key: cord-0698422-gs7w2wjs authors: Solari, H. G.; Natiello, M. A. title: Stochastic model for COVID-19 in slums: interaction between biology and public policies date: 2021-01-06 journal: nan DOI: 10.1101/2021.01.06.21249318 sha: dbafa6dc9e5a064c838ee4d293acd70036fbc233 doc_id: 698422 cord_uid: gs7w2wjs We present a mathematical model for the simulation of the development of an outbreak of COVID-19 in a slum area under different interventions. Instead of representing interventions as modulations of the parameters of a free running epidemic we introduce a model structure that accounts for the actions but does not assume the results. The disease is modelled in terms of the progression of viremia reported in scientific works. The emergence of symptoms in the model reflects the statistics of a nation-wide highly detailed database consisting of more than 62000 cases (about a half of the confirmed by RT-PCR tests) with recorded symptoms in Argentina. The stochastic model displays several of the characteristics of COVID-19 such as a high variability in the evolution of the outbreaks, including long periods in which they run undetected, spontaneous extinction followed by a late outbreak and unimodal as well as bimodal progressions of daily counts of cases (second waves without ad-hoc hypothesis). We show how the relation between undetected cases (including the 'asymptomatic' cases) and detected cases changes as a function of the public policies, the efficiency of the implementation and the timing with respect to the development of the outbreak. We show also that the relation between detected cases and total cases strongly depends on the implemented policies and that detected cases cannot be regarded as a measure of the outbreak, being the dependency between total cases and detected cases in general not monotonic as a function of the efficiency in the intervention method. According to the model, it is possible to control an outbreak with interventions based on the detection of symptoms only in the case when the presence of just one symptom prompts isolation and the detection efficiency reaches about 80% of the cases. Requesting two symptoms to trigger intervention can be enough to fail in the goals. Ever since the emergence of COVID-19 (China CDC, 2020) , mathematical models have been proposed to examine, illustrate and forecast the possible evolution of the pandemic, as well as recommending public measures for managing it. Modelling epidemics has to deal with a variety of diculties at dierent levels and the present pandemic is not an exception. On the general level, there is the issue of what mathematical approach is the most appropriate. A number of publications and even more preprints are based on deterministic (i.e., based on ODE's) SIR approaches at dierent levels of sophistication. There even exists a web-page capable of doing the SIR-ODE calculations, graphics, estimates, etc. 1 However, the relevance of such approaches is dubious, the issue of relying in ODE-models has been criticised already 40-50 years ago (Kurtz, 1970 (Kurtz, , 1971 (Kurtz, , 1976 van Kampen, 1981) . On the other hand, stochastic approaches for population dynamics have developed over the last century (McKendrick, 1914; Feller, 1940; Kendall, 1950; Bartlett, 1964) proving to be better suited for the task, since their basic assumptions resemble the actual problems more closely. On the specic level, the amount of activity is (still) so large that the reviewing process cannot match its pace. Contributions are of varying quality, and inhomogeneous, as is often the case in a new and growing subject. There is no agreement around the basic concepts. For example, a good part of the literature has addressed the phenomena of asymptomatic carriers of SARS-CoV-2 (Bai et al, 2020; Cereda et al, 2020; Huang et al, 2020; Nishiura et al, 2020; Sakurai et al, 2020; Mizumoto et al, 2020; Yang et al, 2020) . Unfortunately, the label asymptomatic has been used with dierent meanings, going from 'not presenting the expected symptoms at the moment of infecting someone else', as in (Huang et al, 2020; Bai et al, 2020) to never, in the course of the infection, presenting symptoms (Böhmer et al, 2020) . In all cases, asymptomatic and pre-symptomatic are considered as objective categories pertaining to the relation between the infected person and the infectious agent, thus excluding from examination the actions of the public health system, and therefore preventing any improvement of these actions. In previous modelling work either asymptomatic carriers of SARS-CoV-2 have not been considered or they have been incorporated using an ad-hoc hypothesis, such as that the ratio between asymptomatic and symptomatic cases is constant (see e.g., Hao et al, 2020) . In contrast, our model incorporates a detection component based in what it is known of detection policies. Another sharp dierence with earlier work is that we model a variable contagiousness and not only a variable contagious period, furthermore, intrinsic stochasticity is included in contrast with the extrinsic stochasticity (added a-posteriori 2 ) included in Hao et al, 2020 and few other works. A search in PubMed 3 with keywords covid-19, mathematical, model oered 540 articles. A renement with keywords covid-19, model, asymptomatic, stochastic ends in 6 scientic publications (as of 2020-09-09) plus one news article (not a research article) for a specialised magazine. Of the later 6, Hao et al, 2020 is the most closely related to our work, hence our decision to indicate only the dierences of the present work with a related one among the pre-existing papers. In this work we will take a complex systems view. We begin by acknowledging that the COVID-19 epidemic is no longer a free-running epidemic but rather one in which there is a strong interaction between the public health system and the population dynamics of the outbreaks. Changes in the evolution of an outbreak trigger changes in the consideration of which characteristics of the COVID-19 cases should (or should not) trigger public action. This indicates that there is a clear interaction between these systems and they cannot be considered independent. To illustrate the point we will use the various criteria of COVID-19 case used in our home country (HGS), Argentina, following recommendations by the World Health Organization (WHO). We will produce compartments that relate to the evolution of the case in medical or biological terms as well as to the categories corresponding to the dierent protocols to be applied to the case. The response to an epidemic requires not only the mobilisation of public resources but the participation of the public as well. To organise the actions required for each individual case COVID-hot-lines and web-servers have been organised world wide. Such help services indicate which measures to take to those that suspect they are developing COVID-19, and prompt ocial actions if needed. Hospitals and health centres, as well as help services, are coordinated in their actions by protocols. A main tool of these protocols is the suspectedcase criterion. The suspected case criterion regulates state intervention and depends on clinical symptoms of the (potential) patient and other circumstances. The criterion constitutes a dicult balance between the administration of resources (for example use of Reverse transcription polymerase chain reaction (RT-PCR) kits and laboratories), the developmental stage of the epidemic, the mortality risk of the case and more. As in any decision taken under real circumstances (limited resources), establishing the suspected-case criterion implies trade-os. When diagnostic resources such as RT-PCR tests are limited, a conict emerges: should we reserve them for individual diagnosis or perhaps use them in epidemiological surveillance (triggering actions) as well? In any intermediate cases: in which proportions? Should the general criterion depend on being a contact of a COVID-19 case? Does it make sense to require weaker symptoms for the population which is aware of having epidemiological contact with COVID-19 cases rather than for the communitarian cases that cannot account for how they could have been infected? Actually, it could make sense if by such measures we were able to achieve a more ecient use of a scarce resource to be reserved for diagnosing related to treatment (a private/individual criteria contrasting to public/epidemic criteria). The question must be put: is it correct to focus our attention in travellers and their contacts at the beginning of the outbreak? Is eciency really boosted by requiring two relevant symptoms of a list for potentially communitarian cases and only one to people with epidemiological contacts? In the context of the propagation of SARS-CoV-2, what are the consequences of such decisions? We will address these questions implementing a model apt for answering them. To set the grounds for our model, we analyse data collected by the Public Health Ministry of Argentina, made available to us through the COVID-19 initiative under the Ministry of Science and Technology. The model incorporates medical ndings regarding the transmission of SARS-CoV-2 as well as actions taken by the health authorities and to a certain extent the social behaviour of the population. We apply the model to small slums (variously called in South-America: villas miseria, villas de emergencia, cantegril, favelas, etc.) where the conditions of homogeneous contact, frequently used to simplify the modelling task, are closer to be fullled. We show how the model predicts epidemic circulation below the detection level for surprisingly long periods of time. Also, we illustrate that average epidemics are not good representatives to grasp the dynamics, and that the undetected (mild, unrecognised, presymptomatic, asymptomatic) cases are in good proportion the result of public policies coupled to the characteristics of the illness. The outcome of three forms of surveillance and public action are comparatively analysed. In Section 2 we describe the model, from its basis supported in both biology and social behaviour all the way to the algorithm implementing a Markov Jump process (Feller, 1940; Kendall, 1950) . Results are presented in Section 3 and discussed in the following Section 4. Section 5 nally sums up the Conclusions. . CC-BY 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review) The copyright holder for this preprint this version posted January 6, 2021. ; https://doi.org/10.1101 https://doi.org/10. /2021 or recent close contact with a conrmed (positive PCR test) or probable case (a PCR-tested suspected case without a conclusive result). By February 22nd 5 severity and hospitalisation were no longer required for (A) and dyspnea was recognised among possible symptoms. By March 9th 6 , the considered situations were three: acute respiratory infection (with at least one among fever, cough and diculty in breathing) without other aetiology and either (A) recent presence in areas of local transmission of the disease or (B) close contact with probable or conrmed cases. The third situation considered (C) cases presenting severe acute respiratory infection (fever and at least one symptom of respiratory disease) requiring hospitalisation and without another aetiology that fully explains the clinical presentation. This new item acknowledges the existence of the illness regardless of any presence in risk zones or close contact with probable or conrmed cases. Also the concept of close contact evolved during the period. By January 31st risk contacts 7 considered only recent (within 14 days) travel or cohabitation with a COVID-19 patient (apart from special considerations for sanitary operators). The concept evolved to that of close contact, becoming highly detailed in what regards social distance (2m, 15 min) and hygiene already by February 27th 8 . At the end of May, specic instructions for contact tracing 9 (already operative, though) had been developed. The criteria for identication of cases shifts focus along the pandemic. At the beginning, the focus is in the virus import from other regions where it is active, while the local diusion becomes relevant only some weeks/months later. The trade-o in the identication generates classes of contagion depending on the criterion. Along with the case criteria, surveillance and control criteria are developed. At the beginning of the pandemic, passive surveillance (i.e., to wait for the spontaneous appearance of patients, except perhaps for travellers) was the most common attitude, while soon after many countries developed dierent degrees of contact-tracing (with varying success), even revealing preexistent aws in the various national health and care systems. In Appendix A we show the evolution of the criteria in Argentina and its relation with Italy's case. The decision of what to consider a suspected case, and when further actions are to be taken, is a critical one. However, it is not clear which is the overall criteria, meta-criteria, adopted by Italy or Argentina, presumably upon recommendations of OMS. It appears that the meta-criterion is to keep an even level of certainty of being a COVID-19 case for each individual case. It is then pertinent to explore whether this goal is achieved or not and if such goal is epidemiologically sound. 5 February 22nd resolution accessed 2020-08-26 6 March 9th resolution accessed 2020-08-26 7 January 31st resolution accessed 2020-08-26 8 February 27th resolution accessed 2020-08-26 9 May 29th resolution accessed 2020-08-26 5 . CC-BY 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review) The copyright holder for this preprint this version posted January 6, 2021. 11 . Health workers can be assumed to be more accurately monitored than other patient groups. At June 6th, the criterion for suspicious case for health workers was changed to presenting one symptom belonging to the set: fever, cough, anosmia, dysgeusia, dyspnea, odynophagia (see Appendix A). In August 4th, the set of symptoms was extended to headache, diarrhoea and vomits. For health workers, 98% of the cases that reported symptoms 12 presented at least one symptom in the extended set. Among them, 48% were diagnosed as COVID-19 cases using RT-PCR. Considering cases reporting at least two symptoms, the number of cases falls by 23% but the positive cases within the group move up only to 50%. If the criterion is fever and one symptom, the case fall is 64% while the positivity within this smaller set raises to 60%. Similar trends are found for the whole patient data set. The data indicates that requiring more symptoms results in missing positive cases. The improvement in positivity rates is outnumbered by the large or very large fall in detected cases, with no signicant improvement in the use of resources. At the early stages of the epidemic only hospitalised patients with pneumonia were considered as possible COVID-19 cases, in such case the detection ability drops to less than 10% of the cases showing symptoms. 