CITE THIS NOTEBOOK: Introduction to SI compartmental model by Athanasios Paraskevopoulos. Wolfram Community DEC 25 2022.

Outline

Introducing Compartment-Based Modeling

Modeling with Differential Equations

Introducing Compartment-Based Modeling

Many of the most interesting dynamics in nature have to do with interactions between organisms . These interactions are often subtle, indirect and difficult to detect . Interactions in which one organism consumes all or part of another . This includes predator - prey, herbivore - plant, and parasite - host interactions . These linkages are the prime movers of energy through food chains . They are an important factor in the ecology of populations, determining mortality of prey and birth of new predators .
​
Mathematical models and logic suggests that a coupled system of predator and prey should cycle : predators increase when prey are abundant, prey are driven to low numbers by predation, the predators decline, and the prey recover, ad infinitum . One of such models that simulates predator - prey interactions is the Lotka - Volterra Model .
We will discuss a behavior of 2 D - subsystems : SI - model .

SI Model (without predator)

We will study a mathematical model that appears in eco - epidemiology .
The SI-model describes the interactions between S-prey with density x and I-prey with density y under the assumption z≡0
◼
  • Break population into compartments.
    • ◼
  • Prey (
    
    )
    • ◼
  • Infected prey (
    ℐ
    )
    • SI Equations :
    x'[t]=r*x[t]-b*x[t]^2-c*x[t]*y[t]-β*x[t]*y[t]/(a+x[t])
    Out[]=
    rx[t]-b
    2
    x[t]
    -cx[t]y[t]-
    βx[t]y[t]
    a+x[t]
    y'[t]=-μ*y[t]+β*x[t]*y[t]/(a+x[t])
    Out[]=
    -μy[t]+
    βx[t]y[t]
    a+x[t]
    ◼
  •  stands for the intrinsic growth rate of S-prey.
    • ◼
  •  defines the intra-class competition in S-prey.
    • ◼
  •  characterizes the inter-class competition between S-prey and I-prey.
    • ◼
  • μ stands for the mortality of the I-prey
    • ◼
  • β
    considered as the bifurcation parameter.
    • ◼
  • Note: The dynamics of system depends on parameter β

    • Modeling with Differential Equations

    SI Model

    ◼
  • We will fix the parameters in the study to have the following values r = 1, b = 1, c = 1/100, μ = 4/10, a=1/2.75.
    • ◼
  • The initial values will be x(0)=0.2, y(0)=0.05
    • Αs mentioned above the dynamics of system depends on parameter β. So we will consider two cases when β=0.75 and β=1
    We will solve the system for the above values of parameters and initial conditions and for a time interval [0,200] ,then we will create its plots

    First situation

    β=0.75
    In[]:=
    r = 1; b = 1; c = 1/100; μ = 4/10;​​f = r*x[t] - b*x[t]^2 - c*x[t]*y[t] - 3/4*x[t]*y[t]/(a + x[t]);​​g = -μ*y[t] + 3/4*x[t]*y[t]/(a + x[t]);​​deq1 = x'[t] == f​​deq2 = y'[t] == g​​equilibrio = Solve[{f == 0, g == 0}, {x[t], y[t]}]​​MatrixForm[jacobian = D[{f, g}, {{x[t], y[t]}}] // Simplify]​​Simplify[MatrixForm[jacobian] /. equilibrio]​​a=1/2.75;
    Out[]=
    ′
    x
    [t]x[t]-
    2
    x[t]
    -
    1
    100
    x[t]y[t]-
    3x[t]y[t]
    4(a+x[t])
    Out[]=
    ′
    y
    [t]-
    2y[t]
    5
    +
    3x[t]y[t]
    4(a+x[t])
    Out[]=
    {x[t]1,y[t]0},x[t]
    8a
    7
    ,y[t]-
    100(-7a+8
    2
    a
    )
    7(35+a)
    ,{x[t]0,y[t]0}
    Out[]//MatrixForm=
    1-
    (75+a+x[t])y[t]
    100(a+x[t])
    +x[t]-2+
    3y[t]
    4
    2
    (a+x[t])
    -
    x[t](75+a+x[t])
    100(a+x[t])
    3ay[t]
    4
    2
    (a+x[t])
    -
    2
    5
    +
    3x[t]
    4(a+x[t])
    Out[]=
    
