Mathematical Modeling of Covid-19 Transmission in Jakarta with Optimal Control Strategy

Abstract

Covid19 has infected many individuals around the world, this virus is spreading rapidly. In the context of controlling and handling the spread of COVID-19, appropriate strategies and policies are needed, mathematics will play a very important role in this problem, especially to provide information about this with an understanding of the dynamics of the transmission of this covid virus. To suppress the spread of the Covid-19 virus which is currently hitting, several countries have implemented large-scale social restrictions. To identify the best approach to reduce of this Covid-19 disease spreading at minimal cost, we developed a mathematical model of the covid 19 virus by implementing large-scale social restrictions and applying optimal control theory. We provide two types of control: the first is in the form of an education campaign about covid-19 and an awareness program, and the second is in the form of a quarantine program. Compared with no optimal control, giving optimal control can provide a more significant reduction in the number of populations S, Sr, O, P, I and increase the number of individual populations that recover more significantly.

Headnote

Covid19 has infected many individuals around the world, this virus is spreading rapidly. In the context of controlling and handling the spread of COVID-19, appropriate strategies and policies are needed, mathematics will play a very important role in this problem, especially to provide information about this with an understanding of the dynamics of the transmission of this covid virus. To suppress the spread of the Covid-19 virus which is currently hitting, several countries have implemented large-scale social restrictions. To identify the best approach to reduce of this Covid-19 disease spreading at minimal cost, we developed a mathematical model of the covid 19 virus by implementing large-scale social restrictions and applying optimal control theory. We provide two types of control: the first is in the form of an education campaign about covid-19 and an awareness program, and the second is in the form of a quarantine program. Compared with no optimal control, giving optimal control can provide a more significant reduction in the number of populations S, Sr, O, P, I and increase the number of individual populations that recover more significantly.

Keywords: Covid-19; Mathematical model; Social distancing; optimal control; Numerical simulation.

Abstract

(ProQuest: ... denotes formulae omitted.)

1.Introduction

2019 was the year the first case of covid was discovered. As soon as this virus emerged, the attention of the whole world was focused on this covid 19 virus outbreak. Everyone around the world was shocked by this news. Covid-19 is a pandemic, this statement has been officially declared by WHO, after the disease has plagued more than 114 countries and regions throughout the world, on March 11, 2020. A year ago, the new coronavirus that causes Covid-19 was first reported to infect humans, marking the start of a global pandemic. According to Chinese government data, the first cases of the coronavirus appeared on November 17, 2019, weeks before the authorities announced a new disease. This disease easily and very quickly spreads to various countries.

The disease is caused by severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) [1]. Covid-19 cases worldwide that have been confirmed as of April 30, 2021 are 151,161,947 with 3,179,961 deaths and 128,572,958 cases recovered. The largest case occurred in the USA, with 33,044,068 confirmed positive cases, 589,207 deaths and 25,641,574 recoveries with this data based on data from Waldometers. Meanwhile in Indonesia, on March 2, 2020, the first case was detected. Where Central Java, West Java and Jakarta are the provinces most affected by the 34 provinces that have also been exposed to this virus until April 9. Indonesia has reported 1,662,868 positive cases, ranking first in Southeast Asia. In terms of mortality, Indonesia is in the third rank in Asia with 45,334 deaths. However, because there are cases that have been tested or confirmed with acute COVID-19 symptoms, the death rate is estimated to be much higher than the data reported, with 100,102 COVID-19 cases still being treated and 1,517,432 confirmed COVID-19 cases.

The incubation period for COVID-19 is on average 5.1 to 5.8 days and on days 11.5 to 15.6 starting to show symptoms [2]. This virus can infect from person to person, one of the ways is when we come into contact with droplets that come out of the mouth when talking or when coughing [3]-[5]. The symptoms of this disease also vary and differ from other diseases, because not all patients have the same symptoms. These symptoms include shortness of breath, cough, fever, and pneumonia in the severe case category [6]. From WHO data, this covid-19 virus can be transmitted to other people even though infected individuals only have mild symptoms or even no symptoms of pain, positive cases of covid19 like this are found around 85% [7]. Until now, vaccines also do not guarantee 100% that people who have been vaccinated will not be exposed to the virus, especially when vaccines in Indonesia have only been given to them and have not been comprehensive. Thus forcing policy makers to take other interventions to suppress the spread of this disease, such as wearing masks, physical / social distancing, quarantine, hospitalization, etc.

