{"paper_id":"0e74c842-5597-4ca2-97a4-e3340ad2894a","body_text":"2021 LATEX template\nRanking the Effectiveness of\nNon-Pharmaceutical Interventions to\nCounter COVID-19 in UK Universities with\nVaccinated Population\nZirui Niu 1 and Giordano Scarciotti 1*\n1EEE, Imperial College London, Exhibition Rd, South\nKensington, London, SW7 2AZ, United Kingdom.\n*Corresponding author(s). E-mail(s): g.scarciotti@imperial.ac.uk;\nContributing authors: z.niu@outlook.com;\nAbstract\nSeveral universities around the world have resumed in-person teach-\ning after successful vaccination campaigns have covered 70/80% of\nthe population. In this study, we combine a new compartmental\nmodel with an optimal control formulation to discover, among dif-\nferent non-pharmaceutical interventions, the best prevention strategy\nto maximize on-campus activities while keeping spread under con-\ntrol. Composed of two interconnected Susceptible-Exposed-Infected-\nQuarantined-Recovered (SEIQR) structures, the model enables staff-\nto-staff infections, student-to-staff cross infections, student-to-student\ninfections, and environment-to-individual infections. Then, we model\ninput variables representing the implementation of different non-\npharmaceutical interventions and formulate and solve optimal control\nproblems for four desired scenarios: minimum number of cases, min-\nimum intervention, minimum non-quarantine intervention, and mini-\nmum quarantine intervention. Our results reveal the particular sig-\nnificance of mask wearing and social distancing in universities with\nvaccinated population (with proportions according to UK data).\nThe study also reveals that quarantining infected students has a\nhigher importance than quarantining staff. In contrast, other mea-\nsures such as environmental disinfection seems to be less important.\nKeywords: COVID-19; Non-pharmaceutical interventions; Optimal control;\nComparison of effectiveness\n1\n\n2021LATEX template\n2 Ranking the Effectiveness of Non-Pharmaceutical Interventions\nThe epidemic ascribed to the virus called Severe Acute Respiratory Syndrome\nCoronavirus 2 (SARS-CoV-2) has greatly impacted society and the economy\naround the world. In December 2019, China reported cases of pneumonia of\nunknown cause in Wuhan. The World Health Organisation (WHO) declared\nthe coronavirus disease 2019 (COVID-19) a pandemic in March 11, 2020 [1].\nBy 3 November 2021, WHO has reported over 246 million cases with more\nthan 5 million deaths around the world [1].\nThe virus generally causes respiratory symptoms such as cough, sneezing,\nshortness of breath, along with other symptoms including fever, headache [2]\nand olfactory or gustatory dysfunctions [3]. Since direct contact and aerosol\ntransmissions are two important ways of infection [4], symptomatic individuals\nare high spreaders. Meanwhile, the virus can also survive on surfaces and\nthen invade the human body through eyes, nose or mouth via touching [5, 6].\nSome carriers are asymptomatic, but they can still infect other susceptible\nindividuals [7, 8].\nMany countries have conducted successful vaccination campaigns. How-\never, vaccination uptake in most of these countries have platooned at around\n70/80% of their total population [9]. Moreover, SARS-CoV-2 has shown a rel-\natively high ability to adapt [10]. Several variants have been reported [11]\nwith four considered to be of main concern: Alpha (B.1.1.7), Beta (B.1.351),\nGamma (P.1) and Delta (B.1.617.2). In particular, the Delta variant has shown\nto be more infectious and more severe, leading to second or third waves in the\nUK, India and South Africa [12], in addition to having partial resistance to\nvaccines [13, 14]. The fast emergence of viral mutations has also raised wide\nconsiderations on whether current vaccines will be effective on new lineages\nappearing in the future [15–17]. Moreover, recent studies also show a decay of\nthe protection that vaccines offer as time from vaccination increases [18, 19],\nwhich prompted several counties to start a booster campaign. Therefore, in\nsuch a complex situation the implementation of non-pharmaceutical interven-\ntions such as mask wearing and social distancing have remained fundamental\nmeasures in containing the disease [20].\nThis work studies the current situation in British universities which have\nre-opened and maintain a combination of non-pharmaceutical measures in\nplace. The objective of universities is to maximise on-campus activities while\nmaintaining the spread of the disease under control. Universities are “small-\nenvironments” which have special features for which general purpose models\nmay be inadequate. For instance, a university is composed of two fundamen-\ntally different populations, the students and the staff, which have different\ndegrees of interaction, vaccination rates and serious symptoms. General pur-\npose models focus mostly on modelling the spread of the disease, but here\nwe are interested in maximising on-campus activity subject to limited spread.\nOur main contribution is two-fold: on one hand we provide a modelling frame-\nwork to maximise safe on-campus activity. On the other hand, a ranking of\n\n2021 LATEX template\nRanking the Effectiveness of Non-Pharmaceutical Interventions 3\nnon-pharmaceutical interventions and their fundamental importance in achiev-\ning the objective naturally emerges from our analysis. Moreover this emergent\nbehaviour is shown to be relative robust to modelling parameters.\nThe first block of our framework is a compartmental model. Compartmen-\ntal modelling is a popular choice for research on COVID-19. Cooper et al. [21]\nused an SIR model with changing total population to estimate the growth of\nthe epidemic in different nations. To explicitly model details such as incuba-\ntion period, hospitalization, and quarantine, Leontitsis et al. [22] proposed an\nSEAHIR model. Giordano et al. [23] created a far more complete SIDARTHE\nmodel, reflecting potential effects of non-pharmaceutical interventions imple-\nmented by the government. Various stochastic versions with higher complexity\nhave also been designed to estimate the development of the pandemic under\nfeasible countermeasures [24–26]. However, most of these models consider the\ninfluence of prevention and control measures by tweaking the model’s param-\neter values. Consequently, the importance of different interventions cannot be\nsystematically evaluated and compared. In this paper, we propose a new deter-\nministic compartmental model for the spreading of COVID-19 in universities\nand investigate the importance of non-pharmaceutical countermeasures using\noptimal control techniques.\nConsisting of a double SEIQR structure, our model distinguishes the epi-\ndemic evolution stages of students from those of staff. All individuals can\npotentially undergo five statuses: susceptible S (including the vaccinated),\nexposed E (asymptomatic), infected I (symptomatic), quarantined Q (hospital-\nized or isolated), and recovered R. Since all members are studying or working\nwithin the same confined space, the coronavirus can also transmit via the\nenvironment. An extra compartment C is used to represent the environmental\nvirus concentration. The overall structure of the model is depicted in Fig. 1.\nThis compartmental model automatically assumes that the total population is\nhomogeneously mixed, which is reasonable because everyone belonging to the\nsame department is following similar daily routines in the common confined\nspaces. The individuals in the compartments also have similar probability of\ninfections due to the virus surviving in the environment. The model does not\ninvolve the isolation of asymptomatic subjects because it is less likely to occur\nin the context of a university population. We still model the vaccinated indi-\nviduals as susceptible but with lower infection rates (i.e. vaccine breakthrough\ninfections). The possibility of re-infection among recovered people is omitted.\nDeath of vaccinated individuals is also negligible.\nTo assess the efficacy of the various intervention strategies under study we\nfirst evaluate a baseline scenario in which no countermeasures are implemented.