key: cord-0857110-px2lrbrl
authors: Melis, Maurizio; Littera, Roberto
title: Undetected infectives in the Covid-19 pandemic
date: 2021-01-09
journal: Int J Infect Dis
DOI: 10.1016/j.ijid.2021.01.010
sha: 4ec2bff0b2642d969a58f5d4136ce37608a3bc29
doc_id: 857110
cord_uid: px2lrbrl

OBJECTIVES: Epidemiological investigations and mathematical models have revealed that the rapid diffusion of Covid-19 can mostly be attributed to undetected infective individuals, who continue to circulate and spread the disease: finding their number would be of great importance in the control of the epidemic. METHODS: The dynamics of an infection can be described by the SIR model, which divides the population into susceptible ([Formula: see text]), infective ([Formula: see text]) and removed ([Formula: see text]) subjects. In particular, we exploited the Kermack-McKendrick epidemic model which can be applied when the population is much larger than the fraction of infected subjects. RESULTS: We proved that the fraction of undetected infectives, in comparison to the total number of infected subjects, is given by [Formula: see text] , where [Formula: see text] is the basic reproduction number. The mean value [Formula: see text] for the Covid-19 epidemic in three Italian regions yielded a percentage of undetected infectives of 52.4% (52.2% - 52.6%) compared to the total number of infectives. CONCLUSIONS: Our results, straightforwardly obtained from the SIR model, highlight the role played by undetected carriers in the transmission and spread of the SARS-CoV-2 infection. Such evidence strongly recommends careful monitoring of the infective population and ongoing adjustment of preventive measures for disease control until a vaccine becomes available for most of the population.

A critical issue in the control of an epidemic is to know the exact number of infective subjects. Current estimates of SARS-CoV-2 infection are significantly hampered by the difficulty to perform largescale diagnostic tests, despite the current awareness that the spread of the Covid-19 pandemic is mostly caused by undetected carriers.

The speed at which an epidemic grows cannot be explained if we only take into account the number of recorded infected patients who, supposably, are immediately removed from the circulating population by hospitalisation or isolation at home.

Undetected infectives can be classified into two categories: 1) paucisymptomatic or asymptomatic individuals, who never develop overt symptoms during the course of infection; 2) presymptomatic subjects, who will eventually develop symptoms.

Undetected infectives are largely responsible for the rapid increase of the epidemic. In order to reliably detect their presence, it would be necessary to test the entire population and not just the symptomatic cases.

The dynamics of an epidemic can be described by an epidemiological model known as the SIR model, which divides the whole population into three classes of subjects: susceptible ( ), infective ( ) and removed ( ) individuals. Kermack and McKendrick (1933) developed a SIR model for the study of epidemics in populations much larger than the infected fraction. Under this assumption, which is fully verified in the Covid-19 epidemic, we proved that the total number of infectives, when an epidemic occurs, is approximately 0 • , where 0 > 1 is the basic reproduction number of the infection and By applying the aforesaid model to the data available on the Covid-19 epidemic in Italy, we obtained that the mean value of the basic reproduction number in three Italian regions (Lombardy, Emilia-Romagna and Sardinia) was 0 = 2.10 (95% confidence interval, 2.09 -2.11). Consequently, the number of undetected cases turned out to be about 0 − 1 = 1.1 times the number of removed cases.

More specifically, we found that the percentage of undetected infectives was about 1 − 1 0 = 52.4% (95% confidence interval, 52.2% -52.6%) of the total number of infectives.

Previous investigations found that the percentages of asymptomatic infectives (i.e. subjects without fever, cough or any other symptoms) were: 43.2% (32.2% -54.7%) in Vo', a small town near Padua in Italy (Lavezzo et al., 2020) ; 50.5% (46.5% -54.4%) on board the Diamond Princess cruise ship in Yokohama, Japan (Mizumoto et al., 2020) ; 47% (38% -56%) in mainland China (Li R et al., 2020) and 52.0% (including paucisymptomatic infectives) in a large sample (64660 subjects) of the Italian population (Italian National Institute of Statistics, 2020).

The data provided by the Italian Ministry of Health and the Civil Protection Department (2020) up to the 3 rd of June 2020 reported about 233800 removed cases in Italy, including either patients hospitalised or isolated at home or recovered or dead. Based on the result found in the present study, the total number of paucisymptomatic, asymptomatic and presymptomatic infectives had to be almost 491000 up to that date. This means that 257200 individuals were not diagnosed as infected although they continued to circulate and spread the virus.

This study confirms that undetected infectives can be considered the key culprits for the rapid spread of SARS-CoV-2 within the population. Consequently, interventions to control the infection will need to be maintained until the complete disappearance of the epidemic.