10 The Argentine Ministry of Health provides on a daily basis an anonymised copy of the data set corresponding to the nation-wide reported cases in epidemic outbreak for the National Science Council (CONICET). 11 There are 493 cases with reported symptoms where none of the symptoms match the HS expectation. 12 Report of symptoms is not an obligation for the sanitary units 6 . CC-BY 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. The copyright holder for this preprint this version posted January 6, 2021. We assume therefore that the viremic load is the biological origin of both the severity of illness for an average infected individual and the capability to transmit the virus. In simpler words, the quantity of virus in each individual regulates how ill she/he is and with which eciency the infection can be passed along. Symptoms, severity and contagiousness are dierent from person to person, but they follow an approximate sequence from zero up to a maximum value, subsequently decaying towards zero again. From the day of clear symptom onset we adopt a model for the viremic load, based in early ndings (Woelfel et al, 2020; Böhmer et al, 2020) from the initial period of the pandemic where individual cases could be traced in detail. We model the viremia from day 5th to 11th using a gamma distribution. The presymptomatic period (a period usually of weak symptoms) is modelled in three stages, a rst non contagious compartment lasting a day in average, followed by a low contagion compartment, lasting on average two days, with the same viremic level than the last day of contagion and nally followed by a compartment with higher contagiousness lasting on average one day. The duration and distribution of the presymptomatic days, from contagion to symptoms, described in this form is supported by the distribution of the times between the appearance of earlier symptoms and the day of diagnostic for the data collected in Argentina (see Figure 2 ). In fact, the observed mean for the data points is 3.86 ± 0.5 days, plus one day without any symptoms, yielding slightly less than ve days before the onset of recognised symptoms and the decision of swabbing. After the presymptomatic period, symptoms usually appear clearly until they gradually decline. We assume the symptomatic compartments to last in average one day each, with viremic levels as in the nal part of Fig. 1 . For the sake of dealing with a pandemic, symptoms in themselves are only an ingredient. They facilitate the possibility of detecting infected patients, especially when the pandemic constrains the sanitary authorities to keep a passive attitude. In any case, the appearance of symptoms on each individual depends not only on the viremic load but also on the individual condition of each patient. On the other hand, regardless of if and when symptoms appear, the two processes driving the evolution of the pandemic are contagiousness and detec-7 . CC-BY 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. The copyright holder for this preprint this version posted January 6, 2021. ; https://doi.org/10.1101/2021.01.06.21249318 doi: medRxiv preprint The modelling prole is summarised in Table 2 . The social component reects the ability of the sanitary authorities to enforce measures in order to (a) detect infected individuals and reduce the chances of contagion (by isolation, hospitalisation, etc.) and (b) eectively inuence social behaviour, aiming to reduce the chances that infected, undetected individuals may transmit the disease. The detection of cases as a function of the surveillance protocol The decision of admitting a case as a probable case of COVID-19 depends not only on the biological/health condition of the case (i.e., the viremic level, presence of symptoms, etc.) but also on the expectations of the health services, HS, as we have discussed in Section 2.1.1 and Appendix A. Since the chances for a contagious person to produce new cases depends on a-priori expectations, the expectations change the removal rate of contagious people (e.g., by isolating the person). Furthermore, the condition of being suspected a-priori is mostly hered-8 . CC-BY 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. The copyright holder for this preprint this version posted January 6, 2021. ; https://doi.org/10.1101/2021.01.06.21249318 doi: medRxiv preprint Figure 2 : Distribution of time between initial symptoms and swabbing calculated using 121223 entries corresponding to the cases during the month of August 2020 (dots) and curve t y(x) = 421627( (histogram). The observed average time in Argentina is of ∼ 3.86 ± 0.5 days. The tted curve is the composition of two exponentially distributed stages, 2.10 and 1.86 days long in average. It is important to understand that the data reects not only a biological matter but it is also aected by public health decisions, the information of the population and self diagnosing of the patient concerning the initial symptoms. As such, the statistical error is not the most relevant error. At the beginning of the outbreak the average time was longer than 5 days. . CC-BY 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. The copyright holder for this preprint this version posted January 6, 2021. itary. The suspicion increases the probability of detection and the detection of a case makes those infected by the case more likely to be detected. Let us call T , traceable, those with larger probabilities of detection a-priori, and U,untraceable, those with smaller probabilities of detection. Let us further consider the limit situation where all T are traced and detected with certainty and no U is ever detected. Such an idealised, limit situation will result in two independent epidemics, for no T can ever produce a U case and reciprocally, no U can produce a D case. No real situation is expected to reach this limit case, hence, in a more accurate description U cases are detected with lesser probability and later than T cases. Also, some T may escape tracing and detection when still contagious. The inheritance of the tracing classes is then imperfect and there is only one, mixed-type, epidemic. We represent this situation by a probability table (written in matrix form) The probabilities, P (X by Y ) indicate the probability for a susceptible person infected by a contagious case of type Y of becoming a case of type X assuming it was eectively infected. The non-negative quantities , s are not new parameters since we have to satisfy that P (T by T ) is exactly equal to the probability of a T case being eectively detected. The same can be said of P (U by U ) with respect to the undetectable cases U . Since all health systems have limited resources and suer dierent epidemic impacts, dierent strategies are likely to appear. One of the goals of this manuscript is to explore the impact of dierent strategies on the (local) evolution of the pandemic. Schematically, we will consider three scenarios, labelled 10 . CC-BY 