key: cord-1026253-t44zt8vc authors: Elías, L. Llamazares; Elías, S. Llamazares; del Rey, A. Martín title: An analysis of contact tracing protocol in an over-dispersed SEIQR Covid-like disease date: 2021-12-14 journal: Physica A DOI: 10.1016/j.physa.2021.126754 sha: abd58e723b028f25ee2e41da117f1ab21e0354c5 doc_id: 1026253 cord_uid: t44zt8vc The aim of this work is to study an over-dispersed SEIQR infectious disease and obtain optimal methods of contact tracing. A prototypical example of such a disease is that of the current SARS-CoV-2 pandemic. In consequence, this study is immediately applicable to the current health crisis. In this paper, we introduce both a discrete and continuous model for various modes of contact tracing. From the continuous model, we derive a basic reproductive number and study the stability of the equilibrium points. We also implement the continuous and discrete models numerically and further analyze the effectiveness of different types of contact tracing and their cost on society. Additionally, through these simulations, we also study the effect that various parameters of the disease have on its evolution. The global health crisis brought about by SARS-CoV-2 has revealed our lack of preparedness to deal with a pandemic. As of today, 7 months since the WHO declared the coronavirus a global pandemic, there have been 112,000,000 J o u r n a l P r e -p r o o f Journal Pre-proof detected infections of COVID19 and 2,500,000 detected deaths [1] and there is 5 still no unified protocol for contact tracing or collection of data. Due to the unreliability and heterogeneities of the collection of data, in this paper, we look to use only the already well-established characteristics of COVID19 to study optimal control efforts. These methods of disease prevention will in turn apply to any disease with a similar structure and qualities to COVID19. 10 There are several well-established methods of infectious disease control such as physical distancing, disinfection of hands and surfaces, and vaccination. In this article, we will focus on contact tracing and how it should be carried out to reduce the spread of over-dispersed infectious diseases. Possible ways to improve contact tracing have recently been the object of study of a growing number of 15 researchers. Tedelay the authors conclude via an agent-based model that the delay between the onset of symptoms and the positive diagnosis is critical to containing the advance of COVID19. In [2] and [3] the writers conclude that bidirectional contact tracing is superior to traditional contact tracing. However, to the extent of our knowledge, and at the time of publication, this result has 20 only been shown using agent-based models or simple approximations of the spread of COVID19 whereas in this work we propose a continuous model, as well as a discrete one, to verify this hypothesis. In addition, we estimate the cost of each contact tracing protocol in terms of the number of individuals that enter quarantine. 25 The mathematical study of the propagation of biological agents plays a fundamental role when it comes to understanding the behavior of infectious diseases. Furthermore, mathematical models help us to both determine (in a theoretical way) and evaluate control measures. Modern mathematical epidemiology goes back to Kermack and McKendrick model [4] . This is a very simple SIR 30 (Susceptible-Infectious-Recovered) compartmental model where the dynamics are modeled in terms of a system of ordinary differential equations. Since then, several models based on this paradigm have been proposed in the scientific literature (see, for example, [5, 6] , and references therein). Of special interest are those works devoted to the study and analysis the basic reproductive number 35 J o u r n a l P r e -p r o o f Journal Pre-proof [7, 8] . During the last two years, the research in this field has increased notably due to the social impact of the COVID19 pandemic [9, 10, 11] To achieve this aim our work is organized as follows: in Section 2 we introduce the two main ways of contact tracing that are currently used and the negative binomial distribution, which will serve as an over-dispersed offspring 40 distribution. In Section 3 and 4 we build a continuous model for an SEIQR disease based on each of the two types of contact tracing introduced in the first section. We will take COVID19 as the prototype for our study as it fits the SEIQR model and is a prototypical example of an over-dispersed disease. In Section 5 we study the stability of the equilibrium points to these models and 45 calculate their basic reproductive number R 0 . In Section 6 we conduct various numerical simulations to study the previously discussed continuous models. We then construct in Section 7 a discrete (stochastic) version of our continuous models, conduct further simulations and analyze the obtained results. Finally, in Section 8 we discuss the conclusions that one may