SIR models
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We consider SIR models for a population of total size
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(1) |
SIR epidemic
We will go through three examples of epidemic SIR models with diminishing transmission coefficients α=0.01, 0.005, 0.003 and fixed removal rate γ=0.25. The fixed removal rate of 0.25 means that the infected individuals are infected for 4 days.
Example 1: Consider the SIR model with transmission coefficient α=0.01 and removal rate γ=0.25
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![proc (P) options operator, arrow; VectorCalculus:-`<,>`(VectorCalculus:-`+`(P[1], VectorCalculus:-`-`(VectorCalculus:-`*`(VectorCalculus:-`*`(0.1e-1, P[1]), P[2]))), VectorCalculus:-`+`(VectorCalculus...](images/SIR_87.gif)
![proc (P) options operator, arrow; VectorCalculus:-`<,>`(VectorCalculus:-`+`(P[1], VectorCalculus:-`-`(VectorCalculus:-`*`(VectorCalculus:-`*`(0.1e-1, P[1]), P[2]))), VectorCalculus:-`+`(VectorCalculus...](images/SIR_88.gif) |
(2) |
has relative removal rate
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and contact number
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)))); 1](images/SIR_91.gif) |
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(4) |
This disease needs at least 25 susceptibles in order to spread. An invidual in this population has 4 potentially contagious contacts over 4 days.
Here is the time plot for the population orbit through P0=<95,5,0>
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![display(timeplot(F, `<,>`(95, 5, 0), 40), implicitplot([y = 25, x = 6.81], x = 0 .. 40, y = 0 .. 100, color = magenta)); 1](images/SIR_93.gif)
![display(timeplot(F, `<,>`(95, 5, 0), 40), implicitplot([y = 25, x = 6.81], x = 0 .. 40, y = 0 .. 100, color = magenta)); 1](images/SIR_94.gif) |
The maximal number of infected occurs when S=ρ after 6-7 time steps.
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![I[max] = max(seq(P[2], P = orbit(F, `<,>`(95, 5, 0), 20))); 1](images/SIR_96.gif) |
![I[max] = 47.31512962](images/SIR_97.gif) |
(5) |
and there remain no
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![S[infinity] = iterate(F, `<,>`(95, 5, 0), 30)[1]; 1](images/SIR_98.gif) |
![S[infinity] = .7920061990](images/SIR_99.gif) |
(6) |
susceptibles in the population after the epidemic.
Here is the directional field in the SI-phase plane
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![display([dirfield(F, 100, 20, N), implicitplot(P[1] = rho(F), P[1] = 0 .. 100, P[2] = 0 .. 20, color = magenta)]); 1](images/SIR_100.gif) |
and the orbits through various points
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![display([seq(phasedia(F, P0, 20), P0 = [seq(`<,>`(VectorCalculus:-`+`(100, VectorCalculus:-`-`(VectorCalculus:-`*`(k, 5))), VectorCalculus:-`*`(k, 5), 0), k = 1 .. 10)]), implicitplot(P[1] = rho(F), P...](images/SIR_102.gif)
![display([seq(phasedia(F, P0, 20), P0 = [seq(`<,>`(VectorCalculus:-`+`(100, VectorCalculus:-`-`(VectorCalculus:-`*`(k, 5))), VectorCalculus:-`*`(k, 5), 0), k = 1 .. 10)]), implicitplot(P[1] = rho(F), P...](images/SIR_103.gif) |
The maximal number of infected lies in the range 45-62 and there are no susceptibles left after the epidemic.
