How Efficient is Contact Tracing in Mitigating the Spread of Covid-19? A Mathematical Modeling Approach
HHow Efficient is Contact Tracing in Mitigating theSpread of Covid-19? A Mathematical ModelingApproach
T. A. Biala* , Y. O. Afolabi , A. Q. M. Khaliq Center for Computational Science and Department of Mathematical Sciences,Middle Tennessee State University, Department of Mathematics,University of Louisiana at Lafayette.
Abstract
Contact Tracing (CT) is one of the measures taken by government and healthofficials to mitigate the spread of the novel coronavirus. In this paper, we inves-tigate its efficacy by developing a compartmental model for assessing its impacton mitigating the spread of the virus. We describe the impact on the repro-duction number R c of Covid-19. In particular, we discuss the importance andrelevance of parameters of the model such as the number of reported cases,effectiveness of tracking and monitoring policy, and the transmission rates tocontact tracing. We describe the terms “perfect tracking”, “perfect monitoring”and “perfect reporting” to indicate that traced contacts will be tracked whileincubating, tracked contacts are efficiently monitored so that they do not causesecondary infections, and all infected persons are reported, respectively. Weconsider three special scenarios: (1) perfect monitoring and perfect tracking ofcontacts of a reported case, (2) perfect reporting of cases and perfect monitoringof tracked reported cases and (3) perfect reporting and perfect tracking of con-tacts of reported cases. Furthermore, we gave a lower bound on the proportionof contacts to be traced to ensure that the effective reproduction, R c , is belowone and describe R c in terms of observable quantities such as the proportionof reported and traced cases. Model simulations using the Covid-19 data ob-tained from John Hopkins University for some selected states in the US suggestthat even late intervention of CT may reasonably reduce the transmission ofCovid-19 and reduce peak hospitalizations and deaths. In particular, our find-ings suggest that effective monitoring policy of tracked cases and tracking oftraced contacts while incubating are more crucial than tracing more contacts.The use of CT coupled with other measures such as social distancing, use of facemask, self-isolation or quarantine and stay-at-home orders will greatly reducethe spread of the pandemic as well as peak hospitalizations and total deaths. Email address: *[email protected] (T. A. Biala* ) Preprint submitted to arXiv August 11, 2020 a r X i v : . [ q - b i o . P E ] A ug eywords: SEIR model, Covid-19, Contact tracing, Time fractional-ordermodels
1. Introduction
Infectious diseases are often spread via direct and indirect contacts such asperson-to-person contact, droplets spread, airborne transmissions and so on.Several studies [1, 2, 3, 4, 5, 6, 7, 8] have shown that the novel coronavirus in-fection spread through these means. Several measures which includes social dis-tancing, stay-at-home orders, self-isolation/quarantine, use of face-masks, con-tact tracing amongst others, have been enforced by authorities in reducing thespread of the virus. In any disease outbreak therefore, contact tracing is an im-portant tool for combating the spread of the outbreak. Contact tracing (CT) isthe process whereby persons who have come in contact with a reported/isolatedcase are traced and monitored so that if they become symptomatic they canbe efficiently isolated to reduce transmissions. Previous outbreaks of infectiousdiseases have been rapidly controlled with contact tracing and isolation, for ex-ample, the Ebola outbreak in West Africa in 2014, see [9]. Furthermore, in anydisease control it is important to evaluate the efficacy of intervention strate-gies such as contact tracing. Thus, the need to explicitly measure how contacttracing can help in mitigating the transmission of coronavirus cannot be over-emphasized. A lot of studies have been conducted on the efficacy of contacttracing in relation to some diseases in the past, see [10, 11, 12, 13, 14, 15, 16].Several mathematical models have been proposed for the dynamics of the novelcoronavirus, see for example [17, 18, 19, 20, 21, 22, 23, 24] and several mod-els have incorporated contact tracing using stochastic modeling approach [25]and networks [26]. However, these studies did not include the effect of contacttracing on the reproduction number of Covid-19 and the expression of this re-production number in terms of observable quantities, a quick and efficient wayof estimating the reproduction number. In 2015, Browne et al. [16] developed adeterministic model of contact tracing for Ebola epidemics which links tracingback to transmissions, and incorporates disease traits and control together withmonitoring protocols. Eikenberry et al. [23] examined the potential of face maskuse by general public to curtail the Covid-19 pandemic. Their findings suggestthat face mask should be adopted nation-wide and be implemented without de-lay, even if most masks are homemade and of relatively low quality. Motivatedessentially by the works of [16] and [23], the goal in this work is to develop adeterministic model to measure the efficacy of contact tracing in mitigating thespread of Covid-19. As noted in [16], explicitly incorporating contact tracingwith disease dynamics presents challenges, and population level effects of con-tact tracing are difficult to determine. Here, we propose a compartmental modelwhich incorporates the disease traits and monitoring protocols. We describe theimpact on the reproduction number R c of Covid-19 and discuss the importanceand relevance of parameters of the model.In particular, we develop a time-fractional compartmental model; a modifica-2ion of the SEIR model similar to the one given in [23] where they divided theinfected population into symptomatic and asymptomatic compartments. Weadopt the use of time-fractional models because they reduce errors which mayarise from neglect of parameters in the model. Moreover, it was pointed out in[17] that the spread of infectious diseases depends not only on its current state,but also on its past state. However, integer order models lack the capacity to in-corporate this history dependence of infectious diseases. The remaining sectionsare organized as follows: In section 2, we begin with a discussion on the basicSEIR model and used this as a building block in deriving the new model withcontact tracing. Thereafter, we consider special cases of the model and calcu-late their effective reproduction numbers. Furthermore, we gave a lower boundon the proportion of reported cases that must be traced to ensure the repro-duction number is below one and express the reproduction number in terms ofobservable quantities such as average number of secondary infected persons pertraced and untraced reported case. In section 3, we perform several numericalexperiments to corroborate our theoretical observations in Section 2. Finally, insection 4, we gave a comprehensive discussion on the impact of contact tracingin mitigating the spread of the virus.
