STABILITY ANALYSIS OF SCPUR MATHEMATICAL MODEL FOR THE SPREAD OF COVID-19 (CORONA VIRUS DISEASE-19)

This article discusses SCPUR mathematical model for the spread of COVID-19 using data of people with COVID-19 in Makassar City. In this model, the population class is divided into five classes: susceptible, asymptomatic infectious, reported symptomatic infectious, unreported symptomatic infectious, and recovered classes. The proportion of body immunity to the increase of infected individuals, the proportion of large-scale social restrictions, and the proportion of quarantine as a healing process are also added. The research begins by determining the equilibrium point of the model, namely the disease-free equilibrium point and the endemic equilibrium point. Then, the stability test is carried out using linearization method and the eigenvalues are determined. The value of the basic reproduction number is obtained using next-generation matrix method, where the initial state of the basic reproduction number value R0 > 1 mean COVID-19 will still exist in Makassar City. Treatment is carried out so that R0 < 1 which means Makassar City will be free of COVID-19.


INTRODUCTION
There are various kinds of disease viruses in the world such as human immunodeficiency virus, ebola virus, dengue virus, etc. One of the viral diseases that is currently being discussed is the novel coronavirus. It is not a new virus in the health sector. The virus was first discovered in 2003 and caused severe acute respiratory (SARS). Then, a new type of the virus was found in 2012 and caused middle east respiratory syndrome-corona virus (MERS-CoV) [1,2]. Lately, this virus mutated to a new type of coronavirus (SARS-CoV-2) and caused the emergence of a disease called coronavirus disease-19   [3,4]. From the first case of COVID-19 in Wuhan, China in late December 2019 [5,6] until July 10, 2020, the virus had infected 12.015.193 individuals, caused 549.247 death, and spread to 216 countries in the world. The highest cases were reported in the Americas with 6.264.626 cases. Asia was in the fourth place with 1.032.167 cases, 70.736 of which were from Indonesia with 32.651 recovered individuals and 3.417 deaths. It has been predicted that this number will increase given the absence of antivirus for this disease [7,8]. Based on these facts, as of March 11, 2020, the world health organization (WHO) decided that COVID-19 is a pandemic disease [9,10].
The field of mathematics is one of the fields of science that can provide solutions in a phenomena, by modeling and formulating the phenomena. The phenomena are transformed into either an equation or an equation system, for example in dynamical populations [11,12,13,14] and the field of epidemiology [15,16,17]. With an assumption that COVID-19 is an epidemiology, the dynamics of COVID-19 spread can also be considered in mathematical modeling [18,19].
Based on studies in [20] discusses the SIRU mathematical model (Susceptible, Asymptomatic Infected, Reported Infected Case, Unreported Symptomatic infected) to predict the cumulative number of COVID-19 cases in China. Furthermore, from [20] researchers will make modification by adding one class of individual namely recovery class, with the consideration that an increasing number of individuals who recovery from disease by quarantine. Furthermore, the assumption is added that individuals who are infected without symptoms and have good immunity will recovery 4084 ALVIONI BANI, SYAMSUDDIN TOAHA, KASBAWATI from the disease without quarantine. Then, another treatment is also added, namely Large-Scale Social Restrictions.