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![I[max] = seq(evalf[3](max(seq(P[2], P = orbit(F, P0, 20)))), P0 = [seq(`<,>`(VectorCalculus:-`+`(100, VectorCalculus:-`-`(VectorCalculus:-`*`(5, k))), VectorCalculus:-`*`(5, k), 0), k = 1 .. 10)]); 1](images/SIR_105.gif)
![I[max] = seq(evalf[3](max(seq(P[2], P = orbit(F, P0, 20)))), P0 = [seq(`<,>`(VectorCalculus:-`+`(100, VectorCalculus:-`-`(VectorCalculus:-`*`(5, k))), VectorCalculus:-`*`(5, k), 0), k = 1 .. 10)]); 1](images/SIR_106.gif) |
![I[max] = (47.3, 49.2, 50.7, 51.3, 54.3, 55.3, 57.5, 59.9, 61.7, 62.5)](images/SIR_107.gif) |
(7) |
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![S[infinity] = seq(evalf[3](iterate(F, P0, 30)[1]), P0 = [seq(`<,>`(VectorCalculus:-`+`(100, VectorCalculus:-`-`(VectorCalculus:-`*`(5, k))), VectorCalculus:-`*`(5, k), 0), k = 1 .. 10)]); 1](images/SIR_108.gif) |
![S[infinity] = (.796, .696, .638, .572, .493, .437, .372, .328, .275, .230)](images/SIR_109.gif) |
(8) |
Example 2: The SIR model with transmission coefficient α=0.005 and removal rate ρ=0.25 models an epidemic that is less contagious.
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![proc (P) options operator, arrow; VectorCalculus:-`<,>`(VectorCalculus:-`+`(P[1], VectorCalculus:-`-`(VectorCalculus:-`*`(VectorCalculus:-`*`(0.5e-2, P[1]), P[2]))), VectorCalculus:-`+`(VectorCalculus...](images/SIR_111.gif)
![proc (P) options operator, arrow; VectorCalculus:-`<,>`(VectorCalculus:-`+`(P[1], VectorCalculus:-`-`(VectorCalculus:-`*`(VectorCalculus:-`*`(0.5e-2, P[1]), P[2]))), VectorCalculus:-`+`(VectorCalculus...](images/SIR_112.gif) |
(9) |
has relative removal rate
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(10) |
and contact number
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(11) |
This disease needs at least 50 susceptibles in order to spread. An invidual in this population has 2 potentially contagious contacts over 4 days.
Here is the time plot for the population orbit through <95,5,0>
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![display(timeplot(F, `<,>`(95, 5, 0), 40), implicitplot([y = 50, x = 11.5], x = 0 .. 40, y = 0 .. 100, color = magenta)); 1](images/SIR_117.gif)
![display(timeplot(F, `<,>`(95, 5, 0), 40), implicitplot([y = 50, x = 11.5], x = 0 .. 40, y = 0 .. 100, color = magenta)); 1](images/SIR_118.gif) |
The maximal number of infected occurs when S=ϱ
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![I[max] = max(seq(P[2], P = orbit(F, `<,>`(95, 5, 0), 30))); 1](images/SIR_120.gif) |
![I[max] = 19.07431229](images/SIR_121.gif) |
(12) |
and there remain
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![S[infinity] = iterate(F, `<,>`(95, 5, 0), 30)[1]; 1](images/SIR_122.gif) |
![S[infinity] = 18.02571494](images/SIR_123.gif) |
(13) |
susceptibles in the population after the epidemic.
Here are several population orbits in the SI-phase plane
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![display([dirfield(F, 100, 29, 100), seq(phasedia(F, P0, 50), P0 = [seq(`<,>`(VectorCalculus:-`+`(100, VectorCalculus:-`-`(VectorCalculus:-`*`(2, k))), VectorCalculus:-`*`(2, k), 0), k = 1 .. 10)]), im...](images/SIR_124.gif)
![display([dirfield(F, 100, 29, 100), seq(phasedia(F, P0, 50), P0 = [seq(`<,>`(VectorCalculus:-`+`(100, VectorCalculus:-`-`(VectorCalculus:-`*`(2, k))), VectorCalculus:-`*`(2, k), 0), k = 1 .. 10)]), im...](images/SIR_125.gif) |
The maximal number of infected lies in the range 19-50 and there are 6-17 susceptibles left after the epidemic.