One of the intervention procedures to reduce the spreading of this disease is to avoid meeting or interacting with many people. Many efforts have been made by various regions to suppress the spread of the COVID-19 virus, ranging from physical distancing, social distancing, to lockdown. Isolating individuals suspected of being exposed to COVID-19 for at least two weeks or 14 days is also one way to prevent this virus from spreading [8]. In the country where the researcher is located, namely Indonesia, the government has issued a policy to maintain a safe distance where the COVID-19 virus cannot spread, which in practice several regions in Indonesia have used this policy in accordance with the direction of the Indonesian government. Citizen awareness and discipline are very much needed in this Indonesian government policy. Henceforth, several provinces in Indonesia, starting with the capital city of Indonesia, namely Jakarta, have enacted regulations regarding large-scale social restrictions, as an action by local governments to suppress the development of the COVID-19 virus.

An important role of mathematical models in helping society understand the behavior of this virus, which in the next steps will be able to help policymakers to develop intervention strategies to properly eradicate Covid-19. Many approaches have been taken to study the spread of Covid19 diseases in Wuhan, Italy and the other countries in the world [9]-[17]. In Indonesia, Nuning et al., expect that the number of cases of COVID-19 disease in Indonesia will reach over 8000 cases by April 2020[18]. In the research of Aldila et al., the usage of medical masks was found to have the largest impact on the number of new infections[19].

In this article we analyze the dynamics of Covid-19 in mathematical model in Jakarta by paying attention to government policies regarding large-scale social restrictions [20] and in contrast to some of the references that have been mentioned, we providing optimal control of the mathematical model to provide mathematical analysis information on how to control the optimal effect is given to the functions used in the covid-19 mathematical model with the application of large-scale social restrictions in Jakarta. We used the S Sr О P I R model by providing control in the form of a campaign / awareness program and quarantine. The rest of this paper is laid out as follows. The methodologies employed in this research are described in Section 2. The numerical simulation is discussed in Section 3. The research's findings are presented in the final section.

2.Methods

A.Model Description

In this study, a mathematical model of the Covid-19 pandemic was formed by implementing large-scale social restrictions [20] and providing two controls in the form of a campaign / awareness program and quarantine, the data used is in Jakarta. In this mathematical model we divide the total population N (ť) into six compartments:

* S represented the population of vulnerable/susceptible individuals

* Sr represented the population of restricted susceptible individuals

* О represented the population of under monitoring individuals

* P represented the population of under surveillance individuals

* / represented the population of confirmed infected individuals

* R represented the population of recovered individuals

Individuals entering communities, such as emigration, immigration, and new births, grow susceptible populations at a constant Л recruitment rate. Contact between susceptible and positively confirmed infected persons can infect vulnerable populations. Individuals who have not screened positive during therapy or while being monitored (P) will be considered as vulnerable individuals who come into touch with infected people before being confirmed positive for infection. ß is the transfer rate from the vulnerable to the observed population. Keeping social separation or PSBB at the rate a and ordinary death at rate p causes defenseless populations to decline as well. People that visit to or have been in areas that COVID-19 was seminated would then be monitored (O). The rate of transition from S to O is given by c in this model.

The confined susceptible persons are created at rate a by applying PSBB or the physical distance regulations to susceptible individuals. Interaction with an infected person can also bring limited vulnerable people into the under surveillance group, providing if physical distance also isn't optimum. The density values of ß has been magnified through (1 - p) in this situation, which 0 < p < 1 has become the tier of transmission efficacy determined by individual discipline in adhering to PSBB regulations, establishing personal space, as well as the government's persistence toward individuals who break the physical distance regulations. It's worth noting that p = 0 denotes that no one follows the PSBB regulations.

People emerge during reconnaissance as a consequence of the association between vulnerable people and contaminated people at the rate ß then, at that point, diminished by the advancement of tainted populaces at a rate y. The reported affected are also from the under-monitored person who screened positive in a rate a. Disease-induced death reduces the established affected population at a rate d. The populations diagnosed infected, under surveillance, and under monitoring are all decreasing at rates of v, p and m, correspondingly. Furthermore, all compartments' populations are supposed to die naturally at a rate p.