\nThe parameters for this scenario are taken from the literature. Some of the\nvalues of the parameters are obtained according to the proportion of vaccinated\nindividuals and vaccine effectiveness reported in the UK [27–29]. A detailed\ndescription of this procedure is reported in the Methods.\n\n2021LATEX template\n4 Ranking the Effectiveness of Non-Pharmaceutical Interventions\nSusceptible\nSy\nExposed\n(Asymptomatic)\nEy\nInfected\n(Symptomatic)\nIy\nβyy, βsy, βcy εy\nRecovered\nRy\nQuarantined\nQy\nλy \nηy \nξy \nStudent Group\nρy\nSusceptible\nSs\nExposed\n(Asymptomatic)\nEs\nInfected\n(Symptomatic)\nIs\nβss, βys, βcs εs\nRecovered\nRs\nQuarantined\nQs\nλs \nηs \nξs \nStaff Group\nρs\nVirus \nEnvironment\nConcentration\nC\nβsyβys\nμy\nβcy\nμs\nβcs\nEnvironment \nDecaying rate \nδ\nFig. 1 Model structure. Flow chart of five epidemic stages among students and staff\nin a university department: S, susceptible (including the vaccinated); E, exposed (asymp-\ntomatic); I, infected (symptomatic); Q, quarantined (hospitalized or mandatorily isolated);\nR, recovered. The subscript “y” stands for students while the subscript “s” denotes staff.\nWe consider five possible non-pharmaceutical interventions that can be\nimplemented by the university after reopening: mask wearing, social dis-\ntancing, environmental disinfection, quarantine on infected students and\nquarantine of infected staff. To study the effectiveness of these measures in the\nmodel, we define five input control variables associated with each of these mea-\nsures. We then use optimal control to study four desired scenarios: minimum\nnumber of cases, minimum intervention, minimum non-quarantine interven-\ntion and minimum quarantine intervention. In all scenarios, it is assumed that\nthe University starts to control the epidemic 14 days after the asymptomatic\ncarriers firstly appear. We also model the desire of keeping the infection under\ncontrol as constraints in the optimisation. In particular, we impose constraints\non the number of infected cases and on the number of days required to extin-\nguish the epidemic. In the minimum-case scenario, the epidemic is controlled\nwith no efforts spared, leading to the strongest minimization of COVID-19\ncases. For minimum intervention, we study the possibility of minimizing the\ntotal effort of all control measures. In the minimum non-quarantine interven-\ntion, we minimise the use of mask wearing, social distancing and environmental\ndisinfection. In the last scenario, minimum quarantine intervention, we mini-\nmize the use of quarantines. By comparing the obtained optimal trajectories\n\n2021 LATEX template\nRanking the Effectiveness of Non-Pharmaceutical Interventions 5\nin different scenarios, we identify an emergent behaviour that shows a ranking\non the importance of the different non-pharmaceutical interventions.\n1 Results\n1.1 Baseline: no interventions\nWe first generate a baseline scenario in which no intervention is performed.\nWe simulate the evolution of the epidemic in one department of the university.\nThe results are shown in Fig. 2. As case study we consider a department where\nthe total number of students is 1200 and the number of members of staff is\n150 (i.e. similar to the EEE department of Imperial College London). Initially,\nwe assume 5 students and 2 staff members start as asymptomatic cases. The\neffective reproduction number R0 on day 0 is 1 .40, which indicates a poten-\ntial outbreak of COVID-19 in this small environment. After two weeks, 1.8%\nof students and 3 .1% of staff have caught the disease. Without countermea-\nsures, increasingly more individuals will get infected during the coming 120\ndays. Simultaneously, Rt keeps decreasing and becomes smaller than 1 after\n76 days. At day 134, 56% of students and 63% of staff have been infected and\nRt = 0.718. In the end (without considering further infections from outside\nthe department), the epidemic last around 250 days, rendering 59% of stu-\ndents and 66% of staff infected. This prediction result reveals the necessity of\nimposing non-pharmaceutical measures at the current UK vaccination levels\n(see Methods for the exact percentages).\n0 50 100 150 200 250\nTime (days)\n0\n0.2\n0.4\n0.6\n0.8\n1\nStudent Case Proportion\nTrajectories of Student Group\nSy(t)\nEy(t)\nIy(t)\nQy(t)\nRy(t)\n0 50 100 150 200 250\nTime (days)\n0\n0.2\n0.4\n0.6\n0.8\n1\nStaff Case Proportion\nTrajectories of Staff Group\nSs(t)\nEs(t)\nIs(t)\nQs(t)\nRs(t)\na\nb\nFig. 2 Prediction in the baseline scenario of no interventions. a Evolution of\nCOVID-19 among students. b Evolution of COVID-19 among staff. Magnitudes are in\nproportion to the total number of students or staff.\n\n2021LATEX template\n6 Ranking the Effectiveness of Non-Pharmaceutical Interventions\n1.2 Optimal interventions for four objectives\nWe now study the effect of non-pharmaceutical interventions. Which inter-\nventions to implement, when and how strongly are all decisions made by the\noptimisation method to optimise the objective. We have selected four different\noptimisation objectives. In all scenarios, interventions are introduced 14 days\nafter the initial exposure of 5 students and 2 staff members. In all scenarios, we\nimpose the constraints that the epidemic must end within 120 days and that\nat least 94% of students and staff are not infected. Thus, the overall timeline\nis 134 days.\nMinimum number of cases: In this scenario the optimisation objective\nis formulated to minimise infections, even though this may require that all\nnon-pharmaceutical interventions are implemented at full strength. The opti-\nmal trajectories are depicted in Fig. 3. The epidemic is completely ended at\naround the 60 th day when the individuals are only in two states: suscepti-\nble and recovered. More than 97% of students and 96% of staff do not get\ninfected in this scenario. From the figure we see that this result is achieved by\nimplementing all interventions unreservedly, from mask wearing to mandatory\nquarantine. This reduces Rt to around 0.217. While initially there is a strong\nneed for all countermeasures, after 30 days the optimal strategy relies mostly\non distancing and masks. All cases have been quarantined by the 45 th day.\nAfter approximately 60 days, the department reaches a steady state and the\ninfection is stopped. Consequently, at this point the optimal strategy eases the\ninterventions because the epidemic has been successfully contained.\n0 20 40 60 80 100 120\nTime (days)\n0\n0.2\n0.4\n0.6\n0.8\n1\nStudent Case Proportion\nTrajectories of Student Group\nSy(t)\nEy(t)\nIy(t)\nQy(t)\nRy(t)\n0 20 40 60 80 100 120\nTime (days)\n0\n0.2\n0.4\n0.6\n0.8\n1\nStaff Case Proportion\nTrajectories of Staff Group\nSs(t)\nEs(t)\nIs(t)\nQs(t)\nRs(t)\n0 20 40 60 80 100 120\nTime (days)\n0\n0.2\n0.4\n0.6\n0.8\n1\nControl Variable Values\nm, d and e\nm(t)\nd(t)\ne(t)\n0 20 40 60 80 100 120\nTime (days)\n0\n0.2\n0.4\n0.6\n0.8\n1\nControl Variable Values\nqy and qs\nqy(t)\nqs(t)\na\nc\nb\nd\nFig. 3 Optimal trajectories for the minimum-case scenario. Optimal trajectories\nwhen the department spares no effort to contain the epidemic. a,b, The epidemic evolution\namong students and staff, respectively. c, The optimal strategies for mask wearing ( κm),\nsocial distancing (κd), and environmental disinfection ( κe). d, The optimal strategies for\nmandatory quarantine on infected students (κqy) and infected staff (κqs).\n\n2021 LATEX template\nRanking the Effectiveness of Non-Pharmaceutical Interventions 7\nMinimum intervention: In this case the objective function is the norm\nof all control variables. As a consequence, the aim here is to minimise the use\nof all interventions (including quarantine) while still satisfying the constraint\nthat at least 94% of the population is not infected. The resulting optimal\ntrajectories are shown in Fig. 4. The epidemic is ended with 4 .3% of students\nand 6% of staff having been infected. With respect to before we can see that\nthere is a decrease in the strength of the interventions. We can also note that\nthere is an emerging ranking between the interventions. Fig. 4c shows that\nthe environmental disinfection is far less important than mask wearing. For\nwhat concerns mandatory quarantine shown in Fig. 4d, we notice again that\nisolation of infected students plays a more significant role in controlling the\nspread of COVID-19 than the isolation of staff. All five control interventions\nare strongest at the beginning of the epidemic, and then their magnitude is\nattenuated gradually over time.