An epidemiological model for the spread of COVID-19: A South African case study
AAn epidemiological model for the spread of COVID-19: A South Africancase study
Laurentz E. Olivier a,b , Ian K. Craig b, ∗ a Moyo Africa, Centurion, South Africa. b University of Pretoria, Pretoria, 0002, South Africa.
Abstract
An epidemiological model is developed for the spread of COVID-19 in South Africa. A variant of the classicalcompartmental SEIR model, called the SEIQRDP model, is used. As South Africa is still in the early phasesof the global COVID-19 pandemic with the confirmed infectious cases not having peaked, the SEIQRDPmodel is first parameterized on data for Germany, Italy, and South Korea – countries for which the numberof infectious cases are well past their peaks. Good fits are achieved with reasonable predictions of where thenumber of COVID-19 confirmed cases, deaths, and recovered cases will end up and by when. South Africandata for the period from 23 March to 8 May 2020 is then used to obtain SEIQRDP model parameters. Itis found that the model fits the initial disease progression well, but that the long-term predictive capabilityof the model is rather poor. The South African SEIQRDP model is subsequently recalculated with thebasic reproduction number constrained to reported values. The resulting model fits the data well, andlong-term predictions appear to be reasonable. The South African SEIQRDP model predicts that the peakin the number of confirmed infectious individuals will occur at the end of October 2020, and that the totalnumber of deaths will range from about 10,000 to 90,000, with a nominal value of about 22,000. All ofthese predictions are heavily dependent on the disease control measures in place, and the adherence to thesemeasures. These predictions are further shown to be particularly sensitive to parameters used to determinethe basic reproduction number. The future aim is to use a feedback control approach together with the SouthAfrican SEIQRDP model to determine the epidemiological impact of varying lockdown levels proposed bythe South African Government.
Keywords:
Compartmental model, COVID-19, Epidemiology, SARS-CoV-2, SEIQRDP model.
1. Introduction
A novel coronavirus emerged in Wuhan, China towards the end of 2019. This virus, which was sub-sequently named SARS-CoV-2 and the disease it causes COVID-19 (World Health Organization, 2020a),has since spread around the world. The WHO characterized COVID-19 as a pandemic on 11 March 2020(World Health Organization, 2020b). The COVID-19 pandemic first took hold in regions of the world thatshare high volumes of air traffic with China (Lau et al., 2020). The worst affected countries are the US,Britain, Italy, France, and Spain who have reported the highest number of COVID-19 deaths to date (TheEconomist, 2020). The importance of “flattening the curve”, i.e. reducing the number of COVID-19 infectedpatients needing critical care to be below the number of available beds in intensive care units, soon becameevident (Stewart et al., 2020).The South African National Institute for Communicable Diseases confirmed the first COVID-19 case inSouth Africa on 5 March 2020, a 38-year-old male who recently returned from Italy (The National Institute ∗ Corresponding author. Address: Department of Electrical, Electronic, and Computer Engineering, University of Pretoria,Pretoria, South Africa.Tel.: +27 12 420 2172; fax: +27 12 362 5000.