Figure 1 shows a graphical representation of the model formed, and here the following system of differential equations from the covid-19 mathematical model with the application of largescale social restrictions with control:

â...ž = Л- aS(t) - ßS(t)I(t) - pS(t) - cS(t) - 8lS(t) (1)

â...žT2 = aS(t) - ß(1 - p)Sr(t)I(t) - pSr(t) + giS(t) (2)

cS(t) - (p + a + ш)О(Р) (3)

â...žp = ßS(t)I(t) + ß(l - p)Sr(ť)I(ť) - (ß + у + v)P(t) (4)

^= YP(t) + oO(ť) - (ß + v + d + g2)I(t) (5)

^ = gP(t) + (v + ß2)I(t) + uO(t) - pR(t) (6)

Where the initial state is 5(0) > 0, Sr (0) > 0,0(0) > 0, P(0) > 0,1(0) > 0 and R(0) > 0. We note that the first control g1(ť) is represents of education campaign about covid-19 and awareness program and the second control g2 (t) is represents of quarantine program.

* a : Restricted coverage rate

* Л : Recruitment rate of susceptible humans

* ß : Transmission rate of susceptible to under surveillance

* p : Natural mortality rate

* C : Transmission rate of susceptible to under monitoring

* a : Recruitment rate of confirmed infected from under monitoring

* p : The efficacy of large-scale social restrictions

* v : Recovery rate of confirmed infected

* ш : Recovery rate of under monitoring

* p : Recovery rate of under surveillance

* у : Recruitment rate of confirmed infected from under surveillance

* d : Mortality rate of confirmed infected.

We run computational modeling utilizing model parameters from data given by the Indonesian government. We focus on information of Jakarta, that was the first province to execute the physical distance strategy. To create more credible computational results, the required parameters are approximated utilizing existing data and several assumptions, as stated below.

The parameter is estimated to be Л = 403 per day. The overall population of Jakarta region is 10,645,000 people, through an average life span of 72.4 years, according to the Central Bureau of Statistics (BPS). As a result, estimated rate of natural death is equal to p = -1- = 0.0000378 per day.

Individuals who are under surveillance, under monitored, and restricted are measured using the parameters c, ß, and a. The values c = 0, ß = 0, and a = 0 can be used throughout the initial stages of the COVID-19 model. Following the government's recognition of positive COVID-19 patients, the government began enforcing physical separation restrictions. As a result, we change the value of c, ß, and a to a constant number. Because no data is available, these numerical values were chosen to guarantee that the outcomes were reasonable. Because identifying over monitoring persons appears be simpler then identifying surveillance persons, and finding confined persons appears to be simpler then identifying over monitoring persons, it is plausible to conclude that a>c>ß. We assume 0.4 >a> 0.01 per day, 1.0 X 10-2 > c > 3.78 X 10-5 per day, and 3.78 X 10-3 > ß > 3.78 X 10-6 per day for this calculation.

Throughout this point, national Task Force with the Improvement of COVID-19 Handling will indeed send updates on mortality data for persons that have a positive verified COVID-19 test. Individuals who are under monitoring and surveillance die as a result of COVID-19 are not taken into account in this model. Furthermore, we believe that those who are under surveillance and monitoring, as well as those who have been confirmed, can recover from this condition. COVID-19 has an typical length of the illness is 21 days and incubation period of 5.1 to 15.6 days[18]. According to historical the survival rate is around 4.53 percent and mortality rate for COVID-19 patients through Jakarta region from April 10, 2020 to April 10, 2020 is around 8.6%. As a result, it's safe to suppose here that gamma and sigma per day are amongst 0.064 and 0.196, so the illness rate of death is between 0.086 > d > 0.045. However, the percentage of those who have completed monitoring is approximately 86 percent, although the percentage of people who have been certified healed is around 54.8 percent. This number is steadily increasing. As a result, we suppose that the daily values of v, ш, and ß are between 0.045 to 0.86. We used the estimated parameter values that shown on Table 1 for simulation purposes.

A. Solutions' Positivity

Theorem 1.1 : If 5(0) > 0,5r(0) > 0,0(0) > 0,P(0) > 0,1(0) > 0, and R(0) > 0, the solutions S(t),Sr(t), O(t),P(t),I(t), and R(t) of system (3) are positive for all t > 0.

Proof:

...

where Z (t) = a + ßl(t) + ß + c. In the last in-equalities, all sides are divided by exp ^f^Z(s)dsJ.

