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Elisha

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Bifurcation Diagram from my system of eq...

Elisha 55 Maple 2018
ode differential-equations bifurcation
July 08 2023
0 0

Please help with the bifurcation diagram for the system and parameter values below

> 

NULL

with(VectorCalculus)

[`&x`, `*`, `+`, `-`, `.`, `<,>`, `<|>`, About, AddCoordinates, ArcLength, BasisFormat, Binormal, ConvertVector, CrossProduct, Curl, Curvature, D, Del, DirectionalDiff, Divergence, DotProduct, Flux, GetCoordinateParameters, GetCoordinates, GetNames, GetPVDescription, GetRootPoint, GetSpace, Gradient, Hessian, IsPositionVector, IsRootedVector, IsVectorField, Jacobian, Laplacian, LineInt, MapToBasis, Nabla, Norm, Normalize, PathInt, PlotPositionVector, PlotVector, PositionVector, PrincipalNormal, RadiusOfCurvature, RootedVector, ScalarPotential, SetCoordinateParameters, SetCoordinates, SpaceCurve, SurfaceInt, TNBFrame, TangentLine, TangentPlane, TangentVector, Torsion, Vector, VectorField, VectorPotential, VectorSpace, Wronskian, diff, eval, evalVF, int, limit, series]

 (1)
> 

interface(imaginaryunit = F)

I

 (2)

M := Pi*theta-S*c__1-S*lambda+S__v*v__2

Pi*theta-S*c__1-S*lambda+S__v*v__2

 (3)

Y := -S__v*c__2*lambda+Pi*b__1+S*v__1-S__v*c__3

-S__v*c__2*lambda+Pi*b__1+S*v__1-S__v*c__3

 (4)

P := S__v*alpha+`&rho;__A`*A+c__4*`&rho;__Q`*Q+I*`&rho;__I`-µ*V

Q*c__4*rho__Q+A*rho__A+I*rho__I+S__v*alpha-V*µ

 (5)

R := S__v*c__2*lambda-E*c__5+S*lambda

S__v*c__2*lambda-E*c__5+S*lambda

 (6)

U := E*a*delta+Q*k*`&rho;__Q`-A*c__6

E*a*delta+Q*k*rho__Q-A*c__6

 (7)

L := c__7*E-I*c__8

E*c__7-I*c__8

 (8)

X := q__E*E+I*q__I-c__9*Q

E*q__E+I*q__I-Q*c__9

 (9)

solve({L = 0, M = 0, P = 0, R = 0, U = 0, X = 0, Y = 0}, {I, A, E, Q, S, S__v, V})