obtain from our study and 50 propose further avenues of research. In this section, we introduce the two types of contact tracing that we will discuss throughout our work. To effectively complete this study we also introduce the binomial distribution, which will be used as the offspring distribution 55 of our over-dispersed disease. The two contact tracing models that will be studied in this work are the following: Forwards contact tracing (FCT). Under this contact tracing protocol, once an 60 infected individual is detected said individual is placed in quarantine and an attempt is made to find all people the infected individual had a potentially dangerous contact with since contracting the disease. The idea behind this method J o u r n a l P r e -p r o o f Journal Pre-proof is to eliminate any potential infections resulting from the infected individual. This is the contact tracing that has been used in many countries in response to 65 COVID19 [12]. In the particular case of COVID19, one attempts to locate all close contacts that occurred two days back from symptom onset [13] . Backwards contact tracing (BCT). With this protocol, once an infected individual is detected said individual is placed in quarantine one attempts to locate the person that first infected the subject (the primary case). Once this is achieved 70 all individuals that had dangerous contact with the primary case are placed in quarantine. This method is designed for diseases that spread mostly through large clusters (super-spreader events). The reasoning behind BCT protocol is that an individual who is detected to be infected was likely infected by a primary case that caused more infections than the detected individual themselves 75 (this is similar to the so-called friendship paradox). Thus, the optimal way to react would be to locate the primary infection and quarantine everyone that they may have infected. Below, in Figure 1 , is a graph showing an example of both BCT and FCT protocol. Some cases, though in theory detectable, remain undiscovered due to 80 human implementation which is inherently imperfect. When carrying out BCT one uses, in addition to the same tests as FCT, some additional tests to find the primary case. In principle, this may seem like it would lead to more testing, which could prove to be an issue if testing capacity is limited. However, if BCT is more effective, it may end up controlling 85 the disease earlier than a FCT protocol. Due to this BCT may lead to even fewer tests overall than FCT. Diseases like the flu spread in a linear and predictable way where the reproductive number R 0 encapsulates almost all the information of the disease. In 90 essence, the assumption that each infected individual will results in exactly R 0 secondary infections is a good one. On the other hand, as current data shows [14, 15, 16] , COVID19 is a representative case of an overly dispersed disease. That is, the distribution of secondary cases caused by an infected individual has a much higher variance 95 and the disease spreads through large clusters of secondary infections. This has important consequences for effective disease control measures. The offspring distribution of a disease is the distribution that assigns a probability p n to the event E n = {the cluster has n infected individuals} . It is often useful when considering the offspring distribution X of an overly dispersed dis-100 ease to assume that X follows a negative binomial distribution [17] . We now introduce this distribution: The negative binomial distribution with parameters k ∈ R + , and 0 < p < 1 is the distribution X with values in N such that where Γ is the gamma function. We symbolize that X follows a negative bi-105 nomial distribution by X ∼ N B(k, p). By manipulating the series that define the mean and variance of X one may show that the mean and the variance of a random variable following binomial distribution X ∼ N B(k, p) are respectively J o u r n a l P r e -p r o o f Journal Pre-proof We deduce by solving the system of equations in (2) for σ 2 that the variance of 110 X is a quadratic function of its mean with The terms µ 2 /k, k are respectively called the over-dispersion and dispersion parameter of x and indicate an over-dispersion of our distribution with respect to the Poisson distribution. To observe this consider the Poisson distribution of mean µ, P oisson(µ). Then P oisson(µ) also has variance equal to µ and it 115 holds that, if we maintain µ fixed, the limit as the over-dispersion goes to zero of X is P oisson(µ). That is: where the above convergence is in distribution and can be obtained in the usual fashion by considering the limit of the probabilities p n := P(X = n). For all this, the negative binomial distribution is useful when considering infectious 120 diseases whose offspring distribution has a high variance. In particular, in [15] it was found that the negative binomial distribution was the distribution that best