MATHEMATICAL ANALYSIS AND ADAPTIVE CONTROL SPREADING OF CORONAVIRUS DISEASE 2019 (COVID-19)

Coronavirus disease 2019 (COVID-19) is an infectious disease caused by severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2), which is spreading all over the world and the main causes of worldwide death. For this reason, the control analysis of this coronavirus disease 2019 has a significant importance prevent the spread of it. In this paper, we present the modeling, mathematical analysis, and adaptive control of spreading coronavirus disease 2019. The mathematical analysis shows that the two fixed points of the coronavirus disease 2019 are globally asymptotically stable and the basic reproduction ratio R0 is obtained, which characterizes the disease transmission. Moreover, an adaptive control is designed to control and treat coronavirus outbreak. The sufficient control conditions are derived for the existence of stable coronavirus disease 2019 free is presented.


INTRODUCTION
Infectious diseases mainly caused by pathogenic microorganisms, such as bacteria, viruses, fungi, and parasites. The diseases can spread directly or indirectly from one person to another or from animals/birds to humans. These diseases are one of the main causes of worldwide death.
Coronavirus disease 2019 (COVID- 19) is an infectious disease caused by severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2). The first known case was identified in Wuhan, China in December 2019 [24]. The disease has since spread worldwide more than 150 countries across the world. On 8 March 2020, the WHO announced COVID-19 as a global pandemic [34]. As of 29 May 2021, there are over 170 million reported cases, and a death toll exceeding 3 million.
Transmission of COVID-19 occurs when people are exposed to virus-containing respiratory droplets and airborne particles exhaled by an infected person. Those particles may be inhaled or may reach the mouth, nose, or eyes of a person through touching or direct deposition such as being coughed. The risk of infection is highest when people are in proximity for a long time, but particles can be inhaled over longer distances, particularly indoors in poorly ventilated and crowded spaces. In those conditions small particles can remain suspended in the air for minutes to hours. Touching a contaminated surface or object may lead to infection. Symptoms of COVID-19 are including fever, cough, headache, fatigue, breathing difficulties, and loss of smell and taste [1,10,16,26,27]. Symptoms may begin one to fourteen days after exposure to the virus. The most 81% develop mild to moderate symptoms, while 14% develop severe symptoms, and 5% suffer critical symptoms.
Lots of research works have been reported. It shows that SIS, SIR and SEIR models can reflect the dynamics of different epidemics well, these models have been used to model the COVID-19 [9, SIR model is commonly used for disease modeling for the COVID-19 analysis [4,8,23], Tang et al. [29] investigated a general SEIR type epidemiological model where quarantine, isolation and treatment are considered. Moreover, there are also other methods for modeling of the COVID-19 [35]. Wang et al. [33] applied the phase-adjusted estimation for the number of coronavirus disease 2019 cases in Wuhan.
Recently, many scholars have studied the optimal control of COVID-19 introducing different control variables and given the corresponding control strategies [12,17,25]. In fact, the optimal control theory can only aim at the systems with known parameters and obtain the system inputs by minimizing the cost function. It is worth noting that there are various uncertainties in the transmission of COVID-19. If there are uncertain parameters in the system, it is impossible for optimal control theory to obtain the desired results, which need to identify the system parameters in advance. Fortunately, adaptive control can update the system parameters by using adaptive laws and guarantee the stability of closed loop system [6].
In this paper, we present the modeling, mathematical analysis, and adaptive control of spreading coronavirus disease. First, we discuss the description of the proposed model. Then, we present its mathematical analysis, the two fixed points of the system are globally asymptotically stable by using Lasalle's theorem and the basic reproduction ratio R0 is obtained, which characterizes the disease transmission. Next, we design adaptive control for globally stabilize a general coronavirus disease 2019 model by using Lyapunov stability theory. Finally, the conclusion this paper is presented.

Consider a dynamical system which satisfies,
where ( , ): is a fixed point of the system if ( , ) = 0.

Definition 2.1
The fixed point is stable if for > 0 there exists > 0 such that, 19) where (0) is unique solution. The fixed point is unstable if it is not stable. The fixed point is asymptotically stable if it is stable and there exists > 0 such that, The fixed point is globally asymptotically stable if it is stable and The following result is the fundamental importance of a Lyapunov stability theorem [13].

Theorem 2.2 (Lyapunov global asymptotically stability theorem or G.A.S)
Suppose there is a function : → such that, then, every trajectory of ̇= ( ) converges to zero as → ∞, that is the system is globally asymptotically stable.

Theorem 2.3 (Lasalle's theorem [20])
Lasalle's theorem allows us to conclude G.A.S. of a system with only ̇( ) ≤ 0, along with an observability type condition. We consider ̇= ( ). Suppose there is a function : → such that, (1) is positive definite, Next, we consider a nonlinear nonautonomous differential equation of the general form, where the state ( ) take values in , ( , ): × → is a given nonlinear function and An adaptive control is an active field in the design of control systems to deal with uncertainties. To design control laws that stabilize of the chaotic system. The control system can 5866 PRISAYARAT SANGAPATE, CHAYAKORN SOMSILA, PAWEENA TANGJUANG be written as where ( ) is the external control.
Definition 2.4 The control system (3) is stabilizable if there exist the control ( ) = ( ( )) such that the system, is asymptotically stable.

MODEL DESCRIPTION
In this section, we formulate a general SEIR model for the spreading of coronavirus disease 2019 . We separate the total population N(t) into four distinct subgroups which are Susceptible ( ) , Exposed ( ) , Infectious ( ) , and recovered ( ) . Group of susceptible individuals ( ) which will be increased based on the birth and decrease because the death and direct contact with an infected individual group , group of the exposed ( ), which will increase with transmission rate , decreased due to natural mortality and have been the incubation period , group of individuals who are infected with coronavirus disease ( ), which will increase with the incubation period , decreased due to natural mortality and have been recovery with the rate , and group of individuals who recover ( ) who increased because of there was the recovery with rate and decreased because of natural mortality .

FIXED POINT AND BASIC REPRODUCTION RATIO
In this section, we will discuss about the fixed point and basic reproduction ratio R0 of SEIR model.
The two fixed points are obtained as follows: 1. The disease-free fixed point of the proposed SEIR model is acquired by setting, From equation (9), we have From equation (8), we have Substituting equation (10) and (11) into equation (7), we get − 1)). The basic reproduction ratio R0 can be used to measure the rate of spreading of a disease, 1. For 0 ≤ 1, the patient could transmit the disease to a person and eventually the disease will disappear, this mean that the epidemic will not happened.
2. For 0 > 1, the patient could infect in more a new patient and eventually the disease will epidemic, this mean that epidemic is happened.

STABILITY ANALYSIS OF FIXED POINT
In this section, we have discussed stability analysis of both fixed points. We used Lasalle's Theorem for both the disease-free fixed point and the endemic fixed point of the proposed model. First, we present the globally asymptotically stable of the disease-free fixed point. We can see that ̇≤ 0 for 0 ≤ 1. Therefore, by the Lasalle's Theorem [20], the disease-free fixed point 0 of the system is globally asymptotically stable on Ω.
where 1 , 2 and 3 are positive constants to be chosen. By taking the derivative of the above function, we obtain We can see that ̇≤ 0. Therefore, by the Lasalle's Theorem [20], the endemic fixed point 1 of the system is globally asymptotically stable on Ω.

CONCLUSIONS
In this paper, we present the modeling, mathematical analysis, and adaptive control of

CONFLICT OF INTERESTS
The author(s) declare that there is no conflict of interests.