Virus spread routes

Image: Renina Katz (Jornal de Resenhas)


A review of the scientific literature and an overview of Brazil's situation in the world.

This text was prepared with the aim of presenting what is known as scientific knowledge, at the present time, about the spread of SARS-CoV-21. The date is the second half of May 2020. The reviews and scientific articles presented here have, as a common theme, the possible routes for the spread of the virus. In addition to this review of the scientific literature, I present an overview of the situation in Brazil on the world stage, as well as a brief and simplified tutorial on models in epidemiology, with the aim of providing basic tools for the interpretation of data by those who do not have traffic in the area. Finally, I leave some specific bibliographical references that may be of interest for further reading. The outline of the text is below.

Aspects of SARS-CoV-2 transmission: possible routes

Gastrointestinal and fecal-oral

Cipriano et al., in a meta-analysis carried out at the beginning of March of the current year, indicate that fecal-oral contamination should be considered a possible route of transmission of the virus (Cipriano et al., 2020). Pan et al. (Pan et al., 2020) report that in 17 patients tested for SARS-CoV-2 in feces, 9 had detectable viral loads (but lower than in the airways). The authors recommend care in the handling of fecal samples, but do not cite fecal-oral contamination as a route.

McDermott and collaborators (McDermott et al., 2020), on the other hand, considering the known spread of SARS-CoV-1 through aerosols from sanitary discharges, consider this to be a possible route of transmission, being important, mainly, in environments hospitals and the like. The discharges form aerosols with drops smaller than 3 mm that can be inhaled going to the respiratory airways (terminal bronchioles). Thus, the authors suggest that research should be directed to this topic and that while there are no results that contradict the hypothesis, this possible transmission route should be kept in mind and due care should be taken preventively (Wong et al., 2020 ).

Still on this topic, Li and collaborators (Li et al., 2020) compare CoVs with noroviruses (NoVs), the latter transmitted by food. The authors point out that CoVs remain with infectious potential after days to weeks in food, but the authors assume that this is not a relevant route of infection. Deng and collaborators found, in Rhesus monkeys, that ocular inoculation can cause mild pulmonary symptoms, and that inoculation through the gastrointestinal tract does not cause infection (Deng et al., 2020).

SARS-CoV-2 binds to angiotensin converting enzyme type-2 receptors to inject its nucleic material into host cells. Thus, both Xiao et al. and Lamers et al demonstrate that the gastrointestinal tract, with the abundant presence of angiotensin type-2 converting enzyme receptors, is a route for both propagation and infection by SARS-CoV-2 (Lamers et al. al., 2020; Xiao et al., 2020).

Given the possibility of the fecal-oral route of transmission, in a meta-analysis published on April 28, La Rosa and collaborators studied the spread of coronavirus (general) by water (La Rosa et al., 2020). The authors point out that coronaviruses appear to be extremely sensitive to oxidizing agents, such as chlorine, and are significantly inactivated faster than other viruses known to be transmissible through water. The meta-analysis points out that there is no evidence of the persistence of coronavirus in water or of transmission through contaminated water.

In conclusion, the fecal-oral route is an open possibility for the transmission of SARS-CoV-2. However, to date, there is no evidence that there are cases originating from this route. On the other hand, aerosol transmission arising from water contaminated by faeces containing SARS-CoV-2 must be seriously taken into account.

Surfaces and temperatures

The persistence time of coronavirus in general on surfaces is 5 to 9 days (Fiorillo et al., 2020), and the persistence time of SARS-CoV-2 is a little shorter (depending on the type of surface, for example, copper has the potential to inactivate viruses within 4 hours) (van Doremalen et al., 2020).

Christophe Batéjat et al. (version posted May 1, 2020 – bioRxiv preprint doi: – Heat inactivation of the Severe Acute Respiratory Syndrome Coronavirus 2), through the estimation of TCID50 (infecting dose of 50% of tissue culture), present data that indicate that SARS-CoV-2 is inactivated in 30 minutes at 56 oC, 15 minutes at 65 oC and 3 minutes at 95 oC (note, however, that the viral RNA persists intact in the particles, even inactivated).

