Epidemiological projections for COVID-19 considering lockdown policies and social behavior: the case of Bolivia
aa r X i v : . [ phy s i c s . s o c - ph ] A ug Epidemiological projections for COVID-19 considering lockdownpolicies and social behavior: the case of Bolivia
M. L. Pe˜nafiel ∗ and G. M. Ram´ırez- ´Avila † CBPF - Brazilian Center for Research in Physics,Xavier Sigaud st. 150, zip 22290-180, Rio de Janeiro, RJ, Brazil. Instituto de Investigaciones F´ısicas,Universidad Mayor de San Andr´es, Casilla 8635, La Paz, Bolivia. (Dated: August 17, 2020)
Abstract
We assess the epidemic situation caused by SARS-CoV-2 using Tsallis’ proposal for deter-mining the occurrence of the peak, and also the Susceptible-Infected-Recovered-Asymptomatic-Symptomatic and Dead (
SIRASD ) compartmental model. Using these two models, we determinea range of probable peak dates and study several social distancing scenarios during the epidemic.Due to the socioeconomic situation and the conflictive political climate, we take for our study thecase of Bolivia, where a national election was originally scheduled to occur on September 6th andrecently rescheduled to be on October 18th. For this, we analyze both electoral scenarios and showthat such an event can largely affect the epidemic’s dynamics. ∗ mpenafi[email protected] † [email protected] . INTRODUCTION COVID-19 pandemic has shook the world. Since the official notification of the detec-tion of the novel corona-virus, a huge scientific effort has been made all over the worldin many respects, e.g. the investigation of a prospective treatment or vaccine [1, 2], theproper mechanisms of the virus and new epidemiological models for the disease spreading[3], among others. On the other hand, the pandemic control around the world has testedthe sanitary and economic policies of almost any country in the world, leading most of thecountries to impose lockdown periods [4]. Underdeveloped or developing countries representa particularly dangerous scenario for the spreading of COVID-19, since both scientific andgovernmental efforts are strongly limited due to the lack of economic and human resources.Latin America has become the new epicentre for the spreading of COVID-19 during thepast month, with specially concerning cases in Brazil, Mexico, Ecuador, Per´u and Chile,with an overall of over 3 million cases as of July, 2020. Bolivia, in its own right, is aparticularly concerning country due to the high percentage of population living on a ba-sis of informal economy , recent political events leading to exacerbated polarization, anda deficient healthcare system. Moreover, a national election was originally scheduled forSeptember 6, this kind of event implies the mobilization of over 7 million people across thecountry (representing about 70% of whole population). Furthermore, the epidemiologicalpeak for Bolivia is expected to occur in late August or early September according to thepublic health authorities , meaning that the national election was going to take place in theworst possible scenario: the epidemiological peak. Based on their predictions; recently, thelocal electoral authorities have re-scheduled the poll day to occur on October 18.Across Latin America, several local efforts (see [6, 7] and the references therein) have beenmade in order to predict and forecast the epidemiological peaks in each country (c.f. [8, 9])taking into account, for instance, several political, economical [10] and social measures [11].These works are of particular importance in the actual crisis due to the need of governmentsfor taking the less harmful actions in order to overcome the current scenario.Specifically, the current situation (as of July 2020) in Bolivia is alarming. Due to theeconomic urgency, the rigid quarantine was lifted on June 1 in most of the country, and Bolivian informal economy reached 62.3% of the Gross Domestic Product (GDP) as of 2018 according tothe International Monetary Fund (IMF) [5]. Nevertheless, no public data about these projections is available. rigid quarantine before and after the epidemic peak, assessingthe potential case reduction and appearance of a second peak following the first one. Forthis respect, in Sec. II we use a fit model proposed by Tsallis and Tirnakli [9] with thepurpose of fitting the Bolivian data for active cases and estimate a time range for the peakdate of the epidemic curve. In Sec. III we review the
SIRASD compartmental modelwhich accounts for a fraction of the infected population being asymptomatic, which, as isknown, is the case for COVID-19. Furthermore, we estimate the epidemiological parametersfor the early evolution of the epidemic in Bolivia and propose several scenarios for theevolution of the epidemic curves. In Sec. IV we discuss the results obtained for the differentepidemic scenarios proposed; finally in Sec. V we conclude and offer perspectives for furtherdevelopments.We emphasize that the intention of the present work does not consist in predictinga date for the occurrence of the epidemiological peak, but rather on the analysis of thesocial distancing policies near the epidemic peak and near an extraordinary event implyinga massive concentration of persons that might dramatically heighten the contagion: theelection day. 3
I. FITTING THE BOLIVIAN DATA
The first case of COVID-19 in Bolivia was diagnosed on March 9th [13] and, up toJuly 12th the local health authorities have accounted for 47200 positive cases. Tsallis andTirnakli [9] inspired on a stock-market model proposed an analytic function that fits several full epidemiological curves for countries such as China, South Korea, France, etc. with theaim of forecasting their epidemiological curves. They have proposed the following functionalform for the behavior of the COVID-19 active cases N = C ( x − x ) α (1 + ( q −
1) ¯ β ( x − x ) ¯ γ ) / (1 − q ) , (1)where C > , α > , ¯ β > , ¯ γ > , q > x accounts for the time elapsed since the firstcase measured in days. Eq. (1) is known as a q-exponential function. The parameters in (1)are interpreted as ( C, x ) being trivial parameters depending on the country’s populationand initial day of the pandemic with respect to China’s data , respectively; ( q, ¯ γ ) beingdisease-specific parameters that may account for the biological aspects of COVID-19 andare quite universal in the many cases analyzed, with q ≈ .