In the SIR epidemic model the population is divided into three distinct classes (Murray, 2002) : the susceptible subjects, , who can catch the disease; the undetected infectives, , who have the disease and can transmit it; and the removed infected subjects, , namely those with a laboratory diagnosis who are either hospitalised, isolated at home, dead or recovered.

We assume that all the individuals diagnosed as infectedeither by nasopharyngeal swab or serological testare immediately isolated, thus passing from the class of infectives to that of the removed infectives . On the contrary, the infected subjects without a positive diagnosis are classified as undetected infectives ( ), who are either still infective ( ) or infected but no longer contagious ( 0 ). The total number of undetected infectives is then given by: = + 0 . where 0 is the basic reproduction number (discussed in Appendix B). Under the assumption 0 • ( )⁄ ≪ 1 (a condition which is certainly verified if the population size is much larger than the number of infected subjects) we can approximate ( ) in the following form:

The total number of infected subjects ( ) at time then becomes

while the undetected infectives ( ) at time turn out to be

The ratio between the removed infected subjects ( ) and ( ) at time is

while the ratio between the undetected infectives ( ) and ( ) at time is

Being 0 • ( )⁄ ≪ 1, the previous four equations can be approximated as

These results are obtained under the assumption 0 ≅ , which implies 0 > 1, i.e. that an epidemic ensues.

The fraction of undetected infectives , in comparison to the total infectives , has been derived straightforwardly from the SIR epidemic model and only depends on the basic reproduction number 0 .

Appendix A provides further details on the SIR model, Appendix B shows how the evaluation of the basic reproduction number 0 was performed, Appendix C presents the numerical fit of the data,

Appendix D shows how the effective reproduction number eff ( ) was computed, Appendix E describes the evaluation of the constants of the fit from the data at the peak of new infectives and Appendix F shows the application of the model to stratified groups.

The data provided by the Italian Ministry of Health and the Civil Protection Department (2020) In Appendix D, the Kermack-McKendrick model was also used to compute the effective reproduction number eff ( ) and to evaluate the time corresponding to the threshold eff ( ) = 1 at which the epidemic starts to decline. The validity of our model was tested by determining the constants 1 , 2 , 3 of the fit from the data on ( peak ) and ′( peak ), as discussed in Appendix E. The results obtained only differ by a maximum of 3% from the values in Table 1 , thereby confirming that our model provides reliable estimates of the main epidemic parameters. conducted in Vo' (Lavezzo et al., 2020) , Japan (Mizumoto et al., 2020) and China (Li R et al., 2020) , with the percentage of undetected infectives in Lombardy, Emilia-Romagna and Sardinia obtained in this study through the SIR model.

The serological investigation conducted in Italy on 64660 subjects from the 15 th of May to the 15 th of July 2020 revealed that the percentage of paucisymptomatic infectives was 24.7% and that of asymptomatic infectives was 27.3%. Therefore, the total percentage of paucisymptomatic and asymptomatic infectives resulted to be 52.0%, as discussed in the preliminary report released by the Italian National Institute of Statistics (2020).

The result obtained with the SIR model (shown in Figure 4 ) seems to be affected by a relatively small error in comparison to the errors of other studies. The reason is that the 95% confidence interval associated to our finding only represents the uncertainty intrinsic to the mathematical model, excluding the error related to the data provided by the Italian Ministry of Health and Civil Protection Department (2020) for removed infectives ( ) at time . These data were probably understimated because of the difficulty to administer swabs or serological tests to all the suspect cases or even to subjects with overt symptoms. However, we only considered the errors associated to the statistical goodness of fit in our model, being unable to evaluate the uncertainty of the data on removed infectives. The assumption that the population size must be larger than the number of infected subjects corresponds to an approximated relative error 0 • ( ) 2 on the undetected fraction of infectives, i.e. a percent error lower than 0.9% in Lombardy, 0.6% in Emilia-Romagna and 0.1% in Sardinia.

Our model can be applied to a sample of infected individuals stratified into two groups by a specific characteristic, e.g. age or gender (Appendix F). The stratification of the undetected infectives in J o u r n a l P r e -p r o o f the two groups turns out to approach that of the whole sample of infected subjects , i.e. 1 2 ≅ ; 1 ; 2 . This conclusion was confirmed by exploiting the data of the investigation conducted in Vo' (Lavezzo et al., 2020) .

The speed at which an infection spreads is strongly influenced by the number of undetected infected individuals who contribute to disseminate the virus without being diagnosed as positive. This study proved that in any epidemic the fraction of undetected infectives, compared to the total number of infections, is given by the approximated expression 1 − 1 0 , which only depends on the basic reproduction number 0 .