Preprint submitted to Elsevier May 25, 2020 a r X i v : . [ q - b i o . P E ] M a y or Communicable Diseases, 2020). Having learnt from elsewhere about the importance of “flattening thecurve”, the South African Government was quick to place the country under strict lockdown on 27 March2020 after only 1,170 confirmed COVID-19 cases and 1 related death (Humanitarian Data Exchange, 2020).The early strict lockdown measures have been successful in stemming the spread of the disease. By 8 May2020 the number of confirmed cases in South Africa were 8,895, a compound daily growth rate of about 5%since the start of the lockdown on 27 March 2020. In contrast, the confirmed COVID-19 cases in Italy wentfrom a similar base of 1,128 cases on 29 February 2020 to 9,172 cases on 9 March 2020, a compound dailygrowth rate of about 23%.The strict lockdown measures in South Africa have been successful from an epidemiological point ofview, but great harm has been done to an economy that was already weak before the COVID-19 pandemicstarted (Arndt et al., 2020). As a result, pressure is building to relax the lockdown measures (Harding,2020). There is thus significant interest in determining the epidemiological impact of the lockdown levelsproposed by the South African Government (South African Government, 2020b). Towards this end, anepidemiological model is developed for South Africa in this work.Epidemiological models are useful in that they can predict the progression of an epidemic, and enhanceunderstanding of the impact that infectious disease control measures, such as vaccination and quarantining,may have on reducing the level of infection in a population (Keeling and Rohani, 2008). A variant of theclassical compartmental epidemiological SEIR model (see e.g. Hethcote and den Driessche (1991)), calledthe SEIQRDP model, is used here to model the spread of COVID-19 in South Africa. This model wasintroduced by Peng et al. (2020) specifically to model the COVID-19 pandemic, and was subsequently usedto model the pandemic in Iraq (Al-Hussein and Tahir, 2020) and the USA (Xu et al., 2020). The model isparameterized from data available from The Humanitarian Data Exchange, compiled by the Johns HopkinsUniversity Center for Systems Science and Engineering (JHU CCSE). Available data include COVID-19confirmed cases, deaths, and recovered cases.South Africa is still in the early phases of the global COVID-19 pandemic compared to many countries inthe Northern Hemisphere. Importantly, confirmed infectious cases in South Africa have yet to peak, whereassuch cases are well past their peaks in countries such as Germany, Italy, and South Korea. The SEIQRDPmodel is therefore first parameterized on data for these three countries in order to test the efficacy of themodel. Good fits are achieved with reasonable predictions of where the number of COVID-19 confirmedcases, deaths, and recovered cases will end up and by when.South African data for the period from 23 March to 8 May 2020 is used to obtain SEIQRDP modelparameters. It is found that the model fits the initial disease progression well, but that the long-termpredictive capability of the model is rather poor. A similar scenario is observed for Germany when onlyinitial data, from 11 March to 28 March 2020, are used to determine where German cases will end up.The South African SEIQRDP model was subsequently recalculated with the basic reproduction numberconstrained to reported values (Lai et al., 2020). The resulting model still fits the data well, and long-term(into 2021) predictions appeared much more reasonable. The study concludes with a sensitivity analysisthat show that the model is particularly sensitive to parameters used to determine the basic reproductionnumber.The aim is to in future work use the South African SEIQRDP model to determine the epidemiolog-ical impact of the different lockdown levels proposed by the South African Government (South AfricanGovernment, 2020b), using a feedback control approach similar to Stewart et al. (2020).
2. SEIQRDP model
The SEIQRDP model is a generalized compartmental epidemiological model with 7 states. The modelwas proposed by Peng et al. (2020) and is an adaptation of the classical SEIR model (see e.g. Hethcote andden Driessche (1991)). A numerical implementation of the SEIQRDP model in MATLAB is provided byCheynet (2020). The implementation used here is similar to Cheynet (2020), but model parameter limitsand some parameter equations are different as presented in the rest of this paper. The model states andmodel parameters that drive transitions between them are shown in Fig. 1. The colours used for Q, R, andD correspond to what is used in the results figures later in the article. The states are described as:2 gkl db
Susceptible (S)Insusceptible (P) Exposed (E)Infectious (I)Quarantined (Q)Recovered (R) Deceased (D)
