for efficientestimation of seasonal transmission profiles using acontinuous differential equation model of the diseasedynamics. While existing discrete time approachesrequire that the reporting interval be an integer fractionof the serial interval of the disease, continuous timemodels allow for the use of case counts in their nativeform. Our estimation formulation is a nonlinearoptimization problem subject to differential equationconstraints arising from the infectious disease model.There are several strategies for solving optimizationproblems subject to differential constraints. Here, weuse a simultaneous approach where the state variablesin the infectious disease model are discretized, and thediscretized differential equations are included as con-straints in the optimization problem. We discretizethe differential equations using high-order Gauss–Lobatto collocation on finite elements, resulting in alarge-scale algebraic nonlinear programming problem.This estimation formulation is then solved using a non-linear primal–dual interior-point method. The use ofgeneral nonlinear programming tools allows for anapproach that is flexible to proposed model changes.We show that this technique is very efficient for the esti-mation of seasonal transmission parameters incontinuous time infectious disease models using casecount data over long time horizons. The technique isdemonstrated using measles case data from three differ-ent locations—London, New York City and Bangkok.These locations form an excellent test bed for the esti-mation given the availability of data, and our resultsstrengthen the evidence that the dynamics of measlesare strongly dependent on school term holidays, butdiffer from recent findings that show the dynamicsbeing captured with large estimates for transmissionparameters [2,3].

A brief review of infectious disease modelling andestimation is given in §2. Section 3 introduces theformulation of our example susceptibleinfectedrecov-ered (SIR) model and the estimation approach.Section 4 gives estimation results for both simulateddata and data from three cities, and in §5 we discussthe significance of these results and offer conclusions.

2. BACKGROUND

Infectious disease spread is typically described by one oftwo fundamental classes of mechanistic models. Agent-based or individual modelling approaches, which havebeen used to propose control strategies [4], can sufferfrom a model parameter space that is too large for theavailable data to successfully specify parameters. Alter-natively, compartment-based modelling approachesdescribe the behaviour with a small number of differen-tial equations and parameters. These models categorize

J. R. Soc. Interface (2012)

the population with respect to various stages of diseaseinfection. For example, individuals can be consideredsusceptible to the disease (S), infected with the disease(I) or recovered from the disease and therefore immune(R). Many additional compartments can be added torepresent other stages, such as an (E) compartmentto represent those individuals who have contractedthe disease but are not yet infectious, or an (M) com-partment to represent individuals with maternalimmunity. The classification of the model is determinedby the progression of the population from one compart-ment to the next [5]. Compartment-based modelling isthe approach used in this work.

It is clear from measles case data that measles inci-dence follows strong seasonal patterns. As early as1929, Soper [6] estimated transmission rates frommonthly measles case data and proposed that the sea-sonality was correlated with school terms. In 1982,Fine & Clarkson [7] used measles data from the years1950 to 1965 in England and Wales to estimate atime-varying transmission profile. Over this time hor-izon, the data show a biennial pattern of alternatinglarge and small measles outbreaks, yet the estimatedtransmission profile remained very similar from yearto year. In addition, the estimated transmission profileappeared to be correlated with school holidays.

Reliable estimation from the available data can bechallenging. Typically, the only disease data availableare the case counts (or incidence) of the disease. How-ever, the number of susceptible individuals in thepopulation also has a significant effect on the dynamicsof disease spread, and little quantitative data are typi-cally available describing this population. In addition,owing to passive collection, the number of reportedcases is often significantly lower than the actualnumber of cases. This can lead to significant under-reporting in the data with some datasets reportingfewer than 5 per cent of the actual number of cases.This has led researchers to consider two-stageapproaches where the susceptible dynamics and report-ing factor are estimated first [1] and are then treated asknown inputs for estimating transmission parameters[8]. In this work, we estimate the degree of under-reporting in case counts using a procedure similar tosusceptible reconstruction, but the susceptibledynamics themselves are estimated simultaneouslywith the transmission parameters.

Finkenstädt & Grenfell [8] developed a time-seriesSIR (TSIR) model as a discrete time model with 26discretizations per year, which is consistent with theserial interval of measles. This model uses a time-vary-ing transmission parameter b with yearly periodicityto give 26 unique values for b to describe the time pro-file. Although the transmission parameter is assumedto have yearly periodicity, no strong assumption ismade regarding the functional form of the parameter.Instead, the parameter profile is treated as anunknown input to be estimated using case countdata by finding a one-step-ahead solution to the esti-mation problem. The estimates from this modelusing data from England and Wales strengthened theclaim that transmission of measles is correlated toschool holidays.

Seasonal parameters in diverse settings D. P. Word et al. 1985

While discrete time models like these have provenuseful, they suffer from significant drawbacks. Thesemodels are discretized over the serial interval of the dis-ease, and the estimation approach requires that thereporting interval of the disease data be an integer frac-tion of the serial interval. The England and Walesmeasles data are reported weekly and the serial intervalis two weeks, making the approach suitable for thisdataset, but most existing datasets have reporting inter-vals longer than the serial interval of the disease.Furthermore, the TSIR model includes a parameter aas an exponent on the incidence (Iiþ1 ¼ bi Iia Si). Thisparameter is difficult to interpret physically, and ithas been conjectured that it simply serves to correctfor the fact that the model is discrete and notcontinuous [9,10].

Continuous models not only are a more naturalframework for modelling the continuous nature of dis-ease dynamics, but also they overcome some of thechallenges inherent in many discrete time models. Con-tinuous time models allow for varying discretizationstrategies and also for the discretization to differ fromthe reporting interval of data and the serial intervalof the disease. This allows the data to be used in itsnative form rather than requiring that the data fit aspecific reporting interval, as is required in the TSIRapproach of Finkenstädt & Grenfell [8]. In addition,for this work, no exponent a is included on the inci-dence term, reducing the number of parameters to beestimated by 1. Despite this reduction in parameters,these models have been shown to capture the observeddisease dynamics comparably [11,12].

