Infectious diseases are often spread via direct and indirect contacts such as person-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 infection spread through these means. Several measures which includes social distancing, stay-at-home orders, self-isolation/quarantine, use of face-masks, contact tracing amongst others, have been enforced by authorities in reducing the spread of the virus. In any disease outbreak therefore, contact tracing is an important tool for combating the spread of the outbreak. Contact tracing (CT) is the process whereby persons who have come in contact with a reported/isolated case are traced and monitored so that if they become symptomatic they can be efficiently isolated to reduce transmissions. Previous outbreaks of infectious diseases have been rapidly controlled with contact tracing and isolation, for example, the Ebola outbreak in West Africa in 2014, see [9] . Furthermore, in any disease control it is important to evaluate the efficacy of intervention strategies such as contact tracing. Thus, the need to explicitly measure how contact tracing can help in mitigating the transmission of coronavirus cannot be overemphasized. A lot of studies have been conducted on the efficacy of contact tracing 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 novel coronavirus, see for example [17, 18, 19, 20, 21, 22, 23, 24] and several models have incorporated contact tracing using stochastic modeling approach [25] and networks [26] . However, these studies did not include the effect of contact tracing on the reproduction number of Covid-19 and the expression of this reproduction number in terms of observable quantities, a quick and efficient way of estimating the reproduction number. In 2015, Browne et al. [16] developed a deterministic model of contact tracing for Ebola epidemics which links tracing back to transmissions, and incorporates disease traits and control together with monitoring protocols. Eikenberry et al. [23] examined the potential of face mask use by general public to curtail the Covid-19 pandemic. Their findings suggest that face mask should be adopted nation-wide and be implemented without delay, even if most masks are homemade and of relatively low quality. Motivated essentially by the works of [16] and [23] , the goal in this work is to develop a deterministic model to measure the efficacy of contact tracing in mitigating the spread of Covid-19. As noted in [16] , explicitly incorporating contact tracing with disease dynamics presents challenges, and population level effects of contact tracing are difficult to determine. Here, we propose a compartmental model which incorporates the disease traits and monitoring protocols. We describe the impact on the reproduction number R c of Covid-19 and discuss the importance and relevance of parameters of the model. In particular, we develop a time-fractional compartmental model; a modifica-tion of the SEIR model similar to the one given in [23] where they divided the infected population into symptomatic and asymptomatic compartments. We adopt the use of time-fractional models because they reduce errors which may arise 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 incorporate this history dependence of infectious diseases. The remaining sections are organized as follows: In section 2, we begin with a discussion on the basic SEIR model and used this as a building block in deriving the new model with contact tracing. Thereafter, we consider special cases of the model and calculate their effective reproduction numbers. Furthermore, we gave a lower bound on the proportion of reported cases that must be traced to ensure the reproduction number is below one and express the reproduction number in terms of observable quantities such as average number of secondary infected persons per traced and untraced reported case. In section 3, we perform several numerical experiments to corroborate our theoretical observations in Section 2. Finally, in section 4, we gave a comprehensive discussion on the impact of contact tracing in mitigating the spread of the virus.
We begin with a basic time-fractional SEIR model consisting of four compartments that represents the susceptible (S), exposed (E), infected (I), recovered (R). We assume that all the infected individuals are unreported and thus not hospitalized. The following system of differential equations models the transmission dynamics of the population:
is the disease transmission rate which takes into account effects of governmental actions such as social distancing, mask usage and so on, t * is the day after which the 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 to the infectious class, γ (1/γ) is the recovery rate (time from infectiousness until recovery) of an infected individual. We note that the parameters of the model are non-negative and have dimensions given by 1/time α . This observation was originally noted in Diethelm [27] . To alleviate this difference in dimensions, we replace the parameters with a power α of new parameters to obtain the new system of equations:
with
The next step in the development of our model is the incorporation of hospitalized 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 either due to unawareness or early recovery or just perceptions of the infected individuals. We note that only the reported cases are being hospitalized during the infectious period and neglect the possibility of transmission of an hospitalized individual since they are not exposed to the general population. Thus, we obtain the following system of time-fractional differential equations:
where C(t) and D(t) represents the number of cumulative infected (both reported and unreported) and the disease-induced deaths, respectively. These numbers can be explicitly calculated as
γ R , γ U , and γ H are the recovery rates of a reported, unreported and hospitalized individuals, respectively. ϕ R is the hospitalization rate of reported infected person and µ H is the disease-induced death rate.