We've got

exp ^J Z(s)ds^ Jį) + Z(s) exp ^J Z(s)ds^ S(t) > 0

subsequently

dS(t)

When this inequality is integrated from 0 to t, the following results are obtained:

In a similar vein, we demonstrate that Sr (0) > 0,0(0) > 0, P(0) > 0,1(0) > 0, and R(0) > 0.

B.Invariant Region

The boundedness of the model (1)-(6) are given in the following theorem.

Theorem 1.2 : The solution of the system (1)-(6) are uniformly bounded [21].

Proof: Given (S(t),Sr(t), O(t), P(t),I(t), R(t), assume that N = S + Sr + O + P + I + R and

...

0 <N<- (11)

So that the model positively invariant region is obtained by:

ů = {ne!+: 0 < N(S, Sr, O, P, I, R) < ^ (12)

C.The Existence of Optimal Control

We consider the objective function is to minimize the problem [22][23][24][25].

3İ9i,g2) = ľ/ [^(t) + lit) - R(ť) + ^kigįiO + ^.øKO] dt (13)

Where k1 and k2 is the positif weight parameters for education campaign about covid-19 and quarantine program. tf is the final time of the simulation. Our main objective function is to

3(g·,g2) = minai,g2et 3(gi,g2) (14)

5 = {gi,g2/0 < gimln < дЛО < gimax < i,° < g2min < g2(t) < g2max < i, te [0, tf]} (15)

Theorem 2: Consider the control problem in system (13)[26][27]. There exists an optimal control pair [28]

g· = (g'·,g2 e О (16)

such that

3(gl g2) = mingi,g2et Яз^ g2) (17)

Proof: To proof the existence of the optimal control, according to the classic literature, we have to check the following steps:

1. The control and state variables is nonnegative values.

2. The objective function is convex in

3. The control space is convex and closed.

4. The integrand of the objective function is concave on ţ

£(t; дт, g2) = s(t) + i(t)-R(t) + T(kigį(t)) + Т(к2дК0) (18)

* Objective functional is follows

£(t;дт, g2) > qd\gi\2 + \g2\2)b/2 - Ч2 (19)

the set of controls and the corresponding state variables of the system () is nonnegative values and condition 1 is satisfied.

* The control set ţ = {(gu g2)/(âi, дг) is measurable,0 < gi mâ...∂ < gfO < gi max < i , 0 < g2 min < g2(t) < g2 max < i, t e [0, tf - i]} is convex and closed by definition.

* All The right side of the state system is bounded above by a sum of bounded control and state, and can be written as a linear function of дт, g2 with coefficients depending on time and state [29].

* The integrand of the objective cost function, S(t) + I(t) - R(t) + 2 (k1g:į(t)) +1 (k2g2(t))

is clearly convex on Since the state solutions are bounded the Lipschitz property of the state system corresponding to the state variables is fulfilled [30]. I.e

...

...

is also true, when we choose ą1 = min ļļI,and for all q2 eR+, b = 2.

D.Maintaining the Integrity of the Specifications

In this section, the Hamiltonian function will be given by:

K = S(t) + I(t) - R(t) +1 k^1(t) + 2k2gį(t) + Пг(Л - aS(t) - ßS(t)I(t) - pS(t) - cS(t) - g1(t)S(t)') + ü2{aS(t) - ß(1 - p)Sr (t)I(t) - pSr (t) + g1(t)S(t)) + ū3{cS(t) - (p + a + of)0(t)) + ü4{ßS(t)I(t) + ß(1 - p)Sr(t)I(t) - (p + у + p)P(t)) + ßs (yP(t) + oO(t) - {p + v + d + д2(0)1(£)) + fieOlP(t) + (v + g2(t))I(t) + ^O(t) - pR(t)) (27)

Theorem 3 : Consider the control problem with the system (1)-(6) [31]. There are two optimal controls (gi,g2) e f2. Given optimal control pair (gf, g2) and S(t),Sr(t), O(t),P(t),I(t),R(t) of the corresponding state system (1)-(6), there exist adjoint variables â...∂ i = 1,2,3,4,5,6 that fulfill:

...(28)

...(29)

...(30)

...(31)

...(32)

...(33)

