Network-Based Analysis of a Small Ebola Outbreak
Mark G. Burch, Karly A. Jacobsen, Joseph H. Tien, Grzegorz A. Rempala
MMATHEMATICAL BIOSCIENCES doi:10.3934/mbe.201X.XX.xxAND ENGINEERINGVolume XX , Number , pp. X–XX
NETWORK-BASED ANALYSIS OF A SMALL EBOLAOUTBREAK
Mark G. Burch
College of Public HealthThe Ohio State UniversityColumbus, OH 43210, USA
Karly A. Jacobsen
Mathematical Biosciences InstituteThe Ohio State UniversityColumbus, OH 43210, USA
Joseph H. Tien
Department of Mathematics and Mathematical Biosciences InstituteThe Ohio State UniversityColumbus, OH 43210, USA
Grzegorz A. Rempa(cid:32)la
Mathematical Biosciences Institute and College of Public HealthThe Ohio State UniversityColumbus, OH 43210, USA
Abstract.
We present a method for estimating epidemic parameters in network-based stochastic epidemic models when the total number of infections is as-sumed to be small. We illustrate the method by reanalyzing the data from the2014 Democratic Republic of the Congo (DRC) Ebola outbreak described inMaganga et al. (2014). Introduction.
The best known models for the spread of infectious disease inhuman populations are based on the classical SIR model of Kermack and McK-endrick [19]. The same system of ordinary differential equations (ODEs) may bederived as the large population limit of a density-dependent Markov jump processusing the methods of Kurtz [20]. This stochastic formulation brings a number ofmathematically attractive properties such as explicit likelihood formulas and ease ofsimulation. Nevertheless, a drawback of the Kermack and McKendrick-type mod-els is that they can be unrealistic in describing the interactions of infectives andsusceptibles as they are based on assumptions of homogeneous mixing [16].In recent years there has been considerable interest in developing alternativesto the classical SIR, for instance, via network-based epidemic models, as reviewedby Pellis et al. [27] and House and Keeling [13]. Pair approximation models have
Mathematics Subject Classification.
Primary: 92B; Secondary: 92C60.
Key words and phrases.
Ebola, Network Epidemic Models, Configuration Model, BranchingProcess, Statistical Inference.This research has been supported in part by the Mathematical Biosciences Institute and theNational Science Foundation under grants RAPID DMS-1513489 and DMS-0931642. a r X i v : . [ q - b i o . P E ] N ov BURCH, JACOBSEN, TIEN, AND REMPALA been formulated to account for spatial correlations while maintaining mathematicaltractability [28, 17, 18]. Much of the work on these models has regarded the dy-namics of the deterministic system obtained in the large population limit, such asthe work of Miller and Volz on edge-based models [31, 24, 25] and that of Altmann[2] on a process with dynamic partnerships.As in the work of Miller and Volz [31, 24, 25], the configuration model (CM)random graph is often chosen to dictate the network structure in epidemic models[4, 23, 5]. CM networks can be viewed as a generalization of the Erd¨os-R´enyirandom graph, i.e. the G ( n, p ) model where n is the size of the graph and edges aredrawn between each pair of nodes independently, with probability p . In the limitof a large graph this results in a Poisson degree distribution; however, in a generalCM graph model framework Poisson may be replaced with any appropriate degreedistribution (see, e.g., [31]). Miller and Volz heuristically formulated the limitingsystem of ODEs that govern the dynamics of the SIR epidemic on a CM graph.Decreusefond et al. [10] and Janson et al. [14] have recently given formal proofsfor the correctness of the Miller-Volz equations as the law of large numbers (LLN)for the stochastic system under relatively mild regularity conditions on the degreedistribution and on the epidemic initial condition. In addition, Volz and Millerhave conjectured the limiting equations for generalizations which include dynamicgraphs (edge formation and breakage), heterogeneous susceptibility, and multipletypes of transmission [25]. To date, the convergence for the stochastic systems inthese cases has not been formally verified but numerical studies done by the authorssuggest that they are correct.As in the case of classical SIR [1] the early behavior of an epidemic on a CMgraph can be approximated by a suitable branching process [4, 14, 6]. Such approx-imation allows one in turn to use the early epidemic data to statistically ascertainthe probability of a major outbreak (large number of infections among the networknodes) as well as to estimate the rate of infection spread and changes in the con-tact network as described below. By and large, there have been limited studies ofstatistical estimation for network-based models [32]. In one of the early papers onthe topic, O’Neill and Britton study Bayesian estimation procedures for the G ( n, p )model when the network is assumed to be of reasonable enough size that the net-work structure may be included as missing data and imputed via a Monte Carloscheme [7]. More recently, Groendyke has extended this approach to non-Markoviandynamics by allowing both the infection time and the recovery time to follow an ar-bitrary gamma distribution [11, 12]. However, these methods are generally tailoredto networks of small size, where posterior sampling is feasible, and may encountervarious difficulties in large networks when the imputation becomes computation-ally expensive and often impractical. The framework presented here, on the otherhand, assumes the proportion of infectives is small relative to the population size.This allows us to avoid explicit imputation of the network and makes the numericalcomplexity of the analysis comparable to that of a small homogenous SIR epidemic[8, 29].The main contribution of the current paper is to present a statistical inferencemethod for analyzing the early stages of an epidemic, or a small outbreak, evolvingaccording to SIR type dynamics on a random graph. A novel aspect of our methodis that we assume the random graph structure evolves in response to the epidemicprogression. This allows us to account for changing contact patterns in response to NALYSIS OF A SMALL EBOLA OUTBREAK 3
Figure 1.
