An Investigation of Tuberculosis Progression Revealing the Role of Macrophages Apoptosis via Sensitivity and Bifurcation Analysis Appendix

An Investigation of Tuberculosis Progression Revealing the Role of Macrophages Apoptosis via Sensitivity and Bifurcation

Analysis Appendix

Wenjing Zhang (wenjing.zhang@ttu.edu), Leif Ellingson, Federico Frascoli, Jane Heffernan

A Appendix

F(y3, b) =−30000000000000000b²y⁹₃−1099by₃¹⁰+ 8999761976040000000000000b²y₃⁸

−44999560408792000by₃⁹−1500y₃¹0 + 79200000000000000000000000000000b³y₃⁶ +450048906483808320000000000000000b²y₃⁷+ 13499620986477773624000000by₃⁸ +599988000000y₃⁹−15839366400000000000000000000000000000000b³y₃⁵

−239994629325261378575040000000000000000000b²y⁶₃

−674935432208343563635168000000000by⁷₃

−74996700036000000000y₃⁸−3168087118099200000000000000000000000000000000000b³y₃⁴ +2249085776421226264275233280000000000000000000000b²y⁵₃

−89998911238251556826680817584000000000000by₃⁶+ 1499820005399952000000000000y₃⁷ +395978615857442534400000000000000000000000000000000000000b³y₃³

+1124968348874508333423914407616640000000000000000000000000b²y₃⁴+ 16874544911123108329962348161600000000000000000000by₃⁵

+749987250157501199976000000000000000y₃⁶

+49501385952677968321267200000000000000000000000000000000000000000b³y²₃

−93748386192870921164651612428752000000000000000000000000000000000b²y³₃− 1012461001684904609196907954515040000000000000000000000000by⁴₃−

104997450022499610002400000000000000000000000y₃⁵

+198000791965944063360000000000000000000000000000000000000000000000000b³y₃− 181355809477836284380751917200000000000000000000000000000000000000000b²y²₃ +28124097279400065066482602257872000000000000000000000000000000000by³₃ +7124887500787517249910000000000000000000000000000000y₃⁴

+197999208000792000000000000000000000000000000000000000000000000000000000b³ +12231525051410247684090000000000000000000000000000000000000000000000000b²y3

+37230879664420088563862404250000000000000000000000000000000000000000by₃²− 262499249921249700001500000000000000000000000000000000000000y₃³−

118800950397624000000000000000000000000000000000000000000000000000000000b²− 3687322047963393616425000000000000000000000000000000000000000000000000by₃ +4687631251125001874990625000000000000000000000000000000000000000000y₃² +17820356401782000000000000000000000000000000000000000000000000000000000b.

(1)

a0(y3, b) = 1.0000(.25000e−32y¹⁵₃ −.15000e−23y₃¹⁴+ (.14998e−15−.1e−15b)y₃¹³ +(.62500e−7 +.49998e−7b)y¹²₃ + (b+ 2.6518)(b−6.1513)y₃¹¹+ (.11248e10−.16000 e10b−.39998e9b²)y₃¹0 + (.14750e18 +.31498e18b+.19996e17b²)y⁹₃+ (−.40872e 26−.17498e26b+.8e25b²)y⁸₃+ (.45792e34−.15751e34b−.94990e33b²)y⁷₃

+(−.31090e42 +.31998e42b+.99935e40b²)y₃⁶+ (.13592e50−.24060e50b+.65e49b²)y⁵₃ +(−.36088e57 +.93112e57b−.49995e57b²)y₃⁴+ (.47252e64−.17183e65b+.

15622e65b²)y₃³+ (.19764e68−.87495e68b+.93745e68b²)y₃²+ (−.14375e72b+. 18750e72b²+.26250e71)y₃+.11250e74 +.12500e75b²−.75002e74b)/(−.5e−16y₃⁵ +.1e−7y₃⁴+ (1.0000b+ 2.0000)y₃³+ (−.99995e8b−.24999e9)y₃²+ (−.