Several continuous time disease models have beendeveloped. Greenhalgh & Moneim [13] examined thestability properties of different transmission formsand used simulations to show that different periodic sol-utions are possible with different types of seasonallyvarying transmission rates. Schenzle [14] developed anage- and time-dependent differential equation modeland used simulations to demonstrate that an age-dependent transmission rate more accurately reproducedactual measles data. Cauchemez & Ferguson [11]developed a stochastic continuous time SIR model andused it to estimate transmission profiles from Londonmeasles data. Cintrón-Arias et al. [15] studied parametersubset selection using a continuous time SEIRS modelwith a sinusoidally periodic transmission parameter.This work explored parameter identifiability usingsynthetic data.

Another challenge of disease modelling is that thetransmission of infectious disease is an inherently sto-chastic process. The use of deterministic models hasproven reasonable for use in large cities above a criticalcommunity size, but in smaller cities fade-out is seenand deterministic models perform poorly. For measles,the critical community size has been estimated to bearound 300 000 people [16,17]. When fade-out isobserved, reappearance of the disease is caused by aninflux of an infected individual into the susceptiblepopulation. Finkenstädt et al. [18] modified theirTSIR model to allow for stochasticity and used MonteCarlo simulations to study the effect of the latentstochastic variability of influx.


Other models have been developed to allow forstochasticity using various Monte Carlo techniques.While useful, Monte Carlo techniques suffer fromhigh cost of computation, making them unreasonablefor large-scale models. For example, Cauchemez &Ferguson [11] presented a stochastic continuous timemodel using a statistical approach to analyse time-series epidemic data. This approach used a dataaugmentation method to overcome difficulties ininference and presented a diffusion process thatmimics the epidemic process. These systems containstochasticity in the model along with unmeasuredstates. Therefore, Cauchemez & Ferguson constructedthe likelihood of the parameters conditional on thedata including an integration over the unknown statespace. This approach provides a guarantee againstbias in the estimated parameters; however, the Metro-polis–Hastings Markov chain Monte Carlo (MCMC)sampling is very computationally expensive, requiring20 h per run [11]. Hooker et al. [3] presented anSEIR model to perform parameter estimation withmeasles data from Ontario using generalized profiling[19]. This approach estimates state variable trajec-tories and model parameters using a sequentialnumerical optimization approach that is muchmore efficient than MCMC techniques, but thisapproach can still require about 2 h per estimation[3], although part of this time was owing to the itera-tive process of setting the smoothing parameter in theproblem formulation.

He et al. [2] demonstrated their plug-and-playmethod as a framework for modelling and inference byperforming estimates using weekly measles case countdata from the 10 largest cities and 10 small cities inEngland and Wales. This work used an SEIR modelwith a seasonally varying transmission parameter.The seasonality of the transmission parameters wasfixed to correspond with school term holidays, but theamplitude of the seasonality was estimated. Addition-ally, the durations of the latent and infectious periodswere estimated. This approach required approximately5 h per estimation, but reductions in the time require-ments could be made by tailoring the procedurespecifically for a given problem.

3. PROBLEM FORMULATION ANDESTIMATION APPROACH

In this paper, we use an SIR compartment modellingframework to develop a continuous time model thatincludes both model and measurement noise. This isthen used to estimate model parameters and seasonaltransmission profiles from measles case count data. Itis assumed the individuals enter the susceptiblecompartment S when born, move to the infected com-partment I upon acquiring the infection and progressto the recovered compartment R upon recovery fromthe infection. After recovery from measles, individualsare assumed to attain lifelong immunity from the dis-ease so that there is no movement of individuals fromthe recovered compartment to the susceptible com-partment [20]. The infection transmission assumes


frequency dependence [5], and the transmission par-ameter is assumed to be seasonal with a periodicity of1 year. The assumption of frequency-dependent trans-mission is not reasonable for all diseases. While it maymake intuitive sense that in larger populations onewould have more contacts in a day, for childhood dis-eases like measles this is often not the case. Giventhat most children are in school systems with similarschool structures, we assume they have a consistentnumber of contacts per day regardless of city size.This assumption is consistent with recent findings formeasles in the UK [21].

3.1. Problem formulation

The differential equations describing the continuoustime seasonal SIR model are

dSdt¼ �bðyðtÞÞSðtÞI ðtÞ

N ðtÞ � 1M ðtÞ þ BðtÞ ð3:1Þ

and

dIdt¼ bðyðtÞÞSðtÞI ðtÞ

NðtÞ � 1M ðtÞ � gI ðtÞ; ð3:2Þ

where S is the number of susceptibles, I is the numberof infectives, N is the total population and b(t) is thetime-varying transmission parameter. The functiony(t) maps the overall horizon time with the elapsedtime within the current year, making b(y(t)) a seaso-nal transmission parameter with periodicity of 1 year.Births into the population, B, and the population, N,are known time-varying system inputs, and the recov-ery rate (g ¼ 1/14 day) is a known scalar input. Thevariable 1M represents multiplicative model noise,which is assumed to be log-normally distributed witha mean of 1. Additive noise was investigated, but, inthose cases, the estimated values for the model noiseshowed an obvious temporal correlation with thedata, indicating an unlikely model structure. However,the technique that we use can easily be modified tosupport different assumptions on the stochasticnoise. Note that while this distribution can be anacceptable approximation for large datasets, wherethe number of cases never nears zero, for small data-sets this distribution becomes invalid, as it does notallow for zero cases. In addition, as the reportednumber of cases must always be non-negative, theassumption of normally distributed measurementnoise is only a reasonable approximation when thereported number of cases is not near zero. Giventhe size of the cities examined in this work and thenumber of reported cases over the given time hori-zons in the datasets, we find these assumptionsacceptable here.

It is important to distinguish between incidenceand prevalence in this problem formulation. The casecount data are available for the incidence or thenumber of new cases reported over a given time inter-val. The model contains a state variable I(t) that isthe prevalence of the disease, or the number of casespresent at a given point in time. In a given reporting


interval, integrating over the number of new casesgives the incidence,

incidence ¼ðti

ti�1

bðyðlÞÞSðlÞI ðlÞNðlÞ � 1M ðlÞ dl: ð3:3Þ

To account for the difference in incidence and pre-valence, a new state variable is introduced into thesystem through the following differential equation:

dQdt¼ bðyðtÞÞSðtÞI ðtÞ

NðtÞ � 1M ðtÞ: ð3:4Þ

Here, Q(t) represents the cumulative incidence at timet and provides a state variable that can be used toevaluate the incidence over a particular interval.

