Uncertainty quanti cation is an important procedure when dealing with errors and discrepancies that are present in any modeling e ort. This work presents a consistent uncertainty quanti cation framework for an epidemiological dynamical system, which is able to construct robust descriptions given a calibrated model. Since arbitrary choices of distributions for the input parameters can provide biased estimates and results, the maximum entropy principle is employed in the construction of the stochastic model to infer the most possibly unbiased probability density functions a ected by the lack of information. The framework is applied on a SEIR-SEI compartmental system for the Brazilian Zika virus outbreak to study a stochastic scenario. Keywords : Zika Virus, nonlinear dynamics, uncertainty quanti cation, maximum entropy principle, epidemic model. 1. Introduction A primordial step in the application of mathematical models for the study of natural and arti cial phenomena is the validation of such models through several calibration process with independent data sets. Unfortu- nately, that process can be poorly in uenced by data that underestimates the reality or due to inaccurate values of the parameters: small variations on the inputs, in some cases, can bring about huge changes in the model response. Propagation of uncertainties introduced in the parameters allows one to observe if the model provides robust results when confronted with multiple sources of errors and discrepancies [1, 2]. This work presents an uncertainty quanti cation (UQ) framework, and showcase its use for an epidemiological compartmental model previously 1eber.paiva@uerj.br 2michel.tosin@uerj.br 3americo@ime.uerj.br91 Received on Feb, 2019 / Accepted on Apr, 2019 doi: 10.6062/jcis.2019.10.02.0163 (Free PDF) Dantas et al. calibrated to describe the outbreak of the Zika virus in Brazil [3]. In order to do a re ned study while considering the more in uential parameters for the response, uncertainties are propagated only for the more sensible inputs, which were determined and analyzed in another previous work [4]. The lack of knowledge regarding the input distribution introduces bias when naive choices are made. The maximum entropy principle [2] is detailed on the application of selecting the parameter's distribution consistently, picking the less biased one available. 2. Computational model The SEIR-SEI nonlinear dynamical system outlined in Figure 1 was pre- viously calibrated for the Brazilian context of Zika infection with values from the literature [3]. This compartmental model divides the human and vector populations into four groups according to their current state (time t) in relation to the Zika fever: susceptible for infection, S(t); exposed to the virus but unable to transmit it, E(t); infectious, I(t), which are infected and capable of transmitting the disease; and recovered humans, Rh(t). Demo- graphic changes are only considered for the vector, with both populations constant. vh h h h v vγ (βvIh)/Nh αv(βhIv)/Nv αh Human population Vector populationδNv δδ δ Figure 1: Diagram of the SEIR-SEI system dynamics. The transmission mechanism embedded in the continuous-time dynam- ical system is the cross-infection, as expressed in the system of di erential equations92 doi: 10.6062/jcis.2019.10.02.0163 (Free PDF) Dantas et al. dSh dt= hShIv Nv;dSv dt=Nv vSvIh NhSv; dEh dt= hShIv Nv hEh;dEv dt= vSvIh Nh( v+)Ev; dIh dt= hEh Ih;dIv dt= vEvIv; dRh dt= Ih;dC dt= hEh;(1) where 1= represents the incubation period, 1 =is the vector lifespan, 1 = the human infection period, hand vthe transmission rates. The total human population is Nh= 206106, the Brazilian population in 2016, and Nv= 1 since the vector population is treated in proportions. The cumulative number of infectious is collected for each time on C(t), in the last equation. The vector of stated obtained from Eqs.(1) reports how humans and vectors are segmented for each time t. The quantities of interest (QoI) are the cumulative number of infectious and the new cases per epidemiological week (EW), computed from C(t) =Zt t0 hEhdt; N=CwCw1;N1=C1; w = 2;;52;(2) whereCwis the value of Cin thew-th EW. The QoI in this way are quanti- ties associated to the epidemiological concepts of prevalence and incidence, which can be compared to real data about the Brazilian Zika infection of 2016 assembled by the Ministry of Health [5]. The computational model uses the input parameters ( h; h; ; v; v; ) in Eqs.(1) and then obtain the QoIs through Eqs.