EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 16, No. 1, 2023, 523-537 ISSN 1307-5543 – ejpam.com Published by New York Business Global Optimal Control of COVID-19 Model with Partial Comorbidity Sub-populations and Two Isolation Treatments in Indonesia Muhammad Abdurrahman Rois1, Fatmawati1,∗, Cicik Alfiniyah1 1 Mathematics Department, Faculty of Science and Technology, Universitas Airlangga, Surabaya 60115, Indonesia Abstract. We applied sensitivity analysis and optimum control to the COVID-19 model in this research. In addition, the basic reproduction number calculated as 1.57 indicates that this illness is widespread across Indonesia. The most important factor in this model is the contact rate with infected people, with or without comorbidity. Optimal control will minimize the number of infected populations without and with comorbidity, and costs. Numerical experiments will be carried out to describe and compare the graphical models of the spread of COVID-19 with and without controls. From the numerical results and cost-effectiveness analysis on the optimal control problem, it is found that applying a combination of controls can give the best results compared to a single control. 2020 Mathematics Subject Classifications: 92D30, 93E20 Key Words and Phrases: COVID-19, comorbidity, sensitivity analysis, optimal control 1. Introduction COVID-19 symptoms are usually mild and appear gradually. COVID-19 symptoms include fever, a dry cough, and tiredness. Other symptoms include chest pain and tender- ness, nasal congestion, headache, conjunctivitis, diarrhea, loss of taste or smell, skin rash, etc [22, 24]. People who have had diabetes, lung, or heart disease in the past are more likely to get a severe disease with stronger COVID-19 symptoms than people who don’t have comorbidity [9, 25]. COVID-19 comorbidity in Indonesia has recorded 12 different diseases, listed from most dangerous to least dangerous: high blood pressure, diabetes, heart disease, pregnancy, lung, kidney, immune disorders, cancer, other respiratory disor- ders, asthma, tuberculosis, and liver disease [15, 21]. ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v16i1.4666 Email addresses: roizmuhammad.math@gmail.com (M. A. Rois), fatmawati@fst.unair.ac.id (Fatmawati), cicik-a@fst.unair.ac.id (C. Alfiniyah) https://www.ejpam.com 523 © 2023 EJPAM All rights reserved. M. A. Rois, Fatmawati, C. Alfiniyah / Eur. J. Pure Appl. Math, 16 (1) (2023), 523-537 524 On March 2, 2020, President Jokowi Widodo directly reported the first case in Indone- sia [1]. According to data from the web worldometer [23], On October 2, 2020, Indonesia ranked 23 out of 215 nations known to be afflicted [16]. The studies on the COVID-19 model are presented as follows: Das et al. [7] adds a sub-population of people infected with comorbidity. This makes the population into five sub-populations. The general congenital disease is the comorbidity of this study. While research from Omame et al. [13] also proposed a comorbidity model of COVID-19 with co-morbidity (especially diabetes mellitus). So, Omame et al. construct a model by dividing the population into eight sub-populations. Another study, by Rois et al.[19], incorporates quarantine and isolation sub-populations; hence, the model splits the pop- ulation into seven sub-populations. The model is also based on the most recent WHO data, which says that susceptible people must be quarantined first to stop the disease from spreading further. Research on COVID-19 was also conducted by Prathumwan et al. [14] by adding quarantine and isolation sub-populations so that the constructed model has six sub-populations. Managing the developed mathematical model is necessary to lower COVID-19 infec- tions. Researchers discussing control issues include Deressa & Duressa [8], Olaniyi et al. [12], and Rois et al. [18]. Deressa & Duressa propose three controls: public education, protecting yourself from COVID-19 infection (such