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Paras31

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0 years, 141 days
Hellenic Open University
Mathematician

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Teacher of Mathematics with a proven track record of working in education management. Proficient in Ease of Adaptation, Course Design, and Instructional Technology. Strong education professional with a Bachelor's degree with an emphasis in Mathematics from the University of Aegean. Currently, he is pursuing a Master's degree in Applied Mathematics at the Hellenic Open University, with a specific focus on Ordinary and Partial Differential equations. His enthusiasm lies in the application of mathematical models to real-world contexts, such as epidemiology and population growth. Aside from his passion for teaching, Athanasios enjoys football, basketball, and spending time with his dogs.

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Exploring the Impact of Infection Rate V...

Paras31 85 Maple 2024
ode dsolve numeric
September 02 2024
1 6

Today in class, we presented an exercise based on the paper titled "Analysis of regular and chaotic dynamics in a stochastic eco-epidemiological model" by Bashkirtseva, Ryashko, and Ryazanova (2020). In this exercise, we kept all parameters of the model the same as in the paper, but we varied the parameter β, which represents the rate of infection spread. The goal was to observe how changes in β impact the system's dynamics, particularly focusing on the transition between regular and chaotic behavior.

This exercise involves studying a mathematical model that appears in eco-epidemiology. The model is described by the following set of equations:

dx/dt = rx-bx^2-cxy-`βxz`/(a+x)-a[1]*yz/(e+y)

dy/dt = -`μy`+`βxy`/(a+x)-a[2]*yz/(d+y)

" (dz)/(dt)=-mz+((c[1 ]a[1])[ ]xz)/(e[]+x)+((c[1 ]a[2])[ ]yz)/(d+y)"

 

where r, b, c, β, α,a[1],a[2], e, d, m, c[1], c[2], μ>0 are given parameters. This model generalizes the classic predator-prey system by incorporating disease dynamics within the prey population. The populations are divided into the following groups:

 

• 

Susceptible prey population (x): Individuals in the prey population that are healthy but can become infected by a disease.
• 

Infected prey population (y): Individuals in the prey population that are infected and can transmit the disease to others.
• 

Predator population (z): The predator population that feeds on both susceptible (x) and infected (y) prey.

 

The initial conditions are always x(0)=0.2, y(0)=0.05, z(0)=0.05,  and we will vary the parameter β.;

 

For this exercise, the parameters are fixed as follows:

 

"r=1,` b`=1,` c`=0.01, a=0.36 ,` a`[1]=0.01,` a`[2]=0.05,` e`[]=15,` m`=0.01,` d`=0.5,` c`[1]=2,` `c[2]==1,` mu`=0.4."

NULL

Task (a)

• 

Solve the system numerically for the given parameter values and initial conditions with β=0.6 over the time interval t2[0,20000].
• 

Plot the solutions x(t), y(t), and  z(t) over this time interval.
• 

Comment on the model's predictions, keeping in mind that the time units are usually days.
• 

Also, plot the trajectory in the 3D space (x,y,z).

 

restart

r := 1; b := 1; f := 0.1e-1; alpha := .36; a[1] := 0.1e-1; a[2] := 0.5e-1; e := 15; m := 0.1e-1; d := .5; c[1] := 2; c[2] := 1; mu := .4; beta := .6

sys := {diff(x(t), t) = r*x(t)-b*x(t)^2-f*x(t)*y(t)-beta*x(t)*y(t)/(alpha+x(t))-a[1]*x(t)*z(t)/(e+x(t)), diff(y(t), t) = -mu*y(t)+beta*x(t)*y(t)/(alpha+x(t))-a[2]*y(t)*z(t)/(d+y(t)), diff(z(t), t) = -m*z(t)+c[1]*a[1]*x(t)*z(t)/(e+x(t))+c[2]*a[2]*y(t)*z(t)/(d+y(t))}

{diff(x(t), t) = x(t)-x(t)^2-0.1e-1*x(t)*y(t)-.6*x(t)*y(t)/(.36+x(t))-0.1e-1*x(t)*z(t)/(15+x(t)), diff(y(t), t) = -.4*y(t)+.6*x(t)*y(t)/(.36+x(t))-0.5e-1*y(t)*z(t)/(.5+y(t)), diff(z(t), t) = -0.1e-1*z(t)+0.2e-1*x(t)*z(t)/(15+x(t))+0.5e-1*y(t)*z(t)/(.5+y(t))}