The analytical expression of 0 found in Appendix B was exploited to compute the basic reproduction number in three Italian regions (Lombardy, Emilia-Romagna and Sardinia); the corresponding mean value 0 = 2.10 (95% confidence interval, 2.09 -2.11) overlaps well with the result 0 = 2.2 (1.4 − 3.9) found in China (Li Q et al., 2020) and the result 0 = 2.28 (2.06 -2.52) obtained in Japan from the data collected on board a cruise ship (Zhanga et al., 2020) .

In Appendix D, the Kermack-McKendrick model was also used to compute the effective reproduction number eff ( ), as previously defined e.g. by Nishiura and Chowell (2009) .

By exploiting the aforesaid mean value of 0 , we found that the percentage of undetected infectives was 1 − 1 0 ≅ 52.4% (95% confidence interval, 52.2% -52.6%) of the total infectives.

The assumption that the population size must be larger than the number of infected subjects corresponds to a percent error lower than 1% on the undetected fraction of infectives.

As shown in Figure 4 , the percentage of undetected infectives obtained in this study overlaps well with the percentages of asymptomatic infectives found in previous investigations (Lavezzo et al., 2020; Mizumoto et al., 2020; Li R et al., 2020) , confirming that the fraction of undetected infectives is considerable and is likely to have a strong influence on the dynamics of the epidemic.

In a study conducted in Vo' (Lavezzo et al., 2020) , a small town in Veneto (Italy), most inhabitants were tested through nasopharyngeal swabs in two consecutive surveys; the mean percentage of asymptomatic infectives corresponded to 43.2% (32.2% -54.7%) of the total of SARS-CoV-2 infections. . Importantly, the nasopharyngeal swabs performed in this study showed no statistically significant differences between the viral load of symptomatic and asymptomatic infections.

Moreover, the viral load tended to peak in the large majority of participants around the day of symptom onset, suggesting an important role for presymptomatic transmission in the spread of the virus. Therefore, both asymptomatic and presymptomatic transmission represent a major threat to epidemiologic control and containment of the infection (Lavezzo et al., 2020) .

Investigation performed on the passengers of the Diamond Princess (Mizumoto et al., 2020) , a cruise ship in Yokohama (Japan), revealed that from the start of the epidemic the percentage of asymptomatic infectives on board the ship was 50.5% (46.5% -54.4%) of the total infectives.

One of the first studies (Li R et al., 2020) cases. However, in this study the transmission rate of undetected infectives was assumed to be = 55% (46% − 62%) of the transmission rate of symptomatic infectives. On the contrary, we assumed that all infected subjectswith or without symptomsmay present an elevated viral load and transmit the virus at the same rate, as confirmed by the investigation in Vo' (Lavezzo et al., 2020) and the systematic literature search conducted by Walsh et al. (2020) . Under this assumption, the effective percentage eff of undetected infectives is given by eff = • = 47% (38% − 56%).

Another study (Yusef et al., 2020) 

The studies (Lavezzo et al., 2020; Mizumoto et al., 2020; Yusef et al., 2020) were based on laboratory tests performed in small communities (the inhabitants of Vo' in Italy, the passengers of a cruise ship in Japan and the attendees of a wedding in Jordan, respectively) where the Covid-19 infection had spread. On the contrary, the study in China (Li R et al., 2020) was based on a mathematical model comparing mobility data and infection diffusion in mainland China after the start of the Covid-19 epidemic.

A serological investigation in the Italian population conducted by the Italian National Institute of Statistics (2020) on 64660 subjects revealed that the percentage of paucisymptomatic and asymptomatic infectives up to mid-July 2020 was 52.0%.

A Review by Oran and Topol (2020) of the available evidence on asymptomatic SARS-CoV-2 infectives found that asymptomatic subjects accounted for approximately from 40% to 45% of the total number of infections and could transmit the virus to others. The authors of the Review also pointed out that the high frequency of asymptomatic infections could at least partly explain the rapid spread of the virus, since infected subjects who feel and look well are likely to have more social interaction compared to symptomatic infectives.

The results obtained in the aforementioned investigations (Lavezzo et al., 2020; Mizumoto et al., 2020; Yusef et al., 2020; Oran and Topol, 2020) concerned asymptomatic infected subjects, while the results found in our study included all the undetected infectives, both asymptomatic subjects and paucisymptomatic or presymptomatic individuals. This can explain why the percentages of asymptomatic infected subjects in those studies turned out to be a bit lower than the percentage we found for all the undetected infectives.

The 95% confidence intervals of the epidemiological parameters reported in Table 1 were only associated to the error intrinsic to the mathematical model considered in this study, while the uncertainty on the data concerning the removed infectives was not included, although the recorded positive cases were probably underestimated as a consequence of the low efficiency in administering swabs and serological tests to the population in most Italian regions. 

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The authors are grateful to Anna Maria Koopmans for translations, professional writing assistance and preparation of the manuscript.

The authors contributed equally to the article.

The authors received no specific funding for this work.

Not applicable.

The authors declare that no competing interests exist.