Figure 1: SEIQRDP model. • S - Portion of the population still susceptible to getting infected, • E - Population exposed to the virus; they are infected but not yet infectious, • I - Infectious population; infectious but not yet confirmed infected, • Q - Population quarantined; confirmed infected, • R - Recovered, • D - Deceased, • P - Insusceptible population.The model equations are given as: dS ( t ) dt = − αS ( t ) − β ( t ) N S ( t ) I ( t ) (1) dE ( t ) dt = − γE ( t ) + β ( t ) N S ( t ) I ( t ) (2) dI ( t ) dt = γE ( t ) − δI ( t ) (3) dQ ( t ) dt = δI ( t ) − ( λ ( t ) + κ ( t )) Q ( t ) (4) dR ( t ) dt = λ ( t ) Q ( t ) (5) dD ( t ) dt = κ ( t ) Q ( t ) (6) dP ( t ) dt = αS ( t ) (7)where N is the total population size, α is the rate at which the population becomes insusceptible (in generaleither through vaccinations or medication). At present there is no vaccine that will allow an individual3o transfer from the susceptible to insusceptible portion of the population (Prompetchara et al., 2020).Consequently α should be considered to be close to zero. β ( t ) is the (possibly time dependent) transmissionrate parameter, γ = 1 /N lat is the inverse of the average length of the latency period before a person becomesinfectious (in days), δ = 1 /N inf is the inverse of the number of days that a person stays infectious without yetbeing diagnosed, λ ( t ) is the recovery rate, and κ ( t ) is the mortality rate. Both λ ( t ) and κ ( t ) are potentiallyfunctions of time, and Peng et al. (2020) notes that λ ( t ) gradually increases with time while κ ( t ) decreaseswith time. As such, the functions shown in (8) and (9) are used to model λ ( t ) and κ ( t ). In (8) it is set that λ ≥ λ such that λ ( t ) ≥ λ ( t ) = λ − λ exp( − λ t ) (8) κ ( t ) = κ exp( − κ t ) . (9) β is often considered to be constant, but is dependent on interventions like social distancing, restrictionson travel, and shutting down of some industries (South African Government, 2020a). This implies that β may also be time dependent. Given that overall restrictions have been made increasingly stringent to curbthe spread of the disease, β ( t ) is modelled using a decreasing function of time of the form β ( t ) = β + β exp( − β t ) . (10)The basic reproduction number, R , which is the expected number of cases directly generated by onecase in the population, is given by Peng et al. (2020) as R = β ( t ) δ (1 − α ) T , (11)where T is the number of days. When α ≈
0, this can be simplified as R ≈ β ( t ) δ . (12)Interventions such as social distancing, restrictions on population movement, wearing of masks (amongothers) can reduce the effective reproduction number mainly through reducing the effective number ofcontacts per person. The effective reproduction number found through modelling may therefore likely belower than what epidemiologists report; e.g. Lai et al. (2020) notes that the reproduction number forCOVID-19 is between 2.24 and 3.58.
3. Parameter estimation
Data are obtained from The Humanitarian Data Exchange , as compiled by the Johns Hopkins UniversityCenter for Systems Science and Engineering (JHU CCSE) from various sources. The data include the numberof confirmed infectious cases, recovered cases, and deceased cases per day from January 2020.In order to get a sense of the applicability of the model and what the parameter values should be,parameter estimations are first carried out to determine SEIQRDP models for Germany, Italy, and SouthKorea. These countries are selected as their outbreaks started earlier than that of South Africa, andconsequently their parameter estimation should be more accurate. They have also had differing approachesunder different circumstances, which means that the different parameters obtained should illustrate how themodel behaves.The parameter values obtained are shown in Table 1, along with the allowed low and high limits. Thelimits are important to ensure that the model not only provides a good fit to the data, but that the behaviouris kept within what is epidemiologically expected. Accessible from https://data.humdata.org/dataset/novel-coronavirus-2019-ncov-cases able 1: Model parameters by region. Param. Min Max Germany Italy South Korea South Africa α − × − − × − × − β R < .
58 0.037 0.343 0.196 0.250 β R < .
58 0.956 0.646 1.298 0.364 β R < .
58 0.087 0.133 0.264 0.003 γ δ λ λ λ κ κ ∞ . × . × . × . × R −∞ α should remainvery close to zero. Lai et al. (2020) notes that the basic reproduction number ( R ) is between 2.24 and 3.58.A dynamic constraint is therefore used to keep R ≈ β ( t ) δ < . γ is therefore limited to γ ∈ [0 . , .
5] to yield a latency period between 2 and 5 days.Lai et al. (2020) notes that the mean incubation period for COVID-19 is between 2.1 and 11.1 days.This implies that 2 . < γ + 1 δ < . < δ < .