Not every individual that becomes infected isreported, which leads to case counts being under-reported. While there are various methods to estimatethe reporting fraction h(t), in this work, we use astraightforward approach to estimate a linearly varyingreporting fraction that is similar to the susceptiblereconstruction approach described in Finkenstädt &Grenfell [8]. At the start of the estimation horizon,we assume that the number of cumulative cases, C0,and cumulative births, Y0, are unknown. Prior towidespread vaccination, almost every individual even-tually contracted the disease. Therefore, on average,the cumulative number of new cases should equal thecumulative number of new births. For a constantreporting fraction, the cumulative cases and births aregiven by

Yt ¼Xti¼1

Bi þ Y0 ð3:5Þ

and

Ct ¼Xti¼1

Rihþ C0; ð3:6Þ

where Bi is the reported number of births at time i, Yt isthe cumulative number of births at time t, Ri is thereported number of cases at time i and Ct is the cumu-lative number of cases at time t. We then minimize thesum-squared error between Yt and Ct to estimatethe reporting fraction h. For the case of our estimationswith data from London and Bangkok, we extend thisbasic formulation to estimate a reporting fraction thatvaries linearly in time.

In our problem formulation, the estimated reportingfraction is treated as a known input for the estimationof the disease model parameters. To obtain a fit,the estimation formulation requires that we minimizesome measure of the model and/or measurement noisesubject to the infectious disease model described bythe differential equations given in equations (3.1),(3.2) and (3.4).

There are two general approaches for the solution oflarge-scale dynamic parameter estimation problemssimilar to that considered here. The sequential approachconsiders only the degrees of freedom as optimizationvariables. This includes the initial conditions for I(t)and S(t), as well as a discretized time profile for the


seasonal transmission parameter, b(t). A completesimulation of the forward problem is performed at eachiteration of the optimization. To make use of moderngradient-based methods, derivative information mustalso be calculated along the entire time-series simu-lation. These derivatives can be expensive to calculate,especially for problems with many degrees of freedom.Furthermore, these derivatives can be noisy unless careis taken to ensure consistency of the integrator betweenruns [22]. Noise in the evaluation of sensitivities throughthe integrator can make these problems very challengingfor the optimization solver. The simultaneous approachcan be used to overcome these difficulties. In the simul-taneous approach, all variables, including the states andthe parameters, are discretized and treated as optimi-zation variables. The entire discretized model isincluded as algebraic constraints in the optimizationproblem. The optimization problem resulting from thesimultaneous approach can be larger than the sequentialapproach. However, this approach can be significantlyfaster than the sequential approach as the differentialequation model is not converged at every iteration ofthe optimizer, but rather it is converged simultaneouslyas constraints to the optimization. Furthermore, accu-rate derivative information is easily obtained usingmodern automatic differentiation tools coupled withexisting modelling frameworks. Recent advancementsin nonlinear programming tools [23] allow efficient sol-ution of sparse problems with hundreds of thousandsof variables and constraints using standard desktopcomputing power [24–27]. In addition to potentialefficiency gains, this simultaneous approach allowsintuitive specification of additional constraints on theparameters and the state variables, including restric-tions on the form of time-varying parameters. Theflexibility of general nonlinear formulations coupledwith the efficiency of large-scale algorithms make thesimultaneous discretization approach coupled withgeneral nonlinear programming tools an appropriateframework for efficient parameter estimation innonlinear infectious disease models.

In this work, we use the simultaneous approach.We use collocation on finite elements to discretize thestates into finite elements with fixed stepsize across theentire time horizon [28]. This converts the continuousdifferential equation model into an algebraic model thatcan be formulated as a nonlinear programming problem.A fifth-degree Gauss–Lobatto collocation technique isused to discretize the dynamics within these finiteelements [29], and the discretized equations are includedas equality constraints in the optimization problem. Theeffect of the instantaneous model noise, 1M(t), on thesystem is approximated by introducing an unknownnoise term between each of the finite elements.

We illustrate this approach by showing the dis-cretization of equation (3.4) within a single finiteelement i. Let ti,j be the time associated with finiteelement i and collocation point j. With the fifthdegree Gauss–Lobatto strategy, there are five colloca-tion points for each finite element i (ti,0 to ti,4). Thetimes ti,0 to ti,4 correspond to the locations of thecollocation points within the finite element, where ti,2is the central collocation point located at the centre


of the finite element. The time at ti,1 is equalto ti;2 �

ffiffiffiffiffiffiffiffi3=7

p12Dti, and the time at t3 is equal to

t2 þffiffiffiffiffiffiffiffi3=7

p12Dti, where Dti is the length of finite element

i. Letting Qi,j be the value of Qðti;jÞ and letting fi;j bethe value of dQ=dtjti;j (equation (3.4)), the collocationequations become