fit the data for the offspring distribution of COVID19. In this section, we will analyze the dynamics of COVID19 in a population 125 which uses as a control measure forward contact tracing (see Subsection 2.1). We begin with a heuristic discussion of the equations that govern our model. It is crucial to realize that any model is only as strong as the set of assumptions that it makes. For this reason, we will carefully lay out all the hypotheses made and explain how they lead logically to each term in the equations that follow. In this way, we hope that the reader may gain full insight into the workings of the models to come, and thus, may be able to adjust them to new data and J o u r n a l P r e -p r o o f Journal Pre-proof information that arises and even to other diseases with similar epidemiological properties. The mathematical model introduced in this work is compartmental where the population is subdivided into five compartments corresponding to susceptible, exposed, infected, quarantined, and recovered individuals. Specifically, a SEIQR model is considered where the dynamics consist of the following process: susceptible individuals become exposed when they have been successfully 140 infected; exposed individuals become infectious once the latency period finishes; susceptible, exposed and infectious individuals can be quarantined; and infectious individuals can recover from infection when the infectious period finishes. Specifically, our model will be made up of the following compartments: • Susceptible (S): These are the members of the population that are at risk 145 of contracting the disease. On becoming infected these individuals pass to compartment E. world reinfection remains possible, it is exceedingly rare [19] . The dynamics of the forward contact tracing model are illustrated in Figure 2 . The equations we will be working with are R = (r s + τ )I s + r a I a + (q E E + q s I s + q a I a ) I s N , (S, E, I s , I a , Q S , R)(0) = (S 0 , E 0 , I s,0 , I a,0 , Q S,0 , R 0 ), where, for brevity in the expression, we have omitted the time t at which the functions S, E, I s , I a , Q s , R are valued (all other terms are constants, i.e. independent of time). This said we now proceed to explain the above equations, 170 clarifying each term in said equations and mentioning the assumptions used to arrive at them. We begin by mentioning that we assume the natural mortality and birth rate to be zero as, due to the short time frame during which the epidemic has taken place, these parameters should not qualitatively affect our model. In what follows we will describe the epidemiological coefficients involved in the proposed model. • Let N be the total amount of individuals in the population. • Set T the average time in days that a susceptible individual remains in 180 quarantine. • Consider µ := E[X] = n np n where p n is the probability that a cluster caused by an infectious individual results in n infections amongst a totally susceptible population. That is p n = P(X = n) where X is the offspring distribution. • Suppose that β E , β s , β a are the rate of clusters caused by one individual of group E, I s , I a respectively in an entirely susceptible population. • Assume that q S , q E , q s , q a are the expected rate of dangerous contacts divided by µ that result in quarantine per member of I s amongst a population made out entirely of members of S, E, I s , I a respectively. The reason 190 why these parameters are not supposed identical is that we allow in theory for different contact rates amongst the separate compartments. • Set γ s , γ a the rates at which a member of E move to I s , I a respectively J o u r n a l P r e -p r o o f Journal Pre-proof and in consequence γ E := γ s + γ a is the rate at which a member of E moves to either I s or I a . • Let r s , r a be the rates of recovery of individuals in I s , I a respectively. • τ is the self-quarantine rate of symptomatic individuals due to awareness of their symptoms. •r s := r s + τ is the sum of r s , τ and indicates the rate at which individuals leave compartment I s due to recovery/quarantine. (1) Susceptible individuals that become infected per unit time. We have that β E E is the number of clusters caused by exposed individuals per unit time. Therefore, p n β E E is the number of clusters we expect to cause n infections in an entirely susceptible population per unit time. In consequence, the sum of 205 the terms np n β E ES/N is the number of infections we expect to be caused by exposed individuals per unit time. A similar discussion goes through for clusters caused by I a , I s with which we arrive at the infinite sum in (5). (2) Susceptible individuals that enter quarantine per unit time. Though in reality susceptible individuals enter quarantine due to close contact with any 210 individual in E, I s , I a who has become detected, due to the great majority of detected cases being due to symptomatic individuals [20] we