Mathematics Model of COVID-19 with Two-Stage Vaccination, Symptomatic, Asymptomatic, and Quarantine Individuals

The COVID-19 mathematical model began to develop since the disease appeared at the end of 2019. This model is used to investigate the characteristics of the spread of a disease. This research developed a model of COVID-19 based on the SEIR model which was further developed by dividing the infected subpopulation into symptomatic and asymptomatic, adding quarantine of infected individuals and vaccination in two steps. Making this model begins with making a compartment diagram of the disease and then forming a system of differential equations. After the model is formed, the disease-free equilibrium point, endemic equilibrium point, and basic reproduction number (R 0 ) are obtained. Analysis of the stability of the disease-free equilibrium point was locally asymptotically stable if R 0 <1 and an endemic equilibrium point existed if R 0 >1. Numerical simulation for the model that has been made is in line with the analysis. Furthermore, the sensitivity analysis of the basic reproduction number obtained that the parameters that have a significant effect on the spread of COVID-19 are the rate of the first dose vaccination, the rate of contact with symptomatic or asymptomatic individuals, and the rate of quarantine of symptomatic infected individuals.


INTRODUCTION
SARS-CoV-2 is the virus that causes Coronavirus Disease 2019 (COVID- 19) and first discovered at Wuhan, Hubei Province, China [1]. The current level of COVID-19 has become a pandemic because the virus has infected almost the entire world, not just in one area [2]. It is recorded that until July 1, 2021, or more than one year since this infection was first discovered, COVID-19 has infected up to 222 countries in the world with 182,989,419 infected cases, 11,452,155 active cases, and 3,962,991 deaths [3]. The first case of COVID-19 in Indonesia was discovered on March 2, 2020, the two positive patients are domiciled in Depok, West Java. After the discovery of the first two positive patients, the positive number of COVID-19 in Indonesia continued to increase [4]. Until July 1, 2021, or more than a year after the first case in Indonesia, it was recorded that this virus had infected 2,203,108 individuals, with 258,826 active cases and 58,995 deaths [5]. In addition to symptomatic infected individuals, some asymptomatic infected individuals have a very large potential to transmit COVID-19 [6]. Research [7] analyzing susceptible individuals who came into contact with infected individuals found that infected assumed to be closed, there is no movement of people out or into the area. (2) Birth and death rates are assumed to be the same with rate , which mean the population is constant. (3) Every individual born is assumed to be in good health but has a risk of infection because it is not immune to disease. (4) Disease transmission occurs through direct contact between susceptible individuals and symptomatic or asymptomatic individuals. (5) Infected individuals are divided into symptomatic and asymptomatic. (6) Symptomatic and asymptomatic individuals who are detected must be quarantined. (7) Asymptomatic infected individuals who are not detected can recover on their own. (8) Vaccination is used to reduce the risk of susceptible individuals being infected. (9) Vaccination is carried out in 2 steps. (10) Death from disease is negligible. Based on the assumption, the model can be made a scheme for the spread of COVID-19 disease with 2 doses of vaccination, symptomatic infection, asymptomatic infection, and quarantine as shown in Figure 1. In Figure 1 the individual population is divided into 8 compartments. Compartment of susceptible individuals ( ), compartment of individuals who have received vaccine dose 1 ( ) , compartment of individuals who have received dose 2 of vaccine ( ) , compartment of latent individuals ( ), compartment of symptomatic infected individuals ( ) , compartment of asymptomatic infected individuals ( ) , compartment of quarantined individuals ( ), and compartment of individuals who have recovered from disease or are immune to disease ( ). Every birth ( ) will be a susceptible individual ( ) with the potential to be infected with COVID-19. To reduce the potential for infection, 2 doses of vaccination are carried out, susceptible individuals will receive 1 dose of vaccine ( ) at a rate of . Individuals who have received vaccine dose 1 still have the potential to be infected by 1 and affected by the proportion of vaccine efficacy by 1 − . Individuals who have received vaccine dose 1 will receive vaccine dose 2 ( ) at a rate of and the proportion of vaccine efficacy is . Individuals who have received dose 2 of the vaccine still have the potential to be infected at a rate of 2 and are affected by the proportion of vaccine efficacy by 1 − . Each susceptible individual, who has received dose 1 of the vaccine and has received dose 2 of the vaccine, who is infected will become a latent individual (E). The rate of susceptible individuals to become latent individuals is when contact with symptomatic infected individuals and when contact with asymptomatic infected individuals. Latent individuals will become symptomatic infected individuals ( ) with a rate of and a proportion of . Latent individuals will become asymptomatic infected individuals ( ) with a rate of and a proportion of 1 − . Symptomatic infected individuals will quarantine ( ) at a rate of 1 . Asymptomatic infected individuals who are detected will also be quarantined at a rate of 2 . Asymptomatic infected individuals who are not detected can recover naturally due to the body's immune factor at a rate of . Individuals who have received 2 doses of the vaccine will get immunity ( ) at a rate of and efficacy of . Individuals who do quarantine will recover ( ) at a rate of . Each compartment has a natural death of .
The mathematical model of COVID-19 with 2 doses of vaccination, symptomatic infected, asymptomatic infected and quarantine is obtained as follows: Where = + + + + + + + then obtained = 0 , thus ( ) = where is positive integer. Since ( ) is constant, then system (1) can be formed into a non-dimensional model in order to simplify the model. The proportion of many individuals in each compartment can be expressed as: Divide equation (1) by ( ) and express them as in (2) to obtain a non-dimensional mathematical model (3). In equation (3), variable r is ignored since it does not affect the other compartments.

