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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
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