An optimal control model of the spread of the COVID19 pandemic in Iraq: Deterministic and chanceconstrained model
Abstract
Many studies have attempted to understand the true nature of COVID19 and the factors influencing the spread of the virus. This paper investigates the possible effect the COVID19 pandemic spreading in Iraq considering certain factors, that include isolation and weather. A mathematical model of cases representing inpatients, recovery, and mortality was used in formulating the control variable in this study to describe the spread of COVID19 through changing weather conditions between 17th March and 15th May, 2020. Two models having deterministic and an uncertain number of daily cases were used in which the solution for the model using the Pontryagin maximum principle (PMP) was derived. Additionally, an optimal control model for isolation and each factor of the weather factors was also achieved. The results simulated the reality of such an event in that the cases increased by 118%, with an increase in the number of people staying outside of their house by 25%. Further, the wind speed and temperature had an inverse effect on the spread of COVID19 by 1.28% and 0.23%, respectively. The possible effect of the weather factors with the uncertain number of cases was higher than the deterministic number of cases. Accordingly, the model developed in this study could be applied in other countries using the same factors or by introducing other factors.
1Introduction
The COVID19 pandemic has resulted in a huge loss of human life and affected economies worldwide. Most nations have entered into a “lockdown” situation, isolating their economies from others in facing the virus and decrease the loss of human life. Several studies have investigated COVID19 to understand the composition of the virus and determine the most suitable treatment. However, the danger of COVID19 is represented by its spread without any clinical symptoms during the early stages of contracting the virus and long period incubation [19]. Chen et al. [2] compared two groups of patients; COVID19 and SARSCoV2negative, where it was revealed that COVID19 patients suffered from high fever and cough more than SARSCoV2 patients. Procalcitonin (PCT) levels of SARSCoV2 patients (approximately 2 out of 5 patients) were shown to be higher compared to COVID19 patients. Also, COVID19 patients had lower creatinine levels compared with SARSCoV2 patients. In the case of those at a young age, the distinction between the two diseases can be diagnosed depending on the fever, cough, urea and creatinine levels, and parameters associated with routine blood workup. COVID19, SARS and MERS have mostly similar pathological features [23]. However, the harm caused by COVID19 may be caused by SARSCoV2 or through liver damage. Ai et al. [1] compared chest CT scans and reversetranscription chain reaction to diagnose the virus in a sample of patients (1,014 patients) in China. Chest CT scans were shown to be highly sensitive in the diagnosis of the virus. X. Chen et al. [3] assessed the state of pregnant women, having a positive test for COVID19, in which fever and cough were the main symptoms that were evident in the pregnant women without vertical transmission of the virus in late pregnancy. The World Health Organisation (WHO) confirmed the effect of COVID19 on the mental health of human; especially children and the elderly [22].
The drug used for the treatment of malaria (Chloroquine phosphate) has found to be effective in the treatment of COVID19 in some 100 patients through clinical trials conducted in hospitals across seven cities in China [6]. Lipsitch et al. [10] discussed the approach adopted in the treatment of an influenza pandemic (2009). While this approach may possibly be used in the treatment of Covid19, it depends on many factors, such as existing surveillance systems. In a separate case, viral RNA samples were collected from survivors and people who had died to explore the factors that caused patients to die in a hospital setting by employing multivariable logistic regression [24].