2. Model Formulation
We begin with a basic time-fractional SEIR model consisting of four compart-ments that represents the susceptible (S), exposed (E), infected (I), recovered(R). We assume that all the infected individuals are unreported and thus nothospitalized. The following system of differential equations models the trans-mission dynamics of the population: D αt S ( t ) = − β ( t ) SIN D αt E ( t ) = β ( t ) SIN − σE D αt I ( t ) = σE − γI D αt R ( t ) = γI where β ( t ) = β ,β exp( − r ( t − t ∗ )) , is the disease transmission rate which takes into account effects of governmentalactions such as social distancing, mask usage and so on, t ∗ is the day after whichthe governmental action begins to have effect on the spread of the virus, σ (1 /σ ) is the transition rate (disease incubation period) from the exposed class tothe infectious class, γ (1 /γ ) is the recovery rate (time from infectiousness untilrecovery) of an infected individual. We note that the parameters of the model3re non-negative and have dimensions given by 1/time α . This observation wasoriginally noted in Diethelm [27]. To alleviate this difference in dimensions, wereplace the parameters with a power α of new parameters to obtain the newsystem of equations: D αt S ( t ) = − β ( t ) SIN D αt E ( t ) = β ( t ) SIN − σ α E D αt I ( t ) = σ α E − γ α I D αt R ( t ) = γ α I (1)with β ( t ) = β α ,β α exp( − r ( t − t ∗ )) . The next step in the development of our model is the incorporation of hos-pitalized compartments (H) and splitting of the infected cases into reported (R)and unreported cases (U). This is necessary as published studies [18, 28, 23]have shown that a considerable number of infected cases go unreported eitherdue to unawareness or early recovery or just perceptions of the infected indi-viduals. We note that only the reported cases are being hospitalized during theinfectious period and neglect the possibility of transmission of an hospitalizedindividual since they are not exposed to the general population. Thus, we obtainthe following system of time-fractional differential equations: D αt S ( t ) = − β ( t ) SN ( I R + I U ) D αt E ( t ) = β ( t ) SN ( I R + I U ) − σ α E D αt I R ( t ) = ησ α E − ( γ α R + ϕ α R ) I R D αt I U ( t ) = (1 − η ) σ α E − γ α U I U D αt H ( t ) = ϕ α R I R − ( γ α H + µ α H ) H D αt R ( t ) = γ α R I R + γ α U I U + γ α H H D αt C ( t ) = σ α E D αt D ( t ) = µ α H H, (2)where C ( t ) and D ( t ) represents the number of cumulative infected (both re-ported and unreported) and the disease-induced deaths, respectively. Thesenumbers can be explicitly calculated as C ( t ) = C (0) + σ α Γ( α ) (cid:90) t ( t − s ) α − E ( s ) ds,D ( t ) = µ α H Γ( α ) (cid:90) t ( t − s ) α − H ( s ) ds. R , γ U , and γ H are the recovery rates of a reported, unreported and hospitalizedindividuals, respectively. ϕ R is the hospitalization rate of reported infectedperson and µ H is the disease-induced death rate. For simplicity, we shall use prior studies to fix several parameters and fitthe other parameters of the model. In particular, we shall fit the parameters β , r, t ∗ , ϕ R , µ H and α using the Covid-19 data obtained from John HopkinsUniversity [29] for some selected states in the US. The inclusion of ϕ R and µ H in the fitting parameters stems from the fact that different states have differ-ent hospitalization and death rates. Prior modeling studies suggest that theeffective transmission rate β ranges between 0.5-1.5 day − [30, 31, 32, 23] andthe incubation period lies in the range between 2–9 days [33, 22, 34, 19]. Theaverage of 5.1 days was estimated by Lauer et al. [35]. The infectious durationseems to have agreeing values of around 7 days for several modeling studies[30, 18, 23, 24, 36, 37]. Lachmann et al. [38] and Li et al. [39] estimated thataround 88% and 86%, respectively, of all infections are undocumented with a95% credible interval. Maugeri et al. [40] estimated that the proportion of un-reported new infections by day ranged from 52.1% to 100% with a total of 91.8%of infections going unreported. Table 1 gives a summary of these values and thedefault values used in our model simulation.Parameters Not. Ranges References DefaultEffective transmission rate β − [37, 30, 32, 23] FittedIntensity of transmission r — — FittedDay after governmental action shows effect t ∗ Varies — FittedIncubation Period σ − η γ R − [19, 37, 30] 1/7Recovery rate (Unreported) γ U − [19, 37, 30] 1/7Recovery rate (Hospitalized) γ H − [41, 36] 1/14Hospitalization rate ϕ R − [37, 41] FittedDisease-induced death rate µ H − [37] FittedTime-fractional order α Table 1: Summary of parameter ranges and default values used in our simulation. “Not”denotes Notations.
