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Section III outlines our modified SEIR model for types [12]. A widely used technique for finding R0, which is Ebola and explains the particle setup and data, and Section based on this definition, is the next generation matrix method IV presents main results of this study. Section V concludes [12]. Applying this technique to the SEIR model above finds β c the paper by suggestions for future research. R0 = γ [10].

II. BACKGROUND B. Sequential Monte Carlo Filter A. Epidemic Modeling Sequential Monte Carlo (SMC) — or particle filter — Mathematical models of infectious disease offer a mech- refers to a class of statistical techniques that estimate un- anistic framework to describe and study the spread of infec- known parameters, namely states in this context, as new tions within human and animal populations, providing deep streaming noisy observations becomes available [13]. In insight into their dynamics and suggesting practical strategies SMC, we iteratively sample from the posterior distribution of to reduce the severity of epidemics [8]. Here, we intro- parameters until the parameters converge to stationary values duce a brief background discussing the susceptible-exposed- [14]. This iterative sampling is updated using a stream of infected-recovered (SEIR) model which is compatible with data, and as such, it enables us to modify our best guesses for our understanding of Ebola virus epidemiology. the states according to actual observations. In the following, In the SEIR model, individuals are assumed to be in we explain the dynamic state–space model and estimation of one of these compartments: susceptible, exposed, infected posterior PDF briefly for the particle filters algorithm. and removed/recovered. In this model, when susceptible a) Dynamic state–space model: The state–space model individuals have contact with an infected person, they enter assumes the , i.e., into the exposed compartment (E) with rate βI. Homogeneity Pr(xk x0:k 1)= Pr(xk xk 1). (2) of the population and how people have contact with each | − | − others in host population is represented by a percentage and describes the distribution of the system state in the next factor c. After the incubation period of disease, of step, as well as the observation, given the current state of 1/λ, they enter into the infected compartment (I). Infectious the system. More rigorously, a state–space model is defined individuals move to the recovered/removed compartment (R) as [13], [15]: at rate γ [9]. The compartmental SEIR model is [9] xk f (xk xk 1,θ), ∼ | − (3) θ dS yk f (yk xk, ), = βcSI, ∼ | dt − where yk represents kth observation, xk represents states dE θ = βcSI λE, corresponding to the kth observation, and represents pa- dt − (1) rameters of the model. Therefore, yk depends only on xk and dI θ θ = λE γI, and xk depends on xk 1 and . dt − b) Estimation of posterior− PDF: Given the observation dR = γI. data y1:k up to time k, the ultimate goal is to define the dt posterior distribution, p(xk,θ yk), which describes the hidden | In this compartmental SEIR, the size of host population is state xk and parameters θ of the . The assumed to remain constant throughout the evolution time, estimation of density function (PDF) i.e., P = S +E +I +R, and demographic effects are ignored. based on Bayes’ theorem is [13]: An important mathematical tool in the study of epi- xk,θk y1:k ∝ f (yk xk,θ)p0(xk,θ) demics is the basic reproductive ratio. Usually the basic | | (4) ∝ f (yk xk,θ)p0(xk,θ y1:k 1). reproductive number is defined as the expected number of | | − secondary individuals produced by a typical single infected The sequential Monte Carlo filter approximates the poste- individual during its infectious period [10]. Thus, R0 is a rior PDF as [13]: dimensionless value that represents the average number of J additional susceptible people to whom an infected person (i) Pr(xk,θk y1:k) ∑ w 1 (i) (i) . (5) k (x ,θ )=(x ,θ ) passes the disease before he/she recovers [11]. For instance, | ≈ i=1 { k k k k } if an infectious person passes the disease on three others on (i) average, during his/her infectious lifetime, then R0 = 3 > 1, where 1 . is the indicator function, xk is a particle and indicating that the number of new infectious individuals (i) { } wk is its weight. The approximation is more accurate if the 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 θ (i) Pr(xk, k y1:k)dxk ∑ wk . (6) increments or decrements of new infected [11]. A