In fact, several studies have formulated mathematical models to simulate actual cases of epidemics and pandemics around the world. In some studies, the optimal control model was widely used to determine the effect of vaccination and isolation. Lee et al. [9] clarified the effect of treatment and isolation on the control of fast transmitting diseases, such as influenza. In their study, they discussed isolation strategies having insufficient antiviral resources. Three models of optimal control, namely, vaccination, isolation and the mixed model of the SIR epidemic were investigated by Hansen and Day [8] to minimise the magnitude of the outbreak. Tuite et al. [18] explored control strategies that could aid in understanding the spreading processes associated with the cholera epidemic in Haiti to reduce the potential effects. Rodrigues et al. [15] discussed a dengue vaccine as a control variable with distinct levels and two methods of treatment. The first method was for paediatric patients, and the second method was for random mass vaccination. A comparison between a different scenario regarding tuberculosis epidemiological features was conducted by P. Rodrigues et al. [16], aiming to reduce total implementation costs and the number of infected cases. Also, regarding tuberculosis, Moualeu et al. [11] formulated a mathematical model that considers infection (diagnosed and undiagnosed), lostsight, and latently infected aspects. Pang et al. [12] formulated a mathematical model that simulated the actual cases of measles transmission in the United States (US) for the period between 1951 and 1962 to determine the optimal strategy for vaccination. The spread of the Ebola virus in West Africa was investigated by Rachah and Torres [13, 14]. The first paper investigated the effect of different cases of vaccination on the virus spreading over time, while the second paper addressed in addition to vaccination, several strategies to reduce the number of infected and exposed individuals. In another study by Gao and Huang, they incorporated three controls as part of a strategy from among several initially developed strategies to minimise intervention costs and reduce the burden of tuberculosis [5].
The structure of this paper is organised into five sections. The first section, already discussed, provided a brief introduction and background information on COVID19. Section 2 provides further information on the spread of COVID19, followed by Section 3 that presents two models; the optimal control model and a model presenting an explicit solution using the Pontryagin maximum principle (PMP). The results and explanation of several models are presented and discussed in Section 4. Lastly, Section 5 presents the overall conclusions and recommendations for future research.
2The spread of COViD19
In December 2019, the first cases of COVID19 surfaced in Wuhan, Hubei, China [2]. Since then, four other Asian countries confirmed a further 282 cases of the virus on 20th January 2020, two cases in Thailand, one case Japan and South Korea, and the remaining number of cases were reported in China. All cases originated from Wuhan City [20].
After several months, the virus spread to other cities in China, and another 33 countries worldwide. On 24th February, the number of deaths reported in China amounted to 2,663, and 33 in other countries [1]. Towards the end of February 2020, the population in Europe and North America showed signs of the virus, albeit a different strain or foci of the virus, including countries in Asia, and the Middle East.
At the same time, the first case of COVID19 was recorded in Africa and Latin America [21]. At the beginning of March 2020, more than 10,000 patients had died from the virus in 10 countries, including Iran, Italy, and South Korea [19]. A significant increase in the harm caused by the was in the Middle East was confirmed during the middle of March. As a result of the rapid spread of the virus, the WHO classified the COVID19 epidemic as a global pandemic [21].
The first case of COVID19 in Iraq appeared in the Najaf province, south of the capital Baghdad on 24th February 2020; an Iranian student studying Islamic science in Najaf. Many Iraqi people travel to Iran to visit many of the holy shrines and enjoy tourist attractions. It was believed that the virus originated from Iran. On 26th February, four new cases were reported with the number increasing to 13 on 1st March, before reporting 93 further cases in the middle of March with nine deaths.
3Optimal control model
This section presents two models having a known and uncertain number of new daily cases; the deterministic model and the chanceconstrained model, respectively.
3.1Deterministic model
Optimal control for optimization is defined by:
(1)
Subject to the state equations of cases, inpatients, recovered cases and death:
(2)
(3)
(4)
(5)
where,
y(t): The control variable.
C(t): Percent of confirmed cases.
I(t): Percent of inpatients.
R(t): Percent of recovered cases.
D(t): Percent of death cases.
β(t): Percent of the new cases to the inpatient cases.
μ(t): Percent of the recover cases to the inpatient cases.
ΔC(t) = C(t)C(t1)
Equation (2) signifies the total number of cases that increased based on new cases daily. The total number of inpatients increases by the addition of new cases and decreases by the number of recovery and deaths each day (Equation 3). Equations (4 & 5) represent the total number of recovered patients and deaths, respectively. The control variable represents the percentage of isolation or each weather factor.