We incorporate CT into the preliminary model by linking the dynamics ofdisease model with actions of contact tracers such as monitoring and tracking.This general modeling framework is similar to a variety of CT models employedin [10, 15, 16]. At first, we describe the four steps of CT for Covid-19 as describedby the Center for Disease Control (CDC) [42]. The Public health officer triesto identify contacts (contact investigation) by working with infected patients5o help recall people they’ve been in contact with while being infectious. Thesecond step (contact tracing) involves notifying and tracing of recorded contactsof the patient. Next (contact support), the officer informs and educates thecontacts on the risk and dangers of being exposed. They also provide supporton the next line of action for the contacts. In the case that a contact is alreadyshowing symptoms, the tracers will call an ambulance to remove/isolate thecontact. Lastly (contact self-quarantine), contacts are encouraged to quarantinefor a minimum of 14 days in case they also become ill.To model the described process, we further make the following assumptions:(a) Only cases that are reported or hospitalized can trigger contact tracing(b) If a traced contact is tracked being infectious, they are immediately iso-lated, otherwise they are monitored for symptoms and possible isolationif symptoms develop.(c) We introduce parameters ρ and ρ that determine the probability orfraction of first or higher order traced contacts who will be incubating andinfectious, respectively, when tracked. We simplify the model by assumingthat ρ = ρ = ρ .Furthermore, we introduce a parameter β M such that 0 ≤ β M ≤ β to controlthe efficacy of monitoring policy of contact tracers and health officers and (cid:15) todenote the fraction of reported cases that will be traced. With these new param-eters and assumptions, we have the following system of differential equations: D αt S ( t ) = − β ( t ) SN ( I R + I U ) − β ( t ) SI T N − β αM SI M N D αt E ( t ) = β ( t ) SN I U + (1 − (cid:15) ) β ( t ) SN I R + (1 − (cid:15) ) β ( t ) SI T N + (1 − (cid:15) ) β αM SI M N − σ α E D αt E IC ( t ) = ρ (cid:15) (cid:18) β ( t ) SN I R + β ( t ) SI T N + β αM SI M N (cid:19) − σ α E IC D αt E IF ( t ) = (1 − ρ ) (cid:15) (cid:18) β ( t ) SN I R + β ( t ) SI T N + β αM SI M N (cid:19) − σ α E IF D αt I R ( t ) = ησ α E − ( γ α R + ϕ α R ) I R D αt I U ( t ) = (1 − η ) σ α E − γ α U I U D αt I M ( t ) = σ α E IC − γ α M I M D αt I T ( t ) = σ α E IF − ( γ α T + ϕ α T ) I T D αt H ( t ) = ϕ α R I R + ϕ α T I T − ( γ α H + µ α H ) H D αt R ( t ) = γ α R I R + γ α U I U + γ α M I M + γ α T I T + γ α H H D αt C ( t ) = σ α E D αt C ( t ) = σ α E IC D αt C ( t ) = σ α E IF D αt D ( t ) = µ α H H, (3)6here E IC and E IF are exposed individuals who will be traced and trackedduring the incubation and infectious stage, respectively. I M are infectious in-dividuals who have been tracked while incubating and are being monitored. I T are infectious individuals who are symptomatic when tracked and will beremoved or isolated. The last four equations in (3) are used to estimate thecumulative total cases (both unreported and reported cases whose contacts arenot being traced), cumulative cases of traced persons who will be tracked whileincubating, cumulative cases of traced persons who are infectious when trackedand the resulting cumulative deaths from the impact of CT. We shall considerthe following three special cases: In this case, we assume that the tracked and monitored contacts do not causesecondary infections, in which case β M = 0 and that all traced contacts will betracked while incubating, that is, ρ = 1 . The effective reproduction number, R c (see Appendix B), is given as R c = R (cid:20) η γ α U γ α R + ϕ α R (1 − (cid:15) ) + (1 − η ) (cid:21) , where R = β ( t ) /γ U is the effective reproduction number of the initial model(no contact tracing or hospitalization of cases). Thus, the contact tracing effortrequired to ensure that the effective reproduction number is below one is: R c < ⇔ η (cid:20) − γ α U γ α R + ϕ α R (1 − (cid:15) ) (cid:21) > − R . In the special case where we have high hospitalization rate and low recoveryrates (see Table 1) such that γ α U = γ α R = ϕ α R , then0 . η (1 + (cid:15) ) > (cid:18) − R (cid:19) . where 0 . η (1+ (cid:15) ) is the critical proportion of the total cases which must be tracedin order for R c <
