In this section, we present the result of numerical simulation using the estimated parameter values that minimize error between the actual COVID-19 data during the Delta and Omicron variant outbreaks with the model’s output. The actual COVID-19 data is data that indicates the weekly number of people infected with COVID-19 recorded by Indonesian government. The output of mathematical model refers to the result of numerical simulation that shows the number of COVID-19 infected people, Ico+Q2, in a week. We examined the effect of quarantine implementation by comparing the dynamic of TB and diabetes compartments in three scenarios: no quarantine, micro quarantine, and macro quarantine. In addition, we suggested the implementation of some control strategies to reduce the risk of tuberculosis transmission and diabetes development in quarantine period. We considered the variations of control parameter separately, and interpreted the simulation’s result for each proposed control.

3.1. Numerical Simulation of Mathematical Model Accomodating Quarantine ProcessThe weekly data of COVID-19 cases during the observation time was fitted with the output of mathematical model to obtain the estimated parameters. Table 2 displays the estimated values for each unobserved parameter with 95% confidence interval obtained from 100 bootstrap realizations. As can be seen in Figure 2a,b, the results of simulation produced a good data fitting in both of Delta period and Omicron period. In order to assess the goodness of fit, we calculated the Pearson correlation coefficient between the actual data and the model’s output, denoted by coefficient rd for Delta and ro for Omicron. The calculation yielded rd=0.9343 and ro=0.8961 with significance level p<0.05, indicating a strong positive relationship between data and simulation result.

For the Delta period, the actual data and model’s output indicate same period of infection peak, the second week of July 2021. The highest number of COVID-19 cases shown in the recorded data was 341,749, while the model resulted 288,169 cases at the peak of infection. The number of COVID-19 cases started to decline significantly in the subsequent weeks. For the Omicron period, the peak of infection shown by data and model’s output is in the third week of February 2022. The number of infected people on this infection peak that recorded in actual data is 385,769 cases, whereas the simulation result indicates that the potential highest case number is only 286,566 cases.

We used the estimated parameter values in Table 2 to simulate the dynamic of TB and diabetes compartments in three scenarios. In Figure 2c,d, we observed that the number of COVID-19 infected people, (Ico+Q2), decreased when the quarantine was implemented during the Delta and Omicron variant outbreaks. The implementation of quarantine not only reduced positive cases but also led to the early occurrence of infection peak; thereby, the emergency period lasted shorter. The summary of simulation results without or with accomodating the implementation of quarantine was given in Table 3. The implementation of micro quarantine reduced 51.48 percent of COVID-19 cases in the Delta period and 69.76 percent in the Omicron period. As expected, a higher decrease in the number of COVID-19 cases was resulted from the implementation of macro quarantine that is 64.17% and 79.60% for the Delta and Omicron period, respectively. When micro and macro quarantine were implemented in Delta period, the number of infected individuals on the peak of infection were 47.40% and 60.76%, respectively, lower than no quarantine scenario. In Omicron period, the percentages of case reduction in the infection peak affected by micro and macro quarantine implementation were 69.19% and 79.27%. The peak of infection shifted two weeks later when the quarantine was not implemented. These implied that quarantine enforced by the government could significantly limiting the spread of COVID-19 infection, particularly during the Delta and Omicron variant outbreaks.In Table 2, we observed that infection rate parameters of quarantined susceptible by contact with infected and diagnosed TB, τ1 and τ2, were higher than the infection rates of susceptible, ρ1 and ρ2, in both the Delta and Omicron periods. This implied a higher number of tuberculosis cases during the quarantine. Figure 3 illustrates how micro and macro quarantine increase the number of TB cases, Itb and Dtb. We considered three values for θ1:0,0.3, and 0.75. We noticed that as the quarantine rate (θ1) increases, the number of Itb and Dtb increase. When micro and macro quarantine were implemented during Delta variant outbreak, the number of active infected TB were 13.24% and 14.09%, respectively, higher than the number Itb of no quarantine scenario at the end of observation time. Similar results were also seen in the Omicron period where Itb during micro and macro quarantine scenarios were 36.18% and 37.5% higher than no quarantine. The significant increase in the number of compartment Itb led to the increment in the number of tuberculosis diagnosed individuals despite the diagnosis ability of health services decreased during pandemic. In the last week of September 2021, the number of diagnosed TB of micro and macro quarantine were 8.04% and 9.91%, respectively, higher than the scenario of no implementation of quarantine. In the last week of April 2022, there were 47.10% and 55.21% higher potential diagnosed TB for micro and macro quarantine, respectively. Table 4 displays the summary of quarantine effect on the increasing number of people with tuberculosis and diabetes during the observation time, the Delta and Omicron periods.In Table 2, the probability of quarantined susceptible developing diabetes, δ2, was higher than the probability of susceptible developing diabetes, δ1. This indicated that the implementation of quarantine possibly caused the increase in the number of people with diabetes. We presented the effect of quarantine rate θ1 to the number of diabetic without complication (Ddm) and diabetic with complications (Cdm) in Figure 4. Here, we also considered three values of susceptible quarantine rate: θ1=0 (no quarantine), θ1=0.3 (micro quarantine), and θ1=0.75 (macro quarantine). We observed that the stricter quarantine implementation, that was, greater values for θ1, the higher number of individuals developing diabetes (see Table 4 for the number of people with diabetes without and with complications). More precisely, at the end of Delta variant observation time, micro quarantine increased the number of diabetes without complications 9.53% higher than no quarantine scenario, and macro quarantine led 10.93% increased cases. On the other hand, the number of diabetes with complications in the last week of September 2021 increased 8.94% and 11.62% in case the government decided to enforce micro and macro quarantine. For the Omicron period, micro quarantine resulted 2.22% and 5.20% higher number of Ddm and Cdm, respectively. When the macro quarantine option was selected, it was possible that the number of diabetic without complications increases 2.96% and the number of diabetic with complications increases 6.40% in the end of observation. 3.2. Effect of Tuberculosis and Diabetes Control Strategies during COVID-19 Quarantine

