Following [
23], we consider three different cost functions,
,
, and
satisfying conditions (
5):
In (20), the coefficients have been chosen to minimize the weighted distance [
23]:
The cost of control,
,
, and
, is infinite as
u approaches its ultimate value 1. For comparison, we also used the cost function
, for which (
5) does not hold. The cost function
is popular in applications of control theory in epidemiology and other fields, since for this function the first-order optimality condition is linear with respect to
u. This is a useful property that simplifies numerical algorithms. However, the cost of control,
, is finite at
, which is not realistic in real-world scenarios. Figures 4, 7 and 12 show that the global minimum,
, of the Hamiltonian,
, did not stay in the range of
when the cost was given by
, especially for small values of
,
. Thus, an explicit constraint
must be imposed in the case of
. Even with the constraint
, the optimal control function,
, often reaches the ultimate level [
17],
, which is not practical.
4.1. Scenario 1: Social Distancing Control Only
In the first scenario, only one (non-medical) control,
, was applied (
Figure 2,
Figure 3 and
Figure 4). As one can see in the figures, when the weight of control
was increased, the effectiveness of the control went down; see also
Table 1 that illustrates how
changes over time for the cost
with different values of
(find similar
Table A13,
Table A14 and
Table A15 for
,
, and
in the
Appendix A). One can conclude from
Figure 2 that control
works by eliminating some cases but also by delaying some of them. Therefore, even though the cumulative number of infections in the controlled environment was significantly less than in the environment with no control, toward the end of the study period, the daily number of infected individuals in the controlled environment may end up being higher.
Figure 2.
Graphs of Susceptible (top on the left), Infected (top on the right), and Recovered (bottom on the left) people, as well as control (bottom on the right) over time for four different cost functions versus No Control when weight is .
Figure 2.
Graphs of Susceptible (top on the left), Infected (top on the right), and Recovered (bottom on the left) people, as well as control (bottom on the right) over time for four different cost functions versus No Control when weight is .
Figure 2,
Figure 3 and
Figure 4 with
equal to
,
, and
, respectively, show the pattern of
decreasing as the values of
went down. Based on these figures and
Table 1, the “flattening of the curve” is evident.
Figure 3.
Graphs of Susceptible (top on the left), Infected (top on the right), and Recovered (bottom on the left) people, as well as control (bottom on the right) over time for four different cost functions versus No Control when weight is .
Figure 3.
Graphs of Susceptible (top on the left), Infected (top on the right), and Recovered (bottom on the left) people, as well as control (bottom on the right) over time for four different cost functions versus No Control when weight is .
Table 1.
Comparison of for cost function in case when only control is applied and varies. As increases, the number of infected individuals, , grows higher on most days.
Table 1.
Comparison of for cost function in case when only control is applied and varies. As increases, the number of infected individuals, , grows higher on most days.
Day | No 2nd Control | No 2nd Control | No 2nd Control | No 2nd Control | No Control |
---|
1 | 200 | 200 | 200 | 200 | 200 |
10 | 85 | 253 | 315 | 387 | 1237 |
20 | 34 | 329 | 527 | 812 | 9228 |
30 | 15 | 432 | 880 | 1691 | 67,606 |
40 | 7 | 571 | 1466 | 3514 | 456,639 |
50 | 4 | 761 | 2447 | 7267 | 1,985,292 |
60 | 3 | 1028 | 4089 | 14,927 | 2,987,989 |
70 | 2 | 1416 | 6873 | 30,388 | 2,015,872 |
80 | 1 | 2003 | 11,678 | 61,024 | 1,023,788 |
90 | 1 | 2953 | 20,264 | 120,429 | 474,813 |
100 | 1 | 4643 | 36,661 | 233,756 | 213,085 |
110 | 1 | 8190 | 72,008 | 453,110 | 94,393 |
120 | 1 | 18,714 | 174,758 | 922,708 | 41,578 |
Figure 4.
Graphs of Susceptible (top on the left), Infected (top on the right), and Recovered (bottom on the left) people, as well as control (bottom on the right) over time for four different cost functions versus No Control when weight is . For the cost function , stayed above the ultimate value, , for more than half of the study period, which is not practical.
Figure 4.