Qi;1 ¼1

686fð39

ffiffiffiffiffi21p

þ 231ÞQi;0 þ 224Qi;2

þ ð�39ffiffiffiffiffi21p

þ 231ÞQi;4 þ Dti½ð3ffiffiffiffiffi21p

þ 21Þfi;0

� 16ffiffiffiffiffi21p

fi;2 þ ð3ffiffiffiffiffi21p

� 21Þfi;4�g ð3:7Þ

Qi;3 ¼1

686fð�39

ffiffiffiffiffi21p

þ 231ÞQi;0 þ 224Qi;2

þ ð39ffiffiffiffiffi21p

þ 231ÞQi;4 þ Dti½ð�3ffiffiffiffiffi21p

þ 21Þfi;0

þ 16ffiffiffiffiffi21p

fi;2 þ ð�3ffiffiffiffiffi21p

� 21Þfi;4�g ð3:8Þ

0 ¼ 1360fð32

ffiffiffiffiffi21p

þ 180ÞQi;0 � 64ffiffiffiffiffi21p

Qi;2

þ ð32ffiffiffiffiffi21p

� 180ÞQi;4 þ Dti½ð9þffiffiffiffiffi21pÞfi;0

þ 98fi;1 þ 64fi;2 þ ð9�ffiffiffiffiffi21pÞfi;4�g ð3:9Þ

and 0 ¼ 1360fð�32

ffiffiffiffiffi21p

þ 180ÞQi;0 þ 64ffiffiffiffiffi21p

Qi;2

þ ð�32ffiffiffiffiffi21p

� 180ÞQi;4 þ Dti½ð9�ffiffiffiffiffi21pÞfi;0

þ 98fi;3 þ 64fi;2 þ ð9þffiffiffiffiffi21pÞfi;4�g: ð3:10Þ

In our formulation, we include model noise betweenthese finite elements so that, while inside a finiteelement, equations (3.7)–(3.10) are exact and betweenfinite elements there can be state discontinuities. Withmodel noise between finite elements, there is littlebenefit in the use of a higher order method. However,we initialize our problem using the deterministic casewhere no model noise is present (model noise termsare fixed to zero), and, for this case, this discretizationis a high-order method.

Our estimation formulation becomes,

min vMXi[Fðlnð1MiÞÞ

2 þ vQXk[Tð1Qk Þ

2

s:t:dSdt¼ �bðyðtÞÞSðtÞI ðtÞ

NðtÞ � 1M ðtÞ þ BðtÞ;

dIdt¼ bðyðtÞÞSðtÞI ðtÞ

N ðtÞ � 1M ðtÞ � gI ðtÞ;

dQdt¼ bðyðtÞÞSðtÞI ðtÞ

N ðtÞ � 1M ðtÞ;

Rwk ¼ hkðQi;k � Qi;k�1Þ þ 1Qk ;

�S ¼

Pi[F

Si

lenðFÞ;

�b ¼

Pi[t

bi

lenðtÞ;

0 � I ðtÞ; SðtÞ � NðtÞand 0 � bðyðtÞÞ;QðtÞ;

9>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>=>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>;

ð3:11Þ


where the differential equations are discretized usingequations (3.7)–(3.10) and are shown here in their differ-ential form for simplicity. The index k is a time pointwithin the set of reporting times, T, while F is the set ofall finite elements used in the discretization.The reportingfraction hk accounts for under-reporting over a given timeinterval spanning k 2 1 to k, Rk

* is the actual reported inci-dence over a given time interval, 1Qk is the measurementnoise, and vM and vQ are weights for the noise terms.Based on the assumption of normality in 1Qk andlnð1MiÞ, the ratio of these weights should be equal to theinverse ratio of the variance in the estimated noiseterms. Since the variances are not known a priori, asimple bisection approach is used to solve for the ratio ofweights until vM=vQ ¼ s2Q=s2M , where s2Q is the calcu-lated variance of the estimated noise terms 1Qk , and s

2M

is the calculated variance of the estimated noise termslnð1MiÞ (using standard mean squares). The estimationformulation is solved completely for each iteration of thebisection method. These weights determine the trade-offbetween model and measurement noise in the objectivefunction. This straightforward approach was used sincesensitivity studies (presented later in the paper) showthat the estimates are relatively insensitive to the selectionof these weights. However, other techniques have beenproposed for determining these weights [3,30] and couldalso be used. Two additional variables are added to theproblem formulation for use in calculating confidenceregions. Here, �S is the average population of susceptiblesover the time horizon and �b is the average value of bacross the yearly set of discretizations t.

It is important to point out that the objectivefunction used in this formulation is the extendedlog-likelihood and only an approximation of the truelog-likelihood for the parameters. The use of this likeli-hood for estimating parameters can cause considerablebias in both the parameters and their uncertainty[31]. In practice, bias may not always be observed, butcare should be taken when evaluating results from thisapproach. The simulation study discussed in §4.1shows no significant bias. However, if bias is a concern,then this efficient formulation and approach can stillbe used to initialize an unbiased approach.

This research focuses on the estimation of the seaso-nal transmission profile b(y(t)). While the case datashow strong seasonality, the functional form of thetransmission profile is unknown, so it is undesirable toforce b to take the form of a particular periodic function(e.g. a sine function). Therefore, we discretize the trans-mission profile along finite-element boundaries,assuming a constant value through the finite element.In previous work, b(y(t)) was further restricted usingtotal variation regularization [12]. However, in thiswork, it was found that regularization was unnecessarywhen the discretization of b matched the reportinginterval of the case data. Some form of regularizationor restriction of b would be necessary if b is discretizedmore finely than the reporting interval.

3.2. Estimation approach

The nonlinear programming formulation describedabove was written in AMPL [32] and solved using


IPOPT [33]. AMPL is an algebraic modelling languagefor optimization problems that provides first- andsecond-order derivatives using automatic differentiation.IPOPT is an open source primal–dual interior-pointalgorithm for solving nonlinear programming problemswith inequality constraints, and is available throughthe COIN-OR foundation. This algorithm considersproblems of the form,

minx[>>>>>=>>>>>>;

ð3:12Þ

where f(x) and c(x) are assumed to be twice differenti-able, dL and dU are lower and upper bounds on ageneral function d, and xL and xU are lower and uppervariable bounds on x. For ease of notation, the algor-ithm is described for the following problem formulation:

minx[>=>>;

ð3:13Þ

Note that general inequalities can be mapped to equalityconstraints and simple variable bounds through theaddition of slack variables.