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![I[max] = seq(evalf[3](max(seq(P[2], P = orbit(F, P0, 50)))), P0 = [seq(`<,>`(VectorCalculus:-`+`(100, VectorCalculus:-`-`(VectorCalculus:-`*`(5, k))), VectorCalculus:-`*`(5, k), 0), k = 1 .. 10)]); 1](images/SIR_127.gif)
![I[max] = seq(evalf[3](max(seq(P[2], P = orbit(F, P0, 50)))), P0 = [seq(`<,>`(VectorCalculus:-`+`(100, VectorCalculus:-`-`(VectorCalculus:-`*`(5, k))), VectorCalculus:-`*`(5, k), 0), k = 1 .. 10)]); 1](images/SIR_128.gif) |
![I[max] = (19.2, 21.8, 24.7, 28.1, 31.4, 34.6, 38.3, 42.0, 46.2, 50.)](images/SIR_129.gif) |
(14) |
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![S[infinity] = seq(evalf[3](iterate(F, P0, 50)[1]), P0 = [seq(`<,>`(VectorCalculus:-`+`(100, VectorCalculus:-`-`(VectorCalculus:-`*`(5, k))), VectorCalculus:-`*`(5, k), 0), k = 1 .. 10)]); 1](images/SIR_130.gif) |
![S[infinity] = (17.1, 15.8, 14.3, 12.7, 11.6, 10.5, 9.26, 8.25, 7.35, 6.38)](images/SIR_131.gif) |
(15) |
Example 3:The SIR model with transmission coefficient α=0.003 and removal rate γ=0.25
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![proc (P) options operator, arrow; VectorCalculus:-`<,>`(VectorCalculus:-`+`(P[1], VectorCalculus:-`-`(VectorCalculus:-`*`(VectorCalculus:-`*`(0.3e-2, P[1]), P[2]))), VectorCalculus:-`+`(VectorCalculus...](images/SIR_133.gif)
![proc (P) options operator, arrow; VectorCalculus:-`<,>`(VectorCalculus:-`+`(P[1], VectorCalculus:-`-`(VectorCalculus:-`*`(VectorCalculus:-`*`(0.3e-2, P[1]), P[2]))), VectorCalculus:-`+`(VectorCalculus...](images/SIR_134.gif) |
(16) |
has relative removal rate
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(17) |
and contact number
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(18) |
This disease needs at least 83 susceptibles in order to spread. An invidual in this population has 1.2 potentially contagious contacts over 4 days.
Here is the time plot for the population orbit through <95,5,0>
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![display(timeplot(F, `<,>`(95, 5, 0), 40), implicitplot([y = rho(F), x = 9], x = 0 .. 40, y = 0 .. 100, color = magenta)); 1](images/SIR_139.gif)
![display(timeplot(F, `<,>`(95, 5, 0), 40), implicitplot([y = rho(F), x = 9], x = 0 .. 40, y = 0 .. 100, color = magenta)); 1](images/SIR_140.gif) |
The maximal number of infected occurs when S=ϱ
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![I[max] = max(seq(P[2], P = orbit(F, `<,>`(95, 5, 0), 30))); 1](images/SIR_142.gif) |
![I[max] = 5.838911423](images/SIR_143.gif) |
(19) |
and there remain
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![S[infinity] = iterate(F, `<,>`(95, 5, 0), 30)[1]; 1](images/SIR_144.gif) |
![S[infinity] = 61.42707396](images/SIR_145.gif) |
(20) |
susceptibles in the population after the epidemic.
Here are several population orbits in the SI-phase plane
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![display([dirfield(F, 100, 22, 100), seq(phasedia(F, P0, 40), P0 = [seq(`<,>`(VectorCalculus:-`+`(100, VectorCalculus:-`-`(VectorCalculus:-`*`(2, k))), VectorCalculus:-`*`(2, k), 0), k = 1 .. 10)]), im...](images/SIR_146.gif)
![display([dirfield(F, 100, 22, 100), seq(phasedia(F, P0, 40), P0 = [seq(`<,>`(VectorCalculus:-`+`(100, VectorCalculus:-`-`(VectorCalculus:-`*`(2, k))), VectorCalculus:-`*`(2, k), 0), k = 1 .. 10)]), im...](images/SIR_147.gif) |
The maximal number of infected lies in the range 5-50 and there remain 18-58 susceptibles in the population after the epidemic.
