The Dynamics of COVID-19 spread: Evidence from Lebanon
TThe Dynamics of COVID-19 spread: Evidence from
Lebanon
Omar El Deeb a,b and Maya Jalloul c a Lebanese University, Faculty of Science b Lebanese International University, Mathematics and Physics Department c Lebanese American University, Department of Economics
Abstract
We explore the spread of the Coronavirus disease 2019 (COVID-19) in Lebanonby adopting two different approaches, namely the STEIR model which is a modifiedSEIR model accounting for the effect of travel, and a model of repeated iterations.We fit available daily data since the first diagnosed case until the end of June 2020and we forecast possible scenarios of contagion associated with different levels ofsocial distancing measures and travel inflows. We determine the initial reproductivetransmission rate in Lebanon and all subsequent dynamics. Our results suggest thatpreserving the current partial mitigation measures would slow down the spread ofthe disease. Nevertheless, relaxing measures and opening the airport would triggera second outbreak of infections, with severity depending on the extent of relaxation.
Keywords:
COVID-19; SEIR model; STEIR model; Mathematical modellingin epidemiology; Compartmental Models.
The Coronavirus disease 2019 (COVID-19) has been widely spreading worldwide since itappeared in the city of Wuhan, China towards the end of December, 2019. The WorldHealth Organization classified the spread as a pandemic in March 2020 [1]. Europeand the United States of America have endured the severest repercussions in terms ofnumber of infections and deaths. The US cases amounted to more than one fourthof total global infections by June 2020 [2]. Governmental and institutional reactionsand measures varied across countries with respect to the time of introduction of socialdistancing measures (SDM, henceforth) and with respect to their degree of severity. In1 a r X i v : . [ phy s i c s . s o c - ph ] J u l pite of some governments being slower in adopting mitigation measures and endorsingthe epidemiological concept of herd immunity [3] to create a resistance to the contagionin the long run at the expense of short term losses while keeping the economy functional,the majority adopted SDM’s that reached countrywide lockdowns.A considerable amount of research has been carried out focusing on the dynamics andextent of the pandemic in different countries notably in the countries that witnessed thefirst cases [4, 5, 6, 7, 8]. In comparison with the most recent deadly mass pandemic ofthe "Spanish flu" that hit the world after World War I during the years of 1918-1919and recurred in two waves, and caused the death of tens of millions of people [9, 10],the extent of the spread of COVID-19 has been far less. The question of the containingCOVID-19 and preventing its spread and expansion into a similar deadly pandemic isa key motivation for the study of various models that describe, simulate and forecastepidemics and dynamics of infections under different reproductive rates and mitigationmeasures.In Lebanon, the spread of COVID-19 coincided with a period of political turmoil, fewmonths of popular uprising and economic collapse finally depicted by the default on debtsin early March 2020 [11]. Lebanon SDM
SDM were adopted at a relatively early stage in Lebanon. While thefirst confirmed case was recorded on February 21, 2020, all academic institutions, namelyschools and universities, were closed starting the first of March. This was followed by aresolution of "Public Mobilization" and ban of public gatherings that imposed closure ofall churches, mosques, shops, restaurants, etc. except for grocery stores and drugstoreson March 22 [12]. Another subsequent measure consisted in constraining vehicles’ mobil-ity to alternating between odd- and even-ending plate numbers, while fully prohibitingmobility on Sundays. Gradual and partial relaxation started on May 10, but the deepeconomic crisis helped in slowing down social activity despite the easing of measures. Thegovernment decided to open the airport with one-fifth of its normal capacity starting July1st, after a period of limited returns for Lebanese people living abroad between April 8and June 30.The source of the first infection was documented before the international travel banto be from a traveler coming from Iran [13] where the spread of the virus had startedearly on [14]. However, it has been discussed that many of the following first cases weretransmitted by travelers coming from the Vatican city during the early stages of thepandemic spread there [15]. The efficacy of SDM’s can be examined by discerning thedaily rate of infection as shown in the daily data of the first 130 days [16]. Our resultswere obtained in relation to data available until June 30, 2020.2 iterature and Methodology
