Impact of asymptomatic COVID-19 carriers on pandemic policy outcomes
IImpact of asymptomatic COVID-19 carriers on pandemic policyoutcomes
WEIJIE PANG β , HASSAN CHEHAITLI and T. R. HURD Mathematics & Statistics, McMaster University, 1280 Main St. West, Hamilton, Ontario, L8S 4L8, Canada
A R T I C L E I N F O
Keywords :Infectious disease modelCOVID-19
A B S T R A C T
This paper provides a mathematical model to show that the incorrect estimation of π , the frac-tion of asymptomatic COVID-19 carriers in the general population, can account for much of theworldβs failure to contain the pandemic in its early phases. The SE(A+O)R model with infec-tives separated into asymptomatic and ordinary carriers, supplemented by a model of the datageneration process, is calibrated to standard datasets for several countries. It is shown that cer-tain fundamental parameters, notably π , are unidentiο¬able with this data. A number of potentialtypes of policy intervention are analyzed. It is found that the lack of parameter identiο¬ability im-plies that only some, but not all, potential policy interventions can be correctly predicted. In anexample representing Italy in March 2020, a hypothetical optimal policy of isolating conο¬rmedcases that aims to reduce the basic reproduction number of the outbreak to π = 0 . assuming π = 10% , only achieves π = 1 . if it turns out that π = 40% .
1. Introduction
From the time of its ο¬rst appearance in late December, 2019 in Wuhan, China, to the time of writing at the endof 2020, COVID-19 continues out of control, and as monitored by JHU [CSSE] has infected more than 74,000,000people in more than 185 countries. Delays in implementing strong mitigation policies allowed the disease to spreadworldwide. In Italy, while the ο¬rst two conο¬rmed cases on January 31, 2020 coincided with the government suspensionof all ο¬ights from China and the announcement of a national emergency, these actions were far from enough to stopthe spread of the virus.In contrast to other coronaviruses, notably SARS and MERS, an essentially new feature of COVID-19 is the preva-lence of asymptomatic infections. This was pointed out early in the pandemic by Al-Tawο¬q [2020], Bai et al. [2020],Chan et al. [2020], Hu et al. [2020], Lai et al. [2020], Tang et al. [2020], Wang et al. [2020]. Li et al. [2020] mentionsthat young carriers tend to show milder symptoms than older patients, while generally socializing more. Therefore,including asymptomatic carriers into COVID-19 epidemic models is crucial. Table 1 shows estimates of π , deο¬nedto be the proportion of asymptomatic carriers among all viral carriers, as provided by various studies Wu and Mc-Googan [2020], Mizumoto et al. [2020], Treibel et al. [2020], Nishiura et al. [2020], Gudbjartsson et al. [2020], Jiaet al. [2020], Oran and Topol [2020]. These estimates of π show no agreement, and consequently our paper aims tostudy how policy interventions are aο¬ected by this uncertainty about the prevalence of asymptomatic carriers. To thisend, we modify the standard SIR compartmental ordinary diο¬erential equation model introduced in Kermack et al.[1927] by splitting the infectious compartment I into A (asymptomatic carriers) and O (ordinary carriers), and includ-ing a compartment E (exposed) to represent the latent phase of the disease before the onset of infectiousness. Detailsabout SIR compartmental models can be found in Bailey et al. [1975], Hethcote [2000].Two distinct points of view are needed to understand infectious diseases such as COVID-19. First we need tounderstand the disease from the virus point of view. That is, the immunological characteristics of the disease must bedetermined: How symptoms are manifested, how transmissible the virus is, etc. Equally important is to understandthe disease from the perspective of human society, addressing questions of how individual behaviour evolves andadapts to the circumstances of the epidemic. Before a vaccine became available, we had limited means to change theimmunological properties of the virus. Instead policy focussed on changing peopleβs behaviour. One of the objectivesof this paper is to carefully disentangle the properties of the virus from human behaviour, and to determine how theeο¬ect of various policy interventions depends on the presence of asymptomatic carriers. β Corresponding author [email protected] (W. PANG)
ORCID (s): (W. PANG); (H. CHEHAITLI); (T.R. HURD)
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Page 1 of 15 a r X i v : . [ q - b i o . P E ] F e b mpact of asymptomatic COVID-19 carriers Society-wide public interventions may dramatically mitigate the course of the disease if implemented early enough.Unfortunately, despite the example of China where the disease originated, most of the worldβs countries were not quickor determined enough in their actions. To clarify the eο¬ectiveness of possible large scale interventions, we introducethe simplifying assumption that no eο¬ective policy measures are undertaken until a certain date, called the policytime π P . Our so-called base model is valid prior to π P , and represents the course of the disease in its βnatural stateβ,where society is behaving normally, unconcerned about the COVID dragon that has been stealthily predating on thepopulation. We imagine that society wakes up suddenly on date π P : The focus of this paper is to determine how theeο¬ectiveness of various alternative policies depends on the uncertain asymptomatic rate, if they are implemented onthat date.Another point is to distinguish the actual state of the disease at any moment (which is never fully observable) fromwhat we know about the state of the disease at that moment. In other words, in addition to modelling the state ofthe disease, we need to make assumptions about our observations about the state of the disease. For example, themost publicly available COVID-19 data are daily time series ( πΆ π‘ , π π‘ , π· π‘ ) of the number of active conο¬rmed cases,removed conο¬rmed cases, and conο¬rmed deaths. Unfortunately, these numbers are the result of often ineο¬cient datagathering procedures that vary dramatically from country to country. Thus our projections may be very imprecise. Inthis paper, to enable βapples to applesβ comparisons, we will assume that each country has in place a system of testingthat generates the time series of ( πΆ π‘ , π π‘ , π· π‘ ) . From this daily data, and known studies of COVID and human behaviour,we will infer the actual state ( π ( π‘ ) , πΈ ( π‘ ) , π ( π‘ ) , π΄ ( π‘ ) , π ( π‘ )) as continuous functions of time π‘ of the disease.Various kinds of public health policies have been proposed and implemented in diο¬erent jurisdictions. Fraser et al.[2004] recommends the isolation of symptomatic patients and their contacts. Wu et al. [2006] provides recommenda-tions for household-based public health interventions. Allred et al. [2020] discusses the eο¬ect of asymptomatic patientson the demand for health care. Bousema et al. [2014] discusses public health tools that can be used to deal with thepresence of asymptomatic carriers. Ferguson et al. [2006] apply large scale agent-based simulations to analyze theeο¬ect of public health policies on the spread of a virus in its early stages. However, few papers focus on public healthpolicy that targets a viral outbreak whose transmission rate is signiο¬cantly impacted by asymptomatic carriers.The structure of the paper is as follows. Section 2 analyzes the properties of the SE(A+O)R model in which infec-tives (I) are separated into asymptomatic carriers (AC) and ordinary carriers (OC). The state process is combined witha model for the observation process in Section 3. The resultant hybrid model is then calibrated to publicly availabledata for the early stages of the disease in two countries, Canada and Italy, prior to any nation-wide policy implementa-tion. It is shown that several parameters, notably π , are not identiο¬able in this calibration. In Section 4, we model fourtypes of public health intervention: isolation of infective patients, social distancing, personal protective equipment,and hygiene. It is shown that the outcome of a policy intervention is predictable (not dependent on the unidentiο¬ableparameters) for some of these policies, but not for others. Finally, in Section 5, we discuss some principles and pitfallsin designing policy interventions using a model with unidentiο¬ed parameters.