1. Another special case is when we have low hospitalizationrate and high recovery rates such that γ α U = γ α R + ϕ α R , then η (cid:15) > (cid:18) − R (cid:19) . This indicates that a larger proportion of reported cases will be traced in theformer (special) case with high hospitalization and low recovery rates than thelatter one with low hospitalization and high recovery rates. Now, let’s rewrite (cid:15) as (cid:15) = Number of traced contacts per reported casesTotal number of contacts reported = (cid:96)n , and let the transmission rate β ( t ) be written as β ( t ) = p c ( t ), where p is theprobability of transmission per contact and c ( t ) is the contact rate. For an7ntraced reported case, n = c ( t ) (cid:18) γ α R + 1 ϕ α R (cid:19) = β ( t ) ( γ α R + ϕ α R ) p γ α R ϕ α R . Let κ be the average number of secondary infected traced contacts identifiedper untraced reported case, then κ := pl = (cid:15) β ( t ) γ α R + ϕ α R γ α R ϕ α R (4)We note that κ can be estimated directly from CT data and records. Also, wedefine the parameter s as the fraction of reported cases which are traced, thatis s = (cid:15) (cid:16) γ αR + ϕ αR (cid:17) (cid:15) (cid:16) γ αR + ϕ αR (cid:17) + η (1 − (cid:15) ) (cid:16) γ αR + ϕ αR (cid:17) + (1 − η ) /γ α U = (cid:15)(cid:15) + η (1 − (cid:15) ) + (1 − η ) γ αR ϕ αR γ αU ( γ αR + ϕ αR ) (5)Using the formulas (4) and (5) for κ and s , respectively, we obtain the formula R c < κ (cid:18) − ss (cid:19) = R ∗ c , where R ∗ c is the product of the average number of the secondary infected tracedcontacts per untraced reported case and the odds that a reported case is nota traced contact. For 100% reporting, s = κ/ ( κ + m ) which implies that areported case causes κ + m secondary infections where κ (or m ) of these casesare traced (or untraced). Thus, R ∗ c = m which is the fraction of secondaryinfected contacts to be traced that are not yet tracked. Here, we consider the case where each traced contact is tracked during theincubation stage and all infected individuals are reported. This implies that η = ρ = 1. The reproduction number in the absence of CT is given as R = β ( t ) / ( γ α R + ϕ α R ), see Appendix A. In a similar manner, the reproduction numberof contact traced (monitored) person is R M = β α M /γ α M . Then θ = R M / R isthe reduction in secondary cases of a traced (monitored) person compared toan untraced person. Thus R c = (1 − (cid:15) ) R + (cid:15) R M and the proportion of casesto be traced so that R c is below one is (cid:15) > (1 − θ ) − (cid:18) − R (cid:19) . Using CT observables, we describe R c by defining κ = (cid:15) R and κ M = (cid:15) R M as the average number of traced infected secondary cases per primary reporteduntraced and traced infected, respectively, with s given as s = (cid:15) , then R c = κ (cid:18) − ss (cid:19) + κ M . .4.3. Perfect Reporting and Monitoring (Imperfect Tracking) Lastly, we consider perfect reporting and monitoring with secondary tracedindividual during the incubation stage ( or infectious stage) with probability ρ (or (1 − ρ )). This implies that β M = 0 and η = 1. The reproduction numberin the absence of CT is R = β ( t ) / ( γ α M + ϕ αR ) and the reproduction number ofcontact traced individual who are incubating or infectious when tracked is R T = β ( t )(1 − ρ ) / ( γ α T + ϕ α T ). Thus, θ = R T / R is the reduction in secondary cases of atraced individual (who will be infectious or incubating when tracked) comparedto an untraced reported case. Thus, the reproduction number R c reduces to R c = (1 − (cid:15) ) R + (cid:15) R T . As in the previous cases, the critical proportion of totalcases which is to be traced for R c < (cid:15) > (1 − θ ) − (cid:18) − R (cid:19) . To describe the reproduction number in terms of CT observables, we let κ T = (cid:15) R T be the average number of traced infected secondary cases per primaryreported traced infected with s = (cid:15) , then R c = κ (cid:18) − ss (cid:19) + κ T .
