📂Dynamics

Dynamics Compartment Model

Overview ¹

Compartmental models in epidemiology serve as models for epidemic outbreaks, incorporating infectious diseases into population dynamics and dividing the ‘population’ into several compartments.

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• Epidemiology is the study of epidemics, unrelated to the mechanics discussed at shrimp sushi restaurants.

Description

Ever since Kermack and McKendrick devised what’s known as the SIR model, there have been numerous modifications and developments. All models that originate from this idea are essentially considered compartmental models in epidemiology. The most well-known model is the initial SIR model mentioned earlier, which divides the total population $N$ into three compartments:

• $S$ Susceptible: Healthy, can become sick
• $I$ Infected: Sick, spreading the disease
• $R$ Recovered: Recovered, immune

These are represented as a simple autonomous system with respect to the transmission rate $\beta > 0$ and the recovery rate $\mu > 0$.

$$ \begin{align*} {{d S} \over {d t}} =& - \beta I S \\ {{d I} \over {d t}} =& \beta S I - \mu I \\ {{d R} \over {d t}} =& \mu I \end{align*} $$

General Structure of Bilinear Systems ²

Beretta and Capasso formalized the general form of many derived models as follows. $$ {{dz} \over {dt}} = \text{diag} (z) (e + A z) + c $$

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• $n$ is the number of compartments.
• $z(t) \in \mathbb{R}^{n}$ is the time-dependent vector of compartments $t$.
• $e, c \in \mathbb{R}^{n}$ is a vector of constants.
• $A \in \mathbb{R}^{n \times n}$ is a constant matrix represented by competitive coefficients.

Why is it represented in this form?

The system is usually derived based on the law of mass action, reasoning that $A$ and $B$ meet and react. They meet and react at a constant ratio in an ideal space, and the extent is proportional to their respective amounts, thus the reacting quantity is also proportional to $[A][B]$. Therefore, at least each compartment must be multiplied twice, but the phenomenon of three compartments coming together and reacting is uncommon in reality. Therefore, most linear models can be represented bilinearly.

Example

Taking the simple SIR model as an example, $$ {{ d } \over { dt }} \begin{bmatrix} S(t) \\ I(t) \\ R(t) \end{bmatrix} = \begin{bmatrix} S(t) & 0 & 0 \\ 0 & I(t) & 0 \\ 0 & 0 & R(t) \end{bmatrix} \left( \begin{bmatrix} 0 \\ -\mu \\ \mu \end{bmatrix} + \begin{bmatrix} 0 & -\beta & 0 \\ \beta & 0 & 0 \\ 0 & 0 & 0 \end{bmatrix} \begin{bmatrix} S(t) \\ I(t) \\ R(t) \end{bmatrix} \right) + \mathbf{0} $$ it can be elaborated as $$ e = \begin{bmatrix} 0 \\ -\mu \\ \mu \end{bmatrix} \\ A = \begin{bmatrix} 0 & -\beta & 0 \\ \beta & 0 & 0 \\ 0 & 0 & 0 \end{bmatrix} \\ c = \mathbf{0} $$