A SIR model assumption for the spread of COVID-19 in different communities
AA SIR model assumption for the spread of COVID-19 in different communities
Ian Cooper a , Argha Mondal b, ∗ , Chris G. Antonopoulos b a School of Physics, The University of Sydney, Sydney, Australia b Department of Mathematical Sciences, University of Essex, Wivenhoe Park, UK
Abstract
In this paper, we study the effectiveness of the modelling approach on the pandemic due to the spreading of the novelCOVID-19 disease and develop a susceptible-infected-removed (SIR) model that provides a theoretical frameworkto investigate its spread within a community. Here, the model is based upon the well-known susceptible-infected-removed (SIR) model with the difference that a total population is not defined or kept constant per se and thenumber of susceptible individuals does not decline monotonically. To the contrary, as we show herein, it can beincreased in surge periods! In particular, we investigate the time evolution of different populations and monitordiverse significant parameters for the spread of the disease in various communities, represented by countries andthe state of Texas in the USA. The SIR model can provide us with insights and predictions of the spread of thevirus in communities that the recorded data alone cannot. Our work shows the importance of modelling the spreadof COVID-19 by the SIR model that we propose here, as it can help to assess the impact of the disease by offeringvaluable predictions. Our analysis takes into account data from January to June, 2020, the period that containsthe data before and during the implementation of strict and control measures. We propose predictions on variousparameters related to the spread of COVID-19 and on the number of susceptible, infected and removed populationsuntil September 2020. By comparing the recorded data with the data from our modelling approaches, we deducethat the spread of COVID-19 can be under control in all communities considered, if proper restrictions and strongpolicies are implemented to control the infection rates early from the spread of the disease.
Keywords:
COVID-19, pandemic, infectious disease, virus spreading, epidemiology, SIR model, forecasting.
1. Introduction
In December 2019, a novel strand of Coronavirus (SARS-CoV-2) was identified in Wuhan, Hubei Province, Chinacausing a severe and potentially fatal respiratory syndrome, i.e., COVID-19. Since then, it has become a pandemicdeclared by World Health Organization (WHO) on March 11, which has spread around the globe [1, 2, 3, 4, 5].WHO published in its website preliminary guidelines with public health care for the countries to deal with thepandemic [6]. Since then, the infectious disease has become a public health threat. Italy and USA are severelyaffected by COVID-19 [7, 8, 9]. Millions of people are forced by national governments to stay in self-isolation and indifficult conditions. The disease is growing fast in many countries around the world. In the absence of availabilityof a proper medicine or vaccine, currently social distancing, self-quarantine and wearing a face mask have beenemerged as the most widely-used strategy for the mitigation and control of the pandemic.In this context, mathematical models are required to estimate disease transmission, recovery, deaths and othersignificant parameters separately for various countries, that is for different, specific regions of high to low reportedcases of COVID-19. Different countries have already taken precise and differentiated measures that are importantto control the spread of the disease. However, still now, important factors such as population density, insufficientevidence for different symptoms, transmission mechanism and unavailability of a proper vaccine, makes it difficultto deal with such a highly infectious and deadly disease, especially in high population density countries such as India ∗ Corresponding author
Email address: [email protected] (Argha Mondal)
Preprint submitted to Elsevier June 19, 2020 a r X i v : . [ q - b i o . P E ] J un
10, 11, 12]. Recently, many research articles have adopted the modelling approach, using real incidence datasetsfrom affected countries and, have investigated different characteristics as a function of various parameters of theoutbreak and the effects of intervention strategies in different countries, respective to their current situations.It is imperative that mathematical models are developed to provide insights and make predictions about thepandemic, to plan effective control strategies and policies [13, 14, 15]. Modelling approaches [8, 16, 17, 18, 19,20, 21] are helpful to understand and predict the possibility and severity of the disease outbreak and, providekey information to determine the intensity of COVID-19 disease intervention. The susceptible-infected-removed(SIR) model and its extended modifications [22, 23, 24, 25], such as the extended-susceptible-infected-removed(eSIR) mathematical model in various forms have been used in previous studies [26, 27, 28] to model the spread ofCOVID-19 within communities.Here, we propose the use of a novel SIR model with different characteristics. One of the major assumptions ofthe classic SIR model is that there is a homogeneous mixing of