Expanded SEIRCQModel Applied to COVID-19 Epidemic Control Strategy Design and Medical Infrastructure Planning

+e rapid spread of COVID-19 has demanded a quick response from governments in terms of planning contingency efforts that include the imposition of social isolation measures and an unprecedented increase in the availability of medical services. Both courses of action have been shown to be critical to the success of epidemic control. Under this scenario, the timely adoption of effective strategies allows the outbreak to be decelerated at early stages. +e objective of this study is to present an epidemic model specially tailored for the study of the COVID-19 epidemics, and the model is aimed at allowing the integrated study of epidemic control strategies and dimensioning of the required medical infrastructure. Along with the theoretical model, a case study with three prognostic scenarios is presented for the first wave of the epidemic in the city of Manaus, the capital city of Amazonas state, Brazil. Although the temporary collapse of the medical infrastructure is hardly avoidable in the state-of-affairs at this time (April 2020), the results show that there are feasible control strategies that could substantially reduce the overload within reasonable time. Furthermore, this study delivers and presents an intuitive, straightforward, free, and open-source online platform that allows the direct application of the model. +e platform can hopefully provide better response time and clarity to the planning of contingency measures.

Introduction
e rapid spread of COVID-19 outbreak has become a very challenging situation due to its substantial demand for medical infrastructure.
e lack of knowledge about the virus at early stages severely limits the effectiveness of available clinical treatments and thus contributes to longer hospitalization periods that might nevertheless result in the loss of a considerable share of patients, as shown in recent clinical publications [1][2][3][4]. As largely explained by political and medical authorities from several countries and the WHO (World Health Organization), the main problem of having a steep epidemic curve is the overload it might cause on health care infrastructure [5]. In this sense, not only medical materials and equipment but also medical personnel suddenly become insufficient, who become especially prone to infection when subjected to an overload of working hours and lack of adequate personal protection equipment [5].
As the world faces the consequences of a largely unknown infectious agent that spreads exponentially, the timely adoption of adequate control and mitigation strategies is of utmost importance. In this context, the mathematical modelling of epidemic dynamics can provide a valuable contribution. Beyond helping to unveil prognostics of future behavior under different scenarios that are unaccessible otherwise, it allows the application of numerical parameters from previous studies and clinical expertise (see, for instance, [1][2][3][4]). Furthermore, while the focus of epidemic modelling is control strategies, the estimated epidemic timeline can help forge ways to optimize the maintenance of economic activities to their fullest yet below levels that are detrimental to epidemic control. Finding this tenuous and fragile equilibrium between epidemic control and economic activity is at the core of several debates [6]. In this sense, mathematical models can provide a rather solid basis for discussion, while their underlying assumptions can be openly scrutinized and updated to evaluate scope and accuracy of the resulting conclusions. Another factor that has gained importance as authorities acknowledged the lack of infrastructure to fight the epidemic is time. If countries can gain time by adopting adequate strategies, they can prepare better to properly handle the epidemic. As such, the study and evaluation of epidemic models can not only provide future scenarios that allow the dimensioning of the infrastructure but also test strategies to flatten the infection curve as much as needed to fit the achievable capacity of the health care system [7][8][9].
Following the seminal works on epidemic dynamics by Kermack and McKendrick [10][11][12], many epidemic models were proposed and studied in the literature, including but not limited to SIR [13,14], SIS [14], SIRS [14,15], SEIR [16,17], SEIS [18,19], and MSEIR [13]. Among those, the SEIR model following classical assumptions is a rather interesting starting point to the study of the COVID-19 since it is a rather simple model for which a substantial body of literature is available.
e SEIR model consists of four population subclasses: S (susceptible), E (exposed, which are infected but not infective), I (infected and infective), and R (removed). e compartment R encompasses both recovered and deceased. e recovered individuals in this compartment represent the fact that once COVID-19 patients recover, they are considered to have built lasting resistance to the infection, at least to the timeframe of the first wave of the epidemic. e SEIR model was used as basis in recent studies about COVID-19 [7,9]. To allow the implementation of epidemic control strategies and dimensioning of medical infrastructure under different scenarios, the model presented in this study was expanded with new compartments. e objective of this study is to present an epidemic model tailored for COVID-19, and the model is aimed at allowing the integrated study of epidemic control strategies and dimensioning of the medical infrastructure required in different scenarios. e model is an expanded version of the well-known SEIR model, in which two new compartments are added: confinement of susceptible (C) and quarantine of infected (Q). ese compartments are aimed at allowing the implementation of two of the epidemic control measures that also include social distancing and vaccination, the latter as a future prospect in the case this scenario extends for a long period. Furthermore, three equations that account severe and critical hospitalization cases and the number of deaths are included to give an account of the burden on health care infrastructure. To ease the application of the model by a general audience, we provide an implementation of the expanded SEIRCQ model as an intuitive, straightforward, free, and open-source online platform (https:// dashseir.herokuapp.com/) that allows the direct application of the model with clinical and physical parameters that are accessible in the literature. To illustrate the application of the model, we also provide a complete case study for the city of Manaus, the capital city of Amazonas State, Brazil. Manaus shall face the collapse of the medical care capacity due the first wave of the coronavirus outbreak [20]. We use the model to evaluate the current state-of-affairs and show that they shall lead the infection curve to overwhelm the current medical care capacity for several months in case they remain unchanged. en, we test possible measures and strategies to mitigate the hazardous effects of the epidemic. As such, the justification for this manuscript is twofold: to introduce the rationale behind the working of the platform and to serve as a reference for its application.

Epidemic Models.
e study of epidemic models dates to 1927 with a set of three publications by Kermack and McKendrick, which were more recently republished in digital versions [10][11][12]. Since then, a number of models were proposed and studied, such as SIR [13,14], SIS [14], SIRS [14,15], SEIR [16,17], SEIS [18,19], and MSEIR [13], among others, which are known as compartmental models. e general idea is that the population is subdivided into compartments, each governed by a dynamical equation. e equations can be either deterministic or stochastic, and each compartment can be further stratified by gender, age, or any other feature that is regarded as relevant for a given study. Furthermore, the model can be either spatially homogeneous or heterogeneous, depending on the relevance of location to the dynamics of the epidemic.
Several models have been proposed to describe the dynamics of the COVID-19 epidemic. Naturally, each modelling approach seeks to describe a set of variables of interest according to specific objectives. A fair share of the recent models appearing in the literature for the study of COVID-19 are extensions of the well-known SEIR (susceptible-exposedinfected-removed) model by Li and Muldowney [17], Roda et al. [9], López and Rodó [21], and Kuniya [7].
A study by Lin et al. [22] took as basis the SEIR model and expanded it by including extracompartments that were aimed at modelling the public perception of risk and the cumulative number of cases (both reported and not reported). In fact, the results presented in Wu and McGoogan [23] suggest that a considerable share of nonreported cases exist. us, modelling the cumulative number of cases, as done by Lin et al. [8], is likely to produce more realistic scenarios and prognostics. Furthermore, a step-like function is applied to model the zoonotic transmission at early stages, from which it was possible to estimate the approximate number of zoonotic infections that presumably occurred in Wuhan and trace the epidemic back to its origin. To deal with the lack of a long enough time series for COVID-19, Lin et al. [8] applied parameters from the influenza epidemic from 1918 in London, UK [24], to fit the model to the official data from Wuhan, China. e main difficulty associated with the application of this model is the definition of control strategy parameters (such as governmental action strength) since they cannot be straightforwardly mapped into collectible real-world parameters.
Roda et al. [9] showed that the confirmed-case data are the main reason to the wide range of variations in model predictions for the COVID-19 epidemic using mathematical models, which pose a difficulty when estimating real transmission rates, since transmission rate coefficients in model calibration might have a very wide range. Roda et al. [9] also point that the confirmed-case data represent only a fraction of the total infected population. us, fitting the model to confirmed-case data is likely to produce misleading prognostics of future scenarios in terms of the peak values.

