An Epidemic Model SIPHERD and its application for prediction of the spread of COVID-19 infection for India and USA
AAn Epidemic Model SIPHERD and its application for prediction of the spread ofCOVID-19 infection for India and USA
Ashutosh Mahajan
Centre for Nanotechnology Research, Vellore Institute of Technology, Vellore-632 014, India ∗ Ravi Solanki
Centre for VLSI and Nanotechnology, Visvesvaraya National Institute of Technology, Nagpur-440 010, India
Namitha A. S.
School of Electronics Engineering, Vellore Institute of Technology, Vellore-632 014, India (Dated: May 13, 2020)We propose an epidemic model SIPHERD in which three categories of infection carriers Symp-tomatic, Purely Asymptomatic, and Exposed are considered with different rates of transmission ofinfection that are taken dependent on the lockdown and social distancing. The rate of detectionof the infected carriers is taken dependent on the tests done per day. The model is applied forthe COVID outbreak in Germany and South Korea to validate its predictive capabilities and thenapplied to India and the United States for the prediction of its spread with different lockdownsituations and testing in the coming months.
I. INTRODUCTION
The outbreak of pandemic Coronavirus disease 2019(COVID-19) has led to more than 4 million total infec-tions and 285 thousand deaths worldwide [1], and seriousefforts are needed for its containment. The CoronavirusSARS-CoV-2 has affected not just the public health butmade a drastic impact on the economy of the world aswell due to the lockdown situations in many countries.Pandemics have hit humanity many times in the pastas well, and mathematical models are already availablefor infectious diseases. Modeling and simulation can helpto predict the extent of the contagious disease and cangive useful inputs on correction measures for its contain-ment. In order to devise the lockdown strategy, it isimportant that the prediction of the disease spread isavailable to the decision-makers. COVID-19 is differentfrom the previously known SARS (Severe acute respira-tory syndrome) infection, such as the existence of purelyasymptomatic cases [2] and the spread of the infectionfrom them as well as from the exposed ones in the in-cubation period [3]. Our proposed mathematical model,SIPHERD incorporates the above facts for the COVID-19 epidemic.Many epidemiological models exist in the literature,and the basic SIR model [4] is the widely used one,which needs to be modified to incorporate the complexi-ties involved in Coronavirus spread and control. An ap-proximate mathematical model of the COVID-19 is ini-tially reported in the literature [5] based on the Between-Countries Disease Spread (Be-CoDiS), which is a spatialepidemiological model for the study of the spread of hu-man diseases between and within the countries. ∗ [email protected] An improved mathematical model for the spread ofCOVID-19 is proposed in [6], by taking into account theinfected and undetected cases. But this study and fore-cast is particularly based only on China.An extended SIR model is proposed in [7], in which theentire people in the country are divided into eight com-partments. Though it is an improved version of the SIRmodel, the study and simulation results are done only forItaly, and the model does not take into account purelyAsymptomatic cases and the role of tests done per day.Another compartmental epidemic model SEIR [8] fore-cast for few countries and the impact of the quarantineon the COVID-19 is investigated. A better adaptive andimproved version of the SIR model is illustrated in [9].In this method, the time dependency of some parametersused for the analysis makes it more robust than the con-ventional SIR method. Some other curve fitting basedmethods are also available in the literature for the fore-cast of COVID-19 in [10], [11] and [12]. Although thesemethods can track the available data correctly, they arenot developed based on the physical insights that affectthe rate of spreading of the disease and also it is ex-tremely sensitive to the initial conditions.In this paper, we formulate the mathematical modelSIPHERD for the COVID-19 epidemic and apply it forforecasting the number of active cases, confirmed cases,daily new cases, and deaths for India and USA, depend-ing on the lockdown strategy and the number of testsperformed per day.