2. The Base Model
This paper will extend the standard SEIR (Susceptible, Exposed, Infectious, Removed) ordinary diο¬erential equa-tion (ODE) model by splitting the infectious compartment into two disjoint sets, asymptomatic viral carriers (AC) andordinary carriers (OC). The base model discussed in this section represents the outbreak only in its initial stages, beforethe ο¬rst major policy disease-speciο¬c intervention such as a country-wide lockdown. Later in the paper, we extend thepicture to account for policy interventions that aim to dramatically alter the progress of the disease. Our model willcarefully distinguish between βactualβ cases and βconο¬rmedβ cases. Conο¬rmed cases are the result of an observationprocess applied to, but not impacting, the actual system.
At the time of writing, there is still a large amount of uncertainty about the prevalence of ACs in the generalpopulation, partly because of diο¬erent deο¬nitions of what is meant by an AC. We adopt clear cut operational deο¬nitionssimilar to those of Oran and Topol [2020] that do not depend on whether or not the case has been tested or otherwiseconο¬rmed:
Deο¬nition 1. An Asymptomatic Carrier (AC) is someone who (a) has been exposed to COVID and is currently infectious; (b) will show no noticeable COVID symptoms for the entire infective period.
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Page 2 of 15mpact of asymptomatic COVID-19 carriers An Ordinary Carrier (OC) is someone who (a) has been exposed to COVID and is currently infectious; (b) will show some noticeable COVID symptoms at some point during the entire infective period.
Remarks 1.
Based on the Deο¬nition 1, presymptomatic carriers and carriers with mild symptoms are included in OC,as long as they eventually show recognizable symptoms.
With Deο¬nition 1, AC individuals are unlikely to be identiο¬ed and conο¬rmed, leading to great uncertainty in theirprevalance. Moreover, various studies deο¬ne the term βasymptomaticβ diο¬erently. These intrinsic diο¬culties make itextremely problematic to determine the key parameter, the asymptomatic fraction π , which we deο¬ne to be the long-timelimiting fraction of individuals who had the disease but were asymptomatic. Note that π is an intrinsic characteristic ofthe infection mechanism, but its value may diο¬er greatly from studies that adopt a diο¬erent deο¬nition. Table 1 displaysestimates of π made in a number of studies, which we see have sharpened somewhat over 2020, but still remain veryuncertain.The most conclusive study on Table 1, Oran and Topol [2020], summarizes their important message: βOn the basisof the three cohorts with representative samplesβIceland and Indiana, with data gathered through random selection ofparticipants, and Voβ, with data for nearly all residentsβthe asymptomatic infection rate may be as high as 40% to 45%.A conservative estimate would be 30% or higher to account for the presymptomatic admixture that has thus far notbeen adequately quantiο¬ed.β Table 1
Estimation of asymptomatic rate π of COVID-19 from various researchers betweenFebruary 2020 to September 2020.Date Sources Estimation of π February 2020 Novel et al. [2020] 1.2%February 2020 Mizumoto et al. [2020] 17.9% (15.5 - 20.2%)May 2020 Treibel et al. [2020] 1.1% - 7.1 %May 2020 Nishiura et al. [2020] 30.8% ( 7.7% - 53.8%)May 2020 Arons et al. [2020] 0% - 6%June 2020 Gudbjartsson et al. [2020] 5.7% - 58.3%August 2020 Jia et al. [2020] 58.9% - 92.5 %September 2020 Oran and Topol [2020] 40% - 45%
In the present section, we specify the basic SEAOR model for a fully homogeneous well-mixed population, appli-cable in a jurisdiction before any signiο¬cant COVID mitigation response has been initiated.
Assumptions 1. The total population π = π ( π‘ ) + πΈ ( π‘ ) + π΄ ( π‘ ) + π ( π‘ ) + π ( π‘ ) is constant. The natural birth rateequals the natural (pre-COVID) death rate. No immigration or emigration is considered from other countries. The removed compartment includes the recovered population who acquire permanent immunity and all COVIDdeaths. No vaccine is yet available for the virus, which means all people are susceptible prior to their ο¬rstexposure. The population is homogeneous and well-mixed, and thus the mass-action principle is assumed for the infectiontransmission. Both the latent period (exposed and not yet infectious) and infectious period are exponentialrandom times.
Remarks 2.