COVID-19 Epidemic Parameters.
e knowledge gained during the crisis in China provides several useful parameters for modelling. For instance, the work by Wu and McGoogan [23] considered over 72, 000 cases from the first wave of the epidemic in China and established that 81% of the cases were mild (all the way from asymptomatic to mild symptoms), 14% were severe (high respiratory frequency, low blood oxygen saturation, and lung infiltrates), and 5% were critical (respiratory failure, septic shock, and/or multiple organ dysfunction or failure). Furthermore, the authors also found the overall case-fatality rate of 2.3%, while the death rate was 49% among critical patients, and no deaths were reported in mild and severe cases. Wu and McGoogan [23] also presented numbers that hint that insufficient testing may have caused case-fatality rates to be overestimated approximately by 7-fold factor. Interestingly, a 7-fold factor was also found to be the factor of subnotification of infected individuals in the State of Rio Grande do Sul, Brazil, as concluded the first large field study based on massive testing of individuals [25]. e basic reproduction number in Japan considering the SEIR model was estimated by Kuniya [7] as R 0 � 2.6 (2.4 to 2.8 in a 95% confidence interval), which means that an infected individual will infect, on average, 2.6 new individuals. Meanwhile, considering the case of Wuhan, Tang et al. [26] found that the basic reproduction number can be as large as 6.47 (5.71 to 7.23, 95% CI) and Li et al. [22] found 2.2 (1.4 to 3.9, 95% CI) on the basis of analysis of 425 laboratory-confirmed cases. Regarding the basic reproduction number after social distancing measures, Wang et al. [3] considered a scenario with R m � 0.9, which later was found to provide an accurate description of the epidemic dynamics after the social distancing measures were implemented in China.
As other important parameters to estimate the load upon medical care services, Chen et al. [27] calculated the average duration of hospitalization of 31 discharged patients and found 12.39 days ( ± 4.77, 95% CI). Meanwhile, Li et al. [22] calculated the average incubation period to be 5.2 days (4.1 to 7.0, 95% CI), while Bi et al. [1] found 4.8 days (4.2 to 5.4, 95% CI) on the basis of 391 cases and 1, 286 of their close contacts. To take advantage of the results from clinical studies, the next section presents the expanded SEIRCQ model in a formulation that allows explicit application of clinical and physical parameters. is is aimed at allowing straightforward application of medical literature about COVID-19.

Materials and Methods
is section gives an account of the proposed model and the epidemic control strategies to be applied. It also presents the region considered in the case study and the methods applied to acquire the model parameters.
3.1. e Expanded SEIRCQ Model. Consider a population with N individuals and the following stratification compartments: susceptible (S), exposed (E), infected (I), removed (R), quarantined infected (Q), and confined susceptible (C), which are represented in Figure 1 and related according to the equations: where R t is the average number of new infections caused by an infected individual, T INF is the average duration of the infection in an individual, R CFN is the rate of confinement of susceptible, T CNF is the average time of confinement of a susceptible individual, R VCN is the rate of vaccination of susceptible individuals, T IMN is the average time for immunization of an individual after vaccination, T INC is the average incubation time of an individual exposed to the infection, T INC R QRT is the rate of infected individuals entering quarantine, T LIC is the average time lag until an infected individual is put in quarantine, and T QRT is the average time of quarantine. As a result, the total population is given by e parameter R t can be a function of time to describe the strategy of lockdown, during which the infection rate tends to be lower. For instance, to describe the transition from free circulation to isolation at time T ISO , one can define R t as a function defined by parts as where R t � R 0 during (−∞, T ISO ] and R t � R m from T ISO on. In this case, it is expected that R 0 > R m . It is supposed that a portion of the infected population will need hospitalization and a smaller portion will require hospitalization in an Intensive Care Unit (ICU). is portion of the infected population causes greater concern due to the prospect of overload in the health care infrastructure. To model the dynamics of individuals requiring hospitalization and individuals requiring intensive care, the model is expanded and two new state variables are defined, H and U, respectively. e rates of change in the number of individuals requiring hospitalization and those requiring intensive care are modeled as where R HSP is the rate of hospitalization in simple hospital beds, T HSP is the average duration of hospitalization in simple hospital beds, R ICU is the rate of critical cases requiring hospitalization in ICUs, and T ICU is the average duration of hospitalization in ICUs. In any case, we consider valid the inequality 1 ≪ R HSP ≪ R ICU , which represents the fact that the number of hospitalizations in common beds is a fraction of the current number of infected individuals, and that the number of hospitalizations in ICU beds is a fraction of the hospitalizations. Besides, the number of mild cases (that do not require hospitalization) can be represented by M and defined as To assess the possible number of deaths in different scenarios, the number of deceased individuals is defined by P. e number of fatal cases is modeled as a fraction of those who require intensive treatment. Note that, in practice, not all patients who require intensive treatment can access it since the number of ICU beds is limited. In the model, we will express this limitation by N ICU , which represents the number of ICU beds available to COVID-19 patients in this system. us, the rate of change in the number of deaths considers two strata of patients who need an ICU: those who have access and those who cannot, each with a different mortality rate. It is expected that the patient who needs intensive care and does not obtain access will have lesser chances of survival as compared to the patient who needs intensive care and obtains access. us, two mortality rates are defined, R DHT1 and R DHT2 , for patients with access and without access to ICU, respectively. e rate of change in the number of deaths is modeled as where U represents the number of patients in need of intensive care and obtains access to an ICU bed. e variable U receives the value of the state variable U up to the limit of ICU vacancies, N ICU , according to function In addition, T FOR defines the average survival time for a patient who needs intensive treatment but cannot access an ICU bed. In this way, a fraction R DHT2 of critical patients not able to find an ICU are removed from U and transferred to compartment P after a period of T FOR . Meanwhile, σ is a function that takes on binary values and controls the second term of equation (6). When U is greater than the number of ICU beds available, the function returns a unit value and the second term of the equation comes into play with its respective mortality rate R DHT2 and mean survival time T FOR . e mortality according to the increased rate R DHT2 is modeled as proportional to the number of people in need of intensive treatment but does not get access to a bed due to overcrowding. us, the σ function is modeled as

Epidemic Control Strategies Using the SEIRCQ Model.
e SEIRCQ model allows the application of four epidemic control strategies. e aim of this section is to present the rationale under each of them and define measures of effectiveness for each one separately and then to all of them combined.

Quarantine of Infected Individuals.
e strategy of quarantine of infected individuals seeks to remove from circulation those who are infected to neutralize their contribution to the spread of the infection. By decreasing the number of circulating infected, I, one has the multiplication R t SI assume lower values, such that the rate of change in the number of individuals exposed to the infection, given as decreases accordingly. To have the epidemic under control, it is necessary that the number of circulating infected individuals decreases, i.e., dI dt Reorganizing the terms in equation (10), one obtains Figure 1: Flowchart of the SEIRCQ model. from which it is possible to observe that T INC and T INF are parameters whose values are characteristic of the infection and thus not controllable. In turn, the quarantine rate R QRT and the time lag between infection and quarantine T LIC are controllable parameters that depend on the agility and effectiveness of identifying and isolating infected individuals.
As one seeks to make the right side of the inequation as large as possible, one shall seek for larger values of R QRT and smaller values of T LIC . e strategy of quarantine of infected individuals usually works by flattening the epidemic curve.
To that end, it is necessary that the number of quarantined infected individuals, Q, reach a proportion of the total number of infected individuals, I, in the population N. e coefficient of quarantine of the infected is defined as for I + Q > 0, and it measures how much a given set of quarantine parameters R QRT , T QRT , and T LIC perform in a given scenario. e coefficient ranges from 0 to 1 that correspond to 0 to 100% quarantine of infected individuals.

Confinement of Susceptible Individuals.
e strategy of confinement of susceptible individuals seeks to remove from circulation a fraction of those who are susceptible to neutralize their exposition to the infection. By decreasing the number of circulating susceptible, S, one has the multiplication R t SI assume lower values, such that the rate of change in the number of individuals exposed to the infection decreases accordingly. is strategy can be thought of as relaying a share of the workforce that is susceptible, for example, to a 70 − 30% proportion of circulating and confined, respectively. e rate of change in the confined susceptible is given by in which susceptible individuals enter according to a rate R CNF and confined individuals leave after being confined for T CNF days. As a confined individual leaves the compartment, he goes back to the compartment of susceptible individuals since during T CNF , the confined susceptible individuals are isolated from infected individuals. e confinement strategy usually works by flattening the epidemic curve. To that end, it is necessary that the number of confined individuals, C, reach a proportion of the total number of susceptible individuals, S, in the population N. e coefficient of confinement of susceptible is defined as for S + C > 0. e coefficient ranges from 0 to 1 that corresponds to 0 to 100% confinement of susceptible individuals.

Social
Distancing. e strategy of social distancing seeks to prevent gatherings, circulation, and exposition to deaccelerate the spread of the epidemic. It includes physical distancing in supermarkets, restaurants, and drugstores, for example, and the use of masks in public places. It is expected that the social distancing will reduce the value of the basic reproduction number, R t by means of restriction of social activities that may induce social interactions, such as circulation of public transport or cultural events, for example. Most commonly, only essential services are allowed to function and nevertheless under considerable changes in regulation. In terms of the equations, the strategy of social distancing works by decreasing the basic reproduction number, R t , such that multiplication R t SI will assume lower values and the rate of change in the number of individuals exposed to the infection decreases accordingly, as shown in equation (9).
Lockdowns are considered energic measures due to the amount of changes they bring to the daily routines. Nevertheless, they were shown to be effective to deaccelerate infection rates although they may also cause undesirable economical side effects.
us, the accurate dimensioning of the duration and severity of social distancing measures and the planning of its time window are fundamental to mitigate its negative effects on the economy while being effective in controlling the surge of infections. e level of what we refer to as social distancing will be calculated by means of the coefficient of social distancing: where R 0 > 0 is the basic reproduction number, i.e., the basic reproduction number before social distancing.