25e17b+.13750e17)y3−.50e20b+.15e20)²/(y3+ 2000.0)/(y₃²−.99998e8y3+.5e16)²; (2)

a₁(y₃, b) = (.125e−37by¹⁶₃ + (−.74990e−29b+.16667e−32b³+.250e−32b²)y₃¹⁵ +(.74940e−21b−.99998e−24b³−.50150e−21b²)y¹⁴₃ + (−.66668e−16b⁴−.

10001e−15b³+.24994e−12b²−.37500e−15 +.31205e−12b)y₃¹³+ (.33332e−7b⁴−. 49948e−2b³−.15017e−1b²+.18750e−6−.11331e−1b)y¹²₃ + (.66668b⁵+ 2.6670b⁴ +.19979e7b³−46.872 +.45055e7b+.59918e7b²)y₃¹¹+ (−.26665e9b⁵−.30665e10b⁴

−.99882e14b³+.74995e10−.89920e15b−.74835e15b²)y¹⁰₃ + (−.26670e17b⁵

+.12997e18b⁴−.39960e23b³−.84370e18 +.11229e24b+.14907e23b²)y₃⁹+ (.17333e 26b⁵+.82332e26b⁴+.47448e31b³+.69370e26−.95388e31b+.74920e31b²)y₃⁸

+(.56685e33b⁵−.48992e34b⁴−.49925e38b³−.42182e34 +.56088e39b−.10483e40b²)y₃⁷ +(−.43332e42b⁵−.81668e42b⁴−.32468e47b³+.18748e42−.22429e47b+.71122 e47b²)y₃⁶+ (−.16670e50b⁵+.47952e50b⁴+.24972e55b³−.58588e49

+.56060e54b−.26200e55b²)y⁵₃+ (.41665e58b⁵−.78035e62b³+.12710e58b⁴+.11717e57

−.70052e61b+.46775e62b²)y⁴₃+ (.26045e66b⁵−.40142e66b³−.27082e66b

−.11715e64−.32715e65b+.26678e66b²)y₃³+ (.15626e70b⁵−.13646e70b⁴+.12901e 70b³−.46872e67 +.54448e69b−.16484e70b²)y²₃+ (.31250e73b⁵−.22083e73b⁴

+.75455e73b³+.12730e73b−.63405e73b²−.46878e70)y₃−.11250e76b⁴+.20833e76b⁵ +.79680e76b³−.47022e76b²+.70700e75b)/(−.50000e−16y⁵₃+.10000e−7y⁴₃

+(1.0000b+ 2.0000)y³₃+ (−.99995e8b−.24999e9)y²₃+ (−.25000e17b+.13750e17)y3

−.50000e20b+.15000e20)²/b/(y₃+ 2000.0)/(y²₃−.99998e8y₃+.50000e16)².

(3)

a₂(y₃, b) = (.12500e−48by₃¹⁷+ (.40438e−38b+.12500e−37b²+.83332e−38b³) y¹⁶₃ + (−.37012e−29b−.55492e−29b³−.91540e−29b²)y₃¹⁵+ (−.15073e−20b

−.17040e−21b³+.74380e−21b²−.18750e−20−.33332e−21b⁴)y₃¹⁴+ (.17753e−12b⁴

−.24999e−7b³−.75000e−7b²+.93748e−12−.56250e−7b)y₃¹³+ (−.14992e−1b⁴+ 9.97 70b³+ 29.988b²−.23436e−3 + 22.499b−.33300e−2b⁵)y₃¹²+ (.13320e7b⁵+.59845

e7b⁴−.49088e9b³+ 37498.−.44998e10b−.37450e10b²)y₃¹¹+ (.13322e15b⁵

+.10227e13b⁴−.20090e18b³−.42185e13 +.56245e18b+.74065e17b²)y¹⁰₃ + (−.86578 e23b⁵−.20961e24b⁴+.23702e26b³+.34685e21−.47808e26b+.37612e26b²)y₃⁹

+(−.28312e31b⁵+.97168e31b⁴−.23522e33b³−.21091e29 +.28122e34b−.52590e34b²)y₃⁸ +(.21644e40b⁵+.22464e40b⁴−.16362e42b³+.93738e36−.11248e42b+.