3. Simulation Experiments and results
We use the infected and cumulative mortality data compiled by the Centerfor Systems and Science Engineering at John Hopkins University (2020) [29]from the day of the first record of infection in a given state to calibrate theparameter set ( β , r, t ∗ , ϕ, µ, α ) and the initial condition E . The other initialconditions are fixed, for example, I R is matched with the first recorded case, I U = (0 . / . I R since 65% of the cases are taken to be unreported and therest are set to zero. The remaining parameters in the model are fixed at defaultvalues given in Table 1. Parameter fittings were performed using a nonlinearleast squares algorithm in python with the limited memory BFGS method.One main benefit of the routine is the use of bounds for fit parameters. Thisallows faster convergence of the algorithm and ensures obtaining meaningfulfit parameters. The fitted parameters are given in Table 2. All numericalsimulations were done with our numerical scheme [43] from which we obtain thesolution of the proposed model at each time step as1. Predictor: 9 p = S j + τ α Γ(1 + α ) F ( t j , S j , E j , I R,j , I
U,j , H j , R j , D j ) + ˜ H ,j E p = E j + τ α Γ(1 + α ) F ( t j , S j , E j , I R,j , I
U,j , H j , R j , D j ) + ˜ H ,j I R,p = I R,j + τ α Γ(1 + α ) F ( t j , S j , E j , I R,j , I
U,j , H j , R j , D j ) + ˜ H ,j I U,p = I U,j + τ α Γ(1 + α ) F ( t j , S j , E j , I R,j , I
U,j , H j , R j , D j ) + ˜ H ,j H p = H j + τ α Γ(1 + α ) F ( t j , S j , E j , I R,j , I
U,j , H j , R j , D j ) + ˜ H ,j R p = R j + τ α Γ(1 + α ) F ( t j , S j , E j , I R,j , I
U,j , H j , R j , D j ) + ˜ H ,j D p = D j + τ α Γ(1 + α ) F ( t j , S j , E j , I R,j , I
U,j , H j , R j , D j ) + ˜ H ,j
2. Corrector: S j +1 = S j + τ α Γ(2 + α ) (cid:16) α F ( t j , S j , E j , I R,j , I
U,j , H j , R j , D j )+ F ( t j +1 , S p , E p , I R,p , I
U,p , H p , R p , D p ) (cid:17) + ˜ H ,j ,E j +1 = E j + τ α Γ(2 + α ) (cid:16) α F ( t j , S j , E j , I R,j , I
U,j , H j , R j , D j )+ F ( t j +1 , S p , E p , I R,p , I
U,p , H p , R p , D p ) (cid:17) + ˜ H ,j ,I R,j +1 = I R,j + τ α Γ(2 + α ) (cid:16) α F ( t j , S j , E j , I R,j , I
U,j , H j , R j , D j )+ F ( t j +1 , S p , E p , I R,p , I
U,p , H p , R p , D p ) (cid:17) + ˜ H ,j ,I U,j +1 = I U,j + τ α Γ(2 + α ) (cid:16) α F ( t j , S j , E j , I R,j , I
U,j , H j , R j , D j )+ F ( t j +1 , S p , E p , I R,p , I
U,p , H p , R p , D p ) (cid:17) + ˜ H ,j ,H j +1 = H j + τ α Γ(2 + α ) (cid:16) α F ( t j , S j , E j , I R,j , I
U,j , H j , R j , D j )+ F ( t j +1 , S p , E p , I R,p , I
U,p , H p , R p , D p ) (cid:17) + ˜ H ,j ,R j +1 = R j + τ α Γ(2 + α ) (cid:16) α F ( t j , S j , E j , I R,j , I
U,j , H j , R j , D j )+ F ( t j +1 , S p , E p , I R,p , I
U,p , H p , R p , D p ) (cid:17) + ˜ H ,j ,D j +1 = D j + τ α Γ(2 + α ) (cid:16) α F ( t j , S j , E j , I R,j , I
U,j , H j , R j , D j )+ F ( t j +1 , S p , E p , I R,p , I
U,p , H p , R p , D p ) (cid:17) + ˜ H ,j , F ( t j , S j , E j , I R,j , I
U,j , H j , R j , D j ) = − β ( t ) S j N ( I R,j + I U,j ) ,F ( t j , S j , E j , I R,j , I
U,j , H j , R j , D j ) = β ( t ) S j N ( I R,j + I U,j ) − σ α E j ,F ( t j , S j , E j , I R,j , I
U,j , H j , R j , D j ) = ησ α E j − ( γ α R + ϕ α R ) I R,j ,F ( t j , S j , E j , I R,j , I
U,j , H j , R j , D j ) = (1 − η ) σ α E j − γ α U I U,j ,F ( t j , S j , E j , I R,j , I
U,j , H j , R j , D j ) = ϕ α R I R,j − ( γ α H + µ α H ) H j ,F ( t j , S j , E j , I R,j , I
U,j , H j , R j , D j ) = γ α R I R,j + γ α U I U,j + γ α H I H,j ,F ( t j , S j , E j , I R,j , I
U,j , H j , R j , D j ) = µ α H E j , and ˜ H ,j = τ α Γ(2 + α ) j (cid:88) l =0 a l,j F ( t l , S l , E l , I A,l , I
S,l , H l , R l , D l ) , ˜ H ,j = τ α Γ(2 + α ) j (cid:88) l =0 a l,j F ( t l , S l , E l , I A,l , I
S,l , H l , R l , D l ) , ˜ H ,j = τ α Γ(2 + α ) j (cid:88) l =0 a l,j F ( t l , S l , E l , I A,l , I
