Sanz, MT.; Caselles, A.; Micó, JC.; Soler, C. (2020). A System Dynamics model to predict the impact of COVID-19 in Spain. 146-151. http://hdl.handle.net/10251/178213S14615

Caselles, A.

Micó, Joan C.

Sanz, M. T.

Soler, C.

English

Sanz, MT.; Caselles, A.; Micó, JC.; Soler, C. (2020). A System Dynamics model to predict the impact of COVID-19 in Spain. 146-151. http://hdl.handle.net/10251/17821314615

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A System Dynamics model to predict the impact ofCOVID-19 in SpainM.T. Sanz ♭1, A. Caselles ♮, J. C. Micó♯ and C. Soler♯♯(♭) Departament de Didàctica de la Matemàtica,Universitat de València, Av. dels Tarongers, 4, 46022 València, Spain,(♮) IASCYS member, Departament de Matemàtica Aplicada,Universitat de València (retired), Dr. Moliner 50, 46100 Burjassot, Spain,(♯)Institut Universitari de Matemàtica Multidisciplinar,Universitat Politècnica de Valéncia, Camı́ de Vera s/n, 46022 Valéncia, Spain,(♯♯)Institut Valencià de Cultura,Generalitat Valenciana, Poĺıgon Accés Ademús s/n, 46100 Burjassot, Spain.1 IntroductionOn 21 January 2020, the World Health Organization published its first report about the situ-ation of the world with the Novel Coronavirus (2019-nCoV), a new human infectious disease.A total number of confirmed cases in that day was 282 and this disease was only present inChina, Japan, Republic of Korea and Thailand. Nowadays, this organization has publishedits 94 report and it reflects that COVID-19 is around the world (Figure 1), the actual casesnumber is 2544792 and there are 175694 deaths.Figure 1: Cases of COVID-19 on 23 April 2020. Obtained from https://www.who.int1e-mail: m.teresa.sanz@uv.es146Modelling for Engineering & Human Behaviour 2020The scientific community concentrates its efforts on obtaining predictive models that can bean instrument to help governments to establish sanitary and social measures [1] or to identifycases as fast as possible [2]. In Spain, the Spanish Mathematics Committee calls on mathe-maticians and statisticians to participate in the construction of a meta-predictor to provide theauthorities with information on the short-term behaviour of important variables in the spreadof the COVID-19. The method uses the predictions from different models or algorithms thatare provided by the participating researchers (https://covid19.citic.udc.es/), and buildoptimized combinations of these, disaggregated by Autonomous Communities. Thus, in orderto help our Government in this fight, the DinSisCovid19 group is created, the members of whichare the authors of this paper.2 The modelThe model presented in this research is an extension of the simplest model of the epidemicoutbreak evolution, the SIR [3]. According to this model, the population can be divided intothree different groups: a) susceptible (S), individuals without immunity to the infectious agentbut they can be infected if exposed to the infectious agent; b) infected (I), individuals whoare infected at any given time and can transmit the infection to susceptible population; andc) recovered (R), individuals who are immune to infection, and consequently the transmissiondoes not affect when they are in contact with other individuals. This model provides a coupledsystem of three differential equations for the cited variables.dS(t)dt= −β · S(t) · I(t), (1)dI(t)dt= β · S(t) · I(t) − γ · I(t), (2)dN(t)dt= γ · I(t). (3)Being β the transmission rate, γ the recovery rate and 1/γ the average recovery period ofthe epidemic. These equations have been generalized in this paper in order to increase theirpredictive power. First, the population of the territory, N(t), is considered. It is calculated asthe initial population minus the deaths flow (4),dR(t)dt= −kf ·E(t)τf+ S(t) · E(t) ·kaτa+ S(t) · R(t) ·kbτb, (4)where, kf is the deaths rate of the sick population (E(t)); τf is the sick population deathsdelay; and S(t) · E(t) · ka/τa+S(t) · R(t) · kb/τb, refers to those infected at each time step, i.e.,the infection flow. The latter expression is used to calculate the infected