A Conjectural Experiment to Observe the Effect of Conditional locked-down in an Epidemic
AA Conjectural Experiment to Observe the Effect ofConditional locked-down in an Epidemic
M.E. Hoque and S.K. Das
Department of Physics, Shahjalal University of Science and TechnologySylhet - 3114, BangladeshE-mail: [email protected]
May 2020
Abstract.
In a pandemic like Covid-19, there are many countries of lower-earningcannot provide a complete locked-down within the duration of the detected case. Thelocked-down may result in famine throughout the region of underdeveloped countriesafter the outbreak. So, a conjectural setup of an epidemic has been studied by applyingspecific period of locked-down (30 days) in 5 different scenarios. The stochasticapproach to the SEIR (Susceptible, Exposed, Infected and Recovered) model has beenused to evaluate the dynamics and the effects of locked-down. It is observed that thereexist a suitable period to apply locked-down where more susceptible escape from theinfection. The effect of the early (as soon as the infected case detected) and late (withrespect to the estimated peak of detected cases for no locked-down) implementationof the locked-down has also been studied and found that the late implementationof locked-down will take the least time to end the epidemic. The CFR (Case FatalityRate) has also been found to be varied from 7.55 to 8.02 for all the considered scenarios.
Keywords : SEIR model, Stochastic individuals, escape out susceptible, Barabasinetwork, social distancing, quarantine, Case Fatality Rate, Infected Fatality Rate,
1. Introduction
In a pandemic like Covid-19 originated at Wuhan, China, the whole world gets stuckall at once for months [1]. The food supply chain gets broken, all the local and globalbusiness falls, financial crisis has intensified. To prevent the damages due to the virulentpandemic, all the regional area require to be locked-down. But most of the countries ofthe world cannot provide food to their people if they get locked-down for months. Theyhave to trade the damages due to the epidemic of their people with foods and economicsfor the people. Besides, a virus outbreak like coronavirus can take years to control andvaccinated. Thus, it is very necessary to know about the implementation strategy ofthe locked-down situation to a country of low and middle income. a r X i v : . [ q - b i o . P E ] M a y conjectural experiment to observe the effect of locked-down in an epidemic et. al. suggest that the quarantine and the isolation methodwill win over the spreading of Coronavirus [9]. All of those estimations and predictionsmade by the deterministic model are impressively comply with the situation. Theseresearches also help to make policy and strategy over controlling the epidemic for manycountries.The stochastic model of the SEIR and its variations are also being using for theprediction of the dynamics of epidemics [10]. The important part of the stochastic modelof epidemics is that it does not require the integration as that of the deterministic model.It is also important to consider the communication structure of the individuals for betterunderstanding of the transmission of disease, the effect of social distancing and contacttracing which can be implemented in the stochastic model. An implementation of thenetwork model in stochastic SEIR model has been developed by S. G. Ryan et. al. where communication distribution of the individuals can be studied [11].In the case of an epidemic (basic reproduction number is larger than unity), thestochastic SEIR model becomes useful to analyze the implementation of various epidemiccontrol policy. In this work, the authors would like the find out the variation in dynamicsdue to applying the locked-down (social distancing) in a region. The authors wouldalso like to find out the number of escape population from the epidemic disease. Thestochastic network model has been described briefly in the method section. The resultsof a conjectural experiment have been discussed in the results and discussion section.The summary of the work has been provided in the conclusion section.
2. Theory
S E I RD E D I F β σ γσ D μ D γ D θ E , ψ E θ I , ψ I Figure 1.