It is common modeling practice to extend the ODE approach to allow for π communities, with thehomogeneous and well-mixed assumption within each community. It is also common practice to model the latent andinfectious periods as random times with a more realistic gamma distribution. W. Pang et al.:
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Table 2
Notation π Susceptible population πΈ Those exposed to COVID-19 but not yet infectious π Ordinary carriers π΄ Asymptomatic carriers π Removed population πΌ π Transmission rate of Category (S) from Category (O) πΌ π΄ Transmission rate of Category (S) from Category (A) π½ Inverse duration time in Category (E) πΎ π Inverse duration time in Category (O) πΎ π΄ Inverse duration time in Category (A) π Fraction of (E) that become (A)
Based on these assumptions and using the notation identiο¬ed in Table 2, the SEAOR model is deο¬ned as follows: ππππ‘ = β( πΌ π π + πΌ π΄ π΄ ) πππΈππ‘ = ( πΌ π π + πΌ π΄ π΄ ) π β π½πΈππ΄ππ‘ = ππ½πΈ β πΎ π΄ π΄ππππ‘ = (1 β π ) π½πΈ β πΎ π πππ ππ‘ = πΎ π π + πΎ π΄ π΄ . (1)
Our objective is to consider targeted interventions that can mitigate the stark outcomes that emerge from the basecontagion model. Knowing the meaning of the model parameters is important in understanding the type of data thatwill be needed to determine them, and how potential policy interventions act.1.
Transmission parameters πΌ : These parameters are a product of more fundamental parameters that arise in βmi-croscaleβ agent-based models (ABMs) and network models. In general, πΌ π΄ , πΌ π are the average daily rate ofnew exposures that occur, per susceptible, per asymptomatic or ordinary infectious carrier. They are naturally aproduct of three factors, as described in Hurd [2020]:β’ π = π π = π π΄ is a constant describing the average number of signiο¬cant social relations per individ-ual. Since it is based on studies of normal social conditions, its value does not change under any policyintervention.β’ π§ is the daily rate of βclose contactsβ per signiο¬cant social relation. It naturally changes when an individualbecomes symptomatic and so we should expect that π§ π΄ > π§ π .β’ Infectivity π is the probability that a close contact with an infective person actually leads to exposure (hencethe disease). This can be reduced by policies that either boost immunity or reduce viral transfer. Whileone might expect that π π΄ < π π , in this paper we will assume the worst case π π΄ = π π .2. COVID-19 studies made during the early stages of the pandemic, notably Colizza et al. [2007], Diekmann andHeesterbeek [2000], suggest that the average latent period is about 5 days and average infectious period is 6 days.Under the exponential time assumption, these values justify the estimators for π½, πΎ π we will use throughout thispaper: Μπ½ = 1 Latent Period = 0 . , ΜπΎ π = 1 Infectious Period = 0 . (2)In view of the diο¬culty to observe asymptomatic cases, the parameter πΎ π΄ will be hard to determine, so it is natural tomake the assumption that πΎ π΄ = πΎ π βΆ= πΎ . W. Pang et al.:
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For the remainder of the paper, we make the assumption that πΎ π΄ = πΎ π βΆ= πΎ . With this condition, the SEAORmodel can be almost fully understood from properties of the following reduced model : ππππ‘ = β πΌ eff ππΌππΈππ‘ = πΌ eff ππΌ β π½πΈππΌππ‘ = π½πΈ β πΎπΌππ ππ‘ = πΎπΌ (3)where πΌ eff βΆ= (1 β π ) πΌ π + ππΌ π΄ . This fact derives from the following easily proved result. Proposition 2.
Suppose πΎ π΄ = πΎ π βΆ= πΎ and the initial conditions for (1) satisfy π΄ (0) = π ( π΄ (0) + π (0)) . Then thesolution of (1) is given by the solution of (3) by setting π΄ ( π‘ ) = ππΌ ( π‘ ) , π ( π‘ ) = (1 β π ) πΌ ( π‘ ) for all π‘ . It will turn out that calibration of the reduced model to conο¬rmed daily new case data will determine πΌ eff , but πΌ π , πΌ π΄ will not be determined separately without additional data. If we deο¬ne π = πΌ π΄ β πΌ π = ( π π΄ π§ π΄ )β( π π π§ π ) , then wehave πΌ π = (1 β π + ππ ) β1 πΌ eff , πΌ π΄ = π (1 β π + ππ ) β1 πΌ eff . (4)In our benchmark models, we will take π = 4 , which would result from the plausible relationships π π΄ = π π , π§ π΄ = 4 π§ π ,in other words, if asymptomatic carriers are four times more sociable and similarly infectious compared to symptomaticcarriers. The linearization of (1) about the disease-free equilibrium ( π, , , , provides a useful starting point to under-stand the early stages of the COVID-19 pandemic. With the restriction πΎ π΄ = πΎ π = πΎ , this can be reduced to thefollowing 3-d linear system with state vector π ( π‘ ) = ( πΈ ( π‘ ) , π΄ ( π‘ ) , π ( π‘ )) β² : ββββ ππΈ β ππ‘ππ΄ β ππ‘ππ β ππ‘ ββββ = ββββ β π½ πΌ π΄ π πΌ π ππ½π β πΎ π½ (1 β π ) 0 β πΎ ββββ ββββ πΈπ΄π ββββ βΆ= π΅ ββββ πΈπ΄π ββββ . (5)Any solution vector π ( π‘ ) = ( πΈ ( π‘ ) , π΄ ( π‘ ) , π ( π‘ )) β² of (5) generates an approximate solution of (1) by setting π ( π‘ ) = π (0) + πΎ β« π‘ ( π΄ ( π ) + π ( π )) ππ , (6) π ( π‘ ) = π β πΈ ( π‘ ) β π΄ ( π‘ ) β π ( π‘ ) β π ( π‘ ) . (7)The linearized approximation (5) will be suο¬ciently accurate as long as π ( π‘ )β π is suο¬ciently close to .The spectral properties of the matrix π΅ can be summarized by the three eigenvalue-eigenvector pairs: π + , π + = ββββ ππ£ + (1 β π ) π£ + ββββ ; π β , π β = ββββ ππ£ β (1 β π ) π£ β ββββ ; β πΎ , π πΎ = ββββ ββββ (8)where π Β± = β π½ + πΎ β ( π½ β πΎ ) πΌ eff π½π, π£ Β± = π½ + π Β± πΌ eff π . (9)Furthermore, the basic reproduction number (βR-naughtβ) is π = πΌ eff ππΎ . (10)