Graphs of Susceptible (top on the left), Infected (top on the right), and Recovered (bottom on the left) people, as well as control (bottom on the right) over time for four different cost functions versus No Control when weight is . For the cost function , stayed above the ultimate value, , for more than half of the study period, which is not practical.
4.2. Scenario 2: Control with Antiviral Medication Only
For the next set of experiments, it was assumed that there was only control
in the system. In
Figure 5,
Figure 6 and
Figure 7, one can see the effect of the weight,
, on different cost functions and, as a result, on state variables
,
, and
over time. Again, as the weight
decreases, the control played a more effective role in reducing the number of infected people (See
Table 2,
Table A16,
Table A17 and
Table A18 for more details).
Figure 5.
Graphs of Susceptible (top on the left), Infected (top on the right), and Recovered (bottom on the left) people, as well as control (bottom on the right) over time for four different cost functions versus No Control when weight is .
Figure 5.
Graphs of Susceptible (top on the left), Infected (top on the right), and Recovered (bottom on the left) people, as well as control (bottom on the right) over time for four different cost functions versus No Control when weight is .
Table 2.
Comparison of for cost function in case when only control is applied and varies. As increases, the number of infected individuals, , grows higher on most days.
Table 2.
Comparison of for cost function in case when only control is applied and varies. As increases, the number of infected individuals, , grows higher on most days.
Day | No 1st Control | No 1st Control | No 1st Control | No 1st Control | No 1st Control | No Control |
---|
1 | 200 | 200 | 200 | 200 | 200 | 200 |
10 | 18 | 181 | 254 | 318 | 377 | 1237 |
20 | 1 | 168 | 332 | 530 | 756 | 9228 |
30 | 0 | 164 | 437 | 880 | 1503 | 67,606 |
40 | 0 | 166 | 581 | 1456 | 2949 | 456,639 |
50 | 0 | 177 | 782 | 2395 | 5693 | 1,985,292 |
60 | 0 | 198 | 1073 | 3938 | 10,735 | 2,987,989 |
70 | 0 | 234 | 1515 | 6481 | 19,547 | 2,015,872 |
80 | 0 | 299 | 2226 | 10,735 | 34,005 | 1,023,788 |
90 | 0 | 420 | 3485 | 18,210 | 56,943 | 474,813 |
100 | 0 | 676 | 6092 | 33,048 | 95,559 | 213,085 |
110 | 0 | 1391 | 13,102 | 70,339 | 177,085 | 94,393 |
120 | 0 | 5265 | 49,724 | 244,998 | 499,616 | 41,578 |
Overall, the effects of controls
and
on the system, when only one control was applied, were similar. However, as one can clearly see from
Table 3, for the same cost and over the same time interval, control
suppressed infections more aggressively than
. Also, there is a significant difference between the results for cost function
and the rest of the cost functions. While for
,
, and
the maximum number of infected people on any given day in the case of “first control only” was 923,332, this number was 1,511,537 for
. Additionally, in the case of “second control only”, the maximum daily number of infected individuals for
exceeded the maximum daily number for other cost functions by 154,151 cases.
Figure 6.
Graphs of Susceptible (top on the left), Infected (top on the right), and Recovered (bottom on the left) people, as well as control (bottom on the right) over time for four different cost functions versus No Control when weight is .
Figure 6.
Graphs of Susceptible (top on the left), Infected (top on the right), and Recovered (bottom on the left) people, as well as control (bottom on the right) over time for four different cost functions versus No Control when weight is .
Figure 7.
Graphs of Susceptible (top on the left), Infected (top on the right), and Recovered (bottom on the left) people, as well as control (bottom on the right) over time for four different cost functions versus No Control when weight is .
Figure 7.
Graphs of Susceptible (top on the left), Infected (top on the right), and Recovered (bottom on the left) people, as well as control (bottom on the right) over time for four different cost functions versus No Control when weight is .
The best performance among all cost functions can be attributed to
in both cases where only control
or only control
was applied. For details, one can see
Table 3 and
Figure 8.
Table 3.
Comparison of for all cost functions in the case when only control is applied () or only control is applied () over time.
Table 3.
Comparison of for all cost functions in the case when only control is applied () or only control is applied () over time.