the infected and susceptible populations and that thetotal population is constant in time. In the classic SIR model, the susceptible population decreases monotonicallytowards zero. However, these assumptions are not valid in the case of the spread of the COVID-19 virus, since newepicentres spring up around the globe at different times. To account for this, the SIR model that we propose heredoes not consider the total population and takes the susceptible population as a variable that can be adjusted atvarious times to account for new infected individuals spreading throughout a community, resulting in an increasein the susceptible population, i.e., to the so-called surges. The SIR model we introduce here is given by the samesimple system of three ordinary differential equations (ODEs) with the classic SIR model and can be used to gaina better understanding of how the virus spreads within a community of variable populations in time, when surgesoccur. Importantly, it can be used to make predictions of the number of infections and deaths that may occurin the future and provide an estimate of the time scale for the duration of the virus within a community. It alsoprovides us with insights on how we might lessen the impact of the virus, what is nearly impossible to discern fromthe recorded data alone! Consequently, our SIR model can provide a theoretical framework and predictions thatcan be used by government authorities to control the spread of COVID-19.In our study, we used COVID-19 datasets from [29] in the form of time-series, spanning January to June, 2020.In particular, the time series are composed of three columns which represent the total cases I dtot , active cases I d and Deaths D d in time (rows). These datasets were used to update parameters of the SIR model to understandthe effects and estimate the trend of the disease in various communities, represented by China, South Korea,India, Australia, USA, Italy and the state of Texas in the USA. This allowed us to estimate the development ofCOVID-19 spread in these communities by obtaining estimates for the number of deaths D , susceptible S , infected I and removed R m populations in time. Consequently, we have been able to estimate its characteristics for thesecommunities and assess the effectiveness of modelling the disease.The paper is organised as following: In Sec. 2, we introduce the SIR model and discuss its various aspects. InSec. 3, we explain the approach we used to study the data in [29] and in Sec. 4, we present the results of ouranalysis for China, South Korea, India, Australia, USA, Italy and the state of Texas in the USA. Section 5 discussesthe implications of our study to the “flattening the curve” approach. Finally, in Sec. 6, we conclude our workand discuss the outcomes of our analysis and its connection to the evidence that has been already collected on thespread of COVID-19 worldwide.
2. The SIR model that can accommodate surges in the susceptible population
The world around us is highly complicated. For example, how a virus spreads, including the novel strand ofCoronavirus (SARS-CoV-2) that was identified in Wuhan, Hubei Province, China, depends upon many factors,among which some of them are considered by the classic SIR model, which is rather simplistic and cannot take intoconsideration surges in the number of susceptible individuals. Here, we propose the use of a modified SIR modelwith characteristics, based upon the classic SIR model. In particular, one of the major assumptions of the classicSIR model is that there is a homogeneous mixing of the infected I and susceptible S populations and that the totalpopulation N is constant in time. Also, in the SIR model, the susceptible population S decreases monotonically2owards zero. These assumptions however are not valid in the case of the spread of the COVID-19 virus, since newepicentres spring up around the globe at different times. To account for this, we introduce here a SIR model thatdoes not consider the total population N , but rather, takes the susceptible population S as a variable that canbe adjusted at various times to account for new infected individuals spreading throughout a community, resultingin its increase. Thus, our model is able to accommodate surges in the number of susceptible individuals in time,whenever these occur and as evidenced by published data, such as those in [29] that we consider here.Our SIR model is given by the same, simple system of three ordinary differential equations (ODEs) with theclassic SIR model that can be easily implemented and used to gain a better understanding of how the COVID-19 virus spreads within communities of variable populations in time, including the possibility of surges in thesusceptible populations. Thus, the SIR model here is designed to remove many of the complexities associated withthe real-time evolution of the spread of the virus, in a way that is useful both quantitatively and qualitatively. Itis a dynamical system that is given by three coupled ODEs that describe the time evolution of the following threepopulations:1. Susceptible individuals , S ( t ): These are those individuals who are not infected, however, could become infected.A susceptible individual may become infected or remain susceptible. As