Bifurcations in a Dynamical Model of the Innate Immune System Response to Initial Pulmonary Infection

Abstract

We revisit a dynamical model of the innate immune system response to initial pulmonary infection. By supposing that large pathogen load may not be able to overcome the innate system, a complete analysis on bifurcations with high codimension is given in this paper. It is shown that the highest codimension of a nilpotent cusp is 3, and a center-type equilibrium is a weak focus with order at most 2. As parameters vary, the model can undergo degenerate Bogdanov–Takens bifurcation of codimension 3 and Hopf bifurcation of codimension 2. Finally, numerical simulations, including the coexistence of a limit cycle and a homoclinic cycle, two limit cycles, tristability, are presented to illustrate the theoretical results. Our results indicate the complexity of the interaction between the innate immune system and initial pulmonary bacterial infection.

Appendix A: The Concrete Expressions in Subsection 3.1

$\overline{F}_2$ in (4.6) is given by

$$\begin{aligned} \overline{F}_2&= -5760 p_1^3p_3^7 x_2^6+17280 p_1^3{p^6_ 3}p_5x_2^6-17280 p_1^3p_3^5{p^2_{{5 }}}x_2^6+5760 p_1^3p_3^{4}p_5^3{x^6_{ 2}} -46080 p_1^3p_3^7 x_2^5 \\&\quad +128640 {p^3_{{ 1}}}p_3^6p_5x_2^5-109440 p_1^3{p^5_{3 }}p_5^2x_2^5+17280 p_1^3p_3^{4}{p^3_{ 5}}x_2^5 +9600 p_1^3p_3^3p_5^{4}{x^5 _2}\\&\quad -207360 p_1^3p_3^7 x_2^{4}+539520 {p^3 _1}p_3^6p_5x_2^{4}-377760 p_1^3{p^5_{ 3}}p_5^2x_2^{4}-30240 p_1^3p_3^{4}{p^3_5}x_2^{4}\\&\quad +72480 p_1^3p_3^3p_5^{4}x_2^{4}+3360 p_1^3p_3^2p_5^5x_2 ^{4}-737280 p_1^3p_3^7 x_2^3 +1839360 {p^3_1}p_3^6p_5x_2^3\\&\quad -1137600 p_1^3p_3^5p_5^2x_2^3-249600 p_1^3p_3^{4}{p^3_{{ 5}}}x_2^3+240000 p_1^3p_3^3p_5^{4}{ x^3_2}+46080 p_1^3p_3^2p_5^5x_2^{3}\\&\quad -960 p_1^3p_3p_5^6x_2^3+668160 {p^3_{1 }}p_3^6p_5x_2^2-1647840 p_1^3{p^5_3 }p_5^2x_2^2+1109280 p_1^3p_3^{4}{p^3_5}x_2^2\\&\quad +1070400 p_1^3p_3^3p_5^{ 4}x_2^2+192000 p_1^3p_3^2p_5^5{x^2_{2 }}+8160 