A model for COVID-19 with isolation, quarantine and testing as control measures
aa r X i v : . [ q - b i o . P E ] M a y A model for COVID-19 with isolation, quarantine and testing ascontrol measures
M.S. Aronna ∗ , R. Guglielmi † , and L.M. Moschen ‡ Escola de Matem´atica Aplicada, FGV EMAp - Rio de Janeiro, BrazilMay 18, 2020
Abstract
In this article we propose a compartmental model for the dynamics of Coronavirus Disease2019 (COVID-19). We take into account the presence of asymptomatic infections and the mainpolicies that have been adopted so far to contain the epidemic: isolation (or social distancing)of a portion of the population, quarantine for confirmed cases and testing. We model isolationby separating the population in two groups: one composed by key-workers that keep workingduring the pandemic and have a usual contact rate, and a second group consisting of people thatare enforced/recommended to stay at home. We refer to quarantine as strict isolation, and it isapplied to confirmed infected cases.In the proposed model, the proportion of people in isolation, the level of contact reductionand the testing rate are control parameters that can vary in time, representing policies thatevolve in different stages. We obtain an explicit expression for the basic reproduction number R in terms of the parameters of the disease and of the control policies. In this way we canquantify the effect that isolation and testing have in the evolution of the epidemic. We presenta series of simulations to illustrate different realistic scenarios. From the expression of R andthe simulations we conclude that isolation (social distancing) and testing among asymptomaticcases are fundamental actions to control the epidemic, and the stricter these measures are andthe sooner they are implemented, the more lives can be saved. Additionally, we show that peoplethat remain in isolation significantly reduce their probability of contagion, so risk groups shouldbe recommended to maintain a low contact rate during the course of the epidemic. In late December 2019, several cases of an unknown pneumonia were identified in the city of Wuhan,Hubei province, China [23]. Some doctors of Wuhan conjectured that it could be severe acuterespiratory syndrome (SARS) cases [24]. Many of the found cases had visited or were related to theHuanan Seafood Wholesale Market. On 31 December 2019, the World Health Organization (WHO)China Country Office was informed of these cases of pneumonia detected in Wuhan City and, up to3 January 2020, a total of 44 patients with this unknown pneumonia were reported to WHO [30].In the beginning of January 2020 Chinese officials ruled out the hypothesis that the cases were ofSARS [17], and a few days later the cause was identified to be a new coronavirus that was namedSARS-CoV-2. The name given to the infectious disease caused by SARS-CoV-2 is COVID-19.The first death due to COVID-19 was reported on 9 January and it was a 61-year-old man inWuhan [39]. After mid January infected cases were reported in Thailand, Japan, Republic of Korea,and other provinces in China [23]. On 22 January the Chinese authorities announced the quarantineof greater Wuhan. From that time on the virus rapidly spread in many Asiatic countries, reached ∗ [email protected] † [email protected] ‡ [email protected] key-workers that keep working during the pandemic and having a usual contact rate, and the other groupconsisting of people that are enforced/recommended to stay at home. Certainly, in the group of peoplethat maintain a high contact rate one can also include people that do not respect social distancingrestrictions, that has lately shown to be significant in some countries. We refer to quarantine asstrict isolation, and it is applied to confirmed infected cases. Testing is supposed to be appliedto all symptomatic cases, and to a portion of the population selected using some of the criteriaadopted by health organizations (see e.g. [9, 11]). The idea to analyze the quantitative effects ofnon-pharmaceutical interventions, such as isolation and social distancing, on the evolution of theepidemic was inspired by the work [10].For the proposed model, we obtain an expression for the basic reproduction number R in terms ofthe parameters of the disease and of the control parameters. In this way we can quantify the effectsthat isolation and testing have on the epidemic. We exhibit a series of simulations to illustratedifferent realistic situations. We compare, in particular, different levels of isolation and testing.From the expression of R and the simulations, we conclude that isolation (social distancing) andtesting among asymptomatic cases are fundamental actions to control the epidemic, and the stricterthese measures are and the sooner they are implemented, the more lives can be saved. Additionally,we show that people that remain in isolation significantly reduce their probability of contagion, sorisk groups should be recommended to maintain a low contact rate during the course of the epidemic.Several mathematical models for COVID-19 have appeared recently in the literature. At the timebeing, the flux of publications is very high, so it is difficult to keep track of everything that is beingpublished. We next mention and describe some of the models more closely related to ours. In [3]they consider a simple model, with infected and reported infected compartments, and they assumethat the transmission rate β is a function of a control u, this is β = β ( u ) . They analyze feedbackcontrol strategies, where the control depends on the number of reported cases. In [8] they considermild and severe cases, the latter having a reduced transmission rate since they are assumed to be in2solation. They use a time-dependent control c of reduction of contacts for the whole population, andoptimize with respect to this control. An SEIR model with quarantine for suspected and infectedcases cases is considered in [37], and in [21] they take into account unreported cases, asymptomaticindividuals and quarantine for identified cases.The article is organized as follows. In Section 2 we introduce the model, and we discuss itsstructure. In Section 3 we show an expression of the basic reproduction number R in terms ofthe parameters and we propose an equivalent threshold. Estimation of realistic parameters andnumerical simulations are given in Section 4, while Section 5 is devoted to the conclusions anda description of possible continuations of this research. Finally, in the Appendix we include theanalytical computations of the expression of R and a sensitivity analysis with respect to the involvedparameters. We set up a model to describe the spread of the virus SARS-CoV-2 through a susceptible population.Building upon a usual SEIR model, we obtain a