SIS-models
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| > |  `(`+`(Pmin, `*`(0.5e-1, `*`(i))), `+`(Qmin, `*`(0.5e-1..." align="center" border="0"> `(`+`(Pmin, `*`(0.5e-1, `*`(i))), `+`(Qmin, `*`(0.5e-1..." align="center" border="0"> `(`+`(Pmin, `*`(0.5e-1, `*`(i))), `+`(Qmin, `*`(0.5e-1..." align="center" border="0"> `(`+`(Pmin, `*`(0.5e-1, `*`(i))), `+`(Qmin, `*`(0.5e-1..." align="center" border="0"> `(`+`(Pmin, `*`(0.5e-1, `*`(i))), `+`(Qmin, `*`(0.5e-1..." align="center" border="0"> `(`+`(Pmin, `*`(0.5e-1, `*`(i))), `+`(Qmin, `*`(0.5e-1..." align="center" border="0"> `(`+`(Pmin, `*`(0.5e-1, `*`(i))), `+`(Qmin, `*`(0.5e-1..." align="center" border="0"> `(`+`(Pmin, `*`(0.5e-1, `*`(i))), `+`(Qmin, `*`(0.5e-1..." align="center" border="0"> `(`+`(Pmin, `*`(0.5e-1, `*`(i))), `+`(Qmin, `*`(0.5e-1..." align="center" border="0"> `(`+`(Pmin, `*`(0.5e-1, `*`(i))), `+`(Qmin, `*`(0.5e-1..." align="center" border="0"> |
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| > |  `(`+`(P[1], `-`(`*`(alpha, `*`(P[1], `*`(P[2])))), `*`(gamma, `*`(P[2]))), `+`(P[2], `*`(alpha, `*`(..." align="center" border="0"> `(`+`(P[1], `-`(`*`(alpha, `*`(P[1], `*`(P[2])))), `*`(gamma, `*`(P[2]))), `+`(P[2], `*`(alpha, `*`(..." align="center" border="0"> `(`+`(P[1], `-`(`*`(alpha, `*`(P[1], `*`(P[2])))), `*`(gamma, `*`(P[2]))), `+`(P[2], `*`(alpha, `*`(..." align="center" border="0"> `(`+`(P[1], `-`(`*`(alpha, `*`(P[1], `*`(P[2])))), `*`(gamma, `*`(P[2]))), `+`(P[2], `*`(alpha, `*`(..." align="center" border="0"> `(`+`(P[1], `-`(`*`(alpha, `*`(P[1], `*`(P[2])))), `*`(gamma, `*`(P[2]))), `+`(P[2], `*`(alpha, `*`(..." align="center" border="0"> `(`+`(P[1], `-`(`*`(alpha, `*`(P[1], `*`(P[2])))), `*`(gamma, `*`(P[2]))), `+`(P[2], `*`(alpha, `*`(..." align="center" border="0"> |
| > |  `(0, 1))[1]); `:=`(a, `+`(1, `-`(F(`<,>`(1, 1))[1]), b)); return [a, b] end proc); -1" align="center" border="0"> `(0, 1))[1]); `:=`(a, `+`(1, `-`(F(`<,>`(1, 1))[1]), b)); return [a, b] end proc); -1" align="center" border="0"> |
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![[alpha, gamma]](images/SIS_62.gif){kind=small} |
(1) |
General theory
| > | ![equi2(proc (P) options operator, arrow; `<,>`((SIS(alpha, gamma))(P)[1], (SIS(alpha, gamma))(P)[2]) end proc); 1](images/SIS_63.gif){kind=small} |
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![[[P[1] = P[1], P[2] = 0], [P[1] = `/`(`*`(gamma), `*`(alpha)), P[2] = P[2]]]](images/SIS_65.gif){kind=small} |
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, Vector[column](%id = 138873452)](images/SIS_67.gif){kind=small} |
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, Vector[column](%id = 140547452)](images/SIS_69.gif){kind=small} |
(2) |
In a SIS-model, I = P[2] follows logistic growth because P[1]+P[2]=N is constant
| > | ![subs(P[1] = VectorCalculus:-`+`(N, VectorCalculus:-`-`(P[2])), (SIS(alpha, gamma))(P)[2]); 1](images/SIS_70.gif){kind=small} |
![`+`(P[2], `*`(alpha, `*`(`+`(N, `-`(P[2])), `*`(P[2]))), `-`(`*`(gamma, `*`(P[2]))))](images/SIS_71.gif){kind=small} |
(3) |
| > | ![% = VectorCalculus:-`+`(P[2], VectorCalculus:-`*`(VectorCalculus:-`*`(alpha, P[2]), VectorCalculus:-`+`(VectorCalculus:-`+`(N, VectorCalculus:-`-`(VectorCalculus:-`*`(gamma, `/`(1, `*`(alpha))))), Vec...](images/SIS_72.gif){kind=small} |
![`+`(P[2], `*`(alpha, `*`(`+`(N, `-`(P[2])), `*`(P[2]))), `-`(`*`(gamma, `*`(P[2])))) = `+`(P[2], `*`(alpha, `*`(P[2], `*`(`+`(N, `-`(`/`(`*`(gamma), `*`(alpha))), `-`(P[2]))))))](images/SIS_73.gif){kind=small} |
(4) |
| > | ![VectorCalculus:-`+`(P[2], VectorCalculus:-`*`(VectorCalculus:-`*`(alpha, P[2]), VectorCalculus:-`+`(VectorCalculus:-`+`(N, VectorCalculus:-`-`(VectorCalculus:-`*`(gamma, `/`(1, `*`(alpha))))), VectorC...](images/SIS_74.gif) |
![`+`(P[2], `*`(alpha, `*`(P[2], `*`(`+`(N, `-`(`/`(`*`(gamma), `*`(alpha))), `-`(P[2])))))) = `+`(P[2], `*`(alpha, `*`(`+`(N, `-`(`/`(`*`(gamma), `*`(alpha)))), `*`(P[2](`+`(1, `-`(`/`(`*`(P[2]), `*`(`...](images/SIS_75.gif) |
(5) |
This is logistic growth with carrying capacity N-γ/α or N-ρ where ρ=γ/α is the relative removal rate.