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![I[max] = seq(evalf[3](max(seq(P[2], P = orbit(F, P0, 40)))), P0 = [seq(`<,>`(VectorCalculus:-`+`(100, VectorCalculus:-`-`(VectorCalculus:-`*`(5, k))), VectorCalculus:-`*`(5, k), 0), k = 1 .. 10)]); 1](images/SIR_149.gif)
![I[max] = seq(evalf[3](max(seq(P[2], P = orbit(F, P0, 40)))), P0 = [seq(`<,>`(VectorCalculus:-`+`(100, VectorCalculus:-`-`(VectorCalculus:-`*`(5, k))), VectorCalculus:-`*`(5, k), 0), k = 1 .. 10)]); 1](images/SIR_150.gif) |
![I[max] = (5.86, 10.4, 15., 20., 25., 30., 35., 40., 45., 50.)](images/SIR_151.gif) |
(21) |
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![S[infinity] = seq(evalf[3](iterate(F, P0, 40)[1]), P0 = [seq(`<,>`(VectorCalculus:-`+`(100, VectorCalculus:-`-`(VectorCalculus:-`*`(5, k))), VectorCalculus:-`*`(5, k), 0), k = 1 .. 10)]); 1](images/SIR_152.gif) |
![S[infinity] = (58.0, 48.3, 41.9, 37.4, 33.3, 29.3, 26.0, 23.3, 20.5, 18.0)](images/SIR_153.gif) |
(22) |
SIR endemic
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 `(`+`(`*`(`+`(1, `-`(mu)), `*`(P[1])), `*`(mu, `*`(N)), `-`(`*`(alph..." align="center" border="0">
 `(`+`(`*`(`+`(1, `-`(mu)), `*`(P[1])), `*`(mu, `*`(N)), `-`(`*`(alph..." align="center" border="0">
 `(`+`(`*`(`+`(1, `-`(mu)), `*`(P[1])), `*`(mu, `*`(N)), `-`(`*`(alph..." align="center" border="0">
 `(`+`(`*`(`+`(1, `-`(mu)), `*`(P[1])), `*`(mu, `*`(N)), `-`(`*`(alph..." align="center" border="0">
 `(`+`(`*`(`+`(1, `-`(mu)), `*`(P[1])), `*`(mu, `*`(N)), `-`(`*`(alph..." align="center" border="0">
 `(`+`(`*`(`+`(1, `-`(mu)), `*`(P[1])), `*`(mu, `*`(N)), `-`(`*`(alph..." align="center" border="0">
 `(`+`(`*`(`+`(1, `-`(mu)), `*`(P[1])), `*`(mu, `*`(N)), `-`(`*`(alph..." align="center" border="0">
 `(`+`(`*`(`+`(1, `-`(mu)), `*`(P[1])), `*`(mu, `*`(N)), `-`(`*`(alph..." align="center" border="0">
 `(`+`(`*`(`+`(1, `-`(mu)), `*`(P[1])), `*`(mu, `*`(N)), `-`(`*`(alph..." align="center" border="0">
 `(`+`(`*`(`+`(1, `-`(mu)), `*`(P[1])), `*`(mu, `*`(N)), `-`(`*`(alph..." align="center" border="0">
 `(`+`(`*`(`+`(1, `-`(mu)), `*`(P[1])), `*`(mu, `*`(N)), `-`(`*`(alph..." align="center" border="0"> |
General theory
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![equi2(proc (P) options operator, arrow; `<,>`(F(P)[1], F(P)[2]) end proc); 1](images/SIR_183.gif) |
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(25) |
Example 1
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![equi2(proc (P) options operator, arrow; `<,>`(F(P)[1], F(P)[2]) end proc); 1](images/SIR_204.gif) |
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![display([timeplot(F, `<,>`(95, 5, 0), 40), plot(rhoend(F), 0 .. 40, color = magenta)]); 1](images/SIR_211.gif) |