p_1^3p_3p_5^6x_2^2 -480 {p^3_{{1}}}p_5^7 x_2^2\\&\quad +2419200 p_1^3p_3^6 p_5x_2-5400960 p_1^3p_3^5p_5^2x_2 +2088000 p_1^3p_3^{4}p_5^3x_2 +3012480 {p^3_{ 1}}p_3^3p_5^{4}x_2\\&\quad +640320 p_1^3{p^2_{{3 }}}p_5^5x_2+40320 p_1^3p_3p_5^6x_ 2+714240 p_1^2p_3^6p_4x_2^2 -2434560 p_1^2p_3^5p_4p_5x_2^2\\&\quad +3240960 {p^2_{1 }}p_3^{4}p_4p_5^2x_2^2+2329920 {p^2_1 }p_3^3p_4p_5^3x_2^2 +337920 p_1^ 2p_3^2p_4p_5^{4}x_2^2 +10560 p_1^2 p_3p_4p_5^5x_2^2\\&\quad +737280 p_1^3{p^6_3 }p_5-958080 p_1^3p_3^5p_5^2 -944640 { p^3_1}p_3^{4}p_5^3 +1544160 p_1^3p_3 ^3p_5^{4}\\&\quad +904800 p_1^3p_3^2p_5^5+ 113760 p_1^3p_3p_5^6+2400 p_1^3p_5 ^7 +2626560 p_1^2p_3^6p_4x_2 -8083200 {p^2_{{1 }}}p^5_3p_4p_5x_2\\&\quad -714240 p_1^2{p^5_{3 }}p_5x_2^2 +6821760 p^2_1p^{4}_3p_4{ p^2_5}x_2 +2434560 p_1^2p_3^{4}p_5^2{x^2 _2}+6045120 p_1^2p_3^3p_4p_5^3x_{ 2}\\&\quad -3240960 p_1^2p_3^3p_5^3x_2^2+ 952320 p_1^2p_3^2p_4p_5^{4}x_2-2329920 p^2_1p^2_3p^{4}_5x_2^2 +35520 {p^2_1 }p_3p_4p^5_5x_2\\&\quad -337920 p_1^2p_3{p^5 _5}x_2^2-10560 p_1^2p_5^6x_2^{2 }-656640 p_1p_3^5p_4^2x_2^2 +2954880 p_{{ 1}}p_3^{4}p_4^2p_5x_2^2 \\&\quad +1736640 p_1{p^3_{ 3}}p_4^2p_5^2x_2^2+164160 p_1{p^2_{{3 }}}p_4^2p_5^3x_2^2 +1474560 p_1^2{ p^6_3}p_4 -3978240 p_1^2p_3^5p_4p_5\\&\quad -2626560 p_1^2p_3^5p_5x_2+2080320 p_1^{ 2}p_3^{4}p_4p_5^2 +8083200 p_1^2p_3^{4 }p_5^2x_2+3504960 p_1^2p_3^3p_4{p^3_{{5 }}}\\&\quad -6821760 p_1^2p_3^3p_5^3x_2+ 1043520 p_1^2p_3^2p_4p_5^{4} -6045120 {p^2_{{ 1}}}p_3^2p_5^{4}x_2+73920 p_1^2p_3p _4p_5^5\\&\quad -952320 p_1^2p_3p_5^5x_2- 35520 p_1^2p_5^6x_2 -2142720 p_1p_3^5 p_4^2x_2+6220800 p_1p_3^{4}p_4^2p_5 x_2 \\&\quad +1313280 p_1p_3^{4}p_4p_5x_2^2 + 3939840 p_1p_3^3p_4^2p_5^2x_2-5909760 p_1p_3^3p_4p_5^2x_2^2 \\&\quad +380160 p_{1 }p_3^2p_4^2p_5^3x_2 -3473280 p_1{p^2_{{3 }}}p_4p_5^3x_2^2-328320 p_1p_3p_4 p_5^{4}x_2^2+933120 p_3^{4}p_4^3x_2^2\\&\quad +466560 p_3^3p_4^3p_5x_2^2 -1474560 { p^2_1}p_3^5p_5+3978240 p_1^2p_3^{4}{p^2 _5}-2080320 p_1^2p_3^3p_5^3\\&\quad -3504960 p_1^2p_3^2p_5^{4} -1043520 