The empirical secondary case distribution in the DRCoutbreak dataset (neglecting the index case), as given by Magangaet al. [22].infection, for example due to population behavioral changes or health interventions(e.g. quarantine).The development of our methods is motivated by the devastating 2013 − DRC dataset.
There were a total of 69 confirmed infections in the 2014 DRCoutbreak. The time series of cumulative case counts was reported by Maganga etal. [22]. However, the analysis method presented here does not require temporaldata and instead utilizes the distribution of secondary cases, also given in [22]. Theindex case is believed to have caused 21 secondary cases, presumably due to herfuneral acting as a super-spreading event [22]. Hence, we assume this data point tobe an outlier and exclude it from our final analysis, as Maganga et al. also did intheir estimation of R . The secondary case distribution for other named contactsis shown in Figure 1. One patient caused three subsequent cases, two patientscaused two additional cases, 30 caused one additional case and 11 patients causedzero additional cases. These may be viewed as observations from the post-indexoffspring distribution of the branching process approximation, i.e. the number ofinfections caused by an individual who is himself not the index case. BURCH, JACOBSEN, TIEN, AND REMPALA
Although, as already mentioned, the temporal data measurements are not ex-plicitly required for parameter estimation in our approach, the available temporalinformation can be directly incorporated into the likelihood function, or used toinform the parameterization of the prior distributions, which is what was done forthe current DRC dataset. Details are given below.3.
Epidemic model.
Consider a graph G = (cid:104) V, E (cid:105) where the vertex set, V , cor-responds to individuals in a population of size n and the edge set, E , correspondsto potentially infectious contacts occurring between such individuals. The graphstructure is given as a realization of a configuration model random graph with aprescribed degree distribution, D , where the probability that a node has degree k is denoted by q k ≡ P ( D = k ). Each node i ∈ V is assigned d i half-edges that aredrawn at random from D . Then, the pool of half-edges is paired off uniformly toform the final network. This link formation model does not exclude the possibilityof self-loops or multiple edges but this has a negligible effect in the large graph limit(see, e.g., discussion in [14]).The general disease framework adopted is the standard compartmental SIRmodel where, for every time t >
0, each node i ∈ V is classified as susceptible (S),infectious (I), or recovered (R) from the infection. Given some number of initiallyinfectious nodes, a node i becomes infective via transmission along an edge fromone of his infectious neighbors. In the Markovian case, we assume that i remainsinfectious for an exponentially distributed amount of time with rate parameter γ ,which we refer to as the recovery rate. More generally, a non-exponential (e.g.,gamma) recovery rate leads to the so-called semi-Markov SIR model [15]. Whileinfectious, the infective i attempts to transmit the infection to all of his susceptibleneighbors according to an exponential distribution with rate β . In the case thatthe realization of the infection “clock” for a particular neighbor “rings” before therecovery “clock”, then this neighbor becomes infected. The epidemic ends whenthere are no more infectives.In addition to these standard SIR dynamics, we allow the network structure tochange due to infection status. Specifically, we assume that infectious individualsdrop each of their contacts according to an exponential distribution with rate δ .The dropped contacts, which could account for behavioral changes due to diseasesuch as isolation or decreased mobility, cannot be reformed.4. Statistical inference.
Index case offspring distribution.