A significant challenge in the solution of these pro-blems is identifying the active and inactive sets ofvariables (i.e. the set of variable bounds that are satis-fied with equality at the solution versus the variablesthat are inside their bounds at the solution). Withinterior point methods, the inequality constraints aremoved into the objective function using a log-barrierterm to form the barrier subproblem,

minx[>=>>;

ð3:14Þ

This barrier subproblem is solved (approximately) for asequence of barrier parameters. It can be shown, undermild conditions, that the sequence of solutions of thebarrier subproblem converges to the solution of theoriginal problem. From the first-order optimality con-ditions for (3.14), the following equations can bederived [33]:

rf ðxÞ � rcðxÞl� n ¼ 0;cðxÞ ¼ 0

and XVe� me ¼ 0;

9>=>; ð3:15Þ

where rf ðxÞ is the gradient of the objective function,rcðxÞ is the transpose of the constraint Jacobian, land n are the Lagrange multipliers for the equality con-straints and inequalities, respectively, X is the diagonalmatrix of xi’s, V is the diagonal matrix of ni’s, and e is avector of ones. A variant of Newton’s method is used tosolve equation (3.15) (with modifications to ensure thatthe directions are descent). Calculating the step

(c)

0197519761977197819791980

year19811982198319841985

200

400

600

800

1000

repo

rted

cas

es

8000(a)

7000

6000

5000

4000

3000

2000

1000

01948 1950 1952 1954 1956

year1958 1960 1962 1964

repo

rted

cas

es(b)

year19441946194819501952195419561958196019621964

12000

10000

8000

6000

4000

2000

0

repo

rted

cas

es

Figure 1. The reported number of cases for (a) London (1948–1964), (b) New York City (1944–1963) and (c) Bangkok (1975–1984).


½DxTDlTDnT� requires the solution of the followinglinear system at each iteration:

H rcðxÞ �IrcðxÞT 0 0

V 0 X

264

375

Dx

Dl

Dn

264

375

¼ �rf ðxÞ � rcðxÞl� n

cðxÞXVe� me

264

375; ð3:16Þ

where H is the Hessian of the Lagrange function andI is the identity matrix. This linear system is solvedby first symmetrizing (3.16) and solving the so-calledaugmented system. Exact Hessian and Jacobian infor-mation is provided through AMPL. A filter-basedline-search strategy is used to ensure global conver-gence. Further details concerning IPOPT can be foundin the literature [33]. This approach has been used effec-tively on numerous large-scale nonlinear optimizationproblems [33–36].

4. ESTIMATION RESULTS

To demonstrate the effectiveness of our approach, wefirst estimate parameters using simulated data froman SIR model. We then perform estimation usingthree real datasets from different settings. In our esti-mations, we use existing measles case count data forLondon [21], New York City [37,38] and Bangkok,which have been made available to us by the ThailandMinistry of Public Health [39]. These datasets alsoinclude yearly birth records and populations. TheLondon dataset was chosen as it has been widely


studied and provides a comparison of our model resultswith literature results. The New York City and Bang-kok datasets are from cities with very different socialsettings and entirely different school holiday schedules.This allows us to compare estimated transmission pro-files on locations with different school term schedules.The New York City data contain monthly reportedcase counts. The Bangkok data include monthly casecounts and annual age distributions. There is regularactive surveillance coupled to the passive surveillancein order to assess the performance of the passive surveil-lance system. The data are fully anonymized andlaboratory confirmation is reported when available.The populations are assumed to vary linearly through-out the year, and the birth rates are assumed to beuniform throughout the year (figure 1).

To perform these estimations, we first format thedata as required by the AMPL input data file format.We formulated the model shown in equation (3.11)with discretized differential equations using the alge-braic modelling language AMPL [32]. Any modellinglanguage coupled with a reliable large-scale nonlinearoptimization solver could be used. We solve the problemusing the open-source nonlinear solver IPOPT [33]. Theweights for the objective function are found using theiterative process discussed previously. These weightsare then fixed to find the confidence intervals andconfidence regions.

Effective initialization is important for successfulsolution of general non-convex nonlinear programmingproblems. Here, all problems were initialized simplyby setting all Si;j ¼ 1� 105, Ii;j ¼ 1� 102, andQi;j ¼ 0 8i [ F and j collocation points, and bi ¼18i [ t. Here, F is the set of finite elements and B isthe set of discretizations of b. While this is a very

15.0

14.5

13.5

12.5b /g

11.5

10.5

10.02 4 6

time (months)8 10 12

11.0

12.0

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14.0

Figure 2. The estimated transmission profile b (triangles) for a single dataset with 95% confidence intervals (dashed line) foundusing likelihoods. The true values of b used in the stochastic SIR simulation (solid line). The 2.5th, 50th and 97.5th quantiles ofthe estimates from the simulation (circles). (Online version in colour.)


crude initialization, the formulation is robust, and wefirst solve the deterministic problem, fixing the modelnoise terms to zero, before solving the formulationwith model noise terms included.

Our estimates showed a strong correlation between �Sand �b, and the quality of our fit to the data differsdramatically outside of a narrow range of values. Confi-dence regions, derived from the likelihood-ratio test[40,41], are constructed as described by Rooney & Biegler[42] to show the region in which these values couldbe expected to lie. The regions are constructed overpairs of parameters by fixing the two parameters andreoptimizing over the remaining variables. In addition,likelihood ratios are used to construct confidence intervalsfor b by fixing each bi independently and allowing optim-ization over all other bi’s. These confidence regions andintervals were calculated using the extended likelihood,which is only an approximation of the true likelihood.The simulation study performed in §4.1 indicates thatthis approximation still provides reasonable confidenceintervals. In the simulation study, the estimated confi-dence intervals are slightly more conservative than whatthe simulations would suggest as necessary.

4.1. Simulation

In order to test the estimation procedure using knownparameter values, we perform stochastic simulationswith an SIR model. The simulations were perfor-med using Matlab. Our simulations used a constantpopulation of 10 002 000, a recovery rate of 1/14 day,a birth rate of 2.95 per cent of the population peryear and a reporting fraction of 1. To generate 20years of case data, the deterministic model was inte-grated for 100 years to achieve a cyclic steady state.The final values from this simulation were used as theinitial values for the stochastic simulation. The simu-lations were performed with the same model as thatused for the estimation except that the time step used


within the Matlab integration routine was a half day.Model noise was drawn from a log-normal distributionwith mean 1 and s ¼ 0.05. Measurement noise wasapplied to the reported cases and was drawn from anormal distribution with mean 0 and an s.d. of 1000.While the assumption of normally distributed noise isnot valid for datasets with a low number of reportedcases, we assume it to be a reasonable distribution forour simulations since, in 10 000 simulations, thereported number of cases never fell below 2000.