p_1^2p_3{ p_5}^5-73920 p_1^2p_5^6-1175040 p_1{p^5_{3}}p_4^2\\&\quad +2799360 p_1p_3^{4}p_4^2p_5 + 4285440 p_1p_3^{4}p_4p_5x_2+2203200 p_1{p^3_ 3}p_4^2p_5^2-12441600 p_1p_3^3p_{ {4}}p_5^2x_2\\&\quad -656640 p_1p_3^3p_5^2{x^2_{ 2}}+371520 p_1p_3^2p_4^2p_5^3 -7879680 p_1p_3^2p_4p_5^3x_2 +2954880 p_{{ 1}}p_3^2p_5^3x_2^2\\&\quad -760320 p_1p_3p_{{4}}p_5^{4}x_2+1736640 p_1p_3p_5^{4}x_2^{2} +164160 p_1p_5^5x_2^2 +1866240 p_3^{4}{p_ 4}^3x_2 \\&\quad +933120 p_3^3p_4^3p_5x_2- 2799360 p_3^3p_4^2p_5x_2^2-1399680 {p^2_{{ 3}}}p_4^2p_5^2x_2^2 +2350080 p_1{p^{4}_{{3 }}}p_4p_5\\&\quad -5598720 p_1p_3^3p_4p_5^{2 }-2142720 p_1p_3^3p_5^2x_2-4406400 p_1{p^2 _3}p_4p_5^3 +6220800 p_1p_3^2p_5 ^3x_2\\&\quad -743040 p_1p_3p_4p_5^{4}+3939840 p_{1 }p_3p_5^{4}x_2+380160 p_1p_5^5x_2 + 933120 p_3^{4}p_4^3 \\&\quad +466560 p_3^3p_4^3p _5-5598720 p_3^3p_4^2p_5x_2-2799360 {p^2_{ 3}}p_4^2p_5^2x_2 +2799360 p_3^2p_{{4 }}p_5^2x_2^2\\&\quad +1399680 p_3p_4p_5^3{x^2_{{ 2}}}-1175040 p_1p_3^3p_5^2+2799360 p_1{p _3}^2p_5^3 +2203200 p_1p_3p_5^{4}\\&\quad +371520 p_1p_5^5-2799360 p_3^3p_4^2p_5- 1399680 p_3^2p_4^2p_5^2 +5598720 p_3^{2 }p_4p_5^2x_2 \\&\quad +2799360 p_3p_4p_5^3x_{{2 }}-933120 p_3p_5^3x_2^2-466560 p_5^{4}{x^2_{ 2}} +2799360 p_3^2p_4p_5^2 \\&\quad +1399680 p_3p _4p_5^3-1866240 p_3p_5^3x_2-933120 {p^{4}_{{5}}}x_2 -933120 p_3p_5^3-466560 p_5^{4}. \end{aligned}$$

$\widetilde{{\mathcal {R}}}_{11}$ and $\widetilde{{\mathcal {R}}}_{12}$ are given by

$$\begin{aligned} \widetilde{{\mathcal {R}}}_{11}&= p_3^2x_2^4+p_1 p_3^2x_2^2-p_1 p_3p_5x_2^2+p_3p_5x_2^3-p_3x_2^4 +2 p_1 p_3^2x_2-2 p_1 p_3p_5x_2-3 p_3^2x_2^2\\&\quad -p_5x_2^3+p_1 p_3p_5-p_1 p_5^2-2 p_3^2x_2-3 p_ 3p_5x_2+3 p_3x_2^2-2 p_3p_5+2 p_3x_2+3 p_5x_2+2 p_5,\\ \widetilde{{\mathcal {R}}}_{12}&= -5760 p_1^3p_3^7x_2^8+17280 p_1^3p_3^6p_5x_2^8-17280 p_1^3p_3^5p_5^2x_2^8 +5760 p_1^3p_3^4p_5^3x_2^8-57600 p_1^3p_3^7x_2^7\\&\quad +163200 p_1^3p_3^6p_5x_2^7-144000 p_1^3p_3^5p_5^2x_2^7 +28800 p_1^3p_3^4p_5^3x_2^7+9600 p_1^3p_3^3p_5^4x_2^7\\&\quad -305280 p_1^3p_3^7x_2^6+814080 p_1^3p_3^6p_5x_2^6 -613920 p_1^3p_3^5p_5^2x_2^6+10080 p_1^3p_3^4p_5^3x_2^6\\&\quad +91680 p_1^3p_3^3p_5^{ 4}x_2^6+3360 p_1^3p_3^2p_5^5x_2 ^6-1198080 p_1^3p_3^7x_2^5+3047040 p_1^3p_3^6p_5x_2^5\\&\quad -2002560 p_1^3{p_3^5 }p_5^2x_2^5-292800 p_1^3p_3^4p_5^3x_2^5+394560 p_1^3p_3^3p_5^4 x_2^5+52800 p_1^3p_3^2p_5^5x_2^5\\&\quad -960 p_1^3p_3p_5^6x_2^5-3985920 p_1^3p_3^7x_2^4+9809280 p_1^3p_3^6 p_5x_2^4-5805120 p_1^3p_3^5p_5^2x_2^4\\&\quad -1585920 p_1^3p_3^4p_5^3x_2 ^4+1284960 p_1^3p_3^3p_5^4x_2^4+ 276960 p_1^3p_3^2p_5^5x_2^4+6240 p_ 1^3p_3p_5^6x_2^4\\&\quad -480 p_1^3p_5 ^7x_2^4+4826880 p_1^2p_3^7x_2^5- 6944640 p_1^2p_3^6p_5x_2^5-232320 p_1^2p_3^5p_5^2x_2^5\\&\quad +2001600 p_1^2 p_3^4p_5^3x_2^5+337920 p_1^2p_3^ 3p_5^4x_2^5+10560 p_1^2p_3^2p_5^5x_2^5-3456000 p_1p_3^7x_2^6\\&\quad +1555200 p_1p_3^6p_5x_2^6+1736640 p_1p_3^5 p_5^2x_2^6+164160 p_1p_3^4p_5^3 x_2^6+933120 p_3^7x_2^7\\&\quad +466560 p_3^6 p_5x_2^7-7153920 p_1^3p_3^7x_2^3+ 17040000 p_1^3p_3^6p_5x_2^3-9012480 p_1^3p_3^5p_5^2x_2^3\\&\quad -3705600 p_1^3p_3^4p_5^3x_2^3+2101440 p_1^3p_3^3p_5^4x_2^3+686400 p_1^3p_3^2{p_ 5^5}x_2^3+45120 p_1^3p_3p_5^6x_2^3\\&\quad -960 p_1^3p_5^7x_2^3+15252480 p_1^2p_3^7x_2^4-16765440 p_1^2p_3^6p _5x_2^4-4826880 p_1^2p_3^6x_2^5\\&\quad - 8104320 p_1^2p_3^5p_5^2x_2^4+6944640 p_1^2p_3^5p_5x_2^5+5910720 p_1^{2 }p_3^4p_5^3x_2^4+232320 p_1^2p_3^4p_5^2x_2^5\\&\quad +3301440 p_1^2p_3^3p_5^4x_2^4-2001600 p_1^2p_3^3p_5^{ 3}x_2^5+394560 p_1^2p_3^2p_5^5x_2^4-337920 p_1^2p_3^2p_5^4x_2^5\\&\quad + 10560 p_1^2p_3p_5^6x_2^4-10560 p_1 ^2p_3p_5^5x_2^5-11854080 p_1p_3^7{x _{2}^5}-1779840 p_1p_3^6p_5x_2^5\\&\quad +6912000 p_1p_3^6x_2^6+9123840 p_1p_3^5{p_{{ 5}}^2}x_2^5-3110400 p_1p_3^5p_5x_2^{6 }+4181760 p_1p_3^4p_5^3x_2^5\\&\quad -3473280 p_{ {1}}p_3^4p_5^2x_2^6+328320 p_1p_3^{ 3}p_5^4x_2^5-328320 p_1p_3^3p_5^3 x_2^6+3732480 p_3^7x_2^6\\&\quad +4665600 p_3^{ 6}p_5x_2^6-2799360 p_3^6x_2^7+1399680 {p _3^5}p_5^2x_2^6-1399680 p_3^5p_5{x_ {2}^7}\\&\quad -5276160 