population I(t) thatis modified in (2) through (5).dI(t)dt= S(t) ·E(t) ·kaτa+ R(t) ·kbτb. (5)147Modelling for Engineering & Human Behaviour 2020The infected population come not only from the sick population but also from the recoveredpopulation (R(t)), and both interact with the susceptible population (S(t)) through the ratioof interaction rate and its corresponding delay, ka/τa y kb/τb, respectively. Regarding thesusceptible population, (1) is transformed into (6). In (1), a fixed initial susceptible populationis changed through the infection flow. In the present model, we start from total populationthat is becoming susceptible due to the susceptibility rate (kq). It is calculated as the newsusceptible population, kq · N(t), minus the infection flow.dS(t)dt= kq · N(t) − S(t) ·E(t) ·kaτa+ R(t) ·kbτb. (6)Regarding the recovered population, equation (2) becomes (7). It is defined as the recoveredrate, kr, multiplied by the sum of the infection flow and the sick population afected by therecovered population delay (τr).dR(t)dt= kr ·S(t) · E(t) ·kaτa+ S(t) · R(t) ·kbτb+E(t)τr. (7)The deaths flow allows calculating the total deaths of the system (8).dD(t)dt= kf ·S(t) · E(t) ·kaτa+ S(t) · R(t) ·kbτb+E(t)τf. (8)Finally, the sick population is calculated through the infected flow minus deaths and recov-ered flow.dE(t)dt=S(t) · (E(t) ·kaτa+ S(t) · R(t) ·kbτb−kf ·S(t) · E(t) ·kaτa+ S(t) · R(t) ·kbτb+E(t)τf−kr ·S(t) · E(t) ·kaτa+ S(t) · R(t) ·kbτb+E(t)τr. (9)It should be noted that two additional variables are added to the model: hospitalized (H)and ICU (U), both are included by the collaboration that our research group carries out withthe Spanish Mathematics Committee (CEMAT).dH(t)dt= kh ·S(t) · E(t) ·kaτa+ S(t) · R(t) ·kbτb+E(t)τh, (10)dU(t)dt= ku ·S(t) · E(t) ·kaτa+ S(t) · R(t) ·kbτb+E(t)τu. (11)Where kh is the hospitalization rate of the sick population and ku is the ICU rate of thehospitalized population. Moreover, τh and τu represent the delay of the flow of both variableswith respect to the sick and hospitalized population, respectively.148Modelling for Engineering & Human Behaviour 20203 Results and discussionFor the calibration of the model, the genetic algorithm (hereinafter GA) implemented in Sigem[1] is used. With this tool, rates and delays are variables that the model calculates from anassumed initial data defined on the chromosome. The genetic algorithm is provided with a min-imum and maximum oscillation of these variables, as well as a percentage that corresponds tothe corresponding search window. In addition, the sum of the absolute values of the differencesbetween the infected, deceased, recovered, hospitalized and admitted to the ICU calculatedby the model and the actual trend of each variable is defined as an objective variable to beminimized by the GA. The aforementioned real trend is temporarily fitted through Gaussiancombinations.These adjustments are the ones that allow extrapolations of the variables contained inthe chromosome to be able to make future predictions. The real data are obtained fromhttps://covid19.isciii.es/ and the time period here studied corresponds to the one fromJanuray 31 to April 22, 2020.Figure 2 shows the validation of the model’s deterministic formulation through the fittingto reality degree of infected, deceased, recovered, hospitalized, admitted to the ICU and sickpopulation. The results show a great reliability of the model, being the determination coeffi-cients higher than 0.99, a good graphic overlap betweeen real and simulated data, and a goodrelative error, that in all cases, is less than 5%.