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Our primary interest will focus on cases of a pandemic with π > which is equivalent to π + > > π β . In suchsituations, the general solution π ( π‘ ) = ( πΈ ( π‘ ) , π΄ ( π‘ ) , π ( π‘ )) β² , π‘ β₯ of (5) with any positive initial small COVID infectionhas the form π ( π‘ ) = π π π + π‘ π + + π π π β π‘ π β + π π β πΎπ‘ π πΎ (11)for coeο¬cients π , π , π , and will exhibit an exponentially fast convergence to a multiple of the dominant eigensolution π + ( π‘ ) = (1 , ππ£ + , (1 β π ) π£ + ) β² π π + π‘ . (12)We emphasize that this dominant solution π + ( π‘ ) corresponds to the early exponentially growing phase of the pandemic.Its rate π + depends on the eο¬ective transmission rate πΌ eff = (1 β π ) πΌ π + ππΌ π΄ , rather than on π, πΌ π , πΌ π΄ separately. InSection 3 we will ο¬nd that π + ( π‘ ) does indeed provide a reasonably good ο¬t for various countries during the early stagesof COVID. Note also that by Proposition 2, (1 , π£ + ) β² π π + π‘ will be a solution of the linearization of the reduced system(3). Let us now consider a country, for example Italy or Canada, and let π‘ = 0 denote time 00:00 on January 1, 2020. Weassume that the ODE model is an acceptable approximation after a time called the pandemic time π when a suο¬cientnumber of cases have been generated: We deο¬ne π to be the start of the ο¬rst day the conο¬rmed cumulative casesexceeded cases in the country under study. Our aim is ο¬rst to study the short period of time [ π , π ] called the pre-policy period , starting at the pandemic time π and ending at the policy time π , deο¬ned to be the time of theο¬rst nation-wide policy intervention. Because policy changes taking place on π will take several days to have anobservable eο¬ect on case numbers, we ο¬rst calibrate our model to the calibration period [ π , π + 5] , which includes5 days following π . In Section 4 we will study the eο¬ect of possible public health policy interventions implementedat the policy time π for the six-weeks long post-policy period [ π , π ] with the end time π = π + 42 .In diο¬erent countries around the world, the pandemic time π and policy time π typically occurred in Februaryand March 2020. In this paper, we focus for illustrative purposes on two countries, Italy and Canada, where these datesare summarized in Table 3. In Canada, the conο¬rmed cumulative cases reached 51 on the day following the pandemictime π = 48 (February 18). On the day following the policy time π = 73 (March 15), Ontario, Canadaβs largestprovince, mandated province-wide public school closures and other provinces made similar announcements. Italyβspandemic time was the same, π = 52 (start of February 22), and on that date the conο¬rmed cumulative cases reached79. On the other hand, Italy imposed a nation-wide lockdown policy on the day following the policy time π = 68 (March 10). Table 3
The pre-policy and post-policy periods in Italy and Canada extend over [ π , π ] and [ π , π ] respectively,where π , π are shown in the table and π = π + 42 , which is six weeks after policy time.Country Pandemic Time ( π ) Policy Time ( π )Italy π = 52 π = 68 (start of February 22, 79 Cases) (start of March 10, Nationwide lockdown)Canada π = 48 π = 73 (start of February 18, 51 Cases) (start of March 15, Ontario School Shutdown)
3. Pre-Policy Calibration
This section will show that the SEAOR model with the parametric restriction πΎ π΄ = πΎ π βΆ= πΎ provides a reasonablygood ο¬t to the disease in its early stages for countries such as Canada and Italy, when calibrated to ο¬t the observeddata, speciο¬cally, the conο¬rmed daily new case data for the period [ π , π + 5] (i.e. the pre-policy period plus 5 days). W. Pang et al.:
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The ODE system (1) captures the dynamics of the unobserved state of the population, and its parameters do notdepend on how the system is observed. Parameters πΌ π , πΌ π΄ combine characteristics of the disease with the social mixingparameters and behaviour, and hence depend strongly on policy and jurisdiction. There are many possible methodsfor observing the state and evolution of the system. We make some simplifying assumptions about the model and theobservation process. Assumptions 2.