S,l , H l , R l , D l ) , ˜ H ,j = τ α Γ(2 + α ) j (cid:88) l =0 a l,j F ( t l , S l , E l , I A,l , I
S,l , H l , R l , D l ) , ˜ H ,j = τ α Γ(2 + α ) j (cid:88) l =0 a l,j F ( t l , S l , E l , I A,l , I
S,l , H l , R l , D l ) , ˜ H ,j = τ α Γ(2 + α ) j (cid:88) l =0 a l,j F ( t l , S l , E l , I A,l , I
S,l , H l , R l , D l ) , ˜ H ,j = τ α Γ(2 + α ) j (cid:88) l =0 a l,j F ( t l , S l , E l , I A,l , I
S,l , H l , R l , D l )are the memory terms of the respective population variables and a l,j = τ α Γ( α + 2) − ( j − α )( j + 1) α + j α (2 j − α − − ( j − α +1 , l = 0 , ( j − l + 2) α +1 − j − l + 1) α +1 + 3( j − l ) α +1 − ( j − l − α +1 , ≤ l ≤ j − , α +1 − α − , l = j. We note that the dynamics of the model for all states considered are similarexcept for Michigan that has seen a significant decrease in the number of infectedcases (see Fig. 1). Thus, we consider results for Michigan and Florida (whichhave similar results with the remaining states) in the following sections.11tate E t ∗ r β ϕ R µ H α California 4999.9 170.0 2.0723e-06 0.2167 0.0011 0.1999 0.9995Florida 4852.5 130.1 8.8287e-09 0.2365 0.0019 0.0602 0.9863Michigan 3000.0 21.4 8.3054e-02 0.8030 0.0034 0.1999 0.6000Tennessee 3531.8 139.9 4.8070e-05 0.1866 0.0010 0.1999 0.9010Texas 1221.9 143.6 4.9999e-03 0.2679 0.1000 0.0013 0.9996Washington 4999.8 176.2 2.3849e-03 0.1912 0.0024 0.0532 0.9997
Table 2: Fitted Parameters to some selected States in the US
We run the simulated model with β M = 0 and ρ = 1 for around 20 monthsunder constant conditions while studying the effect of the number of tracedreported cases on the number of infected, hospitalized and dead. Fig. 2 showsthat the total mortality (as well as infected and hospitalized) increases withno contact traced individual ( (cid:15) = 0) and decreases with increased number oftraced reported cases. Furthermore, we simulate the model with several valuesin (cid:15) × η, η, (cid:15) ∈ [0 ,
1] to observe the effect of reporting and tracing on the model.The outcome of interest is total mortality, peak hospitalization and peak infectedwhich are normalized against their respective maximum and the results arepresented in Fig. 3. The results in this figure show that while high reporting rateis crucial for mitigating the spread of the pandemic, the percentage of tracedreported cases have a more substantial effect on the spread. The results forFlorida are expected since the mortality and hospitalized cases will be more withperfect reporting and no tracing. The results for Michigan is quite surprisingshowing that a moderately high (and not very high) reporting rate gives peakhospitalizations and mortality. This may be attributed to the dynamics of thefitted model for Michigan where the number of infected individuals have beenconsiderably lowered after the first few months. Using the formula given ineqn. 5, we estimate the number of reported cases which will be traced. Theresults are presented in fig. 4. A contour plot of the reproduction number R c as a function of fraction of the infected population reported and proportion ofexposed individuals that is traced is shown in fig. 5. The figure shows that ifat least 45% (65%) of total infected cases are reported and at least 40% (60%)of contacts of reported cases are traced in Florida (Michigan), then diseaseelimination is feasible. We run the simulated CT epidemics by assuming that CT was only intro-duced after some discrete time delay (20, 60 and 100 days). The fraction ofreported cases traced was fixed at 50%. We observe, in fig. 6, that the inter-vention of CT reduces the number of infected, hospitalization and mortalityeven with a late intervention time (100 days). We observe that the results forMichigan with intervention after 60 and 100 days are approximately the same.This is because the turning point t ∗ ≈
21 (day after which governmental actions12 igure 1: Data and model fits for some selected states in the US. begins to be functional) is lower than the CT intervention dates.