consider for simplicity that the rate at which susceptible individuals enter quarantine is proportional only to the product I s S (that is without considering the products I a S or ES). Note that this approximation is additionally bolstered by the fact that the 215 amount of symptomatic individuals I s is predictive of the number of individuals in E, I a . Thus we set the number of susceptible individuals entering quarantine per unit time to q S I s S/N . We also employ similar reasoning to set the next three terms. Journal Pre-proof (3) Exposed that enter quarantine per unit time. q E EI s /N . This term is anal-220 ogous to the previous term in (2) where now we need only replace q S by the quarantine coefficient q E for exposed individuals. (4) Infectious symptomatic that quarantine/recover per unit time. q s I 2 s /N + (r s + τ )I s . Once again, we have a term that indicates quarantine due to close contact with detected symptomatic individuals. In this case this term is q s I 2 s /N . Furthermore, we have the additional summands r s I s , τ I s due to recovery and self detection. (5) Infectious asymptomatic that quarantine/recover per unit time. q a I s I a /N + r a I a . Once more, we have a term due to close contact with detected susceptible individuals and which in this case is q a I s I a /N and a term r a I a due to recovery 230 of members of I a . (6) Susceptible individuals that leave quarantine per unit time. Due to the definition of the parameters this is Q S /T . Current guidelines are to quarantine for 14 days in the case of close contact with an infectious individual [21] . For this reason, we will set T = 14 throughout the remainder of our work. (7) Individuals that become exposed per unit time. By definition of E and S this is the same as the number of susceptible individuals that become infected per unit time (term (1)). (8) Individuals that become I a or I s per unit time. This is equal to γ E E; where γ s E individuals go to compartment I s and γ a E of them go to compartment I a . (9) Individuals that recover per unit time. This is equal to r s I s + r a I a ; where the amount preceding from I s , I a are r s I s , r a I a respectively. (10) Infectious individuals that enter quarantine. This is equal to (3)+(4)+(5). As previously explained these individuals go to compartment R on entering quarantine. J o u r n a l P r e -p r o o f Journal Pre-proof For completeness we note that it is also possible to consider in the above model the inclusion of a vaccine. One approach to doing so would be to, as in [22] : set a vaccination rate and efficacy ξ v and v respectively and to split the susceptible population into two groups; those that have been vaccinated (S v ) and those that are yet unvaccinated (S u ). We would also have to subdivide the compartment Q S into two compartment Q v , and Q u , corresponding to the population of S v , S u respectively that have entered quarantine. The analogous of the model in equations (5)-(10) would now be: However, since our main intent is to study the usage of contact tracing at the beginning of an infectious outbreak we will not use the above model in what remains of our work. In this section, we introduce a similar continuous compartmental model for COVID19 which now incorporates backward contact tracing. J o u r n a l P r e -p r o o f As was previously discussed backward contact tracing is especially advantageous when the offspring distribution X has high variance. It is important to notice that, in the FCT model of Section 3, the only characteristic of the 255 offspring distribution that appears is its mean µ (which appears in the term of susceptible individuals that become infected per unit time in equations (5) and (6)). In the FCT model, no other properties of the offspring distribution are used for quarantine purposes. That is, we are disregarding extra important information, namely the over-dispersion of X. On the other hand, as we shall soon see, the variance and hence the dispersion of the offspring distribution will become important in our BCT model. The only terms that we need to change when considering BCT protocol are the terms involving individuals that enter quarantine. We reason as follows: consider the event D an infectious individual was detected, as before, let E n stand for "the 265 detected individual forms part of a cluster of n infected individuals" and set p n := P(E n ). Then, since {E n } ∞ n=1 partitions the sample space of possible clusters we have that, by Bayes Theorem: If we now make the reasonable supposition that the detection of an infected person in a cluster is proportional to the number of people in said cluster, i.e. P(D|E j ) = jP(D|E 1 ). We obtain that In consequence, once we detect an infected individual, the average amount of infections we detect from the original cluster is (assuming we detect all dangerous contacts): J o u r n a l P r e -p r o o f Journal Pre-proof where the infections