35678e42b²)y₃⁷+ (.83268e47b⁵−.11968e48b⁴+.12521e50b³−.29292e44 +.28120 e49b−.13145e50b²)y⁶₃+ (−.20812e56b⁵−.37480e55b⁴−.38942e57b³+.58580e 51−.35145e56b+.23486e57b²)y⁵₃+ (−.13009e64b⁵−.72080e62b³+.70168e63b⁴

−.58570e58−.81545e59b−.60308e61b²)y₃⁴+ (−.77162e67b⁵−.34762e68b³

+.25388e68b⁴−.19918e62−.19438e67b+.14833e68b²)y₃³+ (−.85678e70b⁵+.83918e 71b⁴−.12267e72b³−.14653e69−.56305e70b+.50018e71b²)y²₃+ (.16698e74b⁵ +.59990e74b⁴−.10281e75b³−.34495e73b+.39748e74b²−.29302e72)y₃

+.26750e77b(b²−.29050b+.99324e−1)(b²−.59998b+.89998e−1))/(−.50000e−16y₃⁵ +.1e−7y₃⁴+ (1.0000b+ 2.0000)y³₃+ (−.99995e8b−.24999e9)y₃²+ (−.25000e17b+. 13750e17)y₃−.50000e20b+.15000e20)²/b/(y₃+ 2000.0)/(y²₃−.99998e8y₃+.50000e 16)².

(4)

a₃(y₃, b) = ((.83332e−49b³+.12500e−48b²+.41250e−49b)y₃¹⁷+ (−.61875e−35b

−.82460e−35b²−.27472e−35b³)y₃¹⁶+ (.83000e−30b⁴+.37125e−26b+.16508e

−26b³+.49492e−26b²−.18750e−31)y¹⁵₃ + (−.11212e−17b−.17164e−18b³−.11258e

−17b²+.10312e−22 +.53838e−19b⁴)y¹⁴₃ + (−.16667e−7b⁵−.75028e−7b⁴−.11258e

−6b³−.56178e−7b²−.28122e−14 +.19217e−9b)y¹³₃ + (30.000b⁴+ 45.010b³+ 22.50 1b²+.49215e−6−.28175e−1b+ 6.6665b⁵)y¹²₃ + (.66675e9b⁵+.19152e7b⁴

−.44965e10b³−60.932 +.43852e7b−.44972e10b²)y₃¹¹+ (−.43332e18b⁵−.10501e19b⁴

−.22552e18b³+.55775e10−.65525e15b+.56195e18b²)y₃¹⁰+ (−.14172e26b⁵

+.48705e26b⁴+.73090e26b³−.38432e18 +.76502e23b−.47785e26b²)y⁹₃+ (.10833e 35b⁵+.11253e35b⁴−.56168e34b³+.19918e26−.65078e31b+.28170e34b²)y₃⁸

+(.41678e42b⁵−.59958e42b⁴+.11216e42b³−.76150e33 +.39075e39b−.11331e42b²)y⁷₃ +(−.10416e51b⁵−.18773e50b⁴+.55892e49b³+.20498e41−.16054e47b

+.28730e49b²)y⁶₃+ (−.65112e58b⁵+.35150e58b⁴−.34802e57b³−.35122e48 +.41250e54b

−.37525e56b²)y⁵₃+ (−.51395e62b⁵−.12449e63b³+.13218e63b⁴+.29205e55

−.53205e61b+.44582e62b²)y₃⁴+ (−.65255e68b⁵−.23248e68b³+.67445e68b⁴ +.16405e60−.35930e65b+.27858e67b²)y³₃+ (−.25840e72b⁵+.23808e72b⁴

−.69210e71b³−.14646e67 +.34702e69b+.51735e70b²)y₃²+ (−.25165e75b⁵+.20339e75b⁴

−.48202e74b³+.55135e72b+.11447e73b²−.29298e70)y3−.79725e76b⁴+.88610e76b⁵ +.24168e76b³−.25450e75b²+.23204e73b)/(−.50000e−16y₃⁵+.10000e−7y₃⁴

+(1.0000b+ 2.0000)y₃³+ (−.99995e8b−.24999e9)y₃²+ (−.25000e17b+.13750e17)y₃

−.50000e20b+.15000e20)²/b/(y3+ 2000.0)/(y₃²−.99998e8y3+.50000e16)².