Example 1
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Relative removal rate and contact number
| > | ![`:=`(rho, VectorCalculus:-`*`(param(F)[2], `/`(1, `*`(param(F)[1])))); 1](images/SIS_77.gif){kind=small} |
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(7) |
| > | ![display([timeplot2(F, `<,>`(95, 5), 25), plot(VectorCalculus:-`+`(100, VectorCalculus:-`-`(rho)), 0 .. 25, color = magenta)]); 1](images/SIS_81.gif){kind=small} |
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Example 2
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Relative removal rate and contact number
| > | ![`:=`(rho, VectorCalculus:-`*`(param(F)[2], `/`(1, `*`(param(F)[1])))); 1](images/SIS_84.gif){kind=small} |
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(8) |
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(9) |
| > | ![display([timeplot2(F, `<,>`(95, 5, 0), 25), plot(VectorCalculus:-`+`(100, VectorCalculus:-`-`(rho)), 0 .. 25, color = magenta)]); 1](images/SIS_88.gif){kind=small} |
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Example 3
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Relative removal rate and contact number
| > | ![`:=`(rho, VectorCalculus:-`*`(param(F)[2], `/`(1, `*`(param(F)[1])))); 1](images/SIS_91.gif){kind=small} |
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(10) |
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(11) |
| > | ![display([timeplot2(F, `<,>`(95, 5, 0), 75), plot(VectorCalculus:-`+`(100, VectorCalculus:-`-`(rho)), 0 .. 75, color = magenta)]); 1](images/SIS_95.gif){kind=small} |
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Differentiated SIS model
| > |  `(`+`(P[1], `*`(af, `*`(`+`(Nf, `-`(P[1])), `*`(P[2]))), `-`(`*`(gf, `*`(..." align="center" border="0"> `(`+`(P[1], `*`(af, `*`(`+`(Nf, `-`(P[1])), `*`(P[2]))), `-`(`*`(gf, `*`(..." align="center" border="0"> `(`+`(P[1], `*`(af, `*`(`+`(Nf, `-`(P[1])), `*`(P[2]))), `-`(`*`(gf, `*`(..." align="center" border="0"> `(`+`(P[1], `*`(af, `*`(`+`(Nf, `-`(P[1])), `*`(P[2]))), `-`(`*`(gf, `*`(..." align="center" border="0"> `(`+`(P[1], `*`(af, `*`(`+`(Nf, `-`(P[1])), `*`(P[2]))), `-`(`*`(gf, `*`(..." align="center" border="0"> |
General theory
| > | ![`:=`(F, SISto(alpha[f], gamma[f], N[f], alpha[m], gamma[m], N[m])); 1](images/SIS_102.gif){kind=small} |
![proc (P) options operator, arrow; VectorCalculus:-`<,>`(VectorCalculus:-`+`(VectorCalculus:-`+`(P[1], VectorCalculus:-`*`(VectorCalculus:-`*`(alpha[f], VectorCalculus:-`+`(N[f], VectorCalculus:-`-`(P[...](images/SIS_103.gif){kind=small} ![proc (P) options operator, arrow; VectorCalculus:-`<,>`(VectorCalculus:-`+`(VectorCalculus:-`+`(P[1], VectorCalculus:-`*`(VectorCalculus:-`*`(alpha[f], VectorCalculus:-`+`(N[f], VectorCalculus:-`-`(P[...](images/SIS_104.gif){kind=small} |
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](images/SIS_106.gif){kind=small} |
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The equilibria
| > | ![simplify(solve({F(P)[1] = P[1], F(P)[2] = P[2]}, [P[1], P[2]])); 1](images/SIS_107.gif){kind=small} |
![[[P[1] = 0, P[2] = 0], [P[1] = `/`(`*`(`+`(`*`(alpha[m], `*`(N[m], `*`(alpha[f], `*`(N[f])))), `-`(`*`(gamma[m], `*`(gamma[f]))))), `*`(alpha[m], `*`(`+`(`*`(N[m], `*`(alpha[f])), gamma[f])))), P[2] =...](images/SIS_108.gif){kind=small} |
(14) |
| > | ![equi2(proc (P) options operator, arrow; `<,>`(F(P)[1], F(P)[2]) end proc); 1](images/SIS_109.gif){kind=small} |
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![[[P[1] = 0, P[2] = 0], [P[1] = `+`(`-`(`/`(`*`(`+`(`-`(`*`(alpha[m], `*`(N[m], `*`(alpha[f], `*`(N[f]))))), `*`(gamma[m], `*`(gamma[f])))), `*`(alpha[m], `*`(`+`(`*`(N[m], `*`(alpha[f])), gamma[f]))))...](images/SIS_111.gif){kind=small} |