Furthermore, for t e [0, tf\, the optimal controls g{ and g2 is follows:

g1 = min ^1,max (о, £i£_â...∂£)1 (34)

g2 = min ^1, max (о, П5/1~Пб/)| (35)

Proof: The Hamiltonian K are defined by:

K(t) = S(t) + I(t) - R(t) + \k1gį(t) + 1k2g2(t) + ZUHWfi(S,Sr, O,P,I,R) (36) Where[32],

f1 (S, Sr, O, P, I, R) = Л - aS(t) -ßS(t)I(t) - pS(t) - cS(t) - g1 (t)S(t) (37)

f2(S, Sr, O, P, I, R) = aS(t) - ß(1 - p)Sr (t)I(t) - pSr (t) + gf^Sf) (38)

f3(S,Sr, O,P, I,R)= cS(t) - (p + a + ш)О(Т) (39)

f, (S, Sr, O, P, I, R) = ßS(t)I(t) + ß(1 - p)Sr(t)I(t) - (p + y + p)P(t) (40)

f5(S,Sr, O,P, I,R) = YP(t) + aO(f) - {p + v + d + g2(t))l(t) (41)

f6(S,Sr, O,P, I,R) = VP(t) + (v + g2(t))I(t) + vO(t) - pR(t) (42)

By using Pontryagin Maximum Principle [33], the transversality conditions and adjoint equations is follows:

H = -¶K = -(1 + H1(-a - ßl - p - c - g-ļ^) + H2(a + gf) + H3c + H4ßl) (43)

H2 = -JKt = -(n2(-ß(1 - p)i - p) + H4(ß(1 - p)I)) (44)

Н'з = - = - (H3 (-p + & + ^0 + H5 о + Н(,ш) (45)

H4 = -d-K = -Ш-Р + Y + v) + + ад (46)

П = -^ = -(1 + n.1(-ßs) + a2(-ß(i - p)st) + a4(ßs + ß(1 - P)st) +

...(47)

П = -â...ž = -(-1 + ^e(-ß)) (48)

For t E [0, tf\, from the optimally condition we can solve the optimal controls gl and g·2,

= k1g1 - a1S + a2S =0 (49)

... (50)

We have,

...

The optimal control system are:

dS1(t) dSy(t) dO·(t) dP·(t) dl·(t) dR·(t) dt ' dt ' dt ' dt ' dt ' dt

П = -(1 + П1(-а - ßl - g - c - gß) + П2(а + д1) + П3с + n4ßl)

П = -(Ü2(-ß(1 - р)1 -д)+ ü4(ß(1 - р)1))

...

П = -(П,4(-р + у + р)+ П.5У + ПбР)

п'5 = -(1 + а^-ßs) + n2(-ß(1 - p)sr) + n4(ßs + ß(1 - p)sr) + n5(g2)

+ Пб(р + g2))

...

gļ = min [1, max (0,^5^-^)}

So, that is easy to show that the theorem is proven [34].

2. Numerical Simulation

In this section, we present the results obtained using the numerical method of Runge-Kutta order 4 in solving the optimality system of the covid 19 mathematical model by applying largescale social constraints and providing optimal control. The control we provide is in the form of g1(t) is represents of education campaign about covid-19 and awareness program and g2(t) is represents of quarantine program with the aim of this research function is to reduce the number of susceptible individuals and infected individuals, as well as to increase the number of recovered individuals. First, for each control we provide an initial condition (u=0 is almost always sufficient), and the values of the parameters used are shown in the table 1.

We present the simulation results that we did for each sub-population in Figure 2.for subpopulations of individuals who are susceptible and restricted susceptible, it shows that after being given control in the last days leading up to time end experienced a decrease that was more significant than those without control.

According to Figure 3, it shows the sub-population of individuals who are under monitoring and infected in the first days it showed a decrease, but after that it continued to increase in the uncontrolled model and after being given control in the days towards the end it showed a decrease. Sub-population of individuals who are under surveillance shows before and after being given control both experienced a decrease but after being given control on the last days the decrease was more significant and headed to 0. The sub-population of individuals who are recovered showed an increase from the first day to the last on systems that were not and have been given control, but on systems that were given control, the increase was more significant.