26 and ¯ γ ≈ α, ¯ β )account for the country’s particular spreading of the disease, which include social distancingpolicies, the adequate control of the isolation of infected persons and might, as well, involveparticular environmental conditions that could alter the disease propagation and its severity[14, 15].Figure 1 shows the active cases data fitted with Eq. (1), the relevant parameters for thisfit are C = 2 . × − , α = 3 . β = 2 . × − while the pair (¯ γ, q ) = (3 , .
26) is fixedto the Chinese parameters since the local epidemic in Bolivia has not reached its peak yet.We observe that Fig. 1 is in good agreement with the available data. It is clear that for theparameter fitting, its quality is improved with the quantity of available data. Interestinglyenough, we can perform such a parameter estimation for available data at different datesand calculate both the epidemic peak date and its height. Figure 2 shows this calculation fordifferent available data. According to the fit function, (1), for the available data proposedin [9], the Bolivian epidemic peak is expected to occur in the range [Aug-6,Sep-6], whichcoincides with local authorities’ information divulgation. Since x represents a shift in the origin for the beginning of the country’s epidemic, we will take x = 0for sake of simplicity. II. MODEL
The compartmental epidemiological model we use in the present work is the so-calledSusceptible-Infected-Recovered for Asymptomatic-Symptomatic and Dead (
SIRASD ) model[16] modified in order to account for the sorting of the total population in two groups [10],those who are not economically obliged to break social distancing measures (i.e., the fractionof the population possessing a formal work), and those who need to break social distancingmeasures due to economical reasons (i.e., the fraction of the population living in the informal sector of the economy). This model describes the evolution of a disease accounting for theinfected population consisting of asymptomatic ( A j ) and symptomatic (infected) individuals( I j ) in each group; which, as is known, is the case for the COVID-19 pandemic. For sakeof simplicity, only the symptomatic fraction of the infected individuals is susceptible to die FIG. 1. Data for the active cases in Bolivia as of July 11th, the fit corresponds to the functionproposed in [9]. According to this fit, the peak of infections is expected to occur on the day 160,i.e. September 5th. IG. 2. Projections for the peak date and height of active cases corresponding to data availableon successive dates according to Tsallis’ fit. S j d t = − X k =1 φ j φ k ( β A A k + β I I k ) S j N , (2a)d A j d t = (1 − p ) X k =1 φ j φ k ( β A A k + β I I k ) S j N − γ A A j , (2b)d I j d t = p X k =1 φ j φ k ( β A A k + β I I k ) S j N − γ I I j , (2c)d R j d t = (1 − r ) γ I I j + γ A A j , (2d)d D j d t = rγ I I j , (2e)d N j d t = − rγ I I j , (2f)where S j accounts for the susceptible fraction of the population, A j for the asymptomaticfraction, I j for the symptomatic fraction (which is more likely to be tested positive), R j for the recovered fraction of the total infected individuals and D j for the fraction of thepopulation that dies from the disease, the sub-index j takes account of the group label;consequently, for our case it can be either 1 or 2. Since this model assumes a portion ofthe population dies from the disease, the total number of individuals is not constant overtime and is given by N ( t ) = N ( t ) + N ( t ) = P j =1 ( S j + I j + A j + R j − D j ). Furthermore,the parameter φ j represents the noncompliance degree of the social distancing measurescorresponding to the group j [10]. Note that Eqs. (2) take account of intragroup interactions( φ j φ j ) as well as interactions between members of different groups ( φ j φ k , j = k )For the purpose of estimating the parameters of the model we first need to use the simpler SIRD model [16], which does not assume an asymptomatic fraction of the population andconsists of the population composed by only one group. This