more structured one, which is tailored on thecurrent experience of the COVID-19 epidemic, and which also allows to convey the effects of thenon-pharmaceutical intervention policies being adopted by several countries to face its outbreak.First of all, we normalize to 1 the total population of N individuals, so that all the compartments(and their sub-compartments) introduced below represent the proportion of individuals of the totalpopulation in such compartment. We will assume the population remains constant over time (i.e., weneglect the natural birth and death rates). We start by splitting the population in the compartmentslisted in Table 1. Compartment Description S susceptible E exposed I infectious A asymptomatic and infectious Q infected in quarantine (including hospitalized) R recoveredTable 1: List of aggregated compartmentsMore specifically, the compartment S collects all the individuals that are susceptible to the virus.Once an individual from S gets exposed to the virus, moves to the compartment E . Let us pointout that individuals in E , though already exposed to the virus, are not contagious yet. After a given latent time , an individual in E becomes infectious, and thus is allocated to the compartment I . Atthis stage, after a suitable time, the individual may either remain infectious but asymptomatic (orwith mild symptoms), in which case moves to the compartment A , or may show clear symptomsonset, thus being tested and then quarantined either at home or at the hospital, and being assignedto the compartment Q . Finally, individuals in A and Q will eventually be removed from thosecompartments and will end up either in the compartment R after a recovery time or dead.We will assume that the fraction of asymptomatic individuals among all infected is given by acertain probability α ∈ (0 , β , which is givenby the product between the transmissibility ν (i.e., probability of infection given contact betweena susceptible and infected individual), and the average rate of contact c between susceptible andinfected individuals. In Tables 2 and 3 we list all the parameters of the model and their description.3he model as described so far takes into account several characteristics of the pathogen and itsspread in a susceptible population. We now want to add further structural features to the modelin order to include the non-pharmaceutical interventions adopted by public policies to contain theepidemic. In particular, we assume the following conditions.i) A part p of the population is in isolation (either voluntarily, or as a result of public safetypolicies). The remaining 1 − p of the population instead gathers all those so-called “key workers”(such as physicians and paramedicals, workers in logistics and distribution, food production,security, and others), that must continue with a regular activity, thus maintaining a largecontact rate and being exposed to a higher risk of infection. We will generically refer to such 1 − p part of population as the active population, as opposed to the population in isolation. In thisgroup we can also include people that simply do not respect social distancing, and thus maintaina high contact rate. A situation like this has been observed in countries were monitoring wasnot strict and a significant percentage of the population did not respect isolation.ii) The fraction 1 − p of active population has an effective contact rate β , whereas the p part ofpopulation in isolation has a contact rate reduced by a factor r , thus its compound contactrate is rβ . We will therefore refer to such portion of the population as in r -isolation .iii) A centralized controller (such as the national health system) may intervene on the system bytesting a portion of the population to check for the infectious pathogen. We assume the testingkit to be reliable, that is, we neglect the possibility of false positive/negative. As a rule, then, anindividual from the compartment S will always test negative, an individual from I or A alwayspositive, while an individual from E will result positive with a probability δ ∈ [0 , E are not contagious, we account for the possibility thatthey might result positive to the test, depending on the stage of development of the pathogenin that specific individual and to the efficacy of the testing kit.Let us notice that, in general, the effective contact rate β depends on a variety of factors, includingthe density of population in a given country/region. However, during a pandemic, even the effectivecontact rate of the individuals not in isolation may be reduced by increased awareness (for example,maintaining the social distancing), or by respecting stricter safety protocols and by availability ofproper Personal Protection Equipment (PPE), including face shields, masks, gloves, soap, and so on.According to the above description, each compartment S , E , I and A is partitioned as follows: S = S f ∪ S r , where S f are susceptible and active, while S r are susceptible and in r -isolation; E = E f ∪ E r , where E f are exposed and active, while E r are exposed and in r -isolation; I = I f ∪ I r ,where I f are infectious and active, I r are infectious and in r -isolation; A = A f ∪ A r , where A f are asymptomatic infectious and active, A r are asymptomatic infectious and in r -isolation. Thecompartment Q collects all the infected individuals who have been tested positive, either after onsetof severe symptoms, or because of a sample test among the population, according to the proceduredescribed in iii) of the above list. Let us stress that, among these compartments, only the individualsin Q are aware of being infected, and thus contagious, hence they are either hospitalized or at home,but in both cases they follow strict procedures to reduce their contact rate to 0. Finally, we willuse the compartments R for the recovered and immune individuals, and D for the disease-induceddeaths. Both these last compartments will be removed from the dynamics and will end up in thecounter system (2). Moreover, we point out that the portion p of the population in r -isolation ispredetermined at the initial time of the evolution, reflecting the public policy in place in that specificperiod of time. Of course, such fraction p may be updated at a later time, accordingly to newer(stricter or looser) public policies.The first set of constants, related to the pathogen itself (assuming no mutation occurs in the timeof epidemic, or if so, the mutation does not affect such parameters of the virus) and its induced disease,are collected in Table 2. A graphical representation of the course of the disease for symptomaticcarriers can be seen in Figure 1. 