1. USCC-E4 01 November 2015
MEMORANDUM FOR Dr. Maria Vega, Battalion Operations.
SUBJECT: Preventing Malaria Outbreak for Soldiers Deployed to Liberia
1. Purpose: Despite our unit’s use of the anti-malarial drug Malarone, there is still not a 100% effective
solution for preventing the spread of malaria. We have identified the main problem to be the fact that our
soldiers could possibly be bitten by mosquitos; therefore, contracting the life-threatening disease—
malaria.
2. Recommendation: As a result, we have evaluated three Courses of Actions (COAs) that are aimed to
help control the mosquito population near our soldier’s living quarters. Before we present our empirical
analysis it is important to note that we optimized in a way that best encompasses the following objectives:
reduction of the mosquito population in the short term (less than 6 months), reduction of the mosquito
population in the long term (greater than 6 months), as well as minimizing our environmental footprint and
harm to our human capital. In addition, we found that there are two common ways to control the
mosquito population: insecticide spray and/or elimination of breeding grounds. Our subject matter
experts concluded that there are about 10 million mosquitos living by our base camp, and our rate
coefficient was believed to 1.2. We made the assumptions that all information given to us by the subject
matter experts is correct because of their background in the field, that no external factors like weather will
cause a drastic variation in the population because of accurate forecasting, and that we have taken all
necessary safety precautions to prevents harm to the environment and people due to our adherence to
said precautions.
a. The methodology we used to evaluate the best possible COA was through the use of Improved
Euler’s Method (Appendix B). In order to evaluate the function of time and population, we used a
logistical model in conjunction with the associated COA data provided by our subject matter experts.
After we modeled the problem, we highlighted the 6 month (short-term) and 12 month (long-term),
respectively, in accordance with the objectives of the commander. Furthermore, after evaluation we
looked at, “p_n” or simply the population of mosquitos, values at the short-term and long-term
benchmarks to see if one method worked better than the other. BLUF: Course of Action B, which
consisted of devoting the majority of resources towards destroying mosquito breeding grounds, worked
the best. With the initial estimate of mosquitos at 10 million, COA B could potentially drop the population
of mosquitos down to 5.99 million after 6 months and 5.96 million after a year (Appendix B).
b. Despite the mathematical conformation of the projected success of COA B, other considerations
such as manpower and the severe reduction in carrying capacity were important to the recommendation
process. Although the destruction of breeding grounds would be physically taxing on the men and
women of the 62nd Engineer Battalion, the soldiers would be better off than risking potential negative
health effects associated with the insecticide spray. Furthermore, the use of personnel to fight the
mosquito problem instead of the reliance on the insecticide might be potentially more cost effective
because draining and filling requires less capital than the alternative. We concluded that the destruction
of breeding grounds proves to be a decision of longevity that will positively affect the 62nd Engineer
Battalion and negatively affect the mosquito populations in our area. All in all, COA B is the most cost-
effective, risk reducing, and effective course of action that is in line with the commander’s objectives.
3 Encls Joseph C. Bosse
1. Appendix A CDT, USCC
2. Appendix B United States Military Academy
DEPARTMENT OF THE ARMY
United States Military Academy
West Point, New York 10996
REPLY TO
ATTENTION OF

2. SSE (Step Size 1) SSE (Step Size 6)
n t_n p_n f(t_n,p_n) p_(n+1) n t_n p_n f(t_n,p_n) p_(n+1) t t p_a(t) p_a(t)
1 0 100 -5.776226505 65.34264097 1 0 100 -5.776226505 94.2237735 0 0 100 100 0 0