\n0 20 40 60 80 100 120\nTime (days)\n0\n0.2\n0.4\n0.6\n0.8\n1\nStudent Case Proportion\nTrajectories of Student Group\nSy(t)\nEy(t)\nIy(t)\nQy(t)\nRy(t)\n0 20 40 60 80 100 120\nTime (days)\n0\n0.2\n0.4\n0.6\n0.8\n1\nStaff Case Proportion\nTrajectories of Staff Group\nSs(t)\nEs(t)\nIs(t)\nQs(t)\nRs(t)\n0 20 40 60 80 100 120\nTime (days)\n0\n0.2\n0.4\n0.6\n0.8\n1\nControl Variable Values\nm, d and e\nm(t)\nd(t)\ne(t)\n0 20 40 60 80 100 120\nTime (days)\n0\n0.2\n0.4\n0.6\n0.8\n1\nControl Variable Values\nqy and qs\nqy(t)\nqs(t)\na\nc\nb\nd\nFig. 4 Optimal trajectories for the minimum intervention scenario. Optimal tra-\njectories when the department would like to minimise the enforcement of control measures.\na,b, The epidemic evolution among students and staff, respectively. c, The optimal strate-\ngies for mask wearing (κ m), social distancing (κd), and environmental disinfection (κe). d,\nThe optimal strategies for mandatory quarantine on infected students ( κqy) and infected\nstaff (κqs).\nMinimum use of non-quarantine interventions: In this case we want\nto minimize the use of masks, social distancing and environmental disinfec-\ntion. As a result we expect an increase of the use of quarantines. The resulting\noptimal trajectories are shown in Fig. 5. As expected the figures show little\nuse of non-quarantine interventions, and a strong use of quarantines. We stress\nthat the primary objective of keeping 94% of the susceptible population infec-\ntion free is maintained. Again, Fig. 5d demonstrates the higher importance of\n\n2021LATEX template\n8 Ranking the Effectiveness of Non-Pharmaceutical Interventions\nquarantine among students with respect to staff. This shows the predominant\nrole played by student quarantine in controlling the epidemic.\n0 20 40 60 80 100 120\nTime (days)\n0\n0.2\n0.4\n0.6\n0.8\n1\nStudent Case Proportion\nTrajectories of Student Group\nSy(t)\nEy(t)\nIy(t)\nQy(t)\nRy(t)\n0 20 40 60 80 100 120\nTime (days)\n0\n0.2\n0.4\n0.6\n0.8\n1\nControl Variable Values\nm, d and e\nm(t)\nd(t)\ne(t)\n0 20 40 60 80 100 120\nTime (days)\n0\n0.2\n0.4\n0.6\n0.8\n1\nStaff Case Proportion\nTrajectories of Staff Group\nSs(t)\nEs(t)\nIs(t)\nQs(t)\nRs(t)\n0 20 40 60 80 100 120\nTime (days)\n0\n0.2\n0.4\n0.6\n0.8\n1\nControl Variable Values\nqy and qs\nqy(t)\nqs(t)\na\nc\nb\nd\nFig. 5 Optimal trajectories for minimum use of non-quarantine interventions.\nOptimal trajectories when the departments would like to minimise use of masks, social\ndistancing and environmental disinfection. a,b, The epidemic evolution among students and\nstaff, respectively. c, The optimal strategies for mask wearing ( κm), social distancing (κ d),\nand environmental disinfection (κe). d, The optimal strategies for mandatory quarantine on\ninfected students (κqy) and infected staff (κqs).\nMinimum quarantine: In this scenario we want to minimise the use of\nquarantine, but we allow a strong use of mask wearing, social distancing and\nenvironmental disinfection. The resulting optimal trajectories are shown in\nFig. 6. As a result, the quarantine control variables are zero and the other\nthree interventions are major tools to resolve the epidemic in this scenario.\nFig. 6c clearly shows the primary role of mask wearing and social distancing in\nkeeping the infection under control. The figure also shows that environmental\ndisinfection in comparison play little role when strong mask wearing and social\ndistancing are in place.\n2 Discussion\nThe figures show an emerging behaviour: non-pharmaceutical interventions\nhave different importance and this importance arises mathematically from\nthe evolution of the epidemic. In a typical university department composed\nof 1200 students and 150 staff, with a vaccination rate of 68% for students\nand 78.8% for staff (see Methods, [27]) we see that the implementation of\nnon-pharmaceutical interventions is still fundamental to reduce the number of\n\n2021 LATEX template\nRanking the Effectiveness of Non-Pharmaceutical Interventions 9\n0 20 40 60 80 100 120\nTime (days)\n0\n0.2\n0.4\n0.6\n0.8\n1\nStudent Case Proportion\nTrajectories of Student Group\nSy(t)\nEy(t)\nIy(t)\nQy(t)\nRy(t)\n0 20 40 60 80 100 120\nTime (days)\n0\n0.2\n0.4\n0.6\n0.8\n1\nControl Variable Values\nm, d and e\nm(t)\nd(t)\ne(t)\n0 20 40 60 80 100 120\nTime (days)\n0\n0.2\n0.4\n0.6\n0.8\n1\nStaff Case Proportion\nTrajectories of Staff Group\nSs(t)\nEs(t)\nIs(t)\nQs(t)\nRs(t)\n0 20 40 60 80 100 120\nTime (days)\n0\n0.2\n0.4\n0.6\n0.8\n1\nControl Variable Values\nqy and qs\nqy(t)\nqs(t)\na\nc\nb\nd\nFig. 6 Optimal trajectories for the minimum quarantine scenario. Optimal tra-\njectories when the department would like to minimise the enforcement of mandatory\nquarantines. a,b, The epidemic evolution among students and staff, respectively. c, The\noptimal strategies for mask wearing (κ m), social distancing (κd), and environmental disin-\nfection (κe). d, The optimal strategies for mandatory quarantine on infected students (κ qy)\nand infected staff (κqs).\ninfections to one tenth of the number of infections appearing in a completely\nuncontrolled scenario.\nThe ranking that arises from the study is as follows: wearing masks is the\nmost effective measure among the considered interventions. Keeping social dis-\ntance is ranked close second. This priority of mask wearing is reasonable in a\nuniversity, where close contact is often unavoidable. Furthermore, we can also\nsee that environmental disinfection seems to be far less necessary if both mea-\nsures are strongly enforced. As for the enforcement of mandatory quarantines,\nthe result yields that quarantine of symptomatic students is more significant\nthan quarantine of staff. This ranking is robust with respect to model param-\neters. This is shown in the sensitivity study presented in Figs. A1 to A8 in\nAppendix A.\nFrom a practical perspective, the university should emphasize mask wearing\nand social distancing when on-campus teaching is resumed, especially among\nstudents. The study also suggests that the university should invest particular\neffort in identifying and quarantining infected students.\nThe proposed model and optimal control framework can be easily used to\nassess other scenarios, e.g. other objectives or other constraints. This tool can\nassist universities in predicting and managing the evolution of the epidemic.\n\n2021LATEX template\n10 Ranking the Effectiveness of Non-Pharmaceutical Interventions\nMoreover, our findings demonstrably reflect the importance of different non-\npharmaceutical interventions and help to tackle the trade-off between high-\nquality teaching and limiting COVID-19 infections. This is crucial at a time\nin which universities are under pressure to increase on-campus activities.\nOur study also shows a gap in the epidemiological research of COVID-\n19 regarding the evolution of the pandemic in small environments. There are\nplenty of small environments, such as schools and companies that have their\nvery unique population structure ( i.e. populations with different degrees of\ninteraction, vaccination rates and serious symptoms) and require design tools\nto assess the best interventions to be implemented in order to maximise in-\nperson activities while keeping the infections under control.\nWe also point out that some factors are not explicitly considered in our\nstudy. Firstly, the model does not consider the infections brought from out-\nside the campus. We omitted this aspect because we wanted to focus on the\nstudy of the priority of different mitigation measures. Another limitation is\nthat the input variables (the interventions) in our optimal control problem are\ncontinuous in magnitude. It may be difficult to give practical significance to\nthe numerical values representing the interventions. However, we stress that\nthe optimal trajectories are used here only to compare the relative impor-\ntance of different interventions. Further research can be done on discretizing\nthe magnitude of the input variables into specific levels that correspond to\nscientifically-defined interpretable practical meanings.