By using the PMP, determining the solution of the optimal control model can be achieved. The Lagrangian function is expressed as follows [17]:
(6)
A Hamiltonian function is expressed as:
(7)
Substituting Equation (7) into Equation (6), gives:
(8)
Deriving Equation (8) concerning C (t) , I (t) , R (t) , D (t), separately, gives:
(9)
(10)
(11)
(12)
By rearranging Equations (9–12), we can determine the adjoint equations as:
(13)
(14)
(15)
(16)
From Equations (13  14), we get:
(17)
(18)
Rearranging Equation (13), yields:
(19)
By substituting Equation (19) into Equation (2), it yields:
(20)
From Equation (20), we can determine the total number of cases over time as follows:
(21)
By substituting Equation (19) into Equation (3), it yields:
(22)
From Equation (22), we then get:
(23)
Then, substituting Equation (21) into Equation (23), it yields:
(24)
Equation (24) represents the total number of inpatients over time. We can determine the total number of recovered cases over time by substituting Equation (24) into Equation (4) as follows:
(25)
Substituting Equation (24) into Equation (5) yields the total number of death cases over time as follows:
(26)
Thus, the value of the control variable is as follows:
(27)
From Equation (27), y (t) = 1 means:
(28)
(29)
(30)
Initially the value of the control variable is zero, which means the model represents the actual cases registered in Iraq with the real percentage of new cases, recovered, and deaths as follows:
(31)
Next, by introducing the effect of wind (for example): changing the value of the control variable and cases depending on the wind degree (W) and transition matrix (φ) (see Appendix) we get:
(32)
(33)
(34)
Where ABS = absolute value.
The elements of the transition matrix can be calculated as follows (for example, humidity):
(35)
Where
Finally, the next tasks include incorporating the value of the control variable into an optimal control model to obtain the solution. The solution of the optimal control model relies on Equations (15–18, 21, 24–26). The solution is found by using the goal seek function in Microsoft Excel with λ(T) = 0. Hence, achieving the condition λ(T) = 0 by changing the value of λ(0).
3.2Chanceconstrained model
The number of new cases reported in Iraq depends on the number of samples tested. Therefore, the actual cases may be is greater than those recorded cases. In this model, the number of new cases is uncertain given by the equation representing chanceconstrained:
(36)
Where α takes values between zero and one (1) and NC is a random variable (new cases).
In determining the solution, chanceconstrained must be converted to deterministic constrained. Here, the value of α is equal to zero or one (1) representing an extremely risky or extremely conservative attitude, respectively. The minimum acceptable to achieve the constraint is α, while (1  α) is the maximum acceptable risks [7].
If NC is a random variable that adheres to a normal distribution with mean
(37)
Where ∅ is the cumulative distribution function (CDF) of the standard normal distribution.
The number of daily cases can be found from Eq. (37) with the initial value of cases C (0). To determine the effect of the weather factors, we apply the deterministic model (Equations 1–5) with the daily cases of chanceconstrained (Equation 37).
4Numerical results
Table 1 (see Appendix) shows the number of COVID19 cases and degrees of temperature, humidity, wind and pressure from 17th March to 15th May. First, we determine the values of (β,μ,N) by using Equation (31) to fit the actual cases reported in Table (1) with a zero (0) value of the control variable (see Table 2 in the Appendix).
Next, we change the control variable value to determine the results that represent the effect of isolation and weather.
The effect of isolation is determined by presenting the value of the control variable given below:
Figure (1) shows the effect of isolation on the results, with the actual cases (A.C.), increased cases (I.C.), and decreased cases (D.C.). From Fig. (1), we can conclude that the number of cases increases, due to increase in the number of people that did not commit to staying at home (y(t) = 0.25), and vice versa.
Fig. 1
The increase in the per cent of people that did not commit to staying at home by 25% led to an increase in the number of COVID19 cases by 118%, while, the proportion of COVID19 cases decreased by 53.4% with an increase in the number of citizens staying at home by 25%. The numbers of inpatients (A.I, I.I, D.I), recoverd (A.R, I.R, D.R) and deaths (A.D, I.D, D.D) follow the number of cases.
According to the WHO, symptoms of infection first appear between 2 and 14 days, usually 5 days. In this case, we take the weather for the last 5 days (see Table 3 in the Appendix). We can then find the transition matrix for each factor of the weather (see Tables 4–7 in the Appendix) from Table (3). The values of the main diagonal are zero (0) (without effect) and the other values are negative (cases decrease) and positive (cases increase). The values of the transition matrix represent an increase (or decrease) of COVID19 cases with an increase (or decrease) for every one (1) degree of the weather.