2 6 65.34264097 -3.774338947 42.69660729 2 1 94.2237735 -5.442578578 88.78119492 1 6 94.38743127 70.71067812 0.026783867 28.81582281
3 12 42.69660729 -2.466252747 27.89909081 3 2 88.78119492 -5.128202912 83.65299201 2 12 89.08987181 50 0.095281427 53.33954505
4 18 27.89909081 -1.611514678 18.23000274 4 3 83.65299201 -4.831986296 78.82100571 3 18 84.08964153 35.35533906 0.190662804 55.59563796
5 24 18.23000274 -1.05300625 11.91196524 5 4 78.82100571 -4.552879823 74.26812589 4 24 79.3700526 25 0.301452487 45.83286287
6 30 11.91196524 -0.688062093 7.78359268 6 5 74.26812589 -4.289895172 69.97823071 5 30 74.91535384 17.67766953 0.41890403 33.24334594
7 36 7.78359268 -0.449597943 5.08600502 7 6 69.97823071 -4.04210111 65.9361296 6 36 70.71067812 12.5 0.536479201 22.24449801
8 42 5.08600502 -0.29377917 3.32333 8 7 65.9361296 -3.808620194 62.12750941 7 42 66.74199271 8.838834765 0.649415343 14.0837311
9 48 3.32333 -0.191963068 2.17155159 9 8 62.12750941 -3.588625665 58.53888374 8 48 62.99605249 6.25 0.754367091 8.56539729
10 54 2.17155159 -0.125433739 1.418949159 10 9 58.53888374 -3.381338518 55.15754523 9 54 59.46035575 4.419417382 0.849110657 5.05290062
11 60 1.418949159 -0.081961717 0.927178855 11 10 55.15754523 -3.186024747 51.97152048 10 60 56.12310242 3.125 0.932300686 2.910609472
12 66 0.927178855 -0.053555951 0.60584315 12 11 51.97152048 -3.001992741 48.96952774 11 66 52.97315472 2.209708691 1.003271148 1.644882782
13 72 0.60584315 -0.034994873 0.395873914 13 12 48.96952774 -2.82859084 46.1409369 12 72 50 1.5625 1.061873082 0.915192328
14 78 0.395873914 -0.022866574 0.258674471 14 13 46.1409369 -2.665205027 43.47573187 13 78 47.19371563 1.104854346 1.108343067 0.502653252
15 84 0.258674471 -0.014941623 0.169024731 15 14 43.47573187 -2.511256747 40.96447512 14 84 44.54493591 0.78125 1.14319727 0.273085184
16 90 0.169024731 -0.009763251 0.110445223 16 15 40.96447512 -2.36620087 38.59827425 15 90 42.04482076 0.552427173 1.167146699 0.146997433
17 96 0.110445223 -0.006379566 0.072167825 17 16 38.59827425 -2.229523748 36.36875051 16 96 39.6850263 0.390625 1.181030007 0.078500708
18 102 0.072167825 -0.004168577 0.047156363 18 17 36.36875051 -2.100741406 34.2680091 17 102 37.45767692 0.276213586 1.185760738 0.041634673
19 108 0.047156363 -0.002723858 0.030813213 19 18 34.2680091 -1.979397824 32.28861128 18 108 35.35533906 0.1953125 1.18228644 0.021950241
20 114 0.030813213 -0.001779841 0.020134167 20 19 32.28861128 -1.865063323 30.42354795 19 114 33.37099635 0.138106793 1.171557457 0.011511912
21 120 0.020134167 -0.001162995 0.013156197 21 20 30.42354795 -1.757333041 28.66621491 20 120 31.49802625 0.09765625 1.154503604 0.006009673
22 126 0.013156197 -0.000759932 0.008596606 22 21 28.66621491 -1.655825504 27.01038941 21 126 29.73017788 0.069053397 1.132017185 0.003124497
23 132 0.008596606 -0.000496559 0.00561725 23 22 27.01038941 -1.560181272 25.45020814 22 132 28.06155121 0.048828125 1.104941126 0.001618575
24 138 0.00561725 -0.000324465 0.003670459 24 23 25.45020814 -1.470061668 23.98014647 23 138 26.48657736 0.034526698 1.074061164 0.000835756
25 144 0.003670459 -0.000212014 0.002398375 25 24 23.98014647 -1.385147576 22.59499889 24 144 25 0.024414063 1.040101224 0.000430297
Step Size 6 Step Size 1 Accepted Value
Appendix A.
1a) Using my derived initial value problem for mass of the drug proguanil, I solved for t using Wolfram
Mathematicia:
Solve[30 == 25 ∗ 2^(2 − 𝑡 12⁄ ), {𝑡}]
Therefore, a solider has to wait 20.84 hours before he or she can expect to have enough proguanil in
their blood to prevent malaria.
As per the IVP, k=-LN[2]/12 for proguanil.
1b) The following is the first 24 hours on proguanil implemented by Euler’s Method using step sizes: h=6
hours and h=1 hour.
After complete iteration of the 168 hour period, the SSE for h=6 is 273.33 and the SSE for h=1 is 37.37.
t 20.84358712999447`

3. 1c. The following is the first 24 hours implemented by Improved Euler’s Method using just step size h=1
hour:
After complete iteration of the 168 hour period, the SSE for h=1 is 0.0146, which proves the accuracy of
the Improved Euler’s Method.