Day | No 2nd Control | No 1st Control | No 2nd Control | No 1st Control | No 2nd Control | No 1st Control | No 2nd Control | No 1st Control |
---|
| | | | |
---|
1 | 200 | 200 | 200 | 200 | 200 | 200 | 200 | 200 |
10 | 387 | 318 | 388 | 317 | 377 | 308 | 401 | 349 |
20 | 812 | 530 | 816 | 527 | 766 | 495 | 918 | 644 |
30 | 1691 | 880 | 1706 | 872 | 1552 | 795 | 2133 | 1178 |
40 | 3514 | 1456 | 3557 | 1439 | 3134 | 1273 | 4939 | 2133 |
50 | 7267 | 2395 | 7379 | 2362 | 6302 | 2034 | 11,366 | 3816 |
60 | 14,927 | 3938 | 15,209 | 3879 | 12,618 | 3258 | 25,851 | 6733 |
70 | 30,388 | 6481 | 31,076 | 6381 | 25,120 | 5256 | 57,953 | 11,672 |
80 | 61,024 | 10,735 | 6,2623 | 10,561 | 49,586 | 8558 | 127,302 | 19,877 |
90 | 120,429 | 18,210 | 123,913 | 17,880 | 96,926 | 14,338 | 266,304 | 33,825 |
100 | 233,756 | 33,048 | 240,619 | 32,336 | 188,180 | 25,768 | 509,874 | 60,100 |
110 | 453,110 | 70,339 | 463,527 | 67,697 | 367,745 | 53,756 | 899,275 | 123,379 |
120 | 922,708 | 244,998 | 923,332 | 229,683 | 760,101 | 183,048 | 1,511,537 | 399,149 |
Figure 8.
Graphs of for different cost functions when only is applied and (shown with solid lines), as well as when only is applied and (shown with dashed line).
Figure 8.
Graphs of for different cost functions when only is applied and (shown with solid lines), as well as when only is applied and (shown with dashed line).
4.3. Scenario 3: Non-Medical and Medical Controls in Combination
For the next step, we applied two controls to the
system,
and
, together with the same weights,
, in order to evaluate their effect on the outbreak (See
Figure 9,
Figure 10,
Figure 11 and
Figure 12). As expected, in terms of its dependence on
, the combination of two controls,
and
, behaved pretty similar to the case of one control in a sense that when the weight
decreased, the controls became more effective, and the daily number of infected humans went down.
Figure 9.
Graphs of Susceptible (top on the left), Infected (top on the right), Recovered (bottom on the left) people, controls shown with solid lines, and with dashed lines (bottom on the right) over time for four different cost functions versus No Control when weights, , and , for both controls are .
Figure 9.
Graphs of Susceptible (top on the left), Infected (top on the right), Recovered (bottom on the left) people, controls shown with solid lines, and with dashed lines (bottom on the right) over time for four different cost functions versus No Control when weights, , and , for both controls are .
Figure 10.
Graphs of Susceptible (top on the left), Infected (top on the right), Recovered (bottom on the left) people, control shown with solid lines, and with dashed lines (bottom on the right) over time for four different cost functions versus No Control when weight, , for both controls is .
Figure 10.
Graphs of Susceptible (top on the left), Infected (top on the right), Recovered (bottom on the left) people, control shown with solid lines, and with dashed lines (bottom on the right) over time for four different cost functions versus No Control when weight, , for both controls is .
Table 4 and
Table 5 show the daily number of infected individuals,
, and the cumulative number of infected individuals up to day
t,
, for different control scenarios. This gives an insight into how the two controls,
and
, compare individually and in combination when subject to the same cost,
, and the same weight,
.
Table 5 illustrates that the cumulative number of infections after applying both controls for 120 days was 454,205, while the “no control” counterpart was 9,397,865. And in the case of the control with antiviral medication,
, after 120 days, there were more than the times fewer cases compared to the case of social distancing control,
(692,160 vs. 2,256,854). Similar tables related to the cost functions
,
, and
can be found in
Appendix A (
Table A1,
Table A2,
Table A3,
Table A4,
Table A5 and
Table A6).
Figure 11.
Graphs of Susceptible (top on the left), Infected (top on the right), Recovered (bottom on the left) people, controls shown with solid lines, and with dashed lines (bottom on the right) over time for four different cost functions versus No Control when weight, , for both controls is .
Figure 11.