Despite this long persistence of virial particles on surfaces (fomites2), there has not yet been observational or experimental evidence that this route of contamination has been responsible for cases in non-hospital settings (see “Summary” section below).

In conclusion, it appears that SARS-CoV-2 can be inactivated at temperatures around 60 oC to 70 oC for a couple of tens of minutes, and one should not confuse the persistence of the viral RNA with the infective capacity of the particle.


Aerosol is the name given to liquid droplets of very small size (this will be explained in more detail in the text below).

“Even today, surprisingly, the literature is undecided on how flu spreads in relation to droplet versus airborne [aerosol] transmission. This discussion is notable, as there is no doubt that the flu is highly infectious and airborne; after my own carpool as a med student, feeling the early stages of a flu. I suggested to the two companions that they take a train, but they insisted on getting in the car. There was no coughing, sneezing or even talking, just breathing the same air for half an hour and they both came down with severe flu two days later. Therefore, there will be variable risk depending on the length of exposure, ventilation of the area and amount of virus circulating. Without knowing these parameters, the risk of infection can be high or low.” (Barr, 2020) – free translation.

According to (Hsiao et al., 2020), the dichotomous differentiation made by the World Health Organization between “droplets” and “aerosols” (“droplets” and “airborne”) causes considerable interpretive problems when referring to the possible routes of dissemination of pathogens. The differentiation is due to the size of the particles, the droplets being larger and “wet”, while the aerosols are small and, due to the evaporation of the water originally present at the release of the material, dry. In this way, and due to the different sizes, the droplets tend to fall by the action of gravity and have a much shorter residence time in the air than the particulate material originated by aerosols. On the other hand, the persistence of active pathogens, in general, tends to be lower in dry particulates than in droplets, and it is the latter that, due to falling, settle on surfaces or the floor itself, while aerosols remain suspended in the air. for hours or days.

In an article published on March 17, van Doremalen and collaborators show that the average persistence period of SARS-CoV-2 in aerosols is 3 hours, with characteristics similar to that of SARS-CoV-1 (van Doremalen et al., 2020 ). This article has been the subject of numerous citations, both to see it as evidence of the need to take precautions against contagion by aerosols and to criticize it in relation to the reality of the spread of SARS-CoV-2 through this route.

For example, Peters et al. (Peters et al., 2020) seek to give a more realistic picture to the issue of similarity between the experiment by van Doremalen et al. and particles created in real situations by talking, coughing or breathing, pointing out that there is a very big difference between the experiment with the Goldberg drum (see Figure 1, below) and what is the result of these human activities. Several other authors comment on the necessary distinction between the experiment and the possible route of contamination (there are many articles and letters in this regard, so I just leave the DOI of the New England Journal of Medicine with a series of these for those who are interested: “Stability and Viability of SARS-CoV-2” – DOI: 10.1056/NEJMc2007942).

The criticisms can therefore be summarized on two levels: (1) there is no parity between what is observed experimentally in a Goldberg drum and what occurs in a non-artificial experimentation environment; (2) how much should be invested, in a situation of scarcity, in more refined equipment to prevent aerosol contamination without more solid evidence about the viability of this route (it is important to note that the authors of the original article never said anything about these two items). Within this perspective, in a meta-analysis article from the beginning of April/2020, Tabula concludes that there is no evidence to assume aerosols as a contamination route (Tabula, Joey. “Is SARS-CoV-2 transmitted by airborne route?. ” – Asia Pacific Center for Evidence Based Healthcare).

However, several other articles have been published in the opposite direction. Morawska & Cao draw attention to aerosol spread as an important route for infection, especially in restricted environments (Morawska and Cao, 2020). Hadei and collaborators agree that the evidence for the transmission of SARS-CoV-2 by aerosols is not complete, however that the observational findings are highly suggestive and, therefore, they consider that the preventive use of masks is justified (see next section) (Hadei et al. al., 2020). In a recent May 11 article, Dancer et al.3 again insist that the aerosol spread route must be considered as real, citing at least two incidents in which this type of spread must have been responsible for the cases that arose (Dancer et al., 2020). In addition to presenting these two emblematic events, the authors point out the problem of the binary distinction made between “droplets” and “aerosols”, as raised earlier in this section. Still in this bias of observational evidence, Galbadage et al. consider that the spread via aerosol is real for SARS-CoV-2 and that, among other preventive measures already established, the use of masks is important (see next section) (Galbadage et al., 2020).

