key: cord-1022042-sysgh7mm
authors: Hadi, Musadaq A.; Ali, Hazem I.
title: Control of COVID-19 System using A Novel Nonlinear Robust Control Algorithm
date: 2020-11-04
journal: Biomed Signal Process Control
DOI: 10.1016/j.bspc.2020.102317
sha: 1c5108c0a43039ee03eea274dc40f9337d5c7d89
doc_id: 1022042
cord_uid: sysgh7mm

COVID-19 has been a worldwide concern since the outbreak. Many strategies have been involved such as suppression and mitigation strategies to deal with this epidemic. In this paper, a new mathematical-engineering strategy is introduced in order to control the COVID-19 epidemic. Thereby, control theory is involved in controlling the unstable epidemic alongside with the other suggested strategies until the vaccine will hopefully be invented as soon as possible. A new robust control algorithm is introduced to compensate the COVID-19 nonlinear system by propose a proper controller after using necessary assumptions and analysis are made. In addition, the Variable Transformation Technique (VTT) is used to simplify the COVID-19 system. Furthermore, the Most Valuable Player Algorithm (MVPA) is applied in order to optimize the parameters of the proposed controller. The simulation results are based on the daily reports of two cities Hubei (China) and Lazio (Italy) since the outbreak. It can be concluded that the proposed control algorithm can effectively compensate the COVID-19 system. In addition, it can be considered as an effective mathematical-engineering strategy to control this epidemic alongside with the other strategies.

In the late of December 2019, a group of pneumonia cases was identified in Wuhan (China). It was discovered that these cases were caused by β-coronavirus or what yet initially named as the 2019-novel Coronavirus (2019-nCoV) on 12 January 2020 by World Health Organization (WHO). Later, WHO officially named the disease as Coronavirus disease 2019 , while Coronavirus Study Group (CSG) suggested to name this virus as SARS-CoV-2, both issued on 11 February 2020. After that, the number of the cases were rapidly increased in Wuhan. Consequently, The Chinese scientists decided to isolate a SARS-CoV-2 from a patient within a short time on 7 January 2020 and came out to genome sequencing of the SARS-CoV-2 (Lu et al. 2020; . In 1

March 2020, a total number of reported cases were 79,968 including 2873 deaths of the COVID-19 were confirmed in China (Coronavirus disease 2020).

Studies estimated the basic reproduction number ( 0 ) of COVID-19 to be around 2.2 or even more (1.4 to 6.5) and familial clusters of pneumonia J o u r n a l P r e -p r o o f outbreaks add to evidence of the epidemic SARS-CoV-2 constantly increasing by human-to-human transmission (Riou and Althaus 2019; Liu et al. 2020; Yu et al. 2019 ).

There are two recommended strategies to handle the COVID-19 outbreak suppression and mitigation. First, suppression which is an approach of apply a strict social distancing policies that have been taken by governments such as applying stay-at-home orders, shut down public places, commercial activities, non-essential industrial activities and schools, stopping all kinds of travels, ..., etc. The aim of this strategy is to decrease the number of the reproduction , which represents the number of the infectious persons; this was applied in China. This approach was followed very thoroughly by China effectively for couple of months (KRR et al. 2020; hunter 2020) . Second, mitigation which is a strategy that means letting the COVID-19 epidemic complete its course in a controlled way such as the idea of herd immunity. This strategy was initially applied by UK government, which becomes ineffective therefore, it was replaced with suppression strategy after the public release of report (James et al. 2020; Rocklov 2020) .

The contribution of this paper is to involve the fundamental of the control theory to provide a control engineering insight strategy in order to reduce the COVID-19 epidemic. Consequently, SEIR model is developed to represents the dynamical COVID-19 nonlinear system after some assumptions are made.

Furthermore, a robust control algorithm is applied to compensate the nonlinearity, instability of the COVID-19 nonlinear system. Eventually, the results of the model were based on the daily reports in both Hubei (China) and

Lazio at the beginning of the COVID-19 outbreak.