Models exploring contagion and particularly spreadof infections were developed and extensively studied in various fields of mathematics,physics, economics and epidemiology [17, 18, 19, 20, 21]. The effect of mitigation measureson the spread of infections was studied in [22, 23, 24].The SEIR model [25, 26, 27] has been extensively used in studies of infectious diseasesincluding COVID-19 worldwide. Here we define a novel STEIR model, which is a modifiedSEIR model that takes into account the effect of travel and then establish our study ofthe dynamics of the COVID-19 spread in Lebanon. We namely use the STEIR and arepeated iterations model presented in [28] (henceforth the RI model) and implementit on the daily data of the Coronavirus in Lebanon provided mainly by the Ministry ofPublic Health [16] over a duration of 130 days. 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I n f ec t i o n s (cid:144) m illi o n i nh a b i t a n t s (cid:244) USA (cid:242)
Iran (cid:236)
Lebanon (cid:224)
Turkey (cid:230)
Italy
Figure 1: Daily infections (per million) in Lebanon, Turkey, Iran, Italy and the USA.The structure of the paper is as follows. In section 2, we present the two models.Results and forecasts are discussed in section 3 and section 4 concludes the paper.
We describe here two models that we use to forecast the path of daily cases and tomeasure the extent of the epidemic in Lebanon. Following an abrupt rise in infectedcases at the start, the rate has fallen to a significantly low level after implementationof severe SDM’s, in comparison with other regional and country-level data [29] beforeslowly rising up again after partial relaxation. Figure 1 illustrates the progression of thenumber of daily infections over time in different countries: Lebanon, Turkey, Iran, Italyand the USA. Day represents the date of the first reported infection. Lebanon has a3elatively low number of infections per capita, despite the fact that the first case wasrecorded relatively early. The proposed models accommodate for the actual data andallow for future predictions. The study of the spread of epidemics can be executed by the fundamental SIR model firstintroduced by Kermack and McKendrick [30], that consists of dividing the population intodifferent compartments and investigating the contagion of the disease by examining therate of change of the sizes of these groups. Later versions were developed to accommodatefor different settings and assumptions [31, 32, 33]. The STEIR model that we use isdescribed as follows. Consider a population N that we initially normalize to size anddivide into five categories of individuals: susceptible S , incoming travellers T , exposed E , infectious I and removed (through recovery or death) R . The cumulative numberof cases is provided by C = I + R , since each recorded case would at a given time beeither infectious or recovered or dead. The rates of change of these categories are givenby dSdt , dTdt , dEdt , dIdt and dRdt respectively. The model can be formally described using thefollowing differential equations: dSdt = − β t SN I + (1 − θ ) τ (1) dEdt = β t SN I − σE (2) dIdt = σE − γI + θτ (3) dRdt = γI (4) dNdt = τ N (5)with β t = R t γ where β t is the rate at which infected individuals bump into others(the SN susceptibles), σ is the rate at which exposed individuals become infected and it isassociated to the mean incubation period; γ is rate of exit by recovery or death per-dayand it is associated to the average illness period. In this sense, R t = β t γ is the ratio ofmeeting rate to exit rate; it determines