key: cord-0927395-zni33t6h authors: Tutsoy, Onder; Polat, Adem title: Linear and non-linear dynamics of the epidemics: System identification based parametric prediction models for the pandemic outbreaks date: 2021-08-09 journal: ISA Trans DOI: 10.1016/j.isatra.2021.08.008 sha: 0ec4d4f218e0f13d828a22f2695f1305594cd65f doc_id: 927395 cord_uid: zni33t6h Coronavirus disease 2019 (COVID-19) has endured constituting formidable economic, social, educational, and phycological challenges for the societies. Moreover, during pandemic outbreaks, the hospitals are overwhelmed with patients requiring more intensive care units and intubation equipment. Therein, to cope with these urgent healthcare demands, the state authorities seek ways to develop policies based on the estimated future casualties. These policies are mainly non-pharmacological policies including the restrictions, curfews, closures, and lockdowns. In this paper, we construct three model structures of the S [Formula: see text] I [Formula: see text] I [Formula: see text] I [Formula: see text] D-N (suspicious S [Formula: see text] , infected I [Formula: see text] , intensive care I [Formula: see text] , intubated I [Formula: see text] , and dead D together with the non-pharmacological policies N) holding two key targets. The first one is to predict the future COVID-19 casualties including the intensive care and intubated ones, which directly determine the need for urgent healthcare facilities, and the second one is to analyse the linear and non-linear dynamics of the COVID-19 pandemic under the non-pharmacological policies. In this respect, we have modified the non-pharmacological policies and incorporated them within the models whose parameters are learned from the available data. The trained models with the data released by the Turkish Health Ministry confirmed that the linear S [Formula: see text] I [Formula: see text] I [Formula: see text] I [Formula: see text] D-N model yields more accurate results under the imposed non-pharmacological policies. It is important to note that the non-pharmacological policies have a damping effect on the pandemic casualties and this can dominate the non-linear dynamics. Herein, a model without pharmacological or non-pharmacological policies might have more dominant non-linear dynamics. In addition, the paper considers two machine learning approaches to optimize the unknown parameters of the constructed models. The results show that the recursive neural network has superior performance for learning nonlinear dynamics. However, the batch least squares outperforms in the presence of linear dynamics and stochastic data. The estimated future pandemic casualties with the linear S [Formula: see text] I [Formula: see text] I [Formula: see text] I [Formula: see text] D-N model confirm that the suspicious, infected, and dead casualties converge to zero from 200000, 1400, 200 casualties, respectively. The convergences occur in 120 days under the current conditions. Even though pharmacological developments like vaccines have expected to yield some successful results, the COVID-19 has continued to be a dreadful threat for societies due to challenges in producing a sufficient number of vaccines [1] . The COVID-19 emerged in December 2019 in Wuhan city of China has spread to over 113 countries with 91,605,941 infected and 1,962,345 dead as of 13 January 2021 [2] . The second peak in the COVID-19 casualties, which is larger than the first peak, has caused considerable challenges for the healthcare providers since the hospitals have been overwhelmed with suspicious, infected, intensive care, intubated, and dead people [3] . To halt the spread of the virus, nonpharmacological policies such as closures, restrictions, and curfews have been re-imposed [4] . International organizations such as the World Health Organization (WHO) and also the state authorities require accurate models to understand the character of the pandemic diseases and also to estimate the future casualties [5] . In this paper, we develop a parametric model called as S p I n I t I b D-N (suspicious S p , infected I n , intensive care I t , intubated I b , and death D together with the non-pharmacological policies N) to predict the future pandemic casualties in the presence of the non-pharmacological policies. J o u r n a l P r e -p r o o f 3 The recent history has witnessed severe acute respiratory syndrome (SARS), Middle East respiratory syndrome (MERS), and the COVID-19 outbreaks and more than 2000, 8000, and 91,000,000 people were infected, respectively [6] . It is reported that the COVID-19 outbreak has brought about a heavier burden than the recent 2009 influenza pandemic and seasonal influenza in terms of the hospital requirements and mortality rates [7] . Although the COVID- 19 is not a new member of the pandemic diseases family as it has emerged from the SARS coronavirus, it has unseen characters including the rapid