II. MODEL
We model the evolution of the COVID-19 disease bydividing the population into different categories as listedbelow. As seen in FIG. 1, the rates of transfer fromone category to another can are the model parameters a r X i v : . [ q - b i o . P E ] M a y FIG. 1. SIPHERD Model and a set of differential equations for the entity in eachcategory can be formed. We write the model equationsthat are independent of the population of the country byconsidering the fraction of the people in each category. • S - fraction of the total population that is healthyand has never caught the infection • E - fraction of the total population that is exposedto infection, transmit the infection and turn intoeither Symptomatic or purely Asymptomatic, andnot detected • I - fraction of the total population infected by thevirus that shows symptoms and undetected • P - fraction of the total population infected by thevirus that doesn’t show symptoms even after theincubation period and undetected. These are thepurely Asymptomatic cases • H - fraction of the total population that are foundpositive in the test and either hospitalized or quar-antined • R - fraction of the total population that has recov-ered from the infection • D - fraction of the total population that are extinctdue to the infection.The SIPHERD model equations are a set of coupledordinary differential equations (1 to 7) for the defined en-tities (S,I,P,H,E,R,D), where initial conditions E(0), P(0)and I(0) are not exactly known. The various rates listedin TABLE I are the parameters of the problem whichare also not known, and only possible range is available.Some of the parameters such as rates of infection ( α , β , γ ) change with time in steps, depending on the conditionssuch as lockdown. dSdt = − S ( αE + βI + γP + δH ) (1) dEdt = S ( αE + βI + γP + δH ) − ( µ + ξ I + ξ P ) E (2) dHdt = µ ( E + P ) + νI − σH ( t − t R ) − τ H ( t − t D ) (3) dIdt = ξ I E − ( ν + ω ) I (4) dPdt = ξ P E − ( µ + η ) P (5) dRdt = ωI + ηP + σH ( t − t R ) (6) dDdt = τ H ( t − t D ) (7)where, t R and t D are the delay associated with the re-covery and death respectively with respect to active cases H . We have taken into account this delay because theactive cases are reported after the testing and admissionto healthcare or quarantine center and the number of re-covery and death of the admitted will not immediatelyfollow the active or H category number. All fractionsadd up to unity that can also be seen from summing theabove equations. ddt ( S + I + P + H + E + R + D ) = 0 (8) TABLE I.Parameter Description α Rate of transmission of infection from E to S β Rate of transmission of infection from I to S γ Rate of transmission of infection from P to S δ Rate of transmission of infection from H to S ξ I,P
Rate of conversion from the E to I,P ω Rate of home recovery of H η Rate of home recovery of P µ Probability of E and P being detected ν Probability of I being detected
The probability of getting the infection is assumed uni-form among the susceptible people, although the diseasespreads localised in hot-spots.The asymptomatic proportion of the infected personsonboard the Diamond Princess cruise ship is estimatedin [2]. Among the 634 tested positive onboard, 328 werefound asymptomatic i.e., more than 50 percent of theconfirmed cases were not showing any specific symptomsof COVID-19. This factor is incorporated in the modelby considering the purely Asymptomatic category. Theratio of purely Asymptomatic (P) to total Asymptomatic(E+P) cases is reported to be 0.35 and the ratio of purelyAsymptomatic to the total infected (E+P+I) is 0.179 [2].These reported numbers are used to fix the proportionbetween ξ P and ξ I as 0.36 and the proportion of initialconditions E (0), I (0) and P (0) as well. In other words,out of 100 exposed cases, after the incubation period, 36will turn to be purely asymptomatic, and 74 will havesymptoms.Coronavirus-nCoV2 has shown particular characteris-tics that the asymptomatic patients do transmit the dis-ease. The infection can be transmitted from the personwho is not showing illness during the incubation period[3]. This can be included in the model by considering E category people and their transmission as well. Hospital-ized and quarantined cases can also transmit the disease,and this small rate is taken as parameter δ .The detection of the Asymptomatic and Symptomaticcases can be taken dependent on the number of tests doneper day ( T P D ). For the Symptomatic cases, the detectionis more probable as the infected person can approach forthe tests and more likely to be tested. The detection ofSymptomatic is taken in two parts, one a constant andanother part proportional to the tests done per day. Thiscan be written in terms of parameters as, ν = ν + ν T P D (9) µ = µ T P D (10)where, µ , ν , and ν are positive constants. Recoveryof Asymptomatic cases is taken faster than the Symp-tomatic cases. The total confirmed cases are the addi-tion of the active cases, extinct cases, and a part of therecovered that were detected. This can be written as T c ( t ) = H ( t ) + D ( t ) + (cid:90) t σH ( τ (cid:48) ) dτ (cid:48) (11) III. NUMERICAL IMPLEMENTATION ANDSIMULATION
The set of coupled ordinary differential equations forthe model can be readily solved numerically for a givenset of parameters and initial values. The non-trivial partis the accurate determination of the parameters that willmimic the situation on the ground. The mathematicalproblem is to take into account the four actual data setsof the total number of confirmed cases, active cases ona particular day, cumulative deaths and tests done perday, and find the set of parameters that will provide thebest possible match between the data and model. Theextraction of the parameters is also to be automated sothat the model can be run on data for various countries.A cost function is written in terms of errors between theactual and solver data sets. The minimizer of the cost canbe found to obtain the optimized set of parameters thatbest fit with the data available till date. The model andthe optimization scheme is implemented in MATLAB.