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![equi2(proc (P) options operator, arrow; `<,>`(F(P)[1], F(P)[2]) end proc); 1](images/SIR_215.gif) |
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Example 2
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![display([timeplot(F, `<,>`(95, 5, 0), 70), plot(rhoend(F), 0 .. 70, color = magenta)]); 1](images/SIR_228.gif) |
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![equi2(proc (P) options operator, arrow; `<,>`(F(P)[1], F(P)[2]) end proc); 1](images/SIR_232.gif) |
Example 3
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![display([timeplot(F, `<,>`(95, 5, 0), 70), plot(rhoend(F), 0 .. 70, color = magenta)]); 1](images/SIR_243.gif) |
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![equi2(proc (P) options operator, arrow; `<,>`(F(P)[1], F(P)[2]) end proc); 1](images/SIR_247.gif) |
![proc (P) options operator, arrow; VectorCalculus:-`<,>`(VectorCalculus:-`+`(VectorCalculus:-`+`(VectorCalculus:-`*`(VectorCalculus:-`+`(1, VectorCalculus:-`-`(mu)), P[1]), VectorCalculus:-`*`(mu, N)),...](images/SIR_180.gif){kind=small}
![proc (P) options operator, arrow; VectorCalculus:-`<,>`(VectorCalculus:-`+`(VectorCalculus:-`+`(VectorCalculus:-`*`(VectorCalculus:-`+`(1, VectorCalculus:-`-`(mu)), P[1]), VectorCalculus:-`*`(mu, N)),...](images/SIR_181.gif){kind=small}
![proc (P) options operator, arrow; VectorCalculus:-`<,>`(VectorCalculus:-`+`(VectorCalculus:-`+`(VectorCalculus:-`*`(VectorCalculus:-`+`(1, VectorCalculus:-`-`(mu)), P[1]), VectorCalculus:-`*`(mu, N)),...](images/SIR_182.gif){kind=small}
![[[P[1] = 100, P[2] = 0], [P[1] = `/`(`*`(`+`(mu, gamma)), `*`(alpha)), P[2] = `+`(`-`(`/`(`*`(mu, `*`(`+`(mu, gamma, `-`(`*`(100, `*`(alpha)))))), `*`(alpha, `*`(`+`(mu, gamma))))))]]](images/SIR_185.gif){kind=small}
, Vector[column](%id = 140259432)](images/SIR_187.gif){kind=small}
, Vector[column](%id = 140259432)](images/SIR_188.gif){kind=small}
, Vector[column](%id = 140259432)](images/SIR_189.gif){kind=small}
, Vector[column](%id = 140259432)](images/SIR_190.gif){kind=small}
, Vector[column](%id = 140259432)](images/SIR_191.gif){kind=small}
, Vector[column](%id = 143569992)](images/SIR_193.gif){kind=small}
, Vector[column](%id = 143569992)](images/SIR_194.gif){kind=small}
, Vector[column](%id = 143569992)](images/SIR_195.gif){kind=small}
, Vector[column](%id = 143569992)](images/SIR_196.gif){kind=small}
, Vector[column](%id = 143569992)](images/SIR_197.gif){kind=small}
![proc (P) options operator, arrow; VectorCalculus:-`<,>`(VectorCalculus:-`+`(VectorCalculus:-`+`(VectorCalculus:-`*`(VectorCalculus:-`+`(1, VectorCalculus:-`-`(0.5e-1)), P[1]), VectorCalculus:-`*`(0.5e...](images/SIR_201.gif){kind=small}