common Z | ≈ i=1 Among particle filter techniques are the bootstrap filter, eliminated. After each round, the surviving particles produce 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 (i) employed is that wk might become very small for some filter, due to its flexibility in modeling of non-linear processes particles and affect the accuracy of the particle filter. A as explained in Section III-B. proper way to decrease the number of particles is to select the importance probability function close to the optimal one [15]. III. SMC FOR EBOLA EPIDEMICS Auxiliary particle filter aims is predicting which particles will In this paper, we use the discrete-time SEIR model which have a small weight and minimizes those particles with small is compatible with the epidemiology of the Ebola virus. weights. With respect to these two filters, the kernel particle A. Modeling of Ebola filter not only minimizes those kinds of particles with small weights, but also estimates the unknown parameters [17]. In The state variables S , E , I , and R denote the fraction of t t t t this method, the main goal is to reduce the mean integrated people who are susceptible, exposed, infected and 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(xt ,θk y1:k), of xt . Here, for each Assumption 1: Since the population is much greater than k | k particle i, θ (i) is the value of parameters at time t which we the number of infected cases of Ebola, we assume S 1. k Therefore, the equation for evolution of E(t) in (1) simplifies≃ only have k observations up to time t. to: We choose reported Ebola data in Guinea, one of the three major West Africa countries that experienced the Ebola Et 1 = Et + βctIt λEt . (7) + − outbreak and we analyze the cumulative cases and death Assumption 2: We assume that ct , representing how the counts. To update parameters and states using kernel particle population is mixed, is dynamic due to different intervention filter, we need data. However, the number of cases and strategies to prevent the spread of Ebola such as social death in Guinea are reported at irregular intervals. To address distancing and quarantining. Specifically, we assume ct is this issue, we implement particle filter only for those time decreasing at rate α, i.e., intervals that we have information and use the last updated parameters and states that are generated by particle filter to ct 1 = ct αct . (8) + − 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 we can write the state–space model required for SMC method disease is disseminated, the transmission rate decays expo- as nentially. xt 1 xt NΩ(g(xt ,θ),Q(θ)), (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 αct − Ebola spread. Et + βctIt λEt  − g(xt ,θ)= It + λEt γIt , ct+1 = ct αct + ξα ,  −  −  Rt + γIt  E = E + βc I λE + ξλ ξβ ,   t+1 t t t t  ϕIt + ϕγIt  − −   (11) It 1 = It + λEt γIt ξλ + ξγ , (9) α + − − 0 0 00 Rt+1 = Rt + γIt ξγ 0 λ + β λ 0 0  − 1 − ϕ ϕ ϕγ ξ ξ Q(θ)= 0 λ λ + γ γ γϕ . Dt+1 = Rt+1 = Rt + It γ + ϕ . P2  − − −  − 0 0 γ γ γϕ  ξ χ α β λ γ ϕ  −  In the above equations, χ where , , , , is 0 0 γϕ γϕ γϕ2  a random component, with zero mean and∈{ given }  −  θ T √χ/P, where P is the population size. The variance of noises where Q( ) is the covariance matrix of xt = [ct ,Et ,It ,Rt ,Dt ] are due to stochasticity of the underlying process [13]. Each according to the state-space model (9) with stochastic fluc- of these component are assumed to be uncorrelated. tuations. Here, NΩ(µ,Σ) represents the truncated normal distribution where Ω = (ct ,Et ,It ,Rt ,Dt ) : ct ,Et ,It ,Rt ,Dt B. Filtering Setup { ≥ 0,Et + It + Rt 1 . 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 estimated by a set of particles and at each round their weights 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 2 3φ 1 2 reported data, we need to estimate cumulative cases and In above, a = 1 h and h = 1 ( 2φ− ) . These two quan- death. To this end, using states I and R in equation 9, we tities control the− smoothness of− kernel , model the observations Y as follows [13]: while φ (0,1) is a discount factor which reduces the chance of failure∈ in the filter. Readers are encouraged to refer to [13] Y N (µY ,ΣY ), (12) φ ∼ for more details. Typically, is assumed to be a number between 95 and 99. where . . ζ log(y ) b (I + R ) I σ 0 C. Data Explanation Y = I,k , µ = I t t ,Σ I . Y ζD Y σ log(yD,k)  bD(Dt )  ∼  0 D Epidemic Curves and Its Approximation (X) (13) 4000 Xh : Approximation of X I I Index D and I in observation