What is a COVID-19 case?

We review the evolution of the definition of ‘case’ along with the development of the pandemic. In many countries, this definition emerges from the national health authorities, following recommendations from the WHO. By 27 January 2020, there were comparatively few cases outside China. Apart from special considerations for sanitary operators, the definition of suspected case from the Italian health authoritiesFootnote ⁴ considered two situations: (A) severe acute respiratory infection (fever, cough and request for hospitalisation) and presence in risk zones a few days before the onset (at that moment mainly Wuhan/Hubei), or (B) acute respiratory infection and either recent presence at the Wuhan live animal market or recent close contact with a confirmed (positive PCR test) or probable case (a PCR-tested suspected case without a conclusive result). By 22 February,Footnote ⁵ severity and hospitalisation were no longer required for (A) and dyspnoea was recognised among possible symptoms. By 9 March,Footnote ⁶ the considered situations were three: acute respiratory infection (with at least one among fever, cough and difficulty in breathing) without other aetiology and either (A) recent presence in the areas of local transmission of the disease or (B) close contact with probable or confirmed 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 confirmed cases.

Also the concept of close contact evolved during the period. By 31 January, ‘risk contacts’Footnote 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 (2 m, 15 min) and hygiene already by 27 February 2020.Footnote ⁸

At the end of May, specific instructions for contact tracing Footnote ⁹ (already operative, although) had been developed.

The criteria for identification of cases shifts focus along the pandemic. At the beginning, the focus is on the ‘virus import’ from other regions where it is active, while the local diffusion becomes relevant only some weeks/months later. The trade-off in the identification 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. Soon after, many countries developed different degrees of contact tracing (with varying success), even revealing preexistent flaws in the various national health and care systems.

In Appendix A, we show the evolution of the criteria in Argentina and its relation to 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 the WHO. 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.

We discuss this issue with data from Argentina.Footnote ¹⁰ In Table 1, we report PCR results for health workers after 6 June, from the dataset of 5th October 2020 with 62 920 cases with symptoms information (29 958 positive and 32 962 negative).Footnote ¹¹ Health workers can be assumed to be more accurately monitored than other patient groups. On 6 June, the criterion for suspicious case for health workers was changed to presenting one symptom belonging to the set: fever, cough, anosmia, dysgeusia, dyspnoea, odynophagia (see Appendix A). In 4 August 2020, the set of symptoms was extended to headache, diarrhoea and vomits.

Table 1. Health workers

Number of confirmed PCR-positive and -negative cases displaying at least one or two symptoms (1s, 2s) from the set given in the text. F +: cases with fever plus another symptom. H: cases requiring hospitalisation. Positivity odds in dataset are 0.9089. P _X( ± ) stands for the probability of having one symptom or more of SARS-CoV-2 being positive (negative) for each category. Odds ratios (ORs) are the ratio of the odds under the symptoms condition to the odds in the full set.

For health workers, 98% of the cases that reported symptomsFootnote 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 dataset.

The data indicate 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 significant 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 cases, the detection ability drops to less than 10% of the cases showing symptoms.

Viraemia, symptoms and contagiousness

An important ingredient of any model concerning the evolution of the disease requires the description of a contagion mechanism at the individual level. It is important to relate when, how much and how long a person is in a contagious condition to the evolution of the disease in the agent.

Upon contagion, the infected individual gradually develops larger and larger levels of virus, in pace with the viral reproduction capabilities in the infected patient. Eventually, a maximum level is reached and the viraemic load subsequently declines along with a recovery from infection. This process may be interrupted at any time because of complications, be them virus-based or any other.

We assume, therefore, that the viraemic 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 efficiency the infection can be passed along.

Symptoms, severity and contagiousness are different 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 viraemic load, based on early findings [Reference Böhmer18, Reference Woelfel21] from the initial period of the pandemic where individual cases could be traced in detail. We model the viraemia from days 5 to 10 using a gamma distribution. The presymptomatic period (a period usually of weak symptoms) is modelled in three stages, a first non-contagious compartment lasting a day in average, followed by a low-contagion compartment, lasting about 2 days, with the same viraemic level than the last day of contagion and finally followed by a compartment with higher contagiousness lasting also about 2 days.

The duration and distribution of the presymptomatic days, from contagion to symptoms, described in this form is supported by the distribution of times between the appearance of earlier symptoms and the day of diagnostic for the data collected in Argentina (see Fig. 2). In fact, the observed mean for the data points is 3.86 ± 0.5 days, plus 1 day without any symptoms, yielding slightly less than 5 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 1 day each, with viraemic levels as in the final part of Figure 1.

Fig. 1. Viraemic levels displaying a 2-day stage with low virus levels, followed by a stage with higher viraemic levels but still no detection symptoms. Bars represent days, not stages.

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 viraemic load, but also on the individual condition of each patient.