From Equations (32–34), and Table (4), we can determine the value of the control variable, according to the effect of humidity. Figure (2) shows the possible effect of humidity on the number of new cases.
Fig. 2
The white colour represents the number of new cases without the effect of humidityt, which means the default number. Meanwhile, the actual number of cases, affected by the humidity, is represented by gray colour. The explanation for Fig. (2) is as follows:
Two colours having the same value signifies no change in humidity, thus having no effect.
The white colour higher than the gray colour means humidity increases, while for new cases is decreases.
The other case means a decrease in humidity and an increase in new cases.
Similarly, we determine the possible effect of other weather factors, as can be seen in Figs. (3–5):
Fig. 3
Fig. 4
Fig. 5
As evident in Figs. (2–5), the wind shows as having the highest effect on the number of new COVID19 cases. Figure (6) shows the possible effect of the weather factors at the end of period.
Fig. 6
Figure (6) shows the possible effect of weather on the number of new COVID19 cases. Wind (W.E) and pressure (P.E.) represent the highest and lowest effect, respectively. However, the effect of isolation (see Fig. 1) is more important compared to the weather factors. Figure (7) shows the results at the end of the period without the effect of the weather.
Fig. 7
From Fig. (7), it can be seen that the increase in wind speed (W) and temperature (T) has led to a decrease in the number of COVID19 cases. For example, the number of cases is 3,346 and 3,204 without the possible effect of both wind and temperature, respectively. Whereas the number of cases slightly increased with an increase in humidity (H) and pressure (P) compared to the number of actual cases (A).
For the chanceconstraint model, the β value is changed according to Eq. (36). At first, we divide the study period into 12 subperiods with 5 days for each subperiod. Next, a normality test of observations of the subperiods is conducted, in which all subperiods adhere to a normal distribution (see Table 8). Eq. (36) is then applied to determine the daily cases with α = 0.90 and ∅^{1} (α) = 1.28. Finally, the same steps of the deterministic model are used to find the solution. From Equations (32–34), and Table (9), we can find the value of the control variable and the possible effect of the weather factors, as shown in Figs. (8–11).
Fig. 8
Fig. 9
Fig. 10
Fig. 11
By observing Figs. (8–11), the pressure signifies the highest effect on the number of new COVID19 cases. Whereas, the wind represents the secondhighest effect, but with an inverse relation to COVID19 cases. Figure (12) below illustrates the possible proportional effect of the weather factors at the end of the period.
Fig. 12
From observing Fig. (12), we can see that the pressure (P.E) and temperature (T.E.) represent the highest and lowest effect, respectively. The effect of the weather in the chanceconstrained model is higher than the deterministic model because the increase in the number of COVID19 cases is according to the chanceconstrained model.
5Conclusion
In this paper, the possible effect of both weather factors and isolation on the spread of COVID19 in the context of Iraq was investigated, finding that temperature, humidity, wind, and pressure had a noticeable effect on the spread of the virus. An optimal control model was developed to describe the spread of COVID19 cases, Deterministic and chanceconstrained models were also developed, and the solution of the model using the PMP was also derived. The transition matrix for each of the factor factors was addressed.
Initially, a control variable y(t) representing the percentage of isolation and weather factors was determined. The zero value of the control variable represented the actual data of COVID19 cases in Iraq, while the optimal solution was determined with y = –1. Next, was determining the value of the control variable with respect to five models, before finally, clarifying the possible effect of isolation and the weather factors on the spread of COVID19.
Accordingly, it was shown that isolation was significant in containing COVID19 from contagion. On the other hand, the number of cases increased by 118% attributed to an increase in the number of people who ignored the need to stay at home, which rose by 25%. In contrast, the cases reduced by 53.4% for the opposite case where people stayed at home.
These statistics also support the nature of the virus in reality since it is highly contagious, spreading from one person to many. Weather factors also had a noticeable effect on the spread of COVID19, though lower than selfisolation. Likewise, both wind speed and temperature had the highest effect compared with other weather factors. Further, the wind speed and temperature had an inverse effect on the spread of COVID19 by 1.28% and 0.23%, respectively, while a positive relationship with humidity and pressure. Humidity and temperature had a similar effect but opposingly. Moreover, increasing the number of daily cases of COVID19, according to the chanceconstrained model, weather factors had a greater effect.