that are tested forward are µ and the ones that are detected 275 backwards correspond to the term σ 2 /µ. In the case that X ∼ N B(k, p) and adding the average forward detection µ we obtain from equations (2) and (3) that we can write: Where the first term corresponds to those detected backwards and the second to those tested forwards. By once again simplifying and considering that individuals quarantine only due to contacts with symptomatic individuals we obtain that the quarantine term (10) of the previous model may be rewritten as: Note that the division by µ is due to our previous terminology in which µ infections lead to q S , q E , q s , q a among the respective populations. Where the 285q E ,q s ,q a have the same meaning as previously. That is, they reflect the ratio of dangerous contacts and the imperfections of the testing and tracing process (the latter now being different, reason for which these terms are also different, as it may be more difficult to trace contacts that occur farther back in time). Similar reasoning allows us to set the term of susceptible individuals entering 290 quarantine per unit time to: To simplify our notation we definê By defining alsoq E ,q s ,q a in a similar fashion we have that our backward contact tracing model is J o u r n a l P r e -p r o o f Journal Pre-proof Note that it is clearly qualitatively of the same type as our forwards contact 295 tracing model with the only difference being that the quarantine coefficientŝ q S ,q E ,q s ,q a are larger. In an identical fashion to Subsection 3.5, it is also possible to consider the case where there exists a vaccination process. In this section, we find the equilibrium points and basic reproductive number 300 for the models of the preceding section. We also study the stability of said equilibrium points. We first study the stability points of our continuous FCT model comprised of equations (5) to (10) . We suppose that all parameters for this model are Note that P is formed by all points of the state set at which everyone in the population is susceptible or has recovered (or, equivalently, system states at which no one is infected or in quarantine). We have that the linear approximation of our system at such a point has the As we can see by the box formed by the columns and rows 2, 3, 4 of the matrix (37), which correspond to the infected individuals E, I s and I a respectively, the new infection matrix F and the transition matrix V are given by The basic reproductive number R 0 is defined as the spectral radius of F V −1 320 (see for example [23] ). From (38) we obtain that γE β E + γs rs β s + γa ra β a βs rs βa ra thus we obtain that F V −1 has as its unique non-zero eigenvalue the first entry of the above matrix, i.e. we obtain an R 0 of This expression just obtained for R 0 is very intuitive in that it provides the relative importance of each infectious compartment E, I a , I s to our disease. Due • R a is completely analogous to R s where the above heuristics now apply to compartment I a instead of I s . Due to all this, we have that R E + R s + R a represent the reproduction number The analogous study of the continuous BCT model is practically identical. This is because, as we already mentioned, the system is qualitatively the same. The only difference is that the BCT's quarantine coefficientsq E ,q s ,q a are larger than the FCT's quarantine coefficients q E , q s , q a . In particular, we obtain the same equilibrium points, stability, and R 0 for this model. In this section, we simulate the state equations obtained in Subsection 3.2 360 and Section 4 for forward and backward contact tracing protocol respectively. We conduct all simulations with Mathematica. One of our continuous simulations will be conducted with N = 4.7 · 10 7 individuals to be representative of the population of Spain. The other continuous simulation will use N = 1000 individuals so as to be more comparable with a 365 discrete implementation developed in later sections. This difference in initial population between the continuous simulations may lead one to hypothesize that the existence of finite size effects makes results between the two incomparable. However, this is not the case due to a nice scale invariance of the solutions to our models. We first obtain a numerical simulation to said state equations with the following parameters: • N = 1000 is the population such that we take 10 people to be initially exposed. • γ s = γ a = 1 10 indicates that the incubation period of the disease is 5 days and that infections are equally distributed between asymptomatic and symptomatic. This will in turn cause the evolution of these two 380 compartments to be almost identical as it is shown later in Figure 5 and • q S = q E = q s = q a = 2 indicate that we expect to find 2 close contacts per symptomatic infection in a wholly susceptible population and likewise for an entirely exposed, asymptomatic, symptomatic population. • τ = 