For simplicity, we shall use prior studies to fix several parameters and fit the other parameters of the model. In particular, we shall fit the parameters β 0 , r, t * , ϕ R , µ H and α using the Covid-19 data obtained from John Hopkins University [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 different hospitalization and death rates. Prior modeling studies suggest that the effective transmission rate β 0 ranges between 0.5-1.5 day −1 [30, 31, 32, 23] and the incubation period lies in the range between 2-9 days [33, 22, 34, 19] . The average of 5.1 days was estimated by Lauer et al. [35] . The infectious duration seems 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 that around 88% and 86%, respectively, of all infections are undocumented with a 95% credible interval. Maugeri et al. [40] estimated that the proportion of unreported new infections by day ranged from 52.1% to 100% with a total of 91.8% of infections going unreported. 
We incorporate CT into the preliminary model by linking the dynamics of disease model with actions of contact tracers such as monitoring and tracking. This general modeling framework is similar to a variety of CT models employed in [10, 15, 16] . At first, we describe the four steps of CT for Covid-19 as described by the Center for Disease Control (CDC) [42] . The Public health officer tries to identify contacts (contact investigation) by working with infected patients to help recall people they've been in contact with while being infectious. The second step (contact tracing) involves notifying and tracing of recorded contacts of the patient. Next (contact support), the officer informs and educates the contacts on the risk and dangers of being exposed. They also provide support on the next line of action for the contacts. In the case that a contact is already showing symptoms, the tracers will call an ambulance to remove/isolate the contact. Lastly (contact self-quarantine), contacts are encouraged to quarantine for 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 isolated, otherwise they are monitored for symptoms and possible isolation if symptoms develop. (c) We introduce parameters ρ 1 and ρ 2 that determine the probability or fraction of first or higher order traced contacts who will be incubating and infectious, respectively, when tracked. We simplify the model by assuming that ρ 1 = ρ 2 = ρ.
Furthermore, we introduce a parameter β M such that 0 ≤ β M ≤ β 0 to control the efficacy of monitoring policy of contact tracers and health officers and to denote the fraction of reported cases that will be traced. With these new parameters and assumptions, we have the following system of differential equations:
where E IC and E IF are exposed individuals who will be traced and tracked during the incubation and infectious stage, respectively. I M are infectious individuals who have been tracked while incubating and are being monitored. I T are infectious individuals who are symptomatic when tracked and will be removed or isolated. The last four equations in (3) are used to estimate the cumulative total cases (both unreported and reported cases whose contacts are not being traced), cumulative cases of traced persons who will be tracked while incubating, cumulative cases of traced persons who are infectious when tracked and the resulting cumulative deaths from the impact of CT. We shall consider the following three special cases:
In this case, we assume that the tracked and monitored contacts do not cause secondary infections, in which case β M = 0 and that all traced contacts will be tracked while incubating, that is, ρ = 1. The effective reproduction number, R c (see Appendix B), is given as
where R 0 = β(t)/γ U is the effective reproduction number of the initial model (no contact tracing or hospitalization of cases). Thus, the contact tracing effort required to ensure that the effective reproduction number is below one is:
In the special case where we have high hospitalization rate and low recovery rates (see Table 1 ) such that γ α
where 0.5η (1+ ) is the critical proportion of the total cases which must be traced in order for R c < 1. Another special case is when we have low hospitalization rate and high recovery rates such that γ α
This indicates that a larger proportion of reported cases will be traced in the former (special) case with high hospitalization and low recovery rates than the latter one with low hospitalization and high recovery rates. Now, let's rewrite as = Number of traced contacts per reported cases Total number of contacts reported = n , and let the transmission rate β(t) be written as β(t) = p c(t), where p is the probability of transmission per contact and c(t) is the contact rate. For an untraced reported case,
Let κ be the average number of secondary infected traced contacts identified per untraced reported case, then
We note that κ can be estimated directly from CT data and records. Also, we define the parameter s as the fraction of reported cases which are traced, that is
Using the formulas (4) and (5) for κ and s, respectively, we obtain the formula
where R * c is the product of the average number of the secondary infected traced contacts per untraced reported case and the odds that a reported case is not a traced contact. For 100% reporting, s = κ/(κ + m) which implies that a reported case causes κ + m secondary infections where κ (or m) of these cases are traced (or untraced). Thus, R * c = m which is the fraction of secondary infected contacts to be traced that are not yet tracked.
