📂Dynamics

SIS Model: Reinfection and Chronic Disease

Overview

The SIS model is one that does not take into account immunity or indifference in the spread of contagions or information. It is primarily applied to diseases that are endemic rather than epidemic, such as colds, influenza, sexually transmitted infections, and malaria.

Model ¹

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

Variables

• $S(t)$: Represents the number of individuals who are susceptible to getting sick at time point $t$.
• $I(t)$: Represents the number of individuals who can transmit the disease at time point $t$. In the context of information spread, it is also referred to by the initial letter of Informed.
• $N(t) = S(t) + I(t)$: Represents the total number of individuals. If vital dynamics are not considered, it is usually treated as a conserved quantity (constant), and when referring to the proportions within the total population, it is often denoted as $N(t) = 1$.

Parameters

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

Basic reproduction number

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

Theorem

The SIS model is fundamentally a logistic growth model.

Proof ²

$$ {{d I} \over {d t}} = {{ \beta } \over { N }} S I - \gamma I $$ Since $S = N - I$, $$ \begin{align*} {{d I} \over {d t}} =& {{ \beta } \over { N }} ( N - I ) I - \gamma I \\ =& \left( (\beta - \gamma) - {{ \beta } \over { N }} I \right) I \end{align*} $$

Logistic growth model: $$ \dot{N} = {{ r } \over { K }} N ( K - N) $$

When looking at infected individuals $I$, their numbers grow logistically.

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