(5)

a4(y3, b) = ((.30528e−48b²+.62500e−48 +.87458e−48b)y₃¹⁷+ (.93745e−35b

−.40625e−39 +.12500e−34b²+.41668e−35b³)y¹⁶₃ + (−.56248e−26b−.25000e−26b³

−.74998e−26b²+.56245e−31)y₃¹⁵+ (.28122e−18b−.87502e−18b³−.56258e−18b² +.13750e−22−.25000e−18b⁴)y₃¹⁴+ (.11666e−9b⁴+.62915e−9b³+.91248e−9b²

−.59842e−14 +.34685e−9b)y¹³₃

+(.29182e−2b⁴−.47805e−1b³−.14999b²+.99132e−6−.99135e−1b)y₃¹² +(−.70000e7b⁴−.17625e8b³−89.678 +.13218e8b+.37370e6b²)y₃¹¹

+(.23741e15b⁴+.22518e16b³+.36868e10−.92332e15b+.21156e16b²)y₃¹⁰ +(.16583e24b⁴+.11564e24b³+.13712e18 +.15464e23b−.23435e24b²)y₃⁹ +(−49358e31b⁴−.25460e32b³−.30915e26 +.34628e31b+.58812e31b²)y₃⁸ +(−.17084e40b⁴+.86908e39b³+.22224e34−.37850e39b+.78908e39b²)y⁷₃ +(.16912e47b⁴+.84770e47b³−.91388e41 +.19861e47b−.86125e47b²)y₃⁶ +(.39062e55b⁴−.68352e55b³+.21769e49−.57998e54b+.36322e55b²)y₃⁵ +(.18713e63b³−.16274e63b⁴−.23901e56 +.80535e61b−.68338e62b²)y₃⁴ +(.13346e67b³−.13020e67b⁴−.46482e60 +.49802e65b−.44918e66b²)y₃³ +(−.32338e70b⁴+.29538e70b³+.41500e67 +.63158e68b−.85175e69b²)y²₃ +(−.25182e73b⁴+.20280e73b³+.74655e70b−.47172e72b²+.58598e70)y₃ +.85938e74b⁴−.77345e74b³+.23205e74b²−.23204e73b)/(−.50000e−16y₃⁵ +.10000e−7y₃⁴+ (1.0000b+ 2.0000)y₃³+ (−.99995e8b−.24999e9)y₃²

+(−.25000e17b+.13750e17)y₃−.50000e20b+.15000e20)²/(y₃+ 2000.0) (y₃²−.99998e8y₃+.50000e16)².

(6)

g11= +.7697e−1u12+.4938e−6u²₁₁−.1384e−5u²₁₂−.9928e−7u²₁₃

−.1898e−12u²₁₄+.1361e−8u²₁₁u₁₂−.1762e−7u²₁₁u₁₃ +.1812e−10u²₁₁u₁₄+.2878e−7u₁₁u²₁₂+.9213e−9u₁₁u²₁₃ +.6328e−14u11u²₁₄−.7214e−7u²₁₂u13+.1972e−10u²₁₂u14

+.4026e−8u12u²₁₃+.4401e−14u12u²₁₄+.2444e−11u²₁₃u14

−.1887e−13u₁₃u²₁₄−.8306e−3u₁₁+.7000e−2u₁₂

+.1003u13−.8968e−6u14+.5252e−4u²₁₁+.3959e−4u11u12

−.1433e−2u11u13+.3397e−7u11u14−.1865e−3u²₁₂

−.3623e−2u₁₂u₁₃−.7877e−7u₁₂u₁₄+.1166e−3u²₁₃+.1111e−5u₁₃u₁₄

−.7778e−10u²₁₄−.1053e−7u³₁₁+.1489e−7u³₁₂−.1111e−9u³₁₃ +.1715e−18u³₁₄−.1230e−5u11u12+.1899e−5u11u13

