key: cord-185125-be11h9wn authors: Baldea, Ioan title: What Can We Learn from the Time Evolution of COVID-19 Epidemic in Slovenia? date: 2020-05-25 journal: nan DOI: nan sha: doc_id: 185125 cord_uid: be11h9wn A recent work (DOI 10.1101/2020.05.06.20093310) indicated that temporarily splitting larger populations into smaller groups can efficiently mitigate the spread of SARS-CoV-2 virus. The fact that, soon afterwards, on May 15, 2020, the two million people Slovenia was the first European country proclaiming the end of COVID-19 epidemic within national borders may be relevant from this perspective. Motivated by this evolution, in this paper we investigate the time dynamics of coronavirus cases in Slovenia with emphasis on how efficient various containment measures act to diminish the number of COVID-19 infections. Noteworthily, the present analysis does not rely on any speculative theoretical assumption; it is solely based on raw epidemiological data. Out of the results presented here, the most important one is perhaps the finding that, while imposing drastic curfews and travel restrictions reduce the infection rate kappa by a factor of four with respect to the unrestricted state, they only improve the k{appa}-value by ~15 % as compared to the much bearable state of social and economical life wherein (justifiable) wearing face masks and social distancing rules are enforced/followed. Significantly for behavioral and social science, our analysis of the time dependence k{appa} = k{appa}(t) may reveal an interesting self-protection instinct of the population, which became manifest even before the official lockdown enforcement. an interesting self-protection instinct of the population, which became manifest even before the official lockdown enforcement. In the unprecedented difficulty created by the COVID-19 pandemic outbreak, 1 mathematical modeling developed by epidemiologists over many decades 2-7 may make an important contribution in helping politics to adopt adequate regulations to efficiently fight against the spread of SARS-CoV-2 virus while mitigating negative economical and social consequences. The latter aspect is of paramount importance 8 also because, if not adequately considered by governments currently challenged to deciding possibly under dramatic circumstances and formidable tight schedule, it can jeopardize the healthcare system itself. As an effort in this direction, we drew recently attention 9 to the general fact that the spread of the SARS-CoV-2 virus in smaller groups can be substantially slowed down as compared to the case of larger populations. In this vein, the time evolution of COVID-19 disease in the two million people Slovenia certainly deserves special consideration, as on 15 May 2020, concluding that this country has the best epidemic situation in Europe, Prime Minister Janez Janška declared the end of the COVID-19 epidemic within Slovenian borders. 10 Subsequent developments (only four new cases between 15 and 24 May, 11 cf. Table 1 ) have fortunately given further support to this declaration. Attempting to understanding and learning from this sui generis circumstance is the very aim of the present paper. Thanks to long standing efforts extending over many decades, a rich arsenal of theoretical methods of analyzing epidemics exists. Most of them trace back to the celebrated SIR model 2-7 wherein the time evolution of the numbers of individuals belonging to various epidemiological classes (susceptible (S), infected (I), recovered (R), etc) classes is described by deterministic differential equations. Unfortunately, those approaches need many input parameters 12,13 that can often be reliably estimated only after epidemics ended, 14 which un-avoidably compromises their ability of making predictions. As an aggravating circumstance, one should also add the difficulty not encountered in the vast majority of previous studies: how do the input parameters needed in model simulations change in time under so many restrictive measures (wearing face masks, social distancing, movement restrictions, isolation and quarantine policies, etc) unknown in the pre-COVID-19 era? Estimating model parameters from data fitting in a certain time interval to make predictions can easily run into a difficulty like that described in the first paragraph of Section 2.3. As shown below, our approach obviates the aforementioned difficulty. We will adopt a logistic growth model in a form which is different from that often employed in the past [15] [16] [17] [18] [19] (see Sections 2.3 and 3 for technical details). This model is considerably simpler than SIR flavors, and already turned out to be an appealing framework in dealing with current COVID-19 pandemic issues. 9,13 Logistic functions (see equation (2) below) were utilized for studying various problems. [20] [21] [22] [23] [24] [25] Studies on population dynamics of epidemic populations [26] [27] [28] [29] [30] [31] [32] were also frequently based on the logistic function. Nevertheless, as anticipated, there is an important difference between the present approach (Section 2.3) and all the other approaches of which we are aware. The latter merely justify the logistic model by the fact that recorded disease numbers followed a sigmoidal curve. Shortcomings of this standpoint are delineated in the beginning of Section 2.3. The strength of the approach presented in Section 2.3 is that we do not use data fitting. Rather, we use raw epidemiological data to validate the logistic growth and straightforwardly extract the time dependent infection rate, which is the relevant model parameter for the specific case considered and makes it possible to compare how efficient different restrictive measures act to mitigate the COVID-19 pandemic, and even to get insight significant for behavioral and social science. To briefly remind, standard logistical growth in time t of an infected population n = n(t) follows an ordinary differential equation containing two constants (input model parameters): the (intrinsic) infection rate κ(> 0) and the so called carrying capacity N. In a given environment, the latter has a fixed value to which the population saturates asymptotically (lim t→∞ n(t) = N). This can be seen by straightforwardly integrating equation (1) with the initial condition n(t)| t=t 0 = n 0 , which is often recast by using the half-time τ ≡ t 0 + 1 κ ln N n 0 − 1 , n(t) t=τ = N/2. Noteworthily for the discussion that follows, equation (2) assumes time-independent model parameters. In epidemiological language, n(t) gives the cumulative number of cases at time t. Plotted as a function of t, the derivative with respect to time (throughout assumed a contiguous variable)ṅ(t) ≡ dn/dt, representing the "daily" number of new infections, is referred to as the epi(demiological) curve. Before proceeding with the data analysis let us briefly summarize relevant public heath measures, social distancing and movement restrictions imposed during the COVID-19 crisis in Slovenia. 