On the contrary, regardless of symptoms (if and when they appear), the two processes driving the evolution of the pandemic are contagiousness and detection. The first one is of course mandatory since there is no pandemic without infections. Both these processes have a social component and a biological component. The biological component was discussed above: we assume both the probability of detection and the probability of contagion to be proportional to the viraemic levels of infected individuals, modelled according to Figures 1 and 2. This assumption rests partly on the observations in [Reference Böhmer18, Reference Woelfel21]. However, a recent study ([Reference Combes22], appeared after this submission) suggests that there are other mechanisms operating as well, depending on the patient's response to the infection and deserving a deeper analysis. The modelling profile is summarised in Table 2. The social component reflects 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) effectively influence social behaviour, aiming to reduce the chances that infected, undetected individuals may transmit the disease.

Fig. 2. Distribution of time between initial symptoms and swabbing calculated using 121 223 entries corresponding to the cases during the month of August 2020 (dots) and curve fit y(x) = 421 627 (histogram). The observed average time in Argentina is of ${\sim} {\kern 1pt} 3.86 \pm 0.5$ days. The fitted 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 reflects not only a biological matter, but it is also affected 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.

Table 2. Normalised viraemic levels describing an average evolution of the disease

The probability of contagion is assumed to be proportional to the viraemic levels along the different stages. The levels enter the modelling of the probability of detection as well (along with other important influences such as sanitary policies). The levels are normalised so that ${\mathop \sum \nolimits_i}V_{i}t_{i} = 1$, where t _i is the duration and V _i is the viraemic levels.

About slums

Slum areas have a very specific social structure. They have high-population density and intense poverty, resulting in homes shared by several generations (sometimes with just one bedroom), larger number of homeless people in comparison with the rest of city, precarious services (water, electricity, sewage, etc.), as well as strong internal ties and social organisations such as community run food assistance and other internal solidarity networks. ‘Stay at home’ policies cannot be sustained, to the point that sometimes the whole slum area has been locked allowing for its internal life to continue undisturbed. The contact structure for individuals in slums is more homogeneous and with larger contact rate than the surrounding cities.

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 viraemic level, presence of symptoms, etc.), but also on the expectations of the health services (HS) as we have discussed in Section ‘What is a COVID-19 case?’ and Appendix A. Since the chances for a contagious person to produce new cases depend 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 hereditary. 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 T case. No real situation is expected to reach this limit case, hence, in a more accurate description U cases are detected with lower 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 the matrix form):

$$\left({\matrix{ {P( {T{\rm \;by\;}T} ) } & {P( {T{\rm \;by\;}U} ) } \cr {P( {U{\rm \;by\;}T} ) } & {P( {U{\rm \;by\;}U} ) } \cr } } \right) = \left({\matrix{ {1- \epsilon } & s \cr \epsilon & {1-s} \cr } } \right)$$

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 effectively infected. The non-negative quantities ȯ, 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 effectively 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 suffer different epidemic impacts, different strategies are likely to appear. One of the goals of this paper is to explore the impact of different strategies on the (local) evolution of the pandemic. Schematically, we will consider three scenarios, labelled passive, intermediate and active representing different 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 viraemic 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 reflects the situation in which the HS become aware of the problem of presymptomatic contagious cases, and begin to track oligosymptomatic 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 lower number and a larger set of symptoms, required for sanitary intervention (isolation). The active intervention consists of one of the 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.Footnote ¹³ We will only model the second case.

Rates and actions

In Table 3, we describe the expressions adopted for the different rates and their action on the population (i.e. the non-zero values of the incidence matrix $\{ \delta \} _i^\alpha$). 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 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 the 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 viraemic level V(k) at each stage (Fig. 1 and Table 2). In this study, K = 9. pr describes the rate of passage to the next viraemic 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 between a susceptible and an infected individual. In principle, β _T and β _U may be different but we have not explored that possibility. Since the viraemic levels are normalised, the weighted infective period is one and the factor β corresponds to the basic reproductive number R ₀ of the SIR (Susceptible, Infected, Recovered) model. The approximate value of 2.5 was taken from early reports from China [Reference Li27] and contrasted against data from the initial days of the first outbreak in slums in Argentina (Barrio Padre Mujica, also known as Villa 31 in Retiro, a neighbourhood of Buenos Aires city).Footnote ¹⁴

The constant ȯ indicates what portion of the individuals infected by a T will not be detected by the HS 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 with long distance travels to/from regions of viral circulation. For the case of slums, casual contagion within the same city but in e.g. different 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 viraemic level and on the HS strategies (giving different 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 are the number of confirmed COVID-19 cases. In the present model, these data are represented by the total number D of detected individuals, i.e. the outcome of all removal events within the infective period. The model provides an estimate of the silent cases, i.e. infected individuals of which the HS has no records. In the model, these non-detected infected individuals ND are given by the identity N = S + D + ND, where N is the population size and S is the number of susceptible individuals.

The outcome of the model is presented by computing a few realisations (typically 100) of the Feller–Kendall algorithm. No matter how parameters are chosen, there exists a non-zero probability of early disease extinction (as it is in any Markov-jump process), particularly when the onset of the epidemics contains very few infected individuals (1 or 2 in a population of a few thousands). The model allows for ruling out early extinctions, considering that the epidemics that are tracked, and concern us, are those that avoid early extinction and come to be notable.