matrix Y, represent total 3500 X : Cumulative number of infectious number of dead and infected individuals at time t, as reported I Xh : Approximation of X R R by WHO. In matrix µY , I represents the estimation of number 3000 X : Cumulative number of death of infected individuals (state I) and R represents estimation R of state R; individuals who are dead or recovered. ζs,σs 2500 and bs for both infected and dead are typically unknown, 2000 but we can predict these values by methods [16]. In particular, bs are multiplicative constants and ζs Population 1500 are power-law exponents which can be calculated based on 1000 the significant of dispersed of data points. Since fluctuations exist in the data, ζs are not estimated precisely. The scaling 500 Σ σ ∝ 1 law for Y , , is . where P is the 0 √P Feb14 Apr14 May14 Jul14 Sep14 Oct14 Dec14 Feb15 Mar15 May15 population size [16]. Time (Days) 3) Kernel density particle filter: Model (12) and (13) Fig. 1: Cumulative cases and death data and their estimation define the likelihood of observations, yk, given xt and θ k by the particle filter in Guinea, reported by [WHO] — p(yk xt ,θ) — as a log normal distribution with mean | k µY , which is two-by-one vector and the standard deviation, ΣY , which is a two-by-two diagonal matrix with diagonal The Ebola virus, commonly known as Ebola, causes a element equal to σI and σD. Initially, particle i which serious illness which is fatal if untreated in most cases [2]. i 1,...,J is sampled from an initial probability density It is transmitted via direct contact with blood, secretions, ∈{ } function (PDF) p(x0). For time step k = 1, weights are equal organs, or other bodily fluids of infected individuals [18]. to 1/J for all particles, and θ0 and x0 are generated by The incubation period, the time interval from infection with random sampling from specific distributions. Suppose k+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(xt ,θ y1:k 1) for k > 1 [13]. and sore throat [2]. Since approximately December 2013, k+1 | + (i) (i) (i) West Africa has been affected by this virus. However, The 1) Calculate m : m = aθ + (1 a)θ¯k. k+1 k+1 k − World Health Organization (WHO) declared the epidemic 2) Compute the expectation of x(i) : tk+1 to be a public health emergency of international concern µ(i) θ (i) (i) on August 8, 2014 [2]. We analyze the cumulative case k+1 = E(xtk+1 k ,xtk ), for all i 1,2,..,J . 3) Compute auxiliary| weights and∈{ normalize} them: and death counts in Guinea, one of the three West Africa g(i) (i) (i) µ(i) (i) (i) k+1 Countries that experienced the Ebola outbreak. Cumulative gk+1 = wk p(yk+1 k+1,mk+1) , gk+1 = J . | ∑ (l) cases are classified into three categories: confirmed, prob- gk+1 l=1 able, suspected cases. Similarly, we have three cases for 4) Sampling: Select an index j randomly and afterwards death counts. Confirmed cases are those individuals who sample x( j) with its weights g(1) ,...,g(J) . k k+1 k+1 are diagnosed by polymerase chain reaction (PCR) method. { (i) }( j) θ 5) Reproduce the parameters: θ Nω (m ,V ), k+1 ∼ k+1 k+1 On the other hand, suspected and probable cases denote where Nω (µ,σ) is a truncated normal distribution. those individuals that have symptoms of Ebola but it is (i) (i) (i) (i) ( j) 6) Sample the x : x p(x θ ,x ), 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) Recompute the expectation of x(i) : This data has been collected from reports of WHO available tk+1 µ(i) = E(x(i) θ (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 (i) θ (i) (i) (i) p(yk+1 xt , ) (i) w w = | k+1 k+1 , w = k+1 . We estimate the states of Ebola propagation from March k+1 µ( j) ( j) k+1 J (l) p(yk+1 ,m ) ∑ | k+1 k+1 wk+1 l=1 23, 2014 to April 30, 2015 having data reported only in Evolution of Basic Reproductive Ratio (R ) 0 T = 170 days. Guinea population in 2015 was estimated to 1.6 be around 12,500,000, however, the population in danger to be infected by the Ebola virus was estimated to be roughly )

t 1.4 ( about P = 1,000,000. 0 We estimate parameters by sampling from a log-normal R distribution using J = 5,000, number of particles. The sen- 1.2 sitivity to changes in discount factor is chosen as φ = 0.95 1 1 and initial weights are all set equal to w0 = J− . To run the particle filter, the initial state, x0, and initial parameters, θ0, are generated randomly. Specification of initial priors 0.8 distributions are reported in Table I and II . ective reproductiion ratio ff

E 0.6 TABLE I: Priors specification for parameters