{A = (a*c__8*c__9*delta+c__7*k*q__I*rho__Q+c__8*k*q__E*rho__Q)*lambda*Pi*(b__1*c__1*c__2+b__1*c__2*lambda+c__2*lambda*theta+c__2*theta*v__1+b__1*v__2+c__3*theta)/(c__6*c__9*c__5*c__8*(c__1*c__2*lambda+c__2*lambda^2+c__1*c__3+c__3*lambda-v__1*v__2)), E = lambda*Pi*(b__1*c__1*c__2+b__1*c__2*lambda+c__2*lambda*theta+c__2*theta*v__1+b__1*v__2+c__3*theta)/(c__5*(c__1*c__2*lambda+c__2*lambda^2+c__1*c__3+c__3*lambda-v__1*v__2)), I = c__7*lambda*Pi*(b__1*c__1*c__2+b__1*c__2*lambda+c__2*lambda*theta+c__2*theta*v__1+b__1*v__2+c__3*theta)/(c__5*c__8*(c__1*c__2*lambda+c__2*lambda^2+c__1*c__3+c__3*lambda-v__1*v__2)), Q = (c__7*q__I+c__8*q__E)*lambda*Pi*(b__1*c__1*c__2+b__1*c__2*lambda+c__2*lambda*theta+c__2*theta*v__1+b__1*v__2+c__3*theta)/(c__9*c__5*c__8*(c__1*c__2*lambda+c__2*lambda^2+c__1*c__3+c__3*lambda-v__1*v__2)), S = Pi*(c__2*lambda*theta+b__1*v__2+c__3*theta)/(c__1*c__2*lambda+c__2*lambda^2+c__1*c__3+c__3*lambda-v__1*v__2), S__v = Pi*(b__1*c__1+b__1*lambda+theta*v__1)/(c__1*c__2*lambda+c__2*lambda^2+c__1*c__3+c__3*lambda-v__1*v__2), V = Pi*(a*b__1*c__1*c__2*c__8*c__9*delta*lambda*rho__A+a*b__1*c__2*c__8*c__9*delta*lambda^2*rho__A+a*c__2*c__8*c__9*delta*lambda^2*rho__A*theta+a*c__2*c__8*c__9*delta*lambda*rho__A*theta*v__1+b__1*c__1*c__2*c__4*c__6*c__7*lambda*q__I*rho__Q+b__1*c__1*c__2*c__4*c__6*c__8*lambda*q__E*rho__Q+b__1*c__1*c__2*c__7*k*lambda*q__I*rho__A*rho__Q+b__1*c__1*c__2*c__8*k*lambda*q__E*rho__A*rho__Q+b__1*c__2*c__4*c__6*c__7*lambda^2*q__I*rho__Q+b__1*c__2*c__4*c__6*c__8*lambda^2*q__E*rho__Q+b__1*c__2*c__7*k*lambda^2*q__I*rho__A*rho__Q+b__1*c__2*c__8*k*lambda^2*q__E*rho__A*rho__Q+c__2*c__4*c__6*c__7*lambda^2*q__I*rho__Q*theta+c__2*c__4*c__6*c__7*lambda*q__I*rho__Q*theta*v__1+c__2*c__4*c__6*c__8*lambda^2*q__E*rho__Q*theta+c__2*c__4*c__6*c__8*lambda*q__E*rho__Q*theta*v__1+c__2*c__7*k*lambda^2*q__I*rho__A*rho__Q*theta+c__2*c__7*k*lambda*q__I*rho__A*rho__Q*theta*v__1+c__2*c__8*k*lambda^2*q__E*rho__A*rho__Q*theta+c__2*c__8*k*lambda*q__E*rho__A*rho__Q*theta*v__1+a*b__1*c__8*c__9*delta*lambda*rho__A*v__2+a*c__3*c__8*c__9*delta*lambda*rho__A*theta+b__1*c__1*c__2*c__6*c__7*c__9*lambda*rho__I+b__1*c__2*c__6*c__7*c__9*lambda^2*rho__I+b__1*c__4*c__6*c__7*lambda*q__I*rho__Q*v__2+b__1*c__4*c__6*c__8*lambda*q__E*rho__Q*v__2+b__1*c__7*k*lambda*q__I*rho__A*rho__Q*v__2+b__1*c__8*k*lambda*q__E*rho__A*rho__Q*v__2+c__2*c__6*c__7*c__9*lambda^2*rho__I*theta+c__2*c__6*c__7*c__9*lambda*rho__I*theta*v__1+c__3*c__4*c__6*c__7*lambda*q__I*rho__Q*theta+c__3*c__4*c__6*c__8*lambda*q__E*rho__Q*theta+c__3*c__7*k*lambda*q__I*rho__A*rho__Q*theta+c__3*c__8*k*lambda*q__E*rho__A*rho__Q*theta+alpha*b__1*c__1*c__5*c__6*c__8*c__9+alpha*b__1*c__5*c__6*c__8*c__9*lambda+alpha*c__5*c__6*c__8*c__9*theta*v__1+b__1*c__6*c__7*c__9*lambda*rho__I*v__2+c__3*c__6*c__7*c__9*lambda*rho__I*theta)/(c__5*c__6*c__8*c__9*µ*(c__1*c__2*lambda+c__2*lambda^2+c__1*c__3+c__3*lambda-v__1*v__2))}

 (10)
> 

``
> 

lambda := beta*(I+`&eta;__A`*A+`&eta;__Q`*Q)/N

beta*(I+eta__A*A+eta__Q*Q)/N

 (11)
> 

``
> 

NULL
> 

k := .15

.15

 (12)
> 

delta := .125

.125

 (13)
> 

mu := 0.464360344e-4

0.464360344e-4

 (14)
> 

pi := .464360344

.464360344

 (15)
> 

delta__Q := 0.6847e-3

0.6847e-3

 (16)
> 

beta := .1086

.1086

 (17)
> 

q__E := 0.18113e-3

0.18113e-3

 (18)
> 

rho__Q := 0.815e-1

0.815e-1

 (19)
> 

a := .16255

.16255

 (20)
> 

v__1 := 0.5e-1

0.5e-1

 (21)
> 

v__2 := 0.5e-1

0.5e-1

 (22)
> 

alpha := 0.57e-1

0.57e-1

 (23)
> 

lambda := 0.765e-2

0.765e-2

 (24)
> 

rho__A := 0.915e-1

0.915e-1

 (25)
> 

rho__I := 0.515e-1

0.515e-1

 (26)
> 

a := .16255

.16255

 (27)
> 

q__I := 0.1923e-2

0.1923e-2

 (28)
> 

q__A := 0.4013e-7

0.4013e-7

 (29)
> 

eta__A := .1213

.1213

 (30)
> 

eta__Q := 0.3808e-2

0.3808e-2

 (31)
> 

w := .5925

.5925

 (32)

Download Bifurcation_Equation.mw

I Encountered Error Message trying to us...