For an index case i with degree d i , atmost d i secondary infections can be produced. Conditional on the recovery timeof that index case, t i , the probability that infection has passed to any particularneighbor is given by p t i ≡ p ( t i ; β, δ ) = ββ + δ (1 − e − ( β + δ ) t i ) (1)where the first term in the product on the right-hand side is the probability that anedge transmitted infection prior to being dropped, while the second term representsthe probability that either an infection or a drop occurred before recovery. There-fore, the total number of secondary infections caused by node i , which we denote NALYSIS OF A SMALL EBOLA OUTBREAK 5 by X i , conditional on the time of recovery t i , is given by P ( X i = x i | t i , d i ) = (cid:18) d i x i (cid:19) p x i t i (1 − p t i ) d i − x i . If the recovery time is not known but assumed to follow a distribution f ( t i ) ≡ P ( T i = t i ), then we may analogously define π d i ( x i ), the conditional probability of x i offspring given degree d i . The law of total probability implies π d i ( x i ) = (cid:90) ∞ (cid:18) d i x i (cid:19) p ( t i ) x i (1 − p ( t i )) d i − x i f ( t i ) dt i . We will assume identical recovery distributions for all individuals and, thus, theoffspring distribution will be the same for all index cases.The final form for the offspring distribution of an index case may be found bysupposing that the degree, d i , of the index case is unknown. Let q d i ≡ P ( D i = d i )denote the probability that the index case has degree d i . Therefore, the law of totalprobability gives P ( X i = x i ) = (cid:88) d i ≥ x i q d i π d i ( x i ) . (2)That is, we sum over all possible degrees that could yield at least x i secondaryinfections and weight them according to the degree distribution.4.2. Post-index case offspring distribution.
We now consider the offspring dis-tribution for a post-index case. By definition, such an individual acquired infectionfrom another individual in the network and, thus, has at least one neighbor. There-fore, some adjustments are needed to account for the fact that post-index caseshave a degree distribution which differs from D . Let q (cid:48) k denote the probability thata given neighbor in the CM network has degree k . Then it is known [26] that q (cid:48) k = kq k µ . Since at least one of the neighbors of a post-index case has already been infected,he may pass the infection to at most k − X (cid:48) i denote theoffspring distribution for a post-index infection i . Similarly to Eq. (2) we derive P ( X (cid:48) i = x (cid:48) i ) = ∞ (cid:88) k>x (cid:48) i q (cid:48) k π k − ( x (cid:48) i ) . (3)For a fixed set of parameters the basic reproductive number, R , can be cal-culated as E ( X (cid:48) v ), i.e. the average number of secondary infections caused by apost-index case. That is, R is given by R = ∞ (cid:88) x (cid:48) i =0 x (cid:48) i ∞ (cid:88) k>x (cid:48) i q (cid:48) k π k − ( x (cid:48) i ) . (4)4.3. Example.
To illustrate, we assume that the degree distribution is Poissonwith mean parameter λ and the recovery distribution is exponential with rate pa-rameter γ . This implies f ( t i ) = γe − γt i and q d i = λ d i e − λ d i ! . BURCH, JACOBSEN, TIEN, AND REMPALA
Therefore, by Eq. (2), the offspring distribution for an index case is given by P ( X i = x i | λ, β, γ, δ ) = ∞ (cid:88) k ≥ x i λ k e − λ k ! (cid:90) ∞ (cid:18) kx i (cid:19) p ( t ; β, δ ) x i (1 − p ( t ; β, δ )) k − x i γe − γt dt, and, by Eq. (3), the post-index case offspring distribution is given by P ( X (cid:48) i = x (cid:48) i | λ, β, γ, δ )= ∞ (cid:88) k>x (cid:48) i λ k − e − λ ( k − (cid:90) ∞ (cid:18) k − x (cid:48) i (cid:19) p ( t ; β, δ ) x (cid:48) i (1 − p ( t ; β, δ )) k − − x (cid:48) i γe − γt dt. This expression has no simple analytical form but it is not hard to approximatethe integral numerically since it can be written as an expectation against the recov-ery distribution. Therefore, a simple Monte Carlo sample from the desired recoverydistribution allows for efficient computation of this term.Using Eq. (4), we can calculate the basic reproductive number in this Markoviancase. For an arbitrary degree distribution, R is given by R = ββ + γ + δ ∞ (cid:88) k =0 ( k − kq k µ , (5)which only differs in the inclusion of δ from the corresponding formula on a staticCM graph [24, 14]. Here µ = E ( D ) < ∞ by assumption. The summation in Eq. (5)represents the expected excess degree, i.e. the degree of a node which is necessarilya neighbor of a node, not counting the known edge. In the particular case here ofa Poisson degree distribution, R is found to be (cf., e.g., [3] chapter 6) R = ββ + γ + δ ∞ (cid:88) k =0 λ k − e − λ k ! k ( k −
1) = β λβ + γ + δ . (6)4.4. Likelihood and estimation.