spread in the human body, high infectious speed, extreme resilience against the environmental conditions, efficient adaptation to the human body, and considerably virulent genetic variant [8] . Therefore, developing models for the COVID-19 outbreak is challenging due to these complex and time-varying dynamics [9] . Modelling of the pandemic diseases can be categorized as parametric and non-parametric where the parametric approaches are mainly based on the system identification and the nonparametric approaches are based on the statistical approaches. With respect to the non-parametric approaches, Zhu and Chen introduced a statistical disease transmission model to predict the early-stage transmissibility of the COVID-19 outbreak in China which yielded 4.2 for the infectious period with a 95% confidence interval [10] . Gupta et al. analysed the relationship between the COVID-19 stemmed mortality and air pollution with the variance and regression models which revealed a positive correlation for the nine Asian cities [11] . Similarly, Redon and Aroca reviewed the role of the climate change and the COVID-19 spread with the generalized linear models which showed that the hot weather impacts on the transmission of the virus are insignificant [12] . Rozenfeld the susceptible-exposed-symptomatic-purely asymptomatic-hospitalized-recovered-deceased (SIPHERD) model to predict the casualties in India [5] . All these models are first-order nonlinear and do not take into account the non-pharmacological policies. Recently, we developed the suspicious-infected-death (SpID) model having second-order linear coupled dynamics learned from the available data [19] . We showed that the SpID model can efficiently represent the second-order dynamics such as the distinct peak and performed eigenvalue-based character analysis. In addition, we proposed the SpID-N model with the parametrized nonpharmacological policies N and extensively analyse the role of each non-pharmacological policy on the reported casualties [20] . This research proved that the second-order dynamics of the COVID-19 occur due to non-pharmacological policies and they are not intrinsic. spread [21] . Similarly, Sahin examined the interactions between the COVID-19 and temperature, dew point, humidity, and wind [22] . Ozer analysed the distance education efforts during the COVID-19 outbreak [23] . Morgul et al. examined the relationship between the COVID-19 outbreak and psychological fatigue as a mental health issue [24] . Satici et al. assessed the COVID-19 stemmed fear and psychological distress and life satisfaction [25] . Based on these expressed gaps in the corresponding literature, we can summarize the contributions of this paper as: 1) We construct three S p I n I t I b D-N model structures; namely, linear S p I n I t I b D-N model, non-linear S p I n I t I b D-N model, and strongly non-linear S p I n I t I b D-N model to reveal the linear and non-linear characters of the COVID-19 casualties under the comprehensive non-pharmacological policies. All the parametric models expressed above have nonlinear structures, but they do not consider the non-pharmacological policies. It is a fact that non-pharmacological policies are the essential tools to control pandemic casualties. To the best of our knowledge, this is the first paper examining the linear and non-linear properties of a pandemic disease under such extensive non-pharmacological policies. 2) We modify the non-pharmacological policies since their characters have changed with the occurrence of the second peak in the COVID-19 casualties. Re-opening of the schools partial-by-partial and imposed self-curfews, for instance, are modelled and incorporated into the S p I n I t I b D-N models. 3) We enrich the SpID-N model with the intensive care I t and intubated I b to estimate the healthcare requirements. All the model parameters are assigned as unknown and learned from the available data by using the batch type least-squares (BLS) approach. In this respect, the S p I n I t I b D-N model is adaptive since it updates its parameters as the new data are available. In addition, further linear and nonlinear model structures can be constructed, but as this paper aims to analyse the character of the pandemic dynamics, we have considered only three S p I n I t I b D-N model structures. In the rest of the paper, Section 2 introduces the proposed model structures, Section 3 provides the modified non-pharmacological policies, Section 4 derives the BLS to learn the unknown parameters, Section 5 analysis the models and predicts the future pandemic casualties and Section 6 summarizes the key contributions of the paper. In this section, we introduce three parametric model structures; namely, a linear S p I n I t I b D-N model, a nonlinear S p I n I t I b D-N model, and a strongly non-linear S p I n I t I b D-N model which are all extensively analysed in terms of representing the