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. The copyright holder for this preprint this version posted January 6, 2021. ; passive, intermediate and active representing dierent policies for the detection process. In the passive policy intervention starts when and if the symptoms are clear. The intensity of the perceived symptoms is assumed to be, on average, proportional to the viremic state. A distinction is made between the T and U , being the HS's more prone to act for the T group than for the U group. The passive policy represents the policies adopted during the early days of the pandemic (mid-February to mid-march 2020 in Europe), where the HS focused attention on imported cases (travellers) and their contacts. The intermediate policy reects the situation in which the HS become aware of the problem of presymptomatic contagious cases, and begin to track oligo-symptomatic cases in the T group (contacts of known cases). At the same time, it has been observed a lowering on the requirements, in terms of a lesser number and a larger set of symptoms, required for sanitary intervention (isolation). The active intervention consists in one of two possibilities: either the T class is substantially enlarged by including in it the contacts of contacts, as was done in Italy or by dropping the distinction between U and T and acting (or strongly exhorting to individual action) on cases presenting any symptom compatible with COVID-19, no matter how weak, as it was the public advise of e.g., the Swedish HS 13 . We will only model the second case. Mathematical/Computational support The general approach is based on a Markov-jump process following the setup of the Feller-Kendall (Feller, 1940; Kendall, 1950) algorithm. The compartments X i , i = 1, · · · , N involved in the process are the dierent classes of individuals taken into account (to be described below) and the stochastic dynamics evolves by expressing the number of individuals on each compartment as a function of time. Transitions between compartments are given by Markov jumps triggered by dierent events and characterised by an event probability rate W α (X), α = 1, · · · , E. The relation between events, compartments (populations) and stochastic dynamics is given by where X 0 i is the initial condition for compartment i, n α (t) indicates the number of occurrences of event α up to time t and δ α i is an integer indicating how each occurrence of event α modies the population in compartment i. For the present problem, δ will take the values −1, 0, 1, meaning that e.g., one infected individual is removed from the contagious process by isolation, etc. The stochastic dynamics proceeds by establishing the behaviour of n α (t) . General properties of Markov jump processes are assumed to hold for this problem, in particular that events are independent of each other (although re-13 March 13th recommendation (in Swedish) General information (English) accessed 2020-1013 11 . CC-BY 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. The copyright holder for this preprint this version posted January 6, 2021. ; https://doi.org/10.1101/2021.01.06.21249318 doi: medRxiv preprint 1. The waiting time to the next event is exponentially distributed with rate 2. At the occurrence time indicated above, the probability of occurrence for A realisation of this stochastic dynamical process requires a good knowledge of the probability rates W α and the computation of one random number (exponentially distributed) for the time of occurrence of the next event and another (uniformly distributed) for selecting the event happening at that time. Upon occurrence of each event, populations and consequently transition rates are updated according to eq.(1). Reported curves are the average of a few realisations of this process. Random numbers were generated with the Double precision SIMD-oriented Fast Mersenne Twister (dSFMT) algorithm (Saito and Matsumoto, 2009) , implemented in C. The algorithm is implemented as a C-programme, fully available from github. The compartmental structure is as follows (see Fig. 3 ): There exist three classes of compartments, namely susceptible S, traceable infected T and untraceable infected U . Infected individuals belong in several sub-compartments describing the degree of evolution of their disease (or rather their infective period). At each stage, they may proceed in the disease to the 12 . CC-BY 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. The copyright holder for this preprint this version posted January 6, 2021. ; https://doi.org/10.1101/2021.01.06.21249318 doi: medRxiv preprint R) , thus ceasing in all such situations to be a source of contagion. Infection may proceed either by contact of T or U individuals with an S individual, or by importing the infection from outside the system in consideration. What regards infection by contact, the tracing of infections is usually not complete, for various reasons. To take this fact into account, we assume that a portion of infections by T individuals (of size < 1 in Table 3 ) may remain undetected and also that a portion of infections by U individuals (of size s < 1 in Table 3 ) will eventually become detected. The quantities s and were discussed in Section 2.1.3 and will be further specied below. Two additional uniformly distributed random numbers r 1 and r 2 (in [0, 1]) are computed to decide these outcomes (double arrows from S in Fig. 3) , representing the probability pairs {1 − , } and {s, 1 − s} respectively for T and U infected individuals. For the imported infections taking place outside the system, the uniform random number r 0 distributes the resulting infected individuals among T and U with proportions {1 − η, η}. In Table 3 we describe the expressions adopted for the dierent rates and their action on the population (i.e., the nonzero values of the incidence matrix {δ} α i ). Considering the nature of the available data, the time-unit is (day) −1 , i.e., transition rates are given per day. In the table, N is the size of the population, typically a neighbourhood or other region that can be safely assumed to behave homogeneously (basically, that any individual may in principle meet any other individual; a natural assumption for working places, schools, etc.). Initial conditions for all simulations is that most individuals (N − c) are in compartment S while the remaining c are in T 0 or U 0 (we assume c is typically around 13 . CC-BY 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. The copyright holder for this preprint this version posted January 6, 2021. ; https://doi.org/10.1101/2021.01.06.21249318 doi: medRxiv preprint 2 for N up to a few thousands). The import rate a S N describes infected individuals undergoing contagion outside the system. We include in this event the possibility of travellers bearing the infection when returning to the system after a temporary absence, a group that has been important in global scale to transfer the disease across continents, but is comparatively small for such stable communities as those we consider. The evolution of the illness is given by stages T k , k = 0, · · · , K − 1 and similarly for U ) describing the viremic level V (k) at each stage ( Figure 1 and Table 2 ). In this work, K = 9. pr describes the rate of passage to the next viremic stage i..e., (pr) −1 is the average