0.3/14 indicates that we expect that 30% of symptomatic individuals will self-quarantine throughout their infectious period. Let us perform another simulation we now take an even more over-dispersed disease and hence a smaller k which we set k = 0.3 as has been estimated for COVID19 [15] . We also take N = 4.7·10 7 and set 1% of the initial population to be infected, keeping all other parameters identical. Figures 9-14 below show the evolution of compartments S, E, I s , I a , Q, and R respectively in a simulation for 420 the above values of the parameters. Once again, above each graph, we include the number of individuals in each compartment after 200 days for each mode of contact tracing. As we can see, we now obtain an even more appreciable difference. Namely, we have that after 200 days a total of 6.1 · 10 6 , 2.1 · 10 6 individuals have been infected for the FCT and BCT respectively. In other 425 words, the BCT model performs 66% better than its counterpart. On the 800th day we have that 9.6 · 10 6 , 2.1 · 4.1 6 individuals are infected for the FCT and BCT respectively. This corresponds to a 57% reduction in infections by using BCT versus FCT protocol. Here we implement four stochastic discrete models for a highly dispersed disease. Firstly we discuss a model with no quarantine, secondly a model with where p E is a previously fixed number between 0 and 1 that reflects the probability of an outbreak amongst an entirely susceptible population and where r S (v), r(v) are the amount of susceptible and total individuals within a distance 450 of 2 from v. Where, by distance we between two nodes, we mean the length of the shortest path in the graph between these two nodes. In this way, we ponder the frequency of an outbreak stemming from v by the frequency of the nearby susceptible population. In an analogous fashion, we define f s and f a as the probability of an outbreak occurring due to an infectious asymptomatic and 455 symptomatic individual respectively by where p s , p a now determine the probability of an outbreak amongst an entirely susceptible population by a symptomatic, asymptomatic individual respectively. (2) Determination of infections. If v causes an outbreak it infects a number of individuals following an offspring distribution X given by the negative binomial 1. Susceptible to exposed: Susceptible individuals infected by an outbreak (see point (2)) move to compartment E. to be the nodes of V 1 that tested positive and to be the nodes at a distance of 1 from elements of V + 1 . We now continue our chain by detecting elements of V 2 with probability p d and testing those detected with the same type of test. This test will 510 be completed the next day, two days from the self-quarantine of v 0 . This process continues until one of the sets V + n is empty. Note that in this case, we continue testing all the way down the (detected) chain of infections. This completes the explanation of our discrete models. It is interesting to note the following. In the continuous model, to not overly complicate the J o u r n a l P r e -p r o o f Journal Pre-proof resulting equations, we imposed quarantining due only to contact with members of I s (see equations (9) and (29)). However, in a discrete implementation, we 525 have been able to consider quarantining that occurs also due to contact with members of E, I a . This kind of quarantining occurs to different degrees in both our BCT models. For each mode of contact tracing, we simulated our discrete models 1000 times using the same parameters as in our first continuous model. Additionally, each simulation spanned two hundred days. To get a better idea of how these different methods of contact tracing compare amongst each other we will study the mean of these simulations. In Figures 15-21 we plot the mean amount of nodes in each of the models' compartments for each one of the considered models. Additionally, we give 95% confidence intervals of said means. These confidence intervals are shown in the following 535 graphs as bands enveloping each curve. The confidence intervals are calculated by using the Mathematica function "FindDistribution" to find a functional form to fit the distribution of the data obtained in the simulations. This distribution is then used to calculate the aforementioned confidence intervals. In Figure 15 the mean amount of susceptible nodes each of the 200 days of 540 the simulations is shown. As we can see, BCT is the best performing method of contact tracing followed by the intermediate and FCT methods. Unsurprisingly, the model without quarantine was the one with the worst performance. As we will see, in Figures 16-20 the same ranking is repeated. The very first few days it may seem, paradoxically, that the method without contact tracing is 545 doing better than its fellows. However, this is simply because in the models with contact tracing a small part of the nodes that are not in the susceptible compartment are in the quarantined compartment. Figure 16 shows the evolution of the change in the average exposed individuals. At the start every model has 10 exposed nodes, this number increases 550 slightly the first 5 days as these exposed nodes infect others before sharply dropping when the 10 initial exposed nodes transition into one of the infected compartments. After this initial stage, the BCT model distinguishes itself from the others as a result of its more effective quarantining process. In Figure 17 , we show the evolution of the infected asymptomatic population. 