The rest of the paper is organized as follows. The robust control algorithm is introduced in Section 3 with COVID-19 mathematical description, Variable Transformation Technique (VTT) and Most Valuable Player Algorithm (MVPA). The simulation results are explained in Section 4 in order to show the J o u r n a l P r e -p r o o f potential effects of the control algorithm. In Section 5, the discussion is made to sum up the results and discuss the final evaluation of the proposed control algorithm. Eventually, the conclusion is presented in Section 6.

In this section, a new robust control algorithm is introduced in order to compensate the COVID-19 nonlinear system. In addition, the mathematical model of the COVID-19 is presented with a necessary assumptions and new techniques/methods that are used alongside with the proposed control algorithm to fit the design procedure.

The basic Susceptible-Exposed-Infectious-Recovered (SEIR) model can be developed as COVID-19 model as follows (Rad 2020) :

where ( ) represents Resistant subjects, ( ) represents Susceptible individuals, ( ) represents Infectious individuals that have infected but are not yet infectious, represents the total population and ( ) represents Exposed individuals. The parameter represents the likelihood of infection per unit time;

refers to the inverse of the average latency time of the disease, and represents the inverse of the average time infectious individuals spend by actually infecting other people.

The following are assumptions of the COVID-19 outbreak in Europe which are helps to reduce the abovementioned system (equations (1) to (4)) to a second order control system (Hethcote 2020):

1) Given low mortality percentage, births and deaths are neglected and the short time spans involved.

J o u r n a l P r e -p r o o f 2) Emigration and immigration are also ignored for simplicity.

3) Adding equations (1) to (4) gives:

( ( )+ ( )+ ( )+ ( )) = (5) Therefore,

4) COVID-19 (or SARS-CoV-2) virus is a new virus therefore, the vast majority of the people around the world have never been exposed to it yet.

There is no effective vaccine or cure for the virus.

6) The positive tested ratio depends mainly on the country and it is ranging from about 2% like Germany to about 10% like Italy, France and Spain.

7) The real mortality ratio corresponding to infected people is very low, since many cases have shown no issues and these are not tested positive for the COVID-19 virus but they are still infectious. The ratio between positive tested cases and really infectious cases is called ratio which is around one order of magnitude, mainly country dependent and uncertain.

In the case of the Hubei province outbreak it was estimated that = 0.05 (Rocklov 2020 This number of the patients can be expanded significantly if action is taken in time but certainly not by orders of magnitude.

10) The initial action of the outbreak dynamics is so fast, with times of reported cases that is doubled of the order of 4 days. 11) In many cases, people are tested only after they are shown serious symptoms of the virus, which made on average days after they were being infectious.

12) The testing procedure also presents a time delay in the process.

Although it is possible to come out the results of the test in a few hours.

However, the time average of the reports is longer because of the limitation in the equipments, for example about 1 week in Italy.

To sum up some assumptions, Item 4 proposes to consider (0) can be assumed in equations (2), (3) that is constant, and almost equals to .

By decoupling equations (2), (3) from equation (1) gives (Hethcote 2020): 

In the next section, the Variable Transformation Technique (VTT) is used to reformulate the COVID-19 system in order to fit the analysis procedure of the proposed control algorithm.

In this subsection, the Variable Transformation Technique (VTT) Step1: Let

Step2: Let

The next steps can be written as follow: where − ( ) can be found from equation (16).

The process is repeated to the number of the states ⋮

Step : Let

where − ( ) can be found from equation (18). Finally, the new system which is equivalent to the actual system is presented as follow:

It's obvious that the system in (19) is simpler than many actual systems which have complex structures. Consequently, the output of the actual system equals to a state or number of states summed together which can be transformed in the same abovementioned technique.

First of all, the VTT is applied in order to simplify the COVID-19 system as follow: 

Substituting equations (20) and (21) in equation (22) gives:

The new system becomes as follow:

Now, the Lyapunov Quadratic Function (LQF) is used to analyze the system and figure out a proper controller to stabilize and control the COVID-19 epidemic.

Since, the number of states in the actual systems is = 2, then, a second order model reference is selected to fit the design procedure as follows (Ali and Hadi 2020) :

where , are positive constants and is step input.