Summary of topics around SARS-CoV-2 transmission routes

It seems to me that the meta-analysis carried out by Brurberg, on May 7, serves as a summary of the current state of knowledge about the routes of contamination of SARS-CoV-2, and I transcribe the conclusion of the analysis:

"Transmission tracing and likely transmission routes

Eight transmission tracking studies were included. All studies conclude that transmission usually occurs between people who are in close contact, but one study reports some cases where transmission may have occurred through contaminated inanimate surfaces. These results can be taken as an indication that SARSCoV-2 is transmitted in the community by a combination of droplets, direct and indirect contact. Studies were not designed to differentiate between multiple transmission routes and are inconclusive as to the relative importance of various transmission routes in the community..” (Brurberg, 2020) – free translation.


Empirical/observational approach

It can be estimated that between 50% and 80% of people are asymptomatic carriers of SARS-CoV-2, as cited by (Esposito et al., 2020), and that the viral load transmitted by these individuals is similar to that of symptomatic individuals. According to these authors, it was originally believed that transmission would only occur by droplets originating from coughing/sneezing, there is now evidence that (1) SARS-CoV-2 is present and potentially infectious in aerosols (van Doremalen et al., 2020), (2) the simple act of speaking produces aerosol discharge (Anfinrud et al., 2020). In this way, the authors suggest that the use of masks, even if they are of low effectiveness such as those made at home, be adopted as a complementary measure to social isolation and hygiene care. Still in this line, the study by (He et al., 2020) suggests that the main phase of transmission occurs during the pre-symptomatic period.

Anderson et al. initially discuss the problem of the binary division between “droplets” and “aerosol”, which leads to artificial separations between the possible means of transmission of SARS-CoV-2 (Anderson et al., 2020). Then, from a risk analysis perspective (which deals with incomplete scientific information), the authors point to 3 lines of evidence regarding the spread of the virus by aerosols: (1) reported cases of asymptomatic people having been the transmitting focus to other individuals; (2) samples of SARS-CoV-1 and SARS-CoV-2 in aerosols both in empirical collections in hospital environments and in experiments; (3) spread, via aerosol, of other pathogens. The authors conclude that there is an urgent need to define the possible routes of transmission of SARS-CoV-2 and that the use of “inhaled protectors” has sufficient evidence to be adopted.

Barr (Barr, 2020), quoted at the opening of the previous section, not only advocates the widespread use of masks, but also suggests that each person should have three masks for daily rotation (given that the information available about the persistence of SARS -CoV-2 in masks is 3 days – (Chin et al., 2020)).

How to understand the following apparent paradox? Surgical masks do not have the filtering efficiency of so-called respirators (masks that fit well to the face and have a filter – usually N95, which reduces by 95% the inhalation of aerosols greater than 3 mm in radius), but countries that have adopted the widespread use of masks , even just surgical or homemade ones, recorded a sharp drop in the spread of SARS-CoV-2. What Hsiao and collaborators propose is that masks, even the simplest ones and with low to very low filtering capacity, play an important role in reducing the speed of expelled air, whether during coughing/sneezing, talking or just breathing. normal. With this decrease in speed, the immediate range of the expelled aerosols and droplets is markedly reduced.4 and, in this way, there is a decrease in the probability of transmission of the virus (Hsiao et al., 2020).

theoretical approach

The basic model used to study the spread of epidemics is that of a population of individuals susceptible to the disease in focus, generally denoted by the letter S, a set of infected individuals, generally denoted by I, who, with the resolution of the disease , become recovered, denoted by R. An individual belonging to group S passes to group I due to contact with an individual from group I (that is, the transmission of the disease), and an individual from group I passes to group R due to the time for the infection to heal. This is a so-called SIR model, which can then be made extremely complex by adding “structures” to the population – for example, division into ages, into asymptomatic carriers, into individuals with previous illnesses, etc.- and/or by addition of space as another variable – that is, the locations of individuals take part in the model5. When space is not explicitly taken into account, the model is said to be “compartmental”, and these may have analytical solutions (that is, it may be possible to determine whether, for example, a disease will be eradicated from the population or will remain as a disease). endemic), depending on the number of equations in the model. The last section of this text presents a brief tutorial about models in epidemiology.

