Dynamic Analysis of the Susceptible-Exposed-Infected-Hospitalized-Critical-Recovered-Dead (SEIHCRD)

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INTRODUCTION
Since the beginning of 2019, the world has been fighting together against COVID-19 which is a new virus.This virus spread very quickly, causing several countries to suffer from a shortage of inpatient beds and Intensive Care Unit (ICU) beds [1], [10], one of them in Indonesia.In June 2021 in Indonesia, every day the number of COVID-19 patients who die without receiving treatment is increasing.This is due to the increasing number of epidemiological rational constraints.[7] predict the long-term dynamic COVID-19 in Indonesia.Covid-19 is a type of virus that infects the respiratory tract or is also known as severe acute respiratory syndrome Corona virus-2 (SARS-CoV-2).Researchers through this study were trying to build a mathematical model of the spread of the Covid-19 virus and analyzed the stability of its critical points [8].[9] propose a framework for stress testing and financial scenario generation of market indicators.[10] develop a dynamic transmission model to investigate the impact of social media, particularly tweets via the social networking platform, Twitter on the number of influenza and COVID-19 cases of infection and deaths.[11] propose a new seven compartmental model Susceptible -Exposed -Infected -Asymptomatic -Quarantined -Fatal -Recovered (SEIAQFR) which is based on classical Susceptible-Infected-Recovered (SIR) model dynamic of infectious disease and considered factors like asymptomatic transmission and quarantine of patients.
In 2020, [1] conducted an analysis of the SEIHCRD model using the Least-Square and Levenberg Marquadt methods in determining scenarios for the spread of  Where this study considers the existence of a Case Fatality Rate based on age and comorbidity categories which will affect the spread scenario.In the following year [18] performed a dynamic analysis on the SEIHCRD model in the Kenyan population.The analysis was carried out to find out how sensitive the basic reproduction number analysis is to the parameters of physical distancing and mass testing, considering the presence of migration in the studied population.Then, the following year [11], [18], [19]  In contrast to these studies, in this study the proposed SEIHCRD model will be modified by adding the natural birth rate and natural death rate parameters which will then be analyzed using dynamic analysis.After that the model will be approached through a numerical method, namely the fourth order Runge-Kutta method to display the results of numerical calculations and their simulations.The data used in this study is original data in Indonesia from Infected, Hospitalized, Critical, Recovered and Dead cases from August to October, sourced from the website of the Ministry of Health and the Indonesian COVID-19 Task Force.Meanwhile, the original data from the Susceptible and Exposed cases are not available in Indonesia.Then this data will be used to find out whether the solution graph from the model can approach the solution graph from the original data.So, based on the background previously described, the author wants to apply the SEIHCRD model in solving the scenario of the spread of COVID-19 in Indonesia using the fourth order Runge-Kutta method.

Data and Data Sources
In this study, the type of data used is secondary data from August to October 2021, in the form of daily COVID-19 cases on the website SATGAS COVID-19 and availability of hospital beds on the website KEMENKES [17].This data was taken from August to October due to the complete data on daily cases of COVID-19 and the availability of hospital beds.

Seihcrd Model After Modification
The following is a modified

Identify Initial Values and Parameters Used
The following are the initial values and parameters in this study:

Determining the Equilibrium Point
The equilibrium points are divided into two, namely the disease-free equilibrium point and the endemic equilibrium point.In determining the two equilibrium points, each equation in the system of equations ( 1) must be zero, or Furthermore, equations (5) − ( 8) are not included in the system (11) because equations (2) − ( 4) do not depend explicitly on  * ,  * ,  * , and  * .Where  * can be found by entering the obtained function () into equation (5), then  * can be found by entering the obtained function () into equation (6), then  * and  * is not explicitly stated in the equation.The following is a system (9) for which we will find the disease-free equilibrium point and the endemic equilibrium point: The disease-free equilibrium point is the point at which there is no disease in the population.So that there are no infected individuals or  * = 0, then the disease-free equilibrium point is obtained as follows: The endemic equilibrium point is a point that indicates the conditions under which there is a spread of disease in the population.So that the endemic equilibrium point is obtained as follows: * = ( * ,