3. Conclusion

In this paper, we create a covid 19 mathematical model with the application of large-scale social restrictions and adding optimal control to the model. Our aim is to provide information about the effectiveness of the controls given to the mathematical model used. We propose two controls the first control g1 (t) is represents of education campaign about covid-19 and awareness program and the second control g2 (t) is represents of quarantine program, we have applied optimal control to our mathematical model. Optimality system is created with the help of the pontryagin maximum principle. From this process we get the results from our simulation. From these results we get the characteristics and behavior of the controls we provide, which in the graph we present in Figure 2 and Figure 3, shows that the control we provide can reduce the individual population of susceptible individuals, restricted susceptible, under monitoring, under surveillance, infected and increase the population of recovered individuals is more significant when compared to without optimal control. This shows that the control given to the mathematical model can work well according to the objective function established in this study.

Nur Ilmayasinta received the Bachelor (BSc-Financial Mathematics) and Masters (MSc-Applied Mathematics) from Department of Mathematics Institut Teknologi Sepuluh Nopember. She is a lecturer in Department of Mathematics Education Universitas Islam Lamongan (Unisla). She has teaches courses Real Analysis, Numerical Method, and Multiple Variable Calculus. Her area of interest are financial mathematics, optimal control, mathematical modelling and partial differential equation.

Prismahardi Aji Riyantoko is a Lecturer in Department of Data Science, Faculty of Computer Science, UPN "Veteran" Jawa Timur. He earned Bachelor (BSc-Computational Mathematics), Masters (MSc-Applied Mathematics) in Department of Mathematics Institut Teknologi Sepuluh Nopember. He has been recognized as a professional IT consultant with around one year of experience in working at Viseo with salesforce practice business partner. His research interest includes partial differential equation, optimal control, modelling system, and data analytics.

Nur Qomariyah Nawafilah received the Bachelor (BSc-Mathematics Education) Universitas Negeri Surabaya and Masters (MSc-Mathematics Education) Universitas Negeri Malang. She is a lecturer in Department of Informatics Engineering Universitas Islam Lamongan (Unisla).

Masruroh received the Bachelor (BSc-Science Education) and Masters (MSc-Science Education) Universitas Negeri Surabaya. She is a lecturer in Department of Informatics Engineering Universitas Islam Lamongan (Unisla).

Rahma Febriyanti received a Bachelor degree in 2016, Mathematics Department, Universitas Negeri Surabaya, Indonesia. She obtained a Master in 2018 respectively in Mathematics Education, Universitas Negeri Surabaya. In 2019, she has been a lecturer at FKIP Universitas Islam Lamongan, Indonesia.

Hesti Rahayu was born in Lamongan, East Java, Indonesia in 2000. She is currently B.Ed student at Department of Mathematics Education Universitas Islam Lamongan.

References

4. References

[1] G. Vargas, L. H. Medeiros Geraldo, N. Gedeäo Salomäo, M. Viana Paes, F. Regina Souza Lima, and F. Carvalho Alcantara Gomes, "Severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) and glial cells: Insights and perspectives," Brain, Behav. Immun. - Heal., vol. 7, no. August, p. 100127, 2020, doi: 10.1016/j.bbih.2020.100127.

[2] S. A. Lauer et al., "The incubation period of coronavirus disease 2019 (CoVID-19) from publicly reported confirmed cases: Estimation and application," Ann. Intern. Med., vol. 172, no. 9, pp. 577-582, 2020, doi: 10.7326/M20-0504.

[3] H. David et al., "The continuing 2019-nCoV epidemic threat of novel coronaviruses to global health - The latest 2019 novel coronavirus outbreak in Wuhan, China," Int. J. Infect. Dis., vol. 91, no. January, pp. 264-266, 2020.

[4] A. R. W. H. O. declares global emergency: A. review of the 2019 novel coronavirus (COVID-19). I. J. S. 2020 A.-76. doi: 10. 1016/j. ijsu. 2020. 02. 034. E. 2020 F. 26. E. in: I. J. S. 2020 M. P. 32112977; P. Sohrabi C, Alsafi Z, O'Neill N, Khan M, Kerwan A, AlJabir A, Iosifidis C, "World Health Organization declares global emergency: A review of the 2019 novel coronavirus (COVID-19)," no. January, 2020.

[5] V. Tiwari, N. Deyal, and N. S. Bisht, "Mathematical Modeling Based Study and Prediction of COVID-19 Epidemic Dissemination Under the Impact of Lockdown in India," Front. Phys., vol. 8, no. November, pp. 1-8, 2020, doi: 10.3389/fphy.2020.586899.

