带有外部治疗的癌症模型的动力学分析.pdf

Advances in Applied Mathematics A?,2023,12(8),3684-3708Published Online August 2023 in Hans.https:/www.hanspub.org/journal/aamhttps:/doi.org/10.12677/aam.2023.128365k?Jw?.?CCCCCC“?vF2023c7?21FF2023c8?13FuF2023c8?24F,Jw%?a)x?-;,u?S3un-:5?K.?|X|n?5Jw?a1,?/?Jw?|y?,5u?J4?.3?x5?A?Jw?.?1,?:?a.,y?2?Hopf|2?Bogdanov-Takens|?35.?L?y?(.cJwO?.-5Allee?AHopf|Bogdanov-Takens|Dynamic Analysis of Cancer Model withExternal TherapyYingying SunSchool of Mathematics,Hangzhou Normal University,Hangzhou ZhejiangReceived:Jul.21st,2023;accepted:Aug.13th,2023;published:Aug.24th,2023:CC.k?Jw?.?J.A?,2023,12(8):3684-3708.DOI:10.12677/aam.2023.128365CCAbstractAs we all know,cancer has always been one of the major diseases threatening lifeand health,and its intrinsic pathogenesis is an important research topic that manyscientists focus on.In this paper,we will make full use of bifurcation theory ofdynamical systems and numerical simulations to study various dynamics related tocancer,striving to fully reveal the medical significance of bifurcation related to cancer,and provide positive help for predicting the future development trend of the disease.We mainly study the dynamics of a two-dimensional cancer model involving tumorcells and immune effector cells.We obtain the number and type of equilibrium points,verify the Hopf bifurcation of codimension 2 and the Bogdanov-Takens bifurcation ofcodimension 2.Finally,the conclusions obtained are verified by numerical simulations.KeywordsCancer Growth Model,Stability,Allee Effect,Hopf Bifurcation,Bogdanov-TakensBifurcationCopyright c?2023 by author(s)and Hans Publishers Inc.This work is licensed under the Creative Commons Attribution International License(CC BY 4.0).http:/creativecommons.org/licenses/by/4.0/1.0?Jw-.5?K,5?O?N?|.?,?*!E,.,!z?!-?U?J.+?K)?,?Jw?E3X.3L?Acp,u?)!u?1?,?E,XdEu).5Xm?pE,?y,?N?5.?Khajanchi?.:E0?,0?,k:E(x,y)kXe/?:J(E)=+y+y y(x)+x+yxy(+y)2yx y+(1 y)!,Kdet(J(E)=xy2+x(+y)2+(+y)(2y 1)+)(+y)(+y)y)(+y)2,tr(J(E)=x 2y+y+y y .DOI:10.12677/aam.2023.1283653686A?CCXJdet(J(E)6=0,oE(x,y)z:;XJdet(J(E).:E0?,0?,N/?en.n1 X(2)k.:E0?,0?.XJ+,oE0V-(:;XJ+=,oE0z:.n2 e+=,KE0?,0?z:,d?,(1)XJ 6=2,oE0?,0?Q(:,)-?/;(2)XJ=2,(a)XJ=()2,oE0?,0?Q(:,)-?/;(b)XJ()2 0,oE0?,0?zQ:;(c)XJ0 ()2,oE0?,0?-?z(:.y?+=,kdet(J(E0)=0 tr(J(E0)=(+)+.k,-(u,v)=?x,y?E0?,0?:,X(2)C?dudt=?