11, 33 The first case of coronavirus was confirmed on March 4, 2020, imported via a returnee traveling from Morocco via Italy. 34 On 10 March, the government banned all incoming flights from Italy, South Korea, Iran, and China; the land border with Italy was closed for all but freight transport; indoor public with more than 100 persons were prohibited, sporting and other events with more than 500 participants were allowed only without audience. The (blue) curve of Figure 2a depicting the evolution of total COVID-19 infections in Slovenia (underlying data are collected in Table 1 ) has an appealing similarity to the logistic S-shaped curve depicted in Figure 1 . One would be therefore tempting to follow numerous previous authors, [26] [27] [28] [29] [30] [31] [32] who claimed that the logistic model applies merely because of the (apparently) good data fitting. Still, to claim that a description based on a model like that of equation (2) is valid, checking that the model parameters do not depend on the fitting range (t 1 , t 2 ) is mandatory. For the specific case considered here, this means that fitting numbers of infected individuals in time range t 1 < t < t 2 should yield, within inherent statistical errors, values of N and κ independent of t 1 and t 2 . And, like in other known cases, 35, 36 this is just the stumbling block for the logistic function approach delineated in Section 2.1. In particular, the infection rate κ should not depend on how broad the is the range (t 1 , t 2 ); however, we checked by straightforward numerical calculations that it does. Given the real epidemic timeline delineated in Section 2.2, the infection rate must indeed depend on time, κ = κ(t). If the contrary was true, all containment measures would be useless. But when κ depends on t, equations (2) and (3) no longer apply; they were deduced by integrating out equation (1) assuming a time-independent κ. Fortunately, rather than merely inquiring how good the fitting curve based on equation (2) is, we are able to directly check (and demonstrate, see below) the validity of a time-dependent logistic model merely based on the real epidemiological reports. To this aim, we recast the differential equation (1), which is the basic definition of the logistic growth (not to be confused with the logistic function of equation (2)), as followṡ When put in this way, one can straightforwardly get insight in how to proceed. One should plot the ratio of the daily new cases to the cumulative number of cases (numerator and denominator in equation (4), respectively) as a function of the cumulative number of cases and inspect whether the curve is linear or departs from linearity. Is the decrease linear (like anticipated in the ideal simulation presented in Figure 1b) , we have the demonstration that the logistic growth model applies. The curve constructed as described above using the COVID-19 epidemic reports for Slovenia (Table 1, Noteworthily, the low-κ regime comprises two periods: lockdown and lockdown easing. To quantify the differences between these two periods, we used equation (4) We believe that especially the two results shown in Figure is that what really matters is not to keep everyone at home ("Italian approach") but rather to impede virus transmission ("German approach"), e.g., by wearing masks, adequate hygiene, and social distancing. Infection transmission does not strongly increase upon easing as long as face masks and social distancing prevent SARS-CoV-2 virus spreading. One should add at this point -an important fact that appears to be currently inadequately understandthat, along with a less pleasant effect of a short-term slight increase of the daily new cases, a moderate increase in the infection rate also has a positive impact. It reduces epidemic duration; compare the right tail the green and orange curves in Figure 3b . (ii) The fact that the carrying capacity N does not change upon lockdown easing is equally important. This is the maximum number of individuals that can be infected in a given environment. Rephrasing, the maximum number of infected individuals does not increase when the lockdown is released; the total carrying capacity of a given environment does not change. From a methodological perspective, one should emphasize the important technical strength of the approach proposed above, which made it possible to arrive at the aforementioned conclusions. It is only the differential form, equation (1), of logistic growth employed that obviates the need for any additional theoretical assumption. The traditional approach of validating the logistic model by blind data fitting using its integral counterpart, equation (2), does not work for COVID-19 pandemic applications because the model parameter κ can and does depend on time. This time dependence κ = κ(t) is essential to properly assess and make recommendations on the efficiency of the restriction measures to be enforced against SARS-CoV-2 virus spread. And just because, in its differential form utilized here, the logistic model merely requires directly "measurable" epidemiological quantities (daily reportsṅ(t) and cumulative number of cases n(t), cf. equation (1)) makes in the present unsusual situation this model an alternative preferable to other more elaborate SIR-based flavors. The latter models contain a series of quantities that cannot be directly accessed "experimentally". Governments confronted to taking decisions under unprecedented time pressure cannot await confirmation of often speculative theoretical hypotheses needed in data processing. Before ending, let us also note that monitoring the κ(t)-timeline allowed us to get insight also relevant for behavioral and social science; the self-protection instinct of the population became manifest even before the official lockdown enforcement (cf. Section 2.3). 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