Boone, Edward L.

Hanif, Aiman

English

ISU ReD: Research and eData

SEIRD MODEL FOR QATAR: A CASE STUDY.Aiman Hanif††Virginia Commonwealth University, Richmond, Virginia 23284An Important ProblemCoronavirus Disease 2019 (COVID-19) is a severe pandemic affecting the wholeworld with a fast spreading regime, requiring to perform strict precautions, suchas social distancing and mask-wearing, to keep it under control. The virus hasspread from China to 196 other countries across the globe. The State of Qatarwas one of the countries that were affected by COVID-19 spreading; the firstinfected case was reported on February 29, 2020, and it could be considered the2nd highest in the Arab World with the number of confirmed cases 28,272 as ofMay 14, 2020. For effectively specifying such security measures, it is essentialto have a real-time monitoring system of the infection, recovery and death rates.We develop, implement and deploy a data-driven forecasting model for use bystakeholders in the State of Qatar to deal with the Covid-19 pandemic. The modelwill focus on infected, deaths and recovered as those are the only data availableat this time.Fig. 1: Covid-19 Infected, Recovered and Deaths for Qatar.ModelLet S(t) be the number of people Susceptible at time t, E(t) be the number ofpeople Exposed at time t, I(t) be the total number of Infected at time t, R(t) bethe cumulative number of recovered at time t and D(t) be the cumulative numberof Deaths at time t. Furthermore, α is the transmission rate (per day×individual2)from Susceptible to Exposed, β is the rate (per day) at which Exposed becomeInfected, γ is the rate (per day) at which Infected become recovered and η is themortality rate (per day) for those Infected. This can be modeled with the followingsystem of ordinary differential equations:dλS(t)dt= −W (t)TαλS(t)λE(t)dλE(t)dt= W (t)TαλS(t)λE(t)− βλE(t)− γλE(t)− βAλE(t)δ(t− τ )dλI(t)dt= βλE(t) + βAλE(t)δ(t− τ )− γλI(t)− ηλI(t)dλR(t)dt= γλI(t)dλD(t)dt= ηλI(t) (1)where λS(t), λE(t), λI(t), λR(t) and λD(t) are the means of S(t), E(t), I(t), R(t)and D(t), respectively and the parameters have the same definition as providedin the system given in the equation. Since there is no data for S(t) and E(t), thesecompartments will be latent variables and will not directly factor into the likelihood.The likelihood for I(t), R(t) and D(t) are given by:I(t) ∼ Poisson (λI(t))R(t) ∼ Poisson (λR(t))D(t) ∼ Poisson (λD(t)) (2)InterventionsThe model needs to be able to handle interventions made by the Government ofthe State of Qatar. The main parameter that policy can influence is α, the rateof transmission from Susceptible to Exposed. One way to implement this is theuse of indicator functions Wk(t) defined as:Wk(t) ={1 if t > tk0 otherwise(3)where tk is the time where the kth intervention is taken and index k = 1, 2, .., K.For each intervention, there needs to be a change to the value of α, de-noted αk, that captures the impact of the intervention. Let the vector W (t) =(1,W1(t),W2(t), ...,WK(t))T be the vector of the values of each Wk(t) at time t.Let α = (α0, α1, ...αK)T . This formulation gives the following transitions ratesbetween S(t) and E(t):α(t) =α0 if 0 < t < t1α0 + α1 if t1 < t < t2α0 + α1 + α2 if t2 < t < t3... ...α0 + α1 + ...αK if tK < t(4)which will require the following constraints due to the fact that α(t) > 0 for all t:α0 > 0α0 + α1 > 0α0 + α1 + α2 > 0... > ...α0 + α1 + ... + αK > 0 (5)Let A be the set defined by the constraints above.In addition to changes in infection rates α, impulse functions can be used tomodel dramatic one time shifts in transitions between states. A Dirac delta func-tion defined byδ(x) = 0, if x 6= 0 (6)which satisfies∫∞−∞ δ(x)dx = 1. This can be integrated in the model to capturespikes in the number of cases. In our case, the State of Qatar data exhibitsthis type of behavior at day 12 where one can clearly see a large jump in thenumber of infections. This is incorporated into the model presented by a Diracdelta function, δ(t − τ ), in transition rate between Exposed and Infected, whichis coupled with a coefficient