Benutzer:GeorgQuaas/Mathematische Modellierung der Epidemiologie

$\lambda (t)=\beta I(t)=\beta Ni(t)$

${\frac {d\lambda }{dt}}=\lambda (t)\beta Ns(t)-\left(\upsilon +\mu \right)\lambda (t)$

$N={\text{konst.}}$

$S(t)=Ns(t)$

$R_{0}=\beta N/(\upsilon +\mu )$

${\frac {d\lambda }{dt}}=\lambda (t)R_{0}s(t)(\upsilon +\mu )-(\upsilon +\mu )\lambda (t)=(\upsilon +\mu )\lambda (t)\left[R_{0}s(t)-1\right]$

${\frac {ds}{dt}}=\mu -\left[\mu +\lambda (t)\right]s(t)$

${\frac {d\lambda }{dt}}=(\upsilon +\mu )\lambda (t)\left[R_{0}s(t)-1\right]$

$s(t)=\exp \left\lbrace -\int _{0}^{t}\lambda (u)du\right\rbrace$.

${\frac {d\lambda }{ds}}=-{\frac {\upsilon (R_{0}s-1)}{s(t)}}$

${\frac {di}{ds}}={\frac {1}{R_{0}s(t)}}-1$

$i(t)=1-s(t)+{\frac {\ln {s(t)}}{R_{0}}}$

$R_{0}=1/s^{*}$

$i_{max}=1-{\frac {1}{R_{0}}}+{\frac {-\ln {R_{0}}}{R_{0}}}=1-{\frac {1+\ln {R_{0}}}{R_{0}}}$

$R_{0}=3$

$i_{max}\approx 0,3$

$\mu =0$

$\upsilon \gg \mu$

$s\approx 1$

$D=1/\upsilon$

$L=1/\mu$

${\frac {d\lambda }{dt}}=\upsilon \left(R_{0}-1\right)\lambda (t)$

$\Lambda =\upsilon \left(R_{0}-1\right)$

$\lambda (t)=\lambda (0)exp{(\Lambda t)}$

$\lambda (t)=\beta I(t)$

$\Delta I$

$R_{eff}=sR_{0}$

$s^{*}=1/{R_{0}}$

${\frac {ds}{dt}}=0=\mu -\left[\mu +\lambda (t)\right]s(t)$

$\mu =\mu s^{*}+\lambda ^{*}s^{*}={\frac {\mu +\lambda ^{*}}{R_{0}}}$

$\lambda ^{*}=\mu (R_{0}-1)$

$\lambda (t)=\beta I(t)$

$\beta I^{*}=\mu (R_{0}-1)$

$i^{*}={\frac {\mu }{\beta N}}(R_{0}-1)$

$R_{0}\equiv \beta N/(\upsilon +\mu )$

$i^{*}={\frac {\upsilon }{\upsilon +\mu }}(1-1/R_{0})$

$s(t)=s^{*}+\zeta (t)=1/R_{0}+\zeta (t)$

$\lambda (t)=\lambda ^{*}+\xi (t)=\mu (R_{0}-1)+\xi (t)$

$\Lambda ^{2}+\mu R_{0}\Lambda +\upsilon \mu (R_{0}-1)=0$

$A={\frac {1}{\mu (R_{0}-1}}\approx {\frac {1}{\mu R_{0}}}$

$D=1/\upsilon$

$\Lambda ^{2}+{\frac {\Lambda }{A}}+{\frac {1}{AD}}=0$

$\Lambda \approx -{\frac {1}{2A}}\pm {\sqrt {-{\frac {1}{AD}}}}$

$\zeta (t)=\exp {\left(-{\frac {1}{AD}}t\right)}\left[\cos {\sqrt {{\frac {1}{AD}}t}}\pm i\sin {\sqrt {{\frac {1}{AD}}t}}\right]$

$M(a)=N(0)exp{(-da)}$

${\frac {\partial {S(a,t)}}{\partial {t}}}+{\frac {\partial {S(a,t)}}{\partial {a}}}=-\left[\lambda (t)+\mu (a)\right]S$

${\frac {\partial {E(a,t)}}{\partial {t}}}+{\frac {\partial {E(a,t)}}{\partial {a}}}=\lambda S-\left[\sigma +\mu (a)\right]E$

${\frac {\partial {I(a,t)}}{\partial {t}}}+{\frac {\partial {I(a,t)}}{\partial {a}}}=\sigma E-\left[\upsilon +\mu (a)\right]I$

${\frac {\partial {R(a,t)}}{\partial {a}}}+{\frac {\partial {R(a,t)}}{\partial {a}}}=\upsilon I-\mu (a)R$

${\frac {\partial {N(a,t)}}{\partial {a}}}+{\frac {\partial {N(a,t)}}{\partial {a}}}=-\mu (a)N$

$\lambda (a,t)=\int {\beta (a,a')I(a',t)da'}$

${\vec {I(t)}}=\left(I_{1}(t),I_{2}(t),I_{3}(t)\right)$

${\boldsymbol {\beta }}={\begin{vmatrix}\beta _{11}&\beta _{12}&\beta _{13}\\\beta _{21}&\beta _{22}&\beta _{23}\\\beta _{31}&\beta _{32}&\beta _{33}\end{vmatrix}}$

${\vec {\lambda (t)}}={\boldsymbol {\beta }}{\vec {I(t)}}$

$\beta _{ik}$

$R_{eff}\leq R_{0}(1-p)$

$p_{c}=1-(1/R_{0})$

$S(a)=(1-p)N(0)l(a)exp{(-\lambda 'a)}$

$\lambda '<\lambda$

$\lambda '=\mu R_{0}(p_{c}-p)$

$\lambda =\beta \int _{0}^{\infty }I(a)\,da$

$I(a)=\lambda N(0)l(a)(e^{-\upsilon a}-e^{-\lambda a})/(\lambda -\upsilon )$

$I={\frac {\lambda \mu N}{(\mu +\upsilon )(\mu +\lambda )}}$

$\lambda =\beta I$

$(\mu +\upsilon )(\mu +\lambda )=\beta \mu N$

$\lambda =\mu \left({\frac {\beta N}{\mu +\upsilon }}-1\right)$

$R_{0}=1+(\lambda /\mu )=1+{\frac {\beta N}{\mu +\upsilon }}-1=\beta N/(\mu +\upsilon )$