[6] S. D. Viana, S. Nunes, and F. Reis, "Since January 2020 Elsevier has created a COVID-19 resource centre with free information in English and Mandarin on the novel coronavirus COVID- 19 . The COVID-19 resource centre is hosted on Elsevier Connect , the company ' s public news and information ," no. January, 2020.

[7] B. Aylward and W. Liang, "Report of the WHO-China Joint Mission on Coronavirus Disease 2019 (COVID-19)," WHO-China Jt. Mission Coronavirus Dis. 2019, vol. 2019, no. February, pp. 16-24, 2020, [Online]. Available: https://www.who.int/docs/defaultsource/coronaviruse/who-china-joint-mission-on-covid-19-final-report.pdf.

[8] Y. Yi, P. N. P. Lagniton, S. Ye, E. Li, and R. H. Xu, "COVID-19: What has been learned and to be learned about the novel coronavirus disease," Int. J. Biol. Sci., vol. 16, no. 10, pp. 1753-1766, 2020, doi: 10.7150/ijbs.45134.

[9] N. Piovella, "Analytical solution of SEIR model describing the free spread of the COVID19 pandemic," Chaos, Solitons and Fractals, vol. 140, p. 110243, 2020, doi: 10.1016/j.chaos.2020.110243.

[10] J. A. M. Gondim and L. Machado, "Optimal quarantine strategies for the COVID-19 pandemic in a population with a discrete age structure," Chaos, Solitons and Fractals, vol. 140, p. 110166, 2020, doi: 10.1016/j.chaos.2020.110166.

[11] N. Feroze, "Forecasting the patterns of COVID-19 and causal impacts of lockdown in top five affected countries using Bayesian Structural Time Series Models," Chaos, Solitons and Fractals, vol. 140, 2020, doi: 10.1016/j.chaos.2020.110196.

[12] S. Djaoue, G. Guilsou Kolaye, H. Abboubakar, A. A. Abba Ari, and I. Damakoa, "Mathematical modeling, analysis and numerical simulation of the COVID-19 transmission with mitigation of control strategies used in Cameroon," Chaos, Solitons and Fractals, vol. 139, 2020, doi: 10.1016/j.chaos.2020.110281.

[13] A. Din, Y. Li, T. Khan, and G. Zaman, "Mathematical analysis of spread and control of the novel corona virus (COVID-19) in China," Chaos, Solitons and Fractals, vol. 141, p. 110286, 2020, doi: 10.1016/j.chaos.2020.110286.

[14] A. A. Gebremeskel, H. W. Berhe, and H. A. Atsbaha, "Mathematical modelling and analysis of COVID-19 epidemic and predicting its future situation in Ethiopia," Results Phys., vol. 22, p. 103853, 2021, doi: 10.1016/j.rinp.2021.103853.

[15] K. S. Nisar, S. Ahmad, A. Ullah, K. Shah, H. Alrabaiah, and M. Arfan, "Mathematical analysis of SIRD model of COVID-19 with Caputo fractional derivative based on real data," Results Phys., vol. 21, p. 103772, 2021, doi: 10.1016/j.rinp.2020.103772.

[16] A. B. Gumel, E. A. Iboi, C. N. Ngonghala, and E. H. Elbasha, "A primer on using mathematics to understand COVID-19 dynamics: Modeling, analysis and simulations," Infect. Dis. Model., vol. 6, no. August 2020, pp. 148-168, 2021, doi: 10.1016/j.idm.2020.11.005.

[17] Abdullah et al., "Mathematical analysis of COVID-19 via new mathematical model," Chaos, Solitons and Fractals, vol. 143, no. April 2020, p. 110585, 2021, doi: 10.1016/j.chaos.2020.110585.

[18] N. Nuraini, K. Khairudin, and M. Apri, "Modeling Simulation of COVID-19 in Indonesia based on Early Endemic Data," Commun. Biomath. Sci., vol. 3, no. 1, pp. 1-8, 2020, doi: 10.5614/cbms.2020.3.1.1.

[19] D. Aldila, M. Z. Ndii, and B. M. Samiadji, "Optimal control on COVID-19 eradication program in Indonesia under the effect of community awareness," Math. Biosci. Eng., vol. 17, no. 6, pp. 6355-6389, 2020, doi: 10.3934/mbe.2020335.