the transmission from susceptible to infected andis a proxy for social distancing measures. τ is the daily rate of incoming travellers relativeto the total population N and it contributes to both, the infected I and the susceptible S . θ is the average infection rate among the travellers. The population size N slightlyincreases due to the influx of travellers whose relative contribution is more apparent in4he increase in the number of infections I . We can safely ignore the natural increase inthe population over this period.There are several ways to model R t . It can be taken as a constant parameter in somecircumstances, or a time dependent function as in [25]. In our model, we introduce anovel parameterization for R t . Using data available from the first 130 days in Lebanon,we parameterize R t by the step function R t = R < t < t R t < t < t R t < t < t (6)after the application of SDM, before any relaxation, then after partial relaxation attime t respectively. t = 130 days, corresponding to the last day of used data Whenthe measures are changed after this period of SDM, then R t would be parameterized asfollows: R t = R < t < t R t < t < t R t < t < t R t > t (7)where R , R , R and R are constant parameters that depend on the severity ofmeasures and commitment to those measures. R is the reproductive transmission rateof the disease in the initial phase, while R and R are the reproductive transmissionfactors under strict and relaxed SDM respectively. R represents the possible future rateunder different possible relaxation schemes. We also take σ = . in relation to an averageperiod of incubation of . days, and γ = in relation to an average period of recovery(or death) of days. We can determine τ from the differential equation of N , and wefind that τ = ln( N f N i )∆ t (8)where N i is the population on April 8 and N f is the population on June 30, after thearrival of air passengers. The time ∆ t is the number of days of incoming flights. Thiscorresponds to τ = 2 . × − / day. From available data we also find that θ = 0 . .Assuming that initially no SDM are applied, R represents the transmission of diseasewith no mitigation measures. In [34] they adopt R = 3 . , while in [35] they take valuesbetween 2.76 and 3.25, and [25] considers different values between 3 and 1.6. Recentstudies reveal that R of COVID-19 can assume higher values up to 5.7 and 6.47 accordingto data analyzed from China [36, 37]. After introduction of measures, R t can assume5alues less than 1 in case extremely severe mitigation measures are applied [38]. It waswell established that COVID-19 has higher R than other infections like SARS [39]. Theestimation of R and R t is essential for forecasting the spread, but their determinationdepends on the available data and the accuracy of the reporting of initial cases and dates.We assume that the initial value of I is I = M , in line with the first initial case inLebanon reported on February 21 and a gross population of 6 million inhabitants. We take E = 12 I to account for the fact that the initial case registered had been in contact withmany people on a flight from Iran which raises the number of initially exposed people.This entails some uncertainty in the initial conditions of the spread. In comparison, in[25], they consider a value I = M with initial cases in the United States whosepopulation stands at around 330 millions, and E = 4 I given individuals wereinitially carrying but not contagious, acknowledging the considerable uncertainty relatedto initial cases in the US.In our model we find that R = 5 . , R = 0 . and R = 1 . for t = 32 days, t = 63 days and t = 130 days respectively, provide the best prediction for registered cumulativecases. Consequently, we simulate four possible future scenarios with R = 1 . , . , . and after t = 130 days. Regarding τ , we inspect four possible cases for t > days,upon the opening of Beirut airport, and the expected increase in the number of travellersentering the country. We simulate the effect of τ multiplied by , and in comparison tothe previous rate of arrivals, and its consequences on the cumulative number of infectedcases. We consider another model that takes into account the most