In practice, outbreak data may not arise solelyfrom a single index case and often m separate chains of infection are tracked. Sup-pose the data { x , ..., x m } corresponds to the number of secondary infections foreach of m independent index cases and the data { x (cid:48) , ..., x (cid:48) m (cid:48) } corresponds to sec-ondary infections caused by each of the m (cid:48) post-index cases.Let Θ = ( β, γ, δ, λ ) denote the vector of parameters where γ and λ representthe parameters of the recovery time and degree distributions, respectively. Theoffspring distributions given in Eqs. (2) and (3) allow for explicit formulation of thelikelihood for Θ which is given by L (Θ | x , ..., x m , x (cid:48) , ..., x (cid:48) m (cid:48) ) = m (cid:89) j =1 ∞ (cid:88) k ≥ x j q k π k ( x j ) × m (cid:48) (cid:89) c =1 ∞ (cid:88) l>x (cid:48) c q (cid:48) l π l − ( x (cid:48) c ) . (7)With the specification of the likelihood, maximum likelihood estimators (MLEs)for the rate parameters can be found by numerical optimization. Given the MLEˆ θ = ( ˆ β, ˆ γ, ˆ δ, ˆ λ ), the corresponding estimator for the basic reproductive number canbe calculated by application of the continuous mapping theorem to the expressionfor R given in Eq. (4). For example, under the assumptions of our example inSection 4.3, the estimator following from Eq. (6) would beˆ R = ˆ β ˆ λ ˆ β + ˆ γ + ˆ δ . (8) NALYSIS OF A SMALL EBOLA OUTBREAK 7
Denote the vector of current parameters as Θ = ( β, γ, δ, λ ). If we denote theparameter prior distribution φ (Θ) then the likelihood function above may be alsoused to compute the Metropolis-Hastings acceptance probability in the Markovchain Monte Carlo (MCMC) sampler. Let x denote the vector of data countsand τ be the transition kernel. The MCMC algorithm for obtaining the posteriordistribution of Θ is then as follows1. Initiate Θ curr = Θ .2. Obtain proposal Θ prop from τ (Θ | Θ cur ).3. Accept (or not) this proposal with Metropolis-Hastings probability given by ρ ( x , Θ cur , Θ prop ) = min (cid:18) , L (Θ prop | x ) φ (Θ prop ) τ (Θ prop | Θ cur ) L (Θ cur | x ) φ (Θ cur ) τ (Θ cur | Θ prop ) (cid:19) . (9)4. Return to 2.5. Repeat until convergence.In this way, after sampler convergence, we obtain an approximate sample fromthe posterior distribution of Θ and may subsequently compute its approximate 1 − α credibility region, given the observed data x . In particular, the credibility intervalof the parameter R given by Eq. (4) may be determined.4.5. Generalizations.
Note that the likelihood formula (7) is valid in a non-Markovian setting such as an arbitrary recovery time distribution and could furtherbe extended to the scenario where the transmission rate varies with time since infec-tion. Note that in the latter case the formula for R would differ from Eq. (4) dueto a form for p t i that differs from Eq. (1) but would remain calculable as E ( X (cid:48) v ).If additional data were available, such as the recovery time or number of contactsof each individual, it could be explicitly incorporated into the likelihood function(7) through the joint distribution of recovery times and degrees.5. Analysis of the DRC dataset.
To illustrate our method we perform theBayesian posterior estimation of the model parameters for the 2014 Ebola outbreakin the DRC based on the data described in Section 2. The specific model consideredis as given in Section 3, where transmission occurs according to an exponentialdistribution with rate parameter β , and the degree distribution is Poisson withparameter λ . Recovery time is assumed to follow a gamma distribution Γ( α, β ).We note that, while incubation periods for Ebola range from two to 21 days [9], ourmethod does not require consideration of latent exposure since it does not dependon infection timing.We perform estimation via the MCMC scheme given in Section 4.4. Prior dis-tributions were set to be minimally informative Gaussian distributions and hyper-parameters were selected based on previous estimates [22, 30]. The prior distributionfor λ was taken to be N (16 ,
14) and for δ was taken to be N ( . , . β was estimated on log-scale under an assumed N (log( . ,
4) prior distribution. Lastly, the gamma distribution for the infectiousperiod was chosen to have prior mode (i.e., ( α − /β ) distributed as N (11 ,
6) andprior standard deviation ( √ α/β ) as N (6 , τ wasused. Central 95% credibility intervals were calculated for the parameters of inter-est as well as for R . Results are summarized in Figure 2 including histograms ofposterior samples.The 95% credibility interval for the basic reproductive number R is found tobe ( . , .