Disease Free Equilibrium Point
A disease-free equilibrium point can be obtained when there are no infected individuals in the population. To fulfill this, it must be = 0 and = 0. The disease-free equilibrium point is obtained as follows:

Basic Reproduction Number ( )
The basic reproduction number can be obtained by finding the maximum eigenvalue of the next generation matrix [13]. The next-generation matrix is obtained from the infected subsystem equation. Take the equation which is the new infection case and also the change in the infected case in the system. Infected subsystem in (3) Next, 0 can be obtained by computing the spectral radius ( ) or the largest absolute value of the eigenvalues of −1 , which can be expressed as: The basic reproduction number is defined as the average number of the second infection that occurred when the first infection started to infect all susceptible population [13]. In general, if 0 < 1 then the disease will disappear and if 0 > 1 then the disease will become epidemic. Further interpretation of 0 on (5) is discussed in the following.

Endemic Equilibrium Point
The endemic equilibrium point occurs when the infected class is not zero or the disease has become epidemic in a population. Must be * > 0 and * > 0. *  Where ).
Proof. The existence of an equilibrium point is indicated with each positive element according to the conditions for the formation of this model. We will prove that equation (12) has at least one positive root. According to Descartes' Rules of Sign [14], a polynomial will have as many positive roots as the change in a sign that occurs in the coefficients of the equation. Then it will be proven that there is at least one sign change in the equation. It is clear that 0 > 0. Take the coefficient 3 and it will be proved that its value is negative. With 0 > 1, the value of 3 < 0 is obtained, then there is at least one positive root in the equation according to [14]. So, Theorem 1 is proven to be true. ∎

Stability Analysis of Disease-Free Equilibrium Point
Theorem 2. Disease equilibrium point 0 asymptotically stable if 0 < 1. Proof. Analysis of the stability of the disease-free equilibrium point can be determined by finding the eigenvalues of the Jacobian matrix around the disease-free equilibrium point 0 [15]. The characteristic equation is obtained as follows: where Based on equation (13)   In the previous equation, it has been proven that 3 > 0 and 2 > 0. Then it's proved that The determinant of the Routh-Hurwitz matrix 1 , 2 , 3 has a positive value if and only if 0 < 1. So, the characteristic equation (13) has a negative real root part. Based on the results obtained, it can be concluded that the disease-free equilibrium point 0 is locally asymptotically stable. ∎

Model Simulation
The mathematical model of COVID-19 with 2 doses of vaccination, symptomatic, asymptomatic, and quarantine has been formed and analyzed and then carried out Based on the parameter values in table 1, the basic reproduction number is 0 = 0.9740939287. Because the value of 0 < 1 then the spread of the disease will slowly decrease and after a certain period time the population will be free from disease.     In graphical it presented in figure 4.