During the pre-policy period [ π , π ] , model parameters are constant. Among the OC population,an expected fraction π π are counted as conο¬rmed cases, typically as a result of either a positive RT-PCR test (βswabtestβ) or a diagnosis by symptoms. In contrast, none of the AC population are counted as conο¬rmed cases ( π π΄ = 0 ). Let us denote the conο¬rmed daily new cases on the day ending at time π + π by Μ DNC π , for π = 1 , , β¦ , πΎ = π + 5 β π . We make the assumption that the data generating process is a random process ο¬uctuating around π ( π‘ ) ,the dominant solution (12) of the linearized SEAOR model: Μ DNC π = ( π π (1 β π ) π½ β« π + ππ + π β1 πΈ ( π ) ππ ) π π π . (13)Here ( π π ) π =1 , , β¦ ,πΎ is an i.i.d. π (0 , π ) sequence of residuals, πΈ ( π ) = πΈ π π + ( π β π ) with πΈ = πΈ ( π ) , and π + is givenby (9). The logarithm of the data generation process (13) is a simple linear regression log Μ DNC π = π + π + π + π π , with π βΆ= log( π π (1 β π ) π½πΈ ( π )(1 β π β π + )β π + ) . This leads to the least square estimates Μπ = ( Μπ , Μπ + ) for two identiο¬ableparameters Μπ = intercept coeο¬cient ; Μπ + = slope coeο¬cient . (14)Given Μπ , Μπ + , the factors of π π = π π (1 β π ) π½πΈ (1 β π β π + )β π + are not separately identiο¬able. The mean squaredregression error Μπ = MSE πΎ is Μπ = MSE πΎ = 1 πΎ πΎ β π =1 | log( Μ DNC π ) β π β π + π | (15)Figure 1 displays values given by the calibrated model and observed data for daily new cases on a log-scale.We note that for both Italy and Canada, the calibrated model gives a reasonably good ο¬t, albeit with a signiο¬cantdegree of noise. Estimators Μπ½, ΜπΎ, Μπ + , Μπ , when combined with (9), lead to estimates for further important parameters ΜπΌ eff , Μπ β , Μπ£ + , π . Table 4 shows those parameter values resulting from the calibration for Canada and Italy. Table 4
Parameter estimates for Italy and Canada during the pre-policy period.Country Μπ Μπ + MSE = Μ π Μπ β ΜπΌ eff Μπ£ + π Italy 1.6974 0.1999 0.015 -0.5666 1.214e-08 0.546 4.40Canada 0.4090 0.2037 0.021 -0.5703 1.988e-08 0.540 4.48
These parameter estimates do not fully determine the model and its initial conditions: π, π π , πΌ π , πΌ π΄ , πΈ are leftundetermined, but constrained by two equations: Μπ = log( π½π π (1 β π ) πΈ (1 β π β Μπ + )β Μπ + ) , (16) ΜπΌ eff = (1 β π ) πΌ π + ππΌ π΄ (17) W. Pang et al.:
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Figure 1:
Simulation and oο¬cial data of daily new cases (DNC) for Italy and Canada in the pre-policy calibration period [ π , π + 5] . Table 5 provides the best-ο¬t values for the actual state of the pandemic on the dates π , π using the linearized solutions πΈ ( π‘ ) = ( π½π π (1 β π )(1 β π β Μπ + )β Μπ + ) β1 π Μπ + Μπ + ( π‘ β π ) (18) πΌ ( π‘ ) = Μπ£ + ( π½π π (1 β π )(1 β π β Μπ + )β Μπ + ) β1 π Μπ + Μπ + ( π‘ β π ) (19)assuming the model may have diο¬erent values of π π (1 β π ) . The removed value π ( π‘ ) can be accurately approximatedby πΎ β« π‘ ββ πΌ ( π ) ππ = πΎπΌ ( π‘ )β Μπ + . Note that to determine the separated compartment populations π΄ ( π‘ ) = ππΌ ( π‘ ) , π ( π‘ ) =(1 β π ) πΌ ( π‘ ) , we also need the value of π . This indistinguishability of models leads to diο¬culty about the eο¬cacy ofdiο¬erent health policy interventions, as we will investigate in the next section. Table 5
The actual state of the pandemic at times π , π obtained from the linearized model, depending on the additionalparameters π π , π . Given π , the asymptomatic and ordinary case numbers will be π΄ ( π‘ ) = ππΌ ( π‘ ) , π ( π‘ ) = (1 β π ) πΌ ( π‘ ) . π π (1 β π ) Country πΈ ( π ) πΌ ( π ) π ( π ) πΈ ( π ) πΌ ( π ) π ( π )
20% Italy 150 82 68 8208 4478 3731Canada 41 22 18 15279 8251 560940% Italy 75 41 34 4104 2239 1865Canada 20 11 9 7639 4125 280460% Italy 50 27 22 2736 1492 1243Canada 13 7 6 5093 2750 186980% Italy 37 20 17 2052 1119 932Canada 10 5 4 3819 2062 1402100% Italy 30 16 13 1641 895 746Canada 8 4 3 3055 1650 1121
4. Public Health Policy Interventions
Policy makers seek to mitigate the societal damage from the disease by adopting actions or policies that reduce thetransmission parameters πΌ π , πΌ π΄ , thereby decreasing the exposure rate of the susceptible population to viral carriers.Other important parameters, notably π½, πΎ, π , are not easily controllable and will be taken as unchanged by such policies.In this section, we consider the example of Italy under a range of hypothetical scenarios where a substantial healthpolicy intervention is made instantaneously at the policy time π , with a constant level of eο¬ort thereafter. Since theparameters π, π are not identiο¬able from the database used in calibration, we explore how the eο¬ectiveness of any policy W. Pang et al.:
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Page 8 of 15mpact of asymptomatic COVID-19 carriers depends on π, π . The non-linear system of ODEs (1) is thus solved for pre-policy parameters for the period [ π , π ] with the initial condition at time π given by Table 5 for diο¬erent values of a product π (1β π ) , and then with post-policyparameters for the period [ π , π ] . The starting conditions at time π have the form ( π ( π ) , πΈ ( π ) , π΄ ( π ) , π ( π ) , π ( π )) with π΄ ( π ) = π ( π΄ ( π ) + π ( π )) , and πΎ π΄ = πΎ π = πΎ at all times. Therefore Proposition 2 applies for the period [ π , π ] and again for the period [ π , π ] and so in the following explorations, we may solve the reduced non-linear system (3)and use the equations π΄ ( π‘ ) = ππΌ ( π‘ ) , π ( π‘ ) = (1 β π ) πΌ ( π‘ ) to obtain the desired solution of (1). To quantify the eο¬ect of policy π , we ο¬rst deο¬ne the maximal eο¬ect of the policy to act on the pair of transmissionparameters ( πΌ π , πΌ π΄ ) leading to new values (1β π£ ππ ) πΌ π , (1β π£ π΄π ) πΌ π΄ , where ( π£ ππ , π£ π΄π ) β [0 , is called the maximal eο¬ectvector . Then, if the policy π is adopted with a partial degree of eο¬ort π π β [0 , , the policy eο¬ect on the transmissionparameters is assumed to lead to new values ( πΌ π , πΌ π΄ ) βΆ ( πΌ ππ , πΌ π΄π ) = ( (1 β π π π£ ππ ) πΌ π , (1 β π π π£ π΄π ) πΌ π΄ ) . (20)At a more fundamental level, diο¬erent policies typically act by directly reducing some of the infectious contactparameters π§ π , π§ π΄ or the infection probabilities π π , π π΄ . By their deο¬nition, π = π π = π π΄ are not changed in shortterm policies. Under the assumptions of Proposition 2, the net eο¬ect of each policy on the impact