p_1^3p_3^7x_2^2+10679040 p_1^3p_3^6p_5x_2^2-2003040 p_1^3{ p_3^5}p_5^2x_2^2-5465760 p_1^3p_3 ^4p_5^3x_2^2\\&\quad +765600 p_1^3p_3^3{p_{{ 5}}^4}x_2^2+1141920 p_1^3p_3^2p_5^5 x_2^2+156480 p_1^3p_3p_5^6x_2^2 + 1920 p_1^3p_5^7x_2^2\\&\quad +16876800 p_1^2{ p_3^7}x_2^3-7171200 p_1^2p_3^6p_5{x _{2}^3}-15252480 p_1^2p_3^6x_2^4-24866880 p_1^2p_3^5p_5^2x_2^3\\&\quad +16765440 {p_{{ 1}}^2}p_3^5p_5x_2^4+5308800 p_1^2{p_{{3 }}^4}p_5^3x_2^3+8104320 p_1^2p_3^4{ p_5^2}x_2^4+8340480 p_1^2p_3^3p_5 ^4x_2^3\\&\quad -5910720 p_1^2p_3^3p_5^3{x_{ 2}^4}+1455360 p_1^2p_3^2p_5^5x_2^{3 }-3301440 p_1^2p_3^2p_5^4x_2^4+56640 p_1^2p_3p_5^6x_2^3\\&\quad -394560 p_1^2 p_3p_5^5x_2^4-10560 p_1^2p_5^6{x_ {2}^4}-14515200 p_1p_3^7x_2^4-18109440 p_{{1 }}p_3^6p_5x_2^4\\&\quad +23708160 p_1p_3^6{x_{ 2}^5}+14428800 p_1p_3^5p_5^2x_2^4+ 3559680 p_1p_3^5p_5x_2^5-3456000 p_1{p_{ 3}^5}x_2^6\\&\quad +14878080 p_1p_3^4p_5^3{x_{ 2}^4}-18247680 p_1p_3^4p_5^2x_2^5 + 1555200 p_1p_3^4p_5x_2^6+3153600 p_1{p_{ 3}^3}p_5^4x_2^4\\&\quad -8363520 p_1p_3^3{p_{{ 5}}^3}x_2^5+1736640 p_1p_3^3p_5^2{x_{{2 }}^6} +164160 p_1p_3^2p_5^5x_2^4-656640 p_1p_3^2p_5^4x_2^5\\&\quad +164160 p_1{p_{{3 }}^2}p_5^3x_2^6+5598720 p_3^7x_2^5+ 13996800 p_3^6p_5x_2^5-11197440 p_3^6{x_ {2}^6} \\&\quad +8398080 p_3^5p_5^2x_2^5 -13996800 p_3^5p_5x_2^6+2799360 p_3^5x_2^7+ 1399680 p_3^4p_5^3x_2^5 \\&\quad -4199040 p_3^{4 }p_5^2x_2^6+1399680 p_3^4p_5x_2^7 -1474560 p_1^3p_3^7x_2+1420800 p_1^3{p_{3}^6}p_5x_2\\&\quad +3320640 p_1^3p_3^5p_5^{ 2}x_2-3913920 p_1^3p_3^4p_5^3x_2 - 276480 p_1^3p_3^3p_5^4x_2+760320 {p_{{1} }^3}p_3^2p_5^5x_2\\&\quad +158400 p_1^3p_3{p _5^6}x_2+4800 p_1^3p_5^7x_2+7925760 {p _1^2}p_3^7x_2^2 +7472640 p_1^2p_3^ 6p_5x_2^2\\&\quad -16876800 p_1^2p_3^6x_2^ 3-25489920 p_1^2p_3^5p_5^2x_2^2 + 7171200 p_1^2p_3^5p_5x_2^3 -449280 {p_{{1 }}^2}p_3^4p_5^3x_2^2\\&\quad +24866880 p_1^2 p_3^4p_5^2x_2^3+8004480 p_1^2{p_3 ^3} p_5^4x_2^2-5308800 p_1^2p_3^3{p_ 5^3}x_2^3 +2380800 p_1^2p_3^2p_5^{5}x_2^2 \\&\quad -8340480 p_1^2p_3^2p_5^4{x_{{2 }}^3}+155520 