PDE SEIR Model

Available identifiers

MaRDI QIDQ6775423

susceptible, exposed, infectious, and removed model employing partial differential equations

Description

Spatial spreading model of SEIR (Susceptible, Exposed, Infectious, and Removed) type in a domain Ω ⊂ ℝ², modeling both the SEIR dynamics and spatial diffusion of infectious individuals. Employing partial differential equations.

List of mathematical expressions with quantities

Allee Effect	$1-\frac{A}{n+n_0}\ge \frac{1}{3}$
$A$ symbol represents Allee Threshold
$n$ symbol represents Population Density
Integral of the Population Density Fraction of Exposed (Initial Condition)	$\underset{{\mathrm{\Omega }}^{(l)}}{\int }e(x,0)n(x)dx={\mathrm{\Sigma }}_{{ℰ}}{\widehat{{ℐ}}}^{(l)}$
${\mathrm{\Sigma }}_{{ℰ}}$ symbol represents Coefficient Scaling Infectious to Exposed
${\widehat{{ℐ}}}^{(l)}$ symbol represents Number of Infectious Individuals
$e(x,0)$ symbol represents Fraction of Population Density of Exposed
$n(x)$ symbol represents Total Population Density
Integral of the Population Density Fraction of Infectious (Initial Condition)	$\underset{{\mathrm{\Omega }}^{(l)}}{\int }i(x,0)n(x)dx={\widehat{{ℐ}}}^{(l)}$
${\widehat{{ℐ}}}^{(l)}$ symbol represents Number of Infectious Individuals
$i(x,0)$ symbol represents Fraction of Population Density of Infectious
$n(x)$ symbol represents Total Population Density
Integral of the Population Density Fraction of Susceptibles (Initial Condition)	$\underset{{\mathrm{\Omega }}^{(l)}}{\int }s(x,0)n(x)dx={\widehat{{𝒩}}}_l-(1+{\mathrm{\Sigma }}_{{ℰ}}){\widehat{{ℐ}}}^{(l)}-{\widehat{{ℛ}}}^{(l)}$
${\mathrm{\Sigma }}_{{ℰ}}$ symbol represents Coefficient Scaling Infectious to Exposed
${\widehat{{ℐ}}}^{(l)}$ symbol represents Number of Infectious Individuals
${\widehat{{𝒩}}}_l$ symbol represents Total Population Size
${\widehat{{ℛ}}}^{(l)}$ symbol represents Number of Removed Individuals
$n(x)$ symbol represents Total Population Density
$s(x,0)$ symbol represents Fraction of Population Density of Susceptibles
Integral of the Total Population Density (Initial Condition)	$\underset{{\mathrm{\Omega }}^{(l)}}{\int }n(x)dx={\widehat{{𝒩}}}_l$
${\widehat{{𝒩}}}_l$ symbol represents Total Population Size
$n(x)$ symbol represents Total Population Density
Isotropic Gaussian Function Formulation	$G^{(l)}(x)\equiv \frac{1}{2\pi {\sigma }^2}\text{exp}(-\frac{||x-x_0^{(l)}|{|}^2}{2{\sigma }^2})$
$G^{(l)}(x)$ symbol represents Isotropic Gaussian Function
$\pi$ symbol represents Pi Number
$\sigma$ symbol represents Variance
$x_0^{(l)}$ symbol represents Center of Province
Fraction of Population Density of Exposed Formulation	$e(x,0)=\sum _{\tilde{l}=1}^{\tilde{L}}{w}_e^{(\tilde{l})}{G}^{(\tilde{l})}(x)$
$G^{(\tilde{l})}(x)$ symbol represents Isotropic Gaussian Function
$e(x,0)$ symbol represents Fraction of Population Density of Exposed
$w_e^{(\tilde{l})}$ symbol represents Density Fraction Coefficient
Fraction of Population Density of Infectious Formulation	$i(x,0)=\sum _{\tilde{l}=1}^{\tilde{L}}{w}_i^{(\tilde{l})}{G}^{(\tilde{l})}(x)$
$G^{(\tilde{l})}(x)$ symbol represents Isotropic Gaussian Function
$i(x,0)$ symbol represents Fraction of Population Density of Infectious
$w_i^{(\tilde{l})}$ symbol represents Density Fraction Coefficient
Fraction of Population Density of Susceptibles Formulation	$s(x,0)=\sum _{\tilde{l}=1}^{\tilde{L}}{w}_s^{(\tilde{l})}{G}^{(\tilde{l})}(x)$
$G^{(\tilde{l})}(x)$ symbol represents Isotropic Gaussian Function
$s(x,0)$ symbol represents Fraction of Population Density of Susceptibles
$w_s^{(\tilde{l})}$ symbol represents Density Fraction Coefficient
Number of Exposed Individuals Formulation	${\widehat{{ℰ}}}^{(l)}={\mathrm{\Sigma }}_{{ℰ}}{\widehat{{ℐ}}}^{(l)}$
${\mathrm{\Sigma }}_{{ℰ}}$ symbol represents Coefficient Scaling Infectious to Exposed
${\widehat{{ℰ}}}^{(l)}$ symbol represents Number of Exposed Individuals