the virus spreads from its source ornew sources occur, more individuals will become infected, thus the susceptible population will increase for aperiod of time (surge period).2. Infected individuals , I ( t ): These are those individuals who have already been infected by the virus and cantransmit it to those individuals who are susceptible. An infected individual may remain infected, and can beremoved from the infected population to recover or die.3. Removed individuals , R m ( t ): These are those individuals who have recovered from the virus and are assumedto be immune, R m ( t ) or have died, D ( t ).Furthermore, it is assumed that the time scale of the SIR model is short enough so that births and deaths (otherthan deaths caused by the virus) can be neglected and that the number of deaths from the virus is small comparedwith the living population.Based on these assumptions and concepts, the rates of change of the three populations are governed by thefollowing system of ODEs, what constitutes our SIR model dS ( t ) dt = − aS ( t ) I ( t ) ,dI ( t ) dt = aS ( t ) I ( t ) − bI ( t ) ,dR m ( t ) dt = bI ( t ) , (1)where a and b are real, positive, parameters of the initial exponential growth and final exponential decay of theinfected population I .It has been observed that in many communities, a spike in the number of infected individuals, I , may occur,which results in a surge in the susceptible population, S , recorded in the COVID-19 datasets [29], what amountsto a secondary wave of infections. To account for such a possibility, S in the SIR model (1), can be reset to S surge at any time t s that a surge occurs, and thus it can accommodate multiple such surges if recorded in the publisheddata in [29], what distinguishes it from the classic SIR model.The evolution of the infected population I is governed by the second ODE in system 1, where a is the transmissionrate constant and b the removal rate constant. We can define the basic effective reproductive rate R e = aS ( t ) /b ,as the fate of the evolution of the disease depends upon it. If R e is smaller than one, the infected population I will decrease monotonically to zero and if greater than one, it will increase, i.e., if dI ( t ) dt < ⇒ R e < dI ( t ) dt > ⇒ R e >
1. Thus, the effective reproductive rate R e acts as a threshold that determines whether aninfectious disease will die out quickly or will lead to an epidemic.At the start of an epidemic, when R e > S ≈
1, the rate of infected population is described by theapproximation dI ( t ) dt ≈ ( a − b ) I ( t ) and thus, the infected population I will initially increase exponentially according3o I ( t ) = I (0) e ( a − b ) t . The infected population will reach a peak when the rate of change of the infected populationis zero, dI ( t ) /dt = 0, and this occurs when R e = 1. After the peak, the infected population will start to decreaseexponentially, following I ( t ) ∝ e − bt . Thus, eventually (for t → ∞ ), the system will approach S → I → S will result in R e >
1, and to another exponential growth of the number of infections I .
3. Methodology
The data in [29] for China, South Korea, India, Australia, USA, Italy and the state of Texas (communities) areorganised in the form of time-series where the rows are recordings in time (from January to June, 2020), and thethree columns are, the total cases I dtot (first column), number of infected individuals I d (second column) and deaths D d (third column). Consequently, the number of removals can be estimated from the data by R dm = I dtot − I d − D d . (2)Since we want to adjust the numerical solutions to our proposed SIR model (1) to the recorded data from [29], foreach dataset (community), we consider initial conditions in the interval [0 ,
1] and scale them by a scaling factor f tofit the recorded data by visual inspection. In particular, the initial conditions for the three populations are set suchthat S (0) = 1 (i.e., all individuals are considered susceptible initially), I (0) = R m (0) = I dmax /f <
1, where I dmax is the maximum number of infected individuals I d . Consequently, the parameters a , b , f and I dmax are adjustedmanually to fit the recorded data as best as possible, based on a trial-and-error approach and visual inspections.A preliminary analysis using non-linear fittings to fit the model to the published data [29] provided at best inferiorresults to those obtained in this paper using our trial-and-error approach with visual inspections, in the sense thatthe model solutions did not follow as close the published data, what justifies our approach in the paper. A primereason for this is that the published data (including those in [29] we are using here) are data from different countriesthat follow different methodologies to record them, with not all infected individuals or deaths accounted for.In this context, S , I and R m ≥ t ≥
0. System (1) can be solved numerically to find how the scaled(by f ) susceptible S , infected I and removed R m populations (what we call model solutions) evolve with time, ingood agreement with the recorded data. In particular, since this system is simple with well-behaved solutions, weused the first-order Euler integration method to solve it numerically, and a time step h = 200 / .