The error between the model reference and the actual system can be defined as follow:

Substituting equation (23) and equation (24) in equation (26) where and are 2 × 2 positive definite, real and symmetric matrix.

When substituting equations (24) and (25) in equation (29), we obtain: 

Substituting equation (31) in Equation (29) gives:

Since ( ( )) is functional which means mathematically a function that depend on function/s not variable/s. In addition, ( ( )) depends on ( ) then the controller may not be considered as a suppression or mitigation control policy.

However, It may contain the vaccine as a cure part to eliminate this epidemic in the future. Therefore, the parameters 1 , 2 , and 3 are considered as the parameters of the cure that will be invented hopefully as soon as possible.

Moreover, to find the optimal parameters of the nonlinear control algorithm, the integral square error performance index (ISE) is used. It is expressed by (Rahmian and Travazoei 2013):

where ( ) is the difference value between the model reference output and the system output.

Most  Team: a group of players who are played a sport game against another group of players.

 Player: a person who is participated in a sport game.

 Championship: a competition tournament to find out the best team/player in a certain sport.

 Franchise player: the best player in any sports team who is played professionally.

 League: a group of sports teams who are all played against each other to acquire points and figure out which team is the best.

 Fixture: an event of sports that is prepared to be happened in a certain date and place.

 Most valuable player: the award that is given to the best player in a sport game/series of sport games throughout a certain season.

In this algorithm, a player and a team which is a group of players both are represented as follows (Bouchekara 2017 

where represents how many players that are played in the league, represents the problem dimension and represents the skills.

Each team own a player who has called a franchise also the best player of the league. An example of two players with their corresponding level of skills for each one is shown in Figure 1 . If is chose to play against and wins the player's performance of are expressed as follow (Bouchekara 2017) :

Otherwise, they are expressed as follow (Bouchekara 2017) : The reason behind using MVPA is that the method is converging faster after compared with 13 well-known optimization methods including Genetic Algorithm (GA), Particle Swarm Optimization (PSO), …, etc (Bouchekara 2017) . Aforementioned phases illustrate the MVPA calculations to find the optimal parameters of the controller. The following explains how this method is working on the system in this paper. First, assign the number of teams, players and the problem dimensions which are the parameters of the controller that need to be optimized. These parameters represent the skills of a player in MVPA.

Next, assign the objective function which represents the cost function used in this paper (Integral Square Error). Then, the players gained skills (parameters) throughout the phases that mentioned previously which are embodied in Matlab code (m-files). After that, the pre-optimized parameters are calculated and applied simultaneously into the controlled system to calculate the measured error. Consequently, the measured error used to find the cost function and compared to the previous cost at each iteration in order to obtain best cost and then the optimal parameters. Finally, this process is repeated until the optimal parameters are obtained after certain number of iterations. Figure 2 shows the process of how the MVPA calculates the optimal parameters and then applied to the controller.

The flowchart of MVPA with the system.

In this section, the results of the open loop system is presented of the COVID-19

system. The results of the closed loop controlled system results is presented in Hubei (China) and Lazio (Italy) in order to show the effectiveness of the proposed control algorithm that involve the control theory to make it as a part of the solution to overcome the COVID-19 epidemic. Table ( 

In this subsection, the results of the controlled COVID-19 system are presented which based on the daily reports of the government in Hubei (China). Figure 5 shows the stabilization properties of the system states ( ( ), ( )) with initial condition ( (0) = 0.25 and (0) = 0.75). J o u r n a l P r e -p r o o f Figure 6 presents the system state trajectories after applying the controller (equation (30)). However, the shadow parts caused by the (•) term of the controller. outbreak. Figure 10 proves that the proposed controller (equation (30)) successfully stabilized the COVID-19 system. 

In this subsection, the results of the controlled COVID-19 system are presented that based on the daily reports of the government in Lazio (Italy). Figure 11 represents the stabilized system states ( ( ), ( )) with initial condition ( (0) = 0.25 and (0) = 0.75. It is shown that the potential effect of the proposed controller (equation (30)) in stabilizing the system effectively. Figure 13 illustrates the tracking property trajectory of the COVID-19 system after applying the proposed controller. Figure 14 explains the phase-plane of the COVID-19 system which proves that the zero point is stable equilibrium point of the system. Furthermore, the shadow parts caused by the (•) term of the controller.