recent available daily data ofconfirmed and recovered (or dead) cases. Here we implement a variation of the repeatediterations method proposed in [28]. Denote the currently infected daily values by I i where i is the index of days and i ∈ [1 , n ] . We take the last m values of I i to determine theaverage arithmetic gross rate in the last m days according to G a = 1 m n (cid:88) i = n − m +1 (cid:18) I i I i − − (cid:19) (9)while the average geometric gross rate of the same set of data is defined by G g = (cid:18) I i I i − m (cid:19) m (10)Each of G a or G g allows us to forecast the number of people infected for i > n bysimply implementing a progression for the following days according to6 i +1 = I i (1 + G a ) (11)using the arithmetic gross infection rate G a or alternatively I i +1 = I i G g (12)using the geometric gross infection rate G g .It is important to note that we also have to take into account the number of peopledead or recovered at later dates. To account for this, we denote the death rate by p ,the recovery rate by − p , the average number of days needed for recovery by h andthe average number of days between infection and death by d . This means that on day i + 1 , the number of deaths will be p ( I i − d − I i − d − ) where the term in brackets representsthe number of people who caught the virus d days ago. Similarly the number of peoplerecovered would be proportional to the number of people who caught the virus h daysago, thus it is given by (1 − p ) ( I i − h − I i − h − ) . Our recursive relation includes the numberof dead or recovered people as predicted from the people who got the infection h and d days ago respectively. The values of p and h vary in the literature and in the availabledata from the specific country considered.Then the repeated iterations model forecasts the net number of infected people by ( I i +1 ) net = I i +1 − p ( I i − d − I i − d − ) − (1 − p ) ( I i − h − I i − h − ) (13)From available data in Lebanon, it is reasonable to take p = 0 . , h = 20 days and d = 24 days. Note that in this iteration, we use the available data to forecast the nextunavailable day, and recalculate G accordingly, hence recursively predicting the futuredevelopment of the rate and the number of infected people. Here we will consider m = 14 previous days, and forecast the next upcoming days as well. The interesting feature ofthe RI model is that as new daily data is revealed, we can easily update our daily futureforecast, hence have a new day future expectation every day. G is a dynamic quantity and it will depend on the rapidness and strictness of thepublic policies of social distancing, ban of public gatherings and curfews, as well as onthe commitment of people to those measures and to health measures (sanitation, wearingprotective masks, gloves...). This is why, to forecast future situations, we have to takeinto account different scenarios and possibilities for the progression of G in relation tothose measures and practices. In our simulations, we explore the predictions based on G a , G g , the maximum attained value of G in the past m days as well as possible absoluteincreases or decreases in those rates. 7 (cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224) C u m u l a t i v e nu m b er o f c a s e s Figure 2: Cumulative number of infections according to the STEIR model in Lebanon,with appropriate parameterization with R = 5 . for the first 32 days, R = 0 . for thenext 63 days and R = 1 . until day 130. The actual cases are represented in black andthe predicted cumulative infections are in brown. In our specific parameterization of the STEIR model for the pattern of cases registeredin Lebanon, we find that the initial value of the reproductive transmission factor is re-markably high and starts at R = 5 . but then significantly decreases to R = 0 . at t = 32 days, before moderately increasing to R = 1 . at t = 90 days due to partialrelaxation. This parameterization supplies an accurate fit with the registered cases forthe first days. In addition to absence of any SDM, a relatively high value of R couldbe attributed to late reporting or under reporting of the early cases, hence the fast surgeof registered results during the first few days of official testing. R here reflects the fastpace of spread of the disease in the early days before taking and implementing socialdistancing measures. The rapid fall of the rate from R to R occurs after the implemen-tation and the social commitment to mitigation measures, hence the rapid decrease in thenumber of daily infections and the slow increase in the cumulative number of infections.The very low rate of R sharply diminishes the number of new infections, and the curveof the cumulative number of infections (Figure 2) starts flattening out slowly, before itsnew rise when the SDM are eased. 8