The study by Eikenberry and collaborators is composed of 14 differential equations and models the use of masks with different efficiencies by infected, asymptomatic and susceptible people, to different degrees (Eikenberry et al., 2020). Figure 2 illustrates part of the model results, highlighting that the use of masks, even if not highly effective and even if not for the entire population, has great potential for reducing both the number of hospitalized people and deaths, and this effect is more pronounced at lower propagation rates (such as those observed after the initial days of outbreaks in each location). For example, if 50% of the population uses masks with 50% effectiveness, a 50% decrease in deaths is estimated (for k = 0,5). Note, however, that for k = 1,5, the decrease in peak hospitalized patients is small and negligible in the decrease in total deaths.

This parameter, k, is the rate of transmission of the disease, and is indicated, indirectly, by the rate of growth in the number of cases each day (see the section “A Brief Tutorial …” below). Social isolation is, so far, the only measure known to reduce the value of k in this SARS-CoV-2 pandemic. In this way, this study by Eikenberry and collaborators not only highlights the relevance of using masks, but also the necessary quarantine or social distancing to contain the pandemic.

In another modeling about the use of masks, made through a compartmental model and through an ABM model6, (Kai et al., 2020) conclude: “Our SEIR and ABM models suggest a substantial impact of universal and early mask use. Without such use, but even with continued social distancing after the lockdown ends, the infection rate will increase and nearly half of the population will be affected.”

Note, therefore, how these theoretical results are in line with the observations of (Hsiao et al., 2020) made earlier and the recommendations cited above for the use of masks by the general population.

In conclusion, both observational and experimental and theoretical studies strongly point out that the use of masks is a complementary factor of great importance for containing the spread of SARS-CoV-2.

How is Brazil in the second week of May?7

In order to have an adequate perspective of the general picture in which Brazil is inserted, we need to be clear which questions we want to answer, and which comparisons seem appropriate.

Data sources: ;

Question 1: Is the number of confirmed cases in Brazil significant in the world scenario?

The answer to this question is yes. Brazil is the 3rd or 4th country in number of confirmed cases8, with 271.628 records, corresponding to 6% of the world's cases. In second place is Russia (299.941 cases – 6%) and, in first place, the United States (1.569.659 cases – 32%). Figure 3A.

Question 2: Is the number of deaths from COVID-19 in Brazil significant in the world scenario?

The answer is, again, yes. Brazil is in 5th place with 17.971 deaths, representing, once again, 6% of the world total. Figure 3B.

Question 3: Is the growth rate in the number of cases in Brazil within the rates observed in other countries?

In the first 50 days of the epidemic in Brazil, the growth rate was in the average of the 10 countries that currently have the most cases (USA, Russia, Brazil, United Kingdom, Spain, Italy, France, Germany, Turkey, Iran). From then on, this rate tended to stabilize around 1,06 to 1,07 (6% to 7% daily growth), and now, around the 85th day of the epidemic in Brazil, the same rate, surpassing that of the other 9 other countries mentioned (in comparative terms, the 85th day of France, the USA and Russia had rates higher than those of Brazil, but were already in clear decline). Figure 3C.

Question 4: Is the growth rate of deaths in Brazil within the rates observed in other countries?

The rate of death growth is the highest among the 10 countries with the highest number of cases today, in addition to not showing the downward trend that was observed in other places. 3D figure.

Question 5: Is the percentage of deaths from COVID-19 among those infected in Brazil within the percentages observed in other countries?

Yes, for the current relative period of the epidemic in Brazil, the percentage of deaths stands at 6,5%, which is on average among the 10 countries with the highest number of cases for the period. Figure 4.

Question 6: within Latin America, is Brazil the country with the highest number of infections if adjustments are made for the total population and for the demographic density of each country in the region?