1d) The following is a table that summarizes all numerical methods:
As per the table, the small the step size, the closer it is to the actual mass. Furthermore, the use of
Improved Euler’s Method is far more accurate compared to the Euler’s Method.
n t_n p_n f(t_n,p_n) p_(n+1) t_n+1 p*_(n+1) f(t_(n+1),p*_(n+1)) p_(n+1) t p_a(t) SSE
1 0 100 -5.77623 94.22377 1 94.22377 -5.442578578 94.3906 0 100 0
2 1 94.3906 -5.45221 88.93838 2 88.93838 -5.137282437 89.09585 1 94.38743 1.00248E-05
3 2 89.09585 -5.14638 83.94947 3 83.94947 -4.849111586 84.0981 2 89.08987 3.57254E-05
4 3 84.0981 -4.8577 79.24041 4 79.24041 -4.577105397 79.3807 3 84.08964 7.16147E-05
5 4 79.3807 -4.58521 74.79549 5 74.79549 -4.320357131 74.92792 4 79.37005 0.000113429
6 5 74.92792 -4.32801 70.59991 6 70.59991 -4.078010908 70.72491 5 74.91535 0.000157901
7 6 70.72491 -4.08523 66.63968 7 66.63968 -3.84925886 66.75767 6 70.71068 0.000202577
8 7 66.75767 -3.85607 62.90159 8 62.90159 -3.633338436 63.01296 7 66.74199 0.000245656
9 8 63.01296 -3.63977 59.37319 9 59.37319 -3.429529858 59.47831 8 62.99605 0.000285861
10 9 59.47831 -3.4356 56.04271 10 56.04271 -3.237153723 56.14193 9 59.46036 0.000322331
11 10 56.14193 -3.24289 52.89905 11 52.89905 -3.055568739 52.9927 10 56.1231 0.000354536
12 11 52.9927 -3.06098 49.93173 12 49.93173 -2.884169589 50.02013 11 52.97315 0.000382198
13 12 50.02013 -2.88928 47.13085 13 47.13085 -2.722384907 47.2143 12 50 0.000405236
14 13 47.2143 -2.7272 44.4871 14 44.4871 -2.569675378 44.56586 13 47.19372 0.000423717
15 14 44.56586 -2.57423 41.99163 15 41.99163 -2.425531942 42.06598 14 44.54494 0.000437812
16 15 42.06598 -2.42983 39.63616 16 39.63616 -2.289474092 39.70633 15 42.04482 0.000447772
17 16 39.70633 -2.29353 37.4128 17 37.4128 -2.161048274 37.47904 16 39.68503 0.000453897
18 17 37.47904 -2.16487 35.31417 18 35.31417 -2.039826377 35.37669 17 37.45768 0.000456518
19 18 35.37669 -2.04344 33.33325 19 33.33325 -1.925404305 33.39227 18 35.35534 0.000455983
20 19 33.39227 -1.92881 31.46346 20 31.46346 -1.817400627 31.51916 19 33.371 0.000452641
21 20 31.51916 -1.82062 29.69855 21 29.69855 -1.71545531 29.75113 20 31.49803 0.000446837
22 21 29.75113 -1.71849 28.03264 22 28.03264 -1.619228516 28.08227 21 29.73018 0.000438905
23 22 28.08227 -1.6221 26.46017 23 26.46017 -1.52839947 26.50702 22 28.06155 0.000429161
24 23 26.50702 -1.53111 24.97591 24 24.97591 -1.442665392 25.02013 23 26.48658 0.000417901
25 24 25.02013 -1.44522 23.57491 25 23.57491 -1.361740483 23.61665 24 25 0.0004054
Accepcted ValueImproved Euler's Method (Proguanil)
Time (Hours) Actual Mass of Proguanil In Blood Euler's Method Step Size 1 Euler's Method Step Size 6 Improved Euler's Method Step Size 1
0 100 100 100 100
24 25 23.98014647 0.003670459 25.02013454
48 6.25 5.750474247 1.34723E-07 6.260071322
72 1.5625 1.378972147 4.94494E-12 1.566278267
96 0.390625 0.330679541 1.81502E-16 0.39188493
120 0.09765625 0.079297438 6.66196E-21 0.098050137
144 0.024414063 0.019015642 2.44525E-25 0.024532276
168 0.006103516 0.004559979 8.97517E-30 0.006138008
1.327271704 666.483153 0.000522864Sumof Squared Errors

4. 1e) The following is the first 24 hours on atovaquone implemented by Improved Euler’s Method using
step size h=1 hour.