50 100 1500123456 days R e p r o du c t i v er a t e R (cid:72) t (cid:76) Figure 3: The reproduction transmission factor R ( t ) as a function of time t in days. Itstarts at R = 5 . then falls down at t = 32 days to R = 0 . after strong mitigationmeasures, then rises to R = 1 . at t = 90 days. At t = 130 days, we inspect four possiblescenarios of R , with values equal to . , . , . and in black, brown, blue and redrespectively.However, if the measures are relaxed at a time t = 130 days, we expect an increase inthe reproductive rate from R to a another constant value R . The exact value of R willstill depend on the extent of relaxation and the public commitment to SDM. We considerhere four possible values of R (Figure 3): Continued partial mitigation measures with R = R = 1 . , a weakly higher relaxation is parameterized by R = 1 . while morepublic social interaction and moderate relaxation is parameterized by R = 1 . . The lastchoice is R = 2 , corresponding to wider relaxation, yet panic or self-awareness amongpeople helps in preventing R from returning back to high values as those of the initialrate R .We find that the cumulative number of infections will rise again at a higher paceas depicted in Figure 4 even for the lowest increase in R . The number of cumulativeinfections can reach a total varying between to more than infections in aperiod of 50 days, depending on the extent of relaxation. A continued commitment toSDM measures on the same levels as R would lead to a total of infections bythen. We also find that even for the modest value of R corresponding to the currentreproductive rate, an increased number of arrivals from abroad would push the number ofinfections further, depending on the rate of travel inflow in the upcoming days (Figure5). In fact, when we simulate the first days with no incoming travellers, we find that9 (cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224) (cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236) (cid:242)(cid:242)(cid:242)(cid:242)(cid:242)(cid:242)(cid:242)(cid:242)(cid:242)(cid:242)(cid:242)(cid:242)(cid:242)(cid:242)(cid:242)(cid:242)(cid:242)(cid:242)(cid:242)(cid:242)(cid:242)(cid:242)(cid:242)(cid:242)(cid:242)(cid:242)(cid:242)(cid:242)(cid:242)(cid:242)(cid:242)(cid:242)(cid:242)(cid:242)(cid:242)(cid:242)(cid:242)(cid:242)(cid:242)(cid:242)(cid:242)(cid:242)(cid:242)(cid:242)(cid:242)(cid:242)(cid:242)(cid:242)(cid:242)(cid:242)(cid:242)(cid:242)(cid:242)(cid:242)(cid:242)(cid:242)(cid:244)(cid:244)(cid:244)(cid:244)(cid:244)(cid:244)(cid:244)(cid:244)(cid:244)(cid:244)(cid:244)(cid:244)(cid:244)(cid:244)(cid:244)(cid:244)(cid:244)(cid:244)(cid:244)(cid:244)(cid:244)(cid:244)(cid:244)(cid:244)(cid:244)(cid:244)(cid:244)(cid:244)(cid:244)(cid:244)(cid:244)(cid:244)(cid:244)(cid:244)(cid:244)(cid:244)(cid:244)(cid:244)(cid:244)(cid:244)(cid:244)(cid:244)(cid:244)(cid:244)(cid:244)(cid:244)(cid:244)(cid:244)(cid:244)(cid:244)(cid:244)(cid:244)(cid:244)
