Difference between revisions of "SIR Modeling"

Latest revision as of 03:56, 14 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 three 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]\beta = \tau \times c[/math] ([math]\tau[/math] transmissibility, [math]c[/math] number of people exposed)

[math]\gamma[/math] the transition rate (the inverse of the mean recovery time, in days).

@@ Line 1: / Line 1: @@
+<!-- https://code-for-philly.gitbook.io/chime/what-is-chime/sir-modeling -->
 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 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.
+The dynamics are given by the following three equations;
 <!-- {{#tag:math|S_{t+1} = S_t - \beta S_t I_t }} -->
 :<math>
 \begin{align}
-g_{\mu \nu} & = [S1] \\[6pt]
+S_{t+1} & = S_t - \beta S_t I_t \\[6pt]
-{R^\mu}_{\alpha \beta \gamma} & = [S2] \\[6pt]
+I_{t+1} & = I_t + \beta S_t I_t - \gamma I_t \\[6pt]
-G_{\mu \nu} & = [S3]
+R_{t+1} & = R_t + \gamma I_t
 \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>
-<center><math>S_{t+1} = S_t - \beta S_t I_t</math></center>
-<center><math>I_{t+1} = I_t + \beta S_t I_t - \gamma I_t</math></center>
-<center><math>R_{t+1} = R_t + \gamma I_t</math></center>
 where;
-<math>S</math> number of susceptible individuals,
+:<math>S</math> number of susceptible individuals,
+:<math>I</math> number of infected 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>R</math> number of recovered individuals,
+::<math>\beta = \tau \times c</math> (<math>\tau</math> transmissibility, <math>c</math> number of people exposed)
+:<math>\gamma</math> the transition rate (the inverse of the mean recovery time, in days).
-<math>\beta</math> the average number of contacts per person per time,
-<math>\gamma</math> the transition rate.