As per the IVP, k=-LN[2]/48 for atovaquone.
n t_n p_n f(t_n,p_n) p_(n+1) t_n+1 p*_(n+1) f(t_(n+1),p*_(n+1))p_(n+1) k -0.01444
1 0 250 -3.61014 246.3899 1 246.3899 -3.55801 246.4159 p_0 250
2 1 246.4159 -3.55839 242.8575 2 242.8575 -3.507 242.8832 h 1
3 2 242.8832 -3.50737 239.3759 3 239.3759 -3.45672 239.4012
4 3 239.4012 -3.45709 235.9441 4 235.9441 -3.40717 235.9691
5 4 235.9691 -3.40753 232.5615 5 232.5615 -3.35832 232.5861
6 5 232.5861 -3.35868 229.2275 6 229.2275 -3.31017 229.2517
7 6 229.2517 -3.31052 225.9412 7 225.9412 -3.26272 225.9651
8 7 225.9651 -3.26306 222.702 8 222.702 -3.21594 222.7256
9 8 222.7256 -3.21628 219.5093 9 219.5093 -3.16984 219.5325
10 9 219.5325 -3.17017 216.3623 10 216.3623 -3.12439 216.3852
11 10 216.3852 -3.12473 213.2605 11 213.2605 -3.0796 213.2831
12 11 213.2831 -3.07993 210.2031 12 210.2031 -3.03545 210.2254
13 12 210.2254 -3.03577 207.1896 13 207.1896 -2.99194 207.2115
14 13 207.2115 -2.99225 204.2193 14 204.2193 -2.94904 204.2409
15 14 204.2409 -2.94935 201.2915 15 201.2915 -2.90676 201.3128
16 15 201.3128 -2.90707 198.4058 16 198.4058 -2.86509 198.4267
17 16 198.4267 -2.86539 195.5613 17 195.5613 -2.82402 195.582
18 17 195.582 -2.82432 192.7577 18 192.7577 -2.78353 192.7781
19 18 192.7781 -2.78383 189.9943 19 189.9943 -2.74363 190.0144
20 19 190.0144 -2.74392 187.2705 20 187.2705 -2.70429 187.2903
21 20 187.2903 -2.70458 184.5857 21 184.5857 -2.66552 184.6052
22 21 184.6052 -2.6658 181.9394 22 181.9394 -2.62731 181.9587
23 22 181.9587 -2.62759 179.3311 23 179.3311 -2.58964 179.3501
24 23 179.3501 -2.58992 176.7601 24 176.7601 -2.55252 176.7788
25 24 176.7788 -2.55279 174.2261 25 174.2261 -2.51592 174.2445
Improved Euler's Method (Atovaquone)

5. 1f) The following graph shows that soldiers should start taking Malarone 10-12 days before leaving to
ensure the drug is built-up in their system. At approximately 216 hours, soldiers will have enough drugs
in their system not to contract malaria.
1g) According to the following table, the soldier can miss one dose and still be fine because the immunity
levels are 30 mg of proguanil and 250 mg of atovaquone, and the projected levels are 33.36 mg and
576.91 mg, respectively.
After missing two doses, the soldier will no longer have immunity:
The 8.35 mg level of proguanil is too low for immunity.
215 216 215 35.35205 -2.04201 33.31003 216 33.31003 -1.92406 33.36901 216 215 585.2918 -8.45195 576.8399 216 576.8399 -8.32989 576.9009
216 217 216 33.36901 -1.92747 31.44154 217 31.44154 -1.81613 31.49721 217 216 576.9009 -8.33078 568.5701 217 568.5701 -8.21047 568.6303
217 218 217 31.49721 -1.81935 29.67786 218 29.67786 -1.71426 29.7304 218 217 568.6303 -8.21134 560.4189 218 560.4189 -8.09277 560.4782
239 240 239 8.84513 -0.51091 8.334215 240 8.334215 -0.4814 8.348971 240 239 413.8689 -5.9765 407.8924 240 407.8924 -5.8902 407.9355
240 241 240 8.348971 -0.48226 7.866716 241 7.866716 -0.4544 7.880644 241 240 407.9355 -5.89082 402.0447 241 402.0447 -5.80575 402.0872
241 242 241 7.880644 -0.4552 7.42544 242 7.42544 -0.42891 7.438587 242 241 402.0872 -5.80637 396.2809 242 396.2809 -5.72252 396.3228

6. Appendix B.
2) The following are evaluations of the three Courses of Action:
The highlight regions at 6 months and 12 months, respectively, represent the short-term and long-term
benchmarks. I assumed that the 12th month mark would represent the long term benchmark.