Graphs of Susceptible (top on the left), Infected (top on the right), Recovered (bottom on the left) people, controls shown with solid lines, and with dashed lines (bottom on the right) over time for four different cost functions versus No Control when weight, , for both controls is .
Table 4.
Comparison of for cost function when there is only , only , and both applied versus No Control case over time when .
Table 4.
Comparison of for cost function when there is only , only , and both applied versus No Control case over time when .
Day | No Control | No 2nd Control | No 1st Control |
|
---|
1 | 200 | 200 | 200 | 200 |
10 | 1237 | 387 | 318 | 300 |
20 | 9228 | 812 | 530 | 472 |
30 | 67,606 | 1691 | 880 | 743 |
40 | 456,639 | 3514 | 1456 | 1167 |
50 | 1,985,292 | 7267 | 2395 | 1834 |
60 | 2,987,989 | 14,927 | 3938 | 2897 |
70 | 2,015,872 | 30,388 | 6481 | 4625 |
80 | 1,023,788 | 61,024 | 10,735 | 7522 |
90 | 474,813 | 120,429 | 18,210 | 12,727 |
100 | 213,085 | 233,756 | 33,048 | 23,355 |
110 | 94,393 | 453,110 | 70,339 | 50,792 |
120 | 41,578 | 922,708 | 244,998 | 173,543 |
In the next series of experiments, controls
and
had different weights,
and
, applied to their respective cost functions. We considered two cases. First, for the cost function
, the weight of control
was less than the weight of control
(
).
Table 6 shows the changes in the daily numbers of infected people,
, for the cost function
, in the case of fixed weight (
) for control
and different weights for control
(
Table A7,
Table A8 and
Table A9 for cost functions
, and
can be found in
Appendix A). As it follows from
Table 6, adding the second control,
, with any weight,
, helped to better contain the outbreak and to decrease the daily number of infected people, as well as the cumulative number of cases. Even for a high effort case of
, the number of daily infections was
cases less than the daily number of infected individuals in the case when there was no control:
. However, when the weight of the second control
increased, the effort required to implement that measure also rose, making it increasingly challenging to execute. When the roles were reversed, that is, for the cost function
, the weight,
, of the second control
was fixed, and the sensitivity of the system to the first control
was observed, the pattern ended up being similar. Namely, adding a non-medical control,
, reduced the daily number of infected people. Even though it was not as consequential as in the case when control
was added, there were still fewer infected people in all cases with two controls as opposed to the case of
only. At the same time, it is evident that the second control,
, is more efficient. Indeed, for the high effort case of
, the number of daily infections was only
cases less than the daily number of infected individuals in the case when there was no control
(as opposed to a
reduction when
was added with the same effort of
). The difference in the daily number of infected individuals between the case of no
(i.e.,
only with weight
) and the case of
with
and
with varying weights ranged from
to
. See
Table 7,
Table A10,
Table A11 and
Table A12 for more details.
Figure 12.
Graphs of Susceptible (top on the left), Infected (top on the right), Recovered (bottom on the left) people, controls shown with solid lines, and with dashed lines (bottom on the right) for four different cost functions versus No Control when weight, , for both controls is . Control for cost function takes unrealistic values above 1 at the early period of the study.
Figure 12.
Graphs of Susceptible (top on the left), Infected (top on the right), Recovered (bottom on the left) people, controls shown with solid lines, and with dashed lines (bottom on the right) for four different cost functions versus No Control when weight, , for both controls is . Control for cost function takes unrealistic values above 1 at the early period of the study.
Figure 13 and
Figure 14 show the behaviors of the controls and their effects on the graphs of
for different cost functions and different weights. As is evident from the graphs, when
and
, the second control,
, was dominant and very efficient. At the same time, when
and
, the two controls,
and
, were about the same, and there were more infected people toward the end of the study period, that is, the control strategy in
Figure 14 is less efficient compared to the case of
Figure 13. The two figures, once again, underline the significance of the second control
.
Table 5.
Cumulative number of infections up to day t, for cost function when there is only , only , and both versus No Control case over time when weight .
Table 5.
Cumulative number of infections up to day t, for cost function when there is only , only , and both versus No Control case over time when weight .