100 120 140 160 18002000400060008000 days C u m u l a t i v e nu m b er o f c a s e s Figure 4: Cumulative number of infections with different possible SDM relaxation sce-narios after t = 130 days. The figure represents a forecast with R = 1 . , . , . and ,plotted in black, brown, blue and red respectively in correspondence with Figure 3.the cumulative number of infections would have reached only instead of the registered cases on June 30.This means that the disease can swiftly spread again once the measures are relaxed,and the pace of the spread would depend on the level of official and public relaxationof mitigation measures. A combination of loosened measures and a higher rate of travelarrivals would trigger a stronger second phase of infection This is a result numericallyspecific to Lebanon, but the general pattern is a universal outcome and means that asecond wave of COVID-19 infections is inevitable in absence or weakening of SDM. The arithmetic and geometric means proposed in equations (9) and (10) of the repeatediterations model assume that all SDM will be maintained at their current levels in theupcoming m days under consideration. But this is not necessarily the case. In order totake into account possible changes we consider the following scenarios that we simulatein figure 6:1. We determine the arithmetic and the geometric means of the last 14 days andthe corresponding future forecasts are plotted in brown and black respectively. Itis clear that with the rate infection registered in Lebanon, and with more people10 (cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224) (cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236) (cid:242)(cid:242)(cid:242)(cid:242)(cid:242)(cid:242)(cid:242)(cid:242)(cid:242)(cid:242)(cid:242)(cid:242)(cid:242)(cid:242)(cid:242)(cid:242)(cid:242)(cid:242)(cid:242)(cid:242)(cid:242)(cid:242)(cid:242)(cid:242)(cid:242)(cid:242)(cid:242)(cid:242)(cid:242)(cid:242)(cid:242)(cid:242)(cid:242)(cid:242)(cid:242)(cid:242)(cid:242)(cid:242)(cid:242)(cid:242)(cid:242)(cid:242)(cid:242)(cid:242)(cid:242)(cid:242)(cid:242)(cid:242)(cid:242)(cid:242)(cid:242)(cid:242)(cid:242)(cid:242)(cid:242)(cid:242)(cid:242)(cid:242)(cid:242)(cid:244)(cid:244)(cid:244)(cid:244)(cid:244)(cid:244)(cid:244)(cid:244)(cid:244)(cid:244)(cid:244)(cid:244)(cid:244)(cid:244)(cid:244)(cid:244)(cid:244)(cid:244)(cid:244)(cid:244)(cid:244)(cid:244)(cid:244)(cid:244)(cid:244)(cid:244)(cid:244)(cid:244)(cid:244)(cid:244)(cid:244)(cid:244)(cid:244)(cid:244)(cid:244)(cid:244)(cid:244)(cid:244)(cid:244)(cid:244)(cid:244)(cid:244)(cid:244)(cid:244)(cid:244)(cid:244)(cid:244)(cid:244)(cid:244)(cid:244)(cid:244)(cid:244)(cid:244)
100 120 140 160 18002000400060008000 days C u m u l a t i v e nu m b er o f c a s e s Figure 5: Cumulative number of infections with four possible travel influx scenarios after t = 130 days. The figure represents a forecast with R = 1 . and τ multiplied by , and , plotted in black, pink, green and red respectively.recovered, the number of currently infected people would slowly increase during theupcoming couple of weeks, and the geometric and arithmetic means considered leadto very similar forecasts.2. The rate of infection decreases by an absolute value of and the number ofrecovered people is also on the rise so the total number of currently infected peopledecreases faster.3. The rate of progression is defined as the maximum of the rates of increase recordedin the past fourteen days G max = Max (cid:110) G g i − ( m − , G g i − ( m − , ....G g i (cid:111) The number of current infections would witness a steep increase despite of recoveriesand deaths.4. The mean rate of infection increases by an absolute rate of , and the numberof currently infected people would rise quickly despite recoveries or deaths fromprevious cases.As in the STEIR future forecasts, the different scenarios depend directly on the officialmeasures as well as on public behavior and social distancing. If social life returns back11 (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230)(cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224)(cid:236) (cid:236) (cid:236) (cid:236) (cid:236) (cid:236) (cid:236) (cid:236) (cid:236) (cid:236) (cid:236) (cid:236) (cid:236) (cid:236) (cid:236) (cid:236) (cid:236) (cid:236) (cid:236) (cid:236) (cid:236) (cid:236) (cid:236) (cid:236) (cid:236) (cid:236) (cid:236) (cid:236) (cid:236) (cid:236) (cid:236) (cid:236) (cid:236) (cid:236) (cid:236) (cid:236) (cid:236) (cid:236) (cid:236) (cid:236) (cid:236) (cid:236) (cid:236)(cid:242) (cid:242) (cid:242) (cid:242) (cid:242) (cid:242) (cid:242) (cid:242) (cid:242) (cid:242) (cid:242) (cid:242) (cid:242) (cid:242) (cid:242) (cid:242) (cid:242) (cid:242) (cid:242) (cid:242) (cid:242) (cid:242) (cid:242) (cid:242) (cid:242) (cid:242) (cid:242) (cid:242) (cid:242) (cid:242) (cid:242) (cid:242) (cid:242) (cid:242) (cid:242) (cid:242) (cid:242) (cid:242) (cid:242) (cid:242) (cid:242) (cid:242) (cid:242)(cid:244) (cid:244) (cid:244) (cid:244) (cid:244) (cid:244) (cid:244) (cid:244) (cid:244) (cid:244) (cid:244) (cid:244) (cid:244) (cid:244) (cid:244) (cid:244) (cid:244) (cid:244) (cid:244) (cid:244) (cid:244) (cid:244) (cid:244) (cid:244) (cid:244) (cid:244) (cid:244) (cid:244) (cid:244) (cid:244) (cid:244) (cid:244) (cid:244) (cid:244) (cid:244) (cid:244) (cid:244) (cid:244) (cid:244) (cid:244) (cid:244) (cid:244) (cid:244)