as wearing masks, washing hands, and keeping a safe distance), and treating COVID-19 patients in hospitals. While Olaniyi et al. and Rois et al. provide two controls: public education, and individual care management in hospitals. There are many other studies related to COVID-19 besides the ones listed above. For example, see the following literature [2–4, 6, 11, 17]. The COVID-19 model will be built in this study by combining the research of Das et al. [7], Rois et al. [19], and Prathumwan et al. [14] with two controls: 1) public education (u1), and 2) individual treatment efforts for infected (u2). Model formulation, model validation, sensitivity analysis, effect parameters, and optimum control are discussed. In addition, a numerical simulation of the model is provided. The final topic is cost-effectiveness. 2. Results and Discussion 2.1. Model formulation and validation The COVID-19 model consist of eight sub-populations: susceptible (S), exposed with- out comorbidity (EN ), exposed with comorbidity (EC), infected without comorbidity (IN ), infected with comorbidity (IC), isolated with treatment (HT ), isolated without treatment (HN ), and recovered (R). Furthermore, the COVID-19 model can be presented in a com- partment diagram in Figure 1. The following system of differential equations is derived from the compartmental dia- gram in Figure 1: dS dt = π − β1SIN N − β2SIC N − µS, M. A. Rois, Fatmawati, C. Alfiniyah / Eur. J. Pure Appl. Math, 16 (1) (2023), 523-537 525 Figure 1: COVID-19 Model with Partial Comorbidity Sub-populations and Two Isolation Treatments dEN dt = α (β1SIN + β2SIC) N − δ1EN − µEN , dEC dt = (1− α) (β1SIN + β2SIC) N − δ2EC − µEC , dIN dt = δ1EN − h1IN − r1IN − d1IN − µIN , (1) dIC dt = δ2EC − h2IC − r2IC − d2IC − µIC , dHT dt = θh1IN + δh2IC − r3HT − d3HT − µHT , dHN dt = (1− θ)h1IN + (1− δ)h2IC − r4HN − d4HN − µHN , dR dt = r1IN + r2IC + r3HT + r4HN − µR. Based on Indonesian data, it can be concluded that the MAPE (Mean Absolute Per- centage Error) for model validation that we present in Figure 2 using the lsqcurvefit com- mand is 0.028579. This time period spans from November 1, 2020 to May 19, 2021. Furthermore, the new parameter values and descriptions are shown in Table 1. 2.2. Sensitivity analysis The basic reproduction number (R0) has a sensitivity index that is differentiation for each of its parameters [5, 20]. R0 for system (1) is found. R0 = ρ (M) = RI +RC , with RI = β1δ1α a1a3 and RC = β2δ2(1−α) a2a4 . Here is the parameter sensitivity index R0, with Z = β1δ1a2a4α+ β2δ2a1a3 (1− α). SR0 β1 = δ1β1αa2a4 Z , SR0 β2 = δ2β2a1a3 (1− α) Z , SR0 d1 = −δ1β1d1a2a4α a3Z , SR0 α = (δ1β1a2a4 − δ2β2a1a3)α Z , SR0 δ1 = β1δ1a2a4α Z ( 1− δ1 a1 ) , M. A. Rois, Fatmawati, C. Alfiniyah / Eur. J. Pure Appl. Math, 16 (1) (2023), 523-537 526 Figure 2: Model validation SR0 δ2 = β2δ2a1a3 (1− α) Z ( 1− δ2 a2 ) , SR0 h1 = −δ1β1h1a1a2a4α a3Z , SR0 r1 = −δ1β1r1a2a4α a3Z , SR0 d2 = −δ2β2d2a1a3 (1− α) a4Z , SR0 h2 = −δ2β2h2a1a3 (1− α) a4Z , SR0 r2 = −δ2β2r2a1a3 (1− α) a4Z , and SR0 µ = − δ1β1a2a4αµ ( 1 a1 + 1 a3 ) + δ2β2a1a3µ (1− α) ( 1 a2 + 1 a4 ) Z . Table 2 displays the parameter sensitivity index, which indicates that β2 and β1 are the most sensitive parameters. 2.3. Effect parameters Using contour plots, the effect of parameters on R0 was investigated. We pick three significant parameters (β1, β2, and h2) and plot them as a function of R0. Figure 3 inves- tigates the impact of some R0 parameters further and shows that increasing parameters β1 and β2 as parameters with a positive index can increase the value of R0. This implies that more personal interaction will accelerate the transmission of COVID-19. Meanwhile, increasing the parameter with a negative index, namely the parameter h2, can reduce the value of R0 and implies that increased isolation will reduce the spread of COVID-19. As a result, It is essential to improve education and isolation to stop the development of