Day | No Control | No 2nd Control | No 1st Control |
|
---|
1 | 200 | 200 | 200 | 200 |
10 | 1756 | 643 | 898 | 772 |
20 | 13,747 | 1649 | 2156 | 1761 |
30 | 101,568 | 3735 | 4239 | 3312 |
40 | 697,572 | 8067 | 7686 | 5748 |
50 | 3,338,392 | 17,006 | 13,356 | 9562 |
60 | 7,032,920 | 35,334 | 22,686 | 15,568 |
70 | 8,627,485 | 72,609 | 37,993 | 25,108 |
80 | 9,121,747 | 147,335 | 63,222 | 40,496 |
90 | 9,292,217 | 294,458 | 105,246 | 66,067 |
100 | 9,358,556 | 579,209 | 178,417 | 111,270 |
110 | 9,386,082 | 1,130,270 | 320,258 | 202,389 |
120 | 9,397,865 | 2,256,854 | 692,160 | 454,205 |
Table 6.
Comparison of the daily number of infected people, , for the cost function with the weight for . The weights for the control are , , , and for the second, third, and fourth columns, respectively, and the fifth column shows the case of No Control over time.
Table 6.
Comparison of the daily number of infected people, , for the cost function with the weight for . The weights for the control are , , , and for the second, third, and fourth columns, respectively, and the fifth column shows the case of No Control over time.
Day |
|
|
|
| No 2nd Control |
---|
1 | 200 | 200 | 200 | 200 | 200 |
10 | 180 | 251 | 300 | 321 | 387 |
20 | 167 | 323 | 472 | 545 | 812 |
30 | 163 | 421 | 743 | 925 | 1691 |
40 | 165 | 554 | 1167 | 1561 | 3514 |
50 | 175 | 737 | 1834 | 2629 | 7267 |
60 | 195 | 1002 | 2897 | 4428 | 14,927 |
70 | 231 | 1404 | 4625 | 7481 | 30,388 |
80 | 291 | 2049 | 7522 | 12,748 | 61,024 |
90 | 403 | 3188 | 12,727 | 22,274 | 120,429 |
100 | 646 | 5554 | 23,355 | 41,465 | 233,756 |
110 | 1306 | 11,909 | 50,792 | 89,010 | 453,110 |
120 | 4879 | 44,363 | 173,543 | 280,668 | 922,708 |
Table 7.
Comparison of the daily number of infected people, , for the cost function with the weight for . The weights for the control are , , , and for the second, third, and fourth columns, respectively, and the fifth column shows the case of No Control over time.
Table 7.
Comparison of the daily number of infected people, , for the cost function with the weight for . The weights for the control are , , , and for the second, third, and fourth columns, respectively, and the fifth column shows the case of No Control over time.
Day |
|
|
|
| No 1st Control
|
---|
1 | 200 | 200 | 200 | 200 | 200 |
10 | 245 | 281 | 300 | 306 | 318 |
20 | 308 | 411 | 472 | 492 | 530 |
30 | 390 | 604 | 743 | 791 | 880 |
40 | 500 | 890 | 1167 | 1268 | 1456 |
50 | 649 | 1318 | 1834 | 2028 | 2395 |
60 | 857 | 1976 | 2897 | 3253 | 3938 |
70 | 1159 | 3016 | 4625 | 5258 | 6481 |
80 | 1621 | 4732 | 7522 | 8613 | 10,735 |
90 | 2383 | 7788 | 12,727 | 14,602 | 18,210 |
100 | 3777 | 13,962 | 23,355 | 26,759 | 33,048 |
110 | 6844 | 29,381 | 50,792 | 57,816 | 70,339 |
120 | 16,608 | 90,271 | 173,543 | 200,015 | 244,998 |
Figure 13.
The proportion of infected people, , for different cost functions and No Control case when (on the top) and controls shown with solid lines, with shown with dashed lines (on the bottom).
Figure 13.
The proportion of infected people, , for different cost functions and No Control case when (on the top) and controls shown with solid lines, with shown with dashed lines (on the bottom).
Figure 14.
The proportion of infected people, , for different cost functions and No Control case when (on the top) and controls shown with solid lines, with shown with dashed lines (on the bottom).
Figure 14.
The proportion of infected people, , for different cost functions and No Control case when (on the top) and controls shown with solid lines, with shown with dashed lines (on the bottom).