λ ( t ) = β I ( t ) = β N i ( t ) {\displaystyle \lambda (t)=\beta I(t)=\beta Ni(t)}
d λ d t = λ ( t ) β N s ( t ) − ( υ + μ ) λ ( t ) {\displaystyle {\frac {d\lambda }{dt}}=\lambda (t)\beta Ns(t)-\left(\upsilon +\mu \right)\lambda (t)}
N = konst. {\displaystyle N={\text{konst.}}}
S ( t ) = N s ( t ) {\displaystyle S(t)=Ns(t)}
R 0 = β N / ( υ + μ ) {\displaystyle R_{0}=\beta N/(\upsilon +\mu )}
d λ d t = λ ( t ) R 0 s ( t ) ( υ + μ ) − ( υ + μ ) λ ( t ) = ( υ + μ ) λ ( t ) [ R 0 s ( t ) − 1 ] {\displaystyle {\frac {d\lambda }{dt}}=\lambda (t)R_{0}s(t)(\upsilon +\mu )-(\upsilon +\mu )\lambda (t)=(\upsilon +\mu )\lambda (t)\left[R_{0}s(t)-1\right]}
d s d t = μ − [ μ + λ ( t ) ] s ( t ) {\displaystyle {\frac {ds}{dt}}=\mu -\left[\mu +\lambda (t)\right]s(t)}
d λ d t = ( υ + μ ) λ ( t ) [ R 0 s ( t ) − 1 ] {\displaystyle {\frac {d\lambda }{dt}}=(\upsilon +\mu )\lambda (t)\left[R_{0}s(t)-1\right]}
s ( t ) = exp { − ∫ 0 t λ ( u ) d u } {\displaystyle s(t)=\exp \left\lbrace -\int _{0}^{t}\lambda (u)du\right\rbrace } .
d λ d s = − υ ( R 0 s − 1 ) s ( t ) {\displaystyle {\frac {d\lambda }{ds}}=-{\frac {\upsilon (R_{0}s-1)}{s(t)}}}
d i d s = 1 R 0 s ( t ) − 1 {\displaystyle {\frac {di}{ds}}={\frac {1}{R_{0}s(t)}}-1}
i ( t ) = 1 − s ( t ) + ln s ( t ) R 0 {\displaystyle i(t)=1-s(t)+{\frac {\ln {s(t)}}{R_{0}}}}
R 0 = 1 / s ∗ {\displaystyle R_{0}=1/s^{*}}
i m a x = 1 − 1 R 0 + − ln R 0 R 0 = 1 − 1 + ln R 0 R 0 {\displaystyle i_{max}=1-{\frac {1}{R_{0}}}+{\frac {-\ln {R_{0}}}{R_{0}}}=1-{\frac {1+\ln {R_{0}}}{R_{0}}}}
R 0 = 3 {\displaystyle R_{0}=3}
i m a x ≈ 0 , 3 {\displaystyle i_{max}\approx 0,3}
μ = 0 {\displaystyle \mu =0}
υ ≫ μ {\displaystyle \upsilon \gg \mu }
s ≈ 1 {\displaystyle s\approx 1}
D = 1 / υ {\displaystyle D=1/\upsilon }
L = 1 / μ {\displaystyle L=1/\mu }
d λ d t = υ ( R 0 − 1 ) λ ( t ) {\displaystyle {\frac {d\lambda }{dt}}=\upsilon \left(R_{0}-1\right)\lambda (t)}
Λ = υ ( R 0 − 1 ) {\displaystyle \Lambda =\upsilon \left(R_{0}-1\right)}
λ ( t ) = λ ( 0 ) e x p ( Λ t ) {\displaystyle \lambda (t)=\lambda (0)exp{(\Lambda t)}}
λ ( t ) = β I ( t ) {\displaystyle \lambda (t)=\beta I(t)}
Δ I {\displaystyle \Delta I}
R e f f = s R 0 {\displaystyle R_{eff}=sR_{0}}
s ∗ = 1 / R 0 {\displaystyle s^{*}=1/{R_{0}}}
d s d t = 0 = μ − [ μ + λ ( t ) ] s ( t ) {\displaystyle {\frac {ds}{dt}}=0=\mu -\left[\mu +\lambda (t)\right]s(t)}
μ = μ s ∗ + λ ∗ s ∗ = μ + λ ∗ R 0 {\displaystyle \mu =\mu s^{*}+\lambda ^{*}s^{*}={\frac {\mu +\lambda ^{*}}{R_{0}}}}
λ ∗ = μ ( R 0 − 1 ) {\displaystyle \lambda ^{*}=\mu (R_{0}-1)}