characteristics of the pandemic diseases, specifically the COVID-19. It is important to note that various alternative structures for the linear and non-linear S p I n I t I b D-N model can be formed. However, since this paper focuses on revealing the existence of linear or non-linear dynamics, we have constructed and analysed only three model structures. The linear S p I n I t I b D-N model architecture is shown in Figure 1 As can be seen from Figure These unknown parameters are learned in Section 0. The linear S p I n I t I b D-N model in Figure 1 can be represented by considering only the arrows leaving out each sub-model as Table introduces the components of the linear S p I n I t I b D-N. All the sub-models of the linear S p I n I t I b D-N model have corresponding internal dynamics and also the coupling dynamics associated with the neighbouring sub-models. These dynamics are represented with the parameters learned from the data (reported pandemic casualties); therefore, the parameters of the pandemic diseases such as the infectious rate and recovery rate are learned implicitly. The next sub-section introduces the non-linear S p I n I t I b D-N model. We can represent the non-linear S p I n I t I b D-N model with five sub-models where the suspicious p S and infected n I sub-models are non-linearly coupled as J o u r n a l P r e -p r o o f The next sub-section introduces the strongly non-linear S p I n I t I b D-N model. Figure 3 shows the strongly non-linear S p I n I t I b D-N model structure. As can be seen from Figure 3 , all the internal dynamics of the sub-models are non-linearly coupled with the neighboring sub-models. The next section reviews the modified non-pharmacological policies which are the external inputs of the S p I n I t I b D-N models. J o u r n a l P r e -p r o o f In this section, we review the parametrized non-pharmacological policies imposed to battle against the pandemic diseases. These non-pharmacological policies are A) Curfews on the people with chronic diseases, age over 65, and age under 20, B) Curfews on the weekends and holidays, C) Closure and re-opening of the schools and universities. Since the data for these non-pharmacological policies are not directly available, it is necessary to develop mathematical models which imitate the response of them. As people with chronic diseases and age over 65 are highly defenceless against the outbreaks, curfews are implemented on them primarily. Even though people age under 20 are resilient against the outbreaks, as they are super spreaders of the virus, they are put under curfews as well. To model such curfews, consider these facts about the pandemics:  It is reported that symptoms of a pandemic diseases can appear in 14 days where the peak point is around day 7 as reported by the WHO [26] . Therefore, a nonpharmacological policy should have a transient ascent part that reaches the peak point around day 7 and a transient descent part that converges to zero at day 14 as shown in Since the duration of the curfews on the weekends and holidays are usually for two days, their response consists of the transient ascent and transient descent parts. Therefore, the response has impulse response properties whose transient ascent part is J o u r n a l P r e -p r o o f The overall response wh k u is , 14 The next sub-section introduces the closure and re-opening of the schools and universities.  is the random uncertainty of the response, Figure 6 shows the response of the model for Turkey where the schools were closed, partially re-opened, and re-closed. The next section presents the parametrized S p I n I t I b D-N models and the batch type least squares (BLS) estimator. In this section, initially, we parametrize the S p I n I t I b D-N models in terms of the known bases and unknown parameters. Then, we use the BLS optimization approach to determine the unknown parameters. The estimated sub-models of the S p I n I t I b D-N models can be represented as The unknown parameters can be determined with Equation (46) by using the constructed bases and the outputs. The next section provides the comparative results of the three S p I n I t I b D-N models. In this section, we analyse the proposed models by using the COVID-19 casualties reported by the Health Ministry of Turkey [2] . Turkey is chosen because the authors are able to reach the data required for the constructed models. Even though the casualties are reported daily by the Health Ministry, the data for the non-pharmacological policies are usually announced verbally, and understanding the written statements is not straightforward. To properly represent the character o the models, especially the non-pharmacological policies, the home country of the authors has been chosen. To optimize the unknown parameters of the constructed models, a batch least-squares (BLS) and a neural network (NN) approaches are considered. Initially, we introduce the