\n\n2021 LATEX template\nRanking the Effectiveness of Non-Pharmaceutical Interventions 11\n3 Methods\n3.1 Uncontrolled University Model\nThe proposed double SEIQR model is described by 11 differential equations,\n5 associated to the epidemic evolution of students, 5 associated to the epi-\ndemic evolution of staff, and 1 representing environmental infection. We\nfirst introduce, describe and analyse the uncontrolled model. In Section 3.3\nwe modify the model by introducing the control variables that represent\nnon-pharmaceutical interventions. The double uncontrolled SEIQR model is\ndescribed by\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\ndSy(t)\ndt =−{βy[Ey(t) +kIy(t) +Es(t) +kIs(t)] +βcyC(t)}Sy(t)\ndSs(t)\ndt =−{βs [Es(t) +kIs(t) +Ey(t) +kIy(t)] +βcsC(t)}Ss(t)\ndEy(t)\ndt ={βy[Ey(t) +kIy(t) +Es(t) +kIs(t)] +βcyC(t)}Sy(t)\n− (εy +ξy)Ey(t)\ndEs(t)\ndt ={βs [Es(t) +kIs(t) +Ey(t) +kIy(t)] +βcsC(t)}Ss(t)\n− (εs +ξs)Es(t)\ndIy(t)\ndt =εyEy(t)− (ηy +ρy)Iy(t)\ndIs(t)\ndt =εsEs(t)− (ηs +ρs)Is(t)\ndQy(t)\ndt =ηyIy(t)−ϕyQy(t)\ndQs(t)\ndt =ηsIs(t)−ϕsQs(t)\ndRy(t)\ndt =ξyEy(t) +ρyIy +ϕyQy(t)\ndRs(t)\ndt =ξsEs(t) +ρsIs +ϕsQs(t)\ndC(t)\ndt =µy(Ey(t) +kIi(t)) +µs(Es(t) +kIs(t))−δC(t)\n(1)\nwhere all model parameters are denoted by Greek letters with specific biolog-\nical meanings. We now provide a detailed explanation of each parameter. We\nstress that this model is uncontrolled, so the values discussed below are for the\nbaseline scenario, i.e. no intervention is implemented. Also, the parameters are\nfirstly introduced for unvaccinated population and then modified according to\nthe UK vaccination proportions.\n\n2021LATEX template\n12 Ranking the Effectiveness of Non-Pharmaceutical Interventions\n• Individual infection rates βy, βs kβy, kβs\nThe individual infection rates represent the average number of susceptible\nindividuals who can be infected by a virus carrier via direct contacts in unit\ntime.βy indicates student-to-student and staff-to-student infection rates.βs\nindicates staff-to-staff and student-to-staff infection rates. In other words,\nwe assume that student-to-student and staff-to-student rates are the same\nand staff-to-staff and student-to-staff rates are the same. βs and βy are the\ninfection rates of the asymptomatic compartments. Considering the age dif-\nference, it is reasonable to expect that βy <β s because staff are more likely\nto get infected.kβs andkβy are the infection rates of the symptomatic group.\nSince sneezing and coughing play a major role in the direct transmission of\nthe virus, the symptomatic carriers generally have larger infection rates, i.e.\nk > 1. Here it is assumed that k has the same value among students and\nstaffs. Referring to the scenario 3 of pandemic planning produced by the\nCDC [30], k is generally 4. However, in this confined environment case, k\nshould be smaller because asymptomatic subjects can spread the virus more\neasily. We selected a value of k = 1.5. According to the study conducted\nby Leontitsis et al. [22], the general infection rate β is 0.1466. This value\nis expected to be larger in a confined space because of the higher number\nof direct contacts between people. In summary, putting together all these\ndata and observations, the parameters have been selected as βy = 0.163,\nβs = 0.225, kβy = 0.2445 and kβs = 0.3375.\n• Environmental infection rates βcy, βcs\nThese parameters represent how many susceptible people are infected by the\ncontaminated environment in unit time. They are properties of the virus in\nthe environment. There is no clear value in the literature and we estimate the\nvalue ofβcy to be 0.171 (based on the expected basic reproduction number).\nMoreover, it is reasonable to expect that the ratio βs/βy equals the ratio\np =βcs/βcy. Then βcs =pβcy = 0.236.\n• Probability of becoming symptomatic εy, εs\nThey are the inverse of the average incubation period. According to [31],\nthis average period is 5 days. Considering that staff are of higher age, we\nset εy = 1/5 = 0.2 and εs = 1/10 = 0.1.\n• Probability of recovery from an asymptomatic state ξy, ξs\nSimilarly, the inverses of these quantities denote the average number of days\nspent by exposed/asymptomatic subjects to recover ( i.e. they present no\nsymptom during the whole period). Referring to [30], this portion accounts\nfor 15%. Since εy = 0.2 and εs = 0.1 (corresponding to 85% of the\npopulation). Then ξy = 0.0353 and ξs = 0.0176.\n• Isolation rates ηy, ηs\nThese parameters denote the proportion of symptomatic individuals who are\nisolated due to serious illness or mandatory quarantine. Since we initially\nmodel the unrestricted situation (i.e. no quarantines), infected individuals\nare at this point isolated mainly due to hospitalization. According to the\n\n2021 LATEX template\nRanking the Effectiveness of Non-Pharmaceutical Interventions 13\nsurvey conducted and reported by [32], the values are ηy = 0.06 and ηs =\n0.106.\n• Recovery rates of infected individuals ρy, ρs\nρ−1\ny and ρ−1\ns indicate the average time for infected people which are not\nisolated to recover. If mandatory quarantine is not implemented, mild cases\nwill not isolate. These two parameters mainly describe the recovery rate of\nthis group. According to [33], the average length of recovery is approximately\n14 days. In mild cases, we set this length at 10 days for students, and 18\ndays for staff. Thus, ρy = 1/10 and ρs = 1/18.\n• Recovery rates of quarantined individuals ϕy, ϕs\nϕ−1\ny and ϕ−1\ns indicate the average time for “quarantined” individuals to\nrecover. In the baseline scenario this refers to hospitalised individuals. It\nmay take six-nine weeks for severe cases to recover [34]. We set ϕy = 1/40\nwhile ϕs = 1/55.\n• Virus shedding rates to the environment µy, µs, kµy, kµs\nThese parameters measure the spread of the virus from asymptomat-\nic/symptomatic individuals to the environment, with the effects brought by\nsymptomatic subjects being higher. Similarly to the case of βcy and βcs,\nthere is no clear value in the literature for these parameters. In this “small\nenvironment” model we expect the values to be µy =µs =µ = 0.25 (based\non the expected basic reproduction number).\n• Virus decaying rate in the environment δ\nThis measures the speed of decay of the virus in the small environment.\nSince the airborne virus could stay in aerosol for up to 1 day and survive on\nthe surface for longer [35–37], this rate δ is set at 0.7.\nWe denoteNt to be the total population in the department. The total number\nof students is represented by Ny and number of staff is labelled by Ns. A\nsummary of the meaning the parameters is given in Table 1.\nThese initial values refer to studies on the COVID-19 epidemic before vac-\ncination. We now describe how the parameters are adapted to a vaccinated\npopulation. According to the UK data provided by [27], about 68% of young\nadults between 18 and 24 years old have been vaccinated by the 1 st Novem-\nber 2021. This quantity become 78.7% among people aged between 25 and 64.\nSince vaccines utilized in the UK can reduce COVID-19 infections by around\n65% [28, 29], we can see that the 68% vaccinated students will have 65% less\nprobability of getting infected. The same happens to the 78.7% of staffs. There-\nfore, the average reductions in βy andβcy are 0.68× (1− 0.65) + 0.32 = 0.558.\nThe average reductions inβs andβcs are 0.787×(1−0.65)+0.213 = 0.488. Con-\nsequently, due to vaccinations in the current situation, infection rates in this\nuniversity model with mixed vaccinated/unvaccinated population becomes:\nβy = 0.0910, βcy = 0.0954, βs = 0.1098, and βcs = 0.1152. Since vaccines\ncan also reduce the probability of symptomatic infections and of severe ill-\nness [38], other parameter values are also tuned accordingly. This adjustment\nchanged the R0 from 2.50 (totally unvaccinated and no interventions) to 1.40\n\n2021LATEX template\n14 Ranking the Effectiveness of Non-Pharmaceutical Interventions\nT able 1 Model Parameter Definitions.