β I ∗ = μ ( R 0 − 1 ) {\displaystyle \beta I^{*}=\mu (R_{0}-1)}
i ∗ = μ β N ( R 0 − 1 ) {\displaystyle i^{*}={\frac {\mu }{\beta N}}(R_{0}-1)}
R 0 ≡ β N / ( υ + μ ) {\displaystyle R_{0}\equiv \beta N/(\upsilon +\mu )}
i ∗ = υ υ + μ ( 1 − 1 / R 0 ) {\displaystyle i^{*}={\frac {\upsilon }{\upsilon +\mu }}(1-1/R_{0})}
s ( t ) = s ∗ + ζ ( t ) = 1 / R 0 + ζ ( t ) {\displaystyle s(t)=s^{*}+\zeta (t)=1/R_{0}+\zeta (t)}
λ ( t ) = λ ∗ + ξ ( t ) = μ ( R 0 − 1 ) + ξ ( t ) {\displaystyle \lambda (t)=\lambda ^{*}+\xi (t)=\mu (R_{0}-1)+\xi (t)}
Λ 2 + μ R 0 Λ + υ μ ( R 0 − 1 ) = 0 {\displaystyle \Lambda ^{2}+\mu R_{0}\Lambda +\upsilon \mu (R_{0}-1)=0}
A = 1 μ ( R 0 − 1 ≈ 1 μ R 0 {\displaystyle A={\frac {1}{\mu (R_{0}-1}}\approx {\frac {1}{\mu R_{0}}}}
Λ 2 + Λ A + 1 A D = 0 {\displaystyle \Lambda ^{2}+{\frac {\Lambda }{A}}+{\frac {1}{AD}}=0}
Λ ≈ − 1 2 A ± − 1 A D {\displaystyle \Lambda \approx -{\frac {1}{2A}}\pm {\sqrt {-{\frac {1}{AD}}}}}
ζ ( t ) = exp ( − 1 A D t ) [ cos 1 A D t ± i sin 1 A D t ] {\displaystyle \zeta (t)=\exp {\left(-{\frac {1}{AD}}t\right)}\left[\cos {\sqrt {{\frac {1}{AD}}t}}\pm i\sin {\sqrt {{\frac {1}{AD}}t}}\right]}
M ( a ) = N ( 0 ) e x p ( − d a ) {\displaystyle M(a)=N(0)exp{(-da)}}
∂ S ( a , t ) ∂ t + ∂ S ( a , t ) ∂ a = − [ λ ( t ) + μ ( a ) ] S {\displaystyle {\frac {\partial {S(a,t)}}{\partial {t}}}+{\frac {\partial {S(a,t)}}{\partial {a}}}=-\left[\lambda (t)+\mu (a)\right]S}
∂ E ( a , t ) ∂ t + ∂ E ( a , t ) ∂ a = λ S − [ σ + μ ( a ) ] E {\displaystyle {\frac {\partial {E(a,t)}}{\partial {t}}}+{\frac {\partial {E(a,t)}}{\partial {a}}}=\lambda S-\left[\sigma +\mu (a)\right]E}
∂ I ( a , t ) ∂ t + ∂ I ( a , t ) ∂ a = σ E − [ υ + μ ( a ) ] I {\displaystyle {\frac {\partial {I(a,t)}}{\partial {t}}}+{\frac {\partial {I(a,t)}}{\partial {a}}}=\sigma E-\left[\upsilon +\mu (a)\right]I}
∂ R ( a , t ) ∂ a + ∂ R ( a , t ) ∂ a = υ I − μ ( a ) R {\displaystyle {\frac {\partial {R(a,t)}}{\partial {a}}}+{\frac {\partial {R(a,t)}}{\partial {a}}}=\upsilon I-\mu (a)R}
∂ N ( a , t ) ∂ a + ∂ N ( a , t ) ∂ a = − μ ( a ) N {\displaystyle {\frac {\partial {N(a,t)}}{\partial {a}}}+{\frac {\partial {N(a,t)}}{\partial {a}}}=-\mu (a)N}
λ ( a , t ) = ∫ β ( a , a ′ ) I ( a ′ , t ) d a ′ {\displaystyle \lambda (a,t)=\int {\beta (a,a')I(a',t)da'}}
I ( t ) → = ( I 1 ( t ) , I 2 ( t ) , I 3 ( t ) ) {\displaystyle {\vec {I(t)}}=\left(I_{1}(t),I_{2}(t),I_{3}(t)\right)}
β = | β 11 β 12 β 13 β 21 β 22 β 23 β 31 β 32 β 33 | {\displaystyle {\boldsymbol {\beta }}={\begin{vmatrix}\beta _{11}&\beta _{12}&\beta _{13}\\\beta _{21}&\beta _{22}&\beta _{23}\\\beta _{31}&\beta _{32}&\beta _{33}\end{vmatrix}}}