1. The limits are set to δ ∈ [0 . ,
1] to yield value between 1 and 10 days.Given that λ and κ are daily rates, they are kept between 0 and 1 to ensure a proper fit to the “recovered”and “deceased” data. Al-Hussein and Tahir (2020) and Peng et al. (2020) found the majority of their λ and κ values to be between 0 and 0.1.Also included in Table 1 are two parameters to quantify the goodness-of-fit to the data. Firstly is themean squared error (MSE) defined by: M SE = 1 n n (cid:88) i =1 (cid:107) y i − f ( x i ) (cid:107) (14)where n is the number of data points, y i the output data, and f ( x i ) the output of the model evaluated atthe parameters x found during the fitting step. Also included is the coefficient of determination ( R ), whichindicates the proportion of the variance in the output data predictable using the independent variables. Germany, Italy, and South Korea have all successfully effected a decrease in the number of daily newcases at the time of writing. This implies enough shape in the curves to obtain a proper fit. The fit of themodel to the data for Germany is shown in Fig. 2, the fit for Italy is shown in Fig. 3, and the fit for SouthKorea is shown in Fig. 4.It is interesting to note from Fig. 2, Fig. 3, and Fig. 4 that the mortality rates for all countries are quitedifferent. The progression of the number of cases as well as the fraction of the population that is projectedto be infected in total is also quite different across these countries. The SEIQRDP model is however ableto fit all the data quite well. 5 ar Apr May Jun Jul Aug Sep Oct Nov Dec Jan
Date Mar 11 Mar 25 Apr 08 Apr 22 May 06 May 20 Jun 03
Date N u m be r o f c a s e s Figure 2: Data fit for Germany using available data between 11 March and 8 May 2020, without a prediction horizon (left) andwith a longer prediction horizon to show the final values (right). The figure shows Total cases in blue, Confirmed infectiouscases (Q) in orange, Recovered cases (R) in purple, and Deceased cases (D) in black.
Mar Apr May Jun Jul Aug Sep Oct Nov Dec Jan
Date Apr May Jun
Date N u m be r o f c a s e s Figure 3: Data fit for Italy using available data between 6 March and 8 May 2020, without a prediction horizon (left) and witha longer prediction horizon to show the final values (right). The figure shows Total cases in blue, Confirmed infectious cases(Q) in orange, Recovered cases (R) in purple, and Deceased cases (D) in black. ebMar Apr May Jun Jul Aug Sep Oct NovDec Jan Date
Date N u m be r o f c a s e s Figure 4: Data fit for South Korea using available data between 22 February and 8 May 2020, without a prediction horizon(left) and with a longer prediction horizon to show the final values (right). The figure shows Total cases in blue, Confirmedinfectious cases (Q) in orange, Recovered cases (R) in purple, and Deceased cases (D) in black.
The number of cases in South Africa only really started to increase in March 2020. As such, the datataken for fitting is only from 23 March to 8 May 2020. The initial result obtained for South Africa is shownin Fig. 5 (MSE = 9 . × ; R = 0 . β -value obtained for the model outputs shown in Fig. 5 was a constant 0.748, which is much higherthan the final β values obtained for Germany, Italy, and South Korea. Consequently the projection for thefinal number of cases is extremely high (24 million). This may be attributed to the fact that South Africais still in the initial phase of the pandemic, where the number of cases increase rapidly. The predictionsobtained from the model during this phase of the pandemic are extremely sensitive to β , and hence also thebasic reproduction number, as given in (12). This fact is illustrated in Section 4.COVID-19 has progressed much further in Germany, Italy, and South Korea than it has in South Africa.One would thus expect that the predictive power of the models obtained for these countries to be betterthan that of the South African model. The first South African model (with β = 0 . . × ; R = 0 . β cannot be accurately determined, the projection is that by mid-May the number of confirmed cases willbe in excess of one million, with the peak in the confirmed cased only occurring by the end of June 2020. Inreality, the confirmed cases in Germany appear to flatten out below 20,000 with the peak having occurredin April 2020 (see Fig. 2). This illustrates that it is very difficult to accurately predict the transmission rateparameter ( β ) during the initial phase of the pandemic.To improve the predictive power of the South African model, β was therefore constrained to have a finalvalue of 0.25 ( β = 0 . β value still fits the data quite well even though the projectionsfor the final number of cases is much lower, 570,000 as opposed to 24 million, than when β = 0 .