 (1)

ics := {x(0) = .2, y(0) = 0.5e-1, z(0) = 0.5e-1}

{x(0) = .2, y(0) = 0.5e-1, z(0) = 0.5e-1}

 (2)

NULL

sol := dsolve(`union`(sys, ics), {x(t), y(t), z(t)}, numeric, range = 0 .. 20000, maxfun = 0, output = listprocedure, abserr = 0.1e-7, relerr = 0.1e-7)

`[Length of output exceeds limit of 1000000]`

 (3)

X := subs(sol, x(t)); Y := subs(sol, y(t)); Z := subs(sol, z(t))

``

plot('[X(t)]', t = 0 .. 20000, numpoints = 350, title = "Trajectory of x(t)", axes = boxed, gridlines, color = ["#40e0d0"])

  

plot('[Y(t)]', t = 0 .. 20000, numpoints = 350, title = "Trajectory", axes = boxed, gridlines, title = "Trajectory of y(t)", color = ["SteelBlue"])

  

``

plot('[Z(t)]', t = 0 .. 20000, numpoints = 350, title = "Trajectory", axes = boxed, gridlines, title = "Trajectory of Z(t)", color = "Black"); with(DEtools)

  

with(DEtools)

DEplot3d(sys, {x(t), y(t), z(t)}, t = 0 .. 20000, [[x(0) = .2, y(0) = 0.5e-1, z(0) = 0.5e-1]], numpoints = 35000, color = blue, thickness = 1, linestyle = solid)

  

Task (b)

• 

Repeat the study in part (a) with the same initial conditions but set β=0.61.

NULL

restart

r := 1; b := 1; f := 0.1e-1; alpha := .36; a[1] := 0.1e-1; a[2] := 0.5e-1; e := 15; m := 0.1e-1; d := .5; c[1] := 2; c[2] := 1; mu := .4; beta := .61

NULL

sys := {diff(x(t), t) = r*x(t)-b*x(t)^2-f*x(t)*y(t)-beta*x(t)*y(t)/(alpha+x(t))-a[1]*x(t)*z(t)/(e+x(t)), diff(y(t), t) = -mu*y(t)+beta*x(t)*y(t)/(alpha+x(t))-a[2]*y(t)*z(t)/(d+y(t)), diff(z(t), t) = -m*z(t)+c[1]*a[1]*x(t)*z(t)/(e+x(t))+c[2]*a[2]*y(t)*z(t)/(d+y(t))}

{diff(x(t), t) = x(t)-x(t)^2-0.1e-1*x(t)*y(t)-.61*x(t)*y(t)/(.36+x(t))-0.1e-1*x(t)*z(t)/(15+x(t)), diff(y(t), t) = -.4*y(t)+.61*x(t)*y(t)/(.36+x(t))-0.5e-1*y(t)*z(t)/(.5+y(t)), diff(z(t), t) = -0.1e-1*z(t)+0.2e-1*x(t)*z(t)/(15+x(t))+0.5e-1*y(t)*z(t)/(.5+y(t))}

 (4)

NULL

ics := {x(0) = .2, y(0) = 0.5e-1, z(0) = 0.5e-1}

{x(0) = .2, y(0) = 0.5e-1, z(0) = 0.5e-1}

 (5)

sol := dsolve(`union`(sys, ics), {x(t), y(t), z(t)}, numeric, range = 0 .. 20000, maxfun = 0, output = listprocedure, abserr = 0.1e-7, relerr = 0.1e-7)

`[Length of output exceeds limit of 1000000]`

 (6)

X := subs(sol, x(t)); Y := subs(sol, y(t)); Z := subs(sol, z(t))

NULL

plot('[X(t)]', t = 0 .. 20000, numpoints = 350, title = "Trajectory of x(t)", axes = boxed, gridlines, color = ["Blue"])

  

plot('[Y(t)]', t = 0 .. 20000, numpoints = 350, title = "Trajectory of Y(t)", axes = boxed, gridlines, color = "Red")

  

plot('[Z(t)]', t = 0 .. 20000, numpoints = 350, title = "Trajectory of Y(t)", axes = boxed, gridlines, color = "Black")

  

NULL

with(DEtools)

