David A. Tanzer, July 10, 2020, in unit Epidemic Models 1.
A menu of compartmental models
Last time, we talked about the idea of compartmental models, where the population is divided into compartments like Susceptible and Infected. If these are the only compartments, the model is called SI. But SI is very simplistic, and cannot express things like recovery, immunity or death.
Richer models are obtained by introducing more compartments.
Inventory of compartmental models
-
SI = $\mathrm{Susceptible} \rightarrow \mathrm{Infected}$
-
SIR = $\mathrm{Susceptible} \rightarrow \mathrm{Infected} \rightarrow \mathrm{Recovered} \ \ \ $ # recovered = immune
-
SIRD = $\mathrm{Susceptible} \rightarrow \mathrm{Infected} \rightarrow \mathrm{Recovered}$ or $\mathrm{Dead}$
-
SIRS = $\mathrm{Susceptible} \rightarrow \mathrm{Infected} \rightarrow \mathrm{Recovered} \rightarrow \mathrm{Susceptible} \ \ \ $ # after immune period, return to susceptible
-
SIS = $\mathrm{Susceptible} \rightarrow \mathrm{Infected} \rightarrow \mathrm{Susceptible} \ \ \ $ # for diseases with no immune period
-
SEIR = $\mathrm{Susceptible} \rightarrow \mathrm{Exposed} \rightarrow \mathrm {Infectious} \rightarrow \mathrm {Recovered} \ \ \ $ # exposed = latency period
The connectivity structure of a model
Besides the compartments themselves, the model shows how they are connected together.
Part of this is shown by the arrows between the compartments, which are “one-way streets” giving the directions by which individuals move from one compartment to another.
Take the SIR model, for example:
- Individuals move from $S$ to $I$ at the time of infection.
- Individuals move from $I$ to $R$, at the time of recovery.
These are the only moves that can take place, so it’s a one-way flow $S \rightarrow I \rightarrow R$.
SIRS has a cyclic flow, due to the return to susceptibility after the immune period:
- Individuals move from $S$ to $I$ at the time of infection.
- Individuals move from $I$ to $R$, at the time of recovery.
- Individuals move from $R$ to $S$, when immunity is lost.
This gives us the cyclic flow $S \rightarrow I \rightarrow R \rightarrow S$.
Towards the dynamics of a model
Recall that a major application of these models is that their dynamics can be simulated on a computer, to obtain curves for how the numbers of susceptible, infected, dead, etc. evolve over time.
But something is missing from the description that we have given so far. Notice that there is nothing that can tell us about rates at which the processes of infection, recovery, etc. are taking place. It is these rates which determine how the sizes of the containers evolve over time.
This leads to a deeper point: at the heart of these models is a description of processes, like infection and recovery, which move individuals from container to container. That is where we will find the rate information.
We will take this up next time.
Copyright © 2020, David A. Tanzer. All Rights Reserved.