permanence of an individual on each stage (second column in table 2). The factor βV (k) describes the contagion rate per each encounter between a susceptible and an infected individual. In principle, β T and β U may be dierent but we have not explored that possibility. indicates what portion of the individuals infected by a T will not be detected by the health services during the contagion phase of that case, while similarly s indicates what portion of the individuals infected by a U will eventually be detected by the control procedures. Similarly, the constant η describes the distribution of imported cases among T and U compartments. Since η is largely unknown we will only consider the extreme cases. We have η = 0 usually associated to long distance travels to/from regions of viral circulation. For the case of slums, casual contagion within the same city but in e.g., dierent neighbourhood is expected to be the most frequent case, hence we adopt η = s. Finally, rem is the rate of removal of an individual from stage k out of the contagion chain. This rate also depends both on the viremic level and on the HS strategies (giving dierent choices for the factors d T (k), d U (k)). This part of the model will be described in detail in the next Subsection. The data usually discussed in the news and websites is the number of con- are tracked, and concern us, are those that avoid early extinction and come to be noticeable. The actual evolution of the pandemic is intrinsically stochastic. Borrowing from the modelling language, there is only one realisation of the real process, namely the one we are currently experiencing. There is no second run, although many weakly coupled contagion chains may be running simultaneously 14 . CC-BY 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. The copyright holder for this preprint this version posted January 6, 2021. ; https://doi.org/10.1101/2021.01.06.21249318 doi: medRxiv preprint within e.g. a larger city. With this in mind, we stress that the averaging of realisations is not a substitute for the real process. It has a limited value, in that it highlights features that are recurrent, while it smears out what is less frequent. Moreover, no realisation of the stochastic process is more true than any other. Predictions based only on the averaging of realisations may serve as a clue about what to do, but policy decisions should take into account the whole picture. We consider in detail the mechanisms of contagion and removal, as well as their relation with both the evolution of the disease in the infective individual and the HS policies. Contagion within the system is taken to be strictly proportional to the viremic levels V (k). The proportionality constants β T, β U may vary according to social strategies and attitudes. The eventual removal of an infected individual in the model is governed by the competition between two mutually exclusive events. Either the individuals evolve to the next stage in the viremic levels (i.e., they are still infected and capable of contagion) or they are removed from the contagion chain for whatever reason (detection, isolation, full recovery or death). At stage k, the probabilities P X m (k) of moving to the next stage in the contagion chain and P X r (k) of being removed from the chain for an individual of class X = {T, U }, can be described (in the notation of Table 3 ) as: where B k X , X = {T, U }, model the HS policy adopted. In the present implementations, the factors B k X are set to zero for an initial subset [0, · · · , k 0 − 1] of stages (k 0 ≥ 1) and take the same positive value B X for the remaining stages, k ∈ [k 0 , K − 1]. B X relates to the probability of X being eectively detected (named 1 − and s in Section 2.1.3) through Eqs.2 below. At the nal stage, K − 1, the overall action of both competing events is a removal from the contagion chain. Individuals that have not been removed at any previous stage, have eectively participated in the contagion chain during all of their contagious period. These individuals were not detected by the HS policies while they still were active in the contagion chain. The overall probability of detection can be computed as follows. Let Q X k be the probability of removal up to and including stage k for infected individuals of class X = {D, U } . Set further, Q X −1 ≡ 0. For any stage i, . CC-BY 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. The copyright holder for this preprint this version posted January 6, 2021. ; https://doi.org/10.1101/2021.01.06.21249318 doi: medRxiv preprint which can be restated as Q X i = 1 − P X r (i) Q X i−1 + P X r (i). The total probability of removal during the infective period is Q X K−1 , while the probability of being detected at some point during the infective period for individuals in class T, U is given by Note that Q T K−1 and Q U K−1 are rational functions, the ratio of two polynomials of degree K − k 0 . Eq. 2 relates the value of the constants B T , B U for the dierent HS policies with the probability of detection. The dierences in dealing with T and U infected individuals follows from the dierences between Q T K−1 and Q U K−1 , being the HS's more prone to act for the T group than for the U group. We distinguish three main policies: Intervention on the U class concerns only severe cases (e.g., requiring hospitalisation) in a situation where the viremic levels of the patient are comparatively high. In the model, intervention for the U class starts at k 0 = 3 (stage 3 in 2). The conditions required for sanitary intervention (isolation) in the U group are broadened in terms of a lesser number and a larger set of symptoms and possibly intervention at an earlier stage. Active (preventive) intervention, as in contact tracing, starting at stage k 0 = 0 (or 1), is implemented for the T group. It reects a situation in which the HS become aware of the problem of presymptomatic contagious cases, and begin to track oligo-symptomatic cases in the T group (contacts of known cases). No distinction is made between U and T regarding actions of the HS. One symptom is enough to trigger sanitary actions. Interventions start at stage k 0 = 0 (or 1). In the next section we discuss a few scenarios based on these policies, relating to data from F + (for fever plus other), corresponds to the same detection probabilities as above for the T group, whereas for the U group the detection probabilities s are set to 0.10, 0.28 and 0.36. The latter corresponds roughly to the proportion of 16 . CC-BY 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. The copyright holder for this preprint this version posted January 6, 2021. Unless otherwise stated, all simulations are performed with 5000 individuals, of which two are initially contagious in the T compartment (it makes only imperceptible dierence to set the initial contagion in the T group or the U group), while the contagion rate is set to β = 2.5 and there is a small rate of external contagion (ext = 0.002). A list with the parameter values used in dierent scenarios can be found in Table 4 . Other necessary input data for running the simulations is: number of realisations (usually 100), length of simulation in days, initial condition for populations S, T, U (usually 4998, 2, 0), random number seed, ag to discard early extinctions (positive integer) and maximal duration to be considered early (usually 19 days). The following results follow from the structure of the model. There is essentially nothing left to prove, just following the construction in 2.3.2. Lemma 1. Q X K−1 is a monotonically increasing function of B X . In modelling language, B X senses the eciency of the detection process. Lemma 2. For xed B X , Q X K−1 is decreasing with increasing k 0 . 