FIG. 2. Daily Tests per day data for different countries ob-tained from [14]
IV. RESULTS AND DISCUSSION
We collected the number of total positive or confirmedcases, present active cases and deaths from [1], [13], andthe number of tests per day from [14], which is plottedin FIG. 2. The day on which lockdown is imposed in acountry is also taken into account as changes in the slopesof the data are observed according to it. The rate oftransmission of infection from the Asymptomatic carrier( α , γ ) for a country is taken higher than the Symptomaticones ( β ) as the Asymptomatic carrier may not be awareof his/her infection, and Susceptible may not be keepingdistance as no symptoms are seen. The mortality rate( τ ) is taken different for different countries as it dependson the immunity and how effectively the critical patientsare taken care of by the hospitals.The home recovery rates of Asymptomatic ( η ) andSymptomatic ( ω ), rate of transmission of infection fromhospitalized and quarantined ( δ ), and rate of self-reporting of the Symptomatic people ( ν ) are taken uni-form for all countries. The transfer rate ξ I is the inverseof the incubation period, whose mean is reported 5.2 days[15]. The parameters determined by our model are listedin TABLE II for the countries we studied. A. Model Validation
We apply the SIPHERD model to South Korea andGermany for exhibiting the predictive capability of ourmodel as the disease has almost reached the end stages inthese countries. We used the data for the first 23 and 40days, respectively, for these counties i.e., till March 9 th and March 30 th and compared the future evolution gen-erated by the model to the actual data as shown in thegrey region in FIG. 3 for South Korea and in FIG.4 forGermany. Model predicted higher deaths for South Ko-rea as the mortality rate 2.2E-3 was higher before March FIG. 3. Model prediction using the South Korea data up-toMarch 9 and comparison with the actual data for the con-firmed, active cases and total deaths.
FIG. 4. Model prediction using the Germany data up-toMarch 30 and comparison with the actual data for the con-firmed, active cases and total deaths.
B. Predictions for India
For the available data till date, we run the model toextract parameters, and then with the extracted param-eters, the model is run for 180 days starting from March2 nd . If the lockdown conditions are relaxed on May 17 th ,the rate of transmission of infection is going to increase.In the relaxed lockdown, the α and β values are assumedto jump by 20%. The prediction for both cases, with a4k increase in tests per day and saturation at 200k tests,is compared in FIG. 5 and in FIG. 6, we plot the predic-tion band for the daily new cases considering two percenterror in the estimation of rate of transmission of infec-tion. We compare the effect of testing on the prediction FIG. 5. Comparison of the Model prediction for India for lock-down and relaxed lockdown conditions after May 17 th with4K increase in tests per day after May 11 th and saturating at200k. in FIG. 7. Total, Active and extinct cases are plotted forthe coming months if tests per day are increased by 4kand 8k per day after May 11 th and saturated at 200k and300k, respectively, taking into account the relaxation oflockdown after May 17 th . FIG. 6. Comparison of the daily new cases prediction for Indiafor lockdown conditions with 4K increase in tests per day byafter May 11 th and saturating at 200K. Shaded area showsthe prediction band for 2 percent error in α , β estimation. C. Predictions for USA
The recovery rate of the H category is found to beslow compared to South Korea or Germany, which maybe attributed to either incorrect reporting of the Activecases or the testing of serious cases only and longer recov-ery time in hospitals compared to quarantined with mild FIG. 7. Comparison of the Model prediction for Confirmed,Active and total Deaths Cases for India. After May 11 th testsare increased by 4k and 8k daily and saturated at 200K and300K respectively, with lockdown relaxed after May 17 th . symptoms. The prediction for the next 240 days, thatis till the end of the year 2020, is plotted in FIG. 8 for10k test per day and saturated at 1 million tests per day.The recovery rate σ is taken improved by 25 % after May4 th and mortality rate is taken improved in steps frominitial 6E-3 to 4E-3 to 2.5E-3 after April 9 th and April19 th . The time evolution of the totally unknown and un-detected part of the infected for USA is plotted in FIG.9. The daily new positive cases data and the predictionare plotted in FIG. 10. TABLE II. Parameters values for the Countries studied
Para.
