Computing the left and right eigenvectors

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Question:Computing the left and right eigenvectors

Posted: Elisha 50 Product: Maple 2018

April 29 2025

Someone please help me with the computation of the right and left eigenvectors. my system of equation is attached below

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Limit(N(t) = N__0*exp(-mu*t)+exp(mu*t)*K/mu, t = infinity)

limit(N(t), t = infinity) = limit(N__0*exp(-mu*t)+exp(mu*t)*K/mu, t = infinity)

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>	#to calculate the disease free equilibrium,

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>	$solve({E1 = 0, E3 = 0}, {S, S__A})$

${S = Lambda__p/Âµ__C, S__A = Lambda__A/Âµ__A}$

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>	#to calculate the Endemic Equilibrium state,

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E1 := `Λ__p`-(`ϕ`*`θ__B`*I__A/N__p+µ__C)*S+`ω__B`*I__B

Lambda__p-(varphi*theta__B*I__A/N__p+Âµ__C)*S+omega__B*I__B

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E2 := `ϕ`*`θ__B`*I__A*S/N__p-`ω__B`*I__B-(`σ__B`+µ__C)*I__B

varphi*theta__B*I__A*S/N__p-omega__B*I__B-(sigma__B+Âµ__C)*I__B

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E3 := `Λ__A`-(µ__A+`ϕ`*`α__B`*I__B/N__p)*S__A+`δ__A`*I__A

Lambda__A-(Âµ__A+varphi*alpha__B*I__B/N__p)*S__A+delta__A*I__A

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E4 := `ϕ`*`α__B`*I__B*S__A/N__p-(µ__A+`δ__A`)*I__A

varphi*alpha__B*I__B*S__A/N__p-(Âµ__A+delta__A)*I__A

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>	$solve({E1 = 0, E2 = 0, E3 = 0, E4 = 0}, {I__A, I__B, S, S__A})$

${I__A = 0, I__B = 0, S = Lambda__p/Âµ__C, S__A = Lambda__A/Âµ__A}, {I__A = -(N__p^2*Âµ__A^2*Âµ__C^2+N__p^2*Âµ__A^2*Âµ__C*omega__B+N__p^2*Âµ__A^2*Âµ__C*sigma__B+N__p^2*Âµ__A*Âµ__C^2*delta__A+N__p^2*Âµ__A*Âµ__C*delta__A*omega__B+N__p^2*Âµ__A*Âµ__C*delta__A*sigma__B-varphi^2*Lambda__A*Lambda__p*alpha__B*theta__B)/(varphi*Âµ__A*theta__B*(N__p*Âµ__A*Âµ__C+N__p*Âµ__A*sigma__B+N__p*Âµ__C*delta__A+N__p*delta__A*sigma__B+varphi*Lambda__p*alpha__B)), I__B = -(N__p^2*Âµ__A^2*Âµ__C^2+N__p^2*Âµ__A^2*Âµ__C*omega__B+N__p^2*Âµ__A^2*Âµ__C*sigma__B+N__p^2*Âµ__A*Âµ__C^2*delta__A+N__p^2*Âµ__A*Âµ__C*delta__A*omega__B+N__p^2*Âµ__A*Âµ__C*delta__A*sigma__B-varphi^2*Lambda__A*Lambda__p*alpha__B*theta__B)/(alpha__B*(N__p*Âµ__A*Âµ__C^2+N__p*Âµ__A*Âµ__C*omega__B+N__p*Âµ__A*Âµ__C*sigma__B+varphi*Âµ__C*Lambda__A*theta__B+varphi*Lambda__A*sigma__B*theta__B)*varphi), S = (N__p*Âµ__A*Âµ__C+N__p*Âµ__A*sigma__B+N__p*Âµ__C*delta__A+N__p*delta__A*sigma__B+varphi*Lambda__p*alpha__B)*Âµ__A*N__p*(Âµ__C+omega__B+sigma__B)/(alpha__B*varphi*(N__p*Âµ__A*Âµ__C^2+N__p*Âµ__A*Âµ__C*omega__B+N__p*Âµ__A*Âµ__C*sigma__B+varphi*Âµ__C*Lambda__A*theta__B+varphi*Lambda__A*sigma__B*theta__B)), S__A = N__p*(N__p*Âµ__A^2*Âµ__C^2+N__p*Âµ__A^2*Âµ__C*omega__B+N__p*Âµ__A^2*Âµ__C*sigma__B+N__p*Âµ__A*Âµ__C^2*delta__A+N__p*Âµ__A*Âµ__C*delta__A*omega__B+N__p*Âµ__A*Âµ__C*delta__A*sigma__B+varphi*Âµ__A*Âµ__C*Lambda__A*theta__B+varphi*Âµ__A*Lambda__A*sigma__B*theta__B+varphi*Âµ__C*Lambda__A*delta__A*theta__B+varphi*Lambda__A*delta__A*sigma__B*theta__B)/(varphi*Âµ__A*theta__B*(N__p*Âµ__A*Âµ__C+N__p*Âµ__A*sigma__B+N__p*Âµ__C*delta__A+N__p*delta__A*sigma__B+varphi*Lambda__p*alpha__B))}$