COVID-19. M. A. Rois, Fatmawati, C. Alfiniyah / Eur. J. Pure Appl. Math, 16 (1) (2023), 523-537 527 Table 1: The new parameter values and descriptions Parameter Value Descriptions π 3783175.865 recruitment or birth rate β1 0.5524 contact rate (without comorbidity) β2 0.55348 contact rate (with comorbidity) α 0.49383 contact proportions δ1 0.028911 progressions rate from exposed to infection (without comorbidity) δ2 0.22241 progressions rate from exposed to infection (with comorbidity) δ 0.2349 proportion of isolation from infection (with comorbidity) θ 0.25353 proportion of isolation from infection (without comorbidity) h1 0.11791 isolation rate from infection (without comorbidity) h2 0.11161 isolation rate from infection (with comorbidity) r1 0.087527 recovery rate r2 3.987× 10−5 recovery rate r3 0.54385 recovery rate r4 0.37245 recovery rate d1 0.036233 COVID-19 death rate d2 0.18549 COVID-19 death rate d3 0.28641 COVID-19 death rate d4 0.34042 COVID-19 death rate µ 0.0138 natural death rate 2.4. Optimal control The COVID-19 model by incorporate control variables u1 (public education) and u2 (individual treatment efforts for infected) is given by dS dt = π − (1− u1) (β1SIN + β2SIC) N − µS, dEN dt = α (1− u1) (β1SIN + β2SIC) N − δ1EN − µEN , dEC dt = (1− α) (1− u1) (β1SIN + β2SIC) N − δ2EC − µEC , dIN dt = δ1EN − (h1 + u2) IN − r1IN − d1IN − µIN , dIC dt = δ2EC − (h2 + u2) IC − r2IC − d2IC − µIC , (2) dHT dt = (θh1 + u2) IN + (δh2 + u2) IC − r3HT − d3HT − µHT , dHN dt = ((1− θ)h1 + u2) IN + ((1− δ)h2 + u2) IC − r4HN − d4HN − µHN , dR dt = r1IN + r2IC + r3HT + r4HN − µR. M. A. Rois, Fatmawati, C. Alfiniyah / Eur. J. Pure Appl. Math, 16 (1) (2023), 523-537 528 Table 2: Sensitivity analysis. Parameter β2 β1 d2 µ h1 h2 Index 0.53996 0.46004 −0.32211 −0.229 −0.21233 −0.19382 Parameter r1 δ1 α d1 δ2 r2 Index −0.15762 0.14864 −0.06675 −0.06525 0.03155 −0.000069 Figure 3: Effect parameters of R0. Over a time range of [0, T ], the function that minimizes the number of infected cases without and with comorbidity can be written as F (u1, u2) = ∫ T 0 v (t,−→x ,−→u ) dt = ∫ T 0 ( IN + IC + 1 2 ( K1u 2 1 +K2u 2 2 )) dt, , (3) where K1 and K2 are the relative cost associated with the controls u1 and u2, and T is the final time. The objective of the control is to reduce the cost function. F (u∗1, u ∗ 2) = min F (u1, u2), subject to the system (2), where 0 ≤ (u1, u2) ≤ 1 and t ∈ (0, T ). 2.5. Optimal Control Analysis The Hamilton function can be defined as follows H = IN + IC + 1 2 ( K1u 2 1 +K2u 2 2 ) + τ1 ( π − (1− u1) (β1SIN + β2SIC) N − µS ) + τ2 ( α (1− u1) (β1SIN + β2SIC) N − δ1EN − µEN ) + τ3 ( (1− α) (1− u1) (β1SIN + β2SIC) N − δ2EC − µEC ) M. A. Rois, Fatmawati, C. Alfiniyah / Eur. J. Pure Appl. Math, 16 (1) (2023), 523-537 529 + τ4 (δ1EN − (h1 + u2) IN − r1IN − d1IN − µIN ) + τ5 (δ2EC − (h2 + u2) IC − r2IC − d2IC − µIC) + τ6 ((θh1 + u2) IN + (δh2 + u2) IC − r3HT − d3HT − µHT ) (4) + τ7 (((1− θ)h1 + u2) IN + ((1− δ)h2 + u2) IC − r4HN − d4HN − µHN ) + τ8 (r1IN + r2IC + r3HT + r4HN − µR) . Deriving the Hamilton function (4) for each co-state variable as (2) yields the equation state. The next step is to realize that the negative value of the Hamilton function (4) derivative for each state variable is the co-state, which is represented by the following equation. dτ1 dt = −∂H ∂S = (τ1 − ατ2 − (1− α) τ3) ( (1− u1) (β1IN + β2IC) N ) + (ατ2 + (1− α) τ3 − τ1) ( (1− u1) (β1SIN + β2SIC) N2 ) + µτ1, dτ2 dt = − ∂H ∂EN = (ατ2 + (1− α) τ3 − τ1) ( (1− u1) (β1SIN + β2SIC) N2 ) + (τ2 − τ4) δ1 + µτ2, dτ3 dt = − ∂H ∂EC = (ατ2 + (1− α) τ3 − τ1) ( (1− u1) (β1SIN + β2SIC) N2 ) + (τ3 − τ5) δ2 + µτ3, dτ4 dt = − ∂H ∂IN = (τ1 − ατ2 − (1− α) τ3) ( (1− u1)β1S N ) + (ατ2 + (1− α) τ3 − τ1) ( (1− u1) (β1SIN + β2SIC) N2 ) + τ4 (d1 + µ) + h1 (τ4 − θτ6 − (1− θ) τ7) + u2 (τ4 − τ6 − τ7) + r1 (τ4 − τ8)− 1, (5) dτ5 dt = − ∂H ∂IC = (τ1 − ατ2 − (1− α) τ3) ( (1− u1)β2S N ) + (ατ2 + (1− α) τ3 − τ1) ( (1− u1) (β1SIN + β2SIC) N2 ) + λ5 (d2 + µ) + h2 (τ5 − δτ6 − (1− δ) τ7) + u2 (τ5 − τ6 − τ7) + r2 (τ5 − τ8)− 1, dτ6 dt = −∂HT ∂H = (ατ2 + (1− α) τ3 − τ1) ( (1− u1) (β1SIN + β2SIC) N2 ) + τ6 (d3 + µ) + r3 (τ6 − τ8) , dτ7 dt = − ∂H ∂HN = (ατ2 + (1− α) τ3 − τ1) ( (1− u1) (β1SIN + β2SIC) N2 ) + τ7 (d4 + µ) + r4 (τ7 − τ8) , dτ8 dt = −∂H ∂R = (ατ2 + (1− α) τ3 − τ1) ( (1− u1) (β1SIN + β2SIC) N2 ) + µτ8. with transverse condition τ1 (T ) = τ2 (T ) = τ3 (T ) = τ4 (T ) = τ5 (T ) = τ6 (T ) = τ7 (T ) = τ8 (T ) = 0. So, the optimal control of u∗1 and u∗2 can be written as u∗1 = (β1S ∗I∗N + β2S ∗I∗C) (ατ2 + (1− α) τ3 − τ1) NK1 , u∗2 = I∗N (τ4 − τ6 − τ7) + I∗C (τ5 − τ6 − τ7) K2 M. A. Rois, Fatmawati, C. Alfiniyah / Eur. J. Pure Appl. Math, 16 (1) (2023), 523-537 530 3. Simulation The forward-backward sweep method is used to solve this optimal control problem [10]. In this numerical simulation, the values of the parameters used are shown in Table 1. These values are based on the COVID-19 case in Indonesia, and the final time (T ) is 100 days. Next, the initial values given are as follows S(0) = 270911990, EN (0) = 1000000, EC(0) = 10000, IN (0) = 412784, IC(0) = 500000, HT (0) = 56899, HN (0) = 200000, and R(0) = 341942, with simulation intervals t ∈ [0, 100] . Following are the findings of the simulation of optimum numerical control: 3.1. Strategy 1 (u1 ̸= 0 and u2 = 0) Figure 4: Optimal control simulation results with u1 ̸= 0. Figure 4 shows the control strategy, which is u1 ̸= 0 and u2 = 0. This result of education gives people a constant sense of caution when interacting with others outside the home. By using this strategy, the number of people who are exposed (with and without comorbidity), infected (with and without comorbidity), and isolated (with and without treatment) is significantly reduced. Furthermore, Figure 5 shows the control strategy profiles u1 ̸= 0 and u2 = 0 for reducing the number of COVID-19 cases during t = 100. The control strategy u1 ̸= 0 and u2 = 0 is given by the maximum from the beginning of the period to t = 94. At the end of the period, u1 ̸= 0 and u2 = 0 decrease by a large amount to reach zero. Control is ended at the end of the period, which means no more control is given. M. A. Rois, Fatmawati, C. Alfiniyah / Eur. J. Pure Appl. Math, 16 (1) (2023), 523-537 531 Figure 5: Optimal control profile with u1 ̸= 0. 3.2. Strategy 2 (u1 = 0 and u2 ̸= 0) Figure 6: Optimal control simulation results with u2 ̸= 0. Figure 6 shows that u1 = 0 and u2 ̸= 0 are the control strategy. Because there is more care for infected people, this strategy can reduce the number of people infected with or without comorbidity. By using this strategy, the number of people who are exposed (with or without comorbidity), infected (with or without comorbidity), and isolated (with or without treatment) is reduced by a large amount. Figure 7 depicts the profiles of u1 = 0 and u2 ̸= 0 control strategies for reducing the number of COVID-19 instances for t = 100. Furthermore, the management strategy u1 = 0 and u2 ̸= 0 are given by maximum from the M. A. Rois, Fatmawati, C. Alfiniyah / Eur. J. Pure Appl. Math, 16 (1) (2023), 523-537 532 beginning to t = 31.4 and then decline gradually until t = 100 approaches zero, indicating that control is terminated at the end of the period. Figure 7: Optimal control profile with u2 ̸= 0. 3.3. Strategy 3 (u1 ̸= 0 and u2 ̸= 0) Figure 8: Combined optimal control simulation results. The combination control approach is depicted in Figure 8. This is the outcome of education urging people to use caution at all times (such as while dealing with others outside the home) and increased care for affected individuals. Combined control tactics M. A. Rois, Fatmawati, C. Alfiniyah / Eur. J. Pure Appl. Math, 16 (1) (2023), 523-537 533 can greatly minimize or limit deployment. In addition, Figure 9 depicts the combined control plan profile for reducing the number of COVID-19 cases during t = 100. The combined control technique incorporates two controls. Control u1 has a maximum value of t = 84.3 and progressively approaches 0 as time progresses. Then, u2 is set to a maximum value of t = 5.4 before declining till t = 100 reaches zero gradually. Both controls expire at the end of time, rendering them powerless over u1 and u2. Figure 9: Combined control profile. 