17 . CC-BY 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. The copyright holder for this preprint this version posted January 6, 2021. Simulation results allow us to compare the outcomes of dierent policies on an equal footing. Before considering averaged results let us sense the spread of outcomes from dierent realisations of the process. In Figure 4 we show the fraction of susceptible individuals as a function of time for 100 realisations of the stochastic process in two dierent congurations. The left panel corresponds to an situation where the probability of detection while still contagious is 21% for the T -group and 9% for the U -group, with β = 2.5. All outcomes display a sharp fall in the number of susceptible individuals. Notice however the spread in time: The fastest and slowest realisations dier in about 40 days, corresponding to 100% at the 0.5 level. The right panel corresponds to a weaker contagion situation (β = 1.75), where the probability of T -detection increases every 60 days, from 0.6 through 0.71 up to 0.79 (all detections starting on stage 1), while the U -detection goes from 0.07 through 0.58 (with detection starting on stage 3) up to 0.79, with detection starting on stage 1. Notice here the spread in the outcome. While some realisations display almost no variation in the fraction of susceptible individuals, some others achieve a fall of over 50%. It is worth to keep in mind that the policy of progressively increasing the detection eort has been the rule in practical cases. . CC-BY 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. The copyright holder for this preprint this version posted January 6, 2021. ; https://doi.org/10.1101/2021.01.06.21249318 doi: medRxiv preprint As in most models with homogeneous contact, the initial growth of the epidemic outbreak is almost exponential and this regime lasts for about two months in the present simulations with about 5000 initial susceptible individuals. However, it is worth to indicate that the growth exponent of infected cases and that of detected cases is not the same, being the latter smaller than the former, specially in less eective regimes as H and F +. As a consequence, basic reproductive numbers inferred from the early development of the pandemic that had assumed that detected cases are roughly proportional to the actual cases underestimate the growth rate. See Figure 5 . Note that the gap between growth rates is larger for the lower detection eort as compared with the higher (red/green vs blue/magenta pairs). The ratio between total cases and detected cases of COVID-19 has been the subject of several works. In particular Malani et al (2020) ; Muñoz et al (2020) address the situation in slums, reporting ratios of 10 : 1 (Malani et al, 2020) and 5 : 1 (Muñoz et al, 2020) . The latter study was performed at least one month after most cases occurred, although with the outbreak still running. The majority of the registered cases had occurred before June 6th., when the tracking method in use was of type F +. After June 6th., the tracking sharpened to any two symptoms, a medium form of I. We show averaged ratios in Figure 6 but it is worth to keep in mind that there are usually large uctuations present. The gure shows that in all situations there is a tendency to a sharp increment of the ratio at the beginning of the outbreak followed by a maximum level and subsequently a monotonic decrement. . CC-BY 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. The copyright holder for this preprint this version posted January 6, 2021. ; https://doi.org/10.1101/2021.01.06.21249318 doi: medRxiv preprint The most remarkable features present in the simulations are the diverse forms in which the stochasticity and the particularities of the contagious process manifest globally. Despite being seeded with two traceable cases, it is not uncommon (probability larger than 0.01) to observe the outbreak to remain with sporadic cases up to 50 days and only then the recognisable bell-shaped of the daily cases begins. We show one of such cases in Figure 7 left panel. In the centre panel we show a two waves outbreak under policy IH and a dierent shape of two waves with a long delay between them. The dierence among realisations suggests that stochastic epidemic outbreaks are not just an average outbreak plus noise. As expected, the epidemic size depends strongly on the policy applied. It is 20 . CC-BY 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. The copyright holder for this preprint this version posted January 6, 2021. While extinctions of the epidemic with a low number of cases are always possible, they are infrequent in the Low intensity case, they begin to be noticeable in the Medium intensity and are dominant in the High intensity situation. This transition is known as the stochastic equivalent of the transition in deterministic equations when the basic reproductive number moves from above one to below one and has been discussed elsewhere (Nåsell, 1996 (Nåsell, , 2002 (Nåsell, , 2001 . The number of detected cases, i.e., people diagnosed as infected with SARS-CoV-2, depend in no trivial form of the detection policy and intensity. We show in Figure 9 that the relation is not monotonic. In general, an increase in detection eciency from medium to high intensity may result in a decrease in the number of cases detected (policies F + and I) but also in an increase (H ). In fact, the design of the model only assures that the probability of detection increases with increasing intensity. If the total number of cases is low, the total number of detected cases will also be low, despite a higher detection probability. We can see as well that the eorts made with a passive policy (H ) produce only little changes in the development of the epidemic. In the present model biological aspects are intertwined with sanitary policies. These policies are not considered in terms of their desired eects traduced as eective parameters of an otherwise free running epidemic but rather mechanistically, changing not only parameter values but the structure of the model as well. By doing so, we allow policies to manifest not only what they were intended for, but also unexpected features. The same can be said with respect to the coupling between intrinsic randomness and dynamics which results not only in the, usually expected, decoration with daily uctuations of the 21 . CC-BY 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. The copyright holder for this preprint this version posted January 6, 2021. The eective coupling of control measures and intrinsic randomness brings us into the realm of possible/predictable developments of outbreaks that cannot be described with the standard (intuited) approach. Although it should be clear from the setup adopted in this work, it is worth recalling that all realisations of a stochastic model are on an equal footing. Any of them respond to the process