of the disease is onlythrough the eο¬ective parameter πΌ eff = (1 β π ) πΌ π + ππΌ π΄ where πΌ π = π π§ π π π , πΌ π΄ = π π§ π΄ π π΄ : diο¬erent interventionswhich result in the same change in πΌ eff will lead to essentially the same impact on the disease. However, such anequivalence depends on the unknown value of π : two policies may lead to the same πΌ eff for one value of π , but not forother values of π . This last point about how the impact of policies on the disease depends on the unobserved parameter π is a central message of this paper.To underline the importance of targeting πΌ eff , note that the basic reproduction number π = πΌ eff ππΎ is in ο¬xedproportion to πΌ eff . From Table 4, we see immediately that to reduce π to below in Italy on the policy date π , whichis the goal of public policy, one needs to reduce πΌ eff by an overall factor exceeding . .To this end, the following types of policy, labeled by π , can be implemented one at a time with varying eο¬orts, orin combinations. Table 6 provides the ad hoc benchmark parameters we use for expository purposes: Finding morerealistic values is deserving of further study. Table 6
Benchmark parameters for the four single policies with maximal eο¬ort. π = 1 π = 2 Social π = 3 Protective π = 4 Isolation Distancing Garments Hygiene π£ ππ π π = (0 . . π£ π΄π π π πΌ ππ β πΌ π πΌ π΄π β πΌ π΄ Isolation of Infective Patients:
This type of policy ( π = 1 ) aims to prevent actively infectious cases fromencountering susceptible people, and directly targets the close contact parameter π§ π . Such a policy does notaο¬ect asymptomatic people, nor ordinary carriers that are not conο¬rmed cases. We suppose that the maximaleο¬ect on any one conο¬rmed carrier is to reduce their close contact rate by a fraction π . Thus the maximal eο¬ectvector of this policy is π£ = ( π π π , , and with eο¬ort π its eο¬ect on the transmission parameters is ( πΌ π , πΌ π΄ ) βΆ ( πΌ π , πΌ π΄ ) = ( (1 β π π π π ) πΌ π , πΌ π΄ ) . (21)2. Social Distancing:
Social distancing ( π = 2 ), such as a policy that requires keeping at least 2m distance inpublic spaces, can be targeted at identiο¬able sub-populations, or applied fairly across the general population. Astrategy that targets the general population equally will have equal impact on the close contact fractions π§ π , π§ π΄ W. Pang et al.:
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Page 9 of 15mpact of asymptomatic COVID-19 carriers for both symptomatic viral carriers and asymptomatic viral carriers. If implemented with eο¬ort π and maximaleο¬ect vector ( π£ , π£ ) , the policy leads to ( πΌ π , πΌ π΄ ) βΆ ( πΌ π , πΌ π΄ ) = ( (1 β π π£ ) πΌ π , (1 β π π£ ) πΌ π΄ ) . (22)3. Protective Garments:
The wearing of personal protective equipment (PPE) ( π = 3 ), including gloves, gowns,masks, face shields and eye protection, can be applied to the general population, and attempts to reduce thetransmission probabilities π π , π π΄ equally. If implemented with eο¬ort π and a maximal eο¬ect vector ( π£ , π£ ) ,this policy leads to ( πΌ π , πΌ π΄ ) βΆ ( πΌ π , πΌ π΄ ) = ( (1 β π π£ ) πΌ π , (1 β π π£ ) πΌ π΄ ) . (23)Some studies are helpful for determining π£ : For example, Li et al. [2006] claim that the eο¬ciency of surgicalmasks is 95%, compared with 97% for N95 masks.4. Hygiene:
Infection via contaminated βfomitesβ (i.e. inanimate surfaces or objects), where active virus is ab-sorbed from surfaces, has been considered an important mode of COVID transmission. Cleanliness ( π = 4 ),particularly frequent handwashing and disinfecting surfaces, is the most important way of reducing spreadingby viral contamination of fomites. If implemented with eο¬ort π and maximal eο¬ect vector ( π£ , π£ ) , a cleanlinesspolicy has the following eο¬ect ( πΌ π , πΌ π΄ ) βΆ ( πΌ π , πΌ π΄ ) = ( (1 β π π£ ) πΌ π , (1 β π π£ ) πΌ π΄ ) . (24)Recent studies summarized in Mondelli et al. [2020] have cast doubt on the overall importance of fomite trans-mission compared to aerosol transmission, and suggest that the maximal vector for this policy is small. Remarks 3.
An eο¬ective vaccine is potentially the most powerful intervention tool: It acts directly on the immunesystem of susceptible individuals to reduce the infection probability π to near zero. The SEAOR framework is inadequateto address vaccination: More preferable is to adopt a SVEAOR variation of our model that we do not pursue here. In this section, we use the calibrated base model for Italy to analyze the eο¬ect of implementing a single strategy.We consider two distinct policy strategies, implemented singly: (a) isolation of all conο¬rmed symptomatic patients(strategy π = 1 ), (b) protective garments for the general population (strategy π = 3 ). We will demonstrate the importantpoint that the eο¬ectiveness of a policy that applies equally to the entire population, as in case (b), does not depend on π .In contrast, the eο¬ectiveness of a policy that targets only conο¬rmed active cases, such as isolation, can not be predictedwithout knowing π , and as we will see, other parameters. We therefore focus on the eο¬ect of these policies for diο¬erentvalues of the asymptomatic rate π . Optimally, let us suppose that when fully implemented, each conο¬rmed case reduces their overall transmission rateby π = 90% in Eq (21).The overall eο¬ectiveness of isolation on the disease itself is determined by the change in πΌ eff = ππΌ π΄ + (1 β π ) πΌ π .This however depends on several other undetermined parameters, speciο¬cally π and the ratio π = πΌ π΄ πΌ π : πΌ eff = ππΌ π΄ + (1 β π ) πΌ π βΆ (1 β π + ππ ) β1 [ (1 β π )(1 β π π π ) + ππ ] πΌ eff . (25)Figure 2 shows the log of the actual cumulative cases and conο¬rmed daily new cases predicted by the model for Italyduring the entire pre-post period [ π , π ] , under the maximal isolation policy. These graphs assume that before anypolicy π§ π΄ = 4 π§ π , π π = π π΄ , and hence π = 4 . With these parameters, what appears to be a strong policy measure willfail outright if the asymptomatic rate π exceeds about 10% and of course the results will be even worse if the eο¬ortparameter is π < .Much more can be deduced from the right hand graph shown in Figure 2. As a consequence of Proposition 2, theisolation policy implemented with any other combination of parameters π β² , π β² , π β² , π β² , π β² that satisfy (1 β π β² + π β² π β² ) β1 [ (1 β π β² )(1 β π β² π β² π β² ) + π β² π β² ] = (1 β π + 2 π ) β1 [(1 β π )(1 β 0 .