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 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.

Contagion, removal and HS policies

We consider in detail the mechanisms of contagion and removal, as well as their relation to 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 viraemic 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 viraemic 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 $P_m^X ( k )$ of moving to the next stage in the contagion chain and $P_r^X ( k )$ of being removed from the chain for an individual of class X = {T, U}, can be described (in the notation of Table 3) as:

$$\eqalign{P_m^X ( k ) = & \displaystyle{{\,pr( k ) } \over {\,pr( k ) + B_X^k \;V( k ) }} \cr P_r^X ( k ) = & \displaystyle{{B_X^k \;V( k ) } \over {\,pr( k ) + B_X^k \;V( k ) }} \cr = & 1-P_m^X ( k ) , \;\quad k = 0, \;\ldots , \;K-1} $$

where $B_X^k$, X = {T, U}, model the HS policy adopted. In the present implementations, the factors $B_X^k$ 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 effectively detected (named 1 − ȯ and s in Section ‘The detection of cases as a function of the surveillance protocol’) through Eq. (2).

At the final 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 effectively 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 $Q_k^X$ be the probability of removal up to and including stage k for infected individuals of class X = {D, U}. Set further, $Q_{{-}1}^X \equiv 0$. For any stage i,

$$Q_i^X = Q_{i-1}^X + ( {1-Q_{i-1}^X } ) P_r^X ( i ) , \;\quad i = 0, \;\ldots , \;K-1, \;$$

which can be restated as $Q_i^X = ( {1-P_r^X ( i ) } ) Q_{i-1}^X + P_r^X ( i )$. The total probability of removal during the infective period is $Q_{K-1}^X$, while the probability of being detected at some point during the infective period for individuals in class T, U is given by

(2)$$Q_{K-1}^T = 1-\epsilon , \;\qquad Q_{K-1}^U = s.$$

Note that $Q_{K-1}^T$ and $Q_{K-1}^U$ are rational functions, the ratio of two polynomials of degree K − k ₀ in B _X.

Equation (2) relates the value of the constants B _T, B _U for the different HS policies with the probability of detection. The differences in dealing with T and U infected individuals follows from the differences between $Q_{K-1}^T$ and $Q_{K-1}^U$, being the HS's more prone to act for the T group than for the U group. We distinguish three main policies:

Passive policy: Intervention on the U class concerns only severe cases (e.g. requiring hospitalisation) in a situation where the viraemic levels of the patient are comparatively high. In the model, intervention for the U class starts at k ₀ = 3 (stage 3 in 2).

Intermediate policy: The conditions required for sanitary intervention (isolation) in the U group are broadened in terms of a lower 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 reflects a situation in which the HS become aware of the problem of presymptomatic contagious cases, and begin to track oligosymptomatic cases in the T group (contacts of known cases).

Active policy: 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).

Simulation scenarios: In the next section, we discuss a few scenarios based on these policies, relating to data from Table 1. We consider three detection efforts, that we may call Low, Medium and High (L, M, H) for each scenario. We identify the T group with health workers, who for practical reasons were better monitored than other individuals. A first scenario, labelled I (for ‘Ideal’) corresponds to the active policy, with three different effort levels, represented by B _X such that the detection probability 1 − ȯ = s takes the values 0.21, 0.61 and 0.79. The latter corresponds to the fraction of confirmed cases among health workers displaying one symptom of the extended list of Section ‘What is a COVID-19 case?’. However, registration of symptoms was optional. Therefore, 0.79 is only a crude lower bound to the ability of detecting cases among health workers (which are subject in part to routine testing). A second scenario of intermediate character, labelled 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 confirmed cases among health workers displaying fever plus another symptom in Table 1. The third scenario, labelled H (for ‘hospital’) is still unaltered for the T group relative to the previous two, while the U group is subject to the passive policy (thus assuming that only highly viraemic cases have a chance of being detected, and only from stage 3), with detection probabilities s set to 0.02, 0.06 and 0.08. The latter corresponds roughly to the proportion of confirmed health worker cases that were hospitalised as in Table 1. Hence, the High intensity level of the three scenarios relate to detection policies adopted by HS's at different periods of time. The lower estimate for the HS actions is roughly 1/4 of the upper estimate and the intermediate estimate was taken to obtain an intermediate level of detection. In the unfortunate situations where the epidemic is out of control, the effectiveness of the HS measures could be even lower.

Unless otherwise stated, all simulations are performed with 5000 individuals, of which two are initially contagious in the T compartment (it makes only imperceptible difference 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 different scenarios is provided in Table 4. Other necessary input data for running the simulations are: number of realisations (usually 100), length of simulation in days, initial condition for populations S, T, U (usually 4998, 2, 0), random number seed, flag to discard ‘early’ extinctions (positive integer) and maximal duration to be considered ‘early’ (usually 19 days).

Table 4. Parameter values used in various simulation scenarios, in the variants low, medium and high respectively

B is called det in the code and k ₀ is called delay. Under the present normalisation, the contagion rate β corresponds to the basic reproduction number R ₀ of the SIR model.