DEplot3d(sys, {x(t), y(t), z(t)}, t = 0 .. 20000, [[x(0) = .2, y(0) = 0.5e-1, z(0) = 0.5e-1]], maxfun = 0, numpoints = 35000, color = blue, thickness = 1, linestyle = solid)

  

The rate of the infection spread is affected by the average number of contacts each person has (β=0.6) and increases depending on the degree of transmission within the population, in particular within specific subpopulations (such as those in rural areas). A detailed epidemiological study showed that the spread of infection is most significant in urban areas, where population density is higher, while in rural areas, the rate of infection remains relatively low. This suggests that additional public health measures are needed to reduce transmission in densely populated areas, particularly in regions with high population density such as cities

``

Download math_model_eco-epidemiology.mw

Join the New LinkedIn Group: Maple Softw...

Paras31 85 Maple , Maple Toolboxes , MapleSim , MapleSim Add-Ons , Maple Calculator , Maple Learn
August 26 2024
1 0

I'm excited to announce the creation of a new LinkedIn group, Maple Software Community! This group is dedicated to discussions about the use of Maple software and is designed to be a valuable resource for undergraduate and graduate students, researchers, and all Maple enthusiasts.

By joining this community, you'll have the opportunity to:

  • Learn about upcoming events and workshops that can enhance your skills.
  • Stay informed on the latest projects that leverage Maple software.
  • Engage in discussions that explore the many uses of Maple across various fields.
  • Connect with Maple ambassadors and users worldwide who are eager to share their knowledge and experience.

Whether you're a seasoned user or just starting out with Maple, your contributions to this group are welcome and encouraged. Let's build a thriving community together!


Looking forward to seeing you there! 

Maple Software Community

Unlock Your Maple Potential: Free Guid...

Paras31 85 Maple
education resources books
August 25 2024
0 0

I'm excited to share some valuable resources that I've found incredibly helpful for anyone looking to enhance their Maple skills. Whether you're just starting, studying as a student, or are a seasoned professional, these guides and books offer a wealth of information to aid your learning journey.

Exploring Discrete Mathematics With Maple

These materials are freely available and can be a great addition to your learning resources. They cover a wide range of topics and are designed to help users at all levels improve their Maple proficiency.

You can add your own sources in a comment!

Happy learning and I hope you find these resources as useful as I have!

Exploring Nonlinear Dynamics: Analysis o...

Paras31 85 Maple 2024
ode chaos differential-equations
August 11 2024
1 8


 

An attractor is called strange if it has a fractal structure, that is if it has a non-integer Hausdorff dimension. This is often the case when the dynamics on it are chaotic, but strange nonchaotic attractors also exist.  If a strange attractor is chaotic, exhibiting sensitive dependence on initial conditions, then any two arbitrarily close alternative initial points on the  attractor, after any of various numbers of iterations, will lead to  points that are arbitrarily far apart (subject to the confines of the attractor), and after any of various other numbers of iterations will  lead to points that are arbitrarily close together. Thus a dynamic system with a chaotic attractor is locally unstable yet globally stable: once some sequences have entered the attractor, nearby points diverge  from one another but never depart from the attractor.


The term strange attractor was coined by David Ruelle and Floris Takens to describe the attractor resulting from a series of bifurcations of a system describing fluid flow. Strange attractors are often differentiable in a few directions, but some are like a Cantor dust, and therefore not differentiable. Strange attractors may also be found  in the presence of noise, where they may be shown to support invariant  random probability measures of Sinai–Ruelle–Bowen type.


Examples of strange attractors include the  Rössler attractor, and Lorenz attractor.

 

 

THOMAS``with(plots); b := .20; sys := diff(x(t), t) = sin(y(t))-b*x(t), diff(y(t), t) = sin(z(t))-b*y(t), diff(z(t), t) = sin(x(t))-b*z(t); sol := dsolve({sys, x(0) = 1.1, y(0) = 1.1, z(0) = -0.1e-1}, {x(t), y(t), z(t)}, numeric); odeplot(sol, [x(t), y(t), z(t)], t = 0 .. 600, axes = boxed, numpoints = 50000, labels = [x, y, z], title = "Thomas Attractor")

  

 

 