Vaccination.
e strategy of vaccination seeks to transfer individuals directly from the compartment of susceptible to the compartment of removed. It can be observed in the equations of the model that vaccination is the only form of getting immunized without being sick (infected). e equation for the rate of change in the number of susceptible individuals is dS dt where the last term of the equation represents the outflow of susceptible from the compartment according to a rate of vaccination R VCN and to a time lag T IMN , which represents the number of days necessary after vaccination until the body produces antibodies enough to fight the disease. When leaving the compartment of susceptible, S, the vaccinated individuals enter the compartment of removed according to rate R VCN /T IMN .

Measuring the Combined Effectiveness of Strategies.
In the last section, the presentation of control strategies showed that they operate upon the reduction of the product R t SI, one at a time. Note that this term appears as a flow of individuals from the compartment of susceptible to the Mathematical Problems in Engineering compartment of exposed and then to the compartment of infected, that is, S ⟶ E ⟶ I. When two or more strategies are applied simultaneously, the combined effect of them can be evaluated by means of what we call the effective isolation coefficient, EIC, as where R 0 > 0 is the basic reproduction number and (S + C) > 0, (I + Q) > 0; that is, the coefficient is defined for the period during the epidemic. e EIC allows the evaluation of the relation between the severity of the isolation measures and the results upon the infected population. When R 0 � R t and Q � C � 0, EIC � 0 and there is no isolation at all, whereas R t � 0 or S � 0 or I � 0 with R 0 (S + C)(I + Q) > 0 yielding EIC � 1, which means that there is no transmission. is coefficient will be applied to the scenarios studied in Section 4. us, we reserve the term social isolation to refer to the combined effects of social distancing, confinement of susceptible, and quarantine of infected individuals.

e Parameters.
e clinical parameters used in the model were adopted from previous clinical studies on the assumption that they are roughly the same everywhere. is assumption might be proven mildly inaccurate in the near future, but considering the lack of consolidated data and results at the current moment (April 2020), this seems to be the most viable way to get to a functional and working model for prognostic and strategy planning. Considering several studies in the literature, the parameters of the model are set to the values shown in Table 1.
e values of R t , R HSP , R ICU , and R DHT1 were calculated from the information on confirmed and active COVID-19 cases provided by the Health Secretary of the State of Amazonas [20]. Currently, the amount of data is insufficient to calculate R DHT2 , and thus, it was arbitrarily set to 30% higher than R DHT1 . e values for R t ( * ) and R t ( * * ) resulted from model calibration to the COVID-19 confirmed positive data from Manaus, made available by Health Secretary of the State of Amazonas [20]. Furthermore, the first research with considerable size and scope in Brazil applying widespread testing of individuals chosen at random concluded that subnotification of COVID-19 is considerable, having found that, for every confirmed case, there are at least seven infected individuals EPICOVID19 [27]. We apply this 7-fold factor to compensate the hospitalization rate values R HSP and R ICU presented in Table 1. e death rate R DHT1 is maintained since it is applied upon the rate of critical individuals, R ICU . Since it was not possible to access consolidated data regarding the number of ICU beds available for COVID-19, it was set to N ICU � 148, which represents about 30% of the total number of ICU beds in the State of Amazonas. is is only a reference number and might not reflect the real availability. e number of simple hospital beds for COVID-19 N HSP was set to 1, 000 for reference under the unavailability of updated information.

3.5.
e Model Dashboard. e model dashboard was implemented using Python 3.7 programming language and the Dash Plotly library [28]. e equations are integrated numerically using the fourth-order Runge-Kutta method, with an integration step △t � 1. e model is presented in the form of an interactive dashboard, and it is available online at https://dashseir.herokuapp.com/. e working of the dashboard is further detailed in the flowchart in Figure 2.
3.6. Model Calibration. Many different groups have delivered relevant research and data concerning the coronavirus outbreak as the epidemic is ongoing and evolving from country to country. It is important to note that official data are hampered by a number of sources of uncertainty: undertesting, lag of test results, and lack of accuracy in infection dating, among others. Furthermore, all research is preliminary in the sense that they deal with rather problematic data sources which are likely to be revised and consolidated in the future. e characteristics of the epidemics and the current stage of promptness of health organs and laboratories worldwide make the infection curve lead the official confirmed cases by a period that might well be of about 10 days or more. us, for now, the calibration of epidemic models based on confirmed infection data can lead to highly inaccurate prognostics.
To overcome this limitation, the model was formulated as much as possible on the basis of parameters that can be directly mapped to those of clinical studies. On the contrary, parameters depending on social behavior were calibrated on the basis of available data, such as the reproduction number R t . e reproduction numbers for the periods before and during lockdown were calculated by fitting the model to the available official data from the Health Secretary of the State of Amazonas [20]. Calibration processes were performed and were aimed at finding the value of R t that minimizes the root mean square error (RMSE) between the data and the model outputs.

Case Study.
Manaus is the capital of Amazonas state and is located on the banks of Negro river in Northwest Brazil. As the largest city of the state, it occupies an area of 11, 401 km 2 , from which 427 km 2 is an urban area, as represented in Figure 3. Manaus currently holds a population of about 2, 15 million people and has recently been pointed as one of the first Brazilian cities that shall face the collapse of health care facilities due to the coronavirus epidemic. Although the actions of the State Government in tackling a possible epidemic have been important to increase the number of available beds for the hospitalization of serious cases and the number of health professionals, the cases of COVID-19 rapidly increased since late March to reach over a thousand cases by April 12th.
Temporary hiring of human resources, purchase of supplies, equipment for laboratory analysis, and personal protective equipment (PPE) are some of the actions that were announced. To expand the supply of health professionals available to deal with the pandemic, the State University of Amazonas (UEA) announced the anticipation of the graduation of 79 doctors, 28 nurses, and 21 pharmacists by six months [20]. e measure aims to optimize the availability of health services and increase the service capacity of public units, especially those receiving suspected and confirmed coronavirus patients.
On March 23rd, the Amazonas state government announced that the state of public calamity, suspended activities from public organs, and oriented novel hygiene and monitoring procedures in industries. However, the rapid growth in the number of cases since then until the first days of April prompted the state to take more forceful measures. On April 4th, the state government announced further restrictive measures to contain the epidemics: schools were closed and classes broadcasted on open TV channels; only essential services remained working; and the transportation of passengers in buses, taxis, and vans was suspended in federal and state roads. Since then, the number of cases and deaths kept growing rapidly and the capital Manaus became a major concern due to the concentration of confirmed cases and deaths of coronavirus patients.
According to official data, from the current 863 active cases, 117 are hospitalized in mild conditions and 77 are critical and occupying an ICU bed. Furthermore, there are currently another 366 hospitalization of cases under investigation, 313 in regular beds and 53 in ICU beds [20]. e evolution of the epidemic in Manaus (AM) is presented in Figure 4.
From the official and confirmed active cases, one can calculate the percentages of hospitalization in regular and ICU beds to be 13.6% and 8.9%, respectively. Although these are preliminary data, it is the major source of information in the current situation, and thus, these parameters will be applied in the model to evaluate the deficit in medical infrastructure in each scenario.

Model Application
e expanded SEIRCQ model was run on the data from Manaus in order allow the calibration of parameter R t that depends on social behavior. e number of ICU beds used in this study corresponds to those informed in publications by    Figure 5. ese curves correspond to the initial phase of the epidemics, before the first governmental restriction decree took effect. Recall that there is a delay between the exposition and symptoms (if any) and another delay between symptoms and testing/confirmation by official organs. e time lag between the first social distancing measures and their visible effects in the epidemic curve was of about 14 days. e epidemic growth already shows a slowly decreasing growth rate. Model calibration for the available data of the    Mathematical Problems in Engineering first social distancing period resulted in reproduction number R t � 1.51. e calibration curves are shown in Figure 6.
As plotted in a single graphic window, the model and the data curves present a sigmoidal shape, which may indicate that the epidemic is reaching a plateau. Is that so? e next section shows that if current social distancing measures are maintained to the same levels, the number of confirmed cases would grow steadily for weeks before starting to decelerate. e calibration showed that the infection curve started to decelerate in the beginning of April, about 14 days after the isolation measures became effective. As expected, there is a time lag between the implementation of isolation measures and its visible effects in the infection curve. is should also be taken into account in planning epidemic control measures. As the model results described the infection curves, all the dates mentioned will be referred to the official confirmed data, which means that they will be presumably delayed relatively to the real infection curve.