Yes. If expressed in total number of cases, Brazil is the country with the highest number of cases. If the adjustment is made for the total population (not adequate, as explained in the Tutorial), Brazil becomes the 5th among the 21 countries. If the adjustment for population density is made, Brazil takes the first place again (this adjustment is adequate, as explained in the Tutorial). Figure 5.


Although this text is not the focus of the clinical issues involving SARS-CoV-2 and the disease it causes, COVID-19, given the situation in which the country finds itself, I think it is opportune to see what have about the use of chloroquine for this disease. Thus, I present, below, two extracts extracted from reviews in scientific journals of extreme renown in the medical field.

“Physicians are treating patients with an unparalleled lack of parsimony, using drugs such as chloroquine, hydroxychloroquine, azithromycin, lopinavir-ritonavir, and interleukin-6 inhibitors outside of their indicated and approved uses, without study protocols, and with little scientific evidence to support their claim. administration beyond extrapolation from studies vitro of its antiviral and anti-inflammatory properties. Aside from the possible side effects of medications such as hydroxychloroquine and interleukin-6 inhibitors, which include fatal cardiac arrhythmias and possible worsening of the infection, respectively, prescribing medications based on case reports does little to help advance science or our ability to combat future coronavirus recurrences. … In these uncertain times, physicians fall prey to cognitive errors and unconsciously rely on limited experiences, whether their own or others, rather than scientific investigations.”(Zagury-Orly and Schwartzstein, 2020) – free translation.

“Hydroxychloroquine has been widely administered to patients with Covid-19 without robust evidence supporting its use… We examined the association between hydroxychloroquine use and intubation or death at a large medical center in New York City. CONCLUSIONS. In this observational study involving patients with Covid-19 who were admitted to the central hospital, administration of hydroxychloroquine was not associated with either a reduced or increased risk of the composite endpoint of intubation or death. Randomized controlled clinical trials of hydroxychloroquine in patients with Covid-19 are needed. (Funding: NIH)”. (Geleris et al., 2020) updated May 14 – free translation.

A brief (and simplified) tutorial on models in epidemiology

What needs to be considered before we start a direct analysis of the data?

We first need to have a sense of how data from an epidemic might behave. This, in essence, is having a model against which comparisons/predictions are made. It is not our objective, in this text, to make a detailed presentation of this type of modeling, however, for the analyzes to make sense, a minimal explanation of part of the process is necessary.

As mentioned above, a simple model of the spread of infectious diseases has three states: susceptible, infected and recovered. As an epidemic generally lasts for a “short” time in view of demographic variations in a population, it is considered that the total population N does not change, that is, the sum S+I+R has a constant value9. Also as mentioned, a susceptible individual becomes infected through contact with another infected individual. These qualitative relationships are presented in Figure 6. Below, I write how a compartmental (simple) model deals with the variation in the number of infected over time.

A function is a function that can be simple or complicated, but it doesn't interest us. We are not interested because we are analyzing the initial stages of the spread of the epidemic, a period in which the number of infected I is small compared to the total population, N. Thus, as I is small, obviously the states resulting from it (such as Recovered) are also values small. With this, practically the entire N population is in the Susceptible state. As I is small, the product can be neglected and, thus, the equation, for the initial periods of the propagation, can be approximated by:

And this equation has, as a solution:

being I0 the initial number of infected and I the number of infected at time t. Note that since we are working with normalized values, S ≅ N = 1, so I have omitted this term from the equation.

Suppose we are measuring time in days: day zero, day one, day two, etc. If we know the number of infected on a certain day X and on the following day, X+1, we can calculate the ratio:

Como e is a constant and k is another constant, this ratio is also a constant. Through it we can estimate two things: (1) the constant k of the growth rate of the number of cases; (2) how long will it take to double the number of cases you have on a given day.

In addition to these two important estimates that we can make, there is an additional advantage in making the ratio between the number of cases in a day in relation to the previous day: this value (given by ek in the above equation) is independent of the total population size. That is, if we are facing a country with 200 million inhabitants or if we are facing a country with 30 million inhabitants, the value of the ratio does not depend on these numbers and, therefore, we can compare countries with different populations. We will return to this subject later.