The simulation was run 10 000 times with a reportingfraction of 1 to generate simulated case data, and esti-mations were run on each of these simulations using 12finite elements per year. Figure 2 demonstrates thatour estimation approach gives an extremely good esti-mate for b using data from the SIR simulations. Thesolid line shows the true parameter values used for all10 000 simulations. The circles show the mean of the esti-mated values for the parameters as well as the 2.5 and97.5 quantiles for the parameters estimated from allthe simulations (giving the 95% confidence intervalsfor these estimates). The triangles show the estimatedparameters from an estimation on a single randomlyselected simulated dataset, and the dashed lines showthe 95% confidence intervals generated for this esti-mation using the likelihood-ratio test. The trueparameter values are included well inside these confi-dence intervals. The confidence intervals calculatedfrom the likelihood-ratio test give more conservativeintervals than what the 10 000 simulations wouldsuggest are necessary and actually cover over97 per cent of the values estimated from all 10 000simulations. This is not unexpected given that thelikelihood-ratio test confidence intervals are determinedby fixing only one parameter at a time, allowing theother parameters to be optimized.

Furthermore, the mean values of the estimatedparameters over all 10 000 simulations agree with thetrue values used in the simulation. There is no

0.8

0 2 4 6 8 10 12 14 16 18 20 22 24 26time (biweeks)

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–0.2

–0.4

–0.6

–0.8

norm

aliz

ed b

0

Figure 3. The transmission profile estimated for London by Finkenstädt & Grenfell [8] (dashed line), Cauchemez & Ferguson [11](dotted line) and this work (solid line).

b /g

20

time (biweeks)10

10

55

20 2515

15

Figure 4. The estimated transmission profile b with confidence intervals. School term holidays are shaded.


significant bias observed in the estimate of the seasonaltransmission parameters for the simulated data used inthis study.

4.2. London

The time horizon studied for London was 1948–1964. Thereporting fraction estimated using our approach varies lin-early from 50.65 to 42.55 per cent over the time horizonstudied. This dataset originally had case counts reportedweekly; however, the dataset used here has combinedthe data so that case counts are given biweekly. Theseaggregated data were also used by Finkenstädt & Grenfell[8]. The estimation for London was performed using 26finite elements per year, and the transmission profilewas discretized so that each bi contained one finiteelement, giving 26 discretizations in b. The normalizedestimated seasonal transmission profile for London isshown in figure 3. This figure also compares this resultwith the findings from Finkenstädt & Grenfell [8] andCauchemez & Ferguson [11]. The time horizon used inthis work and by Cauchemez & Ferguson [11] wasthe years 1948–1964, but the time horizon used byFinkenstädt & Grenfell [8] was the years 1944–1964.Despite the differing time horizons, it is clear that the sea-sonal pattern estimated from our approach is similar tothese other results. The estimated profile appears to becorrelated with school term holidays with the trans-mission profile decreasing during the school breaks thatoccur during the Easter holiday around biweek 8 andthe summer holiday over biweeks 15–18. The Christmas


holiday occurs at biweek 25, but there is no immediateeffect captured in our estimates. The lack of any immedi-ate effect could be owing to delays in reporting over theChristmas holidays [7].

Figure 4 shows the non-normalized pattern for theestimated seasonal transmission profile. The estimatedseasonal transmission parameter b(t) is on a per daybasis, and the mean estimated value is �b ¼ 0:95. Itdoes appear that our estimates may be shifted by onebiweek relative to the school holidays, but this couldbe owing to a slight delay in reporting.

In figure 5a, we show nonlinear confidence regions forthe mean transmission parameter value �b=g against themean susceptible fraction over the mean population,�S=Pop, where the means were taken over the entiretime horizon. As expected, this shows a dependencebetween the estimated values for �b and �S . The shapeof these confidence regions provides some insight intowhy it may be difficult to accurately estimate the absol-ute magnitude of the transmission parameter. Anincrease in �b can be offset by a decrease in �S withlittle change in the objective function value. However,even though there is a difference in the absolute magni-tude of the transmission parameter, we see littledifference in the seasonal pattern estimated for thepoints within the indicated confidence region.

4.3. New York City

The estimation for New York City was performed usingmonthly reported data from 1944 to 1963. The reporting

(a) 14.5

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0.050 0.052 0.054 0.056 0.058 0.060 0.062

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Figure 5. A contour plot showing the contours of the left-hand side of the x2 test coming from the likelihood ratios (dashed lines)and the 95% confidence region (solid lines) for �b and �S divided by the population. Regions represent profile likelihood contourswhere likelihood ratios are calculated based on a re-optimization of all other parameters. The ‘plus symbol’ indicates the optimalestimated value. (a) London, (b) New York City and (c) Bangkok.

b /g

14

12

10

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44 6 8 10 122

time (months)

Figure 6. The estimated transmission profile b for New York City with shaded school term holidays.


fraction estimated using our approach was almost con-stant around 11 per cent throughout the time horizonstudied, and the reporting fraction was assumed con-stant in our estimates. The discretization strategy used12 finite elements per year, and b is again assumed tobe constant within each finite element. This gave 12 dis-cretizations for the seasonal transmission profile b. Theestimated seasonal transmission profile for New YorkCity is shown in figure 6. This profile also shows strongcorrelation with the summer school term holidays thatthe New York City Board of Education reported occur-ring from mid-June to mid-September, or over thefinite elements of approximately 6.5–9.5. There areschool holidays around the end of the year; however,these holidays are much shorter than the reportinginterval of the data.