81) + 2 π ] (26)will have the identical time-development as the curve with rate π . W. Pang et al.:
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Page 10 of 15mpact of asymptomatic COVID-19 carriers a Actual Cumulative Cases b Conο¬rmed Daily New Cases
Figure 2:
ITALY: The model prediction showing the eο¬ect of the maximal isolation policy if implemented on the policytime π (March 9, 2020). Here we ο¬x the pre-policy ratio πΌ π΄ = 4 πΌ π , the transmission reduction factor π = 0 . and theconο¬rmation factor π π = 0 . , and plot over the period [ π , π ] for varying π . The ο¬rst graph shows the actual cumulativecases consisting of all symptomatic and asymptomatic infections, and removed patients. The second graph shows conο¬rmeddaily new cases. Here we consider the eο¬ect of a nation-wide policy of mask wearing where for deο¬niteness we suppose there is amaximal eο¬ect π£ βΆ= π£ π = π£ π΄ = 0 . , and a degree of eο¬ort π = 0 . . In this setting, the result does not dependon π or π . Figure 3 shows the log of the actual cumulative cases and conο¬rmed daily new cases predicted by themodel for Italy during the entire pre-post period [ π , π ] , under the mask wearing policy. Since πΌ eff β (1 β π π£ ) πΌ eff ,we see that under all variations of the model with π£ = 0 . , π = 0 . the pandemic is brought under control, with π βΌ 0 . .
40 = 0 . . When the general population keep social distance (strategy π = 2 ), the eο¬ect is similar to protective garments inthat πΌ π΄ , πΌ π are changed by equal fractions ( π£ , π£ ) , even though it targets π§ π΄ , π§ π instead of π π΄ , π π . This similarity alsoholds for policies of improved hygiene, if applied equally across the general population. If we combine these threestrategies into a single policy we call General Personal Protection , then the logic of the previous analysis for protectivegarments remains unchanged.Very unlike general personal protection policies, the eο¬ectiveness of the policy of isolation of conο¬rmed activecases depends very strongly on the diο¬cult-to-observe parameters π, π and π , and in particular will perform especiallypoorly if π is large. Indeed, under conservative assumptions π = 10% , π π = 0 . , π = 4 , the maximal isolation policyfails completely to halt the pandemic.Contact tracing can be regarded as a policy that improves the eο¬ectiveness of isolation by identifying a larger frac-tion of infectious cases. In other words, it seeks both to increase the parameter π π > π , and to identify a fraction π π΄ > of asymptomatic cases. All these additional cases would then be included in the implementation of isolation policy.Eο¬ectively then, isolation combined with improved contact tracing has a maximal vector ( π£ π , π£ π΄ ) = ( π π π , π π π΄ ) .Figure 4 shows how isolation and social distancing policies lead to very diο¬erent outcomes for the pandemic. Undersome reasonable assumptions on π, π and π the maximal isolation policy has very little impact on the pandemic, whilethe maximal protective garments policy eliminates the disease very quickly. Let us consider how the set of COVID mitigation strategies π β π that are available to the policy maker can beimplemented in combination, following the βswiss cheeseβ metaphor. First note that some care is needed to account W. Pang et al.:
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Page 11 of 15mpact of asymptomatic COVID-19 carriers a Actual Cumulative Cases b Conο¬rmed Daily New Cases
Figure 3:
ITALY: The model prediction showing the eο¬ect of the maximal protective garments if implemented on thepolicy time π (March 9, 2020). Here we ο¬x the policy factors π£ = 0 . , π = 0 . , and show that the pandemic curve overthe period [ π , π ] is independent of π . The ο¬rst graph shows the actual cumulative cases consisting of all symptomaticand asymptomatic infections, and removed patients. The second graph shows conο¬rmed daily new cases. a Isolation b Protective Garments Figure 4:
The eο¬ect of isolation and protective garments compared: These graphs show the logarithm of the active conο¬rmcases in Italy on time π corresponding to April 20, 2020. The left graph shows the dependence on the eο¬ort expendedon isolation, π£ π β [0 , , and on the asymptomatic rate π β [0 , . . The right graph shows the dependence on the eο¬ortexpended on protective garments, π£ π β [0 , , and on the asymptomatic rate π β [0 , . when π = 1 and π = 4 . for possible interference between the eο¬ects of diο¬erent strategies. Under the following assumption, such interferenceeο¬ects have been eliminated: Assumption 3. [Independent policy assumption] The eο¬ect of the set of strategies π β π when implemented in com- W. Pang et al.:
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Page 12 of 15mpact of asymptomatic COVID-19 carriers bination with eο¬orts π = ( π π ) π β π β [0 , π is to map the transmission parameters to new values: ( πΌ π , πΌ π΄ ) βΆ ( πΌ π β π β π (1 β π π π£ ππ ) , πΌ π΄ β π β π (1 β π π π£ π΄π ) ) . (27)Let us take the point of view of the Italian National Health Authority that recognized the potentially devastatingimpact of the pandemic, and implemented a remediation strategy eο¬ective on policy date π , March 9, 2020. Thebest data available at that time indicates that the pandemic has a daily exponential growth rate π + βΌ 0 . , and aneο¬ective reproduction rate π βΌ 4 . . To bring the pandemic under control will therefore require extreme measures:we suppose the authority aims to reduce π to a value less than . , ensuring a reasonably quick resolution of thebreakout. Supposing that the authority fails to recognize the presence of asymptomatic carriers, we now study howtheir strategy fails to produce a desirable outcome.Based on these assumptions, their ο¬rst line of attack will be to expend maximal eο¬ort π = 1 to identify andisolate all known active covid cases, following standard epidemic management policy. Because isolation ( π = 1 ) aloneis insuο¬cient to bring π below 0.8, the authority needs to also consider more general strategies. To this end theyhave identiο¬ed three additional