−.1472e−8u₁₁u₁₄+.4052e−5u₁₂u₁₃−.1497e−8u₁₂u₁₄ +.1947e−8u13u14−.6317e−7u11u12u13+.4239e−10u11u12u14

−.2735e−10u11u13u14−.7167e−10u12u13u14

(7)

g₁₂= −.7697e−1u₁₁−.1455e−7u²₁₁+.2650e−6u²₁₂−.1191e−6u²₁₃+.5358e−13u²₁₄ +.6264e−9u²₁₁u₁₂−.1071e−7u²₁₁u₁₃+.6636e−12u²₁₁u₁₄−.1090e−8u₁₁u²₁₂

+.1711e−8u11u²₁₃−.2193e−16u11u²₁₄−.6798e−7u²₁₂u13−.4931e−11u²₁₂u14

+.4356e−8u₁₂u²₁₃−.2267e−14u₁₂u²₁₄+.2265e−11u²₁₃u₁₄−.1832e−13u₁₃u²₁₄

−.1588e−2u₁₁+.1018e−1u₁₂+.9434e−1u₁₃+.2182e−5u₁₄+.3082e−4u²₁₁

+.1729e−5u11u12−.1359e−2u11u13+.2608e−7u11u14−.1933e−3u²₁₂−.3430e−2u12u13

−.8060e−7u₁₂u₁₄+.1101e−3u²₁₃+.1036e−5u₁₃u₁₄−.7384e−10u²₁₄

+.9762e−10u³₁₁−.3594e−8u³₁₂−.7039e−10u³₁₃−.3361e−18u³₁₄+.4206e−7

g₁₃= −62.13u₁₃+.6573e−5u²₁₁−.2148e−4u²₁₂+.3484e−6u²₁₃−.3182e−11u²₁₄ +.8995e−8u²₁₁u12−.8087e−7u²₁₁u13+.2247e−9u²₁₁u14+.3863e−6u11u²₁₂

−.1151e−7u11u²₁₃+.8197e−13u11u²₁₄−.1130e−8u²₁₂u13+.3220e−9u²₁₂u14

−.7625e−8u₁₂u²₁₃+.8781e−13u₁₂u²₁₄+.5510e−12u²₁₃u₁₄+.7053e−14u₁₃u²₁₄

−.3317e−1u11−.1578e−2u12+.2124e−1u13−.1095e−3u14+.2558e−3u²₁₁

+.4837e−3u11u12−.2638e−4u11u13+.5706e−6u11u14+.2373e−3u²₁₂+.1612e−3u12u13

+.3805e−5u₁₂u₁₄+.3419e−5u²₁₃+.1250e−3u₁₃u₁₄−.1790e−8u²₁₄−.1372e−6u³₁₁ +.2413e−6u³₁₂−.4704e−9u³₁₃+.6811e−17u³₁₄−.1645e−4u11u12+.4302e−5u11u13

−.1894e−7u11u14+.1340e−5u12u13−.2254e−7u12u14−.1305e−8u13u14

−.7873e−7u₁₁u₁₂u₁₃+.5527e−9u₁₁u₁₂u₁₄+.2843e−10u₁₁u₁₃u₁₄+.4008e−10u₁₂u₁₃u₁₄ (9)

g14= −.1005e−1u14−.1512e−5u14+.9649e−1u13+.6805e−2u12+.5355e−3u11

−.5502e−13u²₁₄−.7504e−10u²₁₄+.1077e−5u13u14+.1917e−8u13u14

−.1090e−6u²₁₃+.1117e−3u²₁₃−.7832e−7u₁₂u₁₄−.5361e−9u₁₂u₁₄−.3477e−2u₁₂u₁₃ +.3829e−5u₁₂u₁₃−.4696e−6u²₁₂−.1881e−3u²₁₂+.2937e−7u₁₁u₁₄