The “total cases” and “number of infected” curves

Another aspect that must be made clear before looking at data is the question of which dataset one is analyzing. In the context of the spread of an infectious disease, one may want to know how many individuals have already been infected or one may want to know how many individuals are infected at a given moment. The first case is, therefore, a function that always grows over time, as an individual who has been infected enters the count and no longer leaves it, regardless of having recovered. This function will only stop growing when the entire population has been infected.

On the other hand, the role of individuals who are infected at a given time is different. As the number of infected and recovered increases, the number of susceptibles decreases. This means that that F term, which we ignore for the initial periods of the epidemic, will become important in the equation for the variation in the number of infected people. As can be seen, this term is negative, which implies that with an increase in the number of infected and a decrease in susceptibles, at some point the term F·I becomes greater than the term  k·S·I and then the number of infected starts to decrease. Thus, unlike the “total infected” function, the “number of infected” function has an apex followed by a decline. Figure 7 illustrates these two functions.

Figure 7. Total infected (black line), number of infected (blue line) over time. Note that the total infected is an ever-increasing function while the number of infected at a given instant reaches a peak and drops. In the inset, the reason explained earlier.

Therefore, the expression “flattening the curve” refers to the function “number of infected” (blue line in Figure 7).

What would correspond to this flattening of the curve if we were looking at the “total contaminated” function (black line)? As seen, the total infected will not stop growing until the entire population has been infected. However, the rate at which this total grows is given by the variation of infected people. In this way, through the ratio illustrated above, , it is possible to estimate the containment of the propagation. The closer to the value “1” the ratio is, it means that there are less and less cases of contamination appearing in the population. Thus, what corresponds to “flattening the blue curve” of the number of infected people is, as a function of the total number of infected people, having a ratio close to 1.

using reason  to have an estimate of what will happen in the following days

We are now going to present a table so that you have an idea of ​​the impact of the ratio values ​​on what can be expected in the number of cases in the days following a given calculation. This is important because, as readers may have already read elsewhere, the ratio, often presented as a percentage of growth, for example, “6% growth”, is a value apparently small. This can give the false impression that the disease spreads slowly. Let's see.

Table 1. Multiplier factor as a function of the ratio in the number of cases.

In the first column of Table 1, we have ratio values. In the second column, how these values ​​would be read in percentage. So, for example, a ratio of 1,04 means 4% growth. In the other columns there is the multiplier factor according to the number of days, indicated in the second line, after a certain calculation of the ratio. For example, if, on a given day, 8.000 accumulated cases were registered and the calculated ratio was 1,05 (5%), it is projected that after 10 days there will be 13.040 cases (eight thousand times one point sixty-three).

Note how seemingly low rates, such as 2%, result in high values ​​after a longer period. In the example above, if we had a ratio of 1,02, the 8.000 cases would become 26.240 after 60 days, that is, they would more than triple.

The ratios that have been observed for Brazil are in the range of 1,06 to 1,07 (highlighted in the table), as can be seen in Figure 3C. This means that every 10 days or so, the number of cases doubles, and so does the number of deaths. Thus, between March 17, when the first death from COVID-19 was recorded in the country, and May 10, there were 10.000 deaths. Between the 10th and the day this text is being finalized, May 19th, there are 17.971 records of deaths from COVID-19. In 54 days, 10.000 deaths occurred, and in 9 days there were 7.970 more, that is, in nine days there were almost 80% of new deaths than in the previous fifty-four days. This is the impact of propagation at a rate of 6% to 7%.

Appropriate ways to visualize and analyze the data

The question of population size

As mentioned above, there is a potential problem in visualizing/analyzing the data due to the different sizes of populations involved. For example, Argentina has 45 million inhabitants, while Brazil has 210 million. Thus, it does not seem fair to directly compare the values ​​of number of cases or number of deaths in these two countries. However, as I explain below, these comparisons are, in fact, valid.

When graphs of data about the state of the pandemic in Brazil and in several countries were presented, it was said that the visualization/analysis having the number of cases divided by the population size is the least suitable way to proceed. Why?