p_1^2p_3p_5^6x_2^2-1455360 p_1^2p_3p_5^5x_2^3 -56640 p_1^2{ p_5^6}x_2^3\\&\quad -7292160 p_1p_3^7x_2^3- 27008640 p_1p_3^6p_5x_2^3+29030400 p_1{p _3^6}x_2^4 +3490560 p_1p_3^5p_5^2{x_ {2}^3}\\&\quad +36218880 p_1p_3^5p_5x_2^4 -11854080 p_1p_3^5x_2^5+21496320 p_1p_3^4{p_{ 5}^3}x_2^3 -28857600 p_1p_3^4p_5^2{x_{2}^4}\\&\quad -1779840 p_1p_3^4p_5x_2^5+8605440 p _1p_3^3p_5^4x_2^3-29756160 p_1{p_{3 }^3}p_5^3x_2^4 +9123840 p_1p_3^3{p_5^2}x_2^5\\&\quad +708480 p_1p_3^2p_5^5x_2^ 3-6307200 p_1p_3^2p_5^4x_2^4+4181760 p _1p_3^2p_5^3x_2^5 -328320 p_1p_3{p _5^5}x_2^4\\&\quad +328320 p_1p_3p_5^4x_2^ 5+3732480 p_3^7x_2^4+18662400 p_3^6p_5 x_2^4-16796160 p_3^6x_2^5 \\&\quad +19595520 p_3 ^5p_5^2x_2^4-41990400 p_3^5p_5x_2 ^5+11197440 p_3^5x_2^6+6531840 p_3^4{p_{{5 }}^3}x_2^4 \\&\quad -25194240 p_3^4p_5^2x_2^5 +13996800 p_3^4p_5x_2^6-933120 p_3^4{x_{2}^7}+466560 p_3^3p_5^4x_2^4\\&\quad -4199040 {p_ 3^3}p_5^3x_2^5 +4199040 p_3^3p_5^{ 2}x_2^6-466560 p_3^3p_5x_2^7-737280 {p_{{1}}^3}p_3^6p_5\\&\quad +1845120 p_1^3p_3^5{p_{{ 5}}^2}-1158720 p_1^3p_3^4p_5^3 -224160 {p_{ {1}}^3}p_3^3p_5^4 +232800 p_1^3p_3^2 p_5^5+39840 p_1^3p_3p_5^6\\&\quad +2400 p_1 ^3p_5^7+1474560 p_1^2p_3^7x_2 +6297600 p_1^2p_3^6p_5x_2-7925760 p_1^2{p_{3}^6}x_2^2\\&\quad -10123200 p_1^2p_3^5p_5^2x _{2}-7472640 p_1^2p_3^5p_5x_2^2 -2568000 p_1^2p_3^4p_5^3x_2+25489920 p_1^{ 2}p_3^4p_5^2x_2^2\\&\quad +3125760 p_1^2{p_{{3 }}^3}p_5^4x_2+449280 p_1^2p_3^3{p_5 ^3}x_2^2+1609920 p_1^2p_3^2p_5^5x_{ 2} -8004480 p_1^2p_3^2p_5^4x_2^2\\&\quad + 183360 p_1^2p_3p_5^6x_2-2380800 p_1^{2 }p_3p_5^5x_2^2 -155520 p_1^2p_5^6{x_2^2}-1175040 p_1p_3^7x_2^2\\&\quad -14584320 p_{{ 1}}p_3^6p_5x_2^2 +14584320 p_1p_3^6{x_ {2}^3} -9668160 p_1p_3^5p_5^2x_2^2 +54017280 p_1p_3^5p_5x_2^3\\&\quad -14515200 p_1{p _3^5}x_2^4 +14264640 p_1p_3^4p_5^3{x _{2}^2}-6981120 p_1p_3^4p_5^2x_2^3-18109440 p_1p_3^4p_5x_2^4\\&\quad +9866880 p_1{p_ 3^3}p_5^4x_2^2-42992640 p_1p_3^3{p_ 5^3}x_2^3 +14428800 p_1p_3^3p_5^2{x_ {2}^4}+1296000 p_1p_3^2p_5^5x_2^2\\&\quad - 