−.6556e−9u11u14−.1377e−2u11u13+.1648e−5u11u13+.1851e−4u11u12

−.5230e−6u₁₁u₁₂+.2112e−6u²₁₁+.4011e−4u²₁₁−.1071e−18u³₁₄−.1836e−13u₁₃u²₁₄ +.2319e−11u²₁₃u₁₄−.8765e−10u³₁₃+.7178e−15u₁₂u²₁₄−.7026e−10u₁₂u₁₃u₁₄

+.4161e−8u12u²₁₃+.6065e−11u²₁₂u14−.6907e−7u²₁₂u13+.4648e−8u³₁₂

+.2796e−14u₁₁u²₁₄−.2734e−10u₁₁u₁₃u₁₄+.1341e−8u₁₁u²₁₃+.1859e−10u₁₁u₁₂u₁₄

−.5738e−7u₁₁u₁₂u₁₃+.1218e−7u₁₁u²₁₂+.8402e−11u²₁₁u₁₄

−.1366e−7u²₁₁u13+.9453e−9u²₁₁u12−.4620e−8u³₁₁

(10)

g₂₁= +.94832e−1u₂₂+.47226e−11u₂₂u₂₃u₂₄−.46624e−10u₂₁u₂₄u₂₂ +.19598e−9u₂₁u₂₃u₂₂+.26094e−8u₂₁u₂₃u₂₄−.57003e−8u²₂₁u₂₂ +.28029e−6u²₂₁u23+.26314e−10u²₂₁u24+.73590e−7u21u²₂₃ +.55809e−13u₂₁u²₂₄+.31623e−9u²₂₂u₂₁+.19391e−9u²₂₂u₂₃ +.51902e−9u₂₂u²₂₃+.20438e−11u²₂₂u₂₄−.87955e−13u₂₂u²₂₄

+.34679e−9u²₂₃u24+.60916e−11u23u²₂₄+.34935e−2u21u23+.11667e

−6u₂₁u₂₄−.47165e−4u₂₂u₂₁−.31084e−5u₂₂u₂₃

−.59276e−7u₂₂u₂₄+.28066e−6u₂₃u₂₄+.33933e−8u³₂₁

+.51829e−8u³₂₃+.49901e−16u³₂₄+.19590e−10u³₂₂+.60112e−5u²₂₁ +.45255e−3u²₂₃−.66279e−10u²₂₄+.94343e−6u²₂₂+ (−.92433e−4u₂₁u₂₃ +.18924e−5u₂₂u₂₁−.24869e−7u₂₂u₂₃−.89691e−8u₂₁u₂₄

+.77679e−8u22u24−.43018e−6u23u24+.10059e−88−.17231e−5u²₂₁

−.12126e−4u²₂₃−.94889e−11u²₂₄−.51025e−7u²₂₂−.19778e−2u21

−.57853u₂₃−.17350e−4u₂₄+.81015e−2u₂₂)

(11)

g₂₂= −.94832e−1u₂₁−.46133e−10u₂₂u₂₃u₂₄+.92951e−10u₂₁u₂₄u₂₂

−.54825e−8u21u23u22−.75266e−9u21u23u24

+.13672e−7u²₂₁u22−.75031e−7u²₂₁u23−.14335e−9u²₂₁u24

−.29954e−7u₂₁u²₂₃−.26479e−12u₂₁u²₂₄−.17179e−8u²₂₂u₂₁

−.10534e−8u²₂₂u23−.31017e−8u22u²₂₃−.11103e−10u²₂₂u24

+.10623e−12u22u²₂₄−.16955e−9u²₂₃u24−.19786e−11

u₂₃u²₂₄−.11689e−2u₂₁u₂₃−.13510e−6u₂₁u₂₄+.43586e−4u₂₂u₂₁

−.20031e−4u22u23+.52056e−7u22u24+.12244e−6u23u24

−.19088e−7u³₂₁−.45383e−8u³₂₃−.14957e−15u³₂₄−.10642e−9u³₂₂

−.40797e−4u²₂₁−.18302e−3u²₂₃−.31486e−10u²₂₄−.50390e−5u²₂₂ +(.24637e−4u₂₁u₂₃−.45756e−5u₂₂u₂₁+.86371e−6u₂₂u₂₃