We proposed some control strategies in this work to reduce the risk of tuberculosis and diabetes during the implementation of quarantine that focus at mitigating the COVID-19 disease transmission. We added three control functions (u1,u2,u3) related to the reduction of infected tuberculosis, and two controls (u4,u5) intended to minimize the probability of developing diabetes. We assumed that the controls were continuous defined by a constant rate. Each control simulated both separately and combined. Here, the percentage of reduction in the total number of cases during observation time compared to no quarantine scenario was chosen to illustrate the efficacy of each control strategy.

For the first scenario, we used merely the control u1(t). This control intended to increase public awareness to protect them from tuberculosis risk during quarantine at home. The awareness program were carried out by direct campaigns or by using mass media and social media to inform the citizens about the dangers of TB transmission in the close environment when they stayed at home. Using constant rate u1=0.5, Figure 5a,b display significant reduction of latent TB during Delta and Omicron period. The variations in the value of control u1, ranging 0≤u1≤1, showed the efficacy in reducing latent TB up to 72.28% for Delta variant and 98.01% for Omicron variant (see Figure 5c,d). Also, reduced cases can be seen in the number infected TB (Itb) during both Delta and Omicron variant outbreaks (see Figure 6a,b). In Figure 6c,d, we can see that the efficacy of control u1(t) to reduce the number of active infected TB during the pandemic of Delta and Omicron variant was up to 58.64% and 97.19%, respectively.Next, we used only the control u2(t) for the second scenario. This control represented the proportion of latent individuals Ltb that was identified and received medical treatment. The expansion of the screening test and diagnosis for latent TB or people at high infection risk could be adopted. In Figure 5a,b, using constant rate u2=0.5, we observed that the decrease in the number of latent Ltb in both periods were less significant than the first proposed TB control strategy. Now, by varying control rate value, 0≤u2≤1, the percentage of reduced cases were only up to 6.519% for Delta and only up to 4.463% for Omicron, as can be seen in Figure 5b,d. In addition, control u2 also was not more effective than u1 in reducing the number of infected TB (Itb). In Figure 6c,d, the effectiveness measurement showed that the efficacy of this type of control was only up to 4.938% during Delta period and 6.301% during Omicron period.The third control proposed to reduce tuberculosis risk was u3(t). In this strategy, the diagnosis program for the infected individuals was intensified. A specific team could be formed to continue TB diagnosis and treatment program even though COVID-19 was a priority during the pandemic. This control was focused on the diagnosis of infected, so the latent was not significantly reduced. Even, in the Omicron period there was no decrease in the number of latent TB.In Figure 5c, we observed that the number of latent decreased only up to 0.115%. Although control u3 was not significantly affecting latent TB, this control has potential to reduce the number of infected TB quite notably. The efficacy of this control in reducing Itb was up to 41.17% for Delta period and 48.62% for Omicron period as shown in Figure 6, with the values of control u3 ranging 0,1.For the last scenario of tuberculosis control strategy, we combined all proposed control, u1, u2, and u3. We assumed that all controls were implemented with equal rate, that was, u1=u2=u3. As expected, Figure 5 and