555 Qualitatively, it is very similar to the evolution of the exposed population. It is worth noting that the models that implement worse methods of quarantine have larger confidence intervals as they are more prone to large outbreaks. Figure 17 . This is a result of the fact that in these simulations we set an exposed individual to be 560 equally likely to become symptomatic or asymptomatic. However, a discerning eye may note that on average there are slightly more infected asymptomatic nodes than infected symptomatic nodes. This is a result of a portion of the infected symptomatic nodes self quarantining. One of the most impressive aspects of the model that implements BCT is 565 the fact that the reduction of infections comes despite fewer individuals being quarantined on average. This is reflected in Figure 19 . In Figure 20 , we show the evolution of the average recovered individuals. We remind the reader that, the larger the amount of nodes in the recovered compartment for a given model, the worse this model performs, as these are Perhaps the best way of comparing the models is by using the average nodes that have been infected at some point. This is shown in Figure 21 Recent events have manifested the importance of contact tracing in controlling infectious diseases. In most cases, the predominant method used is forward contact tracing. With this strategy, once an infected individual is detected, the people with whom that person was recently in contact are warned and recom-585 mended to self-quarantine. However, as we have seen throughout this study, backward contact tracing presents notable benefits in comparison to forward contact tracing when the disease in question has a high over-dispersion. These advantages are not only theoretical but have been verified experimentally. In both our discrete and continuous models, backward contact tracing performs 590 better than its counterpart especially when the over-dispersion is greater. This strongly suggests that backward contact tracing should be the preferred method of detection in infectious diseases that present a significant variability of the number of infections caused by an outbreak. The implementation of backward contact tracing is likely more difficult as it 595 is based on detecting contacts that occur further in the past than those detected through forward contact tracing. Furthermore, this adds the logistical difficulty that the number of contacts to be traced through BCT may be significantly J o u r n a l P r e -p r o o f Journal Pre-proof higher than those through BCT. However, since the advantages gained through BCT are in large part due to the detection of large clusters, in practice we 600 recommend that maximum priority is assigned to contacts occurring at events with large agglomerations of people such as those occurring at parties, concerts, sporting events or public transport. These contacts may be easier to trace as attending such as event is more difficult to forget and leverage the main princi- best pair with backward contact tracing while analyzing their respective cost 615 on society. It is all but certain that sometime in the future, humanity will face another pandemic, and it is the hope of the authors that this paper can help to shed light on more effective techniques for controlling and preventing any such disease that may lie ahead. Finally, though it was determined via the stochastic implementation of Sec-620 tion 7, that backward contact tracing is not only more efficient but imposes a lower cost on society than its alternatives, we did not study the number of tests performed in each mode of contact tracing. Since, at least at first, backwards contact tracing requires more tests than other modes of contact tracing, a possible further study could take this into account to determine how the availability 625 of tests influences the relative efficiency of backward contact tracing with respect to other modes of contact tracing. In this sense, an additional analysis of the criticality of the system by means of the finite-size scaling method is also proposed as future work. Furthermore, the analysis of the variability of the J o u r n a l P r e -p r o o f Journal Pre-proof system when the initial conditions (related to BA algorithm) associated with 630 the complex networks used could also be of interest. 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