Exploring the Application Epidemic Control Measures:
Alternative Scenarios and Actions. Next, we present three case studies that consider different scenarios for 180 days following the first efforts of social isolation which occurred in March 2020. ree scenarios are considered as prognostics of the epidemic:

Scenario 1: Epidemic Prognostics under Mild Social
Distancing Measures (as Currently Adopted). e following simulation presents a prognostic of the likely scenario under EIC � 0.57, which roughly corresponds to the calibrated parameters that reflect the current situation by April 2020. Although the current measures helped reduce the rate of infection as compared to the initial stages of the epidemic, the model shows that the sigmoid-shaped curve would not lead to a plateau in the following days, as one might believe by checking Figure 7. Rather, the number of infected people shows a tendency to accelerate in the following weeks. e hospitalization curves for this scenario are shown in Figure 8. Note that, under this scenario, the number of ICU beds would be exceeded in late April and the number of patients continue to rise for about two months. Under this scenario, the epidemic would reach a peak only in the period from mid-June to early July and drag the health care system into a period of overload, in which case the available medical infrastructure would be greatly exceeded. Indeed, Manaus and more broadly the Amazonas state are known to have a low average number of ICU beds of about 1.24 to each 10.000 inhabitants. e problem would be worsened by the insufficient reduction of the basic reproduction number under distancing, which is an effect of social behavior.
For the same scenario, Figures 9 and 10 show the peak of the epidemics expected to happen in mid-June. If hospitalization rates continue at the same levels, the hospitalizations would amount to a couple thousands at the peak, as shown in Figure 10. Note that, under this scenario, the medical infrastructure would be greatly and rapidly exceeded beyond any feasible perspective of building new ICU beds. Figure 11 shows the effective isolation coefficient for scenario 1.
Of course, as the perspective of such a scenario becomes a reality, more strict measures are likely to be taken and social behavior naturally would change. How about alternative scenarios of more severe social distancing measures? How would they impact the overload in the medical infrastructure? is is a question on which the next section attempts to shed some light.

Scenario 2: More Strict Social Distancing Measures.
By making the distancing measures more severe, the epidemic curve would reach a maximum and then start to decline by the end of May. At this time (April 2020), Manaus is facing the beginning of the exhaustion of the medical infrastructure. In a scenario of more intense social distancing measures in which the reproduction number would decrease to R t � 1.10, the curve would reach a peak and then slowly start to decline. As such measures last for about 90 days, they would support a consistent reduction in the number of infected people. Figure 12 shows the prognostics of the numbers of deaths under this scenario, and Figure 13 focuses on hospitalizations.
It is noticeable that even though the epidemic is controlled under 0.70 EIC, the number of ICU beds would be insufficient. Under this scenario, the number of critical patients would reach nearly double of available ICU beds. It is important to note that Manaus concentrates the ICU beds in the State of Amazonas, and thus, the total count of ICU beds required statewide would be considerably higher.       Moreover, under this scenario, about 3 months ahead of April, the number of required ICU beds would be nearly the total capacity, as shown in Figure 14. It should be noted that the model does not take into account that equipment may become unavailable as they break, as well as medical staff might themselves be unavailable as they might be infected during the epidemic. It is noticeable that even under more strict distancing measures, the existing infrastructure would be insufficient. Figure 15 shows the EIC for scenario 2. Under this scenario, a consistent decline in the number of infected is observed and the number of infected individuals would return to the current level after a steep ascent in the last week of April. However, strict social distancing measures might be hard to implement in practice since there are economic issues that have to be taken into account. ere are other alternatives to help handle this situation. e next section gives a hybrid approach in which three epidemic control strategies are simultaneously applied to reduce infection rates while also maintaining the economy running as much as possible.

Scenario 3: Severe Social Distancing and Confinement, Followed by Quarantine of Infected Individuals.
is scenario considers an EIC of about 0.80 during four to six weeks, such that the spread of the virus is further reduced relatively to the previous scenarios. In this scenario, the reproduction number is decreased to R t � 1.10 by more severe social distancing measures.
is period would ideally start as soon as possible (the simulations considered late April) and last for about 45 days. After that, the more strict social distancing measures could be relaxed to levels comparable to those imposed by the government decree from March 23 but added with two other strategies: confinement (of susceptible) and quarantine (of infected). e confinement of the susceptible is at the rate of 2% per day, that is, R CNF � 0.02 and T CNF � 15, which would result in a peak of about 20% of the total susceptible population being confined. About four weeks later, as the more severe social distancing period reaches about 2/3 of its duration, the quarantine of infected individuals would take place with R QRT � 0.10, T QRT � 15 days, and T LIC � 2 days. It shall be remarked that the quarantine of infected individuals heavily relies on testing, clinical diagnostics, and adequate social behavior. e period of more severe social distancing measures would give authorities time to put into practice testing campaigns starting in late May or early June.
Under such conditions, the epidemic would be maintained under control after a peak in mid-May, as shown in Figures 16 and 17. As a result of controlling the epidemic, the curve of hospitalized individuals is found to behave satisfactorily since it remains below the presumable number of available beds. In this case, the peak would overwhelm the medical infrastructure during a short period after which the number of hospitalizations in ICU beds would consistently decrease, as shown in Figure 18. e EIC for scenario 3 is shown in Figure 19.
is strategy would require that widespread testing be performed upon mid-June 2020 upon the mild relaxation of social distancing measures.

e Importance of Mathematical Models in is Epidemic.
As the number of cases of COVID-19 in the world surpasses a couple millions, the effects of exponential growth seem to continue misunderstood. is is quite problematic since most of the success or failure of strategic control measures relies upon how they are implemented in practice by society as a whole. In this sense, models can have a widespread didactic effect upon the individual comprehension of the dynamics of the epidemic and how fast things can deteriorate.
As the world goes through one of the most challenging periods in recent history, the application of all available knowledge that can gain us time is needed to deal with the limitations of health care systems and search for effective treatments that can help reduce the death toll. In this context, gaining time is mainly a matter of strategically dealing with this first wave of the epidemic. To this end, scientific, medical, and political personnel are working to flatten the curve as much as possible and make it fit into the health care system capacity. Naturally, not the same strategy will work everywhere. Meanwhile, there is enormous concern about the impacts of the epidemic and of the control strategies in the economy, as well as about a second wave of the epidemic. As such, humanity is currently living in an ongoing multiobjective optimization problem whose outcome greatly depends on early action.
In this sense, the application of mathematical models can be worth an alternative to gain time to find solutions that can be effective to simultaneously preserve public health and mitigate the deep effects that the epidemic shall have upon   Figure 19: Cumulative effect of the isolation measures. Effective isolation coefficient for the third scenario with three control strategies in use, which would stabilize the epidemic faster and allow greater retake of economical activity (interrupting mild distancing and implementing severe distancing along with confinement of susceptible and then effective quarantine of infected). production, employment, and the economy as a whole. While the obvious best solution from the point of view epidemic control is a complete lockdown for about 4 to 6 weeks, this is not economically affordable nor physically feasible for many cities, states, or countries without severe collateral damage, such as the consequences to economically vulnerable people. As such, alternative strategies must be considered in such cases, and they should be tested and perfected in mathematical models before application.

From the Mathematical Model to the Real Epidemic.
Regarding the results of mathematical models and their application in practice, some matters of utmost importance are (i) how to translate and relate the theoretical control actions and parameters into practical control parameters of the real-world epidemic and (ii) how to evaluate the evolution of the control measures in the real-world epidemic? While the results of control strategies from the models seem promising, they are of no practical interest if we fail to map them to real life. To illustrate how this can be done, let us consider one strategy at a time.

Social Distancing.
Social distancing was shown to have effect in the reproduction number, R t , since the parameter represents the interaction between individuals that will result in exposition to the infection. is study found the basic reproduction number as R 0 � 3.55 for Manaus and then R t � 1.51 after the government decree of March 23rd, imposing a rather mild set of social distancing measures. Social distancing in Brazil has been measured by means of smartphone apps, and for the period under study, the State of Amazonas achieved indices of reduction of mobility of the order of 50%, which reflected in the reduction of the reproduction number in the period. us, although monitoring methodologies may find enhancements in the following weeks, they seem to be sufficiently adequate to allow the verification of the control parameter R t in practice.
While other methodologies might be available, such image processing from aerial images by vants or street surveillance cameras, the individual localization of a fraction of the population by means of smartphone apps seems to give a sharper account of the mobility dynamics of a given geographic area. To this end, the mapping of EIC to isolation indices from smartphone monitoring can help provide a link between the model parameter and the real-world epidemic.