Accordingly, the model developed in this study could be applied in other countries using the same factors or by introducing other factors, such as communication, transportation, and payment.
Appendices
Appendix
Table 1
Date  Cases  Recover  Death  New Cases  Temp.  Wind  Humid.  Pres. 
17 March  165  43  12  11  26  7  40  1015 
18  177  49  12  12  22  13  39  1010 
19  192  49  13  15  18  7  57  1016 
20  208  49  17  16  23  9  32  1014 
21  214  52  17  6  22  7  45  1006 
22  233  57  20  19  17  2  52  1011 
23  266  62  23  33  20  9  46  1015 
24  316  75  27  50  24  4  34  1014 
25  346  103  29  30  27  13  29  1009 
26  382  105  36  36  28  20  30  1010 
27  458  122  40  76  24  11  40  1014 
28  506  131  42  48  24  7  46  1009 
29  547  143  42  41  22  7  50  1014 
30  630  152  46  83  23  9  33  1012 
31  694  170  50  64  26  4  26  1017 
1 April  728  182  52  34  28  26  29  1007 
2  772  202  54  44  26  9  28  1012 
3  820  226  54  48  25  4  33  1016 
4  878  259  56  58  25  9  37  1018 
5  961  259  61  83  26  4  29  1021 
6  1031  344  64  70  29  9  30  1014 
7  1122  373  65  91  26  9  37  1012 
8  1202  452  69  80  30  7  27  1010 
9  1232  496  69  30  26  4  54  1006 
10  1279  550  70  47  25  7  35  1008 
11  1318  601  72  39  25  9  30  1012 
12  1352  640  76  34  26  6  29  1016 
13  1378  717  78  26  27  10  33  1017 
14  1400  766  78  22  27  13  15  1017 
15  1415  812  79  15  29  17  11  1016 
16  1434  856  80  19  30  2  17  1013 
17  1482  906  81  48  32  7  14  1011 
18  1513  953  82  31  31  4  21  1011 
19  1539  1009  82  26  28  7  29  1015 
20  1574  1043  82  35  32  4  21  1013 
21  1602  1096  83  28  33  4  20  1012 
22  1631  1146  83  29  33  4  25  1010 
23  1677  1171  83  46  37  6  18  1004 
24  1708  1204  86  31  29  6  29  1008 
25  1763  1224  87  55  32  13  21  1001 
26  1820  1263  87  57  26  13  33  1008 
27  1847  1286  88  27  28  6  23  1013 
28  1928  1319  90  81  31  9  17  1011 
29  2003  1346  92  75  24  4  29  1008 
30  2085  1375  93  82  24  6  40  1009 
1 May  2153  1414  94  68  31  6  26  1014 
2  2219  1473  95  66  36  2  16  1011 
3  2296  1490  97  77  33  11  20  1013 
4  2346  1544  97  50  26  15  45  1011 
5  2431  1571  102  85  29  11  21  1018 
6  2480  1602  102  49  31  7  20  1016 
7  2543  1626  102  63  37  4  13  1009 
8  2603  1661  104  60  29  9  33  1008 
9  2679  1702  107  76  29  13  20  1008 
10  2767  1734  109  88  29  9  21  1010 
11  2818  1790  110  51  32  6  17  1015 
12  2913  1903  112  95  35  4  13  1013 
13  3032  1966  115  119  39  9  8  1011 
14  3143  2028  115  111  40  11  10  1009 
15  3193  2089  117  50  41  15  10  1008 
The ministry of health/ Iraq.
https://www.timeanddate.com/weather/iraq/baghdad/historic?month=4&year=2020.
