📂Dynamics

SIR Model: The Most Basic Diffusion Model

Overview

The SIR model is one of the simplest and most widely varied compartmental models in epidemiology, offering a straightforward and intuitive explanation of the spread of diseases or information.

Model ¹

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

Variables

• $S(t)$: Represents the number of individuals susceptible to the disease at time $t$.
• $I(t)$: Represents the number of individuals who can transmit the disease at time $t$. In the context of information spread, it also stands for the initial letter of Informed.
• $R(t)$: Represents the number of individuals who have recovered from the disease at time $t$. In the context of information spread, it may also stand for the initial letter of Refractory or Removed, indicating that they no longer respond or are irrelevant in the simulation.
• $N(t) = S(t) + I(t) + R(t)$: Represents the total population. If vital dynamics are not considered, it is usually regarded as a conserved quantity (constant), and if the variables are considered as proportions of the entire population, it is often denoted as $N(t) = 1$.

Parameters

• $\beta>0$: The infection rate.
• $\mu>0$: The recovery rate.

Explanation

The vital dynamics mentioned in Variables literally consider the lifespan of each individual, where the total population itself changes through being born, aging, and dying. This is not usually emphasized unless talking about endemic diseases or analyses over very long periods.

Derivation

Lotka-Volterra Predator-Prey Model: $$ \begin{align*} \dot{x} =& a x - b y \cdot x \\ \dot{y} =& c x \cdot y - d y \end{align*} $$

As a special case of the Lotka-Volterra competition model, the derivation can be considered almost complete. Here, $S$ for the disease acts as the prey $S = x$, and $I$ naturally becomes the predator $I = y$. Assuming the prey has no resistance against the disease, which means $a = 0$, and setting $b = c := \beta / N$, $d = \mu$, we get

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

Simply adding the rate of change of $R$ as $\displaystyle {{d R} \over {d t}} = \mu I$ gives us the system for the SIR model.

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Basic Reproduction Number

$$\mathcal{R}_{0} = {{ \beta } \over { \mu }}$$

To be precise, it would be possible to accurately get the eigenvalues with Jacobian matrices, but skipping the extensive calculations, let’s consider a quick and dirty method. At the beginning of the outbreak, that is at time $S \approx N$, for the epidemic to eventually lead to a major outbreak, it must hold that $\displaystyle {{ d I } \over { d t }} > 0$. In other words, for $I(0) > 0$ $$ {{ \beta } \over { N }} N I - \mu I \approx ( \beta - \mu ) I > 0 $$ mathematically, if $\displaystyle {{ \beta } \over { \mu }} > 1$, then $I$ will keep increasing leading to a major outbreak. From this perspective, $\displaystyle \mathcal{R}_{0} := {{ \beta } \over { \mu }}$ can also be called the Epidemic Threshold².

Variations

SIRS Model: Temporary Immunity ³⁴

Fundamentally, the R state assumes that individuals have acquired permanent immunity against the disease. However, incorporating a term like $\nu R$ allows for reflecting the loss of immunity. Unlike the SIR model, it intuitively deals with endemic diseases.

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

Carriers ³

Carriers refer to individuals who spread the disease but show no clinical symptoms. If their number $C$ is constant, the SIR system could be modified like this:

$$ \begin{align*} {{d S} \over {d t}} =& - {{ \beta } \over { N }} (I + C) S \\ {{d I} \over {d t}} =& {{ \beta } \over { N }} S (I + C) - \mu I \\ {{d R} \over {d t}} =& \mu I \end{align*} $$

Vital Dynamics

Similar to the logistic growth model, introducing the birth rate $r>0$ and mortality rate $\gamma>0$ allows for considering vital dynamics. Here, the mortality rate applies equally regardless of infection status, and the growth rate also similarly affects the current total population $N(t) = S(t) + I(t) + R(t)$ regardless of infection status.

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

Vertical Transmission

Vertical Transmission or Mother-to-Child Transmission refers to infections⁵ directly passed from the mother to the newborn, with Hepatitis B virus being one of the examples. To incorporate this, the system obtained from the above vital dynamics can be modified by giving the vertical transmission probability $q \in (0,1)$ like so:

$$ \begin{align*} {{d S} \over {d t}} =& - \gamma S - {{ \beta } \over { N }} I S + r ( 1- q) N \\ {{d I} \over {d t}} =& - \gamma I + {{ \beta } \over { N }} S I - \mu I + r q N \\ {{d R} \over {d t}} =& - \gamma R + \mu I \end{align*} $$

$r q N$ refers to the term for newborns born with the disease.