Confinement of Susceptible Individuals.
e simulations for scenario 3 applied the confinement rate R CNF � 0.02 (or 2% of individuals per day) and confinement period T CNF � 15 days. While the confinement period is rather intuitive and easy to understand, the confinement rate might be a bit trickier, especially when it comes to translating it into clear instructions on how to proceed in practice. For a big company, it could represent the entering of 2% of its (supposedly) susceptible employees in isolation in a daily basis and leaving isolation after T CNF days. is would create a flow of susceptible to confined and back to susceptible that for these parameter values would reach a peak of 20% of the employees confined simultaneously. Although this can be clear enough as an instruction to employees in a big company, it would be rather confusing and hardly feasible to society as a whole.
us, a different approach would be to confine individuals in the proportion of the peak of the curve. In the example of Manaus, where about 400, 000 would be confined in the peak, this would represent roughly 20% of the population.
us, as a rule of thumb, after checking the strategy in the model, the government could suggest the confinement of 20% of (supposedly) susceptible individuals to be implemented everywhere except basic services. e results of such measure could also be assessed using the monitoring of the location of smartphone devices. It is important to note that this strategy works best upon testing to assure that the individuals being confined are effectively susceptible ones.

Quarantine of Infected Individuals.
e simulations in Section 4.2.3 applied quarantine rate R QRT � 0.10 (or 10% of the infected individuals that were identified during the period T LIC ), quarantine period T QRT � 15 days, and average delay until quarantine of an infected individual as T LIC � 2 days. While the confinement strategy deals with individuals that cannot spread the infection by themselves, the quarantine deals with more hazardous elements of the population that should be monitored more closely. As tests are lacking in Brazil and the ones applied take long until any conclusive result is achieved, this strategy has been applied on the basis of clinical exam of symptomatic individuals.
us, under the current conditions and considering recent studies that found that, for each COVID-19 confirmation, there are about at least 7 asymptomatic or mild positives that were missed, it would be very far from reality to set the quarantine rate to much larger values than 10%. Quarantined infected must be monitored by health agencies during T CNF to guarantee that they comply with the quarantine period under the right conditions to make it effective In practice and with time, this strategy would be equivalent to effectively quarantine about 70% of the infected individuals at the peak since the strategy would greatly reduce the inflow of newly exposed individuals. is requires intensive testing even in asymptomatic individuals, and for this reason, it was left for application from early June such that authorities have time to prepare massive testing campaigns.
As shown in the simulations for the case of Manaus, having this strategy taken into effect would allow that the number of infections progressively decreases along with the number of quarantined individuals after about a month. Under these parameters, the number of quarantined and the number of infected in circulation become roughly the same with time and decrease together to reach values that would clear the overload of the medical infrastructure by the mid-June, 2020.

e Model Results and Infrastructure Planning.
e results indicate that the existing medical infrastructure shall be overwhelmed soon under all scenarios considered in the study. How overwhelmed will fundamentally depend on the course of actions and measures from the last weeks of April on. e first scenario considers the continuity of the current distancing measures reaching an EIC of about 0.57 and, in this case, the number of infected individuals skyrockets. In this case, the number of required simple beds and ICU beds would go well beyond the possibility of the current medical infrastructure and possibly any new infrastructure that might be added in the following weeks. e second scenario considers intensifying the current distancing measures such that the EIC reaches about 0.70 and the maintenance of this scenario for at least ten to twelve weeks. In this case, further maintenance of distancing measures would cause a consistent decrease in the number of infected and hospitalized and bring the number of infections under control a few months later. In this scenario, the number of ICU beds would be overwhelmed by about 100% of capacity as well in the peak of this first wave of infection. Although this is well over the current capacity, newly installed ICU beds can be achieved timely if isolation measures are taken soon and the medical infrastructure is expanded accordingly. e third scenario considers intensifying the current distancing measures such that EIC reaches 0.80 with the confinement of susceptible and quarantine of infected individuals for four to six weeks, along with confinement of susceptible that would reach 20% of confined at peak. After 30 days, the policy of isolating and monitoring infected individuals would be established, and after 45 days, the measures could be relaxed to the levels observed in early April. Under this scenario, ICU beds would be overwhelmed during a much shorter period. is is a deficit more easily reachable by building new infrastructure than the previous ones.

e Effect of Isolation.
From the EIC values obtained in each scenario, the effect of isolation upon the epidemic curves became clear. Compared to one another, the strategy presented in scenario 3 shows far more effectiveness than the other two, yet it allows further relaxation of isolation in the period following the period of severe isolation. is can give some insight into the role of severe social distancing measures to allow a more rapid resume of economical activities.
e EIC for the three scenarios is presented in Figure 20.

Concluding Remarks
e scenarios presented in this study show that the outcomes are highly dependent upon both the strength and timing plus duration of application of epidemic control strategies. As an unprecedented epidemic in recent history and the first worldwide epidemic of large scope to hit the globalized and interconnected world, the coronavirus outbreak is a novel experience and, as such, has demanded the mobilization of efforts from the most varied fields. As some countries and cities are especially prone to the occurrence of major human disasters in the midst of this pandemic, it is urgent that control strategies are meticulously studied and be ever changed as necessary in order to attend their function of controlling the epidemic, minimize deaths, and minimize economic loss which leads to poverty and misery. e case of Manaus is very emblematic since the epidemic is at the beginning (as this manuscript is written in mid-April 2020), and there are forms of avoiding more severe consequences if actions are taken timely and with the adequate severity that the current situation calls for. Many other possible measures and strategies can be tested using the same set of parameters or new ones that eventually become available in the next days in the medical literature. Nevertheless, the current state-of-affairs indicates that the medical infrastructure shall be overwhelmed for at least four weeks regardless of any actions that will be taken to contain the epidemic at the moment. is further illustrates the importance of timely and adequately strong isolation measures. is is especially relevant due to the fact that the epidemic curve is currently leading isolation measures by about two weeks, which means that any actions taken now will reflect on the curves in about two weeks.
is study aimed at presenting a model with simple application for the study of COVID-19. We presented the SEIRCQ model and its dashboard developed and implemented in Python language. e dashboard allows the reproduction of the studies presented in this research and several others, and it is available online free of charge at https://dashseir.herokuapp.com/. We encourage the application of the dashboard by health system authorities for applied studies such that they can draw their own possible scenarios when dealing with strategies to combat the epidemic. Furthermore, a case study is presented of the first wave of the epidemic in Manaus (AM), Brazil, with prognostics for a time window of about 150 days under three hypothetic scenarios. Data Availability e data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest
e authors declare that they have no conflicts of interest.


extfieldsofstudy:
Computer Science
Mathematics

text:
Deep Incomplete Multi-View Multiple Clusterings

Multi-view clustering aims at exploiting information from multiple heterogeneous views to promote clustering. Most previous works search for only one optimal clustering based on the predefined clustering criterion, but devising such a criterion that captures what users need is difficult. Due to the multiplicity of multi-view data, we can have meaningful alternative clusterings. In addition, the incomplete multi-view data problem is ubiquitous in real world but has not been studied for multiple clusterings. To address these issues, we introduce a deep incomplete multi-view multiple clusterings (DiMVMC) framework, which achieves the completion of data view and multiple shared representations simultaneously by optimizing multiple groups of decoder deep networks. In addition, it minimizes a redundancy term to simultaneously %uses Hilbert-Schmidt Independence Criterion (HSIC) to control the diversity among these representations and among parameters of different networks. Next, it generates an individual clustering from each of these shared representations. Experiments on benchmark datasets confirm that DiMVMC outperforms the state-of-the-art competitors in generating multiple clusterings with high diversity and quality.