in its own right. A strength of the present approach is the capability of displaying a variety of possible epidemic outcomes. Indeed, Figure 4 , Right and 7 show that dramatic dierences in epidemic size (for the same stochastic process), second waves and late development of outbreaks are not unlikely to occur. Taking averages over many realisations projects out part of the information given by the stochastic process. Yet, in some cases, averages may be illustrative of general trends. In particular, Figures 5 and 6 show that the ratio of undetected to detected cases is not constant in the course of an epidemic and even worse, the growth rate of undetected cases is larger than that of detected cases. Hence, epidemic size is likely to be underestimated when computed through recorded cases. Finally, Figure 8 illustrates how the stochastic outcomes can be translated into probabilities of e.g., having a given epidemic size. . CC-BY 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. The copyright holder for this preprint this version posted January 6, 2021. ; https://doi.org/10.1101 https://doi.org/10. /2021 One important goal of this work is to assist in the issue of resource allocation when dealing with a pandemic. HSs throughout the world dier in equipment, logistic capabilities, exibility, etc., depending on the preexisting policies and infrastructure. The working conditions dier even locally within the same city, as discussed above. Where should resources go? Will low-cost (and lower eciency) strategies under a longer period of time be preferred to high-cost (and higher eciency) strategies with a shorter time-span? In our model we mimic the HS decisions by considering two groups of individuals: Those that are early identied and recognised as potential patients, T , and the rest, U , of which the HS is initially unaware. We do not deal with nancial costs, but we can compare highly-eective and less eective strategies throughout time. Preventive intervention strategies accrue costs in terms of isolating contagious people and testing. The kind of intervention considered in the present work is based upon tracking oligosymptomatic people but not searching for completely asymptomatic cases, thus a good indicator of costs is the total number of cases detected. The best strategy of all those considered in this respect is the I strategy with an ecient, H, search method, which is able to inhibit the development of outbreaks. The strategy has fewer detected cases and smallest overall size ( see Fig. 8 , right panel and Figure 10 , upper left panel). We compare two distributions of detected cases for equal eciency of detection, see Figure 10 . The comparatively few detected in the I panel of Figure 10 constitute most of the outbreak, while F + adds a larger number of undetected cases. A short side of the I policy is that the eective suocation of an epidemic outbreak in slum areas cannot by itself prevent recurrent late outbreaks triggered by external contagion (see Figure 11 ). Hence, the alert state of the HS will have to be maintained for longer times. However, the advantages of the I strategy under a medium or low detection success, thus having a higher failure rate in avoiding outbreaks, is not so considerable. As it can be seen in the gure, combining a suboptimal policy with a suboptimal tracking (F + M , lower right panel) is expected to be more cost eective than an optimal policy with suboptimal tracking (I M , lower left panel) in terms of detection, although the overall size of the epidemic is expected to be lower in the I situation, while the necessary social eort is larger. Despite our simulations have been seeded in all cases with two infected people and only epidemic outbreaks that do not get extinguished for 19 days have been considered, some runs do not develop an outbreak and some others produce an outbreak only because of external contagion (from outside of the simulated neighbourhood), an eect than can occur in a completely dierent time scale as is shown in 11 Also, we illustrate that average epidemics are not good representatives to grasp the dynamics, and that the undetected (mild, unrecognised, presymptomatic, asymptomatic) cases are in good proportion the result of public policies coupled to the characteristics of the illness. Limitations: As mentioned previously, one of the assumptions of the present model is the homogeneity of contacts through the population. For that reason, 23 . CC-BY 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. The copyright holder for this preprint this version posted January 6, 2021. ; https://doi.org/10.1101/2021.01.06.21249318 doi: medRxiv preprint . CC-BY 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. The copyright holder for this preprint this version posted January 6, 2021. ; https://doi.org/10.1101 https://doi.org/10. /2021 it only makes full sense when applied to small communities. The proper path to surpass this constraint is to raise the level of detail, identifying subpopulations with some common property (e.g., age segregation, mobility, local connement, etc.) that are in weakly mutual interaction. This is a costly approach from the point of view of experimental design, since each new level of detail demands a detailed understanding of the specic interactions. Some eort in this direction has been to identify superspreaders, a possibility that recently became interesting. (Edholm et al, 2018) The modelling goal of this work was to conceive mechanisms for the interplay of the epidemic disease and the adopted social measures. The epidemic is not just biologically given in terms of e.g., a basic reproductive number or a herdimmunity level that are taken to be virus-specic and independent of social organisation. The belief in a biologically given epidemic leads to the false alternative between herd-immunity and vaccine, with the hidden assumption that social behaviour cannot (or must not) be modied. On the contrary, this work suggests that what we (collectively) do inuences the level of risk to which we are exposed. Social behaviour can modify epidemic outcomes (Bavel et al, 2020) . We observe that an increment in the number of daily detected cases does not necessarily imply an improvement on how the epidemic is being managed, nor a worsening of the outbreak. Case-detection cannot be understood separately from the HS policy. Lower detection may be an indicator of success in the proper context. Hence, to translate the statistics for one country to another country is far from straightforward. More locally, the transfer of information from detected (registered) cases to estimated number of cases from seroprevalence studies is not independent of the adopted HS policy and depends as well of the timing with respect to the development of the outbreak. Randomness plays a substantial role in COVID-19 dynamics, a role that departs from the signal+noise analysis framework. Low frequency, or coherent uctuations, are relevant at the level of outbreaks in slums and there is no reason to believe the same is not going to be true in larger, heterogeneous, settings. The immediate consequence is that averaging and uncontrolled approximations to the average outbreak will be aligned with intuitions but misaligned with a reality displaying a largely unpredictable form. The stochastic behaviour