AN OPTIMAL CONTROL OF PREVENTION AND TREATMENT OF COVID-19 SPREAD IN INDONESIA

: In this paper we analyze COVID-19 spread in Indonesia using an epidemio logical model. We consider symptomatic and asymptomatic infections in the model. We analyze the equilibria of the model and their stability which depend on the basic reproduction ratio for symptomatic and asymptomatic infections. Furthermore, we use optimal control in prevention and treatment in decreasing the number of positive COVID-19 patients in Indonesia. Furthermore, we analyze the existence of optimal control using the Pontryagin minimum principle. We also give numerical simulation of COVID-19 spread with and without the control. According to the simulation, COVID-19 spread could be reduced by using prevention and treatment control simultaneously.


INTRODUCTION
Coronavirus Disease  is caused by the severe acute respiratory syndrome corona virus. It can cause respiratory system disorder, ranging from mild symptoms such as flu, to lung infection. The control of COVID-19 spread has been carried out by government and private agencies. The government conducts domestic, regional and inter-country lock downs. Due to COVID-19, many people died and caused economic losses globally.
Mathematical experts have studied models of COVID-19 spread, such as: model of the spread of COVID-19 by considering to asymptomatic, symptomatic infections and waning immunity [1], SIR and SEIR types of COVID-19 models in unreported infected populations with outbreak factors [2], model of the spread of COVID-19 in Ghana by considering government intervention on the community [3], model of the COVID-19 pandemic in Nigeria with regard to social distancing, lock down and non-pharmaceutical interventions [4].
Following models also discuss COVID-19 spread, its treatment and interventions. Model of COVID-19 in the United Kingdom in 2021-2022 with intervention, social distancing, relaxation and no treatment [5]. Stochastic model of COVID-19 spread in Sri Lanka [6].
Model of the dynamics of the spread of COVID-19 by taking into account the isolation subpopulation [7]. Model for predicting the increase and decrease in the second wave of COVID-19 spread in Malaysia [8]. Basnarkov (2021) examines the model of the spread of COVID-19 in the SEAIR type at continuous and discrete times [9]. Nainggolan (2022) analyzes the model of COVID-19 type SIhIcQhQHR with attention congenital disease [10]. Study of the model for the spread of COVID-19 with the assumption of a public knowledge and awareness campaign in Negeria in 2020 [11]. Determining parameter estimates and numerical simulations from a model of the spread of COVID-19 in China by taking into account undetected disease cases [12]. Bifurcation, parameter estimation and prediction of the spread of COVID-19 disease with regard to treatment that cannot predict cure presentation [13].
Stability analysis and numerical simulation, determining the basic reproduction number 3 PREVENTION AND TREATMENT OF COVID- 19 SPREAD IN INDONESIA with the Lyapunov function on the COVID-19 spread model in Indonesia [14]. Mathematical model to determine the basic reproduction number in the dynamics of COVID-19 transmission in Pakistan by paying attention to social distancing [15]. Chen Tian-Mu simulated the basic reproduction number from the phase-based transmission model of COVID-19 [16]. Kim studied the two-patch mathematical model to determine the basic reproduction number on the dynamics of transmission of COVID-19 in South Korea by paying attention to early diagnostic interventions, mobility, cumulative incidence, social distancing [17]. Comparing the basic reproduction ratio of COVID-19 with SARS in Wuhan China and international [18].
Strategies to reduce the spread of disease control efforts can be made. Mathematical model for the spread of the dynamics of COVID-19 in Indonesia with government intervention to control the prevention of social distancing, lock down and treatment [19]. Research on the COVID-19 model type SEQIAHR COVID-19, to find out the dynamics of the disease and strategies to cope with minimizing the pandemic [20]. Optimal control study, sensitivity analysis and detecting the spread of COVID-19 in Indian states [21]. Study of optimal control, model-fitting Basic reproduction number Global stability in preventing the spread of COVID-19 in Nigeria [22]. Optimal control and sensitivity analysis on the dynamics of COVID-19 by considering the decrease in body immunity in West Java, Indonesia [23].
The model studied in this paper, takes into account the quarantined subpopulation and vaccination of the susceptible subpopulation. Next, we examine the optimal control of the prevention and treatment of COVID-19 using data in Indonesia.
Based on the study of the spread of COVID-19 type SEAIR [9], the authors developed the SEAILR model by considering the self-quarantine subpopulation (L), transfer from subpopulation subpopulation A, due to the fact that in Indonesia many COVID-19 patients are not reported to the base. COVID-19 data go.id. This self-quarantine subpopulation (L) is important to note, because real data in Indonesia, individuals infected with COVID-19 without comorbidities, are generally isolated independently in their respective homes because individuals infected with COVID-19 without comorbid diseases can recover with consume nutritious food and drinks, take multi vitamins, maintain health protocols, get enough rest, and keep exercising. So the novelty of the dynamic model on the spread of COVID-19 that is studied in this article by paying attention to the individual self-isolation subpopulation (L), and has not been studied in previous articles.