\nParameters Definitions\nβy, βs Infection rate for asymptomatic students/staff to susceptible\nstudents/staff (including cross infections)\nkβy, kβs Infection rate for symptomatic students/staff to susceptible\nstudents/staff (including cross infections)\nβcy, βcs Infection rate from uncleaned environment to susceptible stu-\ndents/staff\nεy, εs Probability that an asymptomatic student/staff becomes symp-\ntomatic\nξy, ξs Probability that an asymptomatic student/staff recovers without\nsymptoms\nηy, ηs Isolation rate of symptomatic student/staffs\nρy, ρs Recovery rate of infected student/staffs\nϕy, ϕs Recovery rate of quarantined students/staffs\nµy, µs Environmental shedding rate by asymptomatic students/staffs\nkµy, kµs Environmental shedding rate by symptomatic students/staffs\nδ Decaying rate of virus in the environment\n(mixed population but still no intervention). Since it is still greater than 1,\nthe COVID-19 epidemic will still develop in the university model if no other\ncontrol or prevention measures are introduced. The values of the parameters\nof the model before and after vaccinations are listed in Table 2.\nT able 2 Model Parameter Values.\nParameters Before Vaccination After Vaccination\nβy 0.163 0.0910\nβs 0.225 0.1098\nk 1.5 1.5\nβcy 0.171 0.0954\nβcs 0.236 0.1152\nεy 0.2 0.2\nεs 0.1 0.1\nξy 0.0353 0.0857\nξs 0.0176 0.0429\nηy 0.06 0.012\nηs 0.106 0.0212\nρy 0.1 0.125\nρs 0.0556 0.0833\nϕy 0.025 0.0714\nϕs 0.0182 0.0714\nµy 0.25 0.25\nµs 0.25 0.25\nδ 0.7 0.7\n\n2021 LATEX template\nRanking the Effectiveness of Non-Pharmaceutical Interventions 15\n3.2 Analysis of the Uncontrolled University Model\n3.2.1 Equilibrium Points\nDenote x = [Sy, Ss, Ey, Es, Iy, Is, Qy, Qs, Ry, Rs, C]⊤. By equat-\ning all derivatives in (1) to zero, it is easy to determine the equilibria ¯ x =\n[ ¯Sy, ¯Ss, 0, 0, 0, 0, 0, 0, ¯Ry, ¯Rs, 0], where\n¯Sy + ¯Ss + ¯Ry + ¯Rs = 1, ¯Sy≥ 0, ¯Ss≥ 0, ¯Ry≥ 0, ¯Rs≥ 0 (2)\nThese equilibria imply that at the end of pandemic, the individuals are either\nsusceptible to or recovered from the disease.\n3.2.2 Basic Reproduction Number\nThe basic reproduction number is a crucial criterion to measure the average\nnumber of susceptible people that could potentially be infected by a primary\ncase [39]. This parameter is highly dependent on the fraction of the susceptible\npopulation and it provides information about the potential of the epidemic\noutbreak. IfR0 < 1 the disease will gradually disappear. IfR0 > 1 increasingly\nmore people will be infected.\nDerivation of the basic reproduction number for the uncontrolled university\nmodel is based on the next generation matrix method described by [40–42].\nThe university model (1) has five infectious compartments: Ey,Es,Iy,Is and\nC. We collect these in the infection state xif = [Ey,Es,Iy,Is,C ]⊤. Let F\ndenote the rate of increase of secondary cases and V denote the progression\nrate. Accordingly,xif obeys the equation\n˙xif =F−V (3)\nwhereF andV given by\nF =\n\n\n\n\nSy (Cβcy +βy (Es +Ey +Isk +Iyk))\nSs (Cβcs +βs (Es +Ey +Isk +Iyk))\n0\n0\nµs (Es +Isk) +µy (Ey +Iyk)\n\n\n\n\n(4)\nand\nV =\n\n\n\n\nEy (εy +ξy)\nEs (εs +ξs)\nIy (ηy +ρy)−Eyεy\nIs (ηs +ρs)−Esεs\nCδ\n\n\n\n\n. (5)\nWe linearise equation (3) around the equilibrium and we denote the Jacobians\n\n2021LATEX template\n16 Ranking the Effectiveness of Non-Pharmaceutical Interventions\nofF andV as F and V , respectively. Thus, we obtain\n˙xif = (F−V )xif (6)\nwhere\nF =\n\n\n\n\n¯Syβy ¯Syβy ¯Syβyk ¯Syβyk ¯Syβcy\n¯Ssβyp ¯Ssβyp ¯Ssβykp ¯Ssβykp ¯Ssβcyp\n0 0 0 0 0\n0 0 0 0 0\nµy µs kµy kµs 0\n\n\n\n\n(7)\nand\nV =\n\n\n\nεy +ξy 0 0 0 0\n0 εs +ξs 0 0 0\n−εy 0 ηy +ρy 0 0\n0 −εs 0 ηs +ρs 0\n0 0 0 0 δ\n\n\n\n\n. (8)\nDenoting σy = ηy +ρy and σs = ηs +ρs, the next generation matrix K is\ntherefore derived as\nK =FV−1\n=\n\n\n\n\n\n\n\n\n\n\n¯Syβy (σy +εyk)\nσy (εy +ξy)\n¯Syβy (σs +εsk)\nσs (εs +ξs)\n¯Syβyk\nσy\n¯Syβyk\nσs\n¯Syβcy\nδ\n¯Ssβyp (σy +εyk)\nσy (εy +ξy)\n¯Ssβyp (σs +εsk)\nσs (εs +ξs)\n¯Ssβykp\nσy\n¯Ssβykp\nσs\n¯Ssβcyp\nδ\n0 0 0 0 0\n0 0 0 0 0\nµy (σy +εyk)\nσy (εy +ξy)\nµs (σs +εsk)\nσs (εs +ξs)\nkµy\nσy\nkµs\nσs\n0\n\n\n\n\n\n\n\n\n\n\n.\n(9)\nThe obtained matrix K is nonnegative and has rank 2. In particular, it has\nthree zero eigenvalues and two positive eigenvalues. According to [40], R0 is\nthe spectral radius ofK, i.e. its largest eigenvalue. By computing det(λI−K),\nthe characteristic polynomial is\npk(λ) =λ3\n[\nλ2−\n( ¯Syβy (σy +εyk)\nσy (εy +ξy) +\n¯Ssβyp (σs +εsk)\nσs (εs +ξs)\n)\nλ\n− 1\nδ\n( ¯Syβcyµy(σy +εyk)\nσy (εy +ξy) +\n¯Ssβcypµs(σs +εsk)\nσs (εs +ξs)\n)]\n.\n(10)\nWe recall that we have assumed thatµy =µs =µ, which means that students\nand staff have equal rates of spreading the virus into the environment. The\n\n2021 LATEX template\nRanking the Effectiveness of Non-Pharmaceutical Interventions 17\ntwo non-zero eigenvalues can be derived by finding the roots of the polynomial\n˜pk(λ) =λ2−βy\n( ¯Sy (+εyk)\nσy (εy +ξy) +\n¯Ssp (σs +εsk)\nσs (εs +ξs)\n)\nλ\n− µβcy\nδ\n( ¯Sy (σy +εyk)\nσy (εy +ξy) +\n¯Ssp (σs +εsk)\nσs (εs +ξs)\n)\n.\n(11)\nDenote\na =βy\n( ¯Sy (σy +εyk)\nσy (εy +ξy) +\n¯Ssp (σs +εsk)\nσs (εs +ξs)\n)\n, c = µβcy\nδβyy\n. (12)\nThen\n˜pk(λ) =λ2−aλ−ac (13)\nSince the constants a and c are both positive, the quadratic equation has two\nreal roots and R0 will be the larger one. Therefore, we can express R0 as\nR0 = a(1 +\n√\n1 + 4c/a)\n2 . (14)\nIn the next section we show how R0 is related to the stability of the\nequilibrium point.\n3.2.3 Stability Analysis\nProposition 1 If R0 < 1, the equilibrium points ¯x of the uncontrolled university\nmodel (1) is asymptotically stable.\nProof Model (1) can be reformulated into a feedback interconnection. Compartments\nEy, Es, Iy, Is, Qy, Qs, C form a positive linear subsystem with output feedback\ntopology. Defining xl = [Ey,Es,Iy,Is,Qy,Qs,C ]⊤, ys = [Sy,Ss]⊤, yR = [Ry,Rs]⊤,\nthe subsystem can be formulated as\n˙xl(t) =Axl(t) +Bu(t)\n=\n\n\n\n\n\n\n\n\n−ry 0 0 0 0 0 0\n0 −rs 0 0 0 0 0\nεy 0 −σy 0 0 0 0\n0 εs 0 −σs 0 0 0\n0 0 ηy 0 −ϕy 0 0\n0 0 0 ηs 0 −ϕs 0\nµy µs kµy kµs 0 0 −δ\n\n\n\n\n\n\nxl(t) +\n\n\n\n\n\n\n1 0\n0 1\n0 0\n0 0\n0 0\n0 0\n0 0\n\n\n\n\n\n\nu(t)\n(15)\nys(t) =Csxl(t) =\n[βy βy βyk βyk 0 0 βcy\nβs βs βsk βsk 0 0 βcs\n]\nxl(t) (16)\n\n2021LATEX template\n18 Ranking the Effectiveness of Non-Pharmaceutical Interventions\nyR(t) =CRxl(t) =\n[ξy 0 ρy 0 ϕy 0 0\n0 ξs 0 ρs 0 ϕs 0\n]\nxl(t) (17)\nwhere ry = εy +ηy, rs = εs +ηs. Since output yR does not contribute to the\nvariations of the state variable xl, we can represent the overall system dynamics by\nthe output feedback loop as\nu(t) =Ks(t)ys(t) =\n[Sy(t) 0\n0 Ss(t)\n]\nys(t). (18)\nDerivatives of the remaining compartments can be calculated as[ ˙Sy(t)\n˙Ss(t)\n]\n= −\n[Sy(t) 0\n0 Ss(t)\n]\nys(t) (19)\n[ ˙Ry(t)\n˙Rs(t)\n]\n=yR(t). (20)\nNote that the system has a time-varying feedback Ks(t). Around the equilibrium ¯x,\nthe system behaviour is determined by using the constant feedback term ¯Ks as\n¯Ks =\n[ ¯Sy 0\n0 ¯Ss\n]\nys(t). (21)\nTherefore, the dynamics of the original model is equivalent to that of this closed-loop\nsystem. To study its stability, we firstly derive the closed-loop system matrix Acl as\nAcl =A +BKsCs\n=\n\n\n\n\n\n\n¯Syβy −ry ¯Syβy ¯Syβyk ¯Syβyk 0 0 ¯Syβcy\n¯Ssβyp ¯Ssβyp −rs ¯Ssβykp ¯Ssβykp 0 0 ¯Ssβcyp\nεy 0 −σy 0 0 0 0\n0 εs 0 −σs 0 0 0\n0 0 ηy 0 −ϕy 0 0\n0 0 0 ηs 0 −ϕs 0\nµy µs kµy kµs 0 0 −δ\n\n\n\n\n\n\n.\n(22)\nTo determine its closed-loop poles, we compute its characteristic equation, i.e.