4 ar. Description τ inverse of the latent time from exposure to infectiousness onset σ inverse of the time from infectiousness onset to possible symptoms onset θ inverse of mean incubation time (i.e. θ − = τ − + σ − ) α proportion of asymptomatic infections γ recovery rate for asymptomatic or mild symptomatic cases γ recovery rate for severe and critical cases µ mortality rate among confirmed cases δ probability of detection by testing in compartment E Table 2: Parameters of COVID-19exposure infectiousnessonset symptomsonset recovery τ − σ − γ − incubation period: θ − deathFigure 1: Disease timeline for symptomatic casesThe second set of parameters is related to public policies, and consists of the parameters inTable 3. Let us recall at this point that β varies in each territory, depending mainly on populationdensity and behaviour. These constants may be used as control parameters, via the tuned lockdownas decided by the public policies (reflecting on p and partially on r ), the awareness of the populationin respecting the social distancing among individuals and in the widespread use of personal protectionequipment (expressed by β and partially by r ), the availability of testing kits, that results in a higheror lower value of ρ . Par. Description β ( t ) transmission rate at time t (proportional to contact rate) r ( t ) reduction coefficient of transmission ratefor people in isolation at time tρ ( t ) testing rate of people with mild or no symptoms at time tp ( t ) proportion of the population in r -isolationTable 3: Parameters of Public Policies interventionsThe extended state variable of the system thus becomes˜ X = ( E f , E r , I f , I r , A f , A r , Q, S f , S r , R, D ) , where the description of the compartments is given in Table 4.We focus in particular on the evolution of the variable X = ( E f , E r , I f , I r , A f , A r , Q, S f , S r ) , ompartment Description E f exposed, not in isolation, not contagious E r exposed, in r -isolation, not contagious I f infected and contagious, not in isolation I r infected and contagious, in r -isolation A f asymptomatic and contagious, not in isolation A r asymptomatic and contagious, in r -isolation Q infected and tested positive, in enforced quarantine S f susceptible not in isolation S r susceptible in r -isolation R recovered and immune D deadTable 4: List of extended compartmentsthat follows the model˙ E f = β ( t ) S f [ I f + A f + r ( t )( I r + A r )] − ρ ( t ) δE f − τ E f ˙ E r = r ( t ) β ( t ) S r [ I f + A f + r ( t )( I r + A r )] − ρ ( t ) δE r − τ E r ˙ I f = τ E f − σI f − ρ ( t ) I f ˙ I r = τ E r − σI r − ρ ( t ) I r ˙ A f = σαI f − ρ ( t ) A f − γ A f ˙ A r = σαI r − ρ ( t ) A r − γ A r ˙ Q = σ (1 − α )( I f + I r ) + ρ ( t ) (cid:2) δ ( E f + E r ) + I f + I r + A f + A r (cid:3) − γ Q − µQ ˙ S f = − β ( t ) S f [ I f + A f + r ( t )( I r + A r )]˙ S r = − r ( t ) β ( t ) S r [ I f + A f + r ( t )( I r + A r )] (1)while the evolution of the states ˙ R = γ ( A f + A r ) + γ Q ˙ D = µQ (2)only provides counters for the proportion (over the total population) of recovered and dead individ-uals, respectively. See the compartmental diagram associated to this model in Figure 2. S f S r E f E r I f I r A f A r Q DR − p p β τ α σ ( − α ) σ γ γ βr τ α σ ( − α ) σ µδρδρ ρρ γ Figure 2: Model diagram6 emark 2.1 (About the testing rate ρ ) The parameter ρ indicates the proportion of the popu-lation presenting either mild or no symptoms that is tested daily. It can also be thought as the inverseof the mean duration that an infected person passes without being tested. For instance, if the systemmanages to detect, each day, 5% of the asymptomatic infections, then ρ = 0 . . If we are in an ideal“trace and test” situation (see e.g. South Korea [13]), in which for each confirmed infection, his/herrecent contacts are rapidly and efficiently traced and tested, then ρ will be greater and this will havean impact in the basic reproduction number (see Section 3). Recalling that testing is supposed to be applied, at least, to all sufficiently symptomatic cases,we add a counter for the positive tests T ( t ) until time t, which evolves according to the equation˙ T = σ (1 − α )( I f + I r ) + ρ ( t ) (cid:0) δ ( E f + E r ) + I f + I r + A f + A r (cid:1) . Having this quantity, one can estimate the total number of tests in each territory using the testingpositive rate of that location, which is the ratio between reported cases and tests done [29, 45].
Remark 2.2 (About symptoms and quarantine)
In our framework, we assume that all caseswith sufficiently severe symptoms are (tested and) quarantined, and we set the parameter α ∈ (0 , to be the fraction of asymptomatic cases, including the cases with mild symptoms. But we can adaptour model to a scenario in which even severe symptoms are not tested until critical. In this case,with a very large value for α , only a small portion among the symptomatic individuals enters directlyto Q, while the others need to be tested (according to the sampling testing ρ among the population)to be quarantined. System (1) is endowed with the set of initial conditions given by the vector X = ( E f, , E r, , I f, , I r, , A f, , A r, , Q , S f, , S r, ) (3)with components in the interval [0 , C := (cid:8) X = ( x i ) i =1 ,..., ∈ R : x i ∈ [0 , , for i = 1 , . . . (cid:9) , the cube of states with entries between 0 and 1, it is easy to check that C is invariant under the flowof system (1), that is, given an initial condition X ∈ C , the solution X ( t ) to (1)-(3) remains in C for all t > Remark 2.3 (Possible extensions of the model)
We collect here some variations of model (1) that can be formulated in the same framework considered in this paper.1. One might consider a small but not negligible contact rate between susceptible individuals andpeople in the compartment Q , accounting for infections (mainly of medicals and paramedicals)occurred during hospitalization of an infected individual, or for individuals tested positive inenforced quarantine at home, which do not comply strictly to the isolation procedures and endup infecting relatives or other contacts. In this case, the equations for the evolution of thesusceptible compartments shall be completed with additional terms involving ε in the followingway: ˙ S f = − β ( t ) S f [ I f + A f + r ( t )( I r + A r ) + εQ ] , ˙ S r = − r ( t ) β ( t ) S r [ I f + A f + r ( t )( I r + A r ) + εQ ] , and the same terms with opposite sign shall appear in the equation corresponding to ˙ Q .2. At the current stage, it is still not clear how long the immunity of a recovered individual lasts,with a number of findings tending towards a rather long immunization period [1, 38, 41]. Forthis reason, in (1) we assume that a recovered individual will remain immune over the timeframework considered in the different scenarios. However, the model can easily describe the caseof recovered individuals