(a) (b) (c)(d) (e) (f)Figure 2: Validation of the model’s deterministic formulation in the period from January 31 toApril 22, 2020. a) Infected population. R2= 0.999877; b) Deceased population. R2= 0.999779;c) Recovered population. R2= 0.999709; d) Hospitalized population. R2= 0.998582; e) ICUpopulation. R2= 0.998089; f) Sick population. R2= 0.99972. Real data: dots. GA simulateddata: line.The application of the model has been the prediction of the infected flow, deceased flow,recovered flow, hospitalized flow, admitted to the ICU flow, as well as the sicking flow in theperiod from April 23 to May 14, 2020. For this, the chromosome variables are exptrapolated149Modelling for Engineering & Human Behaviour 2020into the mentioned period, and they become input variables of the model designed for thiswork.Figure 3 presents past and future evolution of the first five previously mentioned variables.It can be seen that by the measure of confinement introduced by the Government, the infectionflow has been reduced. However, with the application of massive tests to asymptomatic people,a slight rise in the number of registered cases is expected.Figure 3: Prediction in a trend scenario in the period April 23 to May 14, 2020. Real data:points. Simulated data: line.Regarding the sicking flow (Figure 4), it can be seen that for May 14 the trend continuesgrowing, being observed this increase since April 20, that is the date when the relaxation of thecontainment measures for certain production sectors in the territory took place. In addition, wewant to emphasize that the slope of the forecasted curve is the same as the one that occurredin the last period from March 11 to 31, approximately.Figure 4: Prediction in a trend scenario of the sicking flow in the period from April 23 to May14, 2020. Real data: blue blades. Simulated data: orange line.4 Conclusions and future workIn this report we have analyzed the epidemic data made available to the scientific communityby the Gobierno de España (https://covid19.isciii.es/) and referred to the 31/01/2020 −150Modelling for Engineering & Human Behaviour 202023/04/2020 period. Our results suggest that the containment measures taken by the authoritieshave reduced the spread of COVID-19 among people in Spain. But the introduction of new rapidtests together with the PCR tests that were carried out from the beginning of the epidemic,have led to an increase in detected cases since April 14. This fact does not indicate an increasein new cases, but rather reflects the actual viral reality that our country had. Likewise, thede-escalation process accelerated by the fear to economic collapse from April 20 also causes anincrease in cases. The future work on this model is chalenger. On the one hand, the presentmodel will be compared with models already published by the scientific community. On theother hand, its application to different countries of the world must be done, so that its validitybe ratified. Regarding the improvement of the model, the construction of the correspondingstochastic model formulation will allow giving predictions with a confidence interval. Likewise,the definition of the same by using cohorts and sexes could enable us to evaluate the differentkinds of population and to develop specific measures in both directions. Finally, a possible studyof the past evolution of the parameters included in the cromosome (detected in the calibrationprocess) could teach something about how to influence the future evolucion of the pandemia inSpain.References[1] Fanelli, D., Piazza, F., Analysis and forecast of COVID-19 spreading in China, Italy andFrance Chaos, Solitons & Fractals, Volume(134):1–5, 2020.[2] Rao, A., Vazquez, J., Identification of COVID-19 Can be Quicker through Artifi-cial Intelligence framework using a Mobile Phone-Based Survey in the Populationswhen Cities/Towns Are Under Quarantine Infection Control & Hospital Epidemiology,Volume(41):1–18, 2020.[3] Kermack, W., McKendrick, A. G., Walker, G., A contribution to the mathematical theoryof epidemics Proceedings of the Royal Society, Volume(A115):700–721, 1927.[4] Caselles, A., SIGEM: a realistic models generator expert system. In Trappl (Ed.) Cyber-netics and Systems ’88 (pp. 101-108) Dordrecht, Kluwer Academic Publishers, 1988.151

A System Dynamics model to predict the impact of COVID-19 in Spain

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A System Dynamics model to predict the impact of COVID-19 in Spain

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