MODEL FORMULATION AND ANALYSIS
We assume that human population is homogenous. We divide the population into six subpopulations, which are the susceptible subpopulation S, the exposed subpopulation E, the symptomatic COVID-19 infectious subpopulation A, the asymptomatic COVID-19 infectious subpopulation I, the quarantined COVID-19 subpopulation L and the COVID- We also assume that recovered hosts have permanent immunity. Meanwhile, the quarantine for COVID-19 takes place in hospitals. The quarantined hosts get proper treatment in hospitals. We use transmission diagram as Figure 1 for model construction. Description of parameters could be seen in Table 1. All of the parameters are constant and positive.  α Infection rate 0.95 [26] v Vaccination rate 0,7 [1] µ Natural death rate 0.00712 [26] β Symptomatic transition rate 0.192 [5] σ Asymptomatic transition rate 0.4 [26] δ Quarantine rate 0.1 [5] d COVID-19-death rate 0.002 [1] η Asymptomatic self-recovery rate 0.086 [26] τ Quarantined recovery rate 0.1 [5] γ Symptomatic self-recovery rate 0.075 [26] Based on the assumptions and transmission diagram, we construct a 6-d non-linear dynamical system using ordinary differential equations for capturing the COVID-19 spread in population as following.
where N = S + E + A + I + L + R is the total population. System (1) has the disease-free equilibrium Basic reproduction ratio is an important threshold in mathematical epidemiology. It denotes the expectation number of secondary cases caused by primary cases during its infectious period in a susceptible population [24]. The basic reproduction ratio for COVID-19 spread based on model (1) is The ratio R0a is interpreted as the basic reproduction ratio for asymptomatic COVID-19 infection and the ratio R0s as the basic reproduction ratio for symptomatic COVID-19 infection. The local stability of the equilibrium P0 depends on the basic reproduction ratio.
The local stability is stated in following theorem.
Proof. To analyze the local stability of the equilibrium P0, we linearize system near the equilibrium P0. From the linearization, we get following character istic equation where 2 = + + + + + 3 + , It is clear that the first three terms of equation (3) give three negative roots.
Using the Routh-Hurwitz Criteria, the real part of the roots of the cubic polynomial are negative if R0 < 1. Descartes' rule of sign also confirms that there is no positive root of the cubic polynomial because all of the cubic polynomial's coefficients and constant are positive for R0 < 1.
Proof. Variable R only appears at the last equation of system (1), so we could decouple variable R for the stability analysis of equilibrium P1. We obtain characteristic polynomial as follow. If 0 > 1 and ψ < 1, then b0 > 0, b1 > 0, and b2 > 0. Moreover, if b2 b1 > b0, then according to Routh-Hurtwitz Criteria, the eigen values of the cubic polynomial are negative. Hence, the equilibrium P1 is locally asymptotically stable if R0 > 1. □ Our numerical simulation supports the local stability of equilibria of model (1) which obtained analytically. We use parameters' value as in Table 1 for low transmission of COVID-19.

AN OPTIMAL CONTROL OF COVID-19 SPREAD IN INDONESIA
To optimize the prevention and vaccination of COVID-19 spread, the following controls are given.
1. Control u1 is a vaccination control for susceptible subpopulation S.
In order to minimize the number of individuals who were confirmed positive, preventive controls (u1, u2) and treatment controls (u3, u4) were assigned, weighted administration costs for individuals subpopulation exposed, unreported infected and reported infected, respectively, namely:      Figure 12, preventive controls u1 and u2 better increase the number of individuals who recover from COVID-19 compared to treatment controls u3 and u4. The control u1, u2, u3 and u4 together were better at improving individuals who recovered from COVID-19 compared to the control u1 and u2 or control u3 and u4.   Figure 13 is a preventative control. Optimal control u1 reduces COVID-19 at baseline until time t = 20, after t = 20 decreases until t = 60 days, optimal control u2 reduces COVID-19 from baseline to t = 28, after t = 28 decreased to t = 60 days. Figure 14 is treatment control, optimal control u3 reduces COVID-19 from baseline to t = 35 days, after t = 35 decreases to t = 60 days, optimal control u4 decreased COVID-19 from the beginning to t = 53 days, after t = 53 decreased to t = 60 days.

CONCLUSION
Based on the analysis of the model obtained non-endemic and endemic equilibrium points.
The basic reproduction ratio of the model is obtained using the next generation matrix. The non-endemic equilibrium point in the local asymptotically stable system of equations (1), and 19 PREVENTION AND TREATMENT OF COVID-19 SPREAD IN INDONESIA the endemic equilibrium point in the local asymptotically stable system of equations (1) based on the Routh-Hurwits criteria.
The optimal control characterization analysis obtained the conditions for the existence of optimal control, from the solution of state variables associated with optimal control obtained co-state variables. Based on the Hamiltonian equation, the optimal control for the prevention and treatment of COVID-19 and the existence of co-states for each subpopulation is obtained. Numerical simulation of the basic reproduction number parameter, the value of the reproduction ratio with control is smaller than without control, meaning that by using optimal prevention and treatment controls, the spread of COVID-19 can be reduced.
Preventive control optimal (u1 and u2) were more efficient in reducing the number of individuals infected with COVID-19 compared to using treatment controls (u3 and u4). The control u1, u2, u3 and u4 together were better at improving individuals who recovered from COVID-19 compared to the control u1 and u2 or control u3 and u4.