17210880 p_1p_3^2p_5^4x_2^3+14878080 p_{ {1}}p_3^2p_5^3x_2^4 -1416960 p_1p_3{p_ 5^5}x_2^3+3153600 p_1p_3p_5^4x_2^4\\&\quad +164160 p_1p_5^5x_2^4+933120 p_3^7{x_ {2}^3} +11664000 p_3^6p_5x_2^3-11197440 {p_{{ 3}}^6}x_2^4+22394880 p_3^5p_5^2x_2^{3}\\&\quad -55987200 p_3^5p_5x_2^4 +16796160 p_3^5{ x_2^5}+12130560 p_3^4p_5^3x_2^3- 58786560 p_3^4p_5^2x_2^4\\&\quad +41990400 p_3^ 4p_5x_2^5 -3732480 p_3^4x_2^6+1866240 {p_3^3}p_5^4x_2^3-19595520 p_3^3{p_5 ^3}x_2^4\\&\quad +25194240 p_3^3p_5^2x_2^5 - 4665600 p_3^3p_5x_2^6-1399680 p_3^2{p_{{ 5}}^4}x_2^4+4199040 p_3^2p_5^3x_2^5\\&\quad -1399680 p_3^2p_5^2x_2^6 +1474560 p_1^{ 2}p_3^6p_5-1474560 p_1^2p_3^6x_2- 1628160 p_1^2p_3^5p_5^2\\&\quad -6297600 p_1^{2 }p_3^5p_5x_2 -719040 p_1^2p_3^4{p_{5 }^3}+10123200 p_1^2p_3^4p_5^2x_2+498240 p_1^2p_3^3p_5^4\\&\quad +2568000 p_1^2{p_{{3 }}^3}p_5^3x_2+300480 p_1^2p_3^2{p_5 ^5}-3125760 p_1^2p_3^2p_5^4x_2+73920 {p_1^2}p_3p_5^6\\&\quad -1609920 p_1^2p_3{p_5 ^5}x_2-183360 p_1^2p_5^6x_2-2350080 p_{{1} }p_3^6p_5x_2+2350080 p_1p_3^6x_2^{2}\\&\quad -7292160 p_1p_3^5p_5^2x_2+29168640 p_1{ p_3^5}p_5x_2^2-7292160 p_1p_3^5{x_2 ^3}+3628800 p_1p_3^4p_5^3x_2\\&\quad +19336320 p_{{ 1}}p_3^4p_5^2x_2^2-27008640 p_1p_3^ 4p_5x_2^3+4890240 p_1p_3^3p_5^4x_{{ 2}}-28529280 p_1p_3^3p_5^3x_2^2\\&\quad +3490560 p_1p_3^3p_5^2x_2^3+1123200 p_1{p_{3 }^2}p_5^5x_2-19733760 p_1p_3^2p_5^4 x_2^2+21496320 p_1p_3^2p_5^3x_2^3\\&\quad -2592000 p_1p_3p_5^5x_2^2+8605440 p_1p_{ 3}p_5^4x_2^3+708480 p_1p_5^5x_2^{3}+2799360 p_3^6p_5x_2^2\\&\quad -2799360 p_3^6{x _{2}^3} +12597120 p_3^5p_5^2x_2^2-34992000 p_3^5p_5x_2^3+11197440 p_3^5x_2^{4}\\&\quad +11197440 p_3^4p_5^3x_2^2-67184640 {p_{3}^4} p_5^2x_2^3+55987200 p_3^4p_5{x_{2}^4}-5598720 p_3^4x_2^5\\&\quad +2799360 p_3^3{p_{{ 5}}^4}x_2^2-36391680 p_3^3p_5^3 x_2^{3}+58786560 p_3^3p_5^2x_2^4-13996800 {p_3^3}p_5x_2^5\\&\quad -5598720 p_3^2p_5^4x_2 ^3+19595520 p_3^2p_5^3x_2^4-8398080 {p_{{3 }}^2}p_5^2x_2^5+1399680 p_3p_5^4{x_{2}^4}\\&\quad -1399680 p_3p_5^3x_2^5-1474560 p_1^{ 