3.4. Total infections comparison Different starting values are given for the sub-populations that were exposed. Figure 10 shows the total number of infected sub-populations for four different initial conditions of the exposed sub-populations: EN (0) = 200000, EN (0) = 1000000, EN (0) = 10000000, and EN (0) = 100000000. Figure 10 shows that when the third strategy is used instead of the other strategies, the number of infected sub-populations goes down. The above example of an initial value is meant to help figure out how a disease will spread by showing several ways to stop it. Based on strategies 1 to 3, we can say that strategy 3 is the best way to reduce the number of people in the community infected with COVID-19. 4. Cost evaluation The objective of the cost analysis is to identify the COVID-19 spread control approach with the lowest cost-effectiveness ratio. This study evaluates costs using ACER (Average Cost-Effectiveness Ratio) and ICER (Incremental Cost-Effectiveness Ratio). ACER is mathematically defined as follows, according to the approach of cost-effectiveness analysis: ACER = Objective function (F ) Total number of infections averted The most cost-effective use of ACER is the smallest ACER value. The ACER value is presented in Table 3. M. A. Rois, Fatmawati, C. Alfiniyah / Eur. J. Pure Appl. Math, 16 (1) (2023), 523-537 534 Figure 10: Total subpopulation infected using various strategies. In addition, ICER analyzes two competing intervention choices for the same scarce resource, follows costs and converts them to health benefits. Considering the p and q methods as two competing control intervention techniques, the ICER is defined as follows: ICER = Change in total costs for p and q strategies Control benefits in strategies p and q change . ICER was computed in order to identify the most cost-effective control strategy among the available options. First, calculate the competition for strategy 1 and strategy 2 using the following formula: ICER (1) = 5, 918, 200− 0 53, 996, 000, 000−0 = 0.0001096, ICER (2) = 2, 533, 500− 5, 918, 200 54, 000, 000, 000−53, 996, 000, 000 = −3, 384, 700 4, 000, 000 = −0.846, ICER results for strategy 1 were greater than those for strategy 2, indicating that educa- tional controls alone were more expensive and ineffective than medical care enhancement REFERENCES 535 Table 3: Total infections prevented, total costs, and ACER for strategies 1, 2, and 3. Strategy Infections prevented Total cost (thousands) ACER No strategy 0 0 0 Strategy 1 53,996,000,000 5,918,200 0.0001096 Strategy 2 54,000,000,000 2,533,500 0.0000469 Strategy 3 54,001,000,000 1,738,200 0.0000322 controls. Thus, strategy 1 is eliminated from the list of potential control strategies. The ICER for strategies 2 and 3 is then recalculated as follows: ICER (2) = 2, 533, 500− 0 54, 000, 000, 000−0 = 0.0000469, ICER (3) = 1, 738, 200−2, 533, 500 54, 001, 000, 000−54, 000, 000, 000 = − 795, 300 1, 000, 000 = −0, 795. Strategy 2 has a greater ICER than strategy 3. Due to its cost-effectiveness and ability to prevent the spread of infectious illnesses, strategy 3 (combined control) is the best control plan of all possibilities. 5. Conclusions The COVID-19 model uses public health education and improves the care of infected individuals. The Comorbidity of COVID-19 with other diseases can be controlled by im- plementing public health education and improving the care of infected individuals. The sensitivity analysis of model parameters in this paper shows that contact rate with comor- bidity (β2 = 0.53996) and without comorbidity (β1 = 0.46004) have the biggest effect on R0. 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