−.15757e−7u22u24+.12322e−6u23u24−.35400e−89 +.96839e−5u²₂₁ +.48801e−5u²₂₃+.45217e−10u²₂₄+.27719e−6u²₂₂+.14881e−1u₂₁ +.20029u₂₃+.52580e−4u₂₄−.48270e−2u₂₂+.48858e−7u₂₁u₂₄)

(12)

g23= −62.160u₂₃−.68731e−10u22u23u24+.11820e−9u21u24u22

−.83678e−8u21u23u22+.26304e−9u21u23u24

+.18018e−7u²₂₁u₂₂+.37248e−7u²₂₁u₂₃−.20721e−9u²₂₁u₂₄

−.60719e−8u21u²₂₃−.37880e−12u21u²₂₄−.24827e−8u²₂₂u21

−.15223e−8u²₂₂u23−.45109e−8u22u²₂₃−.16046e−10u²₂₂u24

+.11612e−12u₂₂u²₂₄−.72508e−10u²₂₃u₂₄+.27169e−12u₂₃u²₂₄ +.73283e−4u21u23−.86507e−6u21u24+.41766e−4u22u21

−.17576e−4u22u23−.10559e−5u22u24+.12511e−3u23u24

−.27651e−7u³₂₁−.41819e−8u³₂₃−.20392e−15u³₂₄

−.15380e−9u³₂₂−.59607e−4u²₂₁−.40341e−4u²₂₃+.10168e−8u²₂₄

−.74066e−5u²₂₂+ (−.12447e−4u21u23−.60385e−5u22u21

+.13215e−5u₂₂u₂₃+.70621e−7u₂₁u₂₄−.20110e−7u₂₂u₂₄

−.44702e−7u23u24+.26674e−91 +.14027e−4u²₂₁ +.91471e−6u²₂₃+.64709e−10u²₂₄+.40059e−6u²₂₂

+.19629e−1u₂₁+.64976e−2u₂₃+.12801e−3u₂₄−.94051e−2u₂₂)

(13)

g₂₄= −.10274e−1u₂₄.40234e−11u₂₂u₂₃u₂₄−.17709e−10u₂₁u₂₄u₂₂ +.38363e−9u21u23u22+.72093e−9u21u23u24

−.23052e−8u²₂₁u₂₂+.77087e−7u²₂₁u₂₃+.15514e−10u²₂₁u₂₄ +.20862e−7u₂₁u²₂₃+.30520e−13u₂₁u²₂₄+.18615e−9u²₂₂u₂₁ +.11414e−9u²₂₂u23+.32265e−9u22u²₂₃+.12031e−11u²₂₂u24

−.29210e−13u22u²₂₄+.10003e−9u²₂₃u24+.16965e−11u23u²₂₄ +.97502e−3u₂₁u₂₃+.39108e−7u₂₁u₂₄−.14852e−4u₂₂u₂₁ +.39698e−6u22u23−.17406e−7u22u24−.58486e−7u23u24

+.20372e−8u³₂₁+.16168e−8u³₂₃+.21995e−16u³₂₄+.11532e−10u³₂₂ +.40326e−5u²₂₁+.12822e−3u²₂₃−.16331e−10u²₂₄+.55024e−6u²₂₂ +(−.25415e−4u21u23+.76753e−6u22u21−.58882e−7u22u23

−.52877e−8u21u24+.29669e−8u22u24−.11880e−6u23u24

+.28178e−89−.10339e−5u²₂₁−.34341e−5u²₂₃−.52011e−11u²₂₄

(14)

-400 -200 0 200 400 600 -600

-400 -200 0 200 400 600

R0

[PRCC , p-value] = [-0.022827 , 0.47089].

-400 -200 0 200 400 600

b -600

-400 -200 0 200 400 600

R0

[PRCC , p-value] = [0.49827 , 7.1871e-64].

b

-400 -200 0 200 400 600

-600 -400 -200 0 200 400 600

R0

[PRCC , p-value] = [-0.4511 , 2.7495e-51].