Table 2
Date  β  μ 

17 March  0.066667  0.100000  0.009091 
18  0.067797  0.051724  0.000000 
19  0.078125  0.000000  0.007692 
20  0.076923  0.000000  0.028169 
21  0.028037  0.020690  0.000000 
22  0.081545  0.032051  0.019231 
23  0.124060  0.027624  0.016575 
24  0.158228  0.060748  0.018692 
25  0.086705  0.130841  0.009346 
26  0.094241  0.008299  0.029046 
27  0.165939  0.057432  0.013514 
28  0.094862  0.027027  0.006006 
29  0.074954  0.033149  0.000000 
30  0.131746  0.020833  0.009259 
31  0.092219  0.037975  0.008439 
1 April  0.046703  0.024291  0.004049 
2  0.056995  0.038760  0.003876 
3  0.058537  0.044444  0.000000 
4  0.066059  0.058615  0.003552 
5  0.086368  0.000000  0.007800 
6  0.067895  0.136437  0.004815 
7  0.081105  0.042398  0.001462 
8  0.066556  0.116006  0.005874 
9  0.024351  0.065967  0.000000 
10  0.036747  0.081942  0.001517 
11  0.029590  0.079070  0.003101 
12  0.025148  0.061321  0.006289 
13  0.018868  0.132075  0.003431 
14  0.015714  0.088129  0.000000 
15  0.010601  0.087786  0.001908 
16  0.013250  0.088353  0.002008 
17  0.032389  0.101010  0.002020 
18  0.020489  0.098326  0.002092 
19  0.016894  0.125000  0.000000 
20  0.022236  0.075724  0.000000 
21  0.0174781  0.125295  0.002364 
22  0.0177805  0.124378  0 
23  0.0274299  0.059101  0 
24  0.0181498  0.078947  0.007177 
25  0.0311968  0.044247  0.002212 
26  0.0313186  0.082978  0 
27  0.0146183  0.048625  0.002114 
28  0.0420124  0.063583  0.003853 
29  0.0374438  0.047787  0.003539 
30  0.0393285  0.047001  0.001620 
1 May  0.0315838  0.060465  0.001550 
2  0.0297431  0.090629  0.001536 
3  0.0335365  0.023977  0.002820 
4  0.0213128  0.076595  0 
5  0.0349650  0.035620  0.006596 
6  0.0197580  0.039948  0 
7  0.0247738  0.029447  0 
8  0.0230503  0.041766  0.002386 
9  0.0283687  0.047126  0.003448 
10  0.0318033  0.034632  0.002164 
11  0.0180979  0.061002  0.001089 
12  0.0326124  0.125835  0.002227 
13  0.0392480  0.066246  0.003154 
14  0.035316  0.062  0 
15  0.015659  0.061803  0.002026 
Table 3
Humid. 
 Temp. 
 Wind 
 Pres. 

56–75  50  91–100  55  21–25  40  5026–5035  55 
76–100  51  101–110  23  26–30  52  5036–5045  79 
101–125  60  111–120  31  31–35  54  5046–5055  57 
126–150  56  121–130  53  36–40  41  5056–5065  50 
151–175  67  131–140  51  41–45  59  5066–5075  38 
176–200  48  141–150  58  46–50  61  5076–5085  60 
201–225  28  151–160  62  51–55  53  
226–250  22  161–170  64  56–60  34  
251–275  12  171–180  50 
Table 4
State  75  100  125  150  175  200  225  250  275 
75  0.000  0.030  0.208  0.084  0.170  –0.013  –0.147  –0.160  –0.190 
100  –0.030  0.000  0.386  0.111  0.217  –0.024  –0.182  –0.192  –0.221 
125  –0.208  –0.386  0.000  –0.164  0.132  –0.161  –0.324  –0.307  –0.323 
150  –0.084  –0.111  0.164  0.000  0.428  –0.159  –0.377  –0.343  –0.354 
175  –0.170  –0.217  –0.132  –0.428  0.000  –0.745  –0.780  –0.600  –0.550 
200  0.013  0.024  0.161  0.159  0.745  0.000  –0.815  –0.527  –0.485 
225  0.147  0.182  0.324  0.377  0.780  0.815  0.000  –0.240  –0.320 
250  0.160  0.192  0.307  0.343  0.600  0.527  0.240  0.000  –0.400 