Tracking properties of the system. Figure 14 Phase-plane of the system. Figure 15 shows the control action ( ( ( ))) that is jumped at high point at the day 10 because of the infected cases started to increase in this day in Lazio outbreak. Figure 16 shows and proves that the proposed controller (equation (30)) successfully stabilized the COVID-19 system. (30)). It is shown that 3 iterations are enough to achieve the convergence property to minimize the cost function. It is worth to mention that an iteration is corresponding to the day unit of the COVID-19 epidemic.

The previous results have shown that this strategy (mathematical-engineering strategy) can work in parallel with other existing strategies (suppression and mitigation) to control and reduce the separation of the COVID-19 epidemic.

J o u r n a l P r e -p r o o f However, this strategy may not be effective as the vaccine of the COVID-19 is not been invented yet. In addition, these results are valid for a certain period of time for both cities (Hubei and Lazio) which can be applied for any other place in any time if the required data is available. In addition, the results proved the effectiveness of the proposed control algorithm from control theory prospective which is interpreted by the performance improvements, stability achievement, the optimality of the parameters and the robustness of the controlled COVID-19

system. Eventually, this strategy (robust control algorithm) can be used to control and reduce the separation of any other epidemic outbreak in the future if the required data and model of this epidemic is available.

In this work, a novel robust control algorithm has been proposed in order to involve the control theory concepts to present a solution to the COVID-19 epidemic alongside with suppression and mitigation strategies. In addition, the COVIS-19 nonlinear system has been compensated using the proposed control algorithm. Then, the Variable Transformation Technique (VTT) has been successively applied to simplify the COVID-19 nonlinear system to fit the analysis procedure. After that, the optimal parameters of the proposed control algorithm have been obtained using the Most Valuable Player Algorithm (MVPA). Finally, the simulation results have explained the ability of the proposed control algorithm in compensating the COVID-19 system based on the daily reports in both Hubei (China) and Lazio (Italy).

Please fill in the table below according to following:

-list the authors in order in which they are stated in manuscript. This way each author is assigned with number for further identification.

-each author should sign this document (on designated place in the table). By signing this form authors take full responsibility for all statements it contains.

J o u r n a l P r e -p r o o f

No Competing interests are at stake and there is No Conflict of Interest

Optimal Nonlinear Controller Design for Different Classes of Nonlinear Systems Using Black Hole Optimization Method

Most Valuable Player Algorithm: a novel optimization algorithm inspired from sport

Can the COVID-19 epidemic be controlled on the basis of daily test reports?

Coronavirus disease (COVID-2019) situation reports

The Mathematics of Infectious Diseases

Covid-19 and the Stiff Upper Lip -The Pandemic Response in the United Kingdom

Suppression and Mitigation Strategies for Control of COVID-19 in New Zealand

Non-Pharmaceutical Interventions (NPIs) to Reduce COVID-19 Mortality

The reproductive number of COVID-19 is higher compared to SARS coronavirus

Genomic characterisation and epidemiology of 2019 novel coronavirus: implications for virus origins and receptor binding

Experimental Treatment with Favipiravir for COVID-19: An Open-Label Control Study

A study on control of novel corona-virus (2019-nCoV) disease process by using PID controller

Improving integral square error performance with implementable fractional-order PI controllers

Pattern of early human-to-human transmission of Wuhan

COVID-19 health care demand and mortality in Sweden in response to nonpharmaceutical (NPIs) mitigation and suppression scenarios

Adaptive Neural Control for Nonaffine Pure-Feedback System Based on Extreme Learning Machine

A Familial Cluster of Infection Associated With the 2019 Novel Coronavirus Indicating Possible Person-to-Person Transmission During the Incubation Period

This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors.J o u r n a l P r e -p r o o f

The authors declare that there is no conflict of interest regarding the publication of this paper.