Elisha 55 Maple 2018
ode parameters dsolve
February 11 2023
2 4

I find it difficult to use dsolve to solve system of ordinary differential equations with assigned parameters and initial conditions. The error message "Error, (in dsolve/numeric) 'parameters' must be specified as a list of unique unassigned names" kept coming up.

Pls see the uploaded equation for more understanding

> 

restart
> 

interface(imaginaryunit = F)

I

 (1)
> 

I

I

 (2)
> 

sqrt(-4)

2*I

 (3)
> 

NULL
> 

Suscep := diff(S(t), t) = theta*epsilon+v__2*S__v(t)-S(t)*lambda-S(t)*(µ+v__1)

diff(S(t), t) = theta*varepsilon+v__2*S__v(t)-S(t)*lambda-S(t)*(µ+v__1)

 (4)
> 

Vacc := diff(S__v(t), t) = (1-theta)*epsilon+v__1*S(t)-(µ+alpha+v__2)*S__v(t)-(1-w)*S__v(t)*lambda
> 

Immun := diff(V(t), t) = alpha*S__v(t)+`&rho;__A`*A(t)+(1-k)*`&rho;__Q`*Q(t)+`&rho;__I`*(I)(t)-µ*V(t)
> 

Exp := diff(E(t), t) = S(t)*lambda+(1-w)*S__v(t)*lambda-(q__E+delta+µ)*E(t)
> 

Asymp := diff(A(t), t) = delta*a*E(t)-(`&rho;__A`+µ)*A(t)+k*`&rho;__Q`*Q(t)
> 

Inf := diff((I)(t), t) = delta*(1-a)*E(t)-(`&rho;__I`+q__I+`&delta;__I`+µ)*(I)(t)
> 

Quar := diff((I)(t), t) = q__E*E(t)+q__I*(I)(t)-(`&rho;__Q`+`&delta;__Q`+µ)*Q(t)
> 

init_conds := S(0) = S_0, S__v(0) = S__v*_0, V(0) = V_0, E(0) = E_0, A(0) = A_0, (I)(0) = I_0, Q(0) = Q_0

S(0) = S_0, S__v(0) = S__v*_0, V(0) = V_0, E(0) = E_0, A(0) = A_0, I(0) = I_0, Q(0) = Q_0

 (5)
> 

sys := {Asymp, Exp, Immun, Inf, Quar, Suscep, Vacc, init_conds}
> 

``
> 

sol := dsolve(sys, numeric, parameters = [`&delta;__Q`, `&delta;__I`, a, k, epsilon, v[1], q[E], q[I], q[A], eta[A], eta[Q], rho[A], rho[Q], rho[I], v[2], alpha, mu, delta, alpha, beta, w, lambda, S_0, S__v*_0, V_0, E_0, A_0, I_0, Q_0], method = rkf45)

Error, (in dsolve/numeric) 'parameters' must be specified as a list of unique unassigned names

  
> 

sol(parameters = [delta = .125, `&delta;__Q` = 0.6847e-3, epsilon = .464360344, `&delta;__I` = 0.2230e-8, a = .6255, q[E] = 0.18113e-3, k = .15, v__1 = 0.5e-1, v__2 = 0.6e-1, `&rho;__Q` = 0.815e-1, `&rho;__A` = .1, `&rho;__I` = 0.666666e-1, q__I = 0.1923e-2, q__A = 0.4013e-7, `&eta;__A` = .1213, `&eta;__Q` = 0.3808e-2*alpha and 0.3808e-2*alpha = .4, w = .5925, mu = 0.464360344e-4, lambda = 0.1598643e-7, S_0 = 1.0, S__v*_0 = 0.6e-4, V_0 = 0.35e-4, E_0 = 0.5e-4, I_0 = 0.32e-4, A_0 = 0.15e-4, Q_0 = 0.1e-4])

sol(parameters = [delta = .125, delta__Q = 0.6847e-3, varepsilon = .464360344, delta__I = 0.2230e-8, a = .6255, q[E] = 0.18113e-3, k = .15, v__1 = 0.5e-1, v__2 = 0.6e-1, rho__Q = 0.815e-1, rho__A = .1, rho__I = 0.666666e-1, q__I = 0.1923e-2, q__A = 0.4013e-7, eta__A = .1213, false, w = .5925, mu = 0.464360344e-4, lambda = 0.1598643e-7, S_0 = 1.0, S__v*_0 = 0.6e-4, V_0 = 0.35e-4, E_0 = 0.5e-4, I_0 = 0.32e-4, A_0 = 0.15e-4, Q_0 = 0.1e-4])