275  0.190  0.221  0.323  0.354  0.550  0.485  0.320  0.400  0.000 
Table 5
State  100  110  120  130  140  150  160  170  180 
100  0.000  –3.167  –1.200  –0.070  –0.107  0.053  0.652  0.464  –0.036 
110  3.167  0.000  0.767  2.958  1.370  1.143  1.143  0.803  0.444 
120  1.200  –0.767  0.000  2.191  0.986  0.888  0.787  0.650  0.317 
130  0.070  –2.958  –2.191  0.000  –0.218  0.236  0.318  0.265  –0.058 
140  0.107  –1.370  –0.986  0.218  0.000  0.690  0.587  0.426  –0.018 
150  –0.053  –1.143  –0.888  –0.236  –0.690  0.000  0.484  0.294  –0.254 
160  –0.652  –0.978  –0.787  –0.318  –0.587  –0.484  0.000  0.104  –0.623 
170  –0.464  –0.803  –0.650  –0.265  –0.426  –0.294  –0.104  0.000  –1.350 
180  0.036  –0.444  –0.317  0.058  0.018  0.254  0.623  1.350  0.000 
Table 6
State  25  30  35  40  45  50  55  60 
25  0.000  2.400  1.367  0.036  0.958  0.851  0.446  –0.177 
30  –2.400  0.000  0.333  –1.145  0.477  0.464  0.055  –0.607 
35  –1.367  –0.333  0.000  –2.624  0.549  0.508  –0.015  –0.795 
40  –0.036  1.145  2.624  0.000  3.722  2.074  0.855  –0.337 
45  –0.958  –0.477  –0.549  –3.722  0.000  0.426  –0.578  –1.690 
50  –0.851  –0.464  –0.508  –2.074  –0.426  0.000  –1.582  –2.749 
55  –0.446  –0.055  0.015  –0.855  0.578  1.582  0.000  –3.915 
60  0.177  0.607  0.795  0.337  1.690  2.749  3.915  0.000 
Table 7
State  5035  5045  5055  5065  5075  5085 
5035  0.000  2.350  0.123  –0.158  –0.436  0.090 
5045  –2.350  0.000  –2.104  –1.413  –1.364  –0.475 
5055  –0.123  2.104  0.000  –0.721  –0.995  0.068 
5065  0.158  1.413  0.721  0.000  –1.268  0.463 
5075  0.436  1.364  0.995  1.268  0.000  2.193 
5085  –0.090  0.475  –0.068  –0.463  –2.193  0.000 
Table 8
Date  Cases  β  Mean  S.D. 
17 March  165  0.066667  12.00  3.94 
18  182  0.093407  
19  199  0.085427  
20  216  0.078704  
21  233  0.072961  
22  281  0.170819  33.60  11.19 
23  329  0.145897  
24  377  0.127321  
25  425  0.112941  
26  473  0.10148  
27  558  0.15233  62.40  17.87 
28  643  0.132193  
29  728  0.116758  
30  813  0.104551  
31  898  0.094655  
1 April  975  0.078974  53.40  18.65 
2  1052  0.073194  
3  1129  0.068202  
4  1206  0.063847  
5  1283  0.060016  
6  1378  0.06894  63.60  24.83 
7  1473  0.064494  
8  1568  0.060587  
9  1663  0.057126  
10  1758  0.054039  
11  1797  0.021703  27.20  9.52 
12  1836  0.021242  
13  1875  0.0208  
14  1914  0.020376  
15  1953  0.019969  
16  1999  0.023012  31.80  10.85 
17  2045  0.022494  
18  2091  0.021999  
19  2137  0.021526  
20  2183  0.021072  
21  2236  0.023703  37.80  12.07 
22  2289  0.023154  
23  2342  0.02263  
24  2395  0.022129  
25  2448  0.02165  
26  2542  0.036979  64.40  23.19 
27  2636  0.03566  
28  2730  0.034432  
29  2824  0.033286  
30  2918  0.032214  
1 May  3004  0.028628  69.20  13.14 
2  3090  0.027832  
3  3176  0.027078  
4  3262  0.026364  
5  3348  0.025687  
6  3435  0.025328  67.20  15.09 
7  3522  0.024702  
8  3609  0.024106  
9  3696  0.023539  
10  3783  0.022998  
11  3910  0.032481  85.20  32.84 
12  4037  0.031459  
13  4164  0.0305  
14  4291  0.029597  
15  4418  0.028746 
Table 9
Humid. 