Consider the basic model of the spread of infectious diseases presented above. In it, the growth term of the number of infected depends on the product S·I, which represents the encounter between susceptible and infected. Thus, as is obvious for infectious diseases, there is the assumption of encounters between individuals for the spread of the disease. When trying to correct the potential distortion of population sizes by dividing the number of cases by the total population of the country, it is assumed that all individuals in this population are in contact with each other, as if they were all in a single pot. . And this is not true.

In this way, if you want to make some kind of not very elaborate correction to the data, the most correct thing is to divide by population density of the country, because then there is an index of “proximity” between individuals. For this reason, data corrected for demographic density are presented in Figure 5 for comparisons between Latin American countries.

As I pointed out above, the alleged distortion in raw caseload data is only potential in the early stages of a pandemic. This is due to the fact that local epidemics initially spread in large urban centers, and the world's large cities have very similar demographic characteristics in terms of urban organization and population density. Thus, the raw data reflect, in these initial stages, the transmission in similar centers and the observation of these data directly does not compromise the conclusions that can be reached in these stages.

The numerical artifact of the early days of the epidemic

When we observe Figures 3C and 3D, which show the ratios of infected and deaths, respectively, we notice that the initial days seem to have extremely high propagation rates, which subsequently decrease. Ratios above 2 are found on several days and this occurs in all countries.

A mistaken analysis is to assume that the epidemic is being controlled and that, therefore, the rates (the reasons) decrease, which the virus changes its transmissibility characteristics, and, again, this is why the rates decline.

The correct perspective is that these high rates (ratios) in the early days are nothing more than numerical artifacts that occur due to two factors: (1) small number of cases; (2) detection of cases that were already in the population, but had not yet manifested themselves.

Factor (1) above implies the following. Imagine that there are, on the second day, 10 confirmed cases. On the third day, 8 more cases appear, which results in a ratio of 18/10 = 1,8. That is, basically this number tells us that the number of cases will practically double from one day to the next. But, this is merely the effect of there being few recorded cases. These same 8 cases out of a total of 100 previous ones would result in a ratio of 108/100 = 1,08, a still high value, but much more feasible. And factor (1) is combined with factor (2). The spread of infectious diseases occurs through some type of contact, and when calculating growth rates (as the ratio), the idea is embedded that infected individuals gave rise to new cases. However, at the beginning of the epidemic, this is not what is happening. Most of the cases that appear in the first few days are cases that were already in the population, but had not yet been detected. Thus, these cases do not necessarily have their origin in the cases already registered and, when they “emerge”, they inflate the growth rates of the epidemic. Therefore, as can be seen in the graphs presented, it is only when the initial days are over and the total number of cases becomes “large” that simple calculations of rates, such as the ratio, start to make sense for predicting and diagnosing public policies.

Data on a linear scale and on a logarithmic scale

Figure 8 illustrates the hypothetical situation of spreading an infectious disease in two different places (countries, for example). Panel A presents data on a linear scale, while panel B presents these same data on a logarithmic scale. Since they are the same data, the information given by panel A is the same as that given by panel B. However, visually, the impact of these panels is quite different.

Figure 8. Simulation of the initial stages of the spread of an infectious disease in two different locations. (A) linear scale; (B) logarithmic scale. The simulations had the same spreading constant (0,5 per day) and, in the location represented by the blue line, there is initially 1 infected, while in the location represented by the orange line, there are 10 infected at the initial moment.

We humans evaluate contexts in a basically linear fashion. Thus, when observing panel B, the feeling generated is that the disease is more pronounced in the orange country, but “just a little more pronounced”. But, looking at panel A, you have the real difference between the orange country and the blue country: there are ten times more cases in orange than in blue. On the other hand, precisely because of our linear evaluation bias, when we look at panel A, we have the impression that the disease spreads much more quickly in the orange country than in the blue country. Now, when we look at the data on a logarithmic scale, we see that the scattering rates are the same in both countries.

As can be deduced from everything that has already been presented in this section, the spread of an infectious disease in the early stages of an epidemic has a multiplicative nature. This multiplicative character means that, on a linear scale, some locations have a growth in the number of cases that becomes much and progressively higher than in other regions. However, the logarithm function is a function that deals exactly with multiplications and, therefore, on a logarithmic scale, the multiplicative process becomes linear, facilitating the visualization of the different regions despite the difference in the number of cases of each one. Furthermore, parallel curves on a logarithmic scale indicate that the processes have the same growth rate.