\ndet(λI −Acl). Since all parameters are positive,Acl must have two negative eigenval-\nues at −ϕy and −ϕs. The remaining five eigenvalues are the roots of the polynomial\np5(λ):\np5(λ) =\n\n\n\n\n\n\n\nλ +ry − ¯Syβy − ¯Syβy − ¯Syβyk − ¯Syβyk − ¯Syβcy\n−¯p (λ +ry) λ +rs 0 0 0\n−εy 0 λ +σy 0 0\n0 −εs 0 λ +σs 0\n−µy −µs −kµy −kµs δ +λ\n\n\n\n\n\n\n\n=\n\n\n\n\n\n\n\nλ +ry − ¯Syβy −λ −ry 0 0 − ¯Syβcy\n−¯p (λ +ry) λ +rs + ¯p (λ +ry) −k (λ +rs) 0 0\n−εy εy λ +σy −λ −σy 0\n0 −εs εsk λ +σs 0\n−µ 0 0 0 δ +λ\n\n\n\n\n\n\n\n\n(23)\nwhere we have defined ¯p =\n¯Ssp\n¯Sy\nand used thatµy =µs =µ according to the previous\nanalysis. The polynomial is finally derived as\np5(λ) =λ5 +α4λ4 +α3λ3 + +α2λ2 +α1λ +α0, (24)\n\n2021 LATEX template\nRanking the Effectiveness of Non-Pharmaceutical Interventions 19\nwhere\nα4 =δ +rs +ry +σs +σy − ¯Syβy − ¯Ssβyp\nα3 =δrs +δry +δσs +δσy +rsry +rsσs +rsσy +ryσs +ryσy +σsσy\n− ¯Syβyδ − ¯Syβcyµ − ¯Syβyrs − ¯Syβyσs − ¯Syβyσy − ¯Syβyεyk\n− ¯Ssβyδp − ¯Ssβcyµp − ¯Ssβypry − ¯Ssβypσs − ¯Ssβypσy\n− ¯Ssβyεskp\nα2 =δrsry +δrsσs +δrsσy +δryσs +δryσy +δσsσy +rsryσs +rsryσy\n+rsσsσy +ryσsσy − ¯Syβyδrs − ¯Syβyδσs − ¯Syβyδσy − ¯Syβcyµrs\n− ¯Syβcyµσs − ¯Syβcyµσy − ¯Syβyrsσs − ¯Syβyrsσy − ¯Syβyσsσy\n− ¯Syβyδεyk − ¯Syβcyεykµ − ¯Syβyεykrs − ¯Syβyεykσs\n− ¯Ssβyδpr y − ¯Ssβyδpσ s − ¯Ssβyδpσ y − ¯Ssβcyµpr y − ¯Ssβcyµpσ s\n− ¯Ssβcyµpσ y − ¯Ssβypryσs − ¯Ssβypryσy − ¯Ssβypσsσy\n− ¯Ssβyδεskp − ¯Ssβcyεskµp − ¯Ssβyεskpr y − ¯Ssβyεskpσ y\nα1 =δrsryσs +δrsryσy +δrsσsσy +δryσsσy +rsryσsσy − ¯Syβyδrsσs\n− ¯Syβyδrsσy − ¯Syβyδσsσy − ¯Syβcyµrsσs − ¯Syβcyµrsσy\n− ¯Syβcyµσsσy − ¯Syβyrsσsσy − ¯Syβyδεykrs − ¯Syβyδεykσs\n− ¯Syβcyεykµr s − ¯Syβcyεykµσ s − ¯Syβyεykrsσs − ¯Ssβyδpr yσs\n− ¯Ssβyδpr yσy − ¯Ssβyδpσ sσy − ¯Ssβcyµpr yσs − ¯Ssβcyµpr yσy\n− ¯Ssβcyµpσ sσy − ¯Ssβypryσsσy − ¯Ssβyδεskpr y − ¯Ssβyδεskpσ y\n− ¯Ssβcyεskµpr y − ¯Ssβcyεskµpσ y − ¯Ssβyεskpr yσy\nα0 =δrsryσsσy − ¯Syβyδrsσsσy − ¯Syβcyµrsσsσy − ¯Syβyδεykrsσs\n− ¯Syβcyεykµr sσs − ¯Ssβyδpr yσsσy − ¯Ssβcyµpr yσsσy\n− ¯Ssβyδεskpr yσy − ¯Ssβcyεskµpr yσy\n(25)\nThe conditions to obtain a stable equilibrium point can be determined using the\nRouth-Hurwitz stability criterion. The Routh table is given in Table 3.\nT able 3 Routh Table.\nλ5 1 α3 α1\nλ4 α4 α2 α0\nλ3 b31 = − 1\na4\n(α2 −α4α3) b32 = − 1\na4\n(α0 −α1α4) 0\nλ2 b21 = − 1\nb31\n(α4b32 −α2b31) α0 0\nλ1 b11 = − 1\nb21\n(α0b31 −b32b21) 0 0\nλ0 α0 0 0\nFrom the Routh-Hurwitz stability criterion follows that the equilibrium point is\nasymptotically stable if and only if α4 > 0, b31 > 0, b21 > 0, b11 > 0 and α0 > 0.\nWe evaluate these coefficients numerically for different values of R0. The results are\nshown in Table 4 for R0 < 1 and in Table 5 for R0 > 1. Since α0 > 0 in Table 4 and\n\n2021LATEX template\n20 Ranking the Effectiveness of Non-Pharmaceutical Interventions\nα0 < 0 in Table 5, the Routh-Hurwitz stability criterion confirms that the R0 found\nin (14) is consistent with the expected epidemic dynamics.\nT able 4 Values of the first column of the Routh table when R0 < 1.\nR0 0.9957 0.9966 0.9974 0.9983 0.9992\nα4 1.3087 1.3086 1.3085 1.3084 1.3083\nb31 0.4609 0.4608 0.4607 0.4606 0.4605\nb21 0.0648 0.0648 0.0647 0.0647 0.0646\nb11 0.0036 0.0036 0.0036 0.0036 0.0036\nα0 2.2031e-06 1.7561e-06 1.3091e-06 8.6205e-07 4.1504e-07\nT able 5 Values of the first column of the Routh table when R0 > 1.\nR0 1.0001 1.0009 1.0018 1.0027 1.0036\nα4 1.3082 1.3081 1.3079 1.3078 1.3077\nb31 0.4604 0.4603 0.4602 0.4601 0.4600\nb21 0.0646 0.0646 0.0646 0.0645 0.0645\nb11 0.0036 0.0036 0.0036 0.0036 0.0036\nα0 -3.1972e-08 -4.7898e-07 -9.2600e-07 -1.3730e-06 -1.8200e-06\n□\n3.3 Control of the University Model\n3.3.1 Formulation of the Controlled University Model\nIn this study we consider five non-pharmaceutical interventions that the\nuniversity can implement.\n1. Compulsory mask wearing: how strongly this measure is implemented is\nrepresented by the normalised variable 0≤κm≤ 1.\n2. Keep safe social distance: how strongly this measure is implemented is\nrepresented by the normalised variable 0≤κd≤ 1.\n3. Environment disinfection: how strongly this measure is implemented is\nrepresented by the normalised variable 0≤κe≤ 1.\n4/5. Mandatory quarantines: how strongly these measures are implemented is\nrepresented by the normalised variables 0 ≤ κqy ≤ 1 (for students) and\n0≤κqs≤ 1 (for staff).\nA group of five variables is initially defined to represent the reduction\nfactors in the infection rates, shedding rates, environmental decaying rate, and\nisolation rates. Denoting these factors as u = [up, um, ue, uqy, uqs]⊤, the\n\n2021 LATEX template\nRanking the Effectiveness of Non-Pharmaceutical Interventions 21\nuniversity model at this stage is expressed as\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\ndSy(t)\ndt =−{(1−up)βy[Ey(t) +kIy(t) +Es(t) +kIs(t)]\n− (1−um)βcyC(t)}Sy(t)\ndSs(t)\ndt =−{(1−up)pβy [Es(t) +kIs(t) +Ey(t) +kIy(t)]\n− (1−um)pβcyC(t)}Ss(t)\ndEy(t)\ndt ={(1−up)βy[Ey(t) +kIy(t) +Es(t) +kIs(t)]\n+ (1−um)βcyC(t)}Sy(t)− (εy +ξy)Ey(t)\ndEs(t)\ndt ={(1−up)pβy [Es(t) +kIs(t) +Ey(t) +kIy(t)]\n+ (1−um)pβcyC(t)}Ss(t)− (εs +ξs)Es(t)\ndIy(t)\ndt =εyEy(t)−uqyηyIy(t)\ndIs(t)\ndt =εsEs(t)−uqsηsIs(t)\ndQy(t)\ndt =uqyηyIy(t)−ϕyQy(t)\ndQs(t)\ndt =uqsηsIs(t)−ϕsQs(t)\ndRy(t)\ndt =ξyEy(t) +ϕyQy(t)\ndRs(t)\ndt =ξsEs(t) +ϕsQs(t)\ndC(t)\ndt = (1−um)µ (Ey(t) +kIi(t) +Es(t) +kIs(t))−ueδC (t)\n(26)\nNote that up, um, ue, uqyηy, and uqsηs should vary within [0, 1].\nSince a reduction factor can be influenced by multiple interventions, we\nneed to identify the relationship between the reduction factors ( u’s) and\nintervention variables (κ’s).\n1. Reduction of interpersonal infection rates βy and βs\nThe person-to-person infection rates are directly influenced by two measures:\nwearing masks and keeping social distance. Wearing masks could reduce\nthe probability of infections during contacts while social distancing could\nreduce the number of direct contacts between individuals. Study conducted\nby Karaivanov et al, [43] argued that the mandatory mask-wearing policy\nin confined spaces could reduce the number of infected cases by up to 40%\nweekly. Furthermore, Jarvis et al. [44] conducted a survey which showed\nthat the physical distancing could reduce the number of direct contacts by\n74%. However, since this survey might have selection and recall bias, the\n\n2021LATEX template\n22 Ranking the Effectiveness of Non-Pharmaceutical Interventions\nactual result should be lower than 74% to obtain a conservative estimate.\nIn this case, we set the maximum reduction at 65%. Hence, 1 −up = (1−\n0.4κm)(1− 0.65κm).\n2. Reduction of shedding rates and environment-to-person infection rates: µy,\nµs, βcy, βcs\nShedding rates as well as infections due to unclean environment can be\nreduced by the use of masks. According to laboratory-based investigations\nby [45, 46], masks could block approximately 50% to 70% droplets and\naerosol, greatly reducing the transmission of virus. In this case, we set the\nmaximum reduction of shedding rate to be 60% for a conservative estimate.\nTherefore, 1−um = 1− 0.6κm.\n3. Enhancement of virus environmental decaying rate δ\nThe environmental decaying rate could be magnified by disinfection of the\nconfined space. Recall that the uncontrolled decaying rate was δ = 0.7.\nIt is hard to determine how much this rate is increased. A conservative\nestimate is that the rate is increased at the maximum by 30%, yielding a\nmaximum new rateueδ = 0.91. The linear relationship can be finally defined\nas ue = 1 + 0.3κe.\nWhile the analysis above is derived from considerations extracted from the\nliterature, these still assumed an ideal enforcement of the interventions. Since\nthe university cannot guarantee full compliance, we limit κm, κd, and κe to\n70% of their values.\n4. Quarantine enhancement\nRecall that in model (1) the quarantined populations were simply equiva-\nlent to the populations who developed serious symptoms and their rate were\nηy and ηs. To consider the effects of mandatory quarantines we replace ηy\nandηs byuqyηy anduqsηs, respectively. (uqyηy)−1 and (uqsηs)−1 now indi-\ncate the average time that unisolated symptomatic students/staff stay in\ncampus before being detected by the university. We set these values to 2,\nmeaning that the university takes two days in average to detect and iso-\nlate infected individuals after their symptoms develop. Thus, the maximum\nvalues of (uqsηs)−1 and (uqsηs)−1 are both 0 .5. These values correspond\nto the situation where the enforcement of mandatory quarantines reaches\nthe strongest degree, which means that κqy andκqs are 1. On the contrary,\nwhen mandatory quarantines are not implemented, i.e. κqy = κqs = 0, we\nwant that the resulting quarantine rates uqyηy anduqsηs are stillηy andηs,\nrespectively. These relations are formulated as uqyηy = (0.5−ηy)κqy +ηy\nand uqsηs = (0.5−ηs)κqs +ηs.