Dabras``

with(plots); a := 3.00; b := 2.7; c := 1.7; d := 2.00; e := 9.00; sys := diff(x(t), t) = y(t)-a*x(t)+b*y(t)*z(t), diff(y(t), t) = c*y(t)-x(t)*z(t)+z(t), diff(z(t), t) = d*x(t)*y(t)-e*z(t); sol := dsolve({sys, x(0) = 1.1, y(0) = 2.1, z(0) = -2.00}, {x(t), y(t), z(t)}, numeric); odeplot(sol, [x(t), y(t), z(t)], t = 0 .. 100, axes = boxed, numpoints = 35000, labels = [x, y, z], title = "Dabras Attractor")

  

Halvorsen

NULLwith(plots); a := 1.89; sys := diff(x(t), t) = -a*x(t)-4*y(t)-4*z(t)-y(t)^2, diff(y(t), t) = -a*y(t)-4*z(t)-4*x(t)-z(t)^2, diff(z(t), t) = -a*z(t)-4*x(t)-4*y(t)-x(t)^2; sol := dsolve({sys, x(0) = -1.48, y(0) = -1.51, z(0) = 2.04}, {x(t), y(t), z(t)}, numeric, maxfun = 300000); odeplot(sol, [x(t), y(t), z(t)], t = 0 .. 600, axes = boxed, numpoints = 35000, labels = [x, y, z], title = "Halvorsen Attractor")

  

Chen

 

 

with(plots); alpha := 5.00; beta := -10.00; delta := -.38; sys := diff(x(t), t) = alpha*x(t)-y(t)*z(t), diff(y(t), t) = beta*y(t)+x(t)*z(t), diff(z(t), t) = delta*z(t)+(1/3)*x(t)*y(t); sol := dsolve({sys, x(0) = -7.00, y(0) = -5.00, z(0) = -10.00}, {x(t), y(t), z(t)}, numeric); odeplot(sol, [x(t), y(t), z(t)], t = 0 .. 100, axes = boxed, numpoints = 35000, labels = [x, y, z], title = "Chen Attractor")

  

References

1. 

https://www.dynamicmath.xyz/strange-attractors/
2. 

https://en.wikipedia.org/wiki/Attractor#Strange_attractor

``


 

Download Attractors.mw

Overview of Maple Coding Expert (ChatGp...

Paras31 85 Maple
ai chatgpt
August 07 2024
1 2

 Introduction
Maple Coding Expert is a GPT-based AI tool designed to assist with various mathematical tasks using Maple software. It offers step-by-step guidance and detailed explanations for a range of functions, making it a valuable resource for students, educators, and professionals.

 Core Features and Functions

1.Graph Creation:

   - Function Plotting: Users can plot a wide range of mathematical functions. For instance, to plot the function y = x2, the user would input the command `plot(x^2, x = -10..10);` in Maple. The expert helps in setting up the plotting parameters to visualize the function effectively.
   - Advanced Graphing: Beyond simple functions, the expert can guide users through plotting more complex functions and customizing plots with labels, legends, and different styles.

2. Equation Definition and Manipulation:

   - Defining Equations: The tool assists in defining equations for various calculus operations. For example, to differentiate a function, the command might be `diff(f(x), x);`. This helps in accurately modeling the equations necessary for solving real-world problems.
   - Solving Integrals: For integral calculus, users can get assistance in setting up both definite and indefinite integrals. Commands like `int(f(x), x);` are used to perform integration in Maple.

3. Calculus Problem Solving:
   - Differentiation and Integration: The expert provides guidance on solving derivatives and integrals, which are fundamental operations in calculus. It supports both symbolic and numerical methods, allowing users to choose the best approach for their problem.
   - Differential Equations: Users can solve ordinary and partial differential equations using commands like `dsolve({equations}, {variables});`. The expert offers advice on choosing solution methods and interpreting results.

I recently tried using the Maple Coding Expert for solving some calculus problems. It worked well overall and provided detailed solutions, though sometimes it approached the problems in a more complicated way than expected. Despite this, the accuracy and depth of the explanations were impressive and very helpful for understanding the underlying concepts.

 

Maple Coding Expert stands out as a comprehensive tool for anyone involved with Maple software for mathematical computing. It enhances learning, supports professional tasks, and aids in solving complex mathematical problems with ease.

For more information, you can explore the Maple Coding Expert on [GPTs Hunter](https://www.gptshunter.com/gpt-store/MzExMzI2MzYyMzJlNTAxMjM2) and [YesChat.ai](https://www.yeschat.ai).

 

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