[20] Y. A. Adi and M. Z. Ndii, "Modeling and Prediction of COVID-19 with a Large Scale Social Distancing," J. Fourier, vol. 9, no. 1, pp. 1-9, 2020, doi: 10.14421/fourier.2020.91.1-9.

[21] M. Mandal, S. Jana, S. K. Nandi, A. Khatua, S. Adak, and T. K. Kar, "A model based study on the dynamics of COVID-19: Prediction and control," Chaos, Solitons and Fractals, vol. 136, p. 109889, 2020, doi: 10.1016/j.chaos.2020.109889.

[22] Mardlijah, N. Ilmayasinta, L. Hanafi, and S. Sanjaya, "Optimal Control of Lipid Extraction Model on Microalgae Using Linear Quadratic Regulator ( LQR ) and Pontryagin Maximum Principle ( PMP ) Methods," pp. 129-141, 2018.

[23] Mardlijah, N. Ilmayasinta, and E. A. Irhami, "Optimal control for extraction lipid model of microalgae Chlorella Vulgaris using PMP method," 2019, doi: 10.1088/17426596/1218/1/012043.

[24] N. Ilmayasinta, E. Anjarsari, and M. W. Ahdi, "Optimal Control for Smoking Epidemic Model," Proc. 7th Int. Conf Res. Implementation, Educ. Math. Sci. (ICRIEMS 2020), vol. 528, no. Icriems 2020, pp. 323-328, 2021, doi: 10.2991/assehr.k.210305.046.

[25] L. G. De Pillis et al., "Optimal control of mixed immunotherapy and chemotherapy of tumors," J. Biol. Syst., vol. 16, no. 1, pp. 51-80, 2008, doi: 10.1142/S0218339008002435.

[26] H. T. Alemneh and N. Y. Alemu, "Mathematical modeling with optimal control analysis of social media addiction," Infect. Dis. Model., vol. 6, pp. 405-419, 2021, doi: 10.1016/j.idm.2021.01.011.

[27] A. Kouidere, L. EL Youssoufi, H. Ferjouchia, O. Balatif, and M. Rachik, "Optimal Control of Mathematical modeling of the spread of the COVID-19 pandemic with highlighting the negative impact of quarantine on diabetics people with Cost-effectiveness," Chaos, Solitons and Fractals, vol. 145, p. 110777, 2021, doi: 10.1016/j.chaos.2021.110777.

[28] J. A. Bather, W. H. Fleming, and R. W. Rishel, Deterministic and Stochastic Optimal Control., vol. 139, no. 4. 1976.

[29] M. Derouich, A. Boutayeb, W. Boutayeb, and M. Lamlili, "Optimal control approach to the dynamics of a population of diabetics," Appl. Math. Sci., vol. 8, no. 53-56, pp. 2773-2782, 2014, doi: 10.12988/ams.2014.43155.

[30] E. Bonyah, M. A. Khan, K. O. Okosun, and J. F. Gómez-Aguilar, "Modelling the effects of heavy alcohol consumption on the transmission dynamics of gonorrhea with optimal control," Math. Biosci., vol. 309, no. November 2018, pp. 1-11, 2019, doi: 10.1016/j.mbs.2018.12.015.

[31] A. H. Permatasari, R. H. Tjahjana, and T. Udjiani, "Existence and characterization of optimal control in mathematics model of diabetics population," J. Phys. Conf Ser., vol. 983, no. 1, 2018, doi: 10.1088/1742-6596/983/1/012069.

[32] M. Zamir, T. Abdeljawad, F. Nadeem, A. Wahid, and A. Yousef, "An optimal control analysis of a COVID-19 model," Alexandria Eng. J., vol. 60, no. 3, pp. 2875-2884, 2021, doi: 10.1016/j.aej.2021.01.022.

[33] L. S. Pontryagin, "L.S. Pontryagin Selected Works Volume 4. The Mathematical Theory of Optimal Processes," vol. 4-The Ma, 1986.

[34] C. Mondal, D. Adak, and N. Bairagi, "Optimal control in a multi-pathways HIV-1 infection model: a comparison between mono-drug and multi-drug therapies," Int. J. Control, vol. 0, no. 0, pp. 1-33, 2019, doi: 10.1080/00207179.2019.1690694.