Determining the Basic Reproduction Number (𝑹 𝟎 )
Based on the equation model of the infected subsystem, the Next-Generation matrix will be obtained.The steps in determining the basic reproduction number ( 0 ) are as follows: 1. Take an equation containing infected subsystems, where in this model the infected subsystems are E and I. 2. Linearization of the infected subsystem at the disease-free equilibrium point using the Jacobian matrix as follows: Next, the decomposition of the Jacobian matrix (J) will be carried out.after that the basic reproduction number can be searched using the Next-Generation.
Where  is a transmission matrix that describes the rate of adding cases, while  is a transition matrix that describes the rate of case reduction.So, obtained: Then,  0 =  ( + )

Define Local Stability Analysis
The mathematical model in equation (9) is a system of nonlinear differential equations.So, in looking for stability analysis, linearization will be carried out around the equilibrium point.Here is the jacobi matrix from the results of linearization on Because the values of the roots in the characteristic equation ( 12) are difficult to obtain, using the Routh-Hurwitz criteria will determine the stability of the  0 equilibrium point.So based on these criteria, the equilibrium point  0 will be asymptotically stable if and only if it fulfills the following conditions: Based on the above calculations, it is found that conditions 1-4 are met and all eigenvalues in the characteristic equation for the disease-free equilibrium point are negative, which means that the disease-free equilibrium point in the locally asymptotically stable SEIHCRD model.

Local Stability Analysis of Endemic Equilibrium Points
Based on the disease-free equilibrium point that has been obtained, namely  * = ( So that the characteristic equation is obtained as follows Because the value of the roots of the characteristic equation ( 13) is difficult to obtain, then by using the Routh-Hurwitz criterion, the stability properties of the equilibrium point  * will be known.So based on these criteria, the equilibrium point  * will be asymptotically stable if and only if the following conditions are met: So, by using the parameters in Table 2, we get Based on the above calculations, it was found that conditions 2 and 3 were not fulfilled so that there were eigenvalues in the characteristic equation of the endemic equilibrium point with a positive value, which means that the endemic equilibrium point in the SEIHCRD model was unstable.

Define Global Stability Analysis
Based on the system (9) that S, E and I do not depend on H,C,R and D. hence dynamic analysis using S,E and I )  −  If  0 ≤ 1 so ℒ ′ < 0 for each (, , ) ≠ ( 0 , 0,0).The singularity proved that { 0 } is a set that satisfies the nature ℒ ′ = 0. Based on the principle of Invariant Lasalle equilibrium point  0 globally asymptotic stable.

Numerical Simulation Using the Fourth Order Runge-Kutta Method
Based on the fourth order runge-kutta formula, we get: Dynamic Analysis of the Susceptible-Exposed-Infected-Hospitalized-Critical-Recovered-Dead (SEIHCRD) Juhari 137 With the following set values So that the value in the first iteration is obtained as follows:  In Figure 2, can see that there are 2 graphs of infection cases in the spread of COVID-19 in Indonesia.Where the graph with the blue line is a graph of the original data, while the graph with the dotted line in red is the result of a model simulation.Based on the results of the 2 graphs, it can be concluded that by using the parameters β = 0.06, δ = 0.14,  = 0.95,  = 0.485, and  = 0.25, the SEIHCRD model can capture the peak of infection cases that occur in Indonesia even though an absolute error is obtained.by 28%.Where in the model, the peak of infection cases occurred earlier, namely on the 5th day, while in the original data it occurred on the 7th day.This is of course a good result, where the model can predict earlier when the peak of COVID-19 infection will occur.So that the government and all people in Indonesia can make earlier and more mature preparations   Based on the interpretation of the results in Susceptible, Exposed, and Infected cases above, COVID-19 will subside if the contact of vulnerable individuals with infected individuals is low, by implementing health protocols properly such as wearing masks, maintaining distance and doing vaccines.In addition, the existence of appropriate and fast medical treatment also affects the subsidence of COVID-19 cases.