15) with the R posterior mean of . BURCH, JACOBSEN, TIEN, AND REMPALA
Figure 2.
Estimated posterior densities for the parameters of in-terest for the non-Markovian MCMC sampler. Green line denotesposterior mean.agrees closely with the moment-based estimate given in Maganga et al. [22] andis also consistent with the relatively small total number of confirmed infectionsobserved during the epidemic. We further note that our interval estimate comparesfavorably to the 95% R confidence interval of ( − . , .
06) reported by Magangaet al. [22] indicating that for the DRC data the fully parametric model producesa more precise (shorter) interval. The infection rate is found to have a posteriormean of . . , . . . , . .
18 days with corresponding credibility interval of(6 .
65 days, 27 . . . , . R and infection andrecovery rates are seen to be consistent with a small outbreak behavior observedin the DRC dataset and to agree well with the numerical values reported earlier.However, based on the same data, our parametric model is also seen to yield moreprecise interval estimates. NALYSIS OF A SMALL EBOLA OUTBREAK 9
Figure 3.
Final outbreak size distribution based on 20,000 sim-ulations of the branching processes from the posterior parameterdistribution. The actual outbreak size based on the DRC dataset(black line) is shown for comparison.The final outbreak size of simulated branching processes from the posterior pa-rameter distribution is used as a model diagnostic for fit to the empirical data.Conditioning on the number of infections caused by the index case as the num-ber of independent branches, the posterior parameter samples and correspondingpost-index case offspring distributions were used to simulate branching process real-izations. The final outbreak sizes were calculated and the distribution is presentedin Figure 3. Reasonable agreement is observed between this distribution and theempirical outbreak size of 69 cases [22].6.
Discussion.
We presented here a Bayesian parameter estimation method for aclass of stochastic epidemic models on configuration model graphs. The methodis based on applying the branching process approximation, and is applicable whenthe number of infections is small in relation to the size of the population understudy. In particular, this includes the case when the total outbreak size is small orwhen we are at the onset of a large outbreak. The method are flexible, for exampleallowing for arbitrary degree and recovery distributions, and in principle requiresonly a knowledge of the distribution of secondary cases, although additional datacan be incorporated into the inference procedure, as the likelihood function underbranching approximation remains straightforward to evaluate under a wide rangeof data collection schemes.We illustrated our approach with the analysis of data from the 2014 DRC Ebolaoutbreak which was originally described and analyzed in Maganga et al. [22]. Ourmethod, under only weakly informative prior distributions, is seen to produce aconsiderably tighter credibility interval for R than the moment-based confidenceestimate reported in [22]. This demonstrates the utility of the branching processapproximation for small epidemics in obtaining more precise estimates of R , which is essential in assessing potential risk of a large outbreak and in determining thelevel of control efforts (e.g. vaccination or quarantine) needed to mitigate an out-break. The final size comparison indicates that the observed data is within therange of model predictions, although the direct comparison of observed and modelpredicted offspring distributions indicates some disagreement in the observed fre-quency of zeros and ones. The small sample size prohibits definitive conclusionsbut this may indicate the need to incorporate more complex network dynamics(e.g. distinguishing between multiple types of infectious contacts).As the statistical estimation methods appear essential to inform public health in-terventions, we hope that our work here will help in establishing a broader inferenceframework for epidemic parameters, based on the type of data usually collected inthe course of an outbreak. The Bayesian approach is particularly attractive in thiscontext, as it naturally incorporates any prior or historical information. However,the current approach only addresses the inference problem at the epidemic onsetand, in particular, is not appropriate when the number of infected individuals com-prises a significant portion of the population. We plan to address estimation forsuch large outbreaks, possibly also incorporating more complex network dynamics,in our future work. Acknowledgments.
We thank the Mathematical Biosciences Institute at OSU forits assistance in providing us with space and the necessary computational resources.
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