becoming susceptible again, by adding a transfer term from the compart-ment R to S f and S r , with a coefficient depending on the inverse of the average immunization eriod. Similarly, the model can include the case of reactivation of the virus in an individualpreviously declared recovered (and not newly exposed to the virus), by inserting a transfer termfrom the compartment R into I f and I r , with appropriate coefficients depending on the prob-ability of the reactivation of the virus and on the inverse of the average time of reactivation.However, at the moment there are not strong evidences supporting such possibility [38].3. A crucial issue while coping with the outbreak of the epidemic, which leads to the so-called urgeof flattening the curve , is whether the number of critical cases in need of intensive care (IC)treatment (due to respiratory failure, shock, and multiple organ dysfunction or failure) wouldsaturate the number of available intensive care units (ICUs).This parameter can be estimated directly from model (1) , considering for each country thenumber of available ICUs and the percentage of positive confirmed cases requiring IC treatment.For example, this percentage has been estimated to be about for China [4], and up to for Italy [12, 35]. As an alternative, it would be possible to insert a further compartment C in model (1) counting the number of the individuals needing ICU treatment, by modifying theequations corresponding to the compartments Q and D as follows: ˙ Q = σ (1 − α ) I + ρ ( t ) (cid:2) δE + I + A (cid:3) − γ Q − τ c Q ˙ C = τ c Q − µ c C − γ c C ˙ D = µ c C with suitable coefficients τ c , µ c and γ c denoting the inverse of the time from symptoms onsetto critical symptoms, the mortality of critical cases, and the recovery rate for critical cases,respectively.4. In this paper we have considered the whole population as a fixed number of individuals duringthe time period of the evolution. It is of course possible to consider the case of an evolvingtotal population, by including in the model (1) the natural birth and mortality rate. In particu-lar, newborns of susceptible individuals shall enter the corresponding susceptible compartment,whereas it is not clear whether the offspring of an infectious individual would be infectious, insuch case the newborn shall move directly to the compartment I . On the other hand, the naturalmortality rate shall act on each compartment of system (1) , as well as on R in system (2) . (1) We are interested in determining the basic reproduction number R associated with system (1). Todo this, we assume to fix a time interval [ t , t ] such that the coefficients β ( t ), r ( t ) and ρ ( t ) areconstant over [ t , t ]. This is coherent with the setting of the scenarios simulated in Section 4.2,where we assume such coefficients to be piecewise constant functions, sharing the same switchingtimes, that represent different phases of restrictions and policies. Thus, according to the calculationsgiven in the Appendix A.1 and the parameters in Tables 2 and 3, we obtain that the value of R foreach time interval between two consecutive switching times is given by R = 12 (cid:18) ϕ + r ϕ + 4 σαρ + γ ϕ (cid:19) , (4)with ϕ = βτ (1 − (1 − r ) p )( ρδ + τ )( σ + ρ ) . (5)From this explicit formula for the reproduction number R , we can highlight the qualitative depen-dence of R on each parameter of the system, in particular:- If the effective contact rate β increases, then R increases.8 Focusing on the coefficient 1 − (1 − r ) p , we realize that closer is p to 1, and smaller is r , thatis, as larger is the portion of population in r -isolation and as stricter is the reduction factor r in its contact rate, lower R becomes.- If α increases, that is, if there is a larger proportion of asymptomatic infectious individuals,then R increases.- If σ increases, corresponding to shorter onset time, then R decreases.- If either ρ or γ increase, i.e., either the control action by testing is strengthened, for examplethrough an improved tracing and tracking system, or the recovery rate improves, for example,because of new and more effective treatments, then R decreases.- If δ increases, for example, as a result of improved testing kits able to detect the infection atan earlier stage, then R decreases.Moreover, we can characterize the crucial condition R ≤ Proposition 3.1 (Alternative threshold)
Set T := βτ [1 − (1 − r ) p ]( ρδ + τ )( σ + ρ ) (cid:18) σαρ + γ (cid:19) . Then R is smaller than (respectively, equal to or greater than) 1 if and only if the same relationholds for T . In particular, if ϕ > (see (5) ), then R > and T > . We postpone the proof of Proposition 3.1 to the Appendix A.2. A more quantitative analysis ofthe dependence of the threshold T on the parameters of the model is developed in the Appendix A.3. Remark 3.2 (About no testing among asymptomatic carriers)
If we consider the case of ρ =0 , that is, the situation without sample testing among the asymptomatic population, then the basicreproduction number R is independent of the latent time τ . In particular, T becomes β [1 − (1 − r ) p ] σ (cid:18) σαγ (cid:19) . Remark 3.3 (On the time-dependent reproduction number)
Relation (4) gives an expres-sion of R , that is the reproduction number in a totally susceptible population. As the epidemicevolves, a portion of the population becomes immune to the disease, and this makes the reproductionnumber decrease. More precisely, when p = 0 and all the population has the same contact rate, thetime-dependent reproduction number is given by R ( t ) = S ( t ) R , where S ( t ) is the susceptible portionof the population. In our model, since the groups of active individuals and in r -isolation evolve dif-ferently (see Scenario A and Figure 4 below), the time-dependent reproduction number R ( t ) is givenby the formula (4) where ϕ in (5) is βτ [ S f ( t ) + r S r ( t )]( ρδ + τ )( σ + ρ ) . We do not take into account this time-variation of the reproduction number in our numerical results,since we are only interested in the value of the reproduction number at the beginning of each phase,where S ( t ) is close to 1. Remark 3.4 (On herd immunity)
Herd immunity is defined as the proportion of the populationthat needs to be immunized in order to naturally slow down the spread of the disease. It depends onthe value of the basic reproduction number in the following way: herd immunity level equals − R . So the bigger R , the higher the herd immunity. In connection with above Remark 3.3, we highlightthat herd immunity is achieved at the time t when R ( t ) equals 1. Numerical simulations
In Table 5 we collect some parameter values estimated in the literature, in order to do realisticsimulations. Recall the description of the parameters given in Tables 2-3.