background data and then presents the estimated casualties with developed models. We perform the mean and the standard deviation-based analysis of the models to validate the effectiveness. J o u r n a l P r e -p r o o f Figure 7 presents the real and estimated outputs of the linear S p I n I t I b D-N model with the BLS approach. As can be seen from Figure 7 , all the sub-models of the linear S p I n I t I b D-N model with the BLS are able to follow the real outputs including the peaks. Even though the initial intensive care NN is an iterative approach, whereas the BLS is a batch type approach. It is well known that the batch kind approaches provide accurate results for stochastic optimization problems. As can be seen from Figure 8 , that the suspicious p k S , infected n k I , and intubated b k I casualties have larger fluctuations due to the stochastic nature of the casualties. Henceforth, even though the NN can capture the dynamics of these sub-models, estimates with the NN yield slightly more fluctuations than the BLS estimates in Figure 7 . The next sub-section provides the real and estimated results of the non-linear S p I n I t I b D-N model with the BLS and NN approaches. characteristics, for other cases, these sub-models may provide much more accurate results. In addition, recursive approaches such as the NN can manage to learn the non-linear dynamics with higher accuracy than a batch kind approach. Figure 10 exhibits that the NN approach is able to learn the non-linear parameters with smaller estimation errors. As can be seen from Figure 10 , the recursive estimates of the unknown parameters associated with the non-linear couplings yield a smaller estimation error than the batch type learning shown in Figure 9 . It is known that the NN is a non-parametric modelling approach which can learn any linear or non-linear functions from the input-output data. This result shows that the NN has superior performance for optimizing the non-linear parameters. The next sub-section provides the real and estimated outputs of the strongly non-linear S p I n I t I b D-N model with the BLS and NN approaches. Figure 11 shows the real and estimated outputs of the strongly non-linear S p I n I t I b D-N model with the BLS. All the sub-models of the strongly non-linear S p I n I t I b D-N model have coupled internal dynamics as illustrated by Equations from (11) to (15) Figure 12 illustrates the estimates of the strongly non-linear S p I n I t I b D-N model with the NN approach. As can be seen from Figure 12 , the NN again produces smaller estimation errors but generates larger variations when the data have random characters. This is expected since the recursive approaches consider the instant data; henceforth, the latest variations can be learned while the ones in the past are forgotten. However, the batch type approaches can normalize the data and can learn the average character of them. The next sub-section compares the proposed models in terms of the corresponding mean errors and standard deviations. Figure 13 shows the mean errors and the standard deviations of the estimated outputs by the proposed models. It is clear from Figure 13 , all the sub-models of the linear S p I n I t I b D-N model yield the smallest mean errors and also the smallest standard deviations for the reported casualties of Turkey. Both the mean errors and the standard deviations increase with respect to the non-linear coupling. We can deduct from these results that the linear S p I n I t I b D-N model provides more accurate results for Turkey. However, the character of the non-linear dynamics depends on the healthcare infrastructure of the countries and the properties of the imposed pharmacological or non-pharmacological policies. It is also obvious from Figure 13 Since the linear S p I n I t I b D-N model results in the least estimation errors, we provide the estimated future casualties with the linear S p I n I t I b D-N model. Figure 14 provides predicted This paper constructed three S p I n I t I b D-N model structures consisting of the linear and nonlinear representations of the pandemic dynamic. The research has confirmed that the linear S p I n I t I b D-N model yields more than 10 times smaller mean errors and standard deviations than the nonlinear S p I n I t I b D-N model. The outperformance of the linear S p I n I t I b D-N model can stem from the inclusion of the extensive non-parametric policies, which can dominate the non-linear dynamics. Moreover, the linear S p I n I t I b D-N model predicts that the casualties will reach their minimum of around 120 days under the current conditions. 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Unknown parameters of the non-pharmacological policies for the p S , n I , t I , b I , and D sub-models, respectively. Unknown parameters of the non-pharmacological policies contributing to the p S , n I , t I , b I , and D sub-models, respectively.