I. INTRODUCTION
With the wide-application of Internet of Things, many collected data are naturally represented with multiple feature views. For instance, an image can be encoded by its color, texture, shape and spatial descriptors. These feature views embody the consistent and complementary information of the same image, which spur extensive research of learning on multi-view data [1], [2]. Fusing these feature views can not only form a comprehensive description of the data, but also benefit the learning tasks on them, such as classification [3], clustering [4] and metric learning [5]. This work focuses on multi-view clustering (MVC), which aims to excavate complementary and consensus information across multiple views to identify the essential grouping structure with no requirement of labels from these data.
Various attempts have been made to find essential grouping structures of multi-view data. Some algorithms force the clustering results of different views being consistent with each other via correlation maximization [6], co-regularization [7], This paper was accepted by icdm 2020. * Corresponding author: kingjun@sdu.edu.cn (Jun Wang). This work is supported by NSFC (No. 62072380, 61872300 and 62031003). fusing multiple similarity matrices of individual views [8], or exploring common and diverse information among views [9]. Other approaches assume the clusters of a clustering are embedded in different subspaces and try to explore these subspaces and find clusters therein [10], [11]. More recently, deep learning techniques also have been proposed to extract the high-order correlation and dependency between multi-view data for effective clustering [12], [13]. Besides, some other efforts study how to work on multi-view data with missing views, i.e., some objects are not available for all the views [14], [15].
These mentioned MVC algorithms account for the multiplicity of multi-view data, but focus on generating a single clustering result only. In practice, such multiplicity can also support to group the data in multiple different but meaningful clusterings [16], [17]. For example, a bunch of facial images represented with heterogeneous views can be separately grouped from the perspective of identity, sex and of emotions. All these groupings are different but yet meaningful. These reasonable groupings of the data are potentially useful for some purposes, regardless of whether or not it is optimal according to a specific clustering criterion [18]. Alike traditional clustering that focuses on the quality, multiple clusterings additionally pursue the diversity among alternative clusterings. However, it is a knotty task to balance the diversity and quality of these clusterings [19]. Previous works tried to obtain multiple clusterings in independent (or orthogonal) subspaces [20]- [22], by eliminating redundancy between the clusterings that are generated successively [23], [24], by executing clustering assignment again for the generated base clusterings [18], or by simultaneously gaining multiple clusterings and controlling the redundancy. However, they were designed for single-view data only.
A few efforts have been made toward exploring multiple clusterings on multi-view data. Multi-view multiple clusterings (MVMC) [17] mines the individual and shared information of multi-view data by utilizing self-representation learning [11], and then decomposes the combinations of the individuality feature matrices and commonality feature matrix by semi-nonnegative matrix factorization [25] to obtain multiple clusterings. DMClusts [26] is another multi-view multiple clusterings algorithm based on deep matrix factorization. It decomposes the multi-view data matrices layer-by-layer to obtain multiple common subspaces and generate corresponding clusterings therein. These two efforts still ideally assume all the data views are complete. However, this assumption is often violated in practice for some inevitable reasons [3], [14], such as the temporary failure of sensors or the human caused errors. As a result, the collected multi-view data are often incomplete. A simple strategy is to remove the samples with missing feature views, but this strategy obviously may throw away too much information, especially when with a high missing rate. Incomplete multi-view clustering (IMC) solutions have been proposed to address this practical issue. Some of them resort to matrix factorization to extract the shared subspace [14], [27], fill the missing information [28], or use Generate Adversarial Network (GAN) [29] to replenish the missing data [15], [30]. However, none of existing IMC methods can generate multiple clusterings with both high quality and diversity.
To address the drawbacks mentioned above, we propose a deep incomplete multi-view multiple clusterings framework (DiMVMC, as illustrated in Figure 1). DiMVMC adopts M decoder networks to generate M clusterings from M representational subspaces and to complete the missing features of instances. The input for the m-th decoder network is the m-th shared subspace H m which is randomly initialized at first, while the output is the reconstruction of multi-view data. By alternatively optimizing the decoder deep networks and {H m } M m=1 , DiMVMC can achieve the completeness and M individually shared subspaces {H m } M m=1 simultaneously. Moreover, these decoder networks are not isolated, but additionally controlled by a redundancy term based on Hilbert Schmidt Independence Criterion (HSIC) [31], which further enforces the diversity among subspaces and thus reduces the redundancy between clusterings. The main contributions of our work are summarized as follows: (i) We study how to generate multiple clusterings on multiview data with missing samples, which is an important and practical topic, but more challenging and mostly overlooked by previous solutions. To our knowledge, DiMVMC is the first deep approach to generate multiple clusterings. (ii) DiMVMC discards the ad-hoc encoder part of autoencoder and works in a unsupervised way, as such DiMVMC has a lower network complexity and can more flexibly deal with data incompleteness in different views. In addition, it uses a redundancy quantification term to reduce the overlap among decoder networks for producing less overlapped representational subspaces, and finally generates diverse clusterings in these subspaces. (iii) DiMVMC can find multiple clusterings with higher quality and diversity than the state-of-the-art competitors [17], [21], [24], [26], [32], and it is robust to missing data in a wide range.

II. RELATED WORKS
Our work has close connections with two lines of related works, incomplete multi-view clustering and multiple cluster-ings.

A. Incomplete Multi-view Clustering
Various multi-view clustering solutions have been introduced, most of which focus on extracting the consistent/complementary information from different views to induce a consolidated clustering [6], [33], [34], while others additionally mine the individual information to achieve a more robust clustering [11]. These methods all build on the assumption that all the data views are complete. While in practice, it is more often that some samples are absent in some views. To handle such more challenging incomplete multi-view clustering (IMC), Li et al. [14] presented the first solution (named PVC) based on NMF (Nonnegative Matrix Factorization) [35], which learned common representations for complete instances and private latent representations for incomplete instances with the same basis matrices. Next, PVC used the common and private representations to seek a clustering. Zhao et al. [36] further integrated PVC and manifold learning to learn the global structure of multi-view data. Nevertheless, these NMF-based methods can only deal with two-view data, limiting their application scope. Weighted NMF-based approaches [37], [38] were also proposed to deal with more than two views by filling the missing data and assigning them with lower weights. Wen et al. [39] added an error matrix to compensate the missing data, and combined the original incomplete data matrix with the error matrix to form a completed data matrix for clustering. All these solutions in essence build on NMF, which performs shallow projection that cannot well mine the complex relationships between low-level features of multi-view data.
To mine nonlinear structures and complex correlations among multi-view data, Wang et al. [30] proposed the consistent GAN for the two-view IMC problem, which used one view to generate the missing data of the other view, and then performed clustering on the generated complete data. Xu et al. [15] sought the common latent space of multi-view data and performed missing data inference via combining GAN with autoencoder. Ngiam et al. [40] extracted shared representations by training a two-view deep autoencoder to best reconstruct the two-view inputs. Zhang et al. [41] combined auto-encoder with Bayesian framework to fully exploit partial multi-view data to produce a structured representation for classification. These shallow/deep multi-view clustering solutions still focus on producing a single clustering. Given the multiplicity of multi-view data, it is more desirable to find different clustering results from the same data and each clustering gives an independent grouping of the data.

B. Multiple Clusterings
Multiple clusterings focus on how to generate different clusterings with both high quality and diversity from the same data [19]. It is less well studied than single/multi-view clustering and ensemble clustering [42], [43], due to its requirement on generating multiple groups of results, and the difficulties on guaranteeing the good quality and diversity at the same time. Bae et al. [23] proposed a multiple clusterings solution based on hierarchical clustering (COALA). The main idea of COALA is that instances with higher intra-class similarity still gather in one cluster, while those with lower intra-class similarity are considered to be placed into different clusters for another clustering. Jain et al. presented Dec-kmeans [32], which obtained diverse clusterings simultaneously by finding multiple groups of mutually orthogonal cluster centroids. Unlike COALA and Dec-kmeans that directly control the diversity between clustering results, other solutions control the diversity between clustering subspaces and then generate different clusterings in these subspaces. Cui et al. [20] greedily found orthogonal projection matrices to get different feature representations of the original data and then found clusterings in these orthogonal subspaces. Mautz et al. [21] also tried to explore multiple mutually orthogonal subspaces, along with the optimization of k-means objective function, to find non-redundant clusterings. However, the orthogonal constraint is too strict to generate more than two clusterings. Wang et al. [22] generated multiple independent subspaces with semantic interpretation via independent subspace analysis, and then performed kernel-based clustering in these subspaces to explore diverse clusterings. Yang and Zhang [24] explicitly defined a regularization term to quantify and minimize the redundancy between the already generated clusterings and the to-be-generated one, and then plugged this regularization into the matrix factorization based clustering [25] to find another clustering. Wang et al. [44] and Yao et al. [45] directly minimized the redundancy between all the to-be-generated clusterings to simultaneously find alternative clusterings. Besides, Caruana et al. [18] firstly generated a number of useful high-quality clusterings, and then grouped these clusterings at the meta-level, and thus allowed the user to select a few high-quality and non-redundant clusterings for examination. However, these multiple clusterings methods are still restricted to single-view data.
Given the multiplicity of multi-view data, it is desirable but more difficult to generate multiple clusterings from the same multi-view data. Two approaches have been proposed for attacking this challenging task. MVMC [17] first extends multi-view self-representation learning [11] to explore the individuality information encoding matrices and the commonality information matrix shared across views, and then combines each individuality similarity matrix and the commonality similarity to generate a distinct clustering by matrix factorization. However, given the cubic time complexity of the self-representation learning, MVMC can hardly be applicable on datasets with a large number of samples. To alleviate this drawback, DMClusts extends the deep matrix factorization [13], [46] to collaboratively factorize the multi-view data matrices into multiple representational subspaces layer-bylayer, and seeks a different clustering of high quality per layer. In addition, it introduces a new balanced redundancy quantification term to guarantee the diversity among these clusterings, and thus reduces the overlap between the produced clusterings.
The above-mentioned single/multi-view multiple clusterings solutions assume all instances are complete across views, and project data into linear and shallow subspaces. Therefore, they cannot capture the complex correlations between views and nonlinear clusters in subspaces when data are incomplete. To address these issues, we introduce DiMVMC to mine multiple clusterings from multi-view data with missing instances. DiMVMC can capture the complex correlations among views and complete data by multiple decoder networks, and thus generate multiple nonlinear clusterings with quality and diversity.