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, Vector[column](%id = 135338164)](images/SIS_113.gif) , Vector[column](%id = 135338164)](images/SIS_114.gif){kind=small} , Vector[column](%id = 135338164)](images/SIS_115.gif){kind=small} , Vector[column](%id = 135338164)](images/SIS_116.gif){kind=small} , Vector[column](%id = 135338164)](images/SIS_117.gif){kind=small} , Vector[column](%id = 135338164)](images/SIS_118.gif){kind=small} , Vector[column](%id = 135338164)](images/SIS_119.gif){kind=small} , Vector[column](%id = 135338164)](images/SIS_120.gif){kind=small} , Vector[column](%id = 135338164)](images/SIS_121.gif){kind=small} , Vector[column](%id = 135338164)](images/SIS_122.gif){kind=small} , Vector[column](%id = 135338164)](images/SIS_123.gif){kind=small} , Vector[column](%id = 135338164)](images/SIS_124.gif){kind=small} , Vector[column](%id = 135338164)](images/SIS_125.gif){kind=small} , Vector[column](%id = 135338164)](images/SIS_126.gif){kind=small} , Vector[column](%id = 135338164)](images/SIS_127.gif){kind=small} , Vector[column](%id = 135338164)](images/SIS_128.gif){kind=small} , Vector[column](%id = 135338164)](images/SIS_129.gif){kind=small} |
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, Vector[column](%id = 141281052)](images/SIS_131.gif) , Vector[column](%id = 141281052)](images/SIS_132.gif){kind=small} , Vector[column](%id = 141281052)](images/SIS_133.gif){kind=small} , Vector[column](%id = 141281052)](images/SIS_134.gif){kind=small} , Vector[column](%id = 141281052)](images/SIS_135.gif){kind=small} , Vector[column](%id = 141281052)](images/SIS_136.gif){kind=small} , Vector[column](%id = 141281052)](images/SIS_137.gif){kind=small} , Vector[column](%id = 141281052)](images/SIS_138.gif){kind=small} , Vector[column](%id = 141281052)](images/SIS_139.gif){kind=small} , Vector[column](%id = 141281052)](images/SIS_140.gif){kind=small} , Vector[column](%id = 141281052)](images/SIS_141.gif){kind=small} , Vector[column](%id = 141281052)](images/SIS_142.gif){kind=small} , Vector[column](%id = 141281052)](images/SIS_143.gif){kind=small} , Vector[column](%id = 141281052)](images/SIS_144.gif){kind=small} , Vector[column](%id = 141281052)](images/SIS_145.gif){kind=small} , Vector[column](%id = 141281052)](images/SIS_146.gif){kind=small} , Vector[column](%id = 141281052)](images/SIS_147.gif){kind=small} |
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Example 1
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The equilibria are
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![[[P[1] = 0., P[2] = 0.], [P[1] = 86.63043478, P[2] = 97.19512195]]](images/SIS_151.gif){kind=small} |
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, Vector[column](%id = 142345404)](images/SIS_153.gif){kind=small} |
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, Vector[column](%id = 142339116)](images/SIS_155.gif){kind=small} |
(16) |
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Notice that at the outset the disease is decreasing in one part of the population and increasing in the other part.
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Example 2
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The equlibria are
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![[[P[1] = 0., P[2] = 0.], [P[1] = 97.50000000, P[2] = 97.50000000]]](images/SIS_175.gif){kind=small} |
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, Vector[column](%id = 137697344)](images/SIS_177.gif){kind=small} |
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, Vector[column](%id = 143472596)](images/SIS_179.gif){kind=small} |
(17) |
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| > | ![display(df, seq(phasedia2(F, P0, 10), P0 = [`<,>`(96.5, 96.5), `<,>`(96.5, 97.5)])); 1](images/SIS_183.gif){kind=small} |
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