2}p_3^5p_5+1628160 p_1^2p_3^4p_5^{2 }+719040 p_1^2p_3^3p_5^3-498240 p_1^{2}p_3^2p_5^4\\&\quad -300480 p_1^2p_3p_5^5- 73920 p_1^2p_5^6-1175040 p_1p_3^5{p_{5 }^2}+4700160 p_1p_3^5p_5x_2-1175040 p_1{p_3^5}x_2^2\\&\quad +14584320 p_1p_3^4p_5^2x_{ 2}-14584320 p_1p_3^4p_5x_2^2+803520 p_{{1} }p_3^3p_5^4-7257600 p_1p_3^3p_5^3x_2\\&\quad -9668160 p_1p_3^3p_5^2x_2^2+371520 p_1p_3^2p_5^5-9780480 p_1p_3^2{p_{{ 5}}^4}x_2+14264640 p_1p_3^2p_5^3x_2^{2}\\&\quad -2246400 p_1p_3p_5^5x_2+9866880 p_1p_3 p_5^4x_2^2+1296000 p_1p_5^5x_2^2+2799360 p_3^5p_5^2x_2\\&\quad -8398080 p_3^5p_{{5 }}x_2^2+2799360 p_3^5x_2^3+5132160 p_3 ^4p_5^3x_2-37791360 p_3^4p_5^2x_2^2\\&\quad +34992000 p_3^4p_5x_2^3-3732480 p_3^{4 }x_2^4+1866240 p_3^3p_5^4x_2-33592320 {p_3^3}p_5^3x_2^2\\&\quad +67184640 p_3^3p_5 ^2x_2^3-18662400 p_3^3p_5x_2^4-8398080 p_3^2p_5^4x_2^2+36391680 p_3^2{p_{{5}}^3}x_2^3\\&\quad -19595520 p_3^2p_5^2x_2^{4 }+5598720 p_3p_5^4x_2^3-6531840 p_3p_5^3x_2^4-466560 p_5^4x_2^4\\&\quad +2350080 p_1{ p_3^4}p_5^2-2350080 p_1p_3^4p_5x_2- 7292160 p_1p_3^3p_5^2x_2-1607040 p_1{p_{3}^2}p_5^4\\&\quad +3628800 p_1p_3^2p_5^3x_{{2 }}-743040 p_1p_3p_5^5+4890240 p_1p_3{p_5^4}x_2+1123200 p_1p_5^5x_2\\&\quad +933120 p_3^{4 }p_5^3-8398080 p_3^4p_5^2x_2+8398080 {p_3^4}p_5x_2^2-933120 p_3^4x_2^3+466560 p_3^3p_5^4\\&\quad -15396480 p_3^3p_5^{3 }x_2+37791360 p_3^3p_5^2x_2^2-11664000 {p_3^3}p_5x_2^3-5598720 p_3^2p_5^4x_{2}\\&\quad +33592320 p_3^2p_5^3x_2^2-22394880 {p_{3}^2} p_5^2x_2^3+8398080 p_3p_5^4{x_{{2}}^2}-12130560 p_3p_5^3x_2^3\\&\quad -1866240 {p_5 ^4}x_2^3-1175040 p_1p_3^3p_5^2+803520 p_1p_3p_5^4+371520 p_1p_5^5-2799360 {p_{3}^3}p_5^3\\&\quad +8398080 p_3^3p_5^2x_2- 2799360 p_3^3p_5x_2^2-1399680 p_3^2{p_{{5}}^4}+15396480 p_3^2p_5^3x_2\\&\quad -12597120 {p_{3 }^2}p_5^2x_2^2+5598720 p_3p_5^4x_2- 11197440 p_3p_5^3x_2^2-2799360 p_5^4{x^2_{2}}\\&\quad +2799360 p_3^2p_5^3-2799360 p_3^2{p _5^2}x_2+1399680 p_3p_5^4-5132160 p_3{p^3_{{5}}}x_2-1866240 p_5^4x_2\\&\quad -933120 p_3p_5 ^3-466560 p_5^4. \end{aligned}$$