Figure 5b shows the 95% confidence region for �b and�S . The optimal estimated �S was 11.1 per cent of themean population, and the optimal estimated �b was


0.65. Our approach successfully estimates a seasonaltransmission pattern that shows strong correlation toschool terms.

4.4. Bangkok

In addition to data obtained from locations with asingle large summer break, we also performed estimatesusing measles data for Bangkok, Thailand. Thailandhas two school term holidays—one in the spring andone in the autumn. The estimation for Bangkok wasperformed using monthly reported case count datafrom the years 1975 to 1984. This dataset contains sig-nificant under-reporting with the estimated reportingfraction varying linearly from 1.1 per cent at the startof the time series to 4.5 per cent at the end of thetime series. In addition, case counts are missing forthe year 1979. The discretization strategy used 12finite elements per year with b discretized by finite

1.0

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objective function (×10

7)0.14 0.16 0.180.5

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0

b

Figure 8. The �b estimated for New York City is shown as a function of the reporting fraction (solid line). The objective functionvalue is also shown as a function of the reporting fraction (dotted line). (Online version in colour.)

b /g

10

20

25

15

4 6 8 10 122time (months)

Figure 7. The estimated transmission profile b for Bangkok with shaded school term holidays.


elements, giving 12 discretizations for b. As no data areavailable for the year 1979, these points were excludedin the objective function so that they would not affectthe estimation while still allowing the model to simulatethe states through this year. The estimated seasonaltransmission profile b for Bangkok is shown infigure 7. This profile shows correlation with the twoschool term holidays that occur from the beginning ofMarch until mid-May and the entirety of October, orover the finite elements of approximately 3–5 and10–11. There is an obvious lag between our estimateddrop in b and the start of the school holidays. Thislag is probably owing to a lag in the reporting of casedata. There are consistently extra cases reported in Jan-uary resulting from a backlog of reports that are notprocessed at the end of the year owing to worker holi-days. This suggests that cases are being reported asoccurring when reports are processed rather thanwhen the cases actually occurred, which would causea lag in the estimates.

Figure 5c shows the 95% confidence region for �b and�S : The optimal estimated �S was 5.5 per cent of themean population, and the optimal estimated �b was 1.28.

4.5. Input sensitivity analysis

Our estimates are dependent upon the inputs we use inour model. We use recorded data for birth inputs andfor population inputs, but no data are available forthe reporting fraction and recovery rate, and wewould like to know how sensitive our estimates are tothe values used for these inputs. In addition, we usean iterative approach to set the weights in the objectivefunction, and we would like to know how changingthese weights will affect our estimates.


We first examined the effect of varying the reportingfraction on our estimates of b. To investigate this, wevaried the value of the reporting fraction over a widerange about our estimated value. Using each newreporting fraction, we solved the same problem asbefore. Figure 8 shows the estimated average value ofb for New York and the optimal objective functionvalue as the reporting fraction changes. This plotshows that, even for small changes in the reportingfraction away from the estimated value, there is a signi-ficant decrease in the average value of b. The objectivefunction values show that the best data fit occurs whenthe reporting fraction is where we also get the highest �b,and this reporting fraction is the same as that estimatedusing the approach described in §3.1.

In Bangkok, there is a significant difference in thereporting fraction at the beginning and at the end ofthe time horizon studied and a time-varying reportingfraction is needed. We use a linearly varying report-ing fraction throughout the time horizon, and, for oursensitivity study, we keep the same slope as we estimatedbefore. Figure 9 shows the estimated �b and optimalobjective function values as the initial value of thereporting fraction is varied. Since the reporting fractionestimated previously is so low at the beginning of thetime horizon and must be positive, we are unable tolower the initial value by more than about half of a percent. Wherewe see the minimum in the optimal objectivefunction value is alsowherewe find the initial value of thereporting fraction that we estimated previously.

The recovery rate can also affect the dynamics of thesystem, and different sources use different values (typi-cally 1/13 or 1/14). Figure 10 shows the change in theestimated �b’s for New York City and Bangkok asthe recovery rates are varied. For both cities, the

2.0

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reporting fraction, h

obje

ctiv

e fu

nctio

n

0.070.060.050.040.030.020.010

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Figure 9. The �b estimated for Bangkok is shown as a function of the initial value of the reporting fraction (solid line). The objec-tive function value is also shown as a function of the reporting fraction (dotted line). (Online version in colour.)

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Figure 10. The �b values estimated for Bangkok (dotted line) and New York City (solid line) are shown as functions of the recoveryrate. (Online version in colour.)

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0–7.0 –6.5 –5.5 –4.5 –3.5 –2.5–6.0 –5.0 –4.0 –3.0

b

log10(wf/wM)

Figure 11. The �b values estimated for Bangkok (dotted line) and New York City (solid line) are shown as functions of the log ofthe ratio of the weights on the noise terms in the objective function. (Online version in colour.)


changes in �b are not dramatic at any point, but simplychange slowly throughout the range examined. Thisindicates that, for reasonable values for the recoveryrate, our estimates are not dramatically affected.

Finally, we also show the estimation results as a func-tion of the ratio of the weights in the objective function.Here, we vary this ratio by two orders of magnitude oneither side of the value found using our iterativeprocedure. Figure 11 shows the value of �b for bothNew York City and Bangkok as the weights arevaried. The mean value of the transmission parameterchanges little over a range of values near the estimatedweights. Furthermore, the pattern exhibited by the esti-mated transmission parameter is nearly identical overthis entire range.

5. DISCUSSION AND CONCLUSIONS

Successful estimation of parameters in dynamic modelsfor childhood infectious diseases from time-series datapresents several challenges. Typically, reported cases(the incidence) are the only available data, while thereis little information about the susceptible population.


Therefore, approaches must simultaneously estimatethe prevalence and the unknown susceptible states.Furthermore, the case data are often significantlyunder-reported, the reporting interval is often longerthan the serial interval of the disease and the modelsare highly nonlinear. This paper presented a nonlinearprogramming approach for estimating the unknownstates and the seasonal transmission parameter usinga continuous time model with both measurement andmodel noise.

Continuous time formulations offer several advan-tages over discrete time formulations for estimationof infectious disease models. Data can be handled intheir native form regardless of the reporting interval.This was demonstrated by using biweekly reporteddata from London and monthly reported data fromNew York City and Bangkok. Using data in theirnative form is a significant advantage for diseases withshort serial intervals, where it would be unreasonableto have data reported at the same interval.