Here, we consider the case where each traced contact is tracked during the incubation 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 0 = β(t)/(γ α R + ϕ α R ), see Appendix A. In a similar manner, the reproduction number of contact traced (monitored) person is R M = β α M /γ α M . Then θ 1 = R M /R 0 is the reduction in secondary cases of a traced (monitored) person compared to an untraced person. Thus R c = (1 − )R 0 + R M and the proportion of cases to be traced so that R c is below one is
Using CT observables, we describe R c by defining κ = R 0 and κ M = R M as the average number of traced infected secondary cases per primary reported untraced and traced infected, respectively, with s given as s = , then
Lastly, we consider perfect reporting and monitoring with secondary traced individual during the incubation stage ( or infectious stage) with probability ρ (or (1 − ρ) ). This implies that β M = 0 and η = 1. The reproduction number in the absence of CT is R 0 = β(t)/(γ α M + ϕ α R ) and the reproduction number of contact traced individual who are incubating or infectious when tracked is R T = β(t)(1−ρ)/(γ α T +ϕ α T ). Thus, θ 2 = R T /R 0 is the reduction in secondary cases of a traced individual (who will be infectious or incubating when tracked) compared to an untraced reported case. Thus, the reproduction number R c reduces to R c = (1 − )R 0 + R T . As in the previous cases, the critical proportion of total cases which is to be traced for R c < 1 is
To describe the reproduction number in terms of CT observables, we let κ T = R T be the average number of traced infected secondary cases per primary reported traced infected with s = , then
We use the infected and cumulative mortality data compiled by the Center for 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 the parameter set (β 0 , r, t * , ϕ, µ, α) and the initial condition E 0 . The other initial conditions are fixed, for example, I R0 is matched with the first recorded case, I U 0 = (0.65/0.35)I R0 since 65% of the cases are taken to be unreported and the rest are set to zero. The remaining parameters in the model are fixed at default values given in Table 1 . Parameter fittings were performed using a nonlinear least squares algorithm in python with the limited memory BFGS method. One main benefit of the routine is the use of bounds for fit parameters. This allows faster convergence of the algorithm and ensures obtaining meaningful fit parameters. The fitted parameters are given in Table 2 . All numerical simulations were done with our numerical scheme [43] from which we obtain the solution of the proposed model at each time step as 1. Predictor:
α F 2 (t j , S j , E j , I R,j , I U,j , H j , R j , D j ) + F 2 (t j+1 , S p , E p , I R,p , I U,p , H p , R p , D p ) +H 2,j , I R,j+1 = I R,j + τ α Γ(2 + α) α F 3 (t j , S j , E j , I R,j , I U,j , H j , R j , D j ) + F 3 (t j+1 , S p , E p , I R,p , I U,p , H p , R p , D p ) +H 3,j , I U,j+1 = I U,j + τ α Γ(2 + α) α F 4 (t j , S j , E j , I R,j , I U,j , H j , R j , D j ) + F 4 (t j+1 , S p , E p , I R,p , I U,p , H p , R p , D p ) +H 4,j ,
α F 6 (t j , S j , E j , I R,j , I U,j , H j , R j , D j ) + F 6 (t j+1 , S p , E p , I R,p , I U,p , H p , R p , D p ) +H 6,j ,
a l,j F 1 (t l , S l , E l , I A,l , I S,l , H l , R l , D l ),
a l,j F 4 (t l , S l , E l , I A,l , I S,l , H l , R l , D l ),
a l,j F 5 (t l , S l , E l , I A,l , I S,l , H l , R l , D l ),
a l,j F 6 (t l , S l , E l , I A,l , I S,l , H l , R l , D l ),
a l,j F 7 (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
We note that the dynamics of the model for all states considered are similar except for Michigan that has seen a significant decrease in the number of infected cases (see Fig. 1 ). Thus, we consider results for Michigan and Florida (which have similar results with the remaining states) in the following sections. 