+u?v+v+v?,dvdt=v?+u?+v(1 v)v.(4)?XXeC:u=?X+Y,v=X,t=1,E,t 5L,X(4)3?:NC?1?Vm,kdXdt=a20X2+a11XY+f(X,Y),dYdt=Y+b20X2+b11XY+b30X3+b21X2Y+b40X4+b31X3Y+g(X,Y),(5)f,g u(X,Y)?1w,a20=+()2,a11=1,b20=(2()(+)+)()2)32,b11=(2+)()2,b30=(+)3,b21=2,b40=(+)+2Y)4,b31=23.n5 2,?6=2,E0?,0?Q(:,?-?/.XJ=2,ka20=0.%6/n,b?Y=m1X2+m2X3+o(|X|3),ODOI:10.12677/aam.2023.1283653687A?CCX(5)?1?.LX(5)?1,?m1=()2)(),m2=()2)+2 ()()2 2)2().Y=m1X2+m2X3+o(|X|3)X(5)?1,?dXdt=()2)()X3+O?X3?.?=()2,=()2“X(5)?1,?dXdt=()2X4+O(X)4.n5 2?1,kmC=t,uy?0 ()2 0 ,oE0?,0?(:;?0 ()2()2 0 ,E0?,0?Q:;?=()2,E0?,0?Q(:.R2+=(x,y)|x 0,y 0?X?Ky:E0?,0?.?/,dV-/|.d?,du()2 0,o?/3m.?e5X(2)?:?a.XJE(x,y)X(2)?:,Kx x3+bx2+cx+d=0(6)?.pb=(+)2+2,c=(+1)(+)+)(+)+2)+2,d=(+).?:?a.,-f(x)=x3+bx2+cx+d,f0(x)=3x2+2bx+c.df(x)=0,?=?x x(+x+)(+1)+x+?.(7)DOI:10.12677/aam.2023.1283653688A?CCX(2)3:E(x,y)?zJ(E)=+(+1)+x+(+x+)x?22(+1)+x+)2?+x+x+,KJ(E)?1?,O:det(J(E)=A(+x+)(+1)+x+)2,tr(J(E)=x+(+1)+x+(+x+),A=2x3(+)?2(+1)(+)(+)+2)+2?+2x(+)(+)+2)2)+x2(5 (+4 )+5).(7)“det(J(E),Kdet(J(E)-#Ldet(J(E)=Bf0(x)+f(x)(x )(+1)+x+)2=B(+1)+x+)2f0(x),(8)B=x2+(2 (+2)x+()(+).n?“?,42=22,3=422?k23 3k1k2?+14(k2k3 9k1)2.(9)p2=2?2?2(+2)+()2?+(+)+2?(+)+3+2)+22,k1=(+),k2=(+)+2)2,k3=2(+1)(+)(+)+2)+2.(10)du(6)?dT?,d3?X(2)?:?a.,e:(+1)(+1).DOI:10.12677/aam.2023.1283653689A?CC(I).:E0?,0?,kn?:.1?f(x)?”,ke?n:n3?(+1)x +,X(2)?k?:,kn?:.d?,77n:(1)?3 0 7?IV,X(2)kn?:(a)?ng?n,?b 0,X(2)kn?:E2V-Q:,XJtr(J(Ei)0,oEi(xi,yi)(i=1,3)V-?(:?:;XJtr(J(Ei)=0,oEi(xi,yi)(i=1,3)f?:%,0 x1 x2 x3 +x1 x2 x3.(1(a).(b)3e,X(2)k?:?K:.XJtr(J(E3)0,KE3V-(:?:;XJtr(J(E3)=0,KE3f?:%.duEi(xi,yi)(i=1,2)K:,?.(1(f).(2)?3=0,(a)e2 0,of(x)kn,x?-.i.?b 0,X(2)k?:z:E(x,y)?:E1(x1,y1)(E3(x3,y3).XJtr(J(Ei)0,KE1(x1,y1)(E3(x3,y3)V-(:?:;XJtr(J(Ei)=0,KE1(x1,y1)(E3(x3,y3)f?:%,x1 x(1(b)x1 x3(1(c).ii.XJc 0,X(2)k?:E3(x3,y3).XJtr(J(E3)0,KE3(x3,y3)V-(:?:;XJtr(J(E3)=0,oE3(x3,y3)f?:%.duE(x,y)K:,?.,x 0,f(x)k?E.,Xk?:E3(x3,y3).XJtr(J(E3)0,KE3(x3,y3)V-(:?:;XJtr(J(E3)=0,oE3(x3,y3)f?:%.Ei(xi,yi)(i=1,2)3E,?.(1(e).y d(8),?x +(x2+(2(+2)x+()(+)(+1)+x+)2 0,kf0(x)0,f0(x)0=det(J(E)0 f0(x)=0=det(J(E)=0.DOI:10.12677/aam.2023.1283653690A?CCNwdet(J(Ei)0,det(J(E)=0,det(J(E)=0,dE1,E2E3?:(Ei),(i=1,3)V-Q:,?EEz:.Figure 1.Roots of f(x)=0 when (+1).(a)Three single positive roots x1,x2,x3.(b)(c)Twopositive roots:a double root xand a single rootx1(or x3).(d)A unique triple positive root x.