Par. Value - Range Reference Remark β τ − τ − = θ − − σ − [20] 2 σ − θ − γ γ
15 - 22 days [4, 49] 7 µ [0.03/14,0.1/14] [33, 42] 8 α [0.265,0.643] [5, 19] 3 p [0 ,
1] [14] 9 r [0 ,
1] [14] 9 ρ [0,0.5] [45] 10 δ
1. The parameter β strongly depends on the population behaviour. We take the value of β from [36], where theycalibrated an SEIR model with isolation and estimated the transmission rate β , before lockdown, to be 0 . . , . τ − between 2 and 4 days. More precisely, in [20] they fitted an SEIQR model to the data from Wuhanand estimated a latent period of duration 2.92 days with a 95% CI of (1 . , . . θ to be 6.4 based on travellers returning from Wuhan. In [18] it was estimated to be5.1 days. Other estimates were given in [20].6. The estimate of γ is difficult, since for asymptomatic cases is hard to observe and track the time from exposureto recovery. [16] estimated 9.5 days for asymptomatic cases, while [4] estimated 14 days for mild cases. So itis reasonable to assume γ in the range 7.5 - 12, considering around 2 days between infectiousness onset andsymptoms onset.7. In [49] they measured viral shedding duration, and estimated a median of 20 days, with an interquartile rangeof (17 , . Removing the approximately 2-day period from infectiousness onset to symptoms onset, we get anIQR for γ − of (15 , . These values approximate the duration of quarantine recommended to positive-testedcases.8. The rate µ depends on the percentage of infections that have been detected, since it is proportional to the ratiobetween confirmed cases and deaths. WHO Director-General’s opening remarks at the media briefing of 3 March2020 [33] announced an estimated global death rate of 3.4%. In some countries, like Italy, the ratio betweendeaths and confirmed cases up to May 2020 is larger that 0.1, while in others, like Israel, it is around 0.01.Regarding the time a person takes to die from COVID-19, in [49] they estimated 18 . p and r vary in each country/territory depending on the public policies and the population’scompliance to these measures. A detailed and real-time survey on the percentage of people under lockdown ineach country can be found in [14].
0. As already mentioned in Remark 2.1, ρ represents the proportion of the infected asymptomatic population thatis tested daily. In a realistic scenario, it would not be reasonable to set a too high value of ρ , let us say, over 0.5,because it would account for detecting more than 50% of the infected asymptomatic population daily.11. It is not yet know “at what point during the course of illness a test becomes positive” (see [15]). For thesimulations we set δ to 1 and suppose that the tests detect the infection from exposure. In this subsection we consider several scenarios and show their outcomes. Many of the graphics arein logarithmic scale, given that the values represent portions of the population, and then can assumevery small values.
We consider the four different scenarios with the following characteristics:
Scenario A : no isolation, no testing among asymptomatic people Scenario A : Scenario A : Scenario A : E f (0) + E r (0) = 1 × − , S f (0) + S r (0) = 1 − × − . The remainder of the compartments start with value 0. Results and parameters for Scenario Aare given in Table 6 and graphics in Figure 3. We can observe the effect of the lockdown on theepidemic. The mild lockdown of A reduces more than half of the infections w.r.t. the no actionsituation A , while the strict lockdowns A and A induce a reduction of the order of 10 − in totalrecovered, deaths and positive tests. In particular, comparing A and A we can see that testing andconsequent quarantine for positive-tested asymptomatic cases not only reduces the infections anddeaths more than 66%, but also the duration of the epidemic. Remark 4.1 (About Scenario A ) United States had 15 confirmed infections by February 15th2020 [44]. Many states started their lockdown between March 15th and March 20th, which put around60% of the population in isolation [14]. We can assume that day 0 is February 15th, then day 31would be in the middle point of the interval when lockdowns started. Day 80 (May 6th) US had 72,287confirmed deaths, while in Scenario A gives 91,231, that is larger but not far. Scenario A could bea good approximation of the situation in the US until the beginning of May 2020, and the excess inthe computed deaths in comparison to real data suggests that deaths could be underreported by ,which is coherent with some recent studies on underreported deaths (see e.g. [47]). (a) Scenario A (b) Scenario A (c) Scenario A (d) Scenario A Figure 3: Scenarios A , A , A and A . Par. A A A A β p t ≤ . t >
31 0 if t ≤ . t > r t ≤ . t > ρ δ τ σ θ γ γ µ α R .
51 if t ≤ . t >
31 2 .
51 if t ≤ .
69 if t >
31 1 .
92 if t ≤ .
52 of t > Q ) 76 146 40 38recovered 9 . × − . × − . × − . × − deaths 3 . × − . × − . × − . × − positive tests 5 . × − . × − . × − . × − ending day( Q ≤ − ) 377 >
500 314 225
Table 6: Scenarios A , A , A and A . Parameters and epidemics output12or Scenario A we make a comparison of infections for the two groups: the active one (thatcontinues with the usual contact rate) and the one in r -isolation. By comparing the infections’curves and the cumulative infections, we can give an estimate on the lower chance that people in r -isolation have to get exposed. In this particular scenario, people that are not in isolation havenearly 6 times more chance to be infected. See the graphs in Figure 3, were we show the curves ofinfections and cumulative infections for each group, normalized by the proportions 1 − p , p. Figure 4: Comparison of infections for population in and out isolation
We next consider the following four scenarios in which we vary the values of the portion p of peopleunder lockdown and their level r of restriction of social contacts. Scenario B : Scenario B : Scenario B : Scenario B : ρ among asymptomatic cases.13 ar. B B B B β µ p t ≤ . t >
35 0 if t ≤ .
