SIRD Model: Death and Fatality Rate
Overview
The SIRD model is an epidemiological compartment model that adds death to the SIR model.
Model
$$ \begin{align*} {{d S} \over {d t}} =& - {{ \beta } \over { N }} I S \\ {{d I} \over {d t}} =& {{ \beta } \over { N }} S I - \mu I \\ {{d R} \over {d t}} =& \left( 1 - \delta \right) \mu I \\ {{d D} \over {d t}} =& \delta \mu I \end{align*} $$
Variables
- $S(t)$: Represents the number of individuals who can be susceptible to infection at time $t$.
- $I(t)$: Represents the number of individuals who can transmit the disease at time $t$.
- $R(t)$: Represents the number of individuals who have recovered at time $t$.
- $D(t)$: Represents the number of individuals who have died at time $t$.
- $N(t) = S(t) + I(t) + R(t) + D(t)$: Represents the total population.
Parameters
- $\beta>0$: The infection rate.
- $\mu>0$: The recovery rate.
- $\delta>0$: The fatality rate.
Explanation
When viewed simply as a system of ordinary differential equations, there is no abstract difference between $R$ and $D$ in the model. Even when simulating, the numbers for $D$ are simply proportional to $R$ by $\delta$ and do not generate intrinsically interesting observations on their own. The compartment $D$ becomes meaningful not through simple simulation but through fitting. If one has the time series of infected and deceased cases, then it’s possible to regard $\delta$ as an unknown and estimate it as $\hat{\delta}$. Considering public health strategies, the metric of ‘death’ cannot be overlooked, thus adding $D$ allows for quantification of death and strategic evaluation.
At times, or rather historically, humanity has mostly been unable to overcome pandemics. However, combating pandemics isn’t a competition of who’s the better leader but the actual lives of people living in that era. In reality, victory and defeat are not decided merely on the basis of completely eradicating a disease. Goals of various health strategies could include buying time until a vaccine is developed, minimizing economic costs, or reducing the number of deaths. Among these, death is especially significant, and adding $D$ enables the quantification of death and strategy evaluation.
To put it simply, it introduces options like ‘a strategy that might last longer but reduces the number of deaths’.
Mortality rate, Fatality rate, Lethality
Mortality Rate, Fatality Rate, and Lethality are terms often used interchangeably when discussing death statistics. However, mortality rate is a broad term not limited to disease and doesn’t necessarily imply death caused by a disease, thus it’s not the preferred term in the context of infectious diseases. For the general public, while it might be the most intuitive expression, it also carries ambiguity. More technical literature tends to use fatality rate or lethality (with a preference for fatality rate).
Variations
SIRHDV model: Healthcare Services
$H$ represents hospitals (beds), $V$ represents vaccines as in the SIRV model. Their difference lies in:
- For hospitals, it’s $I \to H$, meaning the hospitalization of the infected,
- For vaccines, it’s $S \to V$, meaning vaccination.
Not only does it reduce the force of infection $\beta SI$, but it also considers the severity of symptoms divided into mild and severe $I = I_{1} + I_{2}$. Such healthcare services are assumed to have an effect on reducing the general severity and even the fatality rate, with their mathematical models being able to show the expected effects in concrete numbers.