Parameters Priors 0.4 Feb14 Apr14 May14 Jul14 Sep14 Oct14 Dec14 Feb15 Mar15 May15 mitigation rate (α) U (.0059,.00593) transmission rate (β ) U (.259,.379) Time (Days) λ latency rate ( ) Beta(78,577) (a) Changes in basic reproductive ratio over disease evolution recovery/remove rate (γ) Beta(21,246) ϕ Evolution of Ebola Disease Parameters fatality rate ( ) Beta(37,15) 0.2

c 0.1 β 0 Based on collected data and expert opinion, some measure- Feb14 Apr14 May14 Jul14 Sep14 Oct14 Dec14 Feb15 Mar15 May15 ments for γ, λ, and ϕ are available. Therefore, we specify 0.15 λ beta distributions for each of parameters, using Beta buster 0.1 and BetaSlicer [19], [20]. Based on collected information, the Feb14 Apr14 May14 Jul14 Sep14 Oct14 Dec14 Feb15 Mar15 May15 1 average incubation period,λ − is less than 21 days with 95% 0.15 γ 0.1 confidence interval and mean around 8 days [2]. Fatality rate, 0.05 ϕ, is less than 80% with 95% confidence interval and Feb14 Apr14 May14 Jul14 Sep14 Oct14 Dec14 Feb15 Mar15 May15 1 71% [2], [21], [22]. The average duration of illness onset to 0.8 1 ϕ death or recovery, γ− , is around 12 days [21], [22]. Since 0.6 we do not have enough information about transmission and Feb14 Apr14 May14 Jul14 Sep14 Oct14 Dec14 Feb15 Mar15 May15 mitigation rates, β 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, we use uniform distributions. For observation constant in SMC filter, we assume that b = .88 and b = .54 and I D Fig. 2: Variation in R and parameters during disease evolu- standard deviation, Σ is a two by two diagonal matrix whose 0 Y tion diagonal elements are σI = .00125 and σD = .00085. Power – law constant ζ, for infected individuals, is .88 and for dead individuals is .68. Fig. 1 shows cumulative cases and deaths data and their estimation by the particle filter in Guinea. In [23], R0(t) is estimated as a single value with confidence interval, while in [6] for the reproduction ratio, R0(t), is our model, the basic reproductive ratio is equal to R0(t)= 1 computed by fitting exponential growth curves to small cβγ− . The result indicates that transmission rate, latency rate and recovery rate are not constant during the disease successive time intervals of the Ebola outbreak. Instead, our method finds the basic reproductive ratio, R0(t), as a evolution. Therefore, as demonstrated in Fig. 2a, R0(t) is not constant neither. continuous function of time during the Ebola evolution and The maximum value of the basic reproductive number is for each time a probability function is represented. Using this 1.51 on March 2014 and it decreases until September 2014 method, we can also see that not only R0(t) changes over time, but also parameters such as β, λ and γ change. which R0 1. Afterward, a pick is occurred on October 2014 and after∼ that it decreases. We can see in Fig. 2b that V. DISCUSSIONANDCONCLUSION transmission rates change during the disease evolution. In Our analysis identifies a correlation between R0(t) tem- poral variations and important events in the 2014 Ebola TABLE II: Priors specification for states outbreak. For instance, a reduction of R0(t) can be seen 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) I U (.000050,.000061) account the introduction of some initial medical support and 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 letters, IEEE, vol. 12, no. 3, pp. 242–245, 2005. [18] P. E. Lekone and B. F. Finkenst¨adt, “ in a stochastic variance in the particle filtering scheme. Furthermore, our epidemic seir model with control intervention: Ebola as a case study,” experience with numerical simulations showed fairly consis- Biometrics, vol. 62, no. 4, pp. 1170–1177, 2006. tent outcomes. However, a more objective quantification of [19] G. Jones and W. O. Johnson, “Prior elicitation: Interactive spreadsheet 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 of California, Davis.” [21] G. Chowell and H. Nishiura, “Transmission dynamics and control of spatial implementation of the particle filter can account for ebola virus disease (evd): a review,” BMC medicine, 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 months of the epidemic and forward projections,” N Engl J Med, vol. 371, no. 16, pp. 1481–95, 2014. The authors would like to thank Joan Salda˜na 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, Nora Bello for fruitful discussions and suggestion currents, vol. 6, 2014. regarding the priors specifications, and Ala Fard for organiz- ing Ebola incidence data for our analysis.