For these reasons, preference is given to presenting data on a logarithmic scale. However, if you are not familiar with this type of graphical representation, it is recommended that you do both, linear and logarithmic, so that you can maintain an intuition about how much the disease has spread in different locations (linear) and how how quickly it spreads across different locations (logarithmic).

As an example, I reproduce, in Figure 9A, the graph of the total number of infected people adjusted for population density in Latin American countries, on a linear scale (Figure 5C) and, in 9B, the same data on a logarithmic scale. Note how, in 9B, we can have a better perception of how quickly the disease spreads in different countries, which is not possible in 9A.

Figure 9. Total confirmed cases divided by the respective demographic densities of Latin American countries. (A) Linear scale. (B) Logarithmic scale.

I end here this brief and simplified tutorial about models for the initial stages of propagation of an infectious disease, and I hope that this tutorial can be of help for a better understanding of the data that has been presented daily about the pandemic caused by SARS- CoV-2.

*José Guilherme Chaui-Berlinck is a professor at the Department of Physiology at the Institute of Biosciences at USP.

Interesting review references

(Bar-On et al., 2020) – They present a great summary of the main aspects of the virus, the disease and the pandemic. This article is worth checking out..

(Mamun et al., 2020) – Another brief summary of the main findings in terms of the pandemic so far

(Fiorillo et al., 2020) – Persistence time of SARS-CoV-2 on different surfaces and temperatures.

(Chin et al., 2020) – Viability of SARS-CoV-2 on different surfaces and different temperatures.

(Netz, 2020) – Physics of droplets and aerosols. This article is worth checking out, but some math background is required.

Bibliographic references

Anderson EL, Turnham P., Griffin JR and Clarke CC (2020). Consideration of the Aerosol Transmission for COVID-19 and Public Health. Risk Anal. 40902-907.

Anfinrud, P., Bax, CE, Stadnytskyi, V. and Bax, A. (2020). Could SARS-CoV-2 be transmitted via speech droplets? medRxiv 2020.04.02.20051177.

Bar-On, YM, Flamholz, A., Phillips, R. and Milo, R. (2020). SARS-CoV-2 (COVID-19) by the numbers. Elife 9,.

barr, gd (2020). The Covid-19 Crisis and the need for suitable face masks for the general population. Chinese J. Med. Res. 328-31.

Brurberg, KG (2020). Transmission of SARS-CoV-2 via contact and droplets, 1st update – a rapid review. norm. Inst. Public Heal. May 71-17.

Chin, AWH, Chu, JTS, Perera, MRA, Hui, KPY, Yen, H.-L., Chan, MCW, Peiris, M. and Poon, LLM (2020). Stability of SARS-CoV-2 in different environmental conditions. The Lancet Microbe 0-4.

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Deng W, Bao L, Gao H, and Qin C. (2020). Ocular conjunctival inoculation of SARS-CoV-2 can cause mild COVID-19 in Rhesus macaques. bioRxiv preprint,.

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[1] SARS: Severe Acute Respiratory Syndrome; CoV-2: coronavirus type 2.

[2] A fomite or fomite is any inanimate object or substance capable of absorbing, retaining, and transporting contagious or infectious organisms (from germs to parasites), from one individual to another. – Source:

[3] It should be noted that Lidia Morawska, quoted at the beginning of this paragraph, is one of the authors of this article.

[4] The authors also argue that for the receptor potential of the released material, the mask reduces the radius of the air to be inspired because it works as an inverted diffuser.

[5] These models in which space becomes one of the components are, in general, impossible to be solved analytically and their results come from numerical simulations.

[6] “Agent Based Modeling” is a numerical simulation model that involves the displacement of simulated individuals.

[7] The data presented here graphically go up to May 18th. The data presented punctually refer to the 19th of May.

[8] The indecision between the 3rd and 4th position stems from the time of data update by different countries.

[9] Note that, for simplification purposes, we can place the individuals who die as part of the recovered, without changing the dynamics of the process, since the total population remains constant.

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