We run the simulated model with β M = 0 and ρ = 1 for around 20 months under constant conditions while studying the effect of the number of traced reported cases on the number of infected, hospitalized and dead. Fig. 2 shows that the total mortality (as well as infected and hospitalized) increases with no contact traced individual ( = 0) and decreases with increased number of traced reported cases. Furthermore, we simulate the model with several values in × η, η, ∈ [0, 1] to observe the effect of reporting and tracing on the model. The outcome of interest is total mortality, peak hospitalization and peak infected which are normalized against their respective maximum and the results are presented in Fig. 3 . The results in this figure show that while high reporting rate is crucial for mitigating the spread of the pandemic, the percentage of traced reported cases have a more substantial effect on the spread. The results for Florida are expected since the mortality and hospitalized cases will be more with perfect reporting and no tracing. The results for Michigan is quite surprising showing that a moderately high (and not very high) reporting rate gives peak hospitalizations and mortality. This may be attributed to the dynamics of the fitted model for Michigan where the number of infected individuals have been considerably lowered after the first few months. Using the formula given in eqn. 5, we estimate the number of reported cases which will be traced. The results 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 of exposed individuals that is traced is shown in fig. 5 . The figure shows that if at 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 disease elimination is feasible.
We run the simulated CT epidemics by assuming that CT was only introduced after some discrete time delay (20, 60 and 100 days) . The fraction of reported cases traced was fixed at 50%. We observe, in fig. 6 , that the intervention of CT reduces the number of infected, hospitalization and mortality even with a late intervention time (100 days). We observe that the results for Michigan with intervention after 60 and 100 days are approximately the same. This is because the turning point t * ≈ 21 (day after which governmental actions begins to be functional) is lower than the CT intervention dates.
We run the simulated CT model with ρ = η = 1 where CT was immediately introduced. The fraction of reported cases traced was fixed at 50%. We examine the effect of monitoring policy on the number of infected, hospitalizations and mortality. Fig. 7 shows that 50% effective monitoring policy reduces the hospitalization and total mortality to approximately half its value. Furthermore, we run several simulations with values in × β M , , β M ∈ [0, 1] and the results are shown in fig. 8 . Similar to previous contour plots, the outcome of interests are relative peak hospitalization and total mortality. We observe that the results for Michigan is quite surprising. The peak hospitalizations and cumulative mortality occur when β M ≈ β 0 and the fraction of traced reported cases is around 20-80% where we would have expected this to be around 0-20%. This shows that, in general, the monitoring policy has a greater effect in reducing the peak values than the fraction of traced reported cases for a state like Michigan where the effects of governmental action is already imminent. A contour plot of the reproduction number R c as a function of the monitoring efficacy and proportion of exposed individuals that is traced is shown in fig. 9 . The figure shows that the disease will die out if traced individual are being monitored so that they are at least half (a quarter) as infectious as an unmonitored or untraced infected case with 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 that every infected case is reported (η = 1) and tracked contacts of reported case are effectively monitored (β M = 0) so that they do not cause secondary infections. We run the simulated CT model under constant conditions with the aim of exploring the effect of ρ (the fraction or probability that a traced reported case is incubating when tracked) on peak hospitalization and mortality. Unsurprisingly, we see that the higher the fraction of tracked contacts who are incubating the lower the number of hospitalizations and deaths. These results are evident in figures 10-12. 