to βA to capture the impact of the jump.Bayesian ComputationTo specify the prior distributions for α, βA, β, γ and η one must incorporate thefollowing constraints α > 0, β > 0, γ > 0 and η > 0. Hence the following priordistributions are set:α ∼ MVN(a1, σ2I)C(A)βA ∼ Exp(1)β ∼ Exp(1)γ ∼ Exp(1)η ∼ Exp(1) (7)where C(A) is an indicator function takes the value 1 if α ∈ A. This serves totruncate the normal distribution in order to keep α in the feasible range of values.The likelihood and prior distributions specifications lead to the following posteriordistribution when a = 1 and σ2 = 1:π (α, βA, β, γ, η|D) ∝ π(α)π(β)π(γ)π(η)L(D|α, βA, β, γ, η)= C(A)e−12(α−1)T (α−1)−βA−β−γ−η×T∏t=1λI(t)I(t)λR(t)R(t)λD(t)D(t)e−λI(t)−λR(t)−λD(t)I(t)!R(t)!D(t)!(8)The posterior distribution does not lend to any analytic solution, hence Markovchain Monte Carlo (MCMC) technique Metropolis-Hastings sampling is used toobtain samples from the posterior distribution. The tuned sampler was used togenerate 5,000 samples from π(α, βA, β, γ, η|D), and trace plots were visuallyexamined for convergence and deemed to be acceptable. All inferences will bemade from these 5,000 samples. The model and sampling algorithm is customprogrammed in the R statistical programming language version 3.6.3. The com-putation takes approximately 290 seconds using an AMD A10-9700 3.50GHzprocessor with 16GB of RAM to obtain 5,000 samples from the posterior distri-bution.ResultsTo apply the model, the following initial conditions are specified: S(0) =2, 782, 000, E(0) = 3, I(0) = 1, R(0) = 0 and D(0) = 0. Furthermore, modelinterventions were placed at days t1 = 12, t2 = 19, t3 = 36, t4 = 40 and t5 = 59with an Dirac delta impulse at time τ = 12.To assess the fit of the model, the posterior predictive distribution was used andis given by:π(Inew(t), Rnew(t), Dnew(t)|D) =∫L(Inew(t), Rnew(t), Dnew(t)|α, βA, β, γ, η)× π (α, βA, β, γ, η|D) dαdβAdβdγdη. (9)Using the 5,000 samples from the posterior distribution, 5,000 samples weregenerated from the posterior predictive distribution. At each time t, the median0.025 and 0.975 quantiles were obtained to form a posterior predictive interval.Figure below shows the model fits for Active Infections, Recovered and Deathswith posterior predictive bands as well as a hold out sample of 10 days. Noticethat the model does quite well at fitting the dynamics of the Active Infectionsincluding the jump at day 12 and captures the plateau and the exponential growthafter the plateau as well. The Recovered model fits well as does the Deaths data.To assess the explained variance, a pseudo-R2 was formed using the medianfrom the posterior predictive distribution at each time as the point estimates.This resulted in a pseudo-R2 of 0.998 which indicates the fitted model explainsapproximately 99.8% of the variance in the data. Based on this, the model isdeemed to fit well.Fig. 2: Covid-19 Infected, Recovered and Deaths for Qatar.DiscussionThis work has demonstrated how to build a SEIRD model for the Covid-19 out-break in the State of Qatar, include interventions, estimate model parametersand generate posterior predictive intervals using a Bayesian framework. Themodeling paradigm is quite flexible for modeling the Covid-19 data as it easilyincorporates interventions into the system and can quantify the impact of theintervention. Future work could be to add an overdispersion parameter into themodel to allow for the more accurate capture of uncertainty. Furthermore, onecan perform simulation studies to better understand how the model may performunder various scenarios. Feature selection methods could be employed to selectwhere the interventions should be placed as well as other forms of interventionscould be included in the model. To address any deviations from the standardmodel, a semi-parametric technique could be studied as well.

SEIRD Model for Qatar COVID-19 Outbreak: A Case Study

https://ir.library.illinoisstate.edu/cgi/viewcontent.cgi?article=1480&amp;context=beer

SEIRD Model for Qatar COVID-19 Outbreak: A Case Study

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