SCPUR MATHEMATICAL MODEL FOR THE SPREAD OF COVID-19
SCPUR model is a development of SIRU model, namely an epidemiological model with compartments consisting of susceptible ( ) class which represents the number of individuals who are susceptible to COVID-19, asymptomatic infectious ( ) class which represents the number of individuals infected with COVID-19 without clinical symptoms, reported symptomatic infectious ( ) class which represents the number of individuals who show symptoms of being infected with COVID-19 and report it, and the unreported symptomatic infectious ( ) class which represents the number of individuals who show symptoms of being infected with COVID-19 but do not report it [20].
In this model, one individual class is added, namely recovery ( ) class which represents the number of individuals who have recovered from COVID-19, so that the individual class who shows symptoms of being infected with COVID-19 and reports it is symbolized as ( ) .
Furthermore, the class of asymptomatic infectious ( ) is termed a carrier ( ) which represents the number of individuals who are infected with COVID-19 but do not show symptoms of infection.
Then, the proportion of the effect of body immunity to the increase of infected individuals, the proportion of large-scale social restrictions, and the proportion of quarantine as a healing process are also added.
The assumptions used in constructing the mathematical model of COVID-19 are as follows: 1. The entire population is assumed to be susceptible to COVID-19 infection  8. Carriers that have a good level of body immunity will recover without being given quarantine treatment.
Based on the above assumptions, a dynamic for the spread of COVID-19 is obtained as shown in   Table 1.    Furthermore, the equation (1) -(5) will be formed into a system of normalized equations by substituting non-dimension variables as follows: So that the following non-dimension nonlinear differential equation system is obtained:

EQUILIBRIUM POINTS AND THEIR STABILITIES
The (15) After obtaining the equilibrium points, disease-free and endemic equilibrium stability analysis will be carried out. The first step is to linearize the equation system for the non-linear spread of COVID-19 using the Jacobi matrix [21]. The Jacobi matrix equation (6) - (10) is Then, we will look for the eigenvalues of the matrix 0 . Characteristic equation of the matrix Based on the routh-Hurwitz criteria [22], the roots of the equation (17)  Because it is assumed that all parameters are positive, then 1 > 0 if 1

BASIC REPRODUCTION NUMBER AND SENSITIVITY ANALYSIS
The basic reproductive number is the threshold for the transmission of a disease caused by infected individuals in a population who are susceptible to infection, which is usually denoted by ). (18) and ( Because the value of the basic reproduction number is the radius spectral of −1 [21]. Then the value of the basic reproduction number is

Furthermore, from equations
Sensitivity of the basic reproduction number is analyzed to determine the effect of parameters on the basic reproduction number.

= ×
where V is the variable to be analyzed and p is the parameter.  Table 1.  Table 1, it can be concluded that some of the parameters have a negative relation to 0 , which means, if the parameter value is increased then the value of 0 will decrease. These parameters are displacement rate due to symptoms ( ) , transition rate from asymptomatic infectious to natural recovering from disease ( ), natural death rate ( ), the proportion of body immunity ( 1 ) and the implementation of large-scale social restrictions ( 2 ). On the other hand, the parameters which have a positive relation to 0 mean that if the parameter value is increased then the value of 0 will also increase. These parameters are the interaction rate ( ), transition rate from asymptomatic infectious to unreported symptom infected individuals ( 2 ), and death rate due to COVID-19 ( ). As an example, the relation between the interaction parameter ( ) and the implementation of large-scale social restrictions ( 2 ) when 0 = 1 is shown in Figure   2. The value of the basic reproduction number is 0 = 1.386. Based on the results obtained, it can be concluded that the disease-free equilibrium point is unstable due to a positive eigenvalue.
Furthermore, for the stable endemic equilibrium point, it can be seen from the eigenvalues which are all negative and 0 > 1. Then, we will observe four parameter values for the implementation of large-scale social restrictions and their effects on the basic reproduction number, namely 2 = 0.08, 0.2, 0.342, 0.6. The results are presented in Table 2.   Table 3. which are prensented in figure 5 and 6. Based on Figure 5 and 6, with different initial values, the graph will converge to the stable equilibrium point. In Figure 5 when we give parameter value of interaction rate is 0.65 or people in Makassar City continues interact intensly without social distancing, the stable equilibrium point is the endemic equilibrium point, meaning that COVID-19 still exist in Makassar City. Otherwise in Figure 6, when we give parameter value of interaction rate is 0.3 or people in Makassar City reduce intraction with other individual, the stable equilibrium point is the free-disease equilibrium point, meaning that COVID-19 will disappear from Makassar City.