 (6)
> 

Evaluate*the*system*at*t = 2
> 

sol(2)

sol(2)

 (7)
> 

sol(1)

sol(1)

 (8)
> 

sol(.1)

sol(.1)

 (9)
> 

sol(.3)

sol(.3)

 (10)
> 

sol(.5)

sol(.5)

 (11)
> 

sol(.7)

sol(.7)

 (12)
> 

sol(.9)

sol(.9)

 (13)
> 

sol(1.1)

sol(1.1)

 (14)
> 

sol(1.3)

sol(1.3)

 (15)
> 

sol(1.5)

sol(1.5)

 (16)
> 

 
> 

 

Download Covid19_Simulation.mw

Comparing RK4 and RK45. How to solve Dif...

Elisha 55 Maple
ode homework dsolve
January 08 2018
1 9

Dear Team, I have used RKF45 to solve my ODE with Maple. now I am required to solve same ODE using RK4 for comparison of solution. Ps help me with an example. Pls find my Parameters, intial values and ODE below:


 

``

> 

ODE*equations

ODE*equations

 (1)
> 

diff(s(t), t) = (1-phi)*epsilon+(1-rho)*a+(1-f)*alpha*v(t)-(lambda+theta[1]+a+epsilon)*s(t)
> 

diff(v(t), t) = phi*epsilon+rho*a+theta[1]*s(t)-((1-f)*alpha+f*theta[2]+a+epsilon)*v(t)
> 

diff(e(t), t) = lambda*s(t)-(delta+a+epsilon)*e(t)
> 

NULL
> 

diff(r(t), t) = eta*i(t)+v(t)*f*theta[2]-(a+epsilon)*r(t)
> 

``
> 

``
> 

My*parameters

My*parameters

 (2)
> 

v(0) := .4

.4

 (3)
> 

NULL
> 

s(0) := 0.6e-1

0.6e-1

 (4)
> 

e[0] := .24

.24

 (5)
> 

i[0] := .17

.17

 (6)
> 

r[0] := .13

.13

 (7)
> 

c := 0.4e-1

0.4e-1

 (8)
> 

f := .4

.4

 (9)
> 

beta := .2

.2

 (10)
> 

epsilon := .8

.8

 (11)
> 

theta[1] := .1

.1

 (12)
> 

theta[2] := .3

.3

 (13)
> 

alpha := .9

.9

 (14)
> 

rho := .7

.7

 (15)
> 

eta := .99

.99

 (16)
> 

delta := .3

.3

 (17)
> 

a := 0.4e-1

0.4e-1

 (18)
> 

phi := 1
> 

``
> 

``

``


 

Download ODE_EQNS.mw

How to Plot Multiple Graphs for Comparis...

Elisha 55 Maple
multiple plotting incomplete-question
December 28 2017
1 8

How can i plot various 3 multiple graphs using Maple to Compare ADM, EXACT SOLUTION AND RKF45 using the table below:

 Time  ADM Solution                                   RKF45 Numerical Solution        EXACT Solution

               ADM Solution                     RKF45 Numerical Solution                EXACT Solution

 t           s               i              r                 s            i                r             s               i                   r

0.1     0.0769540597    0.1477783335     0.1393069312      0.0763361154   0.1477867626   0.1393072151   0.0763361266    0.1477867830     0.1393071934 

0.3     0.1093424148    0.1120401102     0.1509302274      0.1072278404   0.1121142033   0.1509369166   0.1072278479    0.1121142168     0.1509369024 

0.5     0.1392030568    0.0853083077     0.1558066181      0.1353291378   0.0855066806   0.1558355785   0.1353291558    0.08550671332   0.1558355439

0.7     0.1662374563    0.0652194693     0.1562235596      0.1604342222   0.0655908735   0.1562974878   0.1604342352    0.06559089617   0.1562974637 

0.9     0.1903821623    0.0500537619     0.1537887672      0.1825594352   0.0506356391   0.1539346707   0.1825594528    0.05063567192   0.1539346352

1.1     0.2117168860    0.0385555127     0.1496220815      0.2018541863   0.0393727175   0.1498707561   0.2018542024    0.03937274753   0.1498707234 

1.3     0.2303874000    0.0298096396     0.1444864012      0.2185409755   0.0308687065   0.1448804021   0.2185409880    0.03086872986   0.1448803765 

1.5     0.2465077820    0.0231661161     0.1388614678      0.2328759081   0.0244336214   0.1394893808   0.2328759200    0.02443364533   0.1394893543

 

 

 

How can I get an Exact solution for syst...

Elisha 55 Maple 17
ode homework differential-equations
May 15 2017
3 4

I have a system of ODE that I need to get an exact solution to. How can I do this using Maple? I tried doing it but error pops out.

Thanks

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