110 115 120 125 130 135 140200300400500600700800 days C u rre n t l y i n f ec t e dp e o p l e Figure 6: Number of currently infected people according to the RI model. The coloredlines represent the progression of the number of currently infected people in the next14 days according to five possible scenarios. The lower pink line corresponds to anabsolute decrease of in the current rate of infection. The middle black and brownlines almost coincide and they correspond to the continuation of the current geometricand arithmetic average rate of infection. The red line represents a progression with a ratecorresponding to the maximal attained daily rate of the last 14 days, while the uppergreen line corresponds to an absolute increase of in current average rate.to a more normal situation and the precautions are diminished, the number of infectionswill be on the rise again according to scenarios 3 or 4. The continued enforcement ofprevailing measures will help implement scenario 1 or more optimistically scenario 2 incase of more public commitment and resumption of tougher lockdown (Figure 6). The low infection rate in Lebanon in the recorded in the second month of the COVID-19spread in Lebanon could be attributed to public commitment to social distancing and thestrict mitigation measures applied. The number and the distribution of tests conductedcould also have an impact on determining the real scope and rate of infections.To asses this factor, we compare the number of daily tests conducted per millionof residents, in Lebanon and several other countries that witnessed stronger spreads likeItaly, USA, Iran and Turkey. We can find that the percentage of people tested in Lebanonis similar to that conducted in Iran, and tends to rise up with time but it is significantly12 (cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224) (cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224) (cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224) (cid:224)(cid:224)(cid:224) (cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:242)(cid:242)(cid:242)(cid:242) (cid:242)(cid:242)(cid:242)(cid:242)(cid:242)(cid:242)(cid:242) (cid:242)(cid:242)(cid:242)(cid:242)(cid:242)(cid:242)(cid:242)(cid:242)(cid:242)(cid:242)(cid:242)(cid:242)(cid:242) (cid:242)(cid:242)(cid:242)(cid:242)(cid:242)(cid:242)(cid:242)(cid:242)(cid:242)(cid:242)(cid:242)(cid:242)(cid:242)(cid:242)(cid:242) (cid:242)(cid:242)(cid:242)(cid:242)(cid:242)(cid:242)(cid:242)(cid:242)(cid:242) (cid:242)(cid:242)(cid:242)(cid:242)(cid:242)(cid:242)(cid:242)(cid:242)(cid:242)(cid:244)(cid:244)(cid:244)(cid:244)(cid:244)(cid:244)(cid:244)(cid:244)(cid:244)(cid:244)(cid:244)(cid:244)(cid:244)(cid:244)(cid:244)(cid:244)(cid:244)(cid:244)(cid:244)(cid:244)(cid:244)(cid:244)(cid:244)(cid:244)(cid:244)(cid:244)(cid:244)(cid:244)(cid:244)(cid:244)(cid:244)(cid:244)(cid:244)(cid:244)(cid:244)(cid:244)(cid:244)(cid:244)(cid:244)(cid:244)(cid:244)(cid:244)(cid:244)(cid:244)(cid:244)(cid:244)(cid:244)(cid:244)(cid:244)(cid:244)(cid:244)(cid:244)(cid:244)(cid:244)(cid:244)(cid:244)(cid:244)(cid:244)(cid:244)(cid:244)(cid:244)(cid:244)(cid:244)(cid:244)(cid:244)(cid:244)(cid:244)(cid:244)(cid:244)(cid:244)(cid:244)(cid:244)(cid:244)(cid:244)(cid:244)(cid:244)(cid:244)(cid:244)(cid:244)(cid:244)(cid:244)(cid:244)(cid:244)(cid:244)(cid:244)(cid:244)(cid:244)(cid:244)(cid:244)(cid:244)(cid:244)(cid:244)(cid:244)(cid:244)(cid:244)(cid:244)(cid:244)(cid:244)(cid:244)(cid:244)(cid:244)(cid:244)(cid:244)(cid:244)(cid:244)(cid:244)(cid:244)(cid:244)(cid:244)(cid:244)(cid:244)(cid:244)(cid:244)(cid:244)(cid:244) T e s t s (cid:144) m illi o n i nh a b i t a n t s (cid:244) USA (cid:242)