The schematic diagram of the SEIR model where
S, E, I, R, D E and D I represent susceptible, exposed, infected, recovered, detected exposed and detectedinfected respectively. The arrow represents the transition along with transition rate. A schematic diagram of the SEIR compartmental model has been shown in Fig.1. The
S, E, I, D E , D I , R and F are the number of susceptible, exposed, infectious, conjectural experiment to observe the effect of locked-down in an epidemic N . The β, σ, γ, µ I , β D , σ D , γ D , µ D , ξ, µ and ν are the rate of transmission, infection, recovery, infection-related death, transmissionfor the detected case, infection for the detected case, recovery for the detected case,infection-related death for the detected case, re-susceptibility, baseline death andbaseline birth respectively. The θ E , θ I , φ E , φ I , ψ E , ψ I and q are the baseline testing forexposed, baseline testing for infectious, testing when a close contact has tested positivefor exposed, testing when a close contact has tested positive for an infectious, positivetest of exposed state, the positive test of infectious state and the interaction rate ofdetected infection to the susceptible.In the network model of population, a graph represents the state of an individual( S, E, I, D E , D I , R, F ) and their interactions. At a given time, each individualinteracts with close contacts with a probability of (1 − p ) β and the interaction probabilitywith the global network is pβ .Each individual ( i ) in a considered region has a state X i which updates accordingto the following probability transition rates [11] (and Ryan Seamus McGee, personalcommunication, May 8, 2020): P ( X i = S → E ) = (cid:104) p (cid:16) βI + qβ D D I N (cid:17) +(1 − p ) (cid:16) β [ (cid:80) jεC G ( i ) δ X j = I ] + β D [ (cid:80) kεC Q ( i ) δ X k = D I ] | C G ( i ) | (cid:17)(cid:105) δ X i = s (1) P ( X i = E → I ) = σδ X i = E (2) P ( X i = I → R ) = γδ X i = I (3) P ( X i = I → F ) = µ I δ X i = I (4) P ( X i = E → D E ) = (cid:16) θ E + φ E (cid:104) (cid:88) jεC G ( i ) δ X k = D E + δ X k = D I (cid:105)(cid:17) ψ E δ X i = E (5) P ( X i = I → D I ) = (cid:16) θ I + φ I (cid:104) (cid:88) jεC G ( i ) δ X k = D E + δ X k = D I (cid:105)(cid:17) ψ I δ X i = I (6) P ( X i = D E → D I ) = σ D δ X i = D E (7) P ( X i = D I → R ) = γ D δ X i = D I (8) P ( X i = D I → F ) = µ D δ X i = D I (9) P ( X i = any → S ) = ξδ X i = R + νδ X i (cid:54) = F (10)With the state probability, δ X i = A = 1, if the state of X i is A , otherwise it is 0.The C G ( i ) and C Q ( i ) denotes the set of close contacts and quarantine contacts of theindividual i , respectively.The basic reproduction number, for SEIR model of Covid-19 pandemic [4], can begiven by, R = σσ + µ D βγ + µ D (11) conjectural experiment to observe the effect of locked-down in an epidemic µ D → R = βγ (12)which is the expected number of secondary cases produced by a single infectedindividual in a completely susceptible population.
3. Results and Discussions
The conjectural experiment has been designed for a total number of population N ,100000. The magnitude of the parameter β = β D = 1 / . σ = σ D = 1 / . γ = γ D = 1 / .
34 are chosen in such a way that the detected infection has a widthof at least 60 days. The other parameters are chosen as µ I = 0 . µ D = 0 . θ E = θ I = 0 . φ E = φ I = 0 . ψ E = ψ I = 1 . q = 0 . p is taken as 0.1 whereas it is 0.5 for thenormal situation. These intentional parameters produce 2.17 for the basic reproductionnumber ( R ) (since the µ D is very small). F r equen cy o f I nd i v i dua l s NormalDistancingQuarantine
Figure 2.
The distribution of the communication of the individuals in a populationare shown for normal, locked-down (social distancing) and quarantined situation.
In this stochastic model, the structured interaction network, Barabasi-Albert Model[12], has been considered as shown in Fig. 2; where the frequency of individuals is plottedagainst their degrees of communication. The mean degrees of communication for normal,social distancing (locked-down) and quarantine situation is considered as 23.0, 3.3 and1.6 respectively. The quarantine has been applied to the infected individuals after thedetection.The values of the parameters are chosen in such a way that the detected caseof without locked-down has a width of more than 90 days. It is considered that theepidemic region can be locked-down for only 30 days. Thus, it becomes necessary to conjectural experiment to observe the effect of locked-down in an epidemic . . . . . N u m b e r o f ac ti v e d e t ec t e d i n f ec ti on ( i n % ) Lock down: NoLock down: 10 to 40Lock down: 40 to 70Lock down: 70 to 100Lock down: 100 to 130Lock down: 130 to 160
Figure 3.
Active detected cases for the 5 of the locked-down scenario includingwithout locked-down. The percentages of the active infected individuals are takenwith respect to the total population. The vertical lines indicate the starting of thelocked-down for the similar colours of detected cases. apply this 30 days more carefully so that the damage due to epidemic can be minimized.Since the detected case has been found from 30 th day to 160 th day in the case of withoutlocked-down scenario (Fig. 3), the locked-down phase has been chosen form 10 th to 40 th ,40 th to 70 th , 70 th to 100 th , 100 th to 130 th and 130 th to 160 th day (say scenario 1 - 5). Itis to be noted that the increase in the detected case is found after the 44 th day. Theeffect of locked-down for the described scenarios is visualized in Fig. 3 by consideringthe detected active cases ( D E ). In the case of scenario 1 and 2, the shift in the increaseof detected active cases has been observed whereas a steady and decreasing trend hasbeen observed for scenario 3 and 4, respectively. For scenario 5, there is no observableshift whereas the widest shifting is found for scenario 1. It is also observed that thepeak of the detected cases for scenario 3 is lower than 0.25% whereas all other scenariosit crosses the 0.80% of the total population (percentages are taken with respect to thetotal population). It indicates that applying conditional locked-down at a suitable time,can control the number of maximum active infected cases. This will help to use themaximum medical facilities which may contribute to minimize the fatal cases.The simulated active infected cases ( I ) are shown for all of the scenarios in Fig.4. A similar trend of shifting has been observed as described for Fig. 3. But the peakseems to be the same for no locked-down and all the scenarios except 3. In the caseof scenario 3, the peak is found after 79 th day which is 9-days later of the locked-downimplementation day (1 /β + 1 /σ + 1 /γ ≈ th day with respect to the no locked-down situation.Since the detected cases are not inter-related (a random fraction of the infected cases are conjectural experiment to observe the effect of locked-down in an epidemic . . . . . . . . N u m b e r o f ac ti v e i n f ec ti on ( i n % ) Lock down: NoLock down: 10 to 40Lock down: 40 to 70Lock down: 70 to 100Lock down: 100 to 130Lock down: 130 to 160
Figure 4.
