Difference between revisions of "SIR Modeling"
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<!-- {{#tag:math|S_{t+1} = S_t - \beta S_t I_t }} --> | <!-- {{#tag:math|S_{t+1} = S_t - \beta S_t I_t }} --> | ||
− | + | :<math> | |
+ | \begin{align} | ||
+ | g_{\mu \nu} & = [S1] \times \operatorname{diag}(-1,+1,+1,+1) \\[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> | ||
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}
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]
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.
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