III. THE PROPOSED METHOD
It was empirically demonstrated that different data views are complementary to each other, and they carry distinct information for generating diverse clusterings with quality [17], [26]. However, multi-view multiple clusterings is still challenging due to the difficulty in modeling the unknown and complex correlation among different views. Moreover, data with missing views and the required diversity between clusterings further upgrade the difficulty to address the incomplete multi-view multiple clusterings problem. Autoencoder is typically used to reconstruct the data with missing/noisy features [15], [41], [47]. The encoder takes input the incomplete data and learns a compact representation, from which the decoder recovers the missing values. To avoid the ad-hoc design of encoder for the incomplete cases in different views, we skip the encoder and take the shared subspace representation H m as the input for the m-th decoder network, from which the observed data are reconstructed and the missing data are completed, as shown in Fig. 1. In addition, we quantify and minimize the redundancy among these subspaces for generating diverse clusterings therein. The following subsections elaborate on the above procedure.

A. Generating Multiple Representation Subspaces
Suppose a multi-view dataset with V views has N instances. We use x v n ∈ R dv (v = 1, · · · , V ) to denote the feature vector for the v-th view of the n-th instance, where d v is the feature dimension of the v-th view. An indicator matrix Λ ∈ {0, 1} V ×N for all instances is defined as: where each column of Λ is the status (present/absent) of instances for corresponding views. The relation 1 ≤ V v=1 Λ vn ≤ V holds, such that each instance is present in at least one view.
The aim of incomplete multi-view multiple clusterings is to integrate all the incomplete views to generate multiple clusterings. Inspired by cross partial multi-view networks for classification and by multi-view subspace learning [41], [48], we project instances with arbitrary view-missing patterns into the shared representational subspaces in a flexible way, where the subspaces include the information for observed views. That is, each view can be reconstructed by the obtained shared representation. Based on the reconstruction point of view [49], we use the joint distribution to concrete the above idea as follows: where h i ∈ R d is the multi-view shared representation of the ith instance, and can be obtained. However, just alike typical subspace learning methods [50], [51], (2) also optimizes one subspace and a single clustering result therein. Because of the multiplexes, multi-view data has a mix of diverse distributions. Therefore, multiple different subspaces and clusterings can co-exist. To gain multiple (M ) clusterings, we extend (2) as: where h m i is the shared representation of the i-th instance in the m-th shared subspace. Based on different views in S i , we model the likelihood with respect to h m i given the observation x v i as: where ∆ represents the reconstruction loss. Here, we adopt the l 2 -norm for this reconstruction part. f v m is the mapping function from the common subspace H m to the v-th view and Θ v m are decoder network parameters of f v m .
Without loss of generality, suppose the data are independent and identically distributed, we can induce the log-likelihood function as follows: Since maximizing the likelihood is equivalent to minimizing the loss ∆, by considering the missing case, we can obtain the following objective function for the decoder network: Optimizing the above equation can generate M shared representations {H m } M m=1 , each of which will be used to generate a clustering result. Unlike traditional autoencoder based solutions [15], [41], DiMVMC skips the encoder networks, but takes the shared subspace representation H m as the input for the m-th decoder to complete multi-view data, as done by f v m in (6). As such, DiMVMC does not need to specifically consider diverse missing cases of multi-view data, while still makes full use of observed data.

B. Reducing Redundancy between Subspaces
By minimizing (6), we can generate multiple common subspaces from incomplete multi-view data. For multiple clusterings, besides the quality of different clusterings, the diversity between clusterings is also important [19]. The diversity is usually approximately obtained by minimizing the redundancy between these subspaces. Orthogonality is the most common approach that forces two subspaces being orthogonal with each other. Orthogonality based methods may still generate multiple clusterings with high redundancy, since these orthogonal subspaces can still produce clusters with the same structure [26]. Furthermore, orthogonality does not specify which properties of the reference clustering should or should not be retained. Kullback Leibler (KL) divergence was also adopted to find diverse clusterings [52], but KL divergence is not symmetric and not applicable for high-dimensional data, due to its high time and space complexity.
Hilbert-Schmidt Independence Criterion (HSIC) [31] measures the squared norm of the cross-covariance operator over H m and H m in the Hilbert kernel space to estimate the dependency. It is empirically given by: where K m and K m ∈ R N ×N are Gram matrices, defined as an inner product between vectors in a specific kernel space.
In this paper, we adopt the inner product kernel to specify K m = (H m ) T H m . A lower HSIC value means two subspaces are less correlated. This empirical estimation is simpler than any other kernel dependence test, and requires no user-defined regularisation. In addition, it has a solid theoretical foundation, a fast learning rate with guaranteed exponential convergence, and the capability in measuring both linear and nonlinear dependence between variables. For these merits, we adopt HSIC to quantify the overlap between generated subspaces {H m } M m=1 .

C. Unified Model
By integrating (6) with (7), we define the loss function of DiMVMC as: is the normalization factor, λ is the hyperparameter to balance the sought of M subspaces and diversity between them. DiMVMC can generate multiple common subspaces {H m } M m=1 and complete missing data simultaneously via minimizing (8). Since the optimal solution cannot be analytically given, we employ an optimization strategy that alternatively updates Θ v m or H m in an iterative way, while fixing the others. More specifically, Θ v m and H m are randomly initialized at first. The detailed optimization process is given in Algorithm 1. Once the optimization is done, k-means is implemented on each obtained subspace H m , and thus M clusterings with quality and diversity can be accordingly generated.
In the subspace clustering, it is desired that the subspace representation is sparse but captures the group-level semantic information. There are different ways to bring in sparsity. For example, we can add a drop-out layer for the deep learning approach. Here, to make an intuitive implementation, we add a sparsity-induced regularization into the above loss function and define a so-called Sparse DiMVMC: where || · || 1 is the l 1 norm of the matrix. When α = 0, (9) goes back to the plain DiMVMC.

IV. EXPERIMENTAL RESULTS AND ANALYSIS A. Experimental Setup
In this section, we evaluate the performance of our proposed DiMVMC on four benchmark multi-view datasets, as described in Table I. Caltech20 [53] is a subset of Caltech-101 of 6 categories, which contains 2,386 instances and 20 clusters. We utilize 254D CENHIST vector, 512D GIST vector and 928D LBP vector as three views. Handwritten [53] is comprised of 2,000 data points from 0 to 9 digit classes, with 200 data points for each class. There are six public features available. We utilize 240D pixel averages feature, 216D profile correlations feature and 76D LBP feature as three views. Reuters [54] is a textual data set consisting of 111,740 documents in five different languages (English, French, German, Spanish and Italian) of 6 classes. We randomly sample 12000 documents from this collection in a balanced manner. We further do dimensionality reduction on the 12000 samples following the methodology of [55] and represent each view by 256 numeric features. Mirflickr includes 25,000 samples collected from Flicker with an image view and a textual view. Here, we remove textual tags that appear less than 20 times, and then remove samples without textual tags or semantic labels, finally we get 16,738 samples for experiments [17]. We compare DiMVMC against with six representative and recent multiple clusterings algorithms, including Dec-kmeans [32], Nr-kmeans [21], OSC [20], MNMF [24], MVMC [17] and DMClusts [26]. Dec-kmeans is a representative multiple clusterings solution based on orthogonalizing the clustering centroids. Nr-kmeans [21], OSC [20] and MNMF [24] attempt different techniques to seek subspaces and multiple clusterings therein. For this reason, they have close connections with our approach and are used for experimental comparison. MVMC [17] and DMClusts [26] are the only two multi-view multiple clusterings algorithms at present.
None of these compared multiple clusterings algorithms can directly handle missing data, we fill the missing features (instances) with average values for each view at first, and then apply these solutions. For those single-view algorithms, following the solution in [17], [26], we concatenate the feature vectors of multi-view data and then apply them on the concatenated vectors to generate different clusterings. Note, we do not take the deep/incomplete multi-view clustering solutions for experiments, since they can only output a single clustering. In the following experiments, the input parameters of the comparing methods are fixed (or optimized) as the authors suggested in their papers or generously shared codes.
DiMVMC selected the input parameters from the following ranges: λ ∈ {10 −3 , 10 −2 , · · · , 10 3 }, d ∈ {64, 128}, and M = 2. The alternate clusterings are generated by applying k-means algorithm on each shared subspace H m . The number of clusters for alternate clusterings was fixed to K of each dataset, as listed in Table I. In this paper, DiMVMC simply adopts two layers' network structure for each mapping f v m to reconstruct the v-th view from the m-th shared subspace H m . The demo code of DiMVMC is available at http://mlda.swu.edu.cn/codes.php?name=DiMVMC.
Following the evaluation protocol used by the baseline methods [17], [24], [44], we measure the quality of multiple clusterings via the average SC (Silhouette Coefficient) or DI (Dunn Index), and the diversity via the NMI (Normalized Mutual Information) or JC (Jaccard Coefficient). SC and DI quantify the compactness and separation of clusters within a clustering, while NMI and JC quantify the similarity of clusters of two clusterings C 1 and C 2 . We want to remark that unlike traditional clustering problem, a lower value of NMI and JC means the two alternative cluterings are less overlapped, so a smaller value of them is more preferred.