\n\n2021 LATEX template\nRanking the Effectiveness of Non-Pharmaceutical Interventions 23\nIn summary, the following relations hold\n1−up = (1− 0.4· 0.7κm)(1− 0.65· 0.7κd)\n1−um = 1− 0.6· 0.7κm\nue = 1 + 0.3· 0.7κe\nuqyηy = (0.5−ηy)κqy +ηy\nuqsηs = (0.5−ηs)κqs +ηs.\n(27)\nSubstituting these relationships into equation (26), we obtain the final\ncontrolled university model\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\ndSy(t)\ndt =−{(1− 0.28κm)(1− 0.455κd)βy[Ey(t) +kIy(t) +Es(t) +kIs(t)]\n− (1− 0.42κm)βcyC(t)}Sy(t)\ndSs(t)\ndt =−{(1− 0.28κm)(1− 0.455κd)pβy [Es(t) +kIs(t) +Ey(t) +kIy(t)]\n− (1− 0.42κm)pβcyC(t)}Ss(t)\ndEy(t)\ndt ={(1− 0.28κm)(1− 0.455κd)βy[Ey(t) +kIy(t) +Es(t) +kIs(t)]\n+ (1− 0.42κm)βcyC(t)}Sy(t)− (εy +ξy)Ey(t)\ndEs(t)\ndt ={(1− 0.28κm)(1− 0.455κd)pβy [Es(t) +kIs(t) +Ey(t) +kIy(t)]\n+ (1− 0.42κm)pβcyC(t)}Ss(t)− (εs +ξs)Es(t)\ndIy(t)\ndt =εyEy(t)− [(0.5−ηy)κqy +ηy +ρy]Iy(t)\ndIs(t)\ndt =εsEs(t)− [(0.5−ηs)κqs +ηs +ρs]Is(t)\ndQy(t)\ndt = [(0.5−ηy)κqy +ηy]Iy(t)−ϕyQy(t)\ndQs(t)\ndt = [(0.5−ηs)κqs +ηs]Is(t)−ϕsQs(t)\ndRy(t)\ndt =ξyEy(t) +ρyIy(t) +ϕyQy(t)\ndRs(t)\ndt =ξsEs(t) +ρsIs(t) +ϕsQs(t)\ndC(t)\ndt = (1− 0.42κm)µ (Ey(t) +kIi(t) +Es(t) +kIs(t))\n− (1 + 0.21κe)δC (t).\n(28)\nIn this model, the variables κ = [κm, κd, κe, κqy, κqs]⊤ are the control\nvariables that need to be optimised.\n\n2021LATEX template\n24 Ranking the Effectiveness of Non-Pharmaceutical Interventions\n3.3.2 Formulation of the Optimal Control Problem\nNow we formulate the optimal control problem that, once solved, provides the\noptimal trajectories that give the best combination of interventions to contain\nthe epidemic within the university. The optimal control problem is defined as\nJ∗ = min\nx(·),κ(·)\nJ =∥x(tf)−xtar∥2\nH +\n∫ tf\nt0\n∥x(τ)−xtar∥2\nQ +∥κk(τ)∥2\nR dτ\ns.t. ˙ x(t) =f(x(t),κ(t),t),\nxmin≤x(t)≤xmax,\nxf,min≤x(tf)≤xf,max,\nκmin≤κ(t)≤κmax,\nx(0) =x14,\n(29)\nwherextar represents the target state vector,∥·∥H/Q/R indicates the Euclidean\nmatrix norm weighted byH/Q/R and the square matricesH,Q andR contain\nthe weights for the final states, running states and running control variables,\nrespectively. The values of xtar and the weights are changed depending on\nthe scenario that we need to solve (see next section). The problem has also\nconstraints on both states and control variables. In fact, we need that the\nstates lie between 0 and 1. Thus, even without further requirement the state\nconstraints are at least xmin =xf,min = [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]⊤ and\nxmax =xf,max = [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1]⊤. Similarly, the lower bounds\nof the control variables should be κmin = [0, 0, 0, 0, 0]⊤ while their upper\nconstrains are κmax = [1, 1, 1, 1, 1]⊤. The initial condition sets the starting\npoint of the system. We assume that the university starts to react two weeks\nafter the first exposed subjects appear among students or staff. Thus, we first\nrun the simulation of the original un-controlled model for 14 days and get the\nresulting state values on day 14, namely x14. Then this state is used as the\ninitial condition for the optimisation problem, namely x(0) = x14. We finally\nrequire that the epidemic is eliminated within 120 days, sot0 = 0 andtf = 120.\n3.3.3 Implementation of Four Different Scenarios\nBalancing the trade-off between controlling the spread of COVID-19 and\nresuming the normal campus activity is the main question considered in this\nstudy. According to different trade-off’s between these two objectives, four\nscenarios are studied: minimum number of cases, minimum intervention, min-\nimum non-quarantine interventions, and minimum quarantine interventions.\nEach scenario corresponds to different weight matrices and different path\nconstraints.\nFor the first scenario (minimum number of cases), the university does not\nimpose any limitations on the strength of the interventions to lead to the\nfastest mitigation of the epidemic. Mathematically, the controller is designed\nto maximise the number of susceptible people during the whole period, i.e.,\n\n2021 LATEX template\nRanking the Effectiveness of Non-Pharmaceutical Interventions 25\nthe proportions of the susceptible students/staff are maintained as close to one\nas possible. In contrast, the values of the other compartments need to be as\nclose as possible to zero in this scenario. Therefore, we set the target state as\nxtar = [1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0]⊤. (30)\nMeanwhile, the weight matrices are set as\nH =diag([100; 100; 0; 0; 0; 0; 0; 0; 0; 0; 0])\nQ =diag([100; 100; 0; 0; 0; 0; 0; 0; 0; 0; 0])\nR =diag([10−2; 10 −2; 10 −2; 10 −2; 10 −2]).\n(31)\nwhere in the matrix R we use 10−2 instead of 0 to improve numerical stability\nof the solver. The constraints on both states and control variables remain the\nsame as the basic requirements discussed before, namely\nxmin =[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]⊤\nxmax =[1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1]⊤\nxf,min =[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]⊤\nxf,max =[1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1]⊤\nκmin = [0, 0, 0, 0, 0]⊤\nκmax = [1, 1, 1, 1, 1]⊤.\n(32)\nFor the other three scenarios, we require that the number of susceptible\nstudents and staff cannot go below 94% of the total population (i.e. the number\nof infected individuals is equal or below 6%). We can easily achieve this by\nsetting more restrictive state constraints. Consequently, we do not need to\nuse the weights H andQ (also because it is difficult to intuitively understand\nthe meaning of the weights on the states in these scenarios). Hence, we select\nH = 0 and Q = 0 and xtar is not used. Additionally, to ensure that the\nepidemic is completely concluded after 134 days, the number of exposed and\ninfected individuals should become zero at the final state. Recall that Ny,Ns,\nand Nt indicate the total number of student, staff, and the total population,\nrespectively. Then the constraints on states should be\nxmin = [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]⊤\nxmax = [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1]⊤\nxf,min = [0.94Ny\nNt\n, 0.94Ns\nNt\n, 0, 0, 0, 0, 0, 0, 0, 0, 0]⊤\nxf,max = [1, 1, 10−4, 10−4, 10−4, 10−4, 1, 1, 1, 1, 1]⊤,\n(33)\n\n2021LATEX template\n26 Ranking the Effectiveness of Non-Pharmaceutical Interventions\nwhile the input constraints remain unchanged\nκmin = [0, 0, 0, 0, 0]⊤\nκmax = [1, 1, 1, 1, 1]⊤.\n(34)\nThe only difference in the formulation of the other three scenarios is the value\nof the weight matrixR. In the minimum intervention scenario, the solver aims\nto minimize the total norm of the control signal while satisfying the 94%\nconstraint. Thus, all control variables are heavily weighted and the matrix R\nis selected as\nR =diag([100; 100; 100; 100; 100]) . (35)\nIn the scenario of minimisation of non-quarantine interventions, the strength\nof first three interventions (wearing masks κm, keeping social distance κd and\ndisinfecting the environmentκe) are minimised, while the quarantines are not.\nIn this situation, the matrix R becomes\nR =diag([100; 100; 100; 10 −2; 10 −2]). (36)\nIn the minimum quarantine scenario, the norm of quarantine rates for the\ninfected students (κqy) and staff (κqs) are minimised. Since mandatory quaran-\ntine has the largest impact on campus activities, this scenario aims to find how\nthe spread can be minimised while avoiding quarantine enforcement. Hence,\nthe matrix R is\nR =diag([10−2; 10 −2; 10 −2; 100; 100]). (37)\n\n2021 LATEX template\nRanking the Effectiveness of Non-Pharmaceutical Interventions 27\nCode availability\nThe code is available at https://github.com/ZiruiNiu/University Epidemic\nModel with Control.git.\nData availability\nAll model’s parameters taken from the literature are opportunely referenced\nin the main text. No further data is used.