n t_n p_n f(t_n,p_n) t_n+1 p*_(n+1) f(t_(n+1),p*_(n+1)) p_(n+1)
0 0 10 -4.428571429 1 5.571428571 0.352478134 7.961953
1 1 7.961953 -1.671512789 2 6.290440564 -0.118936461 7.066729
2 2 7.066729 -0.760564738 3 6.30616399 -0.130565695 6.621164
3 3 6.621164 -0.375449113 4 6.245714399 -0.086165447 6.390356
4 4 6.390356 -0.193796668 5 6.196559563 -0.050676925 6.268119
5 5 6.268119 -0.102524538 6 6.165594896 -0.028604764 6.202555
6 6 6.202555 -0.054975748 7 6.147579036 -0.015863589 6.167135
7 7 6.167135 -0.02969748 8 6.137437635 -0.008724024 6.147924
8 8 6.147924 -0.016107114 9 6.131817249 -0.004777386 6.137482
9 9 6.137482 -0.008755285 10 6.128726828 -0.002610367 6.131799
10 10 6.131799 -0.004764784 11 6.127034503 -0.001424625 6.128705
11 11 6.128705 -0.002594776 12 6.126109806 -0.000777005 6.127019
12 12 6.127019 -0.00141355 13 6.125605142 -0.000423641 6.1261
n t_n p_n f(t_n,p_n) t_n+1 p*_(n+1) f(t_(n+1),p*_(n+1)) p_(n+1)
0 0 10 -7.461538462 1 2.538461538 1.60268548 7.070574
1 1 7.070574 -1.451847863 2 5.618725646 0.352276139 6.520788
2 2 6.520788 -0.677103719 3 5.843683929 0.123687671 6.24408
3 3 6.24408 -0.329394939 4 5.914684684 0.047661784 6.103213
4 4 6.103213 -0.163242785 5 5.939970261 0.020137127 6.03166
5 5 6.03166 -0.081652218 6 5.950007999 0.009145072 5.995407
6 6 5.995407 -0.041034383 7 5.954372261 0.004354283 5.977067
7 7 5.977067 -0.020671375 8 5.956395219 0.002131232 5.967797
8 8 5.967797 -0.010426041 9 5.957370482 0.001058966 5.963113
9 9 5.963113 -0.005261834 10 5.957851151 0.000530358 5.960747
10 10 5.960747 -0.002656381 11 5.958090866 0.000266703 5.959552
11 11 5.959552 -0.001341257 12 5.958211152 0.000134397 5.958949
12 12 5.958949 -0.00067728 13 5.958271699 6.77972E-05 5.958644
n t_n p_n f(t_n,p_n) t_n+1 p*_(n+1) f(t_(n+1),p*_(n+1)) p_(n+1)
0 0 10 -5.333333333 1 4.666666667 0.82962963 7.748148
1 1 7.748148 -1.805988112 2 5.942160037 0.045825909 6.868067
2 2 6.868067 -0.794925691 3 6.073141356 -0.059226373 6.440991
3 3 6.440991 -0.378722556 4 6.06226846 -0.050331749 6.226464
4 4 6.226464 -0.188009208 5 6.038454655 -0.030960892 6.116979
5 5 6.116979 -0.095407589 6 6.021571224 -0.017319021 6.060616
6 6 6.060616 -0.048982305 7 6.011633203 -0.009324607 6.031462
7 7 6.031462 -0.025301623 8 6.006160429 -0.004933403 6.016345
8 8 6.016345 -0.01311125 9 6.003233289 -0.002588025 6.008495
9 9 6.008495 -0.006805543 10 6.001689358 -0.001351867 6.004416
10 10 6.004416 -0.003535557 11 6.000880639 -0.000704615 6.002296
11 11 6.002296 -0.001837591 12 6.000458519 -0.000366843 6.001194
12 12 6.001194 -0.000955304 13 6.000238589 -0.000190878 6.000621
COA A
COA B
COA C