model is given by the sameequations as (2) with the detail that the compartments A j are removed, as well as theircorresponding parameters β A and γ A [16].The first 2 COVID-19 cases in Bolivia were diagnosed on March 9, 2020. Soon after, thecentral government took several social distancing policies, such as the closure of educationalestablishments at all levels and a partial quarantine, that consisted on a reduction of theworking hours of public and private institutions, including the suspension of cultural, reli-gious and sporting events; however, these measures did not impose, in practice, limits forpotential crowds to gather, e.g., public transportation and political manifestations, among7thers. These measures lasted for 13 days until March 22, where the government announceda rigid quarantine, allowing citizens to stay out of their homes 1 day per week and banningboth public and private transportation. Assuming that this final measure is reflected inthe data with a delay of ∼ SIRASD model) and minimize the square errorwith respect to the relevant parameters [16]. Hence, we estimate the relevant parameters inthe early stage of the epidemics using the
SIRD model as β I ∼ . γ I ∼ . r ∼ . β A ∼ . , γ A ∼ , , p ∼ . SIRASD model us-ing these parameters, additionally we plot the 95% confidence interval for these parameters.Since the scope of the work is to give a qualitative description of social distancing measures,for sake of clarity, the rest of the epidemiological curves are plotted only using the estimatedparameters.With the intention of assessing the behavior of the epidemic in Bolivia we will explorethree scenarios:1. The free evolution of the epidemiological curves, where region I encompasses the first27 days of the epidemic and φ = φ = 1; region II covers the rigid quarantine period,from day 27 to day 67 and we set φ = 0 . φ = 0 .
9. Finally, region
III encompassesthe rest of the epidemic with lax social distancing measures and φ = 0 . φ = 0 . t off after the peak, as in [10]. Regions I and II are the same as in scenario 1, while region III ends at t in = t peak − n, n ∈ N with t peak = 166. Region IV starts at t in andends at t off with t in and t off having variable values. Finally, region V goes from t off to the end of the epidemic. The values of φ j for the first three regions are the sameas in scenario 1, while for region IV we set φ = 0 . , φ = 0 .
94 and region V has φ = 0 . , φ = 0 . IG. 3. Early evolution for infected individuals for the parameters estimated for the
SIRASD model assuming one population only. The red points are the real data for infected cases, the blueline is the estimated infection curve and the green region represents the 95% confidence intervalfor the estimated parameters. to day 181 (and on October 18th, day 223), and no social distancing policies areadopted during the peak. Such a democratic event implies the massive mobilizationof people not only on the election day but also on periods both before and after theevent. For this respect we will assume that massive mobilizations ( φ = φ = 1) takeplace one week before and after the event. For this scenario we have regions I and II unaltered and region III goes up to day 174 for September (216 for October); for region IV we set φ = 1 and it goes up to day 188 for September (230 for October). Finally,region V recovers the values of III for φ and lasts until the end of the evolution. IV. RESULTS
We set Bolivia’s population to be N = 11677580. In order to solve the set of Eqs. (2) weconsider the initial conditions S ( j )0 = f j N − A j (0) − I j (0) − R j (0) − D j (0) according to thefirst reported cases informed by the Ministry of Health. Thus, we set I (0) = 2, A (0) = 2and the rest of the initial conditions to 0. f j is the fraction of the total population which9 IG. 4. Scenario 1. Region I corresponds to φ = φ = 1, region II corresponds to φ = 0 . , φ =0 . III to φ = 0 . , φ = 0 . belongs to each social group j ; therefore we have 1 = f + f . For the Bolivian case, we set f = 0 .
38 and f = 0 .