(2) to output C(t) andNw.93 doi: 10.6062/jcis.2019.10.02.0163 (Free PDF) Dantas et al. 3. Uncertainty Quanti cation The dynamical system from the previous section provides QoI that can be seen as a random quantity due to the variability (randomness) in the system parameters. In this context, the mathematical model de ned by Eqs.(1) and Eqs.(2) can be thought abstractly as an operator M, which maps an input (random) vector Xthat lumps the system random parameters, into an output (random) scalar Y, representing a generic QoI, i.e., Y=M(X): (3) To deal with the uncertainties of the problem, a consistent probabilistic model must be constructed. In general, for lack of information, the joint- distribution of Xcan not be reliably speci ed via non-parametric statistics. In this way, a conservative procedure aims to estimate the least biased dis- tribution, while using the few information known about it. The maximum entropy principle (MaxEnt) is formalized over this premise. The joint-PDF ofX, denoted by pX(x), is obtained by maximizing the Shannon Entropy E(pX) =Z RpX(x) lnpX(x) dx; (4) with the restrictions Z Rg(x)pX(x) dx=b; (5) whereg(x) = 1;g1(x);g2(x);:::;g u(x) is aRu+1-valued function de ned on the support IX, that enforces certain statistical properties b2Ru+1 along with the PDF normalization condition. The general solution for this optimization problem can be written with Lagrange multipliers [2] as pX(x) =1IXexp <;g(x)> ; (6) withbeing theu-dimensional vector of Lagrange multipliers and <;> denoting the dot product. The multipliers are found by solving the nonlin- ear system of equations that appear when substituting Eq.(6) in Eq.(5) to guarantee b[2]. In terms of stochastic calculations, once the input random vector Xdis- tribution is known, the distribution of output Yis determined a posteriori , via Monte Carlo simulation [1, 2].94 doi: 10.6062/jcis.2019.10.02.0163 (Free PDF) Dantas et al. 4. Results Global sensitivity measures developed in a parallel work of the authors [4] suggest that handare the most important inputs when studying how the system response around EW 7 (peak of the outbreak). A stochastic scenario is investigated in Figure 2, where 95%-con dence bands are presented for the system QoI when introducing dispersion in the MaxEnt probability density function. For the construction of this PDF, the parameter's support bounds came from [3], and the mean value was considered the set that provided the calibrated response in [4]. A coecient of variation ( CV) of 10% for hand 15% foris assumed to observe its e ect on the output. 1020304050 time (weeks) 0.0 1.5 3.0 4.5 6.0number of people105 1020304050 time (weeks) 0.0 0.8 1.5 2.3 3.0number of people104 95% prob. mean nominal data Figure 2: The 95%-con dence band for the cumulative number C(t) (left) and new casesNw(right). The graphs in Figure 2 allow to contemplate the robustness of the model in comparison to the epidemic data, showing how the system response keeps its general shape and follows the outbreak evolution around the peak infec- tion until the end, even under some variability in the parameters. Another information regards the initial time, where it is apparent that these param- eters have little control in the QoI at early weeks, due to small width of the con dence bands. This procedure exempli es how the UQ-framework can be employed to study di erent kinds of uncertainties subjected over the model and made in a mathematical consistent manner.95 doi: 10.6062/jcis.2019.10.02.0163 (Free PDF) Dantas et al. 5. Concluding remarks An uncertainty quanti cation framework that uses the the Maximum Entropy Principle to estimate the parameter's distribution was used, evalu- ating the robustness of a SEIR-SEI calibrated model for the context of the 2016 Zika infection in Brazil. A stochastic scenario were tested, estimating the most unbiased input distributions, promoting propagation of uncertain- ties through the system and calculating con dence interval for the response. In future works, the authors intend to apply the uncertainty quanti cation framework in more complex models and also increment the analysis by em- ploying Bayesian updating. Acknowledgments . The authors acknowledge the support given to this research by the funding agencies Carlos Chagas Filho Research Foundation of Rio de Janeiro State (FAPERJ) under grants E-26/010.002.178/2015 and E-26/010.000.805/2018, and Coordena c~ ao de Aperfei coamento de Pessoal de N vel Superior - Brasil (CAPES) - Finance Code 001.