-500 0 500 1000

-600 -400 -200 0 200 400 600

R0

[PRCC , p-value] = [-0.74227 , 1.0491e-175].

-400 -200 0 200 400 600

e B -600

-400 -200 0 200 400 600

R0

[PRCC , p-value] = [0.012661 , 0.68923].

e B

-400 -200 0 200 400 600

e M -600

-400 -200 0 200 400 600 800

R0

[PRCC , p-value] = [-0.021407 , 0.49892].

e M

-400 -200 0 200 400 600

c B -600

-400 -200 0 200 400 600

R0

[PRCC , p-value] = [-0.034241 , 0.27936].

c B

-400 -200 0 200 400 600

c M -600

-400 -200 0 200 400 600

R0

[PRCC , p-value] = [-0.00557 , 0.86036].

c M

-600 -400 -200 0 200 400 600

-600 -400 -200 0 200 400 600

R0

[PRCC , p-value] = [0.60629 , 2.0817e-101].

-500 0 500

N 3 -600

-400 -200 0 200 400 600

R0

[PRCC , p-value] = [-0.77618 , 4.0332e-202].

N₃

-400 -200 0 200 400 600

N 2 -600

-400 -200 0 200 400 600

R0

[PRCC , p-value] = [0.30871 , 1.6022e-23].

N₂

-400 -200 0 200 400 600

N 1 -600

-400 -200 0 200 400 600

R0

[PRCC , p-value] = [0.43409 , 3.275e-47].

N₁

-400 -200 0 200 400 600

K -600

-400 -200 0 200 400 600

R0

[PRCC , p-value] = [0.042171 , 0.1827].

K

-400 -200 0 200 400 600 800

c -600

-400 -200 0 200 400 600

R0

[PRCC , p-value] = [-0.013012 , 0.68109].

c

-400 -200 0 200 400 600 800

T -600

-400 -200 0 200 400 600

R0

[PRCC , p-value] = [0.0078777 , 0.80351].

T

-400 -200 0 200 400 600 800

M -600

-400 -200 0 200 400 600

R0

[PRCC , p-value] = [0.0083056 , 0.79307].

M

-400 -200 0 200 400 600 800

s T -600

-400 -200 0 200 400 600

R0

[PRCC , p-value] = [0.0057221 , 0.85658].

s_T

-400 -200 0 200 400 600 800

-600 -400 -200 0 200 400 600

R0

[PRCC , p-value] = [-0.22642 , 4.3143e-13].

Figure 1: Scatter plots showing either monotonic relationships or no relationships between partial residuals for the output R₀ and parameter inputs. P-values that are greater than 0.05 are not statistically significant.

-1000 -500 0 500 -600

-400 -200 0 200 400 600

Turning Points

[PRCC , p-value] = [0.47086 , 2.5517e-56].

b

b

-1000 -500 0 500

-600 -400 -200 0 200 400 600

Turning Points

[PRCC , p-value] = [-0.23772 , 2.5901e-14].

-1000 -500 0 500

-600 -400 -200 0 200 400 600

Turning Points

[PRCC , p-value] = [0.31239 , 4.4781e-24].

-1000 -500 0 500

-600 -400 -200 0 200 400 600

Turning Points

[PRCC , p-value] = [0.12193 , 0.00011092].

-1000 -500 0 500

N₁ -600

-400 -200 0 200 400 600

Turning Points

[PRCC , p-value] = [0.16357 , 1.9801e-07].

N₁

-1000 -500 0 500

N₂ -600

-400 -200 0 200 400 600

Turning Points

[PRCC , p-value] = [0.043895 , 0.16544].

N₂

-1000 -500 0 500

N₃ -600

-400 -200 0 200 400 600

Turning Points

[PRCC , p-value] = [-0.23825 , 2.2594e-14].

N₃

-1000 -500 0 500

-600 -400 -200 0 200 400 600

Turning Points

[PRCC , p-value] = [0.24549 , 3.4269e-15].