65 if t >
35 0 if t ≤ . t >
35 0 if t ≤ . t > r t ≤ . t >
35 1 if t ≤ . t >
35 1 if t ≤ . t >
35 1 if t ≤ . t > ρ R .
24 if t ≤ .
61 if t >
35 2 .
24 if t ≤ .
23 if t >
35 2 .
24 if t ≤ .
93 of t >
35 2 .
24 if t ≤ .
61 of t > . × − . × − . × − . × − deaths 1 . × − . × − . × − . × − positive tests 5 . × − . × − . × − . × − ending day 436 > >
500 274
Table 7: Scenarios B , B , B and B . Parameters and epidemic outputs −9 −8 −7 −6 −5 −4 −3 −2 −1 p=0.5, r=0.5p=0.65, r=0.4p=0.8, r=0.3p=0.9, r=0.2 (a) Recovered in green, dead in red −9 −8 −7 −6 −5 −4 −3 −2 −1 p=0.5, r=0.5p=0.65, r=0.4p=0.8, r=0.3p=0.9, r=0.2 (b) Infected Figure 5: Scenarios B , B , B and B Scenarios B and B show cases in which the restrictions are not strong enough. Indeed, in bothcases the basic reproduction number R remains above 1 also after the lockdown intervention (seeTable 7), and the infection reaches 71.6% and 43.6% of the population, causing the death of 1.89%and 1.15% of the population, respectively, which is a catastrophic outcome. Comparing these fourscenarios, we shall deduce that, in order to be effective in containing the outbreak, the lockdown shalladdress at least 80% of the population reducing their contact rate to about 30% of their usual contacts.Indeed, in the scenario B , the basic reproduction number becomes 0.93 after day 35, meaning thatloosening the restrictions of this scenario (while keeping all other parameters unchanged) might turnthe R above 1. We now compare two situations, one in which lockdown starts immediately, just 21 days after thefirst confirmed cases, and the other for which lockdown starts four weeks later. More precisely, weconsider the following two scenarios and measure the different outputs:
Scenario C : Scenario C : ar. C C β µ p t ≤ . t >
21 0 if t ≤ . t > r t ≤ . t >
21 1 if t ≤ . t > ρ R .
24 if t ≤ .
61 if t >
21 2 .
24 if t ≤ .
61 if t > . × − . × − deaths 4 . × − . × − positive tests 6 . × − . × − ending day 213 323 Table 8: Scenarios C and C : early lockdown vs. late lockdown. Parameters and epidemic outputsThe parameters and outputs are given in Table 8 and Figure 6. It is evident the impact of delayingthe beginning of lockdown on the final outcome: the numbers of recovered and deaths in the ScenarioC are of the order of 10 times those of the Scenario C . As an example, from Table 8 one noticethat, at the end of the epidemic, the Scenario C counts 4.2 deaths per million inhabitants, while theScenario C faces 1020 deaths per million. Moreover, the epidemic in Scenario C lasts about 110days more than in Scenario C , thus also undergoes worst economic consequences of the lockdown. (a) Scenario C (b) Scenario C Figure 6: Scenarios C and C . We now want to asses the impact of testing timing. For this, we consider the following two scenariosand measure the different outputs:
Scenario D : Scenario D : and D are given in Table 9 and15gures in Figure 7. It can be seen the cost in infection and lives it has to start testing late. It is worthnoticing that, in spite of a higher total number of tests carried out in the Scenario D , the strategyadopted in the Scenario D attains a considerably better outcome: indeed, the infections and deathsof Scenario D are of the order of 10 w.r.t. the ones in Scenario D , and the only difference wasdoing efficient testing at the beginning of the epidemic. Par. D D β µ p t ≤ . t > r t ≤ . t > ρ . t ≤ .
05 if t >
50 0 .
01 if t ≤ . t > R .
53 if t ≤ . t >
50 2 .
37 if t ≤ .
57 if t > . × − . × − deaths 2 . × − . × − positive tests 7 . × − . × − ending day 302 327 Table 9: Scenarios D and D : early efficient testing vs. late massive testing. Parameters andepidemic outputs (a) Scenario D (b) Scenario D Figure 7: Scenarios D and D . Now we fix the parameters β, µ, p, r as in Table 9 and we vary only ρ to take the four differentvalues 0, 0.02, 0.05 and 0.1 over the whole time period. We get the outcome of Figure 8. From thecomparison among these four scenarios, we realize that a high value of ρ , as the result of an efficienttracing and testing strategy, may reduce the number of cumulative infected individuals and deathsof an order of 10 . 16 −9 −8 −7 −6 −5 −4 −3 −2 −1 rho=0rho=0.02rho=0.05rho=0.1 (a) Recovered in green, dead in red (b) Infected Figure 8: Scenarios E , E , E and E The simulations were done with Python and all the codes are in the GitHub repositorygithub.com/lucasmoschen/covid-19-model.