599 as shownin Fig. 5.The final model for South Africa predicts the peak number of infectious cases (about 70,000) to happenin late October (27 October 2020). The total number of cases predicted is about 570,000. These predictionsneed to be taken in light of the sensitivity of the model to the value of β , which in turn is also dependent7 an 2020 Jul 2020 Jan 2021 Jul 2021 Jan 2022 Date Apr May Jun
Date N u m be r o f c a s e s Figure 5: Initial data fit for South Africa where parameter estimation found a constant β ( t ) = 0 .
599 using data from 20 Marchto 8 May 2020, without a prediction horizon (left) and with a longer prediction horizon (right). The figure shows Total casesin blue, Confirmed infectious cases (Q) in orange, Recovered cases (R) in purple, and Deceased cases (D) in black.
Jan 2020 Apr 2020 Jul 2020 Oct 2020 Jan 2021Apr 2021
Date Mar 11 Mar 25 Apr 08 Apr 22 May 06
Date N u m be r o f c a s e s Figure 6: Data fit for Germany only using data from 11 March to 28 March 2020 (i.e. before the number of cases starts toflatten out), without a prediction horizon (left) and with a longer prediction horizon (right). The figure shows Total cases inblue, Confirmed infectious cases (Q) in orange, Recovered cases (R) in purple, and Deceased cases (D) in black. an 2020 Jul 2020 Jan 2021 Jul 2021 Jan 2022 Date Apr May Jun
Date N u m be r o f c a s e s Figure 7: Final data fit for South Africa with R constrained using data from 20 March to 8 May 2020, without a predictionhorizon (left) and with a longer prediction horizon (right). The figure shows Total cases in blue, Confirmed infectious cases(Q) in orange, Recovered cases (R) in purple, and Deceased cases (D) in black. on the number of contacts per person per unit of time. This implies that the number of cases may in futureincrease or decrease dramatically depending on the regulations in place, and the adherence of the populationto those regulations. The model parameter values obtained are shown in Table 1. The total population size ( N ) is simplythe total population of the country. The Germany and Italy data were taken from Eurostat (2020) and theSouth Korea and South Africa data were taken from World Bank (2020). These are included in Table 1 forreference.One criticism of deterministic epidemiological models (see e.g. Britton (2010)) is that if R < R > α > α >
0, which allows individuals to move directly fromthe susceptible portion of the population to the insusceptible portion, seems unlikely. Another possibility isto use an effective population size smaller than the total population of the country like Fanelli and Piazza(2020), which also means that the epidemic will abate sooner. Postnikov (2020) notes this situation, wherethe effective population is much smaller than the actual population, as a “weak outbreak”.Both of these options will however not leave room for much of a second peak, which infectious diseaseexperts warn may occur (Prem et al., 2020) if regulations are relaxed and the number of contacts per personis allowed to increase significantly. Secondary outbreaks have recently occurred in the Chinese provinces ofJilin and Heilongjiang (Reuters, 2020). The option of making β a function of time leaves the susceptiblepopulation in tact, which might be a more realistic scenario.Fig. 8 shows the basic reproduction number ( R ) as a function of time for all 4 countries modelled. Thisfigure is an indication of the swiftness and strictness of preventative measures, as well as adherence to thosemeasures by the citizens of the country concerned. 9 ar 11 Apr 08 May 06 (a) (b) (c) (d) Figure 8: Basic reproduction number ( R ) as a function of time for (a) Germany, (b) Italy, (c) South Korea, and (d) SouthAfrica.Figure 9: Sensitivity analysis for varying values of β . The solid line shows the nominal value, and the filled patch the rangefor 10% smaller and 10% larger values. Total cases are shown in blue, Confirmed infectious cases in orange, Recovered cases inpurple, and Deceased cases in black.