![proc (P) options operator, arrow; VectorCalculus:-`<,>`(VectorCalculus:-`+`(VectorCalculus:-`+`(VectorCalculus:-`*`(VectorCalculus:-`+`(1, VectorCalculus:-`-`(0.5e-1)), P[1]), VectorCalculus:-`*`(0.5e...](images/SIR_202.gif){kind=small}
![proc (P) options operator, arrow; VectorCalculus:-`<,>`(VectorCalculus:-`+`(VectorCalculus:-`+`(VectorCalculus:-`*`(VectorCalculus:-`+`(1, VectorCalculus:-`-`(0.5e-1)), P[1]), VectorCalculus:-`*`(0.5e...](images/SIR_203.gif){kind=small}
![[[P[1] = 100., P[2] = 0.], [P[1] = 30., P[2] = 11.66666667]]](images/SIR_206.gif){kind=small}
, Vector[column](%id = 140304240)](images/SIR_208.gif){kind=small}
, Vector[column](%id = 138651172)](images/SIR_210.gif){kind=small}
![[[P[1] = 100., P[2] = 0.], [P[1] = 30., P[2] = 11.66666667]]](images/SIR_217.gif){kind=small}
, Vector[column](%id = 143279472)](images/SIR_219.gif){kind=small}
, Vector[column](%id = 137453256)](images/SIR_221.gif){kind=small}
![proc (P) options operator, arrow; VectorCalculus:-`<,>`(VectorCalculus:-`+`(VectorCalculus:-`+`(VectorCalculus:-`*`(VectorCalculus:-`+`(1, VectorCalculus:-`-`(0.5e-1)), P[1]), VectorCalculus:-`*`(0.5e...](images/SIR_225.gif){kind=small}
![proc (P) options operator, arrow; VectorCalculus:-`<,>`(VectorCalculus:-`+`(VectorCalculus:-`+`(VectorCalculus:-`*`(VectorCalculus:-`+`(1, VectorCalculus:-`-`(0.5e-1)), P[1]), VectorCalculus:-`*`(0.5e...](images/SIR_226.gif){kind=small}
![proc (P) options operator, arrow; VectorCalculus:-`<,>`(VectorCalculus:-`+`(VectorCalculus:-`+`(VectorCalculus:-`*`(VectorCalculus:-`+`(1, VectorCalculus:-`-`(0.5e-1)), P[1]), VectorCalculus:-`*`(0.5e...](images/SIR_227.gif){kind=small}
![[[P[1] = 100., P[2] = 0.], [P[1] = 60., P[2] = 6.666666667]]](images/SIR_234.gif){kind=small}
, Vector[column](%id = 137599716)](images/SIR_236.gif){kind=small}
, Vector[column](%id = 143517256)](images/SIR_238.gif){kind=small}
![proc (P) options operator, arrow; VectorCalculus:-`<,>`(VectorCalculus:-`+`(VectorCalculus:-`+`(VectorCalculus:-`*`(VectorCalculus:-`+`(1, VectorCalculus:-`-`(0.5e-1)), P[1]), VectorCalculus:-`*`(0.5e...](images/SIR_240.gif){kind=small}
![proc (P) options operator, arrow; VectorCalculus:-`<,>`(VectorCalculus:-`+`(VectorCalculus:-`+`(VectorCalculus:-`*`(VectorCalculus:-`+`(1, VectorCalculus:-`-`(0.5e-1)), P[1]), VectorCalculus:-`*`(0.5e...](images/SIR_241.gif){kind=small}
![proc (P) options operator, arrow; VectorCalculus:-`<,>`(VectorCalculus:-`+`(VectorCalculus:-`+`(VectorCalculus:-`*`(VectorCalculus:-`+`(1, VectorCalculus:-`-`(0.5e-1)), P[1]), VectorCalculus:-`*`(0.5e...](images/SIR_242.gif){kind=small}
![[[P[1] = 100., P[2] = 0.], [P[1] = 85.71428571, P[2] = 2.380952381]]](images/SIR_249.gif){kind=small}
, Vector[column](%id = 141060764)](images/SIR_251.gif){kind=small}
, Vector[column](%id = 145065736)](images/SIR_253.gif){kind=small}