REFERENCES [1] H. Nishiura and G. Chowell, “Early transmission dynamics of Ebola virus disease (EVD), West Africa, March to August 2014,” Euro Surveill, vol. 19, no. 36, p. 20894, 2014. [2] “World Health Organization, Ebola virus disease outbreak.” [Online]. Available: http://www.who.int/csr/disease/ebola/en/ [3] G. Chowell, N. W. Hengartner, C. Castillo-Chavez, P. W. Fenimore, and J. Hyman, “The basic reproductive number of Ebola and the effects of public health measures: the cases of Congo and Uganda,” Journal of Theoretical , vol. 229, no. 1, pp. 119–126, 2004. [4] D. He, E. L. Ionides, and A. A. King, “Plug-and-play inference for disease dynamics: measles in large and small populations as a case study,” Journal of the Royal Society Interface, 2009. [5] C. L. Althaus, “Estimating the reproduction number of Ebola virus (EBOV) during the 2014 outbreak in West Africa,” PLoS currents, vol. 6, 2014. [6] S. Towers, O. Patterson-Lomba, and C. Castillo-Chavez, “Temporal variations in the effective reproduction number of the 2014 West Africa Ebola outbreak,” PLoS currents, vol. 6, 2014. [7] N. Kantas, A. Doucet, S. S. Singh, and J. M. Maciejowski, “An overview of sequential Monte Carlo methods for parameter estimation in general state-space models,” in 15th IFAC Symposium on System Identification (SYSID), Saint-Malo, France.(invited paper), vol. 102, 2009, p. 117. [8] L. Pellis, F. Ball, S. Bansal, K. Eames, T. House, V. Isham, and P. Trapman, “Eight challenges for network epidemic models,” Epi- demics, vol. 10, pp. 58–62, 2015. [9] M. J. Keeling and P. Rohani, Modeling infectious diseases in humans and animals. Princeton University Press, 2008. [10] J. Heffernan, R. Smith, and L. Wahl, “Perspectives on the basic reproductive ratio,” Journal of the Royal Society Interface, vol. 2, no. 4, pp. 281–293, 2005. [11] M. Newman, Networks: an introduction. Oxford University Press, 2010. [12] O. Diekmann, J. Heesterbeek, and J. A. Metz, “On the definition and the computation of the basic reproduction ratio R0 in models for infectious diseases in heterogeneous populations,” Journal of mathematical biology, vol. 28, no. 4, pp. 365–382, 1990. [13] D. M. Sheinson, J. Niemi, and W. Meiring, “Comparison of the performance of particle filter algorithms applied to tracking of a disease epidemic,” Mathematical biosciences, vol. 255, pp. 21–32, 2014. [14] H. Xue, “ based on sequential Monte Carlo methods for dynamic data driven simulation,” Ph.D. dissertation, Georgia State University, 2014. [15] A. Doucet, N. De Freitas, and N. Gordon, “An introduction to sequential Monte Carlo methods,” in Sequential Monte Carlo methods in practice. Springer, 2001, pp. 3–14. [16] A. Skvortsov and B. Ristic, “Monitoring and prediction of an epidemic outbreak using syndromic observations,” Mathematical biosciences, vol. 240, no. 1, pp. 12–19, 2012.