4. Sensitivity analysis
The predictive power of the models obtained in Section 3 rely heavily of the accuracy of the parametersobtained. This is especially true in the South African case where COVID-19 has not progressed far enoughfor the confirmed infectious cases to have peaked. In order to illustrate the sensitivity of the South Africanmodel to the parameters, the base parameters are decreased by 10 % and then increased by 10 %, one at atime, and the simulation over time repeated. This is done for β , γ , δ , λ , and κ . The results are shown inFig. 9 - 13.Fig. 9 shows the impact that varying β has on the total number of cases. It confirms that a small changein the transmission rate over the initial period when the number of cases increases exponentially has a verylarge effect. This also indicates that reducing the possibility of transmission early on should be effective inlimiting the total number of cases.From Fig. 10 it is clear that changing γ has an impact on the number of cases, but the impact is muchsmaller than that of changing β . γ drives the movement of cases from the “Exposed” to the “Infectious”compartment; i.e. the time between being exposed to being infectious. If this period is longer there areslightly fewer cases in the “Infectious” compartment, and hence slightly fewer people that can infect others.It is clear from Fig. 11 that the effect of δ on the number of cases is also very pronounced. Mathematicallythis makes sense as it can be seen from equation (12) that changes in δ have similar effects as changes in β as they affect the basic reproduction number in a similar way. Practically it also makes sense that the longera person remains in the “Infectious” category (with the potential to infect others), without having beenconfirmed infectious such that they can isolate (i.e. move into the “Quarantined” category), the number ofnew cases generated will increase. 10 igure 10: Sensitivity analysis for varying values of γ . The solid line shows the nominal value, and the filled patch the rangefor 10% smaller and 10% larger values. Total cases are shown in blue, Confirmed infectious cases (Q) in orange, Recoveredcases (R) in purple, and Deceased cases (D) in black.Figure 11: Sensitivity analysis for varying values of δ . The solid line shows the nominal value, and the filled patch the rangefor 10% smaller and 10% larger values. Total cases are shown in blue, Confirmed infectious cases (Q) in orange, Recoveredcases (R) in purple, and Deceased cases (D) in black. This is also where testing plays an important role. The more tests that are conducted, the sooner“Infectious” cases will be identified. This means δ increases and the total number of cases generateddecreases.Changing λ or κ does not have any impact on the total number of cases (as can be seen from Fig. 12 and13). The only real effect of these parameters is whether cases will end up in the “Recovered” or “Deceased”compartments. On the scale of the number of “Recovered” cases shown in the figures, the effect is almostnot visible.
5. Conclusion
An epidemiological model was developed for spread of COVID-19 in South Africa. A variant of theclassical compartmental SEIR model, called the SEIQRDP model, was used. The SEIQRDP model wasfirst parameterized on data for Germany, Italy, and South Korea, to test the efficacy of the model. Goodfits were achieved with reasonable predictions of where the number of COVID-19 confirmed cases, deaths,and recovered cases will end up and by when.South African data for the period from 23 March to 8 May 2020 was then used to obtain SEIQRDP modelparameters. It was found that the model fits the initial disease progression well, but that the long-termpredictive capability of the model was rather poor. The South African SEIQRDP model was subsequentlyrecalculated with the basic reproduction number constrained to reported values. The resulting model fit the11 igure 12: Sensitivity analysis for varying values of λ . The solid line shows the nominal value, and the filled patch the rangefor 10% smaller and 10% larger values. Total cases are shown in blue, Confirmed infectious cases (Q) in orange, Recoveredcases (R) in purple, and Deceased cases (D) in black.Figure 13: Sensitivity analysis for varying values of κ . The solid line shows the nominal value, and the filled patch the rangefor 10% smaller and 10% larger values. Total cases are shown in blue, Confirmed infectious cases (Q) in orange, Recoveredcases (R) in purple, and Deceased cases (D) in black. References
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