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Matrix(%id = 18446746854857131062)

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$J := Matrix(4, 4, {(1, 1) = -`ϕ`*`θ__B`*I__A/N__p-µ__C, (1, 2) = `ω__B`, (1, 3) = 0, (1, 4) = -`ϕ`*`θ__B`*S/N__p, (2, 1) = `ϕ`*`θ__B`*I__A/N__p, (2, 2) = -`ω__B`-`σ__B`-µ__C, (2, 3) = 0, (2, 4) = `ϕ`*`θ__B`*S/N__p, (3, 1) = 0, (3, 2) = -`ϕ`*`α__B`*S__A/N__p, (3, 3) = -µ__A-`ϕ`*`α__B`*I__B/N__p, (3, 4) = `δ__A`, (4, 1) = 0, (4, 2) = `ϕ`*`α__B`*S__A/N__p, (4, 3) = `ϕ`*`α__B`*I__B/N__p, (4, 4) = -µ__A-`δ__A`})$

Matrix(%id = 18446746579340105118)

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S := `Λ__p`/µ__C

Lambda__p/Âµ__C

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S__A := `Λ__A`/µ__A

Lambda__A/Âµ__A

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Matrix(%id = 18446746579340107518)

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>	$J := Matrix(4, 4, {(1, 1) = -µ__C, (1, 2) = `ω__B`, (1, 3) = 0, (1, 4) = -`β__1`, (2, 1) = 0, (2, 2) = -`ω__B`-`σ__B`-µ__C, (2, 3) = 0, (2, 4) = -`β__1`, (3, 1) = 0, (3, 2) = -`β__2`, (3, 3) = -µ__A, (3, 4) = `δ__A`, (4, 1) = 0, (4, 2) = `β__2`, (4, 3) = 0, (4, 4) = -µ__A-`δ__A`})$

Matrix(%id = 18446746579417403630)

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Matrix(%id = 18446746579305905318)

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Vector[column](%id = 18446746579445964182)

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Vector[column](%id = 18446746579340117046)

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# to find the trace we

$Matrix(7, 7, {(1, 1) = -beta*lambda-v__1-µ, (1, 2) = v__2, (1, 3) = 0, (1, 4) = 0, (1, 5) = 0, (1, 6) = 0, (1, 7) = 0, (2, 1) = v__1, (2, 2) = beta*(w-1)*lambda-µ-v__2-alpha, (2, 3) = 0, (2, 4) = 0, (2, 5) = 0, (2, 6) = 0, (2, 7) = 0, (3, 1) = 0, (3, 2) = alpha, (3, 3) = -µ, (3, 4) = 0, (3, 5) = `ρ__A`, (3, 6) = `ρ__F`, (3, 7) = -(-1+k)*`ρ__Q`, (4, 1) = beta*lambda, (4, 2) = -beta*(w-1)*lambda, (4, 3) = 0, (4, 4) = -q__E-delta-µ, (4, 5) = 0, (4, 6) = 0, (4, 7) = 0, (5, 1) = 0, (5, 2) = 0, (5, 3) = 0, (5, 4) = a*delta, (5, 5) = -`ρ__A`-q__A-µ, (5, 6) = 0, (5, 7) = k*`ρ__Q`, (6, 1) = 0, (6, 2) = 0, (6, 3) = 0, (6, 4) = -delta*(-1+a), (6, 5) = 0, (6, 6) = -`ρ__F`-q__F-`δ__F`-µ, (6, 7) = 0, (7, 1) = 0, (7, 2) = 0, (7, 3) = 0, (7, 4) = q__E, (7, 5) = q__A, (7, 6) = q__F, (7, 7) = -`ρ__Q`-`δ__Q`-µ})$

Matrix(%id = 36893490965935089652)

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#this shows that trace is negative

#to Achieve stability, the value below must be less than zero

(-q__E-delta-Âµ)*(-rho__F-q__F-delta__F-Âµ)*(-k*q__A*rho__Q+q__A*rho__Q+q__A*Âµ+q__A*delta__Q+rho__A*rho__Q+rho__A*Âµ+rho__A*delta__Q+rho__Q*Âµ+Âµ^2+Âµ*delta__Q)*Âµ < 0