policies ( π = 2 , , ): social distancing (including shutting down some businesses andenforcing distancing rules), mask wearing and improved hygiene (including widespread use of hand sanitizer). We noteagain that if applied to the general population, these three policies have a similar impact on π and can be combinedinto one policy which we call general personal protection (GPP).Assuming the benchmark parameter values shown in Table 6, if the three policies are applied to the general pop-ulation and implemented with eο¬orts ( π , π , π ) , then following (27), their collective maximal and partial eο¬ects willbe multiplicative factors (1 β π£ GPP ) βΆ= Ξ π β{2 , , (1 β π£ π ) , (1 β π GPP π£ GPP ) βΆ= Ξ π β{2 , , (1 β π π π£ π ) , (28)on both πΌ π , πΌ π΄ .Figure 5 shows contour plots of the achieved value of π under a combination of isolation with GPP for a variablelevel of π , π GPP when π£ = 0 . , π£ GPP = 0 . . The ο¬rst assumes π = 40% and the second assumes . We seeclearly from this that if π = 40% , achieving the desired value π = 0 . requires a far greater eο¬ort than the hypotheticalproposed strategy. When π = 40% , only a combination of GPP with π GPP = 90% eο¬ort and maximal isolation policycan control the pandemic. A strict isolation policy combined with GPP with π GPP = 70% eο¬ort that is suο¬cient toachieve π = 0 . if the authority mistakenly assumes π = 10% , only achieves π = 1 . if it turns out that π = 40% .
5. Discussion and Conclusions
The main aim of this paper has been to demonstrate why neglecting the existence of asymptomatic carriers is sucha dangerous error for a disease like COVID-19. When planning health policy to control a pandemic such as this, itis easy to underestimate this eο¬ect, especially during the critical early stages. In hindsight, knowing now that π isconservatively estimated to be larger than , we can argue that indeed apart from parts of East Asia, most healthpolicy in the world has failed outright largely because of this oversight.We have considered two distinct types of intervention policies: Isolation that can only target identiο¬ed active con-ο¬rmed cases and policies that target the general population. Our main contribution is to demonstrate an essentialdiο¬erence between the two, namely that the former type of policy depends critically on a number of additional pa-rameters that the latter do not depend on. The same value of π = πΌ eff π β πΎ will arise from diο¬erent speciο¬cationsof the base model that cannot be distinguished with daily new case data, but these observationally equivalent basemodel speciο¬cations respond very diο¬erently to a policy change that involves isolation of conο¬rmed cases. In modelspeciο¬cations where π and π are large, there will be hard to identify asymptomatic cases, and these cases will be onaverage more infectious than ordinary cases. In such circumstances, the eο¬ective π achievable by a policy of isolationwill be signiο¬cantly underestimated.A second conclusion is that many types of intervention that target the general population can be combined, withresults that do not depend on π and π and are therefore very predictable. To the extent that π GPP is suο¬ciently large, allcombinations of policies involving general personal protection will be similarly eο¬ective in controlling the disease.
W. Pang et al.:
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Page 13 of 15mpact of asymptomatic COVID-19 carriers a π with two policies if π = 40% . b π with two policies if π = 10% . Figure 5:
This contour plot shows the eο¬ective π after policy date π in Italy as a function of π , π GPP , when isolationis applied with eο¬ort π and π£ = 0 . , and general personal protection is applied with eο¬ort π GPP and π£ GPP = 0 . . Thedarkness of the red color denotes the value of π . Plot (a) shows results when π = 40% , (b) assumes π = 10% . With Proposition 2, this paper also makes a mathematical contribution by showing that under a natural assumption πΎ π΄ = πΎ π , the reduced SEIR ODE model will provide almost complete information of the behaviour of the SEAORmodel. This fact provides a very useful simpliο¬cation for studying policy implications.Having a good estimate of the critical parameter π in timely fashion is certainly essential to controlling COVID.Extensive testing accompanied by contact tracing, providing a large representative sample of the general population, isneeded to adequately determine the impact of asymptomatic carriers. Testing and contact tracing also have the eο¬ectof raising the fractions π π , π π΄ , thereby improving the eο¬ectiveness and predictability of isolating known active cases.With extensive testing also comes more granular data that can identify the subpopulations that are most responsiblefor propagating the disease. When such data is used, more speciο¬c targeted interventions become practical and costeο¬ective. Our methods can easily be generalized to analyze such targeted interventions. It is also important to reiteratethat while π may depend on the strain of COVID, its value does not depend on policy or social behaviour and is thereforestable over time and in diο¬erent parts of the world. Thus scientists in every country need to follow carefully the worldliterature for information on this parameter.Many common sense aspects of public policy can be subjected to scrutiny using the techniques discussed in ourpaper. For example, we now know that mask wearing (strategy 3), would have been highly eο¬ective had it been widelyimplemented early in the pandemic. On the other hand, frequent hand washing (strategy 4) that was strongly advocatedand adopted by the general population provided much less protection than was expected. Restricting access to essentialfacilities such as grocery stores can backο¬re because social distancing requires as much space as possible. Finally, theshutting down of parks and open spaces in the early months of COVID was a weak strategy for controlling a diseasewhose transmission is dominated by aerosol and droplets. References
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