In the paper we present an SEIR model with Asymptomatic and Quarantined compartments to de-scribe the recent and ongoing COVID-19 outbreak. Our model is intended to highlight the strengthof three different non-pharmaceutical interventions imposed by public policies in containing the out-break and the total number of disease-induced infections and deaths:- reduction of contact rate for a given portion of the population;- enforced quarantine for confirmed infectious individuals;- testing among the population to detect also asymptomatic infectious individuals.On one hand we show that, as expected, each of these interventions has a beneficial impact onflattening the curve of the outbreak. On the other hand, the comparison among different scenariosshows the remarkable efficacy of an early massive testing approach, when the limited number ofinfected individuals makes easier and more effective the tracing of recent contacts of the individual,as in Scenario D , and of a timely lockdown, although in the presence of few confirmed infectedcases, as in Scenario C . In both situations, the timing of the intervention plays a crucial role on theincisiveness of the public safety policy.In addition, we give an explicit representation of the basic reproduction number in terms of theseveral parameters of the model, which allows to describe its dependence on the features of the virusand on the implemented non-pharmaceutical interventions.This description makes available a valuable tool to tune the public policies in order to control theoutbreak of the epidemic, forcing R below the threshold 1. However, considering the major effectsof an enduring lockdown on the economy of the country that applies it, it is desirable to loosenthe lockdown measures after the containment of the outbreak. Nevertheless, the decision makersand each individual shall be aware that a value of R only barely greater than 1 would lead to anincrease in the number of infected and dead by an order 2 of magnitude, thus provoking the collapseof the relative national health system. This is better explained by the following scenarios: considera situation with constant testing ρ = 0 .
05 and no lockdown where, after the first 35 days of outbreakwith a high R ( ≈ −9 −8 −7 −6 −5 −4 −3 −2 −1 R_0=1R_0=1.1R_0=0.9 (a) Recovered in green, dead in red −9 −8 −7 −6 −5 −4 −3 −2 −1 R_0=1R_0=1.1R_0=0.9 (b) Infected
Figure 9: The impact of small variations on R rate so as to steer R to either 0 .
9, 1 or 1 .
1. Figure 9 illustrates the large deviations among theoutcome of these three different situations.In order to allow the population to circulate with no restrictions, it is necessary that herd immu-nity (see Remark 3.4) is achieved. The value that matters to compute this threshold of immunizationis the basic reproduction number under no social distancing, which has been estimated in this articleand in many others as being, in general, greater than 2.5. So achieving herd immunity would implyto infect at least 60% of the population, which would lead, with the current mortality rates, to 1-5%of the population dying, which is, obviously, a catastrophic unwanted situation. Hence, reinforcingwhat was said in the above paragraph, until a vaccine or treatment is not found, it is necessary tomaintain the value of R below 1. Otherwise, the curve of infections will always be increasing. A Appendix
A.1 Computing R Recalling the model (1), we are able to give an analytic expression of the basic reproduction number R associated to the system.In order to do so, we assume to fix a time interval [ t , t ] such that the coefficients β ( t ), r ( t ) and ρ ( t ) are constant over [ t , t ]. This is coherent with the setting of the Section 4, where we assumesuch coefficients to be piecewise constant functions, sharing the same switching times. Thus, thefollowing procedure allows to evaluate the value of R for each time interval between two consecutiveswitching times.It is well known that the reproduction number R is the crucial parameter to establish whetherDisease Free Equilibria (DFE) are stable or not [7, 40]. We denote by X s the set of DFE, which isgiven by X s = { X ∈ C : E f = E r = I f = I r = A f = A r = Q = 0 } . We can recast system (1) in the compact form˙ X ( t ) = f ( X ( t )) (6)18y introducing f ( X ) = βS f [ I f + A f + r ( I r + A r ) ] − ρδE f − τE f rβS r [ I f + A f + r ( I r + A r ) ] − ρδE r − τE r τE f − σI f − ρI f τE r − σI r − ρI r σαI f − ρA f − γ A f σαI r − ρA r − γ A r σ (1 − α )[ I f + I r ]+ ρ (cid:0) δ ( E f + E r )+ I f + I r + A f + A r (cid:1) − γ Q − µQ − βS f [ I f + A f + r ( I r + A r )] − rβS r [ I f + A f + r ( I r + A r )] . The stability of (6) around a DFE X ∗ is related to the spectral properties of the linearized systemaround X ∗ , whose dynamics is ruled by the Jacobian Df = ( ∂f i /∂x j ) i,j =1 ,..., of f . However, thehigh dimensionality of Df ( X ) makes it difficult to develop an analytical analysis of its spectrum andits stability properties. We will therefore follow a different approach, deducing the value of R fromthe result in [40], which ensures that R is given by the formula R = ρ ( F V − ), where ρ ( A ) denotesthe spectral radius of the matrix A . A comment on the applicability of the results in [40] is given inRemark A.1 below.Since X ∗ is a DFE, we may assume that X ∗ = (0 , , , , , , , − p, p ), for some p ∈ [0 , S r , while the remaining1 − p fraction of the population is in S f . Thus, in our setting, the matrices F and V related to thedynamics (6) are given by F = β (1 − p ) rβ (1 − p ) β (1 − p ) rβ (1 − p ) 00 0 rβp r βp rβp r βp