The percentage (with respect to the population) of the active infected casefor all the locked-down and no locked-down scenarios are shown. detected), the downward shift of the peak between Fig. 3 and Fig. 4 is understandable.All the locked down scenarios have happened as expected for the estimation of infectedcases. S u s ce p ti b l e popu l a ti on ( i n % ) Lock down: NoLock down: 10 to 40Lock down: 40 to 70Lock down: 70 to 100Lock down: 100 to 130Lock down: 130 to 160
Figure 5.
The number of susceptible population (with respect to the total population)are shown in percentage for all the described scenarios.
The dynamics of the susceptible population ( S ) is the greater interest of thisconjectural experiment. Fig. 5 shows the variation in the dynamics of the susceptiblepopulation for all the cases. It is found that there are more than 30% susceptible conjectural experiment to observe the effect of locked-down in an epidemic C a s e F a t a lit y R a t e ( i n % ) Lock down: NoLock down: 10 to 40Lock down: 40 to 70Lock down: 70 to 100Lock down: 100 to 130Lock down: 130 to 160
Figure 6.
Estimation of the Case Fatality Rate (CFR) for various locked-downscenario.
The Case Fatality Rate (CFR) calculation is complicated as it should not be[13, 14, 15]. In this article, the CFR has been calculated as the rate of deceased for therecovered population at the same time [16]. It is observed that the CFR varies from7.55 to 8.02 for all the locked-down scenario including no locked-down case after 130days (Fig. 6). The lowest CFR is found for the scenario 2 whereas it is expected forthe scenario 3 because of the steady observation in detected cases (with respect to Fig.3). The initial fluctuations are well understood for their detection and treatment tothe patient. The observed under or over estimation in the initial CFR has been welldiscussed by Rajgor et.al. [14]. Though the number of case fatality depends on manyof the other factors (like age, immunity, previous medical condition etc.), the variationin the observed CFR indicates that the fatality can also be depended on the applyingconditional locked-down.This simulation has been run until no infected cases for the 100000 population. Itis observed that the longest and shortest epidemic occurs for scenarios 1 and 5. Whichdescribes whether to chose longest or shortest time. conjectural experiment to observe the effect of locked-down in an epidemic
4. Conclusions
A conjectural experiment has been designed to predict the effect of locked-down inan epidemic. The Barabasi-Albert communication network model has been used toproduce deploy the regional normal, social distancing and quarantine communication ofindividuals. The stochastic SEIR model has been intentionally parameterized so thatthe detected active case can be observed for more than 90 days. A 30 days locked-downperiod has been applied in 5 of the specific scenarios. The above mentioned notionalexperiment with intentional parameters can be summaries as:(i) If the allocated 30 days locked-down period has been applied too early (scenario1 and 2) or lately (scenario 5), it barely modifies the dynamics of no locked-downcase.(ii) In the case of scenario 1 and 2, the dynamics only shifted for more than 30 and 20days respectively. So, the locked-down should not be applied too early, if it requiresto maintain a conditional locked-down.(iii) In the study of the dynamics of susceptible population, interestingly scenario 3 and4 has more than 40% susceptible individual remains after the outbreak.(iv) It is observed that the initial fluctuation of CFR becomes steady after the 93 rd day(the day where the peak occurs in the infected case due to no locked-down conditionFig. 4). The lowest CFR is 7.55 which is found for scenario 2.(v) Very interestingly, scenario 5 (where late locked-down has been introduced) takesleast time to reach the end of the epidemic. And the scenario 1 takes the longesttime to reach at the end.Though the verification of the above mentioned experiment is not feasible in reality,the authors believe that the speculated summary will impact very efficiently to makethe policy for controlling the epidemic. Acknowledgements
The authors would like to thank Prof. Dr. Zafar Iqbal, Department of ComputerScience and Engineering, Shahjalal University of Science and Technology, Sylhet - 3114,Bangladesh for his valuable discussions.
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