Sensitivity Analysis
The sensitivity index of a parameter is correlated with the basic reproduction number ( 0 ). This index provides information about the parameters with a significant impact on the value of 0 . Parameters with a high impact on the value of 0 indicate that these parameters have a big responsibility for the spread of COVID-19 [21]. The sensitivity index of a parameter can be calculated as follow: Where p is the parameter for which the sensitivity index will be calculated. Using equation (15) and table 1, the sensitivity index from the parameter will be calculated as follow: The sensitivity index of all parameters is listed in table 3.  Table 3 shows the sensitivity index of each parameter used in this model. In the table, the sensitivity index is ordered based on how much impact the parameter for the value of 0 . The parameter index with a positive value indicates that if the parameter is increased while the other index remains, it will affect the value of 0 which also increases, whereas if the parameter is decreased, the value of 0 will also decrease. The parameter index with a negative value indicates that if the parameter is increased, the value of 0 will decrease, whereas if the parameter is decreased, the value of 0 will increase.

Tabel 3. Sensitivity index parameters
The sensitivity index shows that the parameter (rate of individuals receiving dose 1 vaccine) is the parameter that has the most significant (negative) impact on the spread of COVID-19. t is known that the sensitivity index value of parameter = −0.7561096532, by increasing (or decreasing) the value of parameter by 10%, the value of 0 will decrease (or increase) by 7.561096532%. The sensitivity index of parameter (transmission rate if contact with symptomatic infected individuals) is the parameter that has the most significant (positive) impact on the spread of COVID-19. It is known that the sensitivity index value of the parameter = +0.6944182466 , by increasing (or decreasing) the value of the parameter by 10%, the value of 0 will increase (or decrease) by 6.944182466%.
The results of numerical simulations show that if the value of 0 < 1 then the disease will disappear from the population, but if the value of 0 > 1 then the disease will remain in the population or become endemic. Based on the results of the sensitivity analysis, several actions can be taken to prevent the transmission of COVID-19 by making the value of 0 < 1 based on table 3. (1) Accelerate the rate of dose 1 vaccination ( ). The rate of dose 1 vaccination can be accelerated by facilitating public access to get vaccines. Adding health facilities for vaccination is one way to increase the rate of dose 1 vaccination. (2) Reducing the rate of contact with symptomatic infected ( ) or asymptomatic infected ( ). This can be done by following health protocols and reducing mobility, as has been recommended by the government. (3) Accelerate the quarantine rate for symptomatic infected individuals ( 1 ). The more individuals tested, the greater the probability of detecting infected individuals and this will make the rate at which individuals quarantine themselves will be greater.

CONCLUSIONS
The mathematical model of COVID-19 SVEIQR was obtained where compartment was divided into dose 1 and dose 2, compartment was divided into symptomatic and asymptomatic infected. The model obtained is a system of nonlinear differential equations. It has a disease-free equilibrium point and an endemic equilibrium point. Disease-free equilibrium point 0 = ( + , ( + )( + ) , ( + )( + )( + ) , 0,0,0) locally asymptotically stable when the value of 0 < 1. The endemic equilibrium point 1 ( , , , , , , ) exists when the value of 0 > 1. Based on the model simulation, it is concluded that the disease will disappear if 0 < 1, and the disease will persist in the population if 0 > 1. This is consistent with the existing theorem. Based on the results of the sensitivity analysis, the parameter that has the most influence on the value of 0 is obtained. Several things that can be done to make the disease disappear ( 0 < 1) are to increase the rate of vaccination dose 1 ( ), reduce the rate of contact with symptomatic infected individuals ( ) or asymptomatic infected individuals ( ) and accelerate the quarantine rate for symptomatic infected individuals ( 1 ).