MODEL ANALYSIS OF THE SPREAD OF COVID-19 WITH LOGISTIC GROWTH RECRUITMENT

ABSTRACT


INTRODUCTION
The spread of COVID-19 until 2022 has decreased, but every country is still trying to overcome the disease. Mathematicians, in collaboration with experts in the health sector, continue to conduct research in modeling, analyzing, evaluating, and optimizing the handling of the spread of COVID-19. Din (2020) examines to system analysis of the overall spread of COVID-19 [1]. Mathematical modeling is also studied in the COVID-19 epidemic by taking into account the effects of awareness programs in Nigeria [2]. Analysis of the Suspect-Latent-Asymptomatic-Infectious-Healed COVID-19 pattern of disease spread was studied, determining the threshold and the stability point [3]. Tian (2020) analyzed COVID-19 modeling based on morbidity data in Anhui, China [4]. Epidemic Modeling of SARS-CoV-2 of the SEIR type was studied in Italy using insightful intelligence calculations [5]. COVID-19 dynamics models by taking hereditary diseases into account have also been studied [6]. Fang (2020) analyzes the dynamics of the transmission of COVID-19 with government intervention [7]. Ma (2020) examines the exponential model of a disease and determines the threshold [8]. The COVID-19 pandemic model with logistical growth has also been studied [9], [10]. Logistic transmission patterns in the COVID-19 pandemic have been researched in the Asian region [11], di China [12], [13]. Tene (2021) analyzed the COVID-19 outbreak in Ecuador using a logistical model [14]. Global dynamics research and epidemic control strategies, a model on logistical growth, taking into account the limitations of hospital facilities, have also been carried out [15].
A co-infection model of COVID-19 and tuberculosis with the effect of isolation treatment has been studied [16]. Control of preventing the COVID-19 pandemic in the SIR type has also been researched in Indonesia [17]. Optimal control and simulation on the model of the COVID-19 pandemic have been analyzed by Araz (2021) [18]. Based on the results of the COVID-19 research study above, there has not been any study on recruitment rates with logistical growth, which will be analyzed and simulated in this paper.

RESEARCH METHODS
Population growth is generally in a logistical model. For this reason, it is assumed that the growth in suspected COVID-19 will be in a logistical form. P. F. Verhulst (1838) examine the logistics model [10]: with P, w, and K respectively, are the number of residents, population growth level, and the carrying capacity of the number of residents. dP/dt is the change in the number of residents per unit of day, which is influenced by the rate of population growth and carrying capacity.
The number of residents studied in this study was grouped into five subsections. Individuals who do not yet have the characteristic symptoms of being infected with COVID-19 enter the susceptible subpopulation (S). Individuals who do not yet have symptoms of COVID-19 and have been vaccinated enter the vaccination subpopulation (V). Individuals who have experienced symptoms of COVID-19 enter the E subpopulation. Individuals who have confirmed COVID-19 enter the infected subpopulation (I). People who are healed from COVID-19 enter the R subpopulation. The assumptions from the COVID-19 spread system studied are (1) Recruitment rate in the form of logistical growth. (2) Individuals of the S subpopulation who are in contact with the I subpopulation enter the E subpopulation. (3) Individuals from each subpopulation experience natural death, and I subpopulation individuals may die from COVID-19. (4) Individuals who are confirmed positive for COVID-19 can recover due to increased body resistance through eating/drinking and multivitamins or due to treatment. Deployment dynamics of COVID-19 spread the distribution of the system is analyzed as shown in Figure 1.
Based on the assumptions and schematic diagrams above, a system of differential Equation (1) is obtained: Notation of parameters, description, and value of parameters as shown in Table 1 below: Based on the system model of Equation ( Solving inequality (2), by using the integral factoring method,

Non-Endemic stability point
The non-endemic stability point obtained from the changes in each subpopulation per unit of the day is constant. A mount of non-infected individuals remain, and a mount of infected individuals is zero for t → ∞. The non-endemic stability point of the equations system (1) is: , 0,0,

Threshold
The parameter of threshold, denoted by R0, is a needed tool in an epidemic-spreading system to find out whether the size of the spread is increasing or has decreased. If R0 < 1, then the spread of the illness decreases, and vice versa. If R0 > 1, then the illness increases. Based on the system (1)  . Characteristic roots of ( 0 ), < 0, i = 1, 2, 3, 4, 5 with the provision of 2 > + and + > + √ 2 + (4 + 2 1 + 2 ) + ( 1 + ) 2 . If all the characteristic roots of ( 0 ), of equation system (1) at the nonendemic point P0 are all negative equivalent to R0 < 1 or the endemic stability point P0 is locally asymptotically stable. 

Endemic stability point
The endemic stability point is obtained if the displacement of each subpopulation per day of t is constant, and infected people remain for t → . Based on Equation (1), it is assumed that the logistic growth is constant , obtained endemic point, It can be seen that 1 > 0, 2 > 0, 3 Table 1. Based on the Routh-Hurwitz criteria, 1 > 0, 2 > 0, 3 > 0 , and 1 2 − 3 > 0 , equivalently states that the characteristic roots of the polynomial system (6) all negative or R0 > 1 equivalent to the local asymptotically stable P1 endemic stability point. 

Theorem 3. If the spread of COVID-19 increases, then the endemic stability P1 is globally asymptotically stable.
Proof. To prove Theorem 3, it is sufficient to prove that the differential of the Lyapunov function is negative Apply the invariance Principle by Lasalle [20], if the spread of the disease increases then P1 is globally asymptotically stable.

RESULTS AND DISCUSSION
Writing Based on the system of equation (1), parameter values used in Table 1, by assuming the initial a mount of each, S0 = 26780410, V0 = 48440, E0 = 19750, I0 = 3840, R0 = 280, N = 268000000, and K = 300000000. Solving the model (1) with the 4th order Runge-Kutta method approach. A sensitivity analysis of the recruitment rate in susceptible subpopulations, like in Figure 2, and the vaccination subpopulation, like in Figure 3 below.  Figure 4, increasing the value of the contact rate with infected people and those who are still healthy increases the number of people exposed to COVID-19 from the initial time to t = 60 days. Based on Figure 5, increasing the value of the contact rate between infected people and those who are still healthy increases a mount of people who are confirmed positive for COVID-19 from the initial time to t = 60 days. Based on Figure 6, the greater the value of parameter r, the greater a mount of individuals recovered from COVID-19 from the initial time until t = 60 days.

CONCLUSIONS
The model studied in this paper analyzes the transmission of COVID-19 by taking into account the recruitment rate of logistical growth. Parameter values used in the model are from reference [1], [6], and [11], partly assumed. Based on the analysis of the system of equations (1), it is obtained: (1) the existence of the positivity of each subpopulation at any time. (2) Endemic and non-endemic points of the system of Equation Based on numerical simulations, a mount of susceptible and vaccinated subpopulation individuals increases if the recruitment rate increases. If the coefficient of the vaccination subpopulation increases, then a mount of recovered subpopulation individuals increases. If the value of the contact rate of the parameter between infected individuals and those who are healthy increases, then a mount of infected individuals increases. If the value of the coefficient of the infected subpopulation increases, then a mount of infected subpopulation individuals decreases, while the number of recovered subpopulation individuals increases.