B. Discovering Multiple Clusterings
For the first experiment, we assume the four multi-view datasets are complete without any missing data. We report the average results of ten independent runs and standard deviations of each method on generating two alternative clusterings in Table II.
From Table II, we has the following important observations: (i) Multi-view vs. Single-view: DiMVMC, DMClusts and MVMC can be directly applied on multi-view data, and their generated two clusterings often have a lower redundancy than those generated by other compared methods. That is because these methods lack a redundancy control term or their redundancy strategies are difficult to be optimized. Our following experiment will further analyze the importance of redundancy control. DiMVMC frequently obtains a better quality than compared methods that can only work on the concatenated single view, which suggests that concatenating the feature vectors overrides the intrinsic nature of multi-view data, which helps to generate multiple clusterings with quality. These comparisons prove the effectiveness of our proposed approach in fusing multiple data views to generate multiple clusterings with diversity and quality. (ii) Shallow methods vs. DiMVMC: To our knowledge, DiMVMC is the first deep approach to generate multiple clusterings, and it often performs better on quality metrics (SC and DI), owing to the high-level expression ability of decoder networks. Even though, DiMVMC sporadically has a lower value on SC than some of compared methods. That is due to the widely-recognized dilemma of obtaining alternative clusterings with both high diversity and quality. DiMVMC has a larger diversity. That is explainable, since it can explore diverse nonlinear representation subspaces by decoder networks, while these shallow methods can only obtain lowlevel feature subspaces. Therefore, DiMVMC has a better tradeoff between quality and diversity than these compared methods. Although DiMVMC, DMClusts and MVMC can generate diverse clusterings from the same multi-view data, DiMVMC manifests a better performance than the latter two. That is because DiMVMC can mine the complex correlations between views and features via decoder networks, whereas these compared methods cannot.
In summary, even with complete data across views, DiMVMC outperforms compared methods across different multi-view datasets in terms of quality and diversity.

C. Impact of Missing Data
To study the performance of DiMVMC with missing data views, we define the missing rate as τ = 1 − v,n Λvn V ×N . The instances are randomly selected as missing ones, and the missing views are randomly erased by guaranteeing at least one of them is observed. In this paper, the missing rate is varied from 0 to 50% with an interval as 10%. Figure 2 shows the impact of missing rate of data on the clustering performance of DiMVMC and of compared methods. With the increase of missing rate, the performance of multiple clusterings methods does degrade. Nr-kmeans [21] and OSC [20] are always in a high position in terms of SC at the beginning. This indicates that they can obtain multiple clusterings of high quality under a small rate of missing instances. However, their SC values drop faster than others with the further increase of missing rate, since they do not take into account the missing instances/features. Furthermore, their diversity (1-NMI) is also at a low-level, suggesting their orthogonal subspaces still have a relatively high redundancy.
The performance curves of multi-view methods (MVMC, DMClusts and DiMVMC) drop more slowly than the singleview methods as the increase of missing rate. That is because the correlation between views helps to reduce the impact of missing data, and concatenating features cannot well capture this complementary information. In addition, although the SC curve of DiMVMC is not always in a relatively high level, it always holds better diversity than compared methods. In addition, it holds more stable quality (SC) and diversity (1-NMI) curves than compared methods. This observation again echoes the dilemma of balancing the quality and diversity of multiple clusterings.
Finally, DiMVMC can generate clusterings of diversity controlled by the HSIC term. It is the first multiple clusterings algorithm that considers the missing data views, it can reconstruct the incomplete multi-view dataset to complete the missing data views. As such, DiMVMC is more robust to missing data. By contrast, the simple data complement strategies used by compared methods are not so robust. As a result, the performance of the compared methods is not as stable as DiMVMC is. Figure 2 proves that DiMVMC has a better tradeoff between the quality and diversity of multiple clusterings, and is more competent in dealing with incomplete data than compared methods. That can be attributed to the adopted decoder networks and diversity control term, which can more well capture the correlations among different views and handle diverse missing patterns, and enforce the diversity among subspaces.

D. Parameter Analysis
Parameter λ balances the generation of multiple subspaces and the diversity control among these subspaces. We study the impact of λ by griding it from 10 −3 to 10 3 and plot the variation of quality (SC) and diversity (1-NMI, the larger the better) of DiMVMC on Caltech20 dataset in Figure 3a. We see that: (i) diversity (1-NMI) steadily increases at first and then gradually increases; (ii) the quality (SC) gradually decreases and then keeps relatively stable as λ further increases. Overall, SC keeps relatively stable as λ varies, but always below the starting point (λ = 0, no diversity control). This pattern is explainable, since the promotion of diversity between clusterings is often associated with the scarification of quality. We can conclude that λ indeed helps to boost the diversity between clusterings.
We vary M (number of alternative clusterings) from 2 to 6 on Caltech20 dataset to explore the variation of average quality (SC) and diversity (1-NMI) of multiple clusterings generated by DiMVMC. In Figure 3b, with the increase of M , the average quality (SC) fluctuates in a small range while the diversity (1-NMI) decreases slowly. Overall, DiMVMC can obtain M ≥ 2 alternative clusterings of quality and diversity.
Based on the base DiMVMC, we lead an l 1 norm for each common subspace H m , extend DiMVMC to a sparse DiMVMC. We apply DiMVMC and sparse DiMVMC (with α = 5 × 10 −2 ) on Caltech20 dataset, and report the results in Figure 4a. Although the SC values of these two models are both around 0.007, sparse DiMVMC has an average NMI (the lower the better) as 0.022, which is nearly 12% lower than DiMVMC (average NMI 0.025) . This comparison shows that sparsity helps to generate less correlated subspaces (fewer redundant features), and to better control the diversity of alternative clusterings.

E. Convergence and Complexity Analysis
From Figure 4b, we can find that DiMVMC and Sparse DiMVMC often converge within 15 epochs, while DiMVMC(λ = 0) without diversity control converges in 30 epochs. This trend not only proves the efficiency of our proposed alternative optimization strategy, but also shows that our diversity control term and the added l 1 norm do not increase the complexity.
The memory complexity of DiMVMC can be analyzed by two parts. For simplicity, suppose l is the number of layers, d is the dimension for any   batch optimization technique. Table III gives the run-time of all compared methods. The compared methods are run on a linux server 1 . All methods are implemented by Matlab2014a, except Nr-kmeans, DMClusts and DiMVMC are implemented by Python. We observe that the three fastest methods are OSC, Dec-kmeans and DM-Clusts, respectively. OSC and Dec-kmeans do not consider the correlations between views and work on the concatenated views, so they run faster than others. DMClusts employs the efficient semi-NMF [25] to decompose multi-view data layer by layer and generates multiple clustering simultaneously. Although MNMF also builds on efficient semi-NMF, it is constrained by the reference clustering when seeking the other clustering, and has a larger run-time than DMClusts. Nrkmeans needs to update the clustering center many times, so it has a longer run-time than MNMF. MVMC involves with time demanding self-representation learning and the factorization of multiple representational matrices, so it has a longer runtime than others. Our DiMVMC has the largest run-time, since it has to capture the complex correlations between views and generate multiple clusterings with nonlinear clusters via optimizing multiple decoder networks. However, DiMVMC 1 Configuration: Intel Xeon8163, 1TB RAM with NVIDIA Tesla K80. almost always generate multiple clusterings with better quality and diversity than these compared methods.

V. CONCLUSIONS
In this paper, we introduced the DiMVMC model to explore alternative clusterings from the ubiquitous incomplete multiview data. DiMVMC can complete the missing data via a group of decoder networks, and seek multiple shared but diverse subspaces (clusterings therein) by further reducing the overlaps between subspaces. Experimental results on benchmark datasets confirm the effectiveness of DiMVMC. We will explore deep alternative clusterings by merging prior knowledge of different perspectives.