We run the simulated CT model with ρ = η = 1 where CT was immediatelyintroduced. The fraction of reported cases traced was fixed at 50%. We examinethe effect of monitoring policy on the number of infected, hospitalizations andmortality. Fig. 7 shows that 50% effective monitoring policy reduces the hospi-talization and total mortality to approximately half its value. Furthermore, werun several simulations with values in (cid:15) × β M , (cid:15), β M ∈ [0 ,
1] and the results areshown in fig. 8. Similar to previous contour plots, the outcome of interests arerelative peak hospitalization and total mortality. We observe that the resultsfor Michigan is quite surprising. The peak hospitalizations and cumulative mor-tality occur when β M ≈ β and the fraction of traced reported cases is around20-80% where we would have expected this to be around 0-20%. This shows13 igure 2: Current infected, hospitalized and total mortality with varying fraction of tracedreported cases in a perfect tracking and monitoring case. The first (second) row is for Michigan(Florida) State. that, in general, the monitoring policy has a greater effect in reducing the peakvalues than the fraction of traced reported cases for a state like Michigan wherethe effects of governmental action is already imminent. A contour plot of thereproduction number R c as a function of the monitoring efficacy and proportionof exposed individuals that is traced is shown in fig. 9. The figure shows that thedisease will die out if traced individual are being monitored so that they are atleast half (a quarter) as infectious as an unmonitored or untraced infected casewith at least 40% (60%) of contacts of reported cases being traced in Florida(Michigan). In this case, we consider the numerical experiment where we assume thatevery infected case is reported ( η = 1) and tracked contacts of reported case areeffectively monitored ( β M = 0) so that they do not cause secondary infections.We run the simulated CT model under constant conditions with the aim ofexploring the effect of ρ (the fraction or probability that a traced reported case isincubating when tracked) on peak hospitalization and mortality. Unsurprisingly,we see that the higher the fraction of tracked contacts who are incubating thelower the number of hospitalizations and deaths. These results are evident infigures 10–12. 14 igure 3: Relative peak infected, hospitalizations and total mortality simulated epidemicsunder different reporting and tracing levels. The first (second) row is for Florida (Michigan)State. Figure 4: Estimated number of reported cases traced for Michigan and Florida igure 5: Effect of CT. The first column shows profiles of the control reproduction number asa function of proportion of reported cases ( η ). The second column shows contour plots of thecontrol of reproduction number as a function of proportion of reported cases ( η ) and tracedindividuals ( (cid:15) ). The first (second) row is for Florida (Michigan).Figure 6: CT Intervention after some discrete time delay. First (second) row is for Florida(Michigan) State. igure 7: Efficiency of monitoring policy in CT. The β M are selected to indicate 0%, 50%and 100% (corresponding to β M = β , β / igure 9: Effect of CT. The first column shows profiles of the control reproduction number asa function of monitoring efficacy ( β M ). The second column shows contour plots of the controlof reproduction number as a function of monitoring efficacy ( β M ) and proportion of tracedindividuals ( (cid:15) ). The first (second) row is for Florida (Michigan).Figure 10: Effects of tracking contacts of reported cases when incubating or being infectious.The ρ values are selected to show 0%, 40%, 80% and 100% of traced reported cases areincubating when tracked. Perfect tracking implies ρ = 1. First (second) row is for Florida(Michigan) State. igure 11: Relative peak infected, hospitalizations and total mortality of simulated epidemicsunder different monitoring conditions and fraction of traced reported cases. First (second)row is for Florida (Michigan) State.Figure 12: Effect of CT. The first column shows profiles of the control reproduction numberas a function of tracking efficacy ( ρ ). The second column shows contour plots of the control ofreproduction number as a function of tracking efficacy ( ρ ) and proportion of traced individuals( (cid:15) ). The first (second) row is for Florida (Michigan). . Discussions and Conclusions The novel coronavirus pandemic which began in China has spread acrossthe globe with over 700,000 deaths. Several control measures have been takenby health and government officials to mitigate the spread of the virus. Suchmeasures include social distancing, use of face-masks, stay-at-home orders andcontact tracing. In this work, we focus on studying the efficacy of contact trac-ing on the spread of the virus. In particular, we consider special cases wherewe have perfect tracking and monitoring, perfect reporting and tracking, andperfect reporting and monitoring.We have developed a time-fractional order differential equation model of the con-tact tracing process in the Covid-19 outbreak. Our deterministic model linksthe action of contact tracers such as monitoring and tracking to the number ofreported cases traced. Our framework separates the infected population intounreported and reported, and further splitting the reported cases into fractionwhose contacts will be traced. Additionally, we incorporate the effect of track-ing by