04 thatcorresponds to a final integration time t f of 200 days since January, 2020. This amounts to double the time intervalin the recorded data in [29] and allows for predictions for up to 100 days after January, 2020.Obviously, what is important when studying the spread of a virus is the number of deaths D and recoveries R in time. As these numbers are not provided directly by the SIR model (1), we estimated them by first, plotting thedata for deaths D d vs the removals R dm , where R dm = D d + R d = I dtot − I d and then fitting the plotted data withthe nonlinear function D = D (cid:0) − e − kR m (cid:1) , (3)where D and k are constants estimated by the non-linear fitting. The function is expressed in terms of only modelvalues and is fitted to the curve of the data. Thus, having obtained D from the non-linear fitting, the number ofrecoveries R can be described in time by the simple observation that it is given by the scaled removals, R m fromthe SIR model (1), less the number of deaths, D from Eq. (3), R = R m − D. (4)
4. Results
The rate of increase in the number of infections depends on the product of the number of infected and susceptibleindividuals. An understanding of the system of Eqs. (1) explains the staggering increase in the infection rate around4he world. Infected people traveling around the world has led to the increase in infected numbers and this resultsin a further increase in the susceptible population [14]. This gives rise to a positive feedback loop leading to a veryrapid rise in the number of active infected cases. Thus, during a surge period, the number of susceptible individualsincreases and as a result, the number of infected individuals increases as well. For example, as of 1 March, 2020,there were 88 590 infected individuals and by 3 April, 2020, this number had grown to a staggering 1 015 877 [29].Understanding the implications of what the system of Eqs. (1) tells us, the only conclusion to be drawn usingscientific principles is that drastic action needs to be taken as early as possible, while the numbers are still low,before the exponential increase in infections starts kicking in. For example, if we consider the results of policiesintroduced in the UK to mitigate the spread of the disease, there were 267 240 total infections and 37 460 deathsby 27 May and in the USA, 1 755 803 and 102 107, total infections and deaths, respectively. Thus, even if onestarts with low numbers of infected individuals, the number of infections will at first grow slowly and then, increaseapproximately exponentially, then taper off until a peak is reached. Comparing these results for the UK and USAwith those for South Korea, where steps were taken immediately to reduce the susceptible population, there were11 344 total infections and 269 deaths by 27 May. The number of infections in China reached a peak about 16February, 2020. The government took extreme actions with closures, confinement, social distancing, and peoplewearing masks. This type of action produces a decline in the number of infections and susceptible individuals. Ifthe number of susceptible individuals does not decrease, then the number of infections just gets increased rapidly.As at this moment, there is no effective vaccine developed, the only way to reduce the number of infections is toreduce the number of individuals that are susceptible to the disease. Consequently, the rate of infection tends tozero only if the susceptible population goes to zero.Here, we have applied the SIR model (1) considering data from various countries and the state of Texas in theUSA provided in [29]. Assuming the published data are reliable, the SIR model (1) can be applied to assess the spreadof the COVID-19 disease and predict the number of infected, removed and recovered populations and deaths in thecommunities, accommodating at the same time possible surges in the number of susceptible individuals. Figures1–17 show the time evolution of the cumulative total infections I tot , current infected individuals, I , recoveredindividuals, R , dead individuals, D , and normalized susceptible populations, S for China, South Korea, India,Australia, USA, Italy and Texas in the USA, respectively. The crosses show the published data [29] and the smoothlines, solutions and predictions from the SIR model. The cumulative total infections plots also show a curve forthe initial exponential increase in the number of infections, where the number of infections doubles every five days.The figures also show predictions, and a summary of the SIR model parameters in (1) and published data in [29]for easy comparisons.We start by analysing the data from China and then move on to the study of the data from South Korea, India,Australia, USA, Italy and Texas. The number of infections peaked in China about 16 February, 2020 and since then, it has slowly decreased. Thedecrease only occurs when the susceptible population numbers decrease and this decrease in susceptible numbersonly occurred through the drastic actions taken by the Chinese government. China quarantined and confirmedpotential patients, and restricted citizens’ movements as well as international travel. Social distancing was widelypracticed, and most of the people wore face masks. The actual numbers of infections have decreased at a greaterrate than predicted by the SIR model (see Figs. 1 and 2). Our results in Figs. 1 and 2 provide evidence that theChinese government has done well in limiting the impact of the spread of COVID-19.