${\widehat{{ℐ}}}^{(l)}$ symbol represents Number of Infectious Individuals
Number of Susceptible Individuals Formulation	${\widehat{{𝒮}}}^{(l)}={\widehat{{𝒩}}}^{(l)}-(1+{\mathrm{\Sigma }}_{{ℰ}}){\widehat{{ℐ}}}^{(l)}-{\widehat{{ℛ}}}^{(l)}$
${\mathrm{\Sigma }}_{{ℰ}}$ symbol represents Coefficient Scaling Infectious to Exposed
${\widehat{{ℐ}}}^{(l)}$ symbol represents Number of Infectious Individuals
${\widehat{{𝒩}}}^{(l)}$ symbol represents Total Population Size
${\widehat{{ℛ}}}^{(l)}$ symbol represents Number of Removed Individuals
${\widehat{{𝒮}}}^{(l)}$ symbol represents Number of Susceptible Individuals
Rate of Change of Population Density Fraction of Exposed PDE	${\partial }_{t}e=\mathrm{div}(D\mathrm{\nabla }e)+(1-\frac{A}{n+n_0})s({\beta }_{e}e+{\beta }_{i}i)-\sigma e-{\varphi }_{e}e$
$A$ symbol represents Allee Threshold
$D$ symbol represents Diffusion Coefficient for SEIR Model
${\beta }_e$ symbol represents Asymptomatic Infection Rate
${\beta }_i$ symbol represents Symptomatic Infection Rate
${\varphi }_e$ symbol represents Asymptomatic Recovery Rate
$\sigma$ symbol represents Rate of Becoming Infectious
$e$ symbol represents Fraction of Population Density of Exposed
$i$ symbol represents Fraction of Population Density of Infectious
$n$ symbol represents Population Density
$s$ symbol represents Fraction of Population Density of Susceptibles
$t$ symbol represents Time
Rate of Change of Population Density Fraction of Infectious PDE	${\partial }_{t}i=\mathrm{div}(D\mathrm{\nabla }i)+\sigma e-{\varphi }_{i}i$
$D$ symbol represents Diffusion Coefficient for SEIR Model
${\varphi }_i$ symbol represents Infected Recovery Rate
$\sigma$ symbol represents Rate of Becoming Infectious
$e$ symbol represents Fraction of Population Density of Exposed
$i$ symbol represents Fraction of Population Density of Infectious
$t$ symbol represents Time
Rate of Change of Population Density Fraction of Removed PDE	${\partial }_{t}r=\mathrm{div}(D\mathrm{\nabla }r)+{\varphi }_{i}i+{\varphi }_{e}e$
$D$ symbol represents Diffusion Coefficient for SEIR Model
${\varphi }_e$ symbol represents Asymptomatic Recovery Rate
${\varphi }_i$ symbol represents Infected Recovery Rate
$e$ symbol represents Fraction of Population Density of Exposed
$i$ symbol represents Fraction of Population Density of Infectious
$r$ symbol represents Fraction of Population Density of Removed
$t$ symbol represents Time
Rate of Change of Population Density Fraction of Susceptibles PDE	${\partial }_{t}s=\mathrm{div}(D\mathrm{\nabla }s)-(1-\frac{A}{n+n_0})s({\beta }_{e}e+{\beta }_{i}i)$
$A$ symbol represents Allee Threshold
$D$ symbol represents Diffusion Coefficient for SEIR Model
${\beta }_e$ symbol represents Asymptomatic Infection Rate
${\beta }_i$ symbol represents Symptomatic Infection Rate
$e$ symbol represents Fraction of Population Density of Exposed
$i$ symbol represents Fraction of Population Density of Infectious
$n$ symbol represents Population Density
$s$ symbol represents Fraction of Population Density of Susceptibles
$t$ symbol represents Time
Total Population Density Formulation	$n(x)\equiv \sum _{\tilde{l}=1}^{\tilde{L}}{w}_n^{(\tilde{l})}{G}^{(\tilde{l})}(x)$
$G^{(\tilde{l})}(x)$ symbol represents Isotropic Gaussian Function
$n(x)$ symbol represents Total Population Density
$w_n^{(\tilde{l})}$ symbol represents Density Fraction Coefficient
Zero Flux Condition	${\nu }^{T}D\mathrm{\nabla }s={\nu }^{T}D\mathrm{\nabla }e={\nu }^{T}D\mathrm{\nabla }i={\nu }^{T}D\mathrm{\nabla }r=0$
$D$ symbol represents Diffusion Coefficient
$\nu$ symbol represents Unit Normal Vector
$e$ symbol represents Fraction of Population Density of Exposed
$i$ symbol represents Fraction of Population Density of Infectious
$r$ symbol represents Fraction of Population Density of Removed
$s$ symbol represents Fraction of Population Density of Susceptibles