62 assuming that group 2 consists exclusively on the fraction of thepopulation living on a basis of informal economy according to the IMF data [5].Scenario 1 is shown in Fig. 4. This scenario shows a behavior that is very close to
FIG. 5. (Left) Scenario 3 with t off = 7. For the parameters chosen for region IV there is asubstantial drop in the number of active cases during this stage. (Right) Scenario 3 with t off = 14,for this configurations there is no second peak after region IV . In both plots we set t in = 160. IG. 6. Dependence of the peak height and the day of occurrence of this peak with respect to t in . the official notified cases . In this scenario, the epidemiological peak is expected to occuraround day 166 (i.e., around August 22th) assuming that the overall parameters will remainthe same until the end of the epidemic. This projection for the peak date also coincideswith the range of dates obtained in Sect. II and with official authorities information.Different configurations for t off in scenario 2 for t in = 160 are shown in Fig. 5. For thechosen values of φ for region IV in this scenario, it may be seen that the epidemic peak ofscenario 1 is avoided, and the peak may be shifted to occur before the initial predicted date.Fig. 6 shows this behavior, by varying the starting date of the rigid quarantine around thepeak we show that both the height and the peak day can be lowered by varying t in . In fact,for the given scenario we show that it is possible to engineer an optimized quarantine. Forinstance, in Fig. 6 there is an optimal day for the beginning of the quarantine, where thenew peak shall occur before the initial predicted day, therefore offering for the possibility ofan anticipated economy re-opening, and a considerable reduction in the number of infectedpeople at the peak, allowing for the healthcare system to be able to handle the peak. Fig. 6shows that such value for t in is day 152.The national election scenario is shown in Fig. 7. For the case of the poll day occurringon September 6th, the popular mobilizations before and after the democratic event mightinduce a second peak almost immediately after the first one. This kind of behavior, besides See the site for up to date data. could be a less disadvantageousscenario for the potential spreading of the disease. This behavior is mainly due to the factthat the election day is far away from the epidemic peak. Thus, there are less persons inthe mobilized population that can be potential (symptomatic or asymptomatic) carriers ofthe disease. V. CONCLUSIONS AND PERSPECTIVES
The socioeconomic particularities of Bolivian society play an important role in the dy-namics of the epidemic, since about 62% of the economy relies on informal work, i.e., on a day-to-day based economy. The success rate for any social distancing policy heavily dependson the engagement of the second group at any time of the epidemic.We have shown that the imposition of a rigid quarantine at any time of the epidemic canaffect its dynamics. Furthermore, it is possible to avoid reaching an epidemiological peakby carefully designing a social distancing policy involving the strengthening of all public
FIG. 7. (Left) Scenario 3 for the poll day set on September 6th. There exists a second peak dueto the massive mobilizations near the elections. (Right) Scenario 3 for the election day set onOctober 18th. There exists a slight delay in the diminution of the number of active cases due tothe elections. Based on electoral authorities’ information, this is the last possible date to have an election. soft socialdistancing imposed in the early stage (closure of schools, reduced working hours) but alsofor the non-regulated activities during that period (mainly consisting in crowds gathering inpublic transportation, cultural events, etc.). This fact affects the rest of the evolution of themodel presented in Section III, since φ = 1 at the beginning of the development, this valuerepresents the social scenario of March 2020 and any other scenario with φ < drastically changed due to the social behavior in themidterm. Thus, the peak can easily be shifted for occurring before or after the expectedtime; and even, it might be lowered.The vast heterogeneity of environments and the different palliative actions taken in dis-tinct regions of the country imply that each region possesses a particular dynamical scenariofor the evolution of the epidemics. However, the numerical evidence shows that the out-breaks’ overall behavior can be affected by the imposition of social distancing measures atany time of the epidemic. Of course, it is better to have a rigid quarantine near the peakdue to the overcharging of the healthcare system. We highlight that social distancing policyalone cannot be entirely sufficient if other measures do not accompany it. For instance,those involving the insurance that most of the persons will be able to obey the quarantine(especially involving people belonging to the informal economy group), the strengthening ofthe healthcare system in general including the diagnosis phase, an adequate case trackingand enhancement of treatment resources for the population.13mong further developments of this work, we intend to enhance the epidemiologicalmodel to account for different age groups, imposing various social distancing measures toeach group. For instance, official data reveals that during the early stage of the epidemic,most of the reported cases (and deceases) consisted of elderly persons. In contrast, as forlate July (after having loosened the first rigid quarantine), most of the infected cases accountfor persons in working age. It could also be important to incorporate in future work theeffect of re-contagion. Finally, We consider that the noncompliance degree for each group isan important indicator of social behavior, and it deserves further in-depth analysis. ACKNOWLEDGMENTS
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