(e)A realroot and a pair of conjugate complex roots.(f)Two negative roots x1,x2and a positive root x3.(g)A doublenegtive rootxand a positive root x3 1.?(+1),f(x)=0?.(a)n?x1,x2,x3.(b)(c)?:?-x?x1(x3).(d)?n-x.(e)?E.(f)Kx1,x2?x3.(g)K?-x?x3n4?(+1)x +,X(2)?k?:,kn?:.d?,77n(1)?3 0,KX(2)kn?:(a)?ng?n,?b 0,X(2)kn?:Ei(xi,yi)(i=1,3)-?V-Q:.XJtr(J(Ei)0,oE2V-?(:?:;XJtr(J(Ei)=0,oE2f?:%,x1 x2 x3 0,of(x)kn,x?-.i.?b 0,X(2)k?:z:E(x,y)V-Q:E1(x1,y1)(E3(x3,y3);XJtr(J(Ei)0 KE1(x1,y1)(E3(x3,y3)V-(:?:;XJtr(J(Ei)=0,KE1(x1,y1)(E3(x3,y3)f?:%:,x1 x(1(b)x1 x3(1(c).ii.XJc 0,X(2)k?:E3(x3,y3),E3V-Q:.duE(x,y)K:,vk?,x 0,X(2)k?E.,Xk?:E3(x3,y3),V-Q:.Ei(xi,yi)(i=1,2)3E,?.(1(e).y d(8),?x +,(x2+(2(+2)x+()(+)(+1)+x+)2 0,kf0(x)0=det(J(E)0=det(J(E)0;f0(x)=0=det(J(E)=0.Nwdet(J(Ei)0,(i=1,3),det(J(E2)0,det(J(E)=0,det(J(E)=0 E1,E2E3?:kE2V-Q:,?EEz:.(II)(+1)n5?(+1),X(2)k?:.d?,77n:(1)?3 0,X(2)kn?:(a)?b 0 c 0,f(x)kn?K,X(2)vk?:(2(a).(b)?b 0 c 0,X(2)k?:Ei(xi,yi)(i=2,3).(2(b).i.?x +,E2(x2,y2)V-Q:,XJtr(J(E3)0,oE3(x3,y3)V-(:?:;XJtr(J(E3)=0,oE3(x3,y3)f?:%,x1 0 x2 x3 x1 0 +x2 x3.ii.?x +,E3(x3,y3)V-Q:,XJtr(J(E2)0,oE2(x2,y2)V-(:?:;XJtr(J(E2)=0,oE2(x2,y2)f?:%,x1 0 x2 x3 0,of(x)kn,x?-.i.?c 0 b 0,X(2)vk?:.(2(c)2(d).ii.?b 0 c 0,Kf(x)kK?E.,X(2)vk?:.(2(g).y d(8),?x +(x2+(2(+2)x+()(+)(+1)+x+)2 0,kf0(x)0,f0(x)0=det(J(E)0 f0(x)=0=det(J(E)=0.NDOI:10.12677/aam.2023.1283653692A?CCwdet(J(E2)0,det(J(E)=0,dE2E3?:k(E2)V-Q:,Ez:.?x +,(x2+(2(+2)x+()(+)(+1)+x+)2 0,kf0(x)0=det(J(E)0=det(J(E)0;f0(x)=0=det(J(E)=0.Nwdet(J(E2)0,det(J(E3)0,?A?1,2 uJ.d?,XJA?1,2=i,=h1p2 h2p1.e?n?)Hopf|?.n3 b?F(x)=0,tr(J(E)=0,XJ1oX1 0(1.:-(:(1.603,0).dX(2)?|,X?A?O,?:?dnC(Hopf|:),L5?Ay?5?.5.o(3?,?x5?A?Jw?.?1,?DOI:10.12677/aam.2023.1283653703A?CCFigure 4.The phase portraits of I IV in Figure 3 4.3I IV?:?a.,2?Hopf|,2?Q(|BT|.?L?y?(.?J?,duJw?.?14E,?n?(J?-0,K7II:x1=b?3Y1+3Y2?3a,x2,3=b+?12?3Y1+3Y2?123i?3Y13Y2?3a,(33)Y1,2=Ab+3a2?B B2 4AC?,i2=1.?=B2 4AC=0,7III:x1=ba+K,x2=x3=K2,(34)K=BA,(A 6=0).DOI:10.12677/aam.2023.1283653706A?CC?=B2 4AC 0,1 T 0(d,7II)K).7n5:?A 0(d,7II)K).7n6:?=0,eA=0,K7kB=0(d,7I)K).7n7:?=0,eB 6=0,7III3A 0?(d,7III)K).7n8:?0,7IV 3A 0?(d,7IV)K).7n9:?0,7IV 3T 1T 1?,=T7v1 T 1.n6Zhang et al.2uXdxdt=P2(x,y)dydt=y+Q2(x,y),(39)DOI:10.12677/aam.2023.1283653707A?CCb?O(0,0)X(39)?.:,S(O)O(0,0)NC?,?P2,Q2?u2?S(O)?).d,uv?,3)(x)v(x)+Q2(x,(x)0,|x|0,KO(0,0)-?(:.2.XJm,am 0(0),K?3m().DOI:10.12677/aam.2023.1283653708A?