The estimation approach outlined in this paper ishighly efficient. The estimation formulation usingcontinuous SIR models is a nonlinear optimization

Table 1. Problem sizes and solution times of the London,New York City (NYC) and Bangkok (BKK) estimationproblems studied in this paper.

variables constraints CPU time (s)a

London 16 459 15 963 190NYC 8929 8663 300BKK 4477 4331 76

aAll problems were solved on a 3 GHz Intel Xeon processorand times are reported is seconds.


problem subject to differential equations as constraints.The use of the simultaneous or full-discretizationapproach produces a large-scale algebraic nonlinear pro-gramming problem. Nevertheless, efficient solutions arepossible as the simulation is not converged at each iter-ation. The solution times for all estimations are shownin table 1. These are the full solution times, includingthe times required to initialize the problems and findthe weights to be used in the objective functions. Sig-nificant reductions could be made by initializing theproblem well and by giving good initial guesses forthe objective function weights. Recent work byHooker et al. [3] solves a similar problem formulationin approximately 2 h, the MCMC estimation performedby Cauchemez & Ferguson required approximately 20 hper run [11] and the plug-and-play method of He et al.[2] requires approximately 5 h. None of our estimationstake longer than 5 min. The efficiency of this fully sim-ultaneous approach opens the door to explore manymore model structures efficiently and provides a frame-work that is scalable to large spatially distributedestimations.

Figure 5a–c shows a strong inverse correlationbetween the estimated �b and �S as seen by the narrow,elongated confidence regions. This result is not unex-pected when compared with the approximateexpression relating b to �S given by 1=�S ¼ �b=g [20].This relation gives a curve lying approximately throughthe middle of the 95% confidence region in figure 5a–c.These elongated regions indicate that the estimationis sensitive along this line and that care should betaken when interpreting absolute values for b or S.It should also be noted that the estimations produ-ced nearly identical patterns in b(t) within theseconfidence regions.

More importantly, several recent publications havereported estimated values of the seasonal transmissionparameter, and corresponding R0 values, that arehigher than estimates provided by Anderson & May[20]. For example, the reported estimates of He et al.[2] for London give R0 ¼ 57 with 95% confidence inter-vals of 37 and 60. There is significant complexity infinding R0 values while considering seasonal trans-mission rates, and it is difficult to compare resultsarising from different model structures. Using theapproximate relationship R0 ¼ �bðtÞ=g, we estimateR0 ¼ 13.3 in London with 95% confidence intervals of12.1 and 14.3. Our estimates for New York City(R0 ¼ 9.1) and Bangkok (R0 ¼ 17.9) also give valuesfor R0 that appear consistent with values reported


for measles by Anderson & May for other cities [20]and with values approximated using the average ageof infection.

The estimated transmission profiles from all threecities show strong correlation with school holidaysdespite the very different holiday schedules seenbetween London, New York City and Bangkok. ForBangkok and New York City, there was a lag observedin the estimated transmission profiles that showed thedrop in transmission as occurring after the holidayhad begun. This is probably owing to a lag in reportingcausing cases to be reported well after their occurrenceand the incubation period of measles causing cases to beobserved after the start of the holiday even though theinfection occurred before the holiday.

This overall approach for estimating continuous timeinfectious disease models is reliable, flexible and effi-cient. Although the use of extended likelihood maynot be guaranteed to provide unbiased estimates, thesimulation results showed no evidence of bias. Solutionsto the nonlinear programming problem were possiblewith a general initialization strategy, and effective par-ameter estimates are possible, even in the face ofchallenging sets of data that contain missing years,severe under-reporting and significant noise. It isstraightforward to switch between diseases with differ-ent serial intervals or datasets with different reportingintervals. The approach is independent of model speci-fics. For example, it would be straightforward to addadditional compartments to the model, such asadding an E compartment to make an SEIR modelthat would account for individuals that have beenexposed to a disease but cannot yet infect susceptibles.One could also add a compartment to account for por-tions of the population that were vaccinated against adisease. Furthermore, the approach is highly efficient,making it appropriate for much larger problem formu-lations, or for rapid exploration and comparison ofmultiple model structures. Using this flexible frame-work, we propose to address two important advancesin future work. First, standard assumptions with theSIR model give exponential distributions in age depen-dence of cases. This is contrary to the age-distributedcase data we have for these locations. We propose todevelop an age- and time-discretized model to estimateseasonal age-dependent transmission parameters usingthis approach.

Also, while this work focused on estimating trans-mission parameters for individual large cities, anotherinteresting problem for health officials is looking at aspatial model of disease spread. For accurate estimationof disease dynamics in small cities where fade-out isobserved, information is needed regarding the trans-mission of the disease from a large city where thedisease is endemic to the small city. The approachdescribed in this paper is appropriate for the estimationof large-scale, spatially distributed, nonlinear differen-tial equation models and will be a subject of futureresearch.

The authors gratefully acknowledge financial supportprovided by Sandia National Laboratories and the Office ofAdvanced Scientific Computing Research within the DOE


Office of Science as part of the Applied Mathematicsprogramme. Sandia is a multi-programme laboratoryoperated by Sandia Corporation, a Lockheed MartinCompany, for the US Department of Energy’s NationalNuclear Security Administration under contract DE-AC04-94AL85000. Derek Cummings holds a Career Award at theScientific Interface from the Burroughs Wellcome Fund.D.A.T.C. and D.S.B. were funded by a NIH MIDAS Centerof Excellence grant. D.A.T.C. also received funding from theNational Science Foundation (5 R01 GM 090204).

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A nonlinear programming approach for estimation of transmission parameters in childhood infectious disease using a continuous time modelIntroductionBackgroundProblem formulation and estimation approachProblem formulationEstimation approach

Estimation resultsSimulationLondonNew York CityBangkokInput sensitivity analysis

Discussion and conclusionsThe authors gratefully acknowledge financial support provided by Sandia National Laboratories and the Office of Advanced Scientific Computing Research within the DOE Office of Science as part of the Applied Mathematics programme. Sandia is a multi-programme laboratory operated by Sandia Corporation, a Lockheed Martin Company, for the US Department of Energy’s National Nuclear Security Administration under contract DE-AC04-94AL85000. Derek Cummings holds a Career Award at the Scientific Interface from the Burroughs Wellcome Fund. D.A.T.C. and D.S.B. were funded by a NIH MIDAS Center of Excellence grant. D.A.T.C. also received funding from the National Science Foundation (5 R01 GM 090204).References

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