Iran (cid:236)
Lebanon (cid:224)
Turkey (cid:230)
Italy
Figure 7: The figure shows the number of daily tests conducted per 1 million inhabitantsfor each of Italy (Blue), Turkey (Brown), Lebanon (Green), Iran (Black) and USA (Red).less than the other countries as shown in Figure 7. This suggests that the number ofdetected infections is possibly lower than the actual number of cases.To further asses this effect, it is important to check the ratio of the cumulative numberof infections with respect to the cumulative numbers of tests conducted. This criterionwould give the rate of infected out those of tested, hence eliminates the uncertaintyrelated to under-testing. An examination of the publically available data reveals thatthe cumulative rate of infection among those who were tested in Lebanon was . onJune 24th, 2020, compared to . in the USA, . in Italy, . in Turkey and . in Iran [2]. This is an assertion that the low rate of infection is more related tothe mitigation measures applied in the country, as rates in more affected countries arelarger by several orders of magnitude.However, the relaxation of measures and the rise of the reproductive rate of infectionsdue to increased interaction can easily bring the country back to a high rate of infectionswithin a couple of weeks as the forecast presented in Figure 4 and Figure 6 reveal. This isconfirmed by our future simulations on both of the STEIR and the RI models, followingtwo different methodologies with several possible scenarios. The aforementioned modelsboth forecast a resumption of a quick spread of the disease and an increase in the numberof infected people once the SDM are reduced or abandoned. The severity of the spreaddepends on the extent of the relaxation. The currently achieved pattern of slow thencontrolled spread can slide into a swift and wider spread under looser conditions. Wenote that the RI model, by construction, already accounts for all infected cases, whetherresidents or travellers, but it can only project a short term future behavior of infectious13pread. It is worth mentioning the observation that our data indicates that the mean ageof infected persons has fallen from 43.8 years to 36.7 years old between April 8 until June30, 2020, and this might be related to or caused by the travel inflows during this period,as further analysis may reveal.Under all circumstances, the continued SDM are essential to keep COVID-19 undercontrol until the introduction of effective medications or vaccines, which is estimated totake at least between 12 to 18 months [40], despite ongoing medical and clinical researcharound the globe [41]. This work presented two different models used in the simulation of the spread of infec-tious diseases which are the STEIR model and the RI model. We developed the STEIRmodel, a novel parameterization of the reproductive rate of infection, an improved RImodel and adjusted all relevant parameters to fit the available data from Lebanon. Weanalyzed in detail the current spread and different forecasts for future developments. Wefound out that the rate of infection and the number of infected people fell down quicklydue to the rapid fall in the reproductive number R due to strong mitigation measures.However, relaxing the measures, opening the airport and the resumption of social activ-ity and interaction would swiftly put the infections on a rapid rise again, thus reversingthe temporary success in limiting the spread of COVID-19. Attacking the disease fromdifferent angles allows us to show in different ways that the temporary official and publicSDM have succeeded in halting the spread of the disease, but the resumption of businessand life as usual will put the spread back on track of fast growth. Consequently, SDMshould be maintained in order to safely guarantee controlled spread. Acknowledgements
The authors have no acknowledgements.
Conflict of interests
The authors declare that they have no conflict of interests.
Data availability
All data can be supplied by the authors upon request.14 eferences [1] WHO Director-General’s opening remarks at the media briefingon COVID-19,
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