\n\n2021LATEX template\n28 Ranking the Effectiveness of Non-Pharmaceutical Interventions\nAppendix A Sensitivity Figures\n0 20 40 60 80 100 120\nTime (days)\n0\n0.01\n0.02\n0.03\n0.04\nCurves of Ey(t)\ny Testing Result - Exposed Students\n0 20 40 60 80 100 120\nTime (days)\n0\n0.005\n0.01\nCurves of Es(t)\ny Testing Result - Exposed Staff\n0 20 40 60 80 100 120\nTime (days)\n0\n0.02\n0.04\n0.06\nCurves of Iy(t)\ny Testing Result - Infected Students\n0 20 40 60 80 100 120\nTime (days)\n0\n0.005\n0.01\nCurves of Is(t)\ny Testing Result - Infected Staff\ny = 0.07\ny = 0.075\ny = 0.08\ny = 0.085\ny = 0.09\ny = 0.095\ny = 0.1\ny = 0.105\ny = 0.11\ny = 0.115\na\nc\nb\nd\nFig. A1 Numerical testing of student-to-student infection rate βy. Trajectories of\ncases among students and staffs whenβy varying from 0.07 to 0.115. With differentβy, panels\na and b show the proportion trajectories of exposed students and staffs respectively while\npanels c and d depict the proportion trajectories of infected students and staffs respectively.\n0 50 100 150\nTime (days)\n0\n0.01\n0.02\n0.03\nCurves of Ey(t)\ns Testing Result - Exposed Students\n0 50 100 150\nTime (days)\n0\n2\n4\n6\n8\nCurves of Es(t)\n10-3 s Testing Result - Exposed Staff\n0 50 100 150\nTime (days)\n0\n0.01\n0.02\n0.03\n0.04\nCurves of Iy(t)\ns Testing Result - Infected Students\n0 50 100 150\nTime (days)\n0\n2\n4\n6\n8\nCurves of Is(t)\n10-3 s Testing Result - Infected Staff\ns = 0.08\ns = 0.085\ns = 0.09\ns = 0.095\ns = 0.1\ns = 0.105\ns = 0.11\ns = 0.115\ns = 0.12\ns = 0.125\na\nc\nb\nd\nFig. A2 Numerical testing of student-to-student infection rate βs. Trajectories of\ncases among students and staffs whenβs varying from 0.08 to 0.125. With differentβs, panels\na and b show the proportion trajectories of exposed students and staffs respectively while\npanels c and d depict the proportion trajectories of infected students and staffs respectively.\n\n2021 LATEX template\nRanking the Effectiveness of Non-Pharmaceutical Interventions 29\n0 50 100 150\nTime (days)\n0\n0.01\n0.02\n0.03\n0.04\nCurves of Ey(t)\ncy Testing Result - Exposed Students\n0 50 100 150\nTime (days)\n0\n0.005\n0.01\nCurves of Es(t)\ncy Testing Result - Exposed Staff\n0 50 100 150\nTime (days)\n0\n0.01\n0.02\n0.03\n0.04\nCurves of Iy(t)\ncy Testing Result - Infected Students\n0 50 100 150\nTime (days)\n0\n2\n4\n6\n8\nCurves of Is(t)\n10-3 cy Testing Result - Infected Staff\ncy = 0.075\ncy = 0.08\ncy = 0.085\ncy = 0.09\ncy = 0.095\ncy = 0.1\ncy = 0.105\ncy = 0.11\ncy = 0.115\ncy = 0.12\na\nc\nb\nd\nFig. A3 Numerical testing of student-to-student infection rate βcy. Trajectories\nof cases among students and staffs when βcy varying from 0.075 to 0.12. With different\nβcy, panels a and b show the proportion trajectories of exposed students and staffs respec-\ntively while panels c and d depict the proportion trajectories of infected students and staffs\nrespectively.\n\n2021LATEX template\n30 Ranking the Effectiveness of Non-Pharmaceutical Interventions\n0 50 100 150\nTime (days)\n0\n0.01\n0.02\n0.03\nCurves of Ey(t)\ncs Testing Result - Exposed Students\n0 50 100 150\nTime (days)\n0\n2\n4\n6\n8\nCurves of Es(t)\n10-3 cs Testing Result - Exposed Staff\n0 50 100 150\nTime (days)\n0\n0.01\n0.02\n0.03\n0.04\nCurves of Iy(t)\ncs Testing Result - Infected Students\n0 50 100 150\nTime (days)\n0\n2\n4\n6\n8\nCurves of Is(t)\n10-3 cs Testing Result - Infected Staff\ncs = 0.095\ncs = 0.1\ncs = 0.105\ncs = 0.11\ncs = 0.115\ncs = 0.12\ncs = 0.125\ncs = 0.13\ncs = 0.135\ncs = 0.14\na\nc\nb\nd\nFig. A4 Numerical testing of student-to-student infection rate βcs. Trajectories\nof cases among students and staffs when βcs varying from 0.095 to 0.14. With different\nβcs, panels a and b show the proportion trajectories of exposed students and staffs respec-\ntively while panels c and d depict the proportion trajectories of infected students and staffs\nrespectively.\n0 50 100 150\nTime (days)\n0\n0.01\n0.02\n0.03\nCurves of Ey(t)\ny Testing Result - Exposed Students\n0 50 100 150\nTime (days)\n0\n2\n4\n6\n8\nCurves of Es(t)\n10-3 y Testing Result - Exposed Staff\n0 50 100 150\nTime (days)\n0\n0.01\n0.02\n0.03\n0.04\nCurves of Iy(t)\ny Testing Result - Infected Students\n0 50 100 150\nTime (days)\n0\n2\n4\n6\n8\nCurves of Is(t)\n10-3 y Testing Result - Infected Staff\ny = 0.007\ny = 0.008\ny = 0.009\ny = 0.01\ny = 0.011\ny = 0.012\ny = 0.013\ny = 0.014\ny = 0.015\ny = 0.016\na\nc\nb\nd\nFig. A5 Numerical testing of student-to-student infection rate ηy. Trajectories of\ncases among students and staffs whenηy varying from 0.007 to 0.016. With differentηy, pan-\nels a and b show the proportion trajectories of exposed students and staffs respectively while\npanels c and d depict the proportion trajectories of infected students and staffs respectively.\n\n2021 LATEX template\nRanking the Effectiveness of Non-Pharmaceutical Interventions 31\n0 50 100 150\nTime (days)\n0\n0.01\n0.02\n0.03\nCurves of Ey(t)\ns Testing Result - Exposed Students\n0 50 100 150\nTime (days)\n0\n2\n4\n6\n8\nCurves of Es(t)\n10-3 s Testing Result - Exposed Staff\n0 50 100 150\nTime (days)\n0\n0.01\n0.02\n0.03\n0.04\nCurves of Iy(t)\ns Testing Result - Infected Students\n0 50 100 150\nTime (days)\n0\n2\n4\n6\n8\nCurves of Is(t)\n10-3 s Testing Result - Infected Staff\ns = 0.016\ns = 0.017\ns = 0.018\ns = 0.019\ns = 0.02\ns = 0.021\ns = 0.022\ns = 0.023\ns = 0.024\ns = 0.025\na\nc\nb\nd\nFig. A6 Numerical testing of student-to-student infection rate ηs. Trajectories of\ncases among students and staffs whenηs varying from 0.016 to 0.025. With differentηs, pan-\nels a and b show the proportion trajectories of exposed students and staffs respectively while\npanels c and d depict the proportion trajectories of infected students and staffs respectively.\n0 50 100 150\nTime (days)\n0\n0.01\n0.02\n0.03\n0.04\nCurves of Ey(t)\n Testing Result - Exposed Students\n0 50 100 150\nTime (days)\n0\n0.005\n0.01\nCurves of Es(t)\n Testing Result - Exposed Staff\n0 50 100 150\nTime (days)\n0\n0.02\n0.04\n0.06\nCurves of Iy(t)\n Testing Result - Infected Students\n0 50 100 150\nTime (days)\n0\n0.005\n0.01\nCurves of Is(t)\n Testing Result - Infected Staff\n = 0.5\n = 0.55\n = 0.6\n = 0.65\n = 0.7\n = 0.75\n = 0.8\n = 0.85\n = 0.9\n = 0.95\na\nc\nb\nd\nFig. A7 Numerical testing of student-to-student infection rate δ. Trajectories of\ncases among students and staffs when δ varying from 0.5 to 0.95. With different δ, panels\na and b show the proportion trajectories of exposed students and staffs respectively while\npanels c and d depict the proportion trajectories of infected students and staffs respectively.\n\n2021LATEX template\n32 Ranking the Effectiveness of Non-Pharmaceutical Interventions\n0 50 100 150\nTime (days)\n0\n0.02\n0.04\n0.06\nCurves of Ey(t)\n Testing Result - Exposed Students\n0 50 100 150\nTime (days)\n0\n0.005\n0.01\n0.015\nCurves of Es(t)\n Testing Result - Exposed Staff\n0 50 100 150\nTime (days)\n0\n0.02\n0.04\n0.06\n0.08\nCurves of Iy(t)\n Testing Result - Infected Students\n0 50 100 150\nTime (days)\n0\n0.005\n0.01\n0.015\nCurves of Is(t)\n Testing Result - Infected Staff\n = 0.05\n = 0.1\n = 0.15\n = 0.2\n = 0.25\n = 0.3\n = 0.35\n = 0.4\n = 0.45\n = 0.5\na\nc\nb\nd\nFig. A8 Numerical testing of student-to-student infection rate µ. Trajectories of\ncases among students and staffs when µ varying from 0.05 to 0.5. With different µ, panels\na and b show the proportion trajectories of exposed students and staffs respectively while\npanels c and d depict the proportion trajectories of infected students and staffs respectively.\nReferences\n[1] WHO: Coronavirus disease (COVID-19) (2021). https://www.who.int/\nemergencies/diseases/novel-coronavirus-2019\n[2] Islam, M.A., et al.: Prevalence of headache in patients with coronavirus\ndisease 2019 (covid-19): a systematic review and meta-analysis of 14,275\npatients. Frontiers in neurology 11 (2020)\n[3] Agyeman, A.A., et al.: Smell and taste dysfunction in patients with covid-\n19: a systematic review and meta-analysis. In: Mayo Clinic Proceedings,\nvol. 95, pp. 1621–1631 (2020). 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Aerosol Science and\nTechnology 55(4), 449–457 (2021). https://doi.org/10.1080/02786826.\n2020.1862409\n[46] Ueki, H., et al.: Effectiveness of face masks in preventing airborne trans-\nmission of sars-cov-2. mSphere 5(5), 00637–20 (2020). https://doi.org/10.\n1128/mSphere.00637-20","source_license":"CC-BY-4.0","license_restricted":false}