 Temp. 
 Wind 
 Pres. 

51–75  127  91–100  48  21–25  53  5026–5035  94 
76–100  66  101–110  36  26–30  75  5036–5045  94 
101–125  81  111–120  43  31–35  72  5046–5055  77 
126–150  81  121–130  75  36–40  64  5056–5065  73 
151–175  89  131–140  69  41–45  86  5066–5075  56 
176–200  69  141–150  70  46–50  76  5076–5085  71 
201–225  28  151–160  89  51–55  77  
226–250  25  161–170  84  56–60  43  
251–275  17  171–180  127 
Table 10
State  75  100  125  150  175  200  225  250  275 
75  0.000  –2.430  –0.922  –0.608  –0.380  –0.463  –0.660  –0.583  –0.550 
100  2.430  0.000  0.585  0.303  0.303  0.029  –0.306  –0.275  –0.281 
125  0.922  –0.585  0.000  0.021  0.162  –0.156  –0.529  –0.447  –0.426 
150  0.608  –0.303  –0.021  0.000  0.304  –0.245  –0.712  –0.564  –0.515 
175  0.380  –0.303  –0.162  –0.304  0.000  –0.793  –1.220  –0.853  –0.720 
200  0.463  –0.029  0.156  0.245  0.793  0.000  –1.647  –0.883  –0.696 
225  0.660  0.306  0.529  0.712  1.220  1.647  0.000  –0.120  –0.220 
250  0.583  0.275  0.447  0.564  0.853  0.883  0.120  0.000  –0.320 
275  0.550  0.281  0.426  0.515  0.720  0.696  0.220  0.320  0.000 
Table 11
State  100  110  120  130  140  150  160  170  180 
100  0.000  –1.240  –0.275  0.888  0.530  0.440  0.898  0.596  0.655 
110  1.240  0.000  0.690  3.904  1.679  1.147  1.147  0.972  1.523 
120  0.275  –0.690  0.000  3.214  1.334  0.917  1.174  0.834  1.408 
130  –0.888  –3.904  –3.214  0.000  –0.545  –0.232  0.494  0.239  1.047 
140  –0.530  –1.679  –1.334  0.545  0.000  0.082  1.014  0.501  1.445 
150  –0.440  –1.147  –0.917  0.232  –0.082  0.000  1.946  0.710  1.900 
160  –0.898  –1.347  –1.174  –0.494  –1.014  –1.946  0.000  –0.526  1.877 
170  –0.596  –0.972  –0.834  –0.239  –0.501  –0.710  0.526  0.000  4.280 
180  –0.655  –1.523  –1.408  –1.047  –1.445  –1.900  –1.877  –4.280  0.000 
Table 12
State  25  30  35  40  45  50  55  60 
25  0.000  4.400  1.900  0.703  1.635  0.920  0.813  –0.286 
30  –4.400  0.000  –0.600  –1.145  0.713  0.050  0.095  –1.067 
35  –1.900  0.600  0.000  –1.691  1.369  0.267  0.269  –1.160 
40  –0.703  1.145  1.691  0.000  4.429  1.245  0.922  –1.027 
45  –1.635  –0.713  –1.369  –4.429  0.000  –1.938  –0.832  –2.846 
50  –0.920  –0.050  –0.267  –1.245  1.938  0.000  0.275  –3.300 
55  –0.813  –0.095  –0.269  –0.922  0.832  –0.275  0.000  –6.875 
60  0.286  1.067  1.160  1.027  2.846  3.300  6.875  0.000 
Table 13
State  5035  5045  5055  5065  5075  5085 
5035  0.000  0.000  –0.869  –0.710  –0.943  –0.470 
5045  0.000  0.000  –1.738  –1.065  –1.258  –0.588 
5055  0.869  1.738  0.000  –0.392  –1.017  –0.204 
5065  0.710  1.065  0.392  0.000  –1.643  –0.110 
5075  0.943  1.258  1.017  1.643  0.000  1.423 
5085  0.470  0.588  0.204  0.110  –1.423  0.000 
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