Figure 6 show that this combination decreased the number of Ltb and Itb more significant than three previous single controls. More precisely, the latent individuals decreased up to 73.99% and 98.17% during Delta and Omicron period. For the infected individuals, there were reductions by 77.93% and 98.48% during the observation time starting from June 2021 to September 2021 for Delta variant and from December 2021 to April 2022 for Omicron variant, respectively.One of the important indicators that need to be considered regarding the effectiveness of a TB control strategy was the ratio between Dtb and Itb. The high ratio of diagnosed who receive medical treatment from health services over the infected indicated that the strategy was more effective. In Appendix A, Figure A1 shows the number of Dtb during the observation time, and we observed that the highest ratio of Dtb over Itb was shown by the combination of all controls. The control u3 significantly increased the ratio because it was focused on the increase of diagnosis rate. The control u2, aimed to reduce the number of latent, showed low value of Dtb/Itb and did not significantly influence the diagnosis rate.For the first diabetes reduction scenario, we compared the number of diabetic with complications (Cdm), with and without control u4(t). The goal of this strategy was to increase public awareness to implement healthy lifestyle, set a good diet, and exercise at home regularly in quarantine period. The implementation of constant control u4=0.5 reduced the number of Cdm as shown in Figure 7a,b. Figure 7c,d show that the upper bound of this control efficacy was equal to 61.28% and 59.04% for Delta and Omicron period, respectively. This implied a reduction in the number of people with diabetes of more than half of the total cases when control u4 was not considered.Next, we used only the control u5(t) to reduce the probability of diabetics developing complications during the implementation of quarantine. Diet tracking, regular diet, and healthy lifestyle in the quarantine period could be adopted to prevent the emergence of complications. In Figure 7a,b, we observed that the number of Cdm reduced more significant than control u4. By using constant rate 0≤u5≤1 shown in Figure 7c,d, the percentage of reduced cases was up to 78.11% for Delta and 76.08% for Omicron.In addition, we combined control u4 and control u5. In this strategy, the two controls were applied at the same time in order to obtain better numerical results. We assumed that the controls had equal rate, u4=u5. In Figure 7a,b, we used controls u4=u5=0.5 in the numerical simulation. The uppper bound of the efficacy of this control combination was equal to the upper bound of single control u5, but lower values of this combination yielded higher efficacy than control u5 as shown in Figure 7c,d.In Appendix A, Figure A2 shows that control u4 significantly reduced the number of diabetic without complications but this control did not detain the progression from Ddm to diabetic with complications, Cdm. The implementation of control u5, that was aimed to reduce the probability of developing complications, showed the low percentage of Ddm becoming Cdm. The fatal impact of diabetes with complications could be minimized by implementing strategy u5 during quarantine period. On the other hand, control u5 was not effective to decline the number of diabetic without complications Ddm. The combination of control u4 and u5 could be considered as the best strategy to reduce the risk of diabetes development during quarantine implementation. These strategies decreased the number of Ddm and Cdm, and reduced the possibility of Ddm developing complications.