is aected as well by the social management of the epidemic, coupling two usually neglected contributions and making prediction of outcomes even more dicult. The intervention of health authorities had been from below in most countries. By from below we mean a sequence of interventions going from nonintervention and passing through increasing levels of action until reaching lock downs in desperation. Such an approximation has to be revised, it is an approach that privileges something dierent than people's health. If our model is correct, it is possible to control the outbreaks with interventions that target mostly the symptomatic population. Such a method will have to target 25 . CC-BY 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. The copyright holder for this preprint this version posted January 6, 2021. ; https://doi.org/10.1101/2021.01.06.21249318 doi: medRxiv preprint for isolation of any one presenting a single symptom of those compatible with COVID-19. The cost of more certainties is to lose control of the outbreak, being forced to apply lock downs, thus immobilising the productive forces of the healthy people rather than the comparatively small group that is potentially infected by SARS-CoV-2. The decision of requiring more symptoms to declare a case as COVID-19 suspect whenever the patient has no identied contact with conrmed cases facilitates the circulation of the virus even when a highly ecient detection protocol is used. As an ethical matter, asymptomatic cases are better considered as undetected cases. The term asymptomatic puts the blame on the virus and helps to dispense social failures. In contrast, undetected places the burden on society and should help to x attention in what we can do better. . CC-BY 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. The copyright holder for this preprint this version posted January 6, 2021. CC-BY 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review) The copyright holder for this preprint this version posted January 6, 2021. ; https://doi.org/10.1101/2021.01.06.21249318 doi: medRxiv preprint The early evolution of the criteria (we omit the specications for sanitary operators) for a case to be suspicious in Argentina is as follows (we indicate correspondence with Italy's resolutions): February 29, 2020 (Corresponds to Italy's January resolution) There are two ways to be considered a suspected case: The person has fever and one or more respiratory symptoms (cough, diculty in breathing, odynophagia) without another aetiology that fully explains the clinical presentation, and that in the last 14 days either has been in contact 29 . CC-BY 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review) The copyright holder for this preprint this version posted January 6, 2021. ; https://doi.org/10.1101/2021.01.06.21249318 doi: medRxiv preprint with conrmed or probable COVID-19 cases or has a history of travel or presence in areas with local transmission of SARS CoV-2 March 16, 2020 (Corresponds approximately to Italy's March 9th resolution) In addition to the previous situation, a new suspected case begins to be considered: Any person with severe acute respiratory disease who requires mechanical ventilation due to their respiratory symptoms, without other aetiology that explains it, even without epidemiological link. Travel history to specic countries is substituted by travel abroad. Specications are given for severe acute respiratory disease, dened as pneumonia and one of the following: To the requirements raised on March 5, in the case of symptoms that could accompany fever, the following is added: anosmia/dysgeusia. Anyone presenting fever (37.5ºC or more) and one or more of (cough, odynophagia, diculty in breathing, anosmia, dysgeusia) of recent presentation, without another clinical explanation AND precedents of 30 . CC-BY 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review) The copyright holder for this preprint this version posted January 6, 2021. ; https://doi.org/10.1101/2021.01.06.21249318 doi: medRxiv preprint travelling to (or residence in) places with viral circulation or contact with conrmed cases of COVID-19 Anyone presenting anosmia/dysgeusia is to be observed by 72hs and then tested. Health workers with two or more of the described symptoms. Anyone presenting two or more of (fever -37.5ºC or more-, cough, odynophagia, diculty in breathing, anosmia, dysgeusia) AND (having being present in a zone with viral circulation OR residing in a popular neighbourhood -slum-) OR requiring hospitalization) Health workers and any with close contact with a COVID-19 case presenting ate least one symptom. Three symptoms added to the set: headache and vomits and diarrhoea. Added symptom: myalgia Anyone presenting fever (37.5ºC or more) and one or more of (cough, odynophagia, diculty in breathing, anosmia, dysgeusia) of recent presentation, without another clinical explanation. Anyone presenting anosmia/dysgeusia. For health workers and inhabitants of popular neighbourhoods the requisite is of one symptom. . CC-BY 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review) The copyright holder for this preprint this version posted January 6, 2021. ; https://doi.org/10.1101/2021.01.06.21249318 doi: medRxiv preprint Presumed asymptomatic carrier transmission of covid-19 The relevance of stochastic models for large-scale epidemiological phenomena Using social and behavioural science to support covid-19 pandemic response Investigation of a covid-19 outbreak in germany resulting from a single travel-associated primary case: a case series Limit theorems and diusion approximations for density dependent Markov chains Seroprevalence of sars-cov-2 in slums and non-slums of mumbai, india Studies on the theory of continuous probabilities, with special reference to its bearing on natural phenomena of a progressive nature Estimating the asymptomatic proportion of coronavirus disease 2019 (covid-19) cases on board the diamond princess cruise ship Vigilancia y seroprevalencia: Evaluación de anticuerpos igg para sars-cov2 mediante elisa en el barrio popular villa azul The quasi-stationary distribution of the closed endemic sis model Extinction and quasi-stationarity in the verhulst logistic model Endemicity, persistence and quasi-stationarity Modeling population dynamics based on experimental trials with genetically modied mosquitoes Estimation of the asymptomatic ratio of novel coronavirus infections (covid-19) Monte Carlo and Quasi-Monte Carlo Methods Natural history of asymptomatic sars-cov-2 infection Linear processes in stochastic population dynamics: Theory and application to insect development Clinical presentation and virological assessment of hospitalized cases of coronavirus disease 2019 in a travel-associated transmission cluster Comparison of clinical characteristics of patients with asymptomatic vs symptomatic coronavirus disease 2019 in wuhan, china HGS acknowledges support from the University of Buenos Aires. MAN acknowledges support from Kungliga Fysiograska Sällskapet i Lund (2018-2020). We thank the members of the Red de modelización de enfermedades infecciosas" (CONICET, Argentina) for valuable discussions, and particularly Verónica Simoy for help in tracking public policy in Argentina and Ignacio Simoy who has helped us to analyse the database using the R language.