considering the probability that a traced contact will be incubating (orbeing infectious) when tracked. This inherent structure in the model capturesthe dynamics of contact tracing and enables us to express the reproductionnumber in terms of observable quantities. In particular, under the assumptionthat there is a perfect tracking and monitoring, we gave an upper bound forthe effective reproduction number as R c < κ (1 − s ) /s , where κ is the averagenumber of secondary infected individuals traced per reported untraced case and(1 − s ) /s is the odds that a reported case is not a traced contact. In the case ofperfect tracking with either perfect monitoring or perfect reporting, we obtainthe result R c = κ (1 − s ) /s + κ M , where κ M is the average number of secondaryinfected individuals per reported traced case. With these observable quantities,these formulas can provide a quick and simple estimates of the reproductionnumber in the population. Furthermore, we estimated the proportion of con-tacts that need be traced to ensure that the reproduction number is below one.Although we would have loved to provide daily or weekly estimates of R c fromthe formulas (above) involving observable quantities but we were unable to findCT data for the Covid-19 pandemic. However, we relied on model simulationsto gain insights on the impact of CT with different special cases and duringdifferent stages of the pandemic. In fact, the decline of peak hospitalizationsand total deaths in the simulations of CT model compared to the preliminarymodel shows its efficacy.With the simulated CT model, the efficiency of CT in mitigating the spread ofthe virus and altering the epidemiological outcomes of peak hospitalizations andtotal deaths is a nonlinear function of the fraction of infected cases reported, themonitoring policy and the proportion of traced contacts who are tracked whileincubating (see Fig. 3, 8 and 11). In the first case (”perfect tracking and moni-toring”) and considering that 35% of infected cases are reported with 40%, 80%and 100% of reported cases being traced, the peak hospitalizations are reducedby 50.1% (44.2%), 85.7% (68.2%), and 96.0% (75.8%), respectively, for Florida(Michigan) State. The total mortality is also seen to decline by 30.2% (43.9%),203.2% (67.2%) and 85.2% (75.3%) in Florida (Michigan). Furthermore, we in-vestigated the intervention of CT after some discrete time delay. We observethat early intervention of CT may greatly reduce the peak hospitalizations andtotal mortality. Even with a late intervention (after 100 days), we see that thetotal mortality is reduced by a factor of 38.1% in Florida and a slight decrease ofabout 5.6% in Michigan where governmental actions are seen to be functional.In the second case, where we assumed a perfect reporting of infected cases and50% of these cases are traced and the monitoring policies are implemented with50% and 100% efficiency, we observe the reduction in total mortality (peakhospitalizations) by 47.2% (78.5%) and 95% (96%) in Florida. In Michigan, weobserved a 13.3% (34.5%) reduction with a 50% efficient monitoring policy. Fur-thermore, the contour plots (see fig. 8) show that while both fraction of tracedreported cases and the monitoring strategy are crucial in mitigating the spread,the monitoring strategy or policy is of substantial importance so that trackedreported individuals do not cause secondary infections while being monitored.Similar results are observed in the case of perfect reporting and monitoring.Finally, we showed the effects of the proportion of traced cases ( (cid:15) ), monitoringefficacy ( β M ), and tracking efficacy ( ρ ) on lowering the reproduction number sothat the disease eventually die out after a period of time.In conclusion, our findings suggests that almost all states in the US shouldadopt (if not yet) CT programs. In particular, our findings show that trackinga larger proportion of traced contacts while incubating and perfect monitoringof tracked contacts so that they do not cause secondary infections are highlyimportant for the impact of CT to be seen. Appendix A. Effective Reproduction Number of Preliminary Model
The basic reproduction number R c can be defined using the next generationmatrix [44, 45, 46]. The disease-free equilibrium point of the system is ε =( S , , , ε . By using the notations in [44,45], it follows that the matrices F of new infection terms and V of transfer ofinfection to and from the compartments are given, respectively, as F = β ( t ) β ( t )0 0 00 0 0 , V = σ α − ησ α γ α R + ϕ α R − (1 − η ) σ α γ α U . The effective reproduction number of the model, denoted by R , is given by R = β ( t ) (cid:18) ηγ α R + ϕ α R + 1 − ηγ α U (cid:19) ppendix B. Effective Reproduction Number of Model with CT In a similar manner to the results in Appendix A, the matrix F of newinfections and V of transfer terms are given by F = − (cid:15) ) β ( t ) β ( t ) (1 − (cid:15) ) β α M (cid:15) β ( t )0 0 0 ρ (cid:15) β ( t ) 0 ρ (cid:15) β α M ρ (cid:15) β ( t )0 0 0 (1 − ρ ) (cid:15) β ( t ) 0 (1 − ρ ) (cid:15) β α M (1 − ρ ) (cid:15) β ( t )0 0 0 0 0 0 00 0 0 0 0 0 00 0 0 0 0 0 00 0 0 0 0 0 0 V = σ α σ α σ α − ησ α γ α R + ϕ α R − (1 − η ) σ α γ α U − σ α γ α M
00 0 − σ α γ α T + ϕ α T ) . The effective reproduction number cannot be written explicitly here. However,the given matrices are used to obtain the reproduction numbers for each of thespecial cases given in the text.
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