λ ( t ) → = β I ( t ) → {\displaystyle {\vec {\lambda (t)}}={\boldsymbol {\beta }}{\vec {I(t)}}}
β i k {\displaystyle \beta _{ik}}
R e f f ≤ R 0 ( 1 − p ) {\displaystyle R_{eff}\leq R_{0}(1-p)}
p c = 1 − ( 1 / R 0 ) {\displaystyle p_{c}=1-(1/R_{0})}
S ( a ) = ( 1 − p ) N ( 0 ) l ( a ) e x p ( − λ ′ a ) {\displaystyle S(a)=(1-p)N(0)l(a)exp{(-\lambda 'a)}}
λ ′ < λ {\displaystyle \lambda '<\lambda }
λ ′ = μ R 0 ( p c − p ) {\displaystyle \lambda '=\mu R_{0}(p_{c}-p)}
λ = β ∫ 0 ∞ I ( a ) d a {\displaystyle \lambda =\beta \int _{0}^{\infty }I(a)\,da}
I ( a ) = λ N ( 0 ) l ( a ) ( e − υ a − e − λ a ) / ( λ − υ ) {\displaystyle I(a)=\lambda N(0)l(a)(e^{-\upsilon a}-e^{-\lambda a})/(\lambda -\upsilon )}
I = λ μ N ( μ + υ ) ( μ + λ ) {\displaystyle I={\frac {\lambda \mu N}{(\mu +\upsilon )(\mu +\lambda )}}}
λ = β I {\displaystyle \lambda =\beta I}
( μ + υ ) ( μ + λ ) = β μ N {\displaystyle (\mu +\upsilon )(\mu +\lambda )=\beta \mu N}
λ = μ ( β N μ + υ − 1 ) {\displaystyle \lambda =\mu \left({\frac {\beta N}{\mu +\upsilon }}-1\right)}
R 0 = 1 + ( λ / μ ) = 1 + β N μ + υ − 1 = β N / ( μ + υ ) {\displaystyle R_{0}=1+(\lambda /\mu )=1+{\frac {\beta N}{\mu +\upsilon }}-1=\beta N/(\mu +\upsilon )}
![{\displaystyle {\frac {d\lambda }{dt}}=\lambda (t)R_{0}s(t)(\upsilon +\mu )-(\upsilon +\mu )\lambda (t)=(\upsilon +\mu )\lambda (t)\left[R_{0}s(t)-1\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/13452ebba395b17132a27dc842032c9a3db5ff1f)
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![{\displaystyle \zeta (t)=\exp {\left(-{\frac {1}{AD}}t\right)}\left[\cos {\sqrt {{\frac {1}{AD}}t}}\pm i\sin {\sqrt {{\frac {1}{AD}}t}}\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/eecd16789d7f65436dfd9588c00ae2cf3d5825f0)
![{\displaystyle {\frac {\partial {S(a,t)}}{\partial {t}}}+{\frac {\partial {S(a,t)}}{\partial {a}}}=-\left[\lambda (t)+\mu (a)\right]S}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ed381cabd0d3606f40e9225b6485c413df53c3cd)
![{\displaystyle {\frac {\partial {E(a,t)}}{\partial {t}}}+{\frac {\partial {E(a,t)}}{\partial {a}}}=\lambda S-\left[\sigma +\mu (a)\right]E}](https://wikimedia.org/api/rest_v1/media/math/render/svg/370e7fab4fbceaae606a854e5878635306975f78)
![{\displaystyle {\frac {\partial {I(a,t)}}{\partial {t}}}+{\frac {\partial {I(a,t)}}{\partial {a}}}=\sigma E-\left[\upsilon +\mu (a)\right]I}](https://wikimedia.org/api/rest_v1/media/math/render/svg/39a08008304fe4c0ab50bda4aad189308eb60b03)