From the plots shown in Figs. 3 and 4, it is obvious that the South Korean government has done a wonderfuljob in controlling the spread of the virus. The country has implemented an extensive virus testing program. Therehas also been a heavy use of surveillance technology: closed-circuit television (CCTV) and tracking of bank cardsand mobile phone usage, to identify who to test in the first place. South Korea has achieved a low fatality rate(currently one percent) without resorting to such authoritarian measures as in China. The most conspicuous part ofthe South Korean strategy is simple enough: implementation of repeated cycles of test and contact trace measures.5 igure 1: China: Model predictions for the period from 22 January to 9 August, 2020 with data from January to June, 2020. The datashow a discrete jump in deaths D in mid-April.Figure 2: China: (a) Nonlinear fitting with Eq. (3) using a trial-and-error method to estimate the number of deaths, D from theremoved population, R m (see text for the details). (b) Plots of the number of removals, R m against the cumulative total infections I tot and current active cases I . igure 3: South Korea: Model predictions for the period from 26 February to 13 September, 2020 with data from February to June,2020.Figure 4: South Korea: (a) Nonlinear fitting with Eq. (3) using a trial-and-error method to estimate the number of deaths, D from theremoved population, R m (see text for the details). (b) Plots of the number of removals, R m against the cumulative total infections I tot and current active cases I . igure 5: India: Model predictions for the period from 14 March to 30 September, 2020 with data from March to June, 2020. To match the recorded data from India with predictions from the SIR model (1), it is necessary to include anumber of surge periods, as shown in Fig. 5. This is because the SIR model cannot predict accurately the peaknumber of infections, if the actual numbers in the infected population have not peaked in time. It is most likely thespread of the virus as of early June, 2020 is not contained and there will be an increasing number of total infections.However, by adding new surge periods, a higher and delayed peak can be predicted and compared with future data.In Fig. 5, a consequence of the surge periods is that the peak is delayed and higher than if no surge periods wereapplied. The model predictions for the 30 September, 2020 including the surges are: 330 000 total infections, 700active infections and 7 500 deaths, whereas if there were no surge periods, there would be 130 000 total infections,700 active infections and 6 300 deaths, with the peak of 60 000, which is about 40% of the current number of activecases occuring around 20 May 2020. Thus, the model can still give a rough estimate of future infections and deaths,as well as the time it may take for the number of infections to drop to safer levels, at which time restrictions canbe eased, even without an accurate prediction in the peak in active infections (see Figs. 5 and 6).
A surge in the susceptible population was applied in early March, 2020 in the country. The surge was causedby 2 700 passengers disembarking from the Ruby Princes cruise ship in Sydney and then, returning to their homesaround Australia. More than 750 passengers and crew have become infected and 26 died. Two government enquireshave been established to investigate what went wrong. Also, at this time many infected overseas passengers arrivedby air from Europe and the USA. The Australian government was too slow in quarantining arrivals from overseas.From mid-March, 2020 until mid-May, 2020, the Australian governments introduced measures of testing, contacttracing, social distancing, staying at home policy, closure of many businesses and encouraging people to work fromhome. From Figs. 7 and 8, it can be observed that actions taken were successful as the actual number of infectionsdeclined in accord with the model predictions. There have been no further surge periods. From end of May, 2020,these restrictions are being removed in stages. The SIR model can be used when future data becomes available tosee if the number of susceptible individuals starts to increase. If so, the model can accommodate this by introducingsurge factors. 8 igure 6: India: (a) Nonlinear fitting with Eq. (3) using a trial-and-error method to estimate the number of deaths, D from the removedpopulation, R m (see text for the details). (b) Plots of the number of removals, R m against the cumulative total infections I tot andcurrent active cases I .Figure 7: Australia: Model predictions for the period from 22 January to 9 August, 2020 with data from January to June, 2020. igure 8: Australia: (a) Nonlinear fitting with Eq. (3) using a trial-and-error method to estimate the number of deaths, D from theremoved population, R m (see text for the details). (b) Plots of the number of removals, R m against the cumulative total infections I tot and current active cases I . As of early June, 2020, the peak number of infections has not been reached. When a peak in the data is notreached, it is more difficult to fit the model predictions to the data. In the model, it is necessary to add a fewsurge periods. This is because new epicentres of the virus arose at different times. The virus started spreading inWashington State, followed by California, New York, Chicago and the southern states of the USA. The need to addsurge periods shows clearly that the spread of the virus is not under control.In the USA, by the end of May, 2020, the number of active infected cases has not yet peaked and the cumulativetotal number of infections keeps getting bigger. This can be accounted for in the SIR model by considering how thesusceptible population changes with time in May. During that time, to match the data to the model predictions,surge periods were used where the normalized susceptible population S was reset to 0 . S , varying from about 0 .