CONCLUSIONS
Through the dynamic analysis that has been carried out on the SEIHCRD model, the result is that the number of cases will decrease with increasing time, so that the disease outbreak will end.Then, based on stability analysis at the equilibrium point, it is found that the disease-free equilibrium point is locally asymptotically stable, and the endemic equilibrium point is unstable.Furthermore, based on the results of numerical simulations using the Fourth Order Runge-Kutta method, the value obtained in the first iteration is  = 0.06,  = 0.14,  = 0.95,  = 0.485,  = 0.25, namely  = 272008906,  = 71788,  = 30738,  = 70568,  = 7926,  = 3.9446,  = 1.604.
Then based on the solution graph obtained from the SEIHCRD model, it can be seen that by using  = 0.06,  = 0.14,  = 0.95,  = 0.485,  = 0.25 and other parameters according to Table 2, the solution graph in cases of Infected, Hospitalized, and Critical can catch the sloping trend from the original data in Indonesia.Where the number of cases Dynamic Analysis of the Susceptible-Exposed-Infected-Hospitalized-Critical-Recovered-Dead (SEIHCRD) Juhari 140 will decrease and subside over time, if the following conditions are met: the rate of contact of susceptible individuals with infected individuals () is low, the rate of movement of individuals in the Exposed class to the Infected class is low, the probability of infected individuals being treated hospitalization () is high, the probability of a COVID-19 patient becoming critical and admitted to the Intensive Care Unit (ICU) () is low and the probability of a critical patient dying () is low.Then, the SEIHCRD model can describe the results of the solution graph in the exposed case well.Where the number of people exposed to it will also decrease along with the reduction in existing infection cases.While the solution graphs in the Susceptible, Recovered, and Dead cases cannot be described properly through the SEIHCRD model.So based on the dynamic analysis that has been carried out, COVID-19 will subside if the contact of vulnerable individuals infected individuals is low, by implementing health protocols properly such as wearing masks, maintaining distance, and doing vaccines.In addition, the existence of appropriate and fast medical treatment also affects the easing of covid-19 cases.
also conducted a dynamic analysis of the SEIHCRD model on the spread of COVID-19 in Indonesia.Where the research aims to determine the sensitivity analysis of the parameters of physical distancing and mass testing of basic reproduction numbers.The flow of infection spread in this model includes, among others, susceptible individuals in the Susceptible class will be exposed and enter the Exposed class.Then the exposed individual will be infected and enter the Infectious or Hospitalized class.Furthermore, individuals in the Infectious and Hospitalized class have the possibility of recovering and dying.Where individuals in the Hospitalized class can also experience critical conditions and enter the Critical class.Then finally, the individual in this Critical class will die and enter the Dead class.

Figure 1 .
Figure 1.SEIHCRD Model Compartment Diagram So that the equation in (1) becomes:

Figure 2 .
Figure 2. Infected Graphics on Model and Original Data

Dynamic
Analysis of the Susceptible-Exposed-Infected-Hospitalized-Critical-Recovered-Dead (SEIHCRD) Juhari to deal with the peak of infection cases.Then based on the parameters used, cases of COVID-19 infection in Indonesia can decrease and subside over time, if the contact rate of susceptible individuals with infected individuals () is low, the rate of transfer of individuals in the Exposed class to the Infected class () is low, the individual probability hospitalization () is high, the probability of a COVID-19 patient becoming critical and admitted to the Intensive Care Unit (ICU) () is low, and the probability of a critical patient dying () is low.

Figure 3 .
Figure 3. Hospitalized Graph on Model and Original Data

Figure 4 .
Figure 4. Critical Graph on Model and Original Data

Table 1 .
The Initial Value of the SEIHCRD Model in Indonesia

Table 2 .
Parameter Value of SEIHCRD Model in Indonesia