00 0 0 0 0 0 00 0 0 0 0 0 00 0 σα σα σ (1 − α ) + ρ σ (1 − α ) + ρ ρ ρ ,V = ρδ + τ ρδ + τ − τ σ + ρ − τ σ + ρ ρ + γ ρ + γ − ρδ − ρδ γ + µ . Since V is non-singular, we compute V − = ( ρδ + τ ) − ρδ + τ ) − τ ( σ + ρ )( ρδ + τ ) σ + ρ ) − τ ( σ + ρ )( ρδ + τ ) σ + ρ ) − ρ + γ ) − ρ + γ ) − ρδ ( γ µ )( ρδ + τ ) ρδ ( γ µ )( ρδ + τ ) γ + µ ) − . Thus, one can easily compute the matrix
F V − and check that its characteristic polynomial is givenby p ( λ ) = − λ P ( λ ) , where P ( λ ) is a second order polynomial of the form P ( λ ) = λ − βτ (1 − p + r p )( ρδ + τ )( σ + ρ ) λ − σατ β (1 − p + r p )( ρ + σ )( ρδ + τ )( ρ + γ ) . ( λ ) has one positive and one negative root, given by λ / = 12 (cid:18) βτ (1 − p + r p )( ρδ + τ )( σ + ρ ) ± √ ∆ (cid:19) , with ∆ = (cid:18) βτ (1 − p + r p )( ρδ + τ )( σ + ρ ) (cid:19) + 4 σατ β (1 − p + r p )( ρ + σ )( ρδ + τ )( ρ + γ ) > . Since the term βτ (1 − p + r p )( ρδ + τ )( σ + ρ ) is positive, the value of R coincides with λ , i.e., R = λ = 12 (cid:18) βτ (1 − p + r p )( ρδ + τ )( σ + ρ ) + √ ∆ (cid:19) . (7)This is an analytic expression of R , which shows its explicit dependence on the different parametersof model (1). Proposition 3.1 gives a convenient equivalent condition to ensure the stability of DFE. Remark A.1
In order to directly apply the results in [40], it is required that the eigenvalues of Df ( X ∗ ) have negative values and, under this assumption, the asymptotic stability of the DFE isestablished. In our case, the matrix Df ( X ∗ ) has zero as an eigenvalue of double multiplicity, withassociated eigenvectors in the directions of the last two variables, these being S f and S r . It is nothard to see that the results in [40] hold for our system by simply modifying asymptotic stability tostability in the directions of the susceptible compartments, which has no consequence in the meaningof the threshold R . Alternatively, a way to force the system to comply all the technical assumptionsfrom [40] is adding birth and natural mortality to our model, which has no relevant impact in theresults we showed (since the natural daily birth/death rates are of the order of − , hence negligiblew.r.t. the other parameters). A.2 Proof of Proposition 3.1
We remind that Proposition 3.1 claims the following: for ϕ = βτ [1 − (1 − r ) p ]( ρδ + τ )( σ + ρ ) and T = ϕ (cid:18) σαρ + γ (cid:19) , R is smaller than (respectively, equal to or greater than) 1 if and only if the same relation holds for T . Indeed, by a straightforward computation we realize that R ≤ ⇐⇒ ϕ + r ϕ + 4 σαρ + γ ϕ ≤ ⇐⇒ (0 < ) r ϕ + 4 σαρ + γ ϕ ≤ − ϕ ∗ ⇐⇒ ϕ + 4 σαρ + γ ϕ ≤ (2 − ϕ ) ⇐⇒ σαρ + γ ϕ ≤ − ϕ ⇐⇒ ϕ (cid:18) σαρ + γ (cid:19) ≤ ⇐⇒ T ≤ . Observe that the implication ⇐ = in the equivalence ∗ ⇐⇒ holds true because T ≤ ⇒ ϕ ≤ ρ + γ ρ + γ + σα ≤ , thus | − ϕ | = 2 − ϕ . In particular, the same chain of relations holds with the equal sign. Finally,since R ≤ T ≤
1, then also R > T > ϕ >
1, then both R > T >
1. Indeed, from the definitionof T , since σαρ + γ ≥
0, we have that T ≥ ϕ >
1, and thus also R > .3 Sensitivity analysis of the threshold T The explicit representation (7) of the basic reproduction number R allows to study the sensitivityof R with respect to the several parameters of the model (1). Moreover, thanks to Proposition 3.1,we know that the threshold T = βτ [1 − (1 − r ) p ]( ρδ + τ )( σ + ρ ) (cid:18) σαρ + γ (cid:19) can be used for an equivalent characterization of the condition R <
1. For this reason, it ishandier to develop the sensitivity of T with respect to the parameters of the model, and deduce itsdependence on perturbations of the parameters. We thus compute the normalized sensitivity index S x corresponding to the x parameter, given by S x := x T ∂ T ∂x , and we get that S β = 1 > ,S τ = ρδρδ + τ > ,S p = − (1 − r ) p − (1 − r ) p < ,S r = 2 r p − (1 − r ) p > ,S δ = − ρδρδ + τ < ,S α = σαρ + γ + σα > ,S γ = − σαγ ( ρ + γ )( ρ + γ + σα ) < ,S σ = − σ [ γ + (1 − α ) ρ ]( σ + ρ )( ρ + γ + σα ) < ,S ρ = − ρρ + γ + σα (cid:20) [ δ ( σ + 2 ρ ) + τ ]( ρ + γ + σα )( ρδ + τ )( σ + ρ ) + σαρ + γ (cid:21) < . We thus notice the same qualitative dependence on the parameters already observed in Section 3.In particular, if we increase k times the parameter β , then T increases k times as well. Similardeductions can be made on the other parameters, with the corresponding coefficients obtained byinserting the values of the parameters from Table 5. Moreover, from the expression of S τ we realizethat, if either ρ or δ equal zero, then T does not depend on τ (as it happens for R as well, asnoticed in Remark 3.2). Similarly, if ρ = 0, then S δ = 0, thus T does not depend on δ . Regardingthe parameters p and r , their dependence is mutually related as follows: if p = 0, then T does notdepend on r (since S r = 0), whereas if r = 1 then T does not depend on p . References [1] J. An, X. Liao, T. Xiao, S. Qian, J. Yuan, H. Ye, F. Qi, C. Shen, Y. Liu, L. Wang, et al.Clinical characteristics of the recovered COVID-19 patients with re-detectable positive RNAtest. medRxiv , 2020.[2] J. A. Backer, D. Klinkenberg, and J. Wallinga. Incubation period of 2019 novel coronavirus(2019-nCoV) infections among travellers from Wuhan, China, 20–28 January 2020.
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