06 to 0 . The plots in Figs. 11 and 12 show that the peak in the total cumulative number of infections has not beenreached as early as June, however, the peak is probably not far away. If there are no surges in the susceptiblepopulation, then one could expect that by late September, 2020, the number of infections will have fallen to verysmall numbers and the virus will have been well under control with the total number of deaths in the order of 2000. In mid-May, 2020, some restrictions have been lifted in the state of Texas. The SIR model can be used tomodel some of the possible scenarios if the early relaxation of restrictions leads to increasing number of susceptiblepopulations. If there is a relatively small increase in the future number of susceptible individuals, no series impacts10 igure 9: USA: Model predictions for the period from 22 January to 9 August, 2020 with data from January to June, 2020.Figure 10: USA: (a) Nonlinear fitting with Eq. (3) using a trial-and-error method to estimate the number of deaths, D from the removedpopulation, R m (see text for the details). (b) Plots of the number of removals, R m against the cumulative total infections I tot andcurrent active cases I . igure 11: Texas: Model predictions for the period from 12 March to 28 September, 2020 with data from March to June, 2020.Figure 12: Texas: (a) Nonlinear fitting with Eq. (3) using a trial-and-error method to estimate the number of deaths, D from theremoved population, R m (see text for the details). (b) Plots of the number of removals, R m against the cumulative total infections I tot and current active cases I . igure 13: Texas: Model predictions with a surge period occurring at the end of June, 2020. occur. However, if there is a large outbreak of the virus, then the impacts can be dramatic. For example, at the endof June, 2020, if S was reset to 0.8 ( S = 0 . Figure 15 shows clearly that the peak of the pandemic has been reached in Italy and without further surgeperiods, the spread of the virus is contained and number of active cases is declining rapidly. The plots in panels(a), (b) in Fig. 16 are a check on how well the model can predict the time evolution of the virus. These plots alsoassist in selecting the model’s input parameters.
5. Flattening the curve
The term flattening the curve has rapidly become a rallying cry in the fight against COVID-19, popularised bythe media and government officials. Claims have been made that flattening the curve results in: (i) reduction inthe peak number of cases, thereby helping to prevent the health system from being overwhelmed and (ii) in anincrease in the duration of the pandemic with the total burden of cases remaining the same. This implies that socialdistancing measures and management of cases, with their devastating economic and social impacts, may need tocontinue for much longer. The picture which has been widely shown in the media is shown in Fig. 17(a).The idea presented in the media as shown in Fig. 17(a) is that by flattening the curve, the peak number ofinfections will decrease, however, the total number of infections will be the same and the duration of the pandemicwill be longer. Hence, they concluded that by flattening the curve , it will have a lesser impact upon the demands inhospitals. Figure 17(b) gives the scientific meaning of flattening the curve . By governments imposing appropriatemeasures, the number of susceptible individuals can be reduced and combined with isolating infected individuals,will reduce the peak number of infections. When this is done, it actually shortens the time the virus impacts thesociety. Thus, the second claim has no scientific basis and is incorrect. What is important is reducing the peak13 igure 14: Texas: If a second wave occurs, there could be increase in the number of deaths, D .Figure 15: Italy: Model predictions for the period from 26 February to 13 September, 2020 with data from February to June, 2020. igure 16: Italy: (a) Nonlinear fitting with Eq. (3) using a trial-and-error method to estimate the number of deaths, D from theremoved population, R m (see text for the details). (b) Plots of the number of removals, R m against the cumulative total infections I tot and current active cases I .Figure 17: Flattening the curve: Panel (a): The flattening of the curve diagram used widely in the media to represent a means ofreducing the impacts of COVID-19. Panel (b) If the number of susceptible individuals is reduced, then the peak number of infectionswill be less and the time for the number of infections to fall to low numbers is reduced.