The novel coronavirus pandemic which began in China has spread across the globe with over 700,000 deaths. Several control measures have been taken by health and government officials to mitigate the spread of the virus. Such measures include social distancing, use of face-masks, stay-at-home orders and contact tracing. In this work, we focus on studying the efficacy of contact tracing on the spread of the virus. In particular, we consider special cases where we have perfect tracking and monitoring, perfect reporting and tracking, and perfect reporting and monitoring. We have developed a time-fractional order differential equation model of the contact tracing process in the Covid-19 outbreak. Our deterministic model links the action of contact tracers such as monitoring and tracking to the number of reported cases traced. Our framework separates the infected population into unreported and reported, and further splitting the reported cases into fraction whose contacts will be traced. Additionally, we incorporate the effect of tracking by considering the probability that a traced contact will be incubating (or being infectious) when tracked. This inherent structure in the model captures the dynamics of contact tracing and enables us to express the reproduction number in terms of observable quantities. In particular, under the assumption that there is a perfect tracking and monitoring, we gave an upper bound for the effective reproduction number as R c < κ(1 − s)/s, where κ is the average number 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 of perfect tracking with either perfect monitoring or perfect reporting, we obtain the result R c = κ(1 − s)/s + κ M , where κ M is the average number of secondary infected individuals per reported traced case. With these observable quantities, these formulas can provide a quick and simple estimates of the reproduction number in the population. Furthermore, we estimated the proportion of contacts 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 from the formulas (above) involving observable quantities but we were unable to find CT data for the Covid-19 pandemic. However, we relied on model simulations to gain insights on the impact of CT with different special cases and during different stages of the pandemic. In fact, the decline of peak hospitalizations and total deaths in the simulations of CT model compared to the preliminary model shows its efficacy. With the simulated CT model, the efficiency of CT in mitigating the spread of the virus and altering the epidemiological outcomes of peak hospitalizations and total deaths is a nonlinear function of the fraction of infected cases reported, the monitoring policy and the proportion of traced contacts who are tracked while incubating (see Fig. 3, 8 and 11 ). In the first case ("perfect tracking and monitoring") and considering that 35% of infected cases are reported with 40%, 80% and 100% of reported cases being traced, the peak hospitalizations are reduced by 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%), 63.2% (67.2%) and 85.2% (75.3%) in Florida (Michigan). Furthermore, we investigated the intervention of CT after some discrete time delay. We observe that early intervention of CT may greatly reduce the peak hospitalizations and total mortality. Even with a late intervention (after 100 days), we see that the total mortality is reduced by a factor of 38.1% in Florida and a slight decrease of about 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 and 50% of these cases are traced and the monitoring policies are implemented with 50% and 100% efficiency, we observe the reduction in total mortality (peak hospitalizations) by 47.2% (78.5%) and 95% (96%) in Florida. In Michigan, we observed a 13.3% (34.5%) reduction with a 50% efficient monitoring policy. Furthermore, the contour plots (see fig. 8 ) show that while both fraction of traced reported cases and the monitoring strategy are crucial in mitigating the spread, the monitoring strategy or policy is of substantial importance so that tracked reported 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 ( ), monitoring efficacy (β M ), and tracking efficacy (ρ) on lowering the reproduction number so that the disease eventually die out after a period of time.
In conclusion, our findings suggests that almost all states in the US should adopt (if not yet) CT programs. In particular, our findings show that tracking a larger proportion of traced contacts while incubating and perfect monitoring of tracked contacts so that they do not cause secondary infections are highly important for the impact of CT to be seen.
The basic reproduction number R c can be defined using the next generation matrix [44, 45, 46] . The disease-free equilibrium point of the system is ε 0 = (S 0 , 0, 0, 0). We define a next-generation matrix by considering the linearized system at the disease-free equilibrium point ε 0 . By using the notations in [44, 45] , it follows that the matrices F of new infection terms and V of transfer of infection to and from the compartments are given, respectively, as
The effective reproduction number of the model, denoted by R 0 , is given by
In a similar manner to the results in Appendix A, the matrix F of new infections and V of transfer terms are given by
The effective reproduction number cannot be written explicitly here. However, the given matrices are used to obtain the reproduction numbers for each of the special cases given in the text.