    -1
    -
    76+a
    100(1+a)
    0
    -
    2
    5
    +
    3
    4(1+a)
    ,
    -
    8(-49+161a+3
    2
    a
    )
    21(35+a)
    -
    2
    175
    (35+a)
    -
    7(-7+8a)
    3(35+a)
    0
    ,
    1
    0
    0
    -
    2
    5
    
    MatrixForm[jacobian=D[{f,g},{{x[t],y[t]}}]//Simplify]​​MatrixForm[A=Simplify[jacobian/.equilibrio[[1]]]]​​MatrixForm[B=Simplify[jacobian/.equilibrio[[2]]]]​​MatrixForm[Q=Simplify[jacobian/.equilibrio[[3]]]]
    Out[]//MatrixForm=
    1-
    0.01(75.3636+x[t])y[t]
    0.363636+x[t]
    +x[t]-2+
    0.75y[t]
    2
    (0.363636+x[t])
    -
    0.01x[t](75.3636+x[t])
    0.363636+x[t]
    0.272727y[t]
    2
    (0.363636+x[t])
    0.35(-0.415584+x[t])
    0.363636+x[t]
    Out[]//MatrixForm=
    
    -1.
    -0.56
    0.
    0.15
    
    Out[]//MatrixForm=
    -0.107101
    -0.404156
    0.269923
    -2.49338×
    -17
    10
    Out[]//MatrixForm=
    
    1.
    0.
    0.
    -0.4
    
    We calculate the eigenvalues
    eigen1=Eigenvalues[A]
    Out[]=
    {-1.,0.15}
    The point is unstable.
    eigen2=Eigenvalues[B]
    The point is stable.
    Out[]=
    {-0.0535506+0.325919,-0.0535506-0.325919}
    eigen3=Eigenvalues[Q]
    Out[]=
    {1.,-0.4}
    The point is unstable .
    The only stable point is eigen2
    soln[t_]=NDSolveValue[{deq1,deq2,x[0]==0.2,y[0]==0.05},{x[t],y[t]},{t,0,200}];​​Plot[soln[t],{t,0,200},PlotRangeAll]
    Out[]=
    50
    100
    150
    200
    0.0
    0.2
    0.4
    0.6
    0.8
    Clear["Global`*"]

    Second situation

    β=1
    In[]:=
    r = 1; b = 1; c = 1/100; μ = 4/10;​​f = r*x[t] - b*x[t]^2 - c*x[t]*y[t] - 1*x[t]*y[t]/(a + x[t]);​​g = -μ*y[t] + 1*x[t]*y[t]/(a + x[t]);​​deq1 = x'[t] == f​​deq2 = y'[t] == g​​equilibrio = Solve[{f == 0, g == 0}, {x[t], y[t]}]​​MatrixForm[jacobian = D[{f, g}, {{x[t], y[t]}}] // Simplify]​​Simplify[MatrixForm[jacobian] /. equilibrio]​​a=1/2.75;
    Out[]=
    ′
    x
    [t]x[t]-
    2
    x[t]
    -
    1
    100
    x[t]y[t]-
    x[t]y[t]
    0.363636+x[t]
    Out[]=
    ′
    y
    [t]-
    2y[t]
    5
    +
    x[t]y[t]
    0.363636+x[t]
    We calculate the eigenvalues
    The point is unstable .
    The point is unstable .
    The point is unstable .
    All the crucial points are unstable

    References

    1
    .
    Αλικάκος, N. Δ., and Γ. Η. Καλογερόπουλος. “Συνήθεις διαφορικές εξισώσεις.” Σύγχρονη Εκδοτική, Αθήνα (2003).
    2
    .
    Τραχανάς, Σ. “Mathematica και Εφαρμογές.” Πανεπιστημιακές Εκδόσεις Κρήτης, Ηράκλειο (2004).
    3
    .
    Bashkirtseva, Irina, Lev Ryashko, and Tatyana Ryazanova. "Analysis of regular and chaotic dynamics in a stochastic eco-epidemiological model." Chaos, Solitons & Fractals 131 (2020): 109549.
    4
    .
    Mureşan, Marian. Introduction to Mathematica® with applications. Springer International Publishing, 2017.
    ​