Germany S. Korea India USAPopulation N 8.30E7 5.10E7 1.38E9 3.31E8 T T LD
22 21 21,5 45 α (bf. LD) 0.32 0.39 0.33 0.33 α (af. LD) 0.30,0.2 0.085 0.18,0.23 0.13 β (bf. LD) 0.29,0.11 0.22 0.1 0.26 β (af. LD) 0.24 0.074 0.05,0.18 0.20 γ (bf. LD) 0.32 0.39 0.33 0.33 γ (af. LD) 0.29 0.085 0.18 0.13 δ ξ I ξ P µ ν ν / µ ω η σ τ t R t D V. CONCLUSION
SIPHERD model is developed by considering purelyAsymptomatic category of COVID-19 infected cases inaddition to the Symptomatic, and the disease spread bythe exposed. The effect of lockdown on the rates of trans-mission of infection and the influence of tests per day ondetection rates has been incorporated in the model. TheSIPHERD model is put for trial for the data of SouthKorea and Germany, and with a limited number of daysdata, the model is found to correctly predict the knownevolution. The prediction for India suggests that evenincreasing the rate of infection transmission by 20% dueto relaxation of lockdown leads to around 50k increase inthe total number of cases and 3k increase in total deaths.The prediction for the USA shows that in the absence ofvaccine the infection can last long till the end of this year
FIG. 10. Comparison of the Model prediction for daily newcases for the USA with the increase of the 10k tests per daywith lockdown. and number of deaths could be around 250k if lockdownand social distancing conditions remain the same.
Acknowledgement
AM would like to thank Dr. Shrikant Ambalkar, M.Dfor helpful discussions. [1] .[2] K. Mizumoto, K. Kagaya, A. Zarebski, and G. Chow-ell, “Estimating the asymptomatic proportion of coron-avirus disease 2019 (covid-19) cases on board the dia-mond princess cruise ship, yokohama, japan, 2020,”
Eu-rosurveillance , vol. 25, no. 10, p. 2000180, 2020.[3] C. Rothe, M. Schunk, P. Sothmann, G. Bretzel,G. Froeschl, C. Wallrauch, T. Zimmer, V. Thiel,C. Janke, W. Guggemos et al. , “Transmission of 2019-ncov infection from an asymptomatic contact in ger-many,”
New England Journal of Medicine , vol. 382,no. 10, pp. 970–971, 2020.[4] W. O. Kermack and A. G. McKendrick, “A contributionto the mathematical theory of epidemics,”
Proceedings ofthe royal society of london. Series A, Containing papersof a mathematical and physical character , vol. 115, no.772, pp. 700–721, 1927.[5] B. Ivorra and A. M. Ramos, “Application of the be-codismathematical model to forecast the international spreadof the 2019–20 wuhan coronavirus outbreak,”
Research-Gate http://dx. doi. org/10.13140/RG. 2.2 , vol. 31460,2020.[6] B. Ivorra, M. Ferr´andez, M. Vela-P´erez, and A. Ramos,“Mathematical modeling of the spread of the coronavirusdisease 2019 (covid-19) considering its particular charac-teristics. the case of china,” Technical report, MOMAT,03 2020, Tech. Rep., 2020.[7] G. Giordano, F. Blanchini, R. Bruno, P. Colaneri,A. Di Filippo, A. Di Matteo, M. Colaneri et al. , “Asidarthe model of covid-19 epidemic in italy,” arXiv preprint arXiv:2003.09861 , 2020.[8] D. Efimov and R. Ushirobira, “On an interval predictionof covid-19 development based on a seir epidemic model,”2020.[9] Y.-C. Chen, P.-E. Lu, C.-S. Chang, and T. Liu, “A time-dependent sir model for covid-19 with undetectable in-fected persons,” arXiv preprint arXiv:2003.00122 , 2020.[10] S. Zhao, Q. Lin, J. Ran, S. S. Musa, G. Yang, W. Wang,Y. Lou, D. Gao, L. Yang, D. He et al. , “Preliminaryestimation of the basic reproduction number of novelcoronavirus (2019-ncov) in china, from 2019 to 2020: Adata-driven analysis in the early phase of the outbreak,”
International journal of infectious diseases , vol. 92, pp.214–217, 2020.[11] T. Zeng, Y. Zhang, Z. Li, X. Liu, and B. Qiu, “Predic-tions of 2019-ncov transmission ending via comprehensivemethods,” arXiv preprint arXiv:2002.04945 , 2020.[12] Z. Hu, Q. Ge, L. Jin, and M. Xiong, “Artificial intelli-gence forecasting of covid-19 in china,” arXiv preprintarXiv:2002.07112 , 2020.[13] .[14] https://ourworldindata.org/grapher/full-list-covid-19-tests-per-day .[15] J. Zhang, M. Litvinova, W. Wang, Y. Wang, X. Deng,X. Chen, M. Li, W. Zheng, L. Yi, X. Chen et al. , “Evolv-ing epidemiology and transmission dynamics of coron-avirus disease 2019 outside hubei province, china: a de-scriptive and modelling study,”