Difference between revisions of "SIR Modeling"

Revision as of 13:58, 2 April 2020

Compartmental Modeling

Discrete-time SIR modeling of infections/recovery

The model consists of individuals who are either Susceptible ([math]S[/math]), Infected ([math]I[/math]), or Recovered ([math]R[/math]).

The epidemic proceeds via a growth and decline process. This is the core model of infectious disease spread and has been in use in epidemiology for many years.

The dynamics are given by the following 3 equations.

[math] \begin{align} g_{\mu \nu} & = [S1] \\[6pt] {R^\mu}_{\alpha \beta \gamma} & = [S2] \times \left(\Gamma^\mu_{\alpha \gamma,\beta}-\Gamma^\mu_{\alpha \beta,\gamma}+\Gamma^\mu_{\sigma \beta}\Gamma^\sigma_{\gamma \alpha}-\Gamma^\mu_{\sigma \gamma}\Gamma^\sigma_{\beta \alpha}\right) \\[6pt] G_{\mu \nu} & = [S3] \times \frac{8 \pi G}{c^4} T_{\mu \nu} \end{align} [/math]

[math] \begin{align} S_{t+1} = S_t - \beta S_t I_t) \\[6pt] I_{t+1} = I_t + \beta S_t I_t - \gamma I_t \\[6pt] R_{t+1} = R_t + \gamma I_t \end{align} [/math]

[math]S_{t+1} = S_t - \beta S_t I_t[/math]

[math]I_{t+1} = I_t + \beta S_t I_t - \gamma I_t[/math]

[math]R_{t+1} = R_t + \gamma I_t[/math]

where;

[math]S[/math] number of susceptible individuals,

[math]I[/math] number of infected individuals,

[math]R[/math] number of recovered individuals,

[math]\beta[/math] the average number of contacts per person per time,

[math]\gamma[/math] the transition rate.

@@ Line 12: / Line 12: @@
 :<math>
 \begin{align}
-g_{\mu \nu} & = [S1] \times \operatorname{diag}(-1,+1,+1,+1) \\[6pt]
+g_{\mu \nu} & = [S1] \\[6pt]
 {R^\mu}_{\alpha \beta \gamma} & = [S2] \times \left(\Gamma^\mu_{\alpha \gamma,\beta}-\Gamma^\mu_{\alpha \beta,\gamma}+\Gamma^\mu_{\sigma \beta}\Gamma^\sigma_{\gamma \alpha}-\Gamma^\mu_{\sigma \gamma}\Gamma^\sigma_{\beta \alpha}\right) \\[6pt]
 G_{\mu \nu} & = [S3] \times \frac{8 \pi G}{c^4} T_{\mu \nu}