15n the number of infections and when this is done, it shortens the duration in which drastic measures need to betaken and not lengthen the period as stated in the media and by government officials. Figure 17 shows that thepeak number of infections actually reduces the duration of the impact of the virus on a community.
6. Conclusions
Mathematical modelling theories are effective tools to deal with the time evolution and patterns of diseaseoutbreaks. They provide us with useful predictions in the context of the impact of intervention in decreasing thenumber of infected-susceptible incidence rates [30, 31, 32].In this work, we have augmented the classic SIR model with the ability to accommodate surges in the numberof susceptible individuals, supplemented by recorded data from China, South Korea, India, Australia, USA and thestate of Texas to provide insights into the spread of COVID-19 in communities. In all cases, the model predictionscould be fitted to the published data reasonably well, with some fits better than others. For China, the actualnumber of infections fell more rapidly than the model prediction, which is an indication of the success of themeasures implemented by the Chinese government. There was a jump in the number of deaths reported in mid-April in China, which results in a less robust estimate of the number of deaths predicted by the SIR model. Thesusceptible population dropped to zero very quickly in South Korea showing that the government was quick to actin controlling the spread of the virus. As of the beginning of June, 2020, the peak number of infections in Indiahas not yet been reached. Therefore, the model predictions give only minimum estimates of the duration of thepandemic in the country, the total cumulative number of infections and deaths. The case study of the virus inAustralia shows the importance of including a surge where the number of susceptible individuals can be increased.This surge can be linked to the arrival of infected individuals from overseas and infected people from the RubyPrincess cruise ship. The data from USA is an interesting example, since there are multiple epicentres of the virusthat arise at different times. This makes it more difficult to select appropriate model parameters and surges wherethe susceptible population is adjusted. The results for Texas show that the model can be applied to communitiesother than countries. Italy provides an example where there is excellent agreement between the published data andmodel predictions.Thus, our SIR model provides a theoretical framework to investigate the spread of the COVID-19 virus withincommunities. The model can give insights into the time evolution of the spread of the virus that the data alone doesnot. In this context, it can be applied to communities, given reliable data are available. Its power also lies to thefact that, as new data are added to the model, it is easy to adjust its parameters and provide with best-fit curvesbetween the data and the predictions from the model. It is in this context then, it can provide with estimates of thenumber of likely deaths in the future and time scales for decline in the number of infections in communities. Ourresults show that the SIR model is more suitable to predict the epidemic trend due to the spread of the disease as itcan accommodate surges and be adjusted to the recorded data. By comparing the published data with predictions,it is possible to predict the success of government interventions. The considered data are taken in between Januaryand June, 2020 that contains the datasets before and during the implementation of strict and control measures.Our analysis also confirms the success and failures in some countries in the control measures taken.Strict, adequate measures have to be implemented to further prevent and control the spread of COVID-19.Countries around the world have taken steps to decrease the number of infected citizens, such as lock-down measures,awareness programs promoted via media, hand sanitization campaigns, etc. to slow down the transmission of thedisease. Additional measures, including early detection approaches and isolation of susceptible individuals to avoidmixing them with no-symptoms and self-quarantine individuals, traffic restrictions, and medical treatment haveshown they can help to prevent the increase in the number of infected individuals. Strong lockdown policies can beimplemented, in different areas, if possible. In line with this, necessary public health policies have to be implementedin countries with high rates of COVID-19 cases as early as possible to control its spread. The SIR model used hereis only a simple one and thus, the predictions that come out might not be accurate enough, something that alsodepends on the published data and their trustworthiness. However, as the model data